text
stringlengths 1
1.96M
| meta
dict |
|---|---|
\section{Introduction}
When a Gaussian laser beam heats the interface separating two superimposed optically absorbing liquids, the inhomogeneous heating induces a local variation
of the interfacial tension which in turn generates tangential shear stresses and sets up hydrodynamic flows known as thermocapillary or Marangoni flows \cite{schatz01}.
These laser-induced thermocapillary flows were studied theoretically, experimentally and numerically in various cases, with a particular emphasis
to the free-surface configuration. In the eighties, Loulergue et al. \cite{loulergue81} investigated theoretically the heating of a flat fluid interface by a spatial
sine modulation of an infrared laser beam to produce an infrared image converter.
At the same time, Viznyuk et al. \cite{viznyuk88},
investigated the free-surface deformation of thin layers by Gaussian laser beams and applied their theoretical results to beam
shaping using what they called optocapillarity.
Beyond optical applications, Longtin et al. \cite{longtin99} investigated time dependent
behaviors of free-surfaces by numerical simulation and intended a comparison between predictions for laser pulse heating
with experiments and scaling analyses, the goal being to get new insights in laser melting and welding \cite{mackwood05}.
One can also cite the theoretical analysis of Rivas \cite{rivas91} who studied the effect of the viscous stress on the interface deformation
and the numerical work of Marchuk \cite{marchuk09} who investigated the effect of convection.\\
We know that the temperature variation of the surface tension is usually negative for classical liquids \cite{escobedo98}.
Therefore, fluids are almost always pushed toward coldest regions and the previous experiments showed that laser heating
produces beam centered dimple at the interface. This led Bezuglyi et al. \cite{bezuglyi01} to consider very thin layers
and investigate the possibility to optically control film rupture and hole formation in order to understand new mechanisms
for varnish dewetting, wetting \cite{nagy08} and spreading \cite{garnier03} of liquid drops and microfilms over solid substrates.
However, contrary to usual expectations, Misev \cite{Mizev04} experimentally observed a curvature inversion
of the interface deformation, i.e. a concave to convex transition, when decreasing the thickness of the fluid layer.
Therefore, it appears that dimple formation does not represent a general rule when fluid confinement starts to play a role,
as already supported by classical Marangoni experiments in thin cells \cite{vanhook97}. This transition from concave
to convex interface deformation has been studied recently \cite{karlov05} but the generalization to two-liquid systems
still deserves to be investigated. This is the goal of the present work.\\
We theoretically investigate thermocapillary effects induced by a Gaussian laser beam that locally heats
the interface separating two liquid layers or the interfaces of a thin film bounded by two liquid layers.
Beyond the general description of flow patterns and interface deformations in many different situations,
our investigation also provides new insights for digital optofluidics
applications \cite{delville09},\cite{baroud10}, where light-actuated droplets are naturally confined and often close together, squeezing by the way thin liquid layers.\\
Our study is structured as follows: the theoretical resolution of the flow and the interface deformation
is detailed in section 2. Section 3 reports results and discussion on the dependence of the flow and the deformation to the viscosity and layer thicknesses ratios
in two different configurations. We first consider an interface separating
two different liquid layers and then generalize these results to the case of thin film separating two same liquid layers.\\
\section{Two-Fluid theoretical model}
\begin{figure}[h!!!]
\begin{center}
\includegraphics[scale=0.5]{figure1.eps}
\caption{Schematic representation of the liquid-liquid system heated by a laser beam. See text for other notations.
$\sigma^+$ indicated that the interfacial tension increases with temperature. In this configuration, we arbitrarily consider $\partial \sigma / \partial T>0$, a situation often encountered
in microfluidics due to the presence of large amount of surfactant. }\label{sketch}
\end{center}
\end{figure}
Let us consider two liquid layers separated by an interface initially flat, horizontal and at rest, crossed by a continuous Gaussian laser beam propagating perpendicularly
to the interface (see figure \ref{sketch}).
If at least one of the layers absorbs light, the interface will be locally heated by the laser.
Physical properties of the liquids (denoted $1$ for the bottom liquid and $2$ for the top one)
are their viscosities $\eta_1$, $\eta_2$ and densities $\rho_1$, $\rho_2$.
The interfacial tension is denoted by $\sigma$.
Liquids are enclosed in a cylindrical cell of radius $R>>w$, where $w$ is the radius of the temperature distribution
due to laser heating.\\
Considering the axisymmetry, along the $z$ axis of the exciting beam, of the temperature distribution, cylindrical coordinates (${\bf e_r, e_\theta, e_z})$
with origin $O$ located at the intersection of the beam axis
with the initial flat interface are chosen for this study. A point $\textbf{x}$ is thus referenced by the space
coordinates $(r,\theta,z)$.
Two different configurations are investigated. The first one (a) considers two liquid layers separated by an interface,
and the second one (b) generalizes the two-layer situation to a symmetric system composed of three liquid layers, liquid $1$ being bounded on top and bottom by liquid $2$ (see figure \ref{sketch}).\\
\subsection{Heat equations}
The temperature distribution within the fluid layers due to the heating sources $\Upsilon_i$ obeys the heat equation
\begin{equation}
\frac{\partial T_i}{\partial t}+({\bf u_i\cdot\nabla})T_i=D_{T_i}\nabla^2T_i+\Upsilon_i~~~~i=1,2\label{heat}
\end{equation}
${\bf u_i}$ and $D_{T_i}$ being the velocity and the thermal diffusivity of fluid $i$.\\
Assuming, without the loss of generality, that only fluid $1$ absorbs light, $\Upsilon_2=0$ and
\begin{equation}
\Upsilon_1=\frac{D_{T_1}}{\Lambda_1}a_1 I(r,z)
\end{equation}
where $\Lambda_1$ and $a_1$ are respectively the thermal conductivity and the optical absorption of fluid $i$ at the used optical wavelength.\\
Light intensity distribution is given by
\begin{equation}
I(r,z)=\frac{2P}{\pi \omega_0^2}\exp\left(\frac{-2r^2}{\omega_0^2}-a_1z\right)
\end{equation}
where $P$ is the beam power and $\omega_0$ the radial extension of the laser beam, also called beam waist. $\omega_0$ is always smaller than $w$ due to
the non locality and the strong dependence in boundary conditions of temperature distributions.\\
The keypoint for producing thermocapillary flows is not the overheating itself but the amplitude of the temperature gradient. Laser heating is thus very appealing
because it is extremely easy to produced weak amplitude overheating with large gradients.
Thus we can confidently consider a weak optical absorption in fluid $1$ such as $a_1z \leqslant a_1H_1 \ll 1$, and neglect the axial variation of light intensity and therefore
\begin{equation}
I(r)\simeq \frac{2P}{\pi \omega_0^2}\exp\left(\frac{-2r^2}{\omega_0^2}\right)
\end{equation}
The convective and unsteady terms of equation (\ref{heat}) can be neglected when considering that the thermal P\'eclet number is small
($\displaystyle{Pe_T=\frac{u_0 w}{D_{T_i}} \ll 1}$, where $u_0$ is a characteristic velocity)
and a characteristic heating time $w^2/D_{T_i}$ small compared to the viscous characteristic time $w^2/\nu_i$, where $\nu_i$ is the kinematic viscosity of fluid $i$.\\
An example of experiment fitting these assumptions can be found in a recent publication \cite{rsv12}.\\
Exact solutions of the heat equation using continuity of temperature and heat flux at the interface can be calculated \cite{rsv12}. However for the sake of simplicity we will consider a Gaussian distribution for $T_i(r)$
which is a reasonable approximation of the exact temperature field.\\
\subsection{Fluid equations of motion}
Both fluids obey the mass conservation and Stokes equations, and are coupled through stress balance
in addition to the continuity of velocity at the interface described by its height $h(r)$ and denoted $S_I$. As flows are expected
to be viscous, the boundary value problem can be expressed as\\
\begin{equation}
\nabla . {\bf u_i} = 0~~;~~ i=1,2
\label{mass}
\end{equation}
\begin{equation}
{\bf 0} = -\nabla p_i + \eta_i\Delta {\bf u_i}~~;~~ i=1,2
\label{stokes}
\end{equation}
\begin{equation}
({\bf T_{1}}\cdot {\bf n}- {\bf T_{2}}\cdot {\bf n})=(\sigma \kappa (r) - (\rho_1-\rho_2)g z)
{\bf n}+\frac{\partial \sigma}{\partial s}{\bf t}~~;~~ {\bf x}~ \in~ S_I \label{stressjump}
\end{equation}
\begin{equation}
{\bf u_1} = {\bf u_2}~~;~~ {\bf x}~ \in~ S_I \label{cont_v}
\end{equation}
$p_i$ is the pressure in each fluid and $\displaystyle{{\bf T_i}
= -p_{i} {\bf I} + \eta_i(\nabla {\bf u_i} + ^t\nabla {\bf u_i})}$ is the stress tensor corrected with a gravity term.
The unit vector normal to the interface directed from fluid 1 to fluid 2 is denoted by $\bf{n}$,
$s$ is the arc length on the interface, and ${\bf t}$ is the unit vector tangent to the interface such as ${\bf (t,e_\theta,n)}$ is orthonormal.\\
In Equation (\ref{stressjump}), $\displaystyle{\kappa(r)=\frac{1}{r}
\frac{d}{dr}\frac{r\frac{dh}{dr}}{\sqrt{1+{\frac{dh}{dr}}^2}}}$ is the double mean curvature of the
axisymmetric interface in cylindrical coordinates. $\displaystyle{\frac{\partial \sigma}{\partial s}}$ is the tangential stress due to a spatial variation of the interfacial tension.
When this variation is due to a radially inhomogeneous heating, it is usually called thermocapillary stress.\\
The variation of interfacial tension can be induced by the local heating and by the migration of surface active surfactants molecules at the interface.
Generally, when there is no surfactants, $\partial \sigma /\partial T<0$ and interfacial tension is smaller at the hot spot
but when the interface is charged with surfactants, the effective change of interfacial tension, involving coupling between temperature and concentration, can make interfacial tension larger at the hot spot.
In the present investigation, we consider the effective change of interfacial tension $\partial \sigma /\partial T$ due to both the heating and the solutal effect.
More details on the change of behavior of interfacial tension with temperature, due to the presence of surfactants, can be found in the following references: \cite{rsv12}, \cite{khattari02}.\\
The thermocapillary stress induces local flows on both sides of the interface which lead to its deformation.\\
Considering on the one hand small deformation amplitude of height $h(r)$ (i.e. $\partial h/\partial r<<1$), we can write the tangential stress condition as\\
\begin{equation}
\eta_1\frac{\partial u_{1,r}}{\partial z}|_{z=h}-\eta_2\frac{\partial u_{2,r}}{\partial z}|_{z=h}
=\frac{\partial \sigma}{\partial s}\backsimeq \frac{\partial \sigma}{\partial r}=
\frac{\partial \sigma}{\partial T} \frac{\partial T}{\partial r},\label{tang}
\end{equation}
where $T(r)$ is the temperature distribution at the interface due to laser heating.
On the other hand, assuming a Gaussian axisymmetric temperature distribution such as
\begin{equation}
T(r)=\Delta T_0e^{-r^ 2/w^2},
\end{equation}
This expression of $T(r)$ allows us to solve the hydrodynamics problem using the Fourier-Bessel transform defined such as
\begin{equation}
h(r)=\int_0^{\infty} h^k(k)J_0(kr)kdk\label{fourier-h}
\end{equation}
where $h^k$ is the Fourier-Bessel transform of $h(r)$, $k$ the reciprocal variable, and $J_0(x)$ the 0-order Bessel $J$ function.
We deduce
\begin{equation}
T(r)=\int_0^{\infty}T^k(k)J_0(kr)kdk,
\end{equation}
with
\begin{equation}
T^k(k)=\Delta T_0\frac{w^2}{2}e^{-\frac{w^ 2 k^ 2}{4}} \label{temp}
\end{equation}
As we have $\frac{dJ_0(kr)}{dr}=-kJ_1(kr)$ where $J_1(x)$ is the 1-order Bessel $J$ function, we finally find
\begin{equation}
\frac{\partial T(r)}{\partial r}=-\int_0^{\infty}T^ k(k)J_1(kr)k^ 2dk
\end{equation}
Therefore, we find from equation (\ref{tang}) that $u_{i,r}(r,z)$ necessarily takes the form\\
\begin{equation}
u_{i,r}(r,z)=\int_0^{\infty}u^k_{i,r}(z)J_1(kr)kdk\label{urk}
\end{equation}
Moreover as the mass conservation condition (equation (\ref{mass})) can be written as\\
\begin{equation}
\frac{1}{r}\frac{\partial (r u_{i,r})}{\partial r}+\frac{\partial u_{i,z}}{\partial z}=0\label{cont}
\end{equation}
it yields
\begin{equation}
u_{i,z}(r,z)=\int_0^{\infty}u^k_{i,z}(z)J_0(kr)kdk\label{uzk}
\end{equation}
and the following relation between axial and radial velocities\\
\begin{equation}
\frac{du^k_{i,z}(z) }{d z}=-ku^k_{i,r}(z) \label{rel}
\end{equation}
From Equations (\ref{urk}), (\ref{uzk}) and (\ref{rel}) we can now calculate the velocity field due to localized laser heating.
\subsection{Velocity field}
As far as one considers small interface deformation amplitude, the velocity field can be evaluated for
the initially flat interface $h(r)=0$.\\
The stokes equations (equation (\ref{stokes})) can be written as follows in each liquid layer $i=1,2$\\
\begin{equation}
-\frac{\partial p_i}{\partial r}+\eta_i\left(\frac{\partial^ 2}{\partial r^ 2}
+\frac{1}{r}\frac{\partial}{\partial r}-\frac{1}{r^ 2}+\frac{\partial^ 2}{\partial z^ 2}\right)u_{i,r}(r,z)=0\label{pr}
\end{equation}
\begin{equation}
-\frac{\partial p_i}{\partial z}+\eta_i\left(\frac{\partial^ 2}{\partial r^ 2}
+\frac{1}{r}\frac{\partial}{\partial r}+\frac{\partial^ 2}{\partial z^ 2}\right)u_{i,z}(r,z)=0 \label{pz}
\end{equation}
Remembering the following properties of the Bessel functions\\
\begin{equation}
\frac{d^2J_1(kr)}{dr^ 2}+\frac{1}{r}\frac{dJ_1(kr)}{dr}-\frac{1}{r^ 2}J_1(kr)=-k^ 2J_1(kr)
\end{equation}
\begin{equation}
\frac{d^2J_0(kr)}{dr^ 2}+\frac{1}{r}\frac{dJ_0(kr)}{dr}=-k^ 2J_0(kr)
\end{equation}
we rewrite Equations (\ref{pr}) and (\ref{pz}) as\\
\begin{equation}
-\frac{\partial p_i}{\partial r}+\int_0^{\infty}\eta_i\left(\frac{\partial^ 2}{\partial z^ 2}-k^ 2\right)u^ k_{i,r}(r)J_1(kr)kdk=0
\end{equation}
\begin{equation}
-\frac{\partial p_i}{\partial z}+\int_0^{\infty}\eta_i\left(\frac{\partial^ 2}{\partial z^ 2}-k^ 2\right)u^ k_{i,z}(r)J_0(kr)kdk=0
\end{equation}
Eliminating the pressure and using equation (\ref{rel}), we finally get\\
\begin{equation}
\left(\frac{\partial^ 2}{\partial z^ 2}-k^ 2\right)^ 2u^ k_{i,z}(z)=0
\end{equation}
This $4^{th}$ order linear equation of order $4$ defines the eigen modes of the axial velocity.\\
Its general solution is of the form\\
\begin{equation}
u^ k_{i,z}(z)=A_ie^{kz}+B_ie^{-kz}+C_i\frac{z}{H_i}e^{kz}+D_i\frac{z}{H_i}e^{-kz}
\end{equation}
Using equation (\ref{rel}), the solution for the radial velocity is written\\
\begin{equation}
u^ k_{i,r}(z)=-A_ie^{kz}+B_ie^{-kz}-\frac{C_i}{kH_i}(kz+1)e^{kz}-\frac{D_i}{kH_i}(-kz+1)ke^{-kz}
\end{equation}
where $A_i$, $B_i$, $C_i$ and $D_i$ are constants to be determined from boundary conditions.
We thus have eight unknowns to integrally solve the velocity field. The eight boundary conditions are:\\
Vanishing axial velocities at the interface ($z=0$)
\begin{equation}
u^k_{1,z}(z=0)=u^k_{2,z}(z=0)=0~~~~~B.C.1,2 \label{bc12}
\end{equation}
Continuity of the radial velocity at the interface ($z=0$)
\begin{equation}
u^k_{1,r}(z=0)= u^k_{2,r}(z=0)~~~~~~B.C.3 \label{bc3}
\end{equation}
Vanishing velocities at the top $z=H_2$ and bottom $z=-H_1$ of the container (except for the symmetric configuration (b), where it is $\frac{d u^k_{1,r}}{d z}$ which is null at $z=-H_1$)
\begin{equation}
u^k_{1,z}(z=-H_1)=0~~~~~B.C.4 \label{bc4}
\end{equation}
\begin{equation}
u^k_{1,r}(z=-H_1)=0~~~~~B.C.5a~~~or~~ \frac{d u^k_{1,r}}{d z}(z=-H_1)=0~~~~~B.C.5b\label{bc5}
\end{equation}
\begin{equation}
u^k_{2,z}(z=H_2)=0~~~~~~B.C.6\label{bc6}
\end{equation}
\begin{equation}
u^k_{2,r}(z=H_2)=0~~~~~~B.C.7\label{bc7}
\end{equation}
and finally the tangential stress jump at the interface (equation (\ref{tang})) for $h\simeq 0$
\begin{equation}
\eta_1\frac{\partial u_{1,r}}{\partial z}|_{z=0}-\eta_2
\frac{\partial u_{2,r}}{\partial z}|_{z=0}= \frac{\partial \sigma}{\partial T} \frac{\partial T}{\partial r}~~~~~~B.C.8
\end{equation}
B.C.1 and 2 yield:\\
\begin{equation}
A_1=-B_1~~,~~A_2=-B_2~~~~~B.C.1,2
\end{equation}
We can thus simplify notations and keep only 6 unknowns ($A_1,A_2,C_1,C_2,D_1,D_2$).\\
B.C.3 yield:\\
\begin{equation}
-2A_1-\frac{1}{k}\frac{C_1}{H_1}-\frac{1}{k}\frac{D_1}{H_1}=-2A_2-\frac{1}{k}\frac{C_2}{H_2}-\frac{1}{k}\frac{D_2}{H_2}~~B.C.3
\end{equation}
B.C.4 and 6 yield\\
\begin{equation}
A_1(e^{-kH_1}-e^{kH_1})-C_1e^{-kH_1}-D_1e^{kH_1}=0~~B.C.4
\end{equation}
\begin{equation}
A_2(e^{kH_2}-e^{-kH_2})+C_2e^{kH_2}+D_2e^{-kH_2}=0~~B.C.6
\end{equation}
B.C.5 yield\\
\begin{equation}
-A_1(e^{-kH_1}+e^{kH_1})-\frac{1}{k}\frac{C_1}{H_1}(-kH_1+1)e^{-kH_1}-\frac{1}{k}\frac{D_1}{H_1}(kH_1+1)e^{kH_1}=0~~B.C.5a
\end{equation}
or
\begin{equation}
A_1k(e^{-kH_1}-e^{kH_1})+\frac{C_1}{H_1}(-kH_1+2)e^{-kH_1}-\frac{D_1}{H_1}(kH_1+2)e^{kH_1}=0~~B.C.5b
\end{equation}
and B.C.7 leads to
\begin{equation}
-A_2(e^{kH_2}+e^{-kH_2})-\frac{1}{k}\frac{C_2}{H_2}(kH_2+1)e^{kH_2}-\frac{1}{k}\frac{D_2}{H_2}(-kH_2+1)e^{-kH_2}=0~~B.C.7
\end{equation}
Finally B.C.8 leads to
\begin{equation}
2\frac{\eta_2}{H_2}(C_2-D_2)-2\frac{\eta_1}{H_1}(C_1-D_1)=-\frac{\partial \sigma}{\partial T}kT^ k(k)~~B.C.8
\end{equation}
Boundary conditions $1-8$ entirely determine the velocity field produced by a localized laser heating and allow to
calculate the resulting pressure field.
Note that when $H_1=H_2$, we have $u_r(-z)=u_r(z)$ and $u_z(-z)=-u_z(z)$ whatever $\eta_1/\eta_2$.
\subsection{Pressure field}
The pressure gradient along the $z$-axis (equation (\ref{pz})) can be written as\\
\begin{equation}
\frac{\partial p_i}{\partial z}=\int_0^{\infty}\eta_i\left(\frac{\partial^ 2}{\partial z^ 2}-k^ 2\right)u^ k_{i,z}(z)J_0(kr)kdk
\end{equation}
Integration allows to determine the pressure in fluid $1$ and $2$ by specifying a reference pressure
$p_{H_2}$ at $z=H_2$
\begin{equation}
p_2(r,z)=\int_0^{\infty}\left(p_{H_2}+2\frac{\eta_2}{H_2}(C_2e^{kz}+D_2e^{-kz})\right)J_0(kr)kdk
\end{equation}
and
\begin{equation}
p_1(r,z)=\int_0^{\infty}\left(p_{H_2}+2\frac{\eta_1}{H_1}(C_1e^{kz}+D_1e^{-kz})\right)J_0(kr)kdk
\end{equation}
which finally leads to
\begin{equation}
p_2(r,0)-p_1(r,0)=\int_0^{\infty}\left(2\frac{\eta_2}{H_2}(C_2+D_2)-2\frac{\eta_1}{H_1}(C_1+D_1)\right)J_0(kr)kdk
\end{equation}
\subsection{Interface deflection}
Knowing velocities and pressure fields, the shape of the interface can be deduced from the normal stress equation
(equation (\ref{stressjump})) which can be re-written as
\begin{equation}
\left(-p_1(r,z)+2\eta_1\frac{\partial u_{1,z} }{\partial z }\right)_{z=h}-\left(-p_2(r,z)
+2\eta_2\frac{\partial u_{2,z}}{\partial z }\right)_{z=h}=\sigma \Delta_r h- (\rho_1-\rho_2)g h \label{def}
\end{equation}
Considering as previously the small deformation amplitude case, the left hand side of
equation (\ref{def}) can be approximated by its value at $z=0$. It yields \\
\begin{equation}
\int_0^{\infty}4(\eta_1A_1-\eta_2A_2)kJ_0(kr)kdk=\sigma \Delta_r h- (\rho_1-\rho_2)g h
\end{equation}
Using the definition of the Fourier-Bessel transform (Equation (\ref{fourier-h}))and the relation $\Delta_r h=-\int_0^{\infty}k^ 2h^ k(k)J_0(kr)kdk$, where $\Delta_r$ denotes the radial part
of the Laplacian operator, Equation ($\ref{def}$) becomes\\
\begin{equation}
4(\eta_2A_2-\eta_1A_1)k=(\sigma k^ 2 + (\rho_1-\rho_2)g) h^k(k)
\end{equation}
which leads to the expression of the induced deformation of the interface
\begin{equation}
h(r)= 4\int_0^\infty \frac{\eta_2A_2(k)-\eta_1 A_1(k)}{\sigma k^ 2 + (\rho_1-\rho_2)g} J_0(kr)k^2dk
\end{equation}
Defining the gravitational Bond number as $Bo=\frac{(\rho_1-\rho_2)g w^2}{\sigma}$, we finally find \\
\begin{equation}
h(r)= 4w\frac{\eta_2}{\sigma}\int_0^\infty \frac{A_2(k)-\frac{\eta_1}{\eta_2} A_1(k)}{w^2k^ 2 + Bo} J_0(kr)wk^ 2dk
\end{equation}
In the following, we use the heating length $w\sim10-100\mu m$ as a reference length, $u_0=\frac{1}{\eta_1+\eta_2}|\frac{\partial\sigma}{\partial T}|\Delta T_0$ as a reference velocity
and we define the thermal parameter $\alpha$ such as\\
\begin{equation}
\alpha=\frac{\partial \sigma}{\partial T}\frac{\Delta T_0}{\sigma}
\end{equation}
Assuming $\eta_i\sim 1- 100mPa.s$, $\sigma \sim 10^ {-3} N m^ {-1}$, $\Delta T_0\sim 10K$ and $\frac{\partial\sigma}{\partial T} \sim 10^ {-4}Nm^{-1}K^{-1}$,
we find $\alpha \sim 1$ and $u_0\sim 5-500mm/s$. For all the calculations, we set $Bo=0.05$.\\
In the next section we present the velocity field and the interface deformation predicted analytically for two liquid layers separated by a flat interface (configuration (a)).
We investigate the influence of the layer thickness to the heating spot ratio $H_1/w$, the layer ratio $H_1/H_2$ and the viscosity ratio $\eta_2/\eta_1$ on both thermocapillary flows and interface deformation.
\section{Results and discussion}
\subsection{Double layer system}
\begin{figure}[h!!!]
\begin{center}
\includegraphics[scale=0.5]{figure2.eps}
\caption{Steady flow pattern in a double layer configuration for $H_1=5H_2=5w$, $\alpha=1$ and $\eta_2=\eta_1$.}\label{chp_vit_mur}
\end{center}
\end{figure}
\begin{figure}[h!!!]
\begin{center}
\includegraphics[scale=0.6]{figure3.eps}
\caption{Reduced radial velocity $u_r/u_0$ (Top left), reduced axial velocity $u_z/u_0$ (Top right) and interface deformation (Bottom) for
$H_1=5H_2=5w$, $\alpha=1$ and $\eta_2=\eta_1$.}\label{vit_h_mur}
\end{center}
\end{figure}
\subsubsection{Flow pattern}
An example of velocity field produced by the laser heating at the interface of a double layer configuration is reported in figure \ref{chp_vit_mur} for the case $\alpha>1$.
We observe the development of two contrarotative toroidal eddies induced by the tangential stress at the interface and bounded vertically by the interface and the walls.
The reduced radial and axial velocity, respectively $u_r/u_0$ and $u_z/u_0$, are reported in figure \ref{vit_h_mur} as a function of $z$.
We can notice that the maximum radial velocity is located at the interface ($z=0$) and that the axial velocity is larger in the bottom layer which is thicker.
Figure \ref{vit_h_mur} also shows that the deformation of the interface is directed towards the thicker layer when fluid viscosities are the same. In the case $\alpha<0$, flows
and interface deformation are reverted.\\
\subsubsection{Influence of the heating length}
\begin{figure}[h!!!]
\begin{center}
\includegraphics[scale=0.6]{figure4.eps}
\caption{Variations of the radial (continuous) and the axial (dashed with symbols) velocities versus the fluid layer reduced thicknesses $H_1/w$ ($H_1=H_2$ and $r=0.1w$).
The right inset shows the reduced location $z_0/w$ of null radial velocity and maximum axial velocity ($z_0=z(u_r=0)=z(u_{z max})$). $\alpha=1$ and $\eta_2=\eta_1$.
The left inset shows the reduced deformation height $h/w$ as a function of $H_1/w$ for $\eta_2=10\eta_1$. }\label{H-w}
\end{center}
\end{figure}
Figure \ref{H-w} shows that the maximum radial and axial velocity are increasing functions of $H_1/w$ for $\eta_2=\eta_1$. Note that when $H_1=H_2$ and $\eta_1=\eta_2$,
there is no deformation of the interface because the resulting thermocapillary stress is null. We observe that when $H_1<<w$, both components depend on $H_1/w$, as the radial velocity $u_r$ scales like $u_0 H_1/w$ and the axial velocity $u_z$ scales like $u_0 (H_1/w)^2$.
However, when $H_1>>w$ both radial and axial components saturate and scale like $u_0$, the characteristic thermocapillary velocity.\\
The characteristic lengths of the flow pattern are reported in the right inset of figure \ref{H-w}. We notice that the largest amplitude of the axial velocity always coincides with a null radial velocity.
The location of this characteristic point of the flow $z_0$ scales like $H_1$ when $H_1<<w$ as the liquid is bounded by a wall while it scales like $w$ when $w<<H_1$ as the flow magnitude significantly decreases beyond the heating spot size $w$.
The left inset of figure \ref{H-w} shows the variation of the interface deformation amplitude as a function of $H_1/w$ when the liquid viscosities are contrasted ($\eta_2=10 \eta_1$).
We can first notice that when $\alpha>0$ and $H_1=H_2$ the deformation is always directed towards the less viscous fluid. Moreover, the deformation amplitude decreases when increasing the layers thickness $H_1$ so that $H_1>>w$ leads to $h\rightarrow0$.
Even though not strictly quantitative we can approximate this decrease so that $|h|/w \sim w/H_1$ when $H_1<<w$.\\
\subsubsection{Influence of the liquid layer thickness ratio}
\begin{figure}[h!!!]
\begin{center}
\includegraphics[scale=0.5]{figure5.eps}
\caption{Ratio of maximum axial velocities in each fluid $u_{z1}/u_{z2}$ as a function of $H_1/H_2$ in a double layer system.
Inset shows the reduced deformation height $h/H_1$ as a function of $H_1/H_2$. $H_1=w$, $r=0.1w$ and $\eta_2=\eta_1$.}\label{H1-H2}
\end{center}
\end{figure}
Figure \ref{H1-H2} shows the variation of the maximum axial velocity ratio $u_{z1}/u_{z2}$ as a function of $H_1/H_2$. We chose $H_1=w$ to compare one of the heights to the length scale
of the thermal forcing.
We first observe a saturation of $u_{z_1}/u_{z2}$ when $H_1>>H_2$ or $H_2>>H_1$ and that the axial velocity is always
larger in the thicker layer.
The inset of figure \ref{H1-H2} shows that when $\eta_1=\eta_2$, the deformation is directed towards the thicker layer.
When $H_1<<H_2$ (i.e. the heating characteristic length $w=H_1$ is much smaller than $H_2$), $h$ only depends on $w$, while in the opposite case, when $w>>H_2$, the reduced deformation amplitude $h/w$ strongly depends on $H_1/H_2=w/H_2$.
At the same time, the deformation amplitude $h$ increases as $H_2$ decreases as previously shown in figure \ref{H-w}.\\
\subsubsection{Influence of the viscosity ratio}
An interesting feature of the thermocapillary flow in our configuration is the symmetrical properties of the velocities when $H_1=H_2$.
In that case, we have $u_r(-z)=u_r(z)$ and $u_z(-z)=-u_z(z)$ for any value of $\eta_2/\eta_1$. Figure \ref{eta2-eta1} shows that $u_{max} \sim u_0$
and therefore $u_{max} \sim |\frac{\partial\sigma}{\partial T}|\Delta T_0/(2<\eta>)$.
This results can be retrieved using equation (\ref{tang}) and scaling arguments for $H_1>>w$. In this case $-\partial u_{r1} /\partial z \sim \partial u_{r2} /\partial z \sim u_{max} / w$
and $-\partial \sigma / \partial s \sim \frac{\partial\sigma}{\partial T}\Delta T_0/w$.
Similarly, using scaling arguments and equation (\ref{def}), we find that $\Delta_r h \sim h/w^2 \sim 2(\eta_1-\eta_2) u_{max} /(w\sigma)$ and therefore
$h/w \sim \alpha (\eta_1-\eta_2)/<\eta>$ as illustrated in the inset of figure \ref{eta2-eta1}.\\
\begin{figure}[h!!!]
\begin{center}
\includegraphics[scale=0.5]{figure6.eps}
\caption{Reduced maximum velocity $u_{max}/u_0$ as a function of viscosity ratio $\eta_2/\eta_1$ with $\eta_1=cste$.
The inset shows the reduced deformation height $h/w$ as a function of $1/(1+\eta_2/\eta_1)$. $H_1=H_2$.}\label{eta2-eta1}
\end{center}
\end{figure}
Many features of the flows and interface deformation induced by the laser heating when $\partial \sigma / \partial T>0$ can be concluded from this section.
First, the flow magnitude increases with the layer thickness to the heating length ratio $H_1/w$ when $H_1<<w$ before reaching a saturation when $H_1>>w$.
When viscosities are equal in both layers, the velocity is larger in the thicker one and the interface deformation
is always directed towards this same layer.\\
Moreover, when layer thicknesses are equal, $H_1=H_2$, the deformation is always directed towards
the less viscous fluid. Its amplitude increases when decreasing the layers thicknesses as the vertical confinement increases the efficiency of the localized thermocapillary forcing.
Finally, keeping $H_1=H_2$, the dependence of the velocity magnitude and interface deformation can be predicted using simple scaling arguments that yield
respectively to $u_{max} \sim |\frac{\partial\sigma}{\partial T}|\Delta T_0/(2<\eta>)$ and $h/w \sim \frac{\partial \sigma}{\partial T}\frac{\Delta T_0}{\sigma}(\eta_1-\eta_2)/<\eta>$.\\
In the next section, we investigate the laser heating of two fluid interface in a triple layer symmetric configuration where a thin film of liquid is bounded by two layers of a different liquid.
\subsection{Triple layer system}
\begin{figure}[h!!!]
\begin{center}
\includegraphics[scale=0.37]{figure7.eps}
\caption{Steady flow pattern in a triple layer configuration for $H_1=0.2w$, $H_2=w$, $\alpha=1$ and $\eta_2=\eta_1$.}\label{chp_vit_sym}
\end{center}
\end{figure}
The flow pattern showing the generation of eddies, in a system consisting of a thin film bounded by two external liquid layers of same thickness, is reported in figure \ref{chp_vit_sym}.
We can observe the production of two toroidal contrarotative eddies in the central film in addition to a toroidal eddy in each of the external liquid layers.
The continuity of the radial velocity makes each pair of adjoining eddies contrarotative.\\
In this section, we investigate the influence of the film thickness to the external layer thickness ratio $H_1/H_2$ on the produced thermocapillary flows and interface deformations.\\
\subsubsection{Influence of the film thickness}
The maximum radial velocity reduced by its value at the interface as a function of $H_1/H_2$ is reported
in figure \ref{sym_H1-H2} for equal viscosities ($\eta_1=\eta_2$).
We notice that the maximal radial velocity is constant for thin films ($H_1<<H_2$) as $u_{r max}=0.5 u_r(z=0)$
while this velocity decreases when the film thickness becomes comparable to the external layer thickness.\\
The inset of figure \ref{sym_H1-H2} shows that the deformation is directed towards the external layers when $H_1<<H_2$ and its amplitude $h$ depends on $H_1$ while
its direction changes for $H_1/H_2 \simeq 0.7$.\\
\begin{figure}[h!!!]
\begin{center}
\includegraphics[scale=0.5]{figure8.eps}
\caption{Reduced maximum velocity $u_{r1 max}/u_r(z=0$) as a function of $H_1/H_2$ in a triple layer system. Inset shows the reduced deformation height $h/H_1$ as a function of $H_1/H_2$. $H_1=w$ and $\eta_2=\eta_1$.}\label{sym_H1-H2}
\end{center}
\end{figure}
It is interesting to study this system in both cases where the viscous
tangential stress is directed towards the heating spot ($\partial \sigma / \partial r<0$) or in the opposite
direction ($\partial \sigma / \partial r>0$).
In the first case, the deformation of the interface is directed towards the external layers inducing a
dimple (figure \ref{dimple}) while in the second case, the deformation is directed towards the thin
film inducing two noses (figure \ref{nose}).\\
\begin{figure}[h!!!]
\begin{center}
\includegraphics[scale=0.5]{figure9.eps}
\caption{Reduced radial velocity $u_r/u_0$ (Top left), reduced axial velocity $u_z/u_0$ (Top right) and interface deformation (Bottom) in a three-layer system for
$H_1=0.1w$, $H_2=2w$, $\alpha=0.1$ and $\eta_2=\eta_1$. }\label{dimple}
\end{center}
\end{figure}
\subsubsection{Inducing a dimple}
When the effective interfacial tension increases with temperature, the tangential stress is directed towards the heating spot. Assuming liquid layers
of equal viscosities $\eta_1=\eta_2$, the deformation of the interface reported in figure \ref{dimple} is directed towards the external liquid layers of larger thicknesses.
This prediction has been observed experimentally in a system similar to our configuration. In a recent investigation, Dixit et al. \cite{Dixit09} used an infrared laser beam to
induce the coalescence between two SDS-coated water drops in Decanol. The laser beam heats the rear of one of the drops and thermocapillary stresses force its migration towards the other drop.
The laser heating produces an accumulation of surfactant at the rear of the drop \cite{baroud07} and therefore a deficit at the front which is in contact with the second non-heated drop.
This deficit increases interfacial tension and therefore creates tangential stresses directed towards the front of the drop near the second drop.
Coalescence would then occur at dimple edges location. Even though not strictly comparable, we believe that the dimple observed in this experiment
is similar to that predicted by our analytical model and can be explained using the same arguments.
The coalescence of liquid drops has also been observed experimentally in a different configuration \cite{baroud07b} where
the heating laser intercepts the front and rear interfaces of water drops in contact, flowing in oil inside a microchannel,
however the mechanism of the coalescence was not investigated in details. We believe that a dimple is formed in the thin film separating
the drops before coalescence, similarly to the observation of Dixit et al. \cite{Dixit09}.\\
\begin{figure}[h!!!]
\begin{center}
\includegraphics[scale=0.5]{figure10.eps}
\caption{Reduced radial velocity $u_r/u_0$ (Top left), reduced axial velocity $u_z/u_0$ (Top right) and interface deformation (Bottom) in a three-layer system for
$H_1=0.1w$, $H_2=2w$, $\alpha=-0.075$ and $\eta_2=\eta_1$. }\label{nose}
\end{center}
\end{figure}
\subsubsection{Inducing noses}
In the case where the effective interfacial tension decreases with temperature,
the tangential stress is directed from the hot region to the non-heated area of the interface.
The direction of the flow is inverted and the deformations are directed towards the thin film inducing a nose (or hump) at each interface (see figure \ref{nose}).
This could also lead to drop coalescence by forming a liquid bridge similar to the observation made in the experimental
investigation led by Bremond et al. \cite{Bremond08} even though the mechanism at work here is quite different.
\section{Conclusion}
We investigated the flows and the deformation of a liquid-liquid interface induced by the heating of a Gaussian laser beam in confined multi-layer systems.
A first part was dedicated to an interface separating two liquid layers in the case where the effective interfacial tension increases with temperature.
When the effective interfacial tension decreases with temperature, the direction of velocity and interface deformation is simply reverted.
The analytical resolution showed that the flow magnitude increases when increasing the thickness of the liquid layers
before reaching a saturation when the thickness is much larger than the heating lengths scale.
In the case of equal layer thicknesses, the deformation is always directed towards
the less viscous fluid and its amplitude increases when decreasing the layers thicknesses.
When the viscosities of the layers are equal, the velocity is always larger in the thicker layer and the deformation
is always directed towards this same layer. In the special case where both viscosities and thicknesses are equal, there is no deformation
of the interface.
A second part was dedicated to a thin film separating two layers of the same liquid.
We showed that decreasing the ratio of film thickness to the external layer thickness increases the deformation which is directed towards the external
layer as far as the film is much thinner than the external layers.
Moreover, depending on the variation of interfacial tension with temperature, we showed that
a dimple or a couple of opposite noses can be formed in the thin film,
giving some insights for the explanation of drop coalescence observed in recent experiments.
Although simple, this analytical resolution is predictive and therefore advances an interesting tool to explain and predict the features taking place in
optofluidics experiments in which fluids are naturally confined.
\bibliographystyle{unsrt}
|
{
"timestamp": "2012-03-09T02:02:51",
"yymm": "1203",
"arxiv_id": "1203.1789",
"language": "en",
"url": "https://arxiv.org/abs/1203.1789"
}
|
\section{Introduction}
\label{Introduction}
\vspace{-0.35cm}
Some lattice groups have already started investing effort in studies involving charm-quark vacuum polarization in Lattice QCD simulations~\cite{lat10:gregorio}. The main goal of these is to exclude noticeable corrections due to charm-quark loops. One, for instance, could think of the non-perturbative computation of the quark masses and the $\Lambda$-parameter in QC
. If in a such a computation we can only use the three-flavor theory then the connection to the four-flavor theory can be done just perturbatively. The ALPHA-collaboration has already started computations in $N_f=4$ using as a first observable the running-coupling in the massless \srdg functional scheme\cite{alpha:nf4}. However, in order to set the scale, observables in the massive theory have to be computed. A typical lattice spacing is $a \sim (2 {\rm GeV})^{-1}$. Hence, the question of whether such computations are feasible is arising.
Our main worry occurs from the fact that in this case the charm-quark mass in lattice units is as large as $am_c = 1/2$. The main problem results from the findings of \cite{zastat:pap2,hqet:pap2} according to which in the valence-quark sector of the $\rmO(a)$-improved theory, although for masses up to $am_c = 1/2$ the cutoff effects are sizeable and obey approximately a quadratic behaviour in $a$, above this mass the Symanzik analysis and improvement of cutoff effects breaks down.
The purpose of our work is to investigate how big the cutoff effects in charm-quark vacuum polarization effects are. We, therefore, expand perturbatively a few observables in the renormalized coupling and investigate the dependence of the first non-trivial expansion coefficient on the lattice spacing and the quark mass. We make use of our previous work with H. Panagopoulos~\cite{pertforce:andreasharis} and that by Rainer Sommer and S. Sint~\cite{pert:1loop} in order to extract the lattice cutoff effects due to fermions in the ${\rm q{\overline q}}$-force $F(r)$ and the \srdg functional coupling respectively. Using the latter we defined three different observables, namely the step-scaling function $\Sigma$ and the renormalized quantities $\vbar$ and $\rho$; for these quantities we extract the cutoff effects. This script is a compact version of Ref.~\cite{alpha:potsf}.
\vspace{-0.65cm}
\section{Lattice Formulation}
\label{latticeformulation}
\vspace{-0.35cm}
We begin by commenting briefly on the most important features of the O($a$)-improved theory. All the relevant information is described in detail in~\Ref{impr:pap1}. According to Wilson's lattice regularization the total action $S=S_g+S_f$ is given by:
\vspace{-0.25cm}
\begin{eqnarray}
S_g[U] = \frac{1}{g_0^2}\sum_{p}w(p)\,\tr\,\{1-U(p)\} \ \ \ \ {\rm and} \ \ \ \ S_f[U,\bar\psi,\psi] = a^4\sum_{x}\sum_{i=1}^{\nf}\bar\psi_i(D+\mibare)\psi_i , \label{e:SgSf}
\end{eqnarray}
\vskip -0.25cm
\noindent
where the gluonic part $S_g$ is expressed as a sum over all oriented plaquettes $p$ with $U(p)$ the path-ordered product of the gauge fields around $p$. We consider an infinite lattice and the weight factors $w(p)$ are set to unity for now. The Dirac operator $D$ reads:
\vspace{-0.25cm}
\begin{equation}
D = \frac12 \sum_{\mu=0}^3
\{\gamma_\mu(\nabla_\mu^\ast+\nabla_\mu^{})- a\nabla_\mu^\ast\nabla_\mu^{}\}
+ \csw\,\frac{ia}{4}\sum_{\mu,\nu=0}^3
\,\sigma_{\mu\nu}\hat F_{\mu\nu}, \label{e:Dlat}
\end{equation}
\vskip -0.25cm
\noindent
with $\nabla_\mu$ and $\nabla_\mu^{\ast}$ the forward and backward covariant derivatives respectively and $\csw=1+\rmO\left(g_0^2\right)$. One can perform a systematic investigation of lattice cutoff effects after renormalizing the theory; hence, the cutoff effects depend upon the renormalization conditions. We, therefore, first use a massless renormalization scheme with scale $\mu$ and then a massive scheme at scale $\mu$ in order to investigate cutoff effects in lattice perturbation theory.
At one-loop order, the renormalized coupling and quark masses\footnote{At higher orders in perturbation theory, the sructure of the renormalization and the improvement is more complicated when all quark masses are considered. More information can be found in~\Ref{impr:nondeg}.} are given by:
\vspace{-0.25cm}
\bes
\gbar^2(\mu)=\gtilde^2\zg \left(\gtilde^2,a\mu\right)
\quad {\rm and} \quad
\mir(\mu)=\mqitilde + \rmO\left(g_0^2\right),
\ees
\vskip -0.25cm
\noindent
respectively in terms of the improved bare coupling and improved bare mass \cite{pert:1loop,impr:pap1} :
\vspace{-0.25cm}
\bes
\gtilde^2=g_0^2\left(1+{0.01200(2)}g_0^2 a\sum^{N_f}_{i=1} \mibare\right) + \rmO\left(g_0^6\right)
\quad {\rm and} \quad
\mqitilde=\mibare\left(1-\frac12a\mibare\right)\,.
\ees
\vskip -0.15cm
\noindent
For a complete ${\rm O}(a)$-improvement \`a la Symanzik one has to use the modified-bare coupling and modified-quark-mass in a massless renormalization scheme.
\vspace{-0.35cm}
\section{A systematic way of investigating the cutoff effects}
\vspace{-0.25cm}
\label{cutoffeffects}
It is usually normal to consider observables the perturbative expansion of which in the minimal-subtraction scheme $\gbarMSbar$ is known in the continuum. We, therefore, renormalize in the $\msbar$ scheme. We do so by first moving from the modified-bare coupling to the lattice-minimal-subtraction~(lat) scheme and then to the $\msbar$ scheme. The lattice-minimal-subtraction scheme is defined in such a way so that we subtract the logarithmic divergences in $a$ order by order in perturbation theory. For the perturbative order we are interested in:
\vspace{-0.25cm}
\begin{eqnarray}
\glat^2(\mu)=\gtilde^2\zlat\left(\gtilde^2,a\mu\right) \quad {\rm with} \quad \zlat\left(\gtilde^2,a\mu\right) = 1 -2b_0 \gtilde^2 \log(a\mu) + \rmO\left(\gtilde^4\right)\,,
\end{eqnarray}
\vskip -0.25cm
\noindent
where {\small ${b_0=\left( {11 N_c}/{3} - {2N_f}/{3} \right) / (4 \pi)^2}$}. We use observables $O$ that depend on a single length scale, let us say, $r$ and on the masses $\mir$. These have the following perturbative expansion:
\vspace{-0.25cm}
\bes
O = g_0^2 + O^{(1)}(\vecz,a/r)\,g_0^4 + \ldots\,,
\label{e:Oexpansion1}
\ees
\vskip -0.25cm
\noindent
where $\vecz = (z_1,\ldots,z_{\nf})$ and $z_i=\mir\cdot r$. After moving from the bare to the modified-bare-coupling and then to the lattice-minimal-subtraction scheme, the expansion of the observable takes the form:
\vspace{-0.75cm}
\bes
O = \tilde O_\mrm{cont}\left(r\mu,\vecz,\glat^2(\mu)\right)
\left(1 + \tilde\delta_O(r\mu,\vecz,\glat^2(\mu),a/r)\right),
\label{e:Oexpression1}
\ees
\vskip -0.25cm
\noindent
written as a continuum part and a lattice part $\tilde\delta_O(r\mu,\vecz,\glat^2(\mu),a/r)$. The latter is expressed as:
\vspace{-0.25cm}
\bes
\tilde\delta_O\left(r\mu,\vecz,\glat^2(\mu),a/r\right) =
\tilde\delta_O^{(0)}(r\mu,\vecz,a/r) +
\tilde\delta_O^{(1)}(r\mu,\vecz,a/r) \glat^2(\mu) + \ldots \,.
\ees
\vskip -0.25cm
\noindent
From the expression (\ref{e:Oexpression1}) $\tilde\delta_O(r\mu,\vecz,\glat^2(\mu),a/r = 0) =0$. We can now move to the $\msbar$ by applying the finite scheme transformation \cite{pert:2loopLW,pert:1loop,Bode:2001uz}:
\vspace{-0.25cm}
\bes
\glat^2(\mu) &=& \gbarMSbar^2 (\mu)
- \frac{c_{1}^{\rm lat,\MSbar}}{4 \pi} \gbarMSbar^4 (\mu)
+ \rmO \left( \gbarMSbar^6 (\mu) \right),
\label{e:defc1}
\\
c_1^{\rm lat,\MSbar}&=& -\frac{ \pi }{2 \nc} + 2.135730074078457(2) \nc -0.39574962(2) \nf \,.\quad
\ees
\vskip -0.25cm
\noindent
The expansion of the relative cutoff effects is written as:
\vspace{-0.25cm}
\bes
\delta_O\left(r\mu,\vecz,\gbarMSbar^2(\mu),a/r\right) &\equiv&
\left.{O - O_\mrm{cont} \over O_\mrm{cont}}\right|_{\gbarMSbar,\mir} = \delta_O^{(1)}(r\mu,\vecz,a/r) \gbarMSbar^2(\mu) + \ldots \,.
\label{e:arteexp}
\ees
\vskip -0.25cm
\noindent
Since the theory is renormalized by minimal-subtraction, these lattice artifacts are intrinsically perturbative and have no non-perturbative extension. Nevertheless, one can use combinations of $\delta_O^{(i)}$ from different observables in order to obtain the expansion coefficients of the non-perturbative cutoff effects. In the longer write-up~\cite{alpha:potsf} we demonstrate how by renormalizing i
\ the \srdg functional coupling $\gbarSF^2(L)$ for massless quarks we obtain the non-perturbative lattice artifacts at small couplings. Their fermionic part is approximately the same as that of Eq.~(\ref{e:arteexp}) i.e. $\sim \delta_O^{(1,f)}(r\mu,\vecz,a/r)$.
However, it is more convenient to introduce a massive renormalization scheme for non-perturbative computations in QCD with a charm-quark. This is due to the fact that the mass of the charm-quark is larger than the typical QCD scale, thus, it has reduced vacuum polarization effects which is most efficiently implemented by using a finite mass renormalisation scheme.
We can define a massive scheme with scale $\mu=1/r_0$ through a particular observable $O_0$:
\vspace{-0.25cm}
\bes
\gbar^2_{m}(\mu,\vecmr) \equiv O_0 = g_0^2 + O_0^{(1)}(\vecz,a/r)\,g_0^4 + \ldots\,.
\label{e:Oexpansion}
\ees
\vskip -0.25cm
\noindent
It is straightforward to show~\cite{alpha:potsf} that for an observable $O$ at a different length scale $r$ the relative artifacts in the massive scheme are given as a combination of those in the massless:
\vspace{-0.25cm}
\bes
\label{e:deltamassive}
\delta_{O_{\rm m}} \hspace{-1.0mm} \left( \hspace{-0.5mm} r/r_0,\vecz,\gbarm^2(\mu,\vecmr\hspace{-0.5mm}),a/r \hspace{-0.5mm} \right)
\equiv
\left[\hspace{-0.5mm} \delta_O^{(1)} \hspace{-0.5mm} ( \hspace{-0.5mm} r\mu,r\vecmr,a/r \hspace{-0.5mm} ) \hspace{-0.5mm} - \hspace{-0.5mm}
\delta_{O_0}^{(1)} \hspace{-0.5mm} ( \hspace{-0.5mm} r\mu,r_0\vecmr,a/r_0 \hspace{-0.5mm})\hspace{-0.5mm} \right]\, \hspace{-1.0mm}
\gbarm^2(\mu,\vecmr \hspace{-0.5mm}) \hspace{-0.5mm} + \hspace{-0.5mm} \ldots \,.
\ees
\vskip -0.25cm
\vspace{-0.45cm}
\section{The ${\rm q {\bar q}}$-static-Force.}
\label{Force}
\vspace{-0.25cm}
We start our investigation by considering the force $F(r)= \frac{d}{dr} V(r)$ between static quarks. If one uses on-axis potentials as we do here, the most natural choice to define the derivative would be: $F_\mrm{naive}(r_\mrm{naive}) = \frac1a \left[ V(r,0,0)-V(r-a,0,0) \right]$ with $r_\mrm{naive}=r-\frac{a}{2}$. However, the force can also be defined in terms of an improved ${\rm q {\bar q}}$-separation $r_I$ according to which the force has no effects at tree-level order in perturbation theory~\cite{pot:intermed}, thus, {\small $F_{\rm tree}(r_I)={\cf}/{4\pi \rI^2}$} with {\small $\cf = (N_c^2-1)/2N_c$}. The improved separations $r_I$ for on-axis potentials were calculated in~\cite{pot:intermed}. It is worth mentioning that the non-perturbative force in the pure gauge theory when defined in terms of $r_I$ has much smaller lattice artifacts compared to $r_{\rm naive}$. The results presented below are expressed in terms of $r_I$ while we have in addition looked at all quantities for $r_{\rm naive}$ with no any worth reporting changes; hence, from here on we have $r=r_I$.
The one-loop corrections to the force are calculated as described in~\cite{pertforce:andreasharis}. The expression of the force in the $\msbar$ scheme after setting the renormalization scale to the natural choice $\mu=1/r$ is given by:
\vspace{-0.25cm}
\bes
F = {\cf \alphaMSbar(1/r) \over r^2}
\left\{1 + f_1(\vecz,a/r) \alphaMSbar(1/r)
+ \rmO(\alphaMSbar^2) \right\}\,.
\ees
\vskip -0.25cm
\noindent
The term $f_1$ is split into a gluonic~($g$) and fermionic~($f$) contributions such as:
\vspace{-0.25cm}
\bes
f_1&=& f_{1,g}(a/r) + \sum^{N_f}_{i=1} f_{1,f} (z_i,a/r)\,,\quad
\ees
\vskip -0.25cm
\noindent
with continuum expressions~\cite{pot:Hoang} ($\gamma_E=0.57721566\dots$):
\vspace{-0.25cm}
\bes
f_{1,g}(0) &=& {\nc \over \pi} \left[ - \frac{35}{36} +\frac{11}{6} \gamma_{E} \right],
\label{e:cng} \\
f_{1,f} (z,0) & = & \frac{1}{2\pi} \left[ \frac{1}{3} {\rm log}(z^2)
+ \frac{2}{3} \int_{1}^{\infty} \rmd x
\frac{1}{x^2} \sqrt{x^2-1}\left(1+\frac{1}{2x^2} \right)
\left( 1+2 z x \right)e^{-2zx} \right] \,.
\label{e:cnf}
\ees
\vskip -0.25cm
\noindent
The relative lattice artifacts of the force are defined as:
\vspace{-0.25cm}
\bes
\left.{F - F_\mrm{cont} \over F_\mrm{cont}}\right|_{\gbarMSbar, \mir} =
\delta_F^{(1)}(\vecz,a/r) \gbarMSbar^2(1/r)+\dots,
\ees
\vskip -0.25cm
\noindent
with\footnote{The pre-factor $4 \pi$ on $\delta_F^{(1)}$ appears because we multiply with $\alpha=\gbar^2/(4\pi)$.}:
\vspace{-0.25cm}
\bes
4 \pi \delta_F^{(1)}(\vecz,a/r) = 4 \pi \delta_F^{(1,g)}(a/r) +
4 \pi \sum_{i=1}^{\nf} \delta_F^{(1,f)}(z_i,a/r),
\ees
\vskip -0.25cm
\noindent
The gluonic contribution to the force at finite $a/r$ is not our main concern. However, we need it for a complete picture on the size of the lattice cutoff effects. We, therefore, provide numbers for $\delta_F^{(1,g)}(a/r)$ extracted from~\Ref{pot:pertBB}. Concerning the fermionic piece, we perform a re-evaluation of $\delta_F^{(1,f)}(z,a/r)$ for the relevant bare-masses according to~\Ref{pertforce:andreasharis}. For more details on the procedure adopted to extract $\delta_F^{(1,f)}(z,a/r)$ see~\Ref{alpha:potsf}.
In order to obtain a first glimpse on the relevant size of the lattice artifacts let us view the gluonic part $4 \pi \delta_F^{(1,g)}(a/r)$. For $r/a=2.277$, $3.312$ and $4.319$, $4\pi \delta_F^{(1,g)}(a/r) = -0.232(6)$, $-0.190(19)$ and $-0.151(42)$ respectively. In Fig.~\ref{f:df1f} we display results for the fermionic contribution to the lattice artifacts $4 \pi \delta_F^{(1,f)}(z,a/r)$. The scale of the y-axis is about a factor two smaller than the pure gauge artifacts. Hence, the absolute value of the fermionic piece of the cutoff effects for a single flavor is much less than the gluonic. Concerning the mass dependence, we see that the fermionic cutoff effects depend very little on the mass.
\begin{figure}[tb!]
\vspace{-0.65cm}
\centerline{\scalebox{0.7}{\input{forceproc.tex}}}
\vspace*{-3mm}
\caption[]{\label{f:df1f}
The fermionic cutoff effects $4\pi \delta_F^{(1,f)}(z,a/r)$ for $z=0$~({\LARGE\Blue{$\diamond$}}),
$z=1$~({\small \Red{$\triangle$}}) and
$z=3$~($\triangledown$).}
\vspace{-0.45cm}
\end{figure}
\vspace{-0.35cm}
\section{\srdg Functional}
\label{schrodingerfunctional}
\vspace{-0.35cm}
We call \srdg functional the field theory in a finite space-time with volume $L^4$, Dirichlet boundary conditions along the time direction and periodic in space up to a phase $\theta$ for the quark fields. According to Monte-Carlo simulations~\cite{pert:1loop} the advantageous value of $\theta$ is $\theta=\pi/5$ while $\theta=0$ is the natural, and more aesthetically appealing, alternative. The action has the form~(\ref{e:SgSf}) with all fields being zero outside $0 \le x_0 \le L$, the gauge fields having fixed values at the boundaries $x_0=0$ and $L_0=0$ and the fermionic fields being zero at these boundaries. For a complete ${\rm O}(a)$-improvement one has to modify the weight factor of the timelike plaquettes attached to the boundaries $w(p)= \ct=1-[0.08900 + 0.019141\nf \pm 0.00005]g_0^2 +\ldots$; for more details we refer to \Refs{impr:pap1,SF:LNWW,pert:1loop}.
For our perturbative calculation we use the definition of the coupling appearing in \Refs{SF:LNWW,pert:1loop}. The \srdg functional coupling depends upon $L$ playing the r\^ole of the inverse renormalization scale, the phase $\theta$ and the dimensionless parameter $\nu$ which appears on the fixed boundary gauge fields and is usually set to zero. In a non-perturbative computation of the running of the coupling the main object needed is the step-scaling-function defined as:
\vspace{-0.25cm}
\bes
\Sigma(u,\vecz,a/L) \equiv \gbarSF^2(2L,2z)|_{\gbarSF^2(L,\vecz)=u, \mir L=z_i}
= u + \Sigma_1(\vecz,a/L)u^2 + \ldots \,.
\ees
\vskip -0.25cm
\noindent
However, in order to have a more general picture we can also look at other quantities such as:
\vspace{-0.25cm}
\bes
\Omega(u,\vecz,a/L) \equiv \vbar(L,\vecz)|_{\gbarSF^2(L,\vecz)=u, \mir L=z_i}
= \vbar_1(\vecz,a/L)+\rmO(u)\,,
\ees
\vskip -0.25cm
\noindent
with the quantity $\vbar$ defined explicitly by:
\vspace{-0.25cm}
\bes
{1\over \gbar^2_\nu(L,\nu,\vecz)} = {1\over\gbarSF^2(L,\vecz)}-\nu\, \vbar(L,\vecz)\,,
\ees
\vskip -0.25cm
\noindent
and
\vspace{-0.25cm}
\bes
\rho(u,\vecz,a/L) \equiv
\frac{\gbarSF^2(L,\vecz) - \gbarSF^2(L,\veczero)}{\gbarSF^2(L,\veczero)} \biggr\vert_{\gbarSF^2(L,0)=u, \mir L=z_i} =
\rho_1(\vecz,a/L)u + \rmO(u^2)\,.
\ees
\vskip -0.25cm
\noindent
For the step-scaling function ${\Sigma}$, we consider the relative lattice artifacts:
\vspace{-0.25cm}
\bes
\delta_\Sigma(u,\vecz,a/L) =
{\Sigma(u,\vecz,a/L) - \Sigma(u,\vecz,0) \over \Sigma(u,\vecz,0)}
= \delta_\Sigma^{(1)}(\vecz,a/L) u + \ldots \,,
\ees
\vskip -0.25cm
\noindent
while for both $\rho$ and ${\vbar}$ the absolute artifacts:
\vspace{-0.25cm}
\bes
\Delta_{\vbar}(u,\vecz,a/L) &\equiv& \Omega(u,\vecz,a/L) - \Omega(u,\vecz,0)
=
\Delta_{\vbar}^{(1)}(\vecz,a/L) + \rmO(u) \,,
\\
\Delta_\rho(u,\vecz,a/L) &\equiv& \rho(u,\vecz,a/L) - \rho(u,\vecz,0)
=
\Delta_\rho^{(1)}(\vecz,a/L)u + \rmO(u^2) \,.
\ees
\vskip -0.25cm
\noindent
The leading perturbative terms $\delta_\Sigma^{(1)}(\vecz,a/L)$, $\Delta_{\vbar}^{(1)}(\vecz,a/L)$ and $\Delta_\rho^{(1)}(\vecz,a/L)$ can be decomposed into gluonic and fermionic pieces:
\vspace{-0.25cm}
\bes
\delta_\Sigma^{(1)}(\vecz,a/L)
&=& \delta_\Sigma^{(1,g)}(\vecz,a/L) + {\rm \sum_{i=1}^{\nf}} \delta_\Sigma^{(1,f)}(z_i,a/L),
\\
\Delta_{\vbar}^{(1)}(\vecz,a/L)
&=& \Delta_{\vbar}^{(1,g)}(a/L) + \sum_{i=1}^{\nf}\Delta_{\vbar}^{(1,f)}(z_i,a/L),
\\
\Delta_{\rho}^{(1)}(\vecz,a/L)
&=& \sum_{i=1}^{\nf}\Delta_\rho^{(1,f)}(z_i,a/L)\,.
\ees
\vskip -0.25cm
\noindent
\begin{figure}[tb!]
\vskip -0.65cm
\centerline{\scalebox{1}{\scalebox{0.55}{\input{Stepscaling1.tex}} \hspace{0.0cm} \scalebox{0.55}{\input{Stepscaling2.tex}}}}
\centerline{\scalebox{1}{\scalebox{0.55}{\input{upsilon.tex}} \hspace{0.0cm} \scalebox{0.55}{\input{rho.tex}}}}
\vspace{-0.25cm}
\caption{{\underline{Upper-Left:} The cutoff effects $4\pi \delta_\Sigma^{(1,g)}(a/L)$ ({\large $\blacktriangledown$}) and $4\pi \delta_\Sigma^{(1,f)} (z,a/L)$ for $z=0$~({\LARGE \Red{$\bullet$}}), $z~=~1$~({\large\Magenta{$\blacktriangle$}}) and $z=3$~({\Blue{$\blacksquare$}}) extracted for $\theta=\pi/5$.} {\underline{Upper-Right:} $4\pi \delta_\Sigma^{(1,f)} (z,a/L)$ for $z=0$~({\LARGE\Red{$\bullet$}}), $z=1$~({\large\Magenta{$\blacktriangle$}}) and $z=3$~({\Blue{$\blacksquare$}}) extracted for $\theta=0$.} {\underline{Lower-Left:} The cutoff effects
$\Delta_{\vbar}^{(1,g)}(a/L)$ ({\small $\blacktriangledown$}) and $\Delta_{\vbar}^{(1,f)}(z,a/L)$ for $z=0$ ({\LARGE \Red{$\bullet$}}~({\scriptsize\Magenta{$\bigcirc$}})) and
$z=3$ ({\Blue{$\blacksquare$}} ({\Green{$\square$}})) extracted for $\theta=\pi/5 \ (\theta=0)$.} {\underline{Lower-Right:} The cutoff effects $4\pi \Delta_{\rho}^{(1,f)}(z,a/L)$ for $z=1$~({\LARGE $\bullet$} ({\scriptsize $\bigcirc$})), $z=2$~({\large\Red{$\blacktriangle$}} ({\small\Red{$\triangle$}}))
and $z=3$~({\Blue{$\blacksquare$}} ({\Blue{$\square$}})) extracted for $\theta=\pi/5$ ($\theta=0$).}}
\label{sfunctional}
\vspace{-0.35cm}
\end{figure}
In the upper-left and upper-right plots of Fig.~\ref{sfunctional} we show the artifact $4\pi \delta_\Sigma^{(1,f)} (z,a/L)$ for $\theta=\pi/5$ and $\theta=0$ respectively. The cutoff effects for an individual ${\rm O}(a)$-improved fermion are much smaller than those for the gluonic part. They do appear to grow with $z$, however, not very much. Of course one should always bear in mind that for more than one fermion flavor the cutoff effects add up accordingly.
If we now move to the cutoff effects of the ${\vbar}$ which appears in the lower-left plot of Fig.~\ref{sfunctional}, we observe that are bigger compared to the step-scaling function (given that the overall magnitude is $\vbar_1 \approx 0.1$). In contrast to what we observed before, here, the cutoff effects decrease with the mass.
Finally, in the lower-right plot of Fig.~\ref{sfunctional} we present the absolute lattice artifacts in $\rho$. These have to be compared to the continnum values of $4\pi {\rho}(z,0) $ which range from 0.095 (0.086) for $z=1$ up to 0.188 (0.170) for $z=3$ at $\theta=\pi/5$ ($\theta=0$).
\vspace{-0.35cm}
\section{Conclusion}
\label{conclusions}
\vspace{-0.35cm}
We have investigated the cutoff effects of several observables in two rather different setups from including an ${\rm O}(a)$-improved Wilson charm-quark at one-loop order in lattice perturbation theory. We always restrict ourselves to $am_c<1/2$. We observed that these are comparable to the cutoff effects of the pure gluonic part; for instance see the lower-left plot of Fig.~\ref{sfunctional}. As a matter of fact, in some cases the fermionic pieces of the cutoff effects are even smaller than their pure gluonic ones. This is demonstrated clearly in the upper-left plot of Fig.~\ref{sfunctional}. The reason of studying these was to check whether they become very large for very-massive quarks and more specifically for masses associated with the charm-quark. In contrast to our expectations of seeing large cutoff effects, we observe that they do not grow much with the mass and sometimes even decrease for larger masses. The overall conclusion is that the lattice artifacts remain small as we increase the mass and, thus, the inclusion of charm-quarks in dynamical fermion simulations does not seem to be a problem with the current available lattice spacings. However, non-perturbative investigations of lattice artifacts have to be carried out.
\vspace{-0.35cm}
\section*{Acknowledgements}
\vspace{-0.35cm}
We \hspace{\stretch{1}} would \hspace{\stretch{1}} like \hspace{\stretch{1}} to \hspace{\stretch{1}} thank \hspace{\stretch{1}} Rainer Sommer, \hspace{\stretch{1}} Haris Panagopoulos \hspace{\stretch{1}} and \hspace{\stretch{1}} Ulli \hspace{\stretch{1}} Wolff \hspace{\stretch{1}} for \hspace{\stretch{1}} usefull \hspace{\stretch{1}} discussions \hspace{\stretch{1}} and \hspace{\stretch{1}} Matthias \hspace{\stretch{1}} Steinhauser \hspace{\stretch{1}} for \hspace{\stretch{1}} pointing \hspace{\stretch{1}} out \hspace{\stretch{1}} Eq.~(\ref{e:cnf}). \hspace{\stretch{1}} The numerical computations were carried out on the compute farm at DESY, Zeuthen. This work is supported by the SFB/TR 09 of the Deutsche Forschungsgemeinschaft. \hspace{\stretch{1}}
\vspace{-0.35cm}
|
{
"timestamp": "2012-03-13T01:00:21",
"yymm": "1203",
"arxiv_id": "1203.2197",
"language": "en",
"url": "https://arxiv.org/abs/1203.2197"
}
|
\section{Introduction and statement of results}
A celebrated result due to Erd\"os and Tur\'an says that, for a
univariate polynomial over $\mathbb{C}$ whose middle coefficients are not too
big with respect to its extremal coefficients, the arguments of its roots are
approximately equidistributed~\cite{ET50}. Combined with a recent
result of Hughes and Nikeghbali~\cite{HN08}, this shows that the roots
of such a polynomial cluster near the unit circle.
We introduce some notation to make this result precise. Let $Z$ be an effective
cycle of~$\mathbb{C}^{\times}=\mathbb{C}\setminus \{0\}$ of dimension 0, that is, a formal finite sum
\begin{displaymath}
Z=\sum_{\xi} m_{\xi}[\xi]
\end{displaymath}
with $\xi\in \mathbb{C}^{\times}$ and $m_{\xi}\in \mathbb{N}$ with $m_\xi=0$ for all
but finitely many $\xi$, as in \cite[\S1.3]{Fulton:IT}. The degree of
$Z$, denoted $\deg(Z)$, is defined as the sum of its multiplicities
$m_{\xi}$. We assume that $Z\ne 0$ or, equivalently, that
$\deg(Z)\ge1$.
For each $ -\pi \le \alpha<\beta\le \pi$, consider the cycle
\begin{displaymath}
Z_{\alpha,\beta}= \sum_{\alpha < \arg(\xi) \le \beta} m_{\xi}[\xi] ,
\end{displaymath}
where $\arg(\xi)$ denotes the argument of $\xi$. The {\em angle
discrepancy} of $Z$ is defined as
\begin{equation*}
\Delta_{\rm ang}(Z)= \sup_{-\pi \le \alpha<\beta\le \pi} \bigg| \frac{\deg(Z_{\alpha,\beta})}{\deg(Z)}- \frac{\beta-\alpha}{2\pi}\bigg|.
\end{equation*}
For example, when $Z$ is the zero set of $x^d-1$ in
$\mathbb{C}^\times$, we have that $\Delta_{\rm ang}(Z)=\frac{1}{d}$.
For $0<\varepsilon <1$, consider also the cycle
\begin{displaymath}
Z_{\varepsilon}= \sum_{1-\varepsilon <
|\xi| < ({1-\varepsilon})^{-1}} m_{\xi}[\xi].
\end{displaymath}
The {\em radius discrepancy} of $Z$ with
respect to $\varepsilon$ is defined as
$$
\Delta_{\rm rad}(Z, \varepsilon)=1- \frac{\deg(Z_{\varepsilon})}{\deg(Z)}.
$$
For example, when $Z$ is the zero set of
$x^d-1$ in $\mathbb{C}^\times$, we have that $\Delta_{\rm rad}(Z,\varepsilon)=0$ for all $\varepsilon$.
For a polynomial $f\in \mathbb{C}[x]\setminus\{0\}$, we denote by $Z(f)$ the
0-dimensional effective cycle of $\mathbb{C}^{\times}$ defined by its roots
and their corresponding multiplicities. We also set
$\|f\|_{\sup}=\sup_{|z|=1}|f(z)|$.
\begin{theorem}\label{mtunivariate}
Let $f= a_0+\cdots+a_dx^d \in \mathbb{C}[x] $
with $d\ge1$ and $a_0a_d \ne 0,$ and $0<\varepsilon <1$. Then
\begin{equation*}
\Delta_{\rm ang}(Z(f)) \le c\, \sqrt{\frac{1}{d}\log\bigg(\frac{\|f\|_{\sup}}{\sqrt{|a_0a_d|}}\bigg)},\quad
\Delta_{\rm rad}(Z(f),\varepsilon) \le \frac{2}{\varepsilon d} \, \log\bigg(\frac{\|f\|_{\sup}}{\sqrt{|a_0a_d|}}\bigg),
\end{equation*}
with $c=\sqrt{2\pi/G}= 2.5619\dots$, where
$G=\sum_{m=0}^\infty
\frac{(-1)^{m}}{(2m+1)^2}=0.915965594\dots$ is Catalan's constant.
\end{theorem}
The more interesting (and hardest) part is the bound for the angle discrepancy.
The original Erd\"os-Tur\'an result states \cite{ET50}
$$
\Delta_{\rm ang}(Z(f)) \le 16\, \sqrt{\frac{1}{d}\log\bigg(\frac{\sum_{j}|a_{j}|}{\sqrt{|a_0a_d|}}\bigg)}.
$$
A few years after that paper, Ganelius \cite{Gan54} replaced the
$\ell^{1}$-norm $\sum_{j}|a_{j}|$ by the smaller quantity $\Vert
f\Vert_{\sup}$ and improved the value of the constant to $c\le
\sqrt{2\pi/G}$. On the other hand, Amoroso and Mignotte
\cite{AM96} showed that the optimal value of $c$ cannot be smaller
than $\sqrt{2}$. The bound for the radius discrepancy is due to Hughes
and Nikeghbali~\cite{HN08}.
\medskip Here, we study the distribution of the solutions of a system
of multivariate polynomial equations in the algebraic torus
$(\mathbb{C}^\times)^n$. For instance, consider the following
system of bivariate polynomials:
\begin{equation}\label{33}
f_{1}=x_{1}^{13}+x_{1}x_{2}^{12}+x_{2}^{13}+1, \quad
f_{2}=x_{1}^{12}x_{2}-x_{2}^{13}-x_{1}x_{2}+1 \quad \in \mathbb{C}[x_{1},x_{2}].
\end{equation}
These are polynomials with moderate degree and small integer
coefficients. By direct computation, we can verify that the solutions
in $(\mathbb{C}^{\times})^{2}$ of the system of equations $f_{1}=f_{2}=0$ are
aproximately equidistributed near the unit polycircle $S^{1}\times S^1 $
(Figure~\ref{fig:6}).
\begin{figure}[htpb]
\begin{tabular}{ccc}
\includegraphics[scale=0.30]{13angular}&\hspace*{6mm}& \includegraphics[scale=0.30]{13radial}
\end{tabular}
\vspace{-3mm}\caption{Angle and radius distribution of the zeros of
the system \eqref{33}}\label{fig:6}
\end{figure}
This example and others of the same kind suggest that Theorem
\ref{mtunivariate} has an extension to higher dimensions.
The study of the distribution of the solutions of a system of
multivariate polynomial equations has been addressed from different
perspectives. For instance, Khovanskii's theorem on complex fewnomials
~\cite[\S 3.13, Theorem~2]{Kho91} gives an estimate for the
distribution of the arguments of these solutions in terms of the
number of monomials and the Newton polytopes of the input
system. There are also several interesting results by Shiffman,
Zelditch and Bloom on the asymptotic distribution of the solutions of
a random system of polynomial equations, see for instance \cite{SZ04,
BS07} and the references therein.
Our purpose in this text is to obtain an extension of
Theorem \ref{mtunivariate} to systems of Laurent polynomials
with a given support. For $i=1,\dots,n$, let ${\mathcal A}_i$ be a non-empty
finite subset of $\mathbb{Z}^{n}$ and $Q_{i}=\operatorname{conv}({\mathcal A}_{i})\subset\mathbb{R}^{n}$ its convex
hull. Set $D=\operatorname{MV}_{\mathbb{R}^{n}}(Q_{1},\dots,Q_{n})$ for the mixed volume of these
lattice polytopes, and assume that $D\ge1$. For each $i$,
let $f_{i}$ be a Laurent polynomial with support contained in
${\mathcal A}_{i}$, that is,
\begin{displaymath}
f_i=\sum_{{\boldsymbol{a}}\in {\mathcal A}_{i}}\alpha_{i,{\boldsymbol{a}}} \, {\boldsymbol{x}}^{{\boldsymbol{a}}}\in
\mathbb{C}[x_1^{\pm1},\dots, x_n^{\pm1}]
\end{displaymath}
with $\alpha_{i,{\boldsymbol{a}}}\in \mathbb{C}$ and ${\boldsymbol{x}}^{{\boldsymbol{a}}}=x_{1}^{a_{1}}\dots
x_{n}^{a_{n}}$ for each ${\boldsymbol{a}}=(a_{1},\dots,a_{n})\in {\mathcal A}_{i}$. We
write~${\boldsymbol{f}}=(f_{1},\dots, f_{n})$ for short. We assume that, for all vectors
${\boldsymbol{v}}\in \mathbb{Z}^{n}$, the directional resultant
$\operatorname{Res}_{{\mathcal A}_{1}^{{\boldsymbol{v}}},\dots,{\mathcal A}_{n}^{{\boldsymbol{v}}}}(f_{1}^{{\boldsymbol{v}}},\dots,f_{n}^{{\boldsymbol{v}}})$
(Definition \ref{def:1}) is nonzero. This condition holds for a
generic choice of~${\boldsymbol{f}}$ in the space of coefficients and, by
Bernstein's theorem \cite[Theorem~B]{Ber75}, it implies that all the
solutions of $f_{1}=\dots=f_{n}=0$ are isolated and that their number,
counted with multiplicities, is equal to $D$.
For a vector ${\boldsymbol{w}}\in S^{n-1}$ in the unit sphere of $\mathbb{R}^n$, we denote
by ${\boldsymbol{w}}^{\bot}\subset \mathbb{R}^{n}$ its orthogonal subspace and by
$\pi_{{\boldsymbol{w}}}\colon \mathbb{R}^{n}\to {\boldsymbol{w}}^{\bot}$ the corresponding orthogonal
projection. We denote by $\operatorname{MV}_{{\boldsymbol{w}}^{\bot}}$ the mixed volume of convex
bodies of ${\boldsymbol{w}}^{\bot}$ induced by the Euclidean measure on
${\boldsymbol{w}}^{\bot}$ and, for $i=1,\dots,n$, we set
\begin{displaymath}
D_{{\boldsymbol{w}},i}=\operatorname{MV}_{{\boldsymbol{w}}^\bot}\big(\pi_{\boldsymbol{w}}(Q_1),\dots,
\pi_{\boldsymbol{w}}(Q_{i-1}),\pi_{\boldsymbol{w}}(Q_{i+1}),\dots,\pi_{\boldsymbol{w}}(Q_n)).
\end{displaymath}
We then define the {\em Erd\"os-Tur\'an size} of ${\boldsymbol{f}}$ as
\begin{equation} \label{eq:1}
\eta({\boldsymbol{f}})= \frac{1}{D} \sup_{{\boldsymbol{w}}\in S^{n-1}}
\log\Bigg(\frac{\prod_{i=1}^{n}\|f_i\|_{\sup}^{D_{{\boldsymbol{w}},i}}}
{ \prod_{{\boldsymbol{v}}}|\operatorname{Res}_{{\mathcal A}_{1}^{{\boldsymbol{v}}},\dots,{\mathcal A}_{n}^{{\boldsymbol{v}}}}(f_{1}^{{\boldsymbol{v}}},\dots,f_{n}^{{\boldsymbol{v}}})|^{\frac{|\langle
{\boldsymbol{v}},{\boldsymbol{w}} \rangle|}{2}}}\Bigg),
\end{equation}
where the second product is over all primitive vectors ${\boldsymbol{v}}\in\mathbb{Z}^n$
that is, vectors whose coordinates do not have a non-trivial common
factor, and $\langle \cdot,\cdot\rangle$ is the standard inner product
of $\mathbb{R}^n$. This product is finite because
$\operatorname{Res}_{{\mathcal A}_{1}^{{\boldsymbol{v}}},\dots,{\mathcal A}_{n}^{{\boldsymbol{v}}}}(f_{1}^{{\boldsymbol{v}}},\dots,f_{n}^{{\boldsymbol{v}}})\neq
1$ only if~${\boldsymbol{v}}$ is an inner normal to a facet of the Minkowski sum
$Q_{1}+\cdots +Q_{n}$.
The Erd\"os-Tur\'an size is a generalization
to the multivariate case of the quantity
$\frac{1}{d}\log\Big(\frac{\|f\|_{\sup}}{\sqrt{|a_0a_d|}}\Big)$ that
appears in Theorem \ref{mtunivariate} since, for $n=1$, it is easily
checked that $\eta({\boldsymbol{f}})$ is exactly the preceding quantity
(Proposition \ref{prop:2}).
Let $Z({\boldsymbol{f}})$ denote the 0-dimensional effective cycle
of~$(\mathbb{C}^\times)^n$ defined by the roots of~${\boldsymbol{f}}$ and their multiplicities. The
angle and radius discrepancies of cycles of $(\mathbb{C}^{\times})^{n}$ are
the obvious generalization of those for the univariate case (see
Definition~\ref{def:3}).
Our main result is the following:
\begin{theorem} \label{thm:1} For $n\ge2$, let ${\mathcal A}_{1},\dots,{\mathcal A}_{n}$ be non-empty
finite subsets of $ \mathbb{Z}^{n} $, set $Q_{i}=\operatorname{conv}({\mathcal A}_{i})$ and assume
that $\operatorname{MV}_{\mathbb{R}^{n}}(Q_{1},\dots,Q_{n})\ge1$. Let $f_{1},\dots, f_{n}\in
\mathbb{C}[x_{1}^{\pm1},\dots, x_{n}^{\pm1}]$ with $\operatorname{supp}(f_{i})\subset
{\mathcal A}_{i}$ and such that
$\operatorname{Res}_{{\mathcal A}_{1}^{{\boldsymbol{v}}},\dots,{\mathcal A}_{n}^{{\boldsymbol{v}}}}(f_{1}^{{\boldsymbol{v}}},\dots,f_{n}^{{\boldsymbol{v}}})\ne
0$ for all ${\boldsymbol{v}}\in \mathbb{Z}^{n}\setminus \{{\boldsymbol{0}}\}$. Then
\begin{equation*}
\Delta_{\rm ang}(Z({\boldsymbol{f}}))\leq 66\,n\, 2^{n} ( 18+
\log^{+}({\eta({\boldsymbol{f}})^{-1}}))^{\frac23(n-1)} \eta({\boldsymbol{f}})^{\frac13}
\end{equation*}
with $\log^{+}(x)=\log(\max(1,x))$ for $x>0$.
Also, for $0<\varepsilon <1,$
\begin{equation*}
\Delta_{\rm rad}(Z({\boldsymbol{f}}),\varepsilon) \le \frac{2n}{\varepsilon} \eta({\boldsymbol{f}}).
\end{equation*}
\end{theorem}
Theorem \ref{mtunivariate} shows that these bounds for the angle and the
radius discrepancy also hold in the case $n=1$. By analogy with
the one-dimensional case, it is natural to ask if, in the setting of
our result, a stronger inequality of the form
\begin{displaymath}
\Delta_{\rm ang}(Z({\boldsymbol{f}}))\leq
c(n)\,\eta({\boldsymbol{f}})^{\frac12}
\end{displaymath}
holds, with $c(n)>0$ not depending on ${\boldsymbol{f}}$. It would be interesting to settle this question.
\medskip
Theorem \ref{thm:1} has several consequences. For instance, we can
derive from it a bound for the number of positive real solutions of a
system of polynomial equations, in terms of its Erd\"os-Tur\'an
size. For a cycle $Z=\sum_{{\boldsymbol{\xi}}}m_{{\boldsymbol{\xi}}}[{\boldsymbol{\xi}}] $ of $(\mathbb{C}^{\times})^{n}$,
set
\begin{displaymath}
Z_{+}= \sum_{{\boldsymbol{\xi}} \in (\mathbb{R}_{>0})^{n}}m_{{\boldsymbol{\xi}}}[{\boldsymbol{\xi}}].
\end{displaymath}
The following statement follows immediately from Theorem \ref{thm:1} and
the definition of the angle discrepancy.
\begin{corollary}
\label{cor:3}
Let notation be as in Theorem \ref{thm:1}. Then
\begin{displaymath}
\deg(Z({\boldsymbol{f}})_{+})\leq 66\,n\, 2^{n} ( 18+
\log^{+}({\eta({\boldsymbol{f}})^{-1}}))^{\frac23(n-1)} \eta({\boldsymbol{f}})^{\frac13}
\, \deg(Z({\boldsymbol{f}})).
\end{displaymath}
\end{corollary}
We can also apply our result to study the asymptotic distribution of
the roots of a sequence of systems of polynomials over $\mathbb{Z}$ with
growing supports and whose coefficients are not too big. To be more
precise, let $Q_i$, $i=1,\dots, n$, be lattice polytopes in $\mathbb{R}^{n}$
such that $\operatorname{MV}_{\mathbb{R}^{n}}(Q_1,\ldots, Q_n)\ge1$. For each integer $\kappa\ge 1$
and $i=1,\dots, n$, consider the finite subset of $\mathbb{Z}^{n}$ given by
\begin{equation}
\label{eq:3}
{\mathcal A}_{\kappa, i}=\kappa Q_i\cap\mathbb{Z}^n .
\end{equation}
For a Laurent
polynomial $f\in \mathbb{C}[x_{1}^{\pm1},\dots,x_{n}^{\pm1}]$, we denote by
$\operatorname{supp}(f)$ its support, defined as the subset of $\mathbb{Z}^{n}$ of its
exponent vectors. We also set
\begin{displaymath}
\|f\|_{\sup}=\sup_{|w_{1}|=1,\dots, |w_{n}|=1}|f(w_{1},\dots,w_{n})|.
\end{displaymath}
For a nonzero cycle $Z=\sum_{{\boldsymbol{\xi}}}m_{{\boldsymbol{\xi}}}[{\boldsymbol{\xi}}]$ of
$(\mathbb{C}^{\times})^{n}$, we consider the discrete probability measure on
$(\mathbb{C}^{\times})^{n}$ defined by
$$
\delta_{Z}= \frac{1}{\deg(Z)}\sum_{{\boldsymbol{\xi}}}m_{{\boldsymbol{\xi}}} \delta_{\boldsymbol{\xi}} ,
$$
where $\delta_{\boldsymbol{\xi}}$ is the Dirac measure supported on the point
${\boldsymbol{\xi}}$. Let $\nu_{\text{\rm Haar}}$ be the measure on~$(\mathbb{C}^{\times})^{n}$ supported on
$(S^{1})^{n}$ and whose restriction to this polycircle coincides
with its Haar measure of total mass 1.
Recall that a sequence of measures $(\nu_{\kappa})_{\kappa\in\mathbb{N}}$
on~$(\mathbb{C}^{\times})^{n}$ converges weakly to $\nu_{\text{\rm Haar}}$ if, for every
continuous function with compact support~$h\colon (\mathbb{C}^{\times})^{n}\to
\mathbb{R}$, it holds
\begin{displaymath}
\lim_{\kappa\to\infty}\int_{(\mathbb{C}^{\times})^{n}}h \, \d\nu_{\kappa}=\int_{(\mathbb{C}^{\times})^{n}}h \, \d\nu_{\text{\rm Haar}}.
\end{displaymath}
If this is the case, we write $\lim_{\kappa\to \infty}\nu_{\kappa}=\nu_{\text{\rm Haar}}$.
\begin{theorem} \label{cor:2} For $\kappa\ge 1$, let ${\boldsymbol{f}}_{\kappa}=(f_{\kappa, 1},\dots,
f_{\kappa, n}) $ be a family of Laurent polynomials in $\mathbb{Z}[x_1^{\pm1},\dots,
x_n^{\pm1}] $ such that $ \operatorname{supp}(f_{\kappa, i})\subset \kappa Q_{i}$, $
\log\|f_{\kappa, i}\|_{\sup}= o(\kappa)$, and
$\operatorname{Res}_{{\mathcal A}_{\kappa, 1}^{{\boldsymbol{v}}},\dots,{\mathcal A}_{\kappa, n}^{{\boldsymbol{v}}}}(f_{\kappa, 1}^{{\boldsymbol{v}}},\ldots,f_{\kappa, n}^{{\boldsymbol{v}}})\ne0$
for all ${\boldsymbol{v}}\in \mathbb{Z}^{n}\setminus \{{\boldsymbol{0}}\}$. Then
\begin{displaymath}
\lim_{\kappa\to\infty}\delta_{Z({\boldsymbol{f}}_{\kappa})} = \nu_{\text{\rm Haar}}.
\end{displaymath}
\end{theorem}
This result admits a quantitative version giving information on the
rate of convergence, which we state in Proposition \ref{prop:7}.
Theorem~\ref{cor:2} is related to Bilu's equidistribution theorem for
the Galois orbit of algebraic points in $(\mathbb{C}^{\times})^{n}$ of small
height \cite{Bil97} which, at least for $n=1$, also admits
quantitative versions \cite{Pet05,FR06}.
\medskip We can also apply Theorem \ref{thm:1} to study the
distribution of the roots of a random system of Laurent polynomials
over $\mathbb{C}$. We will show that, under some mild conditions and
without assuming any independence or equidistribution condition on the
coefficients of the system, these roots tend to cluster uniformly near
$(S^{1})^{n}$.
To state this result, let us keep notation as above and set
${\boldsymbol{\mathcal A}}_{\kappa}=({\mathcal A}_{\kappa,1},\dots, {\mathcal A}_{\kappa,n})$ with ${\mathcal A}_{\kappa,i}$ as
in~\eqref{eq:3}. Each point of the projective space
$\P(\mathbb{C}^{{\boldsymbol{\mathcal A}}_{\kappa}})$ can be identified with a system
${\boldsymbol{f}}_\kappa=(f_{\kappa,1},\dots,f_{\kappa,n})$ of Laurent polynomials such that
$\operatorname{supp}(f_{\kappa,i})\subset \kappa Q_{i}$, $i=1,\dots, n$, modulo a
multiplicative scalar. The associated cycle $Z({\boldsymbol{f}}_\kappa)$ is
well-defined, since it does not depend on this multiplicative scalar.
Let $\mu_{\kappa}$ be the normalized Fubini-Study measure on
$\P(\mathbb{C}^{{\boldsymbol{\mathcal A}}_{\kappa}})$ of total mass 1, and $\lambda_{\kappa}$ a probability
density function on $ \P(\mathbb{C}^{{\boldsymbol{\mathcal A}}_{\kappa}})$ (see \S \ref{sec:equid-prob}
for details). Let ${\boldsymbol{f}}_\kappa$ be a random system of Laurent polynomials
with $\operatorname{supp}(f_{\kappa,i})\subset \kappa Q_{i}$, $i=1,\dots, n$, distributed
according to the probability law given by $\lambda_{\kappa}$ with respect to
$\mu_{\kappa}$. The \emph{expected zero density measure} of ${\boldsymbol{f}}_\kappa$ is the
measure on $(\mathbb{C}^{\times})^{n}$ defined, for a Borel subset $U$, as
\begin{displaymath}
\mathbb{E}(Z({\boldsymbol{f}}_\kappa);
\lambda_\kappa)(U)=\int_{\P(\mathbb{C}^{{\boldsymbol{\mathcal A}}_\kappa})}{\deg(Z({\boldsymbol{f}}_\kappa)|_{U})}\, \lambda_{\kappa}({\boldsymbol{f}}_\kappa)\,\d\mu_\kappa,
\end{displaymath}
where $Z({\boldsymbol{f}}_\kappa)|_{U}$ denotes the cycle $\sum_{{\boldsymbol{\xi}}\in V({\boldsymbol{f}}_\kappa)_{0}\cap U}m_{\boldsymbol{\xi}}[{\boldsymbol{\xi}}].$
\begin{theorem} \label{thm:3} For $\kappa\ge1$, let $\lambda_{\kappa}$ be
a probability density function on $\P(\mathbb{C}^{{\boldsymbol{\mathcal A}}_{\kappa}})$ with
respect to the measure $\mu_{\kappa}$, and
${\boldsymbol{f}}_\kappa=(f_{\kappa,1},\dots,f_{\kappa,n})$ a random system of
Laurent polynomials with $\operatorname{supp}(f_{\kappa,i})\subset \kappa Q_{i}$,
$i=1,\dots, n$, distributed according to the probability law given
by $\lambda_{\kappa}$. Assume that the sequence
$(\lambda_{\kappa})_{\kappa\ge1}$ is uniformly bounded. Then
\begin{displaymath}
\lim_{\kappa\to\infty}\frac{\mathbb{E}(Z({\boldsymbol{f}}_\kappa);{\lambda_{\kappa}})}{\kappa^{n}\operatorname{MV}_{\mathbb{R}^{n}}(Q_{1},\dots,Q_{n})} =\nu_{\text{\rm Haar}}.
\end{displaymath}
\end{theorem}
As an application, consider a random system of Laurent polynomials
${\boldsymbol{f}}_\kappa$ with $\operatorname{supp}(f_{\kappa, i})\subset \kappa Q_{i}$ whose coefficients are
independent complex Gaussian random variables with mean 0 and variance
1. The random cycle $Z({\boldsymbol{f}}_\kappa )$ might be described by the uniform
distribution on $\P(\mathbb{C}^{{\boldsymbol{\mathcal A}}_{\kappa}})$ (see Example \ref{exm:1} for
details). Then, Theorem~\ref{thm:3} implies that the roots of
${\boldsymbol{f}}_\kappa $ converge weakly to the equidistribution on $(S^{1})^{n}$, and
we recover in this way a result of Bloom and Shiffman
\cite[Example~3.5]{BS07}.
\medskip Our strategy for proving Theorem \ref{thm:1} consists of
reducing to the univariate case. In \S \ref{sec:angle-radi-distr}, we
consider the problem of studying the angle and radius discrepancies of
an arbitrary 0-dimensional effective cycle $Z$ in $(\mathbb{C}^{\times})^{n}$
in terms of the angle and radius discrepancies of its direct images
under all monomial projections of $(\mathbb{C}^{\times})^{n}$
onto~$\mathbb{C}^{\times}$. By applying a tomography process based on Fourier
analysis, we show that the distribution of $Z$ can be controlled in
terms of the distribution of its projections (Theorem~\ref{thm:2}).
In~\S \ref{sec:bounds-discr-terms}, we consider cycles defined by a
system of Laurent polynomials with given support and we compute their
direct image under monomial projections, in terms of sparse
resultants. Theorem \ref{thm:1} then follows by applying
Erd\"os-Tur\'an's theorem combined with Theorem~\ref{thm:2}, and the
basic properties of the sparse resultant.
In \S~\ref{sec:equid-prob}, we study the asymptotic distribution of
the roots of a sequence of systems of Laurent polynomials over $\mathbb{Z}$ and of
random systems of Laurent polynomials over~$\mathbb{C}$.
In both situations, the key step consists of bounding from below the size of
the relevant directional resultants. In the case of systems over $\mathbb{Z}$, this
is trivial since these directional resultants are nonzero integer
numbers. In the case of random systems over $\mathbb{C}$, the result follows
from an estimate of the volume of a tube around an algebraic
variety due to Beltr\'an and Pardo~\cite{BP:edcnsm}.
\vspace{3mm}
\noindent{\bf Acknowledgments.}
We thank Carlos Beltr\'an and Michael Shub for useful discussions and
pointers to the literature. Part of this work was done while the
authors met at Universitat de Barcelona and Universit\'e de
Nice--Sophia Antipolis. We thank these institutions for their
hospitality.
\section{Angle and radius distribution in the multivariate case}\label{sec:angle-radi-distr}
In this section, we show that the angle and radius discrepancies of an
effective 0-dimensional cycle in the algebraic torus $(\mathbb{C}^{\times})^{n}$ can be
bounded in terms of the angle discrepancy of its image under monomial
maps from $(\mathbb{C}^{\times})^{n}$ to $\mathbb{C}^{\times}$.
Let $Z$ be a nonzero effective 0-dimensional
cycle of $(\mathbb{C}^{\times})^{n}$, which we write as a finite sum
\begin{displaymath}
Z=\sum_{{\boldsymbol{\xi}}}m_{{\boldsymbol{\xi}}}[{\boldsymbol{\xi}}]
\end{displaymath}
with $m_{{\boldsymbol{\xi}}}\in\mathbb{N}$ and ${\boldsymbol{\xi}}\in (\mathbb{C}^\times)^n$.
The \emph{support} of $Z$ is the finite subset of $(\mathbb{C}^{\times})^{n}$
defined as $|Z|=\{{\boldsymbol{\xi}}\mid m_{{\boldsymbol{\xi}}}\ge 1\}$, and the \emph{degree} of $Z$ is the positive number
$\deg(Z)=\sum_{\boldsymbol{\xi}} m_{\boldsymbol{\xi}}.$
\begin{definition} \label{def:3}
Let $ Z=\sum_{{\boldsymbol{\xi}}}m_{{\boldsymbol{\xi}}}[{\boldsymbol{\xi}}] $ be a nonzero effective 0-dimensional
cycle of $(\mathbb{C}^{\times})^{n}$.
For each ${\boldsymbol{\alpha}}=(\alpha_{1},\dots, \alpha_{n})$ and
${\boldsymbol{\beta}}=(\beta_{1},\dots, \beta_{n})$ with $-\pi\le
\alpha_{j}<\beta_{j}\le \pi$, $j=1,\dots, n$, consider the cycle
\begin{displaymath}
Z_{{\boldsymbol{\alpha}},{\boldsymbol{\beta}}}=\sum_{-\alpha_{j}<\arg(\xi_{j})\le \beta_{j}} m_{{\boldsymbol{\xi}}}[{\boldsymbol{\xi}}].
\end{displaymath}
The
{\em angle discrepancy} of $Z$ is defined as
\begin{equation*}
\Delta_{\rm ang}(Z)= \sup_{{\boldsymbol{\alpha}}, {\boldsymbol{\beta}}}\bigg| \frac{\deg(Z_{{\boldsymbol{\alpha}}, {\boldsymbol{\beta}}})}{\deg(Z)}-
\prod_{j=1}^{n}\frac{\beta_{j}-\alpha_{j}}{2\pi}\bigg|.
\end{equation*}
Let $0<\varepsilon <1$ and consider also the cycle
\begin{displaymath}
Z_{\varepsilon}= \sum_{1-\varepsilon<|\xi_j|<(1-\varepsilon)^{-1}} m_{{\boldsymbol{\xi}}}[{\boldsymbol{\xi}}],
\end{displaymath}
where $\xi_{j}$ is the $j$-th coordinate of ${\boldsymbol{\xi}}$.
The {\em radius discrepancy} of $Z$ with
respect to $\varepsilon$ is defined as
$$
\Delta_{\rm rad}(Z, \varepsilon)=1- \frac{\deg(Z_{\varepsilon})}{\deg(Z)}.
$$
\end{definition}
We have $0< \Delta_{\rm ang}(Z)\leq1$. Observe also that $0\le
\Delta_{\rm rad}(Z,\varepsilon)\leq1$ and $\Delta_{\rm rad}(Z, \varepsilon)=0$ for
all $\varepsilon$ if and only if $|Z|\subset (S^{1})^{n}$.
For a lattice point ${\boldsymbol{a}}=(a_{1},\dots,a_{n})\in \mathbb{Z}^{n}$ we denote by
$\chi^{{\boldsymbol{a}}}\colon (\mathbb{C}^{\times})^{n}\to \mathbb{C}^{\times}$ the associated
character, defined
as $\chi^{{\boldsymbol{a}}}({\boldsymbol{\xi}})=\xi_{1}^{a_{1}}\dots\xi_{n}^{a_{n}}$ for ${\boldsymbol{\xi}}\in(\mathbb{C}^{\times})^{n}$.
The direct image of $Z$ under $\chi^{{\boldsymbol{a}}}$ is the cycle of
$\mathbb{C}^{\times}$ given by
\begin{displaymath}
\chi^{{\boldsymbol{a}}}_{*}(Z)= \sum_{{\boldsymbol{\xi}}} m_{{\boldsymbol{\xi}}}\chi^{{\boldsymbol{a}}}({\boldsymbol{\xi}}).
\end{displaymath}
We also set
\begin{equation}\label{theta}
\theta(Z)=\sup_{{\boldsymbol{a}}\in \mathbb{Z}^{n}\setminus \{{\boldsymbol{0}}\}}
\frac{\Delta_{\rm ang}(\chi^{{\boldsymbol{a}}}_{*}(Z))}{\|{\boldsymbol{a}}\|_{2}^{\frac12}}, \quad
\rho(Z,\varepsilon)= \sum_{j=1}^{n}\Delta_{\rm rad}(\chi^{{\boldsymbol{e}}_{j}}_{*}(Z), \varepsilon),
\end{equation}
where ${\boldsymbol{e}}_{j}$ denotes the $j$-th vector in the standard basis of
$\mathbb{Z}^{n},$ and $\|{\boldsymbol{a}}\|_2$ is the Euclidean norm of the vector ${\boldsymbol{a}}\in\mathbb{Z}^n$.
We have $0<\theta(Z)\leq1$ and $ 0\le \rho(Z,\varepsilon)\le 1$.
\begin{theorem} \label{thm:2}
Let $Z$ be a nonzero effective 0-dimensional
cycle of $(\mathbb{C}^{\times})^{n}$. Then
$$
\Delta_{\rm ang}(Z)\le 22n\bigg(\frac83\bigg)^{n}
(9-\log(\theta(Z)))^{\frac23(n-1)} \, \theta(Z)^{\frac23}
$$
and, for $0<\varepsilon <1$,
\begin{displaymath}
\Delta_{\rm rad}(Z ,\varepsilon)
\le \rho(Z,\varepsilon).
\end{displaymath}
\end{theorem}
The rest of this section is devoted to the proof of this result.
Given two vectors
${\boldsymbol{u}}=(u_{1},\dots,u_{n}),{\boldsymbol{v}}=(v_{1},\dots,v_{n})\in \mathbb{R}^{n}$ we write
$\langle{\boldsymbol{u}},{\boldsymbol{v}}\rangle=\sum_{j=1}^{n}u_{j}v_{j}$ for their standard
inner product, and for ${\boldsymbol{\xi}}\in(\mathbb{C}^\times)^n$ we set $\arg({\boldsymbol{\xi}})=
(\arg(\xi_{1}),\dots, \arg(\xi_{n}))\in (-\pi,\pi]^n$.
\begin{lemma}
\label{lemm:1}
Let $Z$ be a nonzero effective 0-dimensional
cycle of $(\mathbb{C}^{\times})^{n}$ and ${\boldsymbol{a}}\in \mathbb{Z}^{n}\setminus \{{\boldsymbol{0}}\}$. Then
\begin{displaymath}
\Bigg|\frac{1}{\deg(Z)} \sum_{{\boldsymbol{\xi}}} m_{{\boldsymbol{\xi}}}\operatorname{e}^{\i \langle {\boldsymbol{a}},
\arg({\boldsymbol{\xi}})\rangle}\Bigg| \le 2\pi\Delta_{\rm ang}(\chi^{{\boldsymbol{a}}}_{*}(Z)).
\end{displaymath}
\end{lemma}
Note that for ${\boldsymbol{a}}={{\boldsymbol{0}}}$ we get $\frac{1}{\deg(Z)}\sum_{{\boldsymbol{\xi}}} m_{{\boldsymbol{\xi}}}\operatorname{e}^{\i \langle {{\boldsymbol{a}}}, \arg({\boldsymbol{\xi}})\rangle}=1$.
\begin{proof}
Set $D_{0}=\#|Z|$ and $D=\deg(Z)$ for short. Let $-\pi\leq \nu_{k}<\pi$,
$k=1,\dots, D_{0}$, denote the inner products $\langle {\boldsymbol{a}},
\arg({\boldsymbol{\xi}})\rangle$ modulo $2\pi$ for the different points ${\boldsymbol{\xi}}$ in the support
of $Z$, and let $m_{k}$ denote their corresponding multiplicity. We
suppose that these numbers are arranged in increasing order, that
is, $\nu_{1}\le \dots\le \nu_{D_{0}}$.
For $-\pi<\nu\le \pi$ set
$$
N(\nu)=\sum_{k\mid \nu_k\leq\nu}m_{k}.
$$
We have
\begin{multline} \label{eq:2}
\int_{-\pi}^{\pi}N(\nu)\operatorname{e}^{\i\nu}d\nu =\sum_{k=1}^{D_{0}}\int_{\nu_{k}}^{\nu_{k+1}}
\bigg(\sum_{l\le k}m_{l}\bigg)\operatorname{e}^{\i\nu} d\nu \\=
-\i\sum_{k}\bigg(\sum_{l\le k}m_{l}\bigg)\,\operatorname{e}^{\i\nu}\bigg|_{\nu_{k}}^{\nu_{k+1}}=\i\Big(D+\sum_{k}m_{k}\operatorname{e}^{\i\nu_k}\Big),
\end{multline}
where we have set $\nu_{D_{0}+1}=\pi$.
On the other hand, an easy calculation shows that
\begin{equation}\label{dos}
\int_{-\pi}^{\pi}\frac{\nu+\pi}{2\pi}
\operatorname{e}^{\i\nu}d\nu=\i.
\end{equation}
Combining (\ref{eq:2}) and (\ref{dos}), we deduce that
$$
\frac{1}{D} \sum_{{\boldsymbol{\xi}}} m_{{\boldsymbol{\xi}}}\operatorname{e}^{\i \langle {\boldsymbol{a}},
\arg({\boldsymbol{\xi}})\rangle}=\frac{1}{D}\sum_{k=1}^{D_{0}}m_{k}\operatorname{e}^{\i\nu_k}=\i\int_{-\pi}^{\pi}\left(\frac{\nu+\pi}{2\pi}-\frac{N(\nu)}{D}\right)\operatorname{e}^{\i\nu}d\nu.
$$
Hence
\begin{multline*}
\left|\frac{1}{D} \sum_{{\boldsymbol{\xi}}} m_{{\boldsymbol{\xi}}}\operatorname{e}^{\i \langle {\boldsymbol{a}},
\arg({\boldsymbol{\xi}})\rangle}\right|\leq\int_{-\pi}^{\pi}\left|\frac{\nu+\pi}{2\pi}-\frac{N(\nu)}{D}\right|d\nu
\\=\int_{-\pi}^{\pi}\left|\frac{\nu+\pi}{2\pi}-\frac{\deg(\chi^{{\boldsymbol{a}}}_{*}(Z)_{-\pi,\nu})}{\deg(\chi^{{\boldsymbol{a}}}_{*}(Z))}\right|d\nu\le 2\pi\Delta_{\rm ang}(\chi^{{\boldsymbol{a}}}_{*}(Z)),
\end{multline*}
which concludes the proof.
\end{proof}
Let $\alpha,\beta,\tau\in\mathbb{R}$ such that $\alpha\leq\beta$ and
$\tau>0$. We consider the function
$h_{\alpha,\beta,\tau}\colon \mathbb{R}\to \mathbb{R}$ defined, for $x\in \mathbb{R}$, by
$$
h_{\alpha,\beta,\tau}(x)=
\begin{cases}
0& \mbox{if }\, x\leq\alpha-\tau,\\
g\big(\frac{x-\alpha+\tau}{\tau}\big)&\mbox{if}\,\alpha-\tau<x\leq\alpha,\\
1&\mbox{if }\, \alpha<x\leq\beta,\\
g\big(\frac{\beta+\tau-x}{\tau}\big) &\mbox{if }\,\beta<x\leq\beta+\tau,\\
0&\mbox{if }\, \beta+\tau<x,
\end{cases}
$$
with $g(x)=-2x^3+3x^2$.
Lemma \ref{univariate} below shows that $h_{\alpha,\beta,\tau}$ is an approximation of the
characteristic function of the interval $[\alpha, \beta]$.
For $m\in\mathbb{N}$, we denote by ${\mathcal C}^m(\mathbb{R})$ the space of functions $f\colon \mathbb{R}\to\mathbb{R}$ having $m$~continuous derivatives.
\begin{lemma}\label{univariate}
Let $\alpha,\beta,\tau\in\mathbb{R}$ such that $\alpha\leq\beta$ and
$\tau>0$. Then
\begin{enumerate}
\item \label{item:1} $h_{\alpha,\beta,\tau}\in{\mathcal C}^1(\mathbb{R})$;
\item \label{item:2} $h_{\alpha,\beta,\tau}(x)=1$ for $x\in
[\alpha, \beta]$, $h_{\alpha,\beta,\tau}(x)=0$ for $x\in
(-\infty, \alpha - \tau] \cup [\beta+\tau, \infty)$,
and
$0\leq h_{\alpha,\beta,\tau}(x)\leq1$ for all $x\in\mathbb{R}$;
\item \label{item:3} $\int_{-\infty}^{\infty}
h_{\alpha,\beta,\tau}dx=\beta-\alpha+\tau$ and,
moreover,
$\int_{\alpha-\tau}^\alpha
h_{\alpha,\beta,\tau}dx=\int_\beta^{\beta+\tau}h_{\alpha,\beta,\tau}dx=\frac{\tau}{2}$;
\item \label{item:4} $\int_{-\infty}^{\infty}|h'_{\alpha,\beta,\tau}|dx=2$;
\item \label{item:5} $\int_{-\infty}^{\infty}|h''_{\alpha,\beta,\tau}|dx=\frac{6}{\tau}.$
\end{enumerate}
\end{lemma}
\begin{proof}
By a direct calculation, we verify that the function $g$ satisfies the following properties:
\begin{itemize}
\item $g(x)\geq0$ for all $x\in[0,1]$;
\item $g(0)=g'(0)=0,\ g(1)=1,\,g'(1)=0$;
\item $\int_0^1g\,dx=\frac12$;
\item $\int_0^1|g'|\,dx=1$;
\item $\int_0^1|g''|\,dx=3.$
\end{itemize}
The claim follows easily from these properties and the definition of
$h_{\alpha,\beta,\tau}$.
\end{proof}
Suppose furthermore that $\beta-\alpha+2\tau<2\pi.$ The support of
$h_{\alpha,\beta,\tau}$ is then contained in an interval of length
bounded by $2\pi$, and so this function can be regarded as a function
on $\mathbb{R}/2\pi\mathbb{Z}$. For $a\in \mathbb{Z}$ set
$c_{a}=\frac{1}{2\pi}\int_{-\pi}^{\pi}h_{\alpha,\beta,\tau}(x)\operatorname{e}^{-\i
ax}\d x$, so that its Fourier series is given by
$$\sum_{a\in\mathbb{Z}}c_{a}\operatorname{e}^{\i ax}.
$$
\begin{lemma}\label{acotacionFourier} Let
$\alpha\le \beta$ and $\tau>0$ such that
$\beta-\alpha+2\tau<2\pi$. Then $\sum_{a\in\mathbb{Z}}c_{a}\operatorname{e}^{\i ax}$
converges absolutely and uniformly on $\mathbb{R}/2\pi\mathbb{Z}$ to
$h_{\alpha,\beta,\tau}$. Moreover,
$c_0=\frac{\beta-\alpha+\tau}{2\pi}, $ and, for $a\neq0$,
\begin{equation}\label{eq:7}
|c_a|\le \min \Big\{ \frac{1}{\pi\,a}, \frac{3}{\pi\,\tau\,a^2}\Big\}.
\end{equation}
\end{lemma}
\begin{proof}
Lemma \ref{univariate}\eqref{item:1} implies that the series
$\sum_{a\in\mathbb{Z}}c_{a}\operatorname{e}^{\i ax}$ converges absolutely and uniformly on
$\mathbb{R}/2\pi\mathbb{Z}$ to the function $h_{\alpha,\beta,\tau}$. The computation
of $c_0$ follows from Lemma
\ref{univariate}\eqref{item:3}. Integrating by parts, we deduce for
$a\in\mathbb{Z}\setminus \{0\}$ that
$$
c_a=\frac{1}{-2\pi
\i a}\int_{-\pi}^{\pi}h'_{\alpha,\beta,\tau}(x)\operatorname{e}^{-\i
ax}dx=\frac{1}{2\pi(-\i a)^2}\int_{-\pi}^{\pi}h''_{\alpha,\beta,\tau}(x)\operatorname{e}^{-\i
ax}dx.
$$
Hence, $|c_a|\le \frac{1}{2\pi
a}\int_{-\pi}^{\pi}|h'_{\alpha,\beta,\tau}|dx$ and also $|c_a|\le
\frac{1}{2\pi a^2}\int_{-\pi}^{\pi}|h''_{\alpha,\beta,\tau}|dx$. Then \eqref{eq:7}
follows by bounding these integrals with Lemma
\ref{univariate}(\ref{item:4}-\ref{item:5}).
\end{proof}
Next, we apply Fourier analysis to control the angle discrepancy of $Z$
in terms of the angle discrepancy of its direct image under monomial
projections.
\begin{lemma}\label{est33}
Let $n\ge 2,\, q \in \mathbb{Z}_{\ge 1}$, ${\boldsymbol{\alpha}}=(\alpha_{1},\dots, \alpha_{n})$ and
${\boldsymbol{\beta}}=(\beta_{1},\dots, \beta_{n})$ with $\alpha_{j},\beta_{j}\in
\mathbb{R}$ such that $-\pi\leq\alpha_j<\beta_j<\pi$ and
$\beta_{j}-\alpha_{j}+\frac{2}{q }< 2\pi$. Then
\begin{displaymath}
\left|\frac{\deg(Z_{{\boldsymbol{\alpha}},{\boldsymbol{\beta}}})}{\deg(Z)}-\prod_{j=1}^n\frac{\beta_j-\alpha_j}{2\pi}\right|
\le 2\, n\, \theta(Z) +\frac{3\, n}{2\, \pi\, q }
+n\frac{2^{n+3}\sqrt{3}}{\pi^{n-1}}
q ^{\frac12}(9+
\log({q }))^{n-1} \, \theta(Z).
\end{displaymath}
\end{lemma}
\begin{proof}
Set $\tau= \frac{1}{q }$ and
$h_{{\boldsymbol{\alpha}},{\boldsymbol{\beta}},\tau}({\boldsymbol{\nu}})=\prod_{j=1}^nh_{\alpha_j,\beta_j,\tau}(\nu_j)$
for ${\boldsymbol{\nu}}=(\nu_{1},\dots, \nu_{n})\in \mathbb{R}^{n}$. Set also $D=\deg(Z)$ and
\begin{align*}
\Sigma_1&=\left|\frac{\deg(Z_{{\boldsymbol{\alpha}},{\boldsymbol{\beta}}})}{D}-\frac{1}{D}\sum_{{\boldsymbol{\xi}}}m_{{\boldsymbol{\xi}}}h_{{\boldsymbol{\alpha}},{\boldsymbol{\beta}},\tau}(\arg({\boldsymbol{\xi}}))\right|,\\
\Sigma_2&=\left|\frac{1}{D}\sum_{{\boldsymbol{\xi}}}m_{{\boldsymbol{\xi}}}h_{{\boldsymbol{\alpha}},{\boldsymbol{\beta}},\tau}(\arg({\boldsymbol{\xi}}))-\prod_{j=1}^n\frac{\beta_j-\alpha_j+\tau }{2\pi}\right|,\\
\Sigma_3&=\left|\prod_{j=1}^n\frac{\beta_j-\alpha_j+\tau }{2\pi}
- \prod_{j=1}^n\frac{\beta_j-\alpha_j}{2\pi}\right|.
\end{align*}
We will bound each of these quantities. For $\Sigma_{1}$,
we consider the subset of $\mathbb{R}^{n}$ given by
$I_{{\boldsymbol{\alpha}},{\boldsymbol{\beta}},\tau}=\prod_{j=1}^n[\alpha_{j}-\tau,\beta_{j}+\tau]\setminus\prod_{j=1}^n
[\alpha_{j},\beta_{j}]$. Then
$$
\Sigma_1=\Bigg|\frac{1}{D}\sum_{\arg({\boldsymbol{\xi}})\in I_{{\boldsymbol{\alpha}},{\boldsymbol{\beta}},\tau}} m_{{\boldsymbol{\xi}}}
h_{{\boldsymbol{\alpha}},{\boldsymbol{\beta}},\tau}(\arg({\boldsymbol{\xi}}))\Bigg|.
$$
For each ${\boldsymbol{\xi}}$ such that $\arg({\boldsymbol{\xi}}) \in I_{{\boldsymbol{\alpha}},{\boldsymbol{\beta}},\tau}$, there is $1\leq j\leq n$ such that either $\alpha_j-\tau <
\arg(\xi_{j})\le \alpha_{j}$ or
$\beta_{j}<\arg(\xi_{j})<\beta_{j}+\tau.$
Since $0\leq h_{{\boldsymbol{\alpha}},{\boldsymbol{\beta}},\tau}({\boldsymbol{\nu}})\leq 1$ for all
${\boldsymbol{\nu}},$ we have that~$\Sigma_{1}$ is bounded from above by
$$
\frac{1}{D}\sum_{j=1}^n\big(\deg(\chi^{{\boldsymbol{e}}_{j}}_{*}(Z)_{\alpha_{j}-\tau ,\alpha_{j}})+
\deg(\chi^{{\boldsymbol{e}}_{j}}_{*}(Z)_{\beta_{j}, \beta_{j}+\tau })\big).
$$
By using the definition of $\Delta_{\rm ang}$ and Lemma \ref{lemm:1}, we get
\begin{displaymath}
\frac{1}{D}\deg(\chi^{{\boldsymbol{e}}_{j}}_{*}(Z)_{\alpha_{j}-\tau ,\alpha_{j}})\le
\Delta_{\rm ang}(\chi^{{\boldsymbol{e}}_{j}}_{*}(Z))+ \frac{\tau }{2\pi}\le
\theta(Z)+ \frac{\tau }{2\pi}=\theta(Z)+ \frac{1}{2\pi q },
\end{displaymath}
and a similar bound holds for $\deg(\chi^{{\boldsymbol{e}}_{j}}_{*}(Z)_{\beta_{j},
\beta_{j}+\tau })$. Hence,
\begin{equation}\label{s1}
\Sigma_1\leq 2n\theta(Z) +\frac{n}{\pi q }.
\end{equation}
Now we turn to $\Sigma_{2}$. Due to the conditions imposed on $\tau$, we can
regard~$h_{{\boldsymbol{\alpha}},{\boldsymbol{\beta}},\tau}$ as a function on~$\mathbb{R}^{n}/2\pi
\mathbb{Z}^{n}\simeq (-\pi,\pi]^{n}$. Let $\sum_{{\boldsymbol{a}}\in\mathbb{Z}^n}c_{\boldsymbol{a}}
\operatorname{e}^{\i\langle{\boldsymbol{a}},{\boldsymbol{\nu}}\rangle}$ be its multivariate Fourier
series. For $j=1,\ldots, n,$ we denote with
$\sum_{a_{j}\in\mathbb{Z}}c_{j,a_{j}}\operatorname{e}^{\i a_{j}\nu_j}$ the Fourier series
of $h_{\alpha_j,\beta_j,\tau }$. Then, for each ${\boldsymbol{a}}=(a_1,\ldots,a_n)\in \mathbb{Z}^{n},$
$$c_{\boldsymbol{a}}=\prod_{j=1}^{n}c_{j,a_j}.
$$
In particular, $c_{{\boldsymbol{0}}}=\prod_{j=1}^n\frac{\beta_j-\alpha_j+\tau }{2\pi}$.
The Fourier series of each $h_{\alpha_j,\beta_j,\tau }$ converges
absolutely to this function, and so the same holds for the Fourier series of
$h_{{\boldsymbol{\alpha}},{\boldsymbol{\beta}},\tau}$. Hence,
$$h_{{\boldsymbol{\alpha}},{\boldsymbol{\beta}},\tau}({\boldsymbol{\nu}})
=\sum_{{\boldsymbol{a}}\in\mathbb{Z}^n}c_{\boldsymbol{a}} \operatorname{e}^{\i\langle{\boldsymbol{a}},{\boldsymbol{\nu}}\rangle}
$$
for ${\boldsymbol{\nu}}\in(-\pi,\pi]^{n}$.
Applying Lemma \ref{lemm:1}, we obtain
\begin{multline} \label{cota1}
\Sigma_2=\bigg|\frac{1}{D}\sum_{{\boldsymbol{\xi}}}m_{{\boldsymbol{\xi}}}\sum_{{\boldsymbol{a}}\neq{{\boldsymbol{0}}}}c_{\boldsymbol{a}} \operatorname{e}^{\i\langle{\boldsymbol{a}},\arg({\boldsymbol{\xi}})\rangle}\bigg|
\leq\sum_{{\boldsymbol{a}}\neq0}|c_{\boldsymbol{a}}|\bigg|\frac{1}{D}\sum_{{\boldsymbol{\xi}}}m_{{\boldsymbol{\xi}}}\operatorname{e}^{\i\langle{\boldsymbol{a}},\arg({\boldsymbol{\xi}})\rangle}\bigg|\\
\leq 2\pi\,\theta(Z) \sum_{{\boldsymbol{a}}\neq0}|c_{\boldsymbol{a}}|{\|{\boldsymbol{a}}\|_2^{\frac12}}
\leq 2^{n+1}\pi\,\theta(Z)\sum_{{\boldsymbol{a}}\geq{{\boldsymbol{0}}}}\bigg(\sum_{s=1}^{n} \sqrt{a_s}\bigg)|c_{\boldsymbol{a}}|
\end{multline}
For $s=1$,
\begin{align*}
\sum_{{\boldsymbol{a}}\geq{{\boldsymbol{0}}}}\sqrt{a_1}|c_{\boldsymbol{a}}|
=\bigg(\sum_{a_1\geq0}\sqrt{a_1}|c_{1,a_1}|\bigg)\prod_{j=2}^n \bigg(\sum_{a_j\geq0}|c_{j,a_j}|\bigg).
\end{align*}
Using the bounds in~\eqref{eq:7} we get, for $j=2,\ldots,n,$
\begin{align}\label{mijack}
\nonumber \sum_{a_j\geq0}|c_{j,a_j}|&\leq|c_{j,0}|+\sum_{a_j=1}^{3q }
\frac{1}{a_{j}\pi}
+\sum_{a_j=3q +1}^\infty\frac{3}{\pi\tau a_{j}^2}\\
\nonumber &\leq
\frac{\beta_{j}-\alpha_{j}+\tau}{2\pi}+ \bigg(1+ \frac{1}{\pi}
\int_{1}^{3q }\frac{dx}{x} \bigg) +
\frac{3}{\pi\tau} \int_{3q }^{\infty}\frac{dx}{x^{2}}
\\
\nonumber &\leq \frac{2\pi}{2\pi}+
1+\frac{1}{\pi}\log\Big(\frac{3}{\tau }\Big)+\frac{3}{\pi\tau } \frac{\tau}{3}\\
\leq\frac{1}{\pi}\big(9+\log q \big).
\end{align}
Similarly, we now bound
\begin{align}\label{roboc}
\nonumber
\sum_{a_1\geq0}\sqrt{a_1}|c_{1,a_1}|&\leq\sum_{a_1=1}^{3q }\frac{1}{\pi\sqrt{a_1}}
+\sum_{a_1=3q +1}^\infty\frac{3}{\pi\tau\, a_{1}\sqrt{a_1}}\\
\nonumber &\le \frac{1}{\pi} \int_{0}^{3q }{x^{-\frac12}}{dx} +
\frac{3}{\pi\tau} \int_{3q }^{\infty}{x^{-\frac32}}{dx}
\\
\nonumber &\leq
\frac{2}{\pi}\bigg(\frac{\tau}{3}\bigg)^{-\frac12}+\frac{6}{\pi\tau}\bigg(\frac{\tau}{3}\bigg)^{\frac12}\\
&\le \frac{4\sqrt{3}}{\pi}{q }^{\frac12}.
\end{align}
It follows from \eqref{cota1}, \eqref{mijack} and \eqref{roboc} that
\begin{equation}\label{s2}
\Sigma_2\leq n\frac{2^{n+3}\sqrt{3}}{\pi^{n-1}}
q ^{\frac12}(9+
\log({q }))^{n-1} \, \theta(Z).
\end{equation}
Next we consider $\Sigma_{3}$. Set
$\phi(t)=\prod_{j=1}^n\frac{\beta_{j}-\alpha_{j}+\tau t}{2\pi}$ for
$t\in \mathbb{R}$. There exists $0<t_{0}<1$ such that
$\Sigma_{3}=|\phi(1)-\phi(0)| = |\phi'(t_{0})|$ and so
\begin{displaymath}
\Sigma_{3}\le \sup_{0<t_{0}<1}|\phi'(t_{0})| \le
\sum_{j=1}^{n}\frac{\tau }{2\pi}\prod_{\ell\neq
j}\frac{\beta_{\ell}-\alpha_{\ell}+\tau}{2\pi}
\le {n\tau}\frac{(2\pi)^{n-1}}{(2\pi)^{n}}=\frac{n}{2\pi q }.
\end{displaymath}
Finally, we collect (\ref{s1}), (\ref{s2}) and the above inequality to get
\begin{align*}
\bigg|\frac{\deg(Z_{{\boldsymbol{\alpha}},{\boldsymbol{\beta}}})}{\deg(Z)}-\prod_{j=1}^n\frac{\beta_j-\alpha_j}{2\pi}\bigg|
\leq & \Sigma_1+\Sigma_2+\Sigma_3\\
\le & 2n\theta(Z) +\frac{n}{\pi q }
+n\frac{2^{n+3}\sqrt{3}}{\pi^{n-1}}
q ^{\frac12}(9+
\log({q }))^{n-1} \, \theta(Z)
+ \frac{n}{2\pi q }\\
= & 2n\theta(Z) +\frac32\frac{n}{\pi q }
+n\frac{2^{n+3}\sqrt{3}}{\pi^{n-1}}
q ^{\frac12}(9+
\log({q }))^{n-1} \, \theta(Z).
\end{align*}
\end{proof}
\begin{proof}[Proof of Theorem \ref{thm:2}]
For the radius discrepancy, we have
$$
|Z_{\varepsilon}|
= \bigcap_{j=1}^n
\Big\{{\boldsymbol{\xi}} \in |Z|\mid 1-\varepsilon < |\xi_j| < \frac{1}{1-\varepsilon} \Big\}.
$$
By taking complements in this equality and considering the corresponding
multiplicities, we deduce that
\begin{displaymath}
\deg(Z) - \deg(Z_{\varepsilon})\le
\sum_{j=1}^{n}\deg(\chi^{{\boldsymbol{e}}_{j}}_{*}(Z))- \deg(\chi^{{\boldsymbol{e}}_{j}}_{*}(Z)_{\varepsilon}).
\end{displaymath}
Hence,
\begin{math}
\Delta_{\rm rad}(Z,\varepsilon) \le\sum_{j=1}^n
\Delta_{\rm rad}(\chi^{{\boldsymbol{e}}_{j}}_{*}(Z,\varepsilon)\big)= \rho(Z, \varepsilon),
\end{math}
as stated.
We now consider the bound for the angle discrepancy. For $n=1$,
$\Delta_{\rm ang}(Z)\le \theta(Z)$ and so the
bound in the claim is trivial. Hence, we suppose that $n\geq2$.
Put then $\zeta(Z)=
(9 -\log(\theta(Z)))^{\frac23(n-1)}\, \theta(Z)^{2/3}\in \mathbb{R}_{>0}$ for
short and set
\begin{displaymath}
q =\left\lfloor \frac{9 ^{\frac23(n-1)}}{\zeta(Z)}\right\rfloor .
\end{displaymath}
Suppose also that $q \ge 1$. Then
\begin{equation}\label{eq:12}
\frac{9 ^{\frac23(n-1)}}{2\zeta(Z)}< q \leq\frac{9 ^{\frac23(n-1)}}{\zeta(Z)}\le\frac{1}{\theta(Z)}.
\end{equation}
Let ${\boldsymbol{\alpha}}=(\alpha_{1},\dots, \alpha_{n})$ and
${\boldsymbol{\beta}}=(\beta_{1},\dots, \beta_{n})$ with $-\pi\le
\alpha_{j}<\beta_{j}\le \pi$. Consider first the case where
$\beta_j-\alpha_j\leq\pi$. In particular,
$\beta_j-\alpha_j+\frac{2}{q }<2\pi$. Applying Lemma \ref{est33}
and the inequalities \eqref{eq:12}, we deduce that the quantity
$\Big|\frac{\deg(Z_{{\boldsymbol{\alpha}},{\boldsymbol{\beta}}})}{\deg(Z)}
-\prod_{j=1}^n\frac{\beta_j-\alpha_j}{2\pi}\Big|$ is bounded from above by
\begin{displaymath}
2n\frac{\zeta(Z)}{9 ^{\frac23(n-1)}}
+\frac{3\, n}{2\, \pi}\frac{2\, \zeta(Z)}{9 ^{\frac23(n-1)}}
+n\frac{2^{n+3}\sqrt{3}}{\pi^{n-1}}
\bigg(\frac{9 ^{\frac23(n-1)}}{\zeta(Z)}\bigg)^{\frac12}(9 -
\log(\theta(Z)))^{n-1} \, \theta(Z) .
\end{displaymath}
Since $n\ge2$, this quantity can be bounded by
\begin{multline*}
n \bigg(\frac{2}{9 ^{\frac23(n-1)}} +\frac{3}{9 ^{\frac23(n-1)}\pi}+\frac{2^{n+3}9 ^{\frac13(n-1)}\sqrt{3}}{\pi^{n-1}}\bigg)
\zeta(Z) \\
\le n\bigg(1+\frac{2^{3}\sqrt{3}\pi}{9 ^{\frac13}}\bigg(\frac{2\cdot9 ^{\frac13}}{\pi}\bigg)^{n} \bigg) \zeta(Z)
\leq 22\,n\left(\frac43\right)^n\zeta(Z),
\end{multline*}
as it can be easily verified that
$\frac{2\cdot9 ^{\frac13}}{\pi}<\frac43$ and
$\frac{2^{3}\sqrt{3}\pi}{9 ^{\frac13}}<21$.
If $q =0$, then $\frac{9 ^{\frac23(n-1)}}{\zeta(Z)} <1$, which implies
that $ \Delta_{\rm ang}(Z)\le 1\le \zeta(Z)$. Hence, in the case where $-\pi\le
\alpha_{j}<\beta_{j}\le \pi$ for all $j$, we have
\begin{equation} \label{eq:15}
\Big|\frac{\deg(Z_{{\boldsymbol{\alpha}},{\boldsymbol{\beta}}})}{\deg(Z)}
-\prod_{j=1}^n\frac{\beta_j-\alpha_j}{2\pi}\Big|\le 22\,n\left(\frac43\right)^n\zeta(Z).
\end{equation}
Now, if $\beta_j-\alpha_j>\pi$ for some $j$, we subdivide each of those
intervals $(\alpha_{j},\beta_{j}]$ into two subintervals of length
$\leq \pi$. We can then decompose $Z_{{\boldsymbol{\alpha}},{\boldsymbol{\beta}}}$ as the sum
of at most $2^n$ cycles of the form
$Z_{{\boldsymbol{\nu}}_{0,\sigma},{\boldsymbol{\nu}}_{1,\sigma}}$ where the $j$th coordinate of
${\boldsymbol{\nu}}_{0,\sigma}$ (respectively ${\boldsymbol{\nu}}_{1,\sigma}$) is either
$\alpha_{j}$ (respectively $\frac{\alpha_{j}+\beta_{j}}{2}$) or
$\frac{\alpha_{j}+\beta_{j}}{2}$ (respectively $\beta_{j}$).
Also, we can expand the product
$$\prod_{j=1}^n\frac{\beta_{j}-\alpha_{j}}{2\pi}
$$
as the sum of the volumes of the sets
$\prod_{j=1}^{m}(\frac{\nu_{0,\sigma,j}}{2\pi},\frac{{\boldsymbol{\nu}}_{1,\sigma,j}}{2\pi}]$.
From here, we easily get that
$$
\left|\frac{\deg(Z_{{\boldsymbol{\alpha}},{\boldsymbol{\beta}}})}{\deg(Z)}-\prod_{j=1}^n\frac{\beta_j-\alpha_j}{2\pi}\right|
\le \sum_{\sigma}
\left|\frac{\deg(Z_{{\boldsymbol{\nu}}_{0,\sigma},{\boldsymbol{\nu}}_{1,\sigma}})}{\deg(Z)}-\prod_{j=1}^n\frac{\nu_{1,\sigma,j}-\nu_{0,\sigma,j}}{2\pi}\right|.
$$
The claim follows applying the bound \eqref{eq:15}, which has to be multiplied by $2^n$. Altogether, we get
$$
\Delta_{\rm ang}(Z)\le 22\,n\bigg(\frac83\bigg)^{n} \zeta(Z),$$
which concludes the proof.
\end{proof}
\section{Bounds for the discrepancy in terms of sparse
resultants} \label{sec:bounds-discr-terms}
In this section, we consider cycles defined by a system of Laurent
polynomials with given support. We compute their direct image under
monomial projections, in terms of sparse resultants, and we derive Theorem
\ref{thm:1} from the Erd\"os-Tur\'an's
theorem and the results in the previous section. We also establish some
basic properties of the Erd\"os-Tur\'an size.
\medskip
We first recall the definition of the sparse
resultant following~\cite{DS:srmet}.
Let ${\mathcal A}_{0},\dots,
{\mathcal A}_{n}$ be a family of $n+1$ non-empty finite subsets of $\mathbb{Z}^{n}$ and put
${\boldsymbol{\mathcal A}}=({\mathcal A}_{0}, \dots, {\mathcal A}_{n})$. Let ${\boldsymbol{u}}_{i}=\{u_{i,{\boldsymbol{a}}}\}_{ a\in
{\mathcal A}_{i}}$ be a group of $\# {\mathcal A}_{i}$ variables, $i=0,\dots,n$, and
set ${\boldsymbol{u}}=\{{\boldsymbol{u}}_{0},\dots, {\boldsymbol{u}}_{n}\}$. For each $i$, let $F_i$ be
the general polynomial with support ${\mathcal A}_{i}$, that is
\begin{equation}\label{Fi}
F_{i}=\sum_{{\boldsymbol{a}}\in {\mathcal A}_{i}}u_{i,{\boldsymbol{a}}} {\boldsymbol{x}}^{{\boldsymbol{a}}}\in
\mathbb{C}[{\boldsymbol{u}}][x_{1}^{\pm1},\dots, x_{n}^{\pm1}],
\end{equation}
and consider the incidence variety
\begin{displaymath}
W_{{\boldsymbol{\mathcal A}}}=\Big\{({\boldsymbol{x}},{\boldsymbol{u}}) \in (\mathbb{C}^{\times})^{n}\times
\prod_{i=0}^{n}\P(\mathbb{C}^{{\mathcal A}_{i}}) \, \Big|\
F_{i}({\boldsymbol{u}}_{i},{\boldsymbol{x}})=0\Big\}.
\end{displaymath}
The
direct image of $W_{{\boldsymbol{\mathcal A}}}$ under the projection \begin{math}
\pi\colon (\mathbb{C}^{\times})^{n}\times \prod_{i=0}^{n}\P(\mathbb{C}^{{\mathcal A}_{i}})\rightarrow
\prod_{i=0}^{n}\P(\mathbb{C}^{{\mathcal A}_{i}})
\end{math}
is the Weil divisor of
$\prod_{i=0}^{n}\P(\mathbb{C}^{{\mathcal A}_{i}})$ given by
\begin{displaymath}
\pi_{*}(W_{{\mathcal A}})=
\begin{cases}
\deg(\pi|_{W_{{\boldsymbol{\mathcal A}}}}) \ov{\pi(W_{{\boldsymbol{\mathcal A}}})} & \text{ if } \operatorname{codim}(\ov{\pi(W_{{\boldsymbol{\mathcal A}}})}) =1,\\
0 & \text{ if } \operatorname{codim}(\ov{\pi(W_{{\boldsymbol{\mathcal A}}})}) \ge 2.
\end{cases}
\end{displaymath}
The \emph{sparse resultant} associated to ${\boldsymbol{\mathcal A}}$, denoted
$\operatorname{Res}_{{\boldsymbol{\mathcal A}}}$, is defined as any primitive equation in $\mathbb{Z}[{\boldsymbol{u}}]$ of
this Weil divisor. It is well-defined up to a sign.
According to this definition, sparse resultants are not necessarily
irreducible. If we denote by $\operatorname{Elim}_{{\boldsymbol{\mathcal A}}}$ what
is classically called the sparse resultant \cite{GKZ94,CLO05,PS93}, then $\operatorname{Res}_{{\boldsymbol{\mathcal A}}}\ne1$ if
and only if $\operatorname{Elim}_{{\boldsymbol{\mathcal A}}}\ne1$ and, if this is the case,
$$
\operatorname{Res}_{{\boldsymbol{\mathcal A}}}=\pm\operatorname{Elim}_{{\boldsymbol{\mathcal A}}}^{\deg(\pi|_{W_{{\boldsymbol{\mathcal A}}}})}.
$$
For instance, for ${\mathcal A}_{0}=\{0\}, {\mathcal A}_{1}=\{0,1,2\}\subset \mathbb{Z}$, we
have that
\begin{displaymath}
\operatorname{Res}_{{\boldsymbol{\mathcal A}}}= \pm
u_{0,0}^{2}, \quad \operatorname{Elim}_{{\boldsymbol{\mathcal A}}}= \pm u_{0,0},
\end{displaymath}
see \cite[Example 3.14]{DS:srmet}.
To recall the basic properties of the sparse resultant that we will need in the
sequel, we need to introduce some definitions.
Let $H$ be a linear subspace of $\mathbb{R}^{n}$ of dimension $m$ and $P_{i}$,
$i=1,\dots, m$, convex bodies of $H$. The \emph{mixed volume} of these
convex bodies is defined as
\begin{equation*}
\operatorname{MV}_{H}(P_{1},\dots,P_{m})=\sum_{j=1}^{m}(-1)^{m-j}\sum_{1\le
i_{1}<\dots<i_{j}\le m}\operatorname{vol}_{H}(P_{i_{1}}+\dots+P_{i_{j}}),
\end{equation*}
where $\operatorname{vol}_{H}$ denotes the Euclidean volume of $H$. We refer to
\cite{CLO05} for further background on the mixed volume of convex
bodies.
Write $\mathbb{C}[{\boldsymbol{x}}^{\pm1}]=\mathbb{C}[x_{1}^{\pm1},\dots, x_{n}^{\pm1}]$ for
short. The \emph{height} of a Laurent polynomial $f=\sum_{{\boldsymbol{a}}\in
\mathbb{Z}^{n}}\alpha_{{\boldsymbol{a}}}{\boldsymbol{x}}^{{\boldsymbol{a}}}\in \mathbb{C}[{\boldsymbol{x}}^{\pm1}]$ is defined as
\begin{displaymath}
\operatorname{h}(f)= \log \big(\max_{{\boldsymbol{a}}}|\alpha_{{\boldsymbol{a}}}|\big).
\end{displaymath}
Given a finite subset ${\mathcal B}$ of $\mathbb{Z}^{n}$, we denote by $\operatorname{conv}({\mathcal B})$ its
convex hull, which is a lattice polytope of $\mathbb{R}^{n}$.
\begin{proposition} \label{prop:6} Let ${\mathcal A}_{0},\dots, {\mathcal A}_{n}\subset
\mathbb{Z}^{n}$ be a family of $n+1$ non-empty finite subsets and set
$Q_{i}=\operatorname{conv}({\mathcal A}_{i})$. Then
\begin{equation*}
\deg_{{\boldsymbol{u}}_{i}}(\operatorname{Res}_{{\boldsymbol{\mathcal A}}})= \operatorname{MV}_{\mathbb{R}^{n}}(Q_{0}, \dots,
Q_{i-1},Q_{i+1},\dots, Q_{n}), \quad i=0,\dots, n,
\end{equation*}
and
\begin{displaymath}
\operatorname{h}(\operatorname{Res}_{{\boldsymbol{\mathcal A}}})\le \sum_{i=0}^{n}\operatorname{MV}_{\mathbb{R}^{n}}(Q_{0}, \dots,
Q_{i-1},Q_{i+1},\dots, Q_{n}) \log (\#{\mathcal A}_{i}).
\end{displaymath}
\end{proposition}
\begin{proof}
The formula for the partial degrees is classical, see for
instance~\cite{GKZ94}. The bound for the height is given by \cite[Theorem
1]{Som04} for the case where the resultant depends on all the groups
of variables ${\boldsymbol{u}}_0,\ldots,\,{\boldsymbol{u}}_n$. The general case can be found
in \cite[Proposition 3.15]{DS:srmet}.
\end{proof}
For a family of Laurent polynomials $f_i\in \mathbb{C}[{\boldsymbol{x}}^{\pm1}]$ with
$\operatorname{supp}(f_{i})\subset {\mathcal A}_i$, $i=0,\dots,n$, we write
\begin{displaymath}
\operatorname{Res}_{{\boldsymbol{\mathcal A}}}(f_0,\dots, f_{n})
\end{displaymath}
for the evaluation of the resultant at their coefficients. The
following is the multiplicativity formula for sparse resultants.
\begin{proposition}\label{additivity}
Let $0\le i\le n$ and consider a family of non-empty finite subsets
${\mathcal A}_0,\ldots,{\mathcal A}_n,{\mathcal A}_{i}'\subset\mathbb{Z}^{n}$. Let $f_j\in
\mathbb{C}[{\boldsymbol{x}}^{\pm1}]$, be a Laurent polynomial with support contained in
${\mathcal A}_j$, $j=0,\dots, n$, and $f_i'\in \mathbb{C}[{\boldsymbol{x}}^{\pm1}]$ a further Laurent
polynomial with support contained in ${\mathcal A}'_i$. Then
\begin{multline*}
\operatorname{Res}_{{\mathcal A}_{0},\dots, {\mathcal A}_{i}+{\mathcal A}_{i}', \dots, {\mathcal A}_{n}}(f_{0},\dots,
f_{i}f_{i}', \dots, f_{n})\\ =\pm \operatorname{Res}_{{\mathcal A}_{0},\dots, {\mathcal A}_{i}, \dots, {\mathcal A}_{n}}(f_{0},\dots,
f_{i}, \dots, f_{n})\operatorname{Res}_{{\mathcal A}_{0},\dots, {\mathcal A}_{i}', \dots, {\mathcal A}_{n}}(f_{0},\dots,
f_{i}', \dots, f_{n})
\end{multline*}
\end{proposition}
\begin{proof}
The validity of this formula, with some restrictions, has been
stablished first in \cite[Proposition 7.1]{PS93}. The general case
can be found in \cite[Corollary 4.6]{DS:srmet}.
\end{proof}
The \emph{support function} of a compact subset $P\subset \mathbb{R}^n$ is the
function $h_{P}\colon \mathbb{R}^n\to \mathbb{R}$ defined, for ${\boldsymbol{v}} \in\mathbb{R}^n$, as
\begin{displaymath}
h_{P}({\boldsymbol{v}} )= \inf_{{\boldsymbol{a}}\in P}\langle {\boldsymbol{a}},{\boldsymbol{v}} \rangle,
\end{displaymath}
where $\langle\cdot,\cdot\rangle$ denotes the inner product of
$\mathbb{R}^{n}$.
Let ${\mathcal B}\subset \mathbb{Z}^{n}$ be a finite subset and $f= \sum_{{\boldsymbol{b}}\in
{\mathcal B}}\beta_{{\boldsymbol{b}}}{\boldsymbol{x}}^{{\boldsymbol{b}}}$ a Laurent polynomial with support
contained in ${\mathcal B}$.
For ${\boldsymbol{v}} \in \mathbb{R}^n$, we set
\begin{displaymath}
{\mathcal B}^{{\boldsymbol{v}} }=\{ {\boldsymbol{b}}\in {\mathcal B}\mid \langle {\boldsymbol{b}},{\boldsymbol{v}} \rangle =
h_{\operatorname{conv}({\mathcal B})}({\boldsymbol{v}} )\}, \quad f^{{\boldsymbol{v}} }= \sum_{{\boldsymbol{b}}\in
{\mathcal B}^{{\boldsymbol{v}} }}\beta_{{\boldsymbol{b}}}x^{{\boldsymbol{b}}}.
\end{displaymath}
\begin{definition} \label{def:1} Let ${\mathcal A}_{1},\dots,
{\mathcal A}_{n}\subset\mathbb{Z}^{n}$ be a family of $n$ non-empty finite subsets,
${\boldsymbol{v}} \in\mathbb{Z}^n\setminus \{0\}$, and ${\boldsymbol{v}} ^{\bot}\subset\mathbb{R}^{n}$ the
orthogonal subspace. Then $\mathbb{Z}^n\cap {\boldsymbol{v}} ^{\bot}$ is a lattice of
rank $n-1$ and, for $i=1,\dots,n$, there exists ${\boldsymbol{b}} _{i,{\boldsymbol{v}} }\in
\mathbb{Z}^n$ such that ${\mathcal A}_{i}^{{\boldsymbol{v}} }-{\boldsymbol{b}}_{i,{\boldsymbol{v}} } \subset \mathbb{Z}^n\cap
{\boldsymbol{v}} ^{\bot}$. The \emph{resultant of ${\mathcal A}_{1},\dots, {\mathcal A}_{n}$ in
the direction of ${\boldsymbol{v}} $}, denoted $\operatorname{Res}_{{\mathcal A}_{1}^{{\boldsymbol{v}}
},\dots,{\mathcal A}_{n}^{{\boldsymbol{v}} }}$, is defined as the resultant of the
family of finite subsets ${\mathcal A}_{i}^{{\boldsymbol{v}} }-{\boldsymbol{b}}_{i,{\boldsymbol{v}} }\subset
\mathbb{Z}^n\cap {\boldsymbol{v}} ^{\bot}$.
Let $f_{i}\in \mathbb{C}[x_1^{\pm1},\ldots, x_n^{\pm1}]$, $i=1,\dots,n$,
with $\operatorname{supp}(f_{i})\subset{\mathcal A}_{i}$. For each $i$, write
$f_{i}^{{\boldsymbol{v}} }={\boldsymbol{x}}^{{\boldsymbol{b}}_{i,{\boldsymbol{v}} }}g_{i,{\boldsymbol{v}} }$ for a Laurent polynomial $g_{i,{\boldsymbol{v}} }\in
\mathbb{C}[\mathbb{Z}^n\cap {\boldsymbol{v}} ^{\bot}]\simeq\mathbb{C}[y_1^{\pm1},\ldots,y_{n-1}^{\pm1}]$
with $\operatorname{supp}(g_{i,{\boldsymbol{v}} })\subset {\mathcal A}_{i}^{{\boldsymbol{v}} }-{\boldsymbol{b}}_{i,{\boldsymbol{v}} }$. The expression
\begin{displaymath}
\operatorname{Res}_{{\mathcal A}_{1}^{{\boldsymbol{v}} },\dots,{\mathcal A}_{n}^{{\boldsymbol{v}} }}(f_{1}^{{\boldsymbol{v}} },\dots,f_{n}^{{\boldsymbol{v}} })
\end{displaymath}
is defined as the evaluation of this directional resultant at the
coefficients of the $g_{i,{\boldsymbol{v}} }$. These constructions are
independent of the choice of the vectors ${\boldsymbol{b}}_{i,{\boldsymbol{v}} }$.
\end{definition}
We have that
$\operatorname{Res}_{{\mathcal A}_{1}^{{\boldsymbol{v}} },\dots,{\mathcal A}_{n}^{{\boldsymbol{v}} }}\ne 1$ only if ${\boldsymbol{v}} $ is
an inner normal to a facet of the Minkowski sum $\sum_{i=1}^{n}\operatorname{conv}({\mathcal A}_{i})$.
In particular, the number of non-trivial directional resultants of the
family ${\mathcal A}_{1},\dots, {\mathcal A}_{n}$ is finite.
With notation as in Definition \ref{def:1}, write ${\boldsymbol{f}}=(f_{1},\dots,
f_{n})$ for short. We denote by $V({\boldsymbol{f}})_{0}\subset \big(\mathbb{C}^\times\big)^n$ the set of isolated
solutions in the algebraic torus of the system of equations $f_{1}=\dots=
f_{n}=0$ and we set
\begin{displaymath}
Z(f_{1},\dots,
f_{n}) = \sum_{{\boldsymbol{\xi}}\in V({\boldsymbol{f}})_{0}} \operatorname{mult}({\boldsymbol{\xi}}|{\boldsymbol{f}}) [{\boldsymbol{\xi}}]
\end{displaymath}
for the associated 0-dimensional cycle, where $ \operatorname{mult}({\boldsymbol{\xi}}|{\boldsymbol{f}})$ denotes
the intersection multiplicity of ${\boldsymbol{f}}$ at a point ${\boldsymbol{\xi}}$.
For $f_{0}\in \mathbb{C}[{\boldsymbol{x}}^{\pm1}]$, we set
\begin{displaymath}
f_{0}(Z(f_{1},\dots,
f_{n}))= \prod_{{\boldsymbol{\xi}}}
f_{0}({\boldsymbol{\xi}})^{ \operatorname{mult}({\boldsymbol{\xi}}|{\boldsymbol{f}})}.
\end{displaymath}
The following result is known as the Poisson formula for sparse resultants.
\begin{proposition} \label{prop:5} Let ${\boldsymbol{\mathcal A}}=({\mathcal A}_0,\ldots,{\mathcal A}_n)$ be
a family of non-empty finite subsets of $\mathbb{Z}^n$ and $f_i\in \mathbb{C}[{\boldsymbol{x}}^{\pm1}]$ a
Laurent polynomial with $\operatorname{supp}(f_{i})\subset {\mathcal A}_i$, $i=0,\dots,n$. Suppose that
$\operatorname{Res}_{{\mathcal A}_{1}^{{\boldsymbol{v}} },\dots,{\mathcal A}_{n}^{{\boldsymbol{v}} }}(f_{1}^{{\boldsymbol{v}} },\dots,f_{n}^{{\boldsymbol{v}} })\ne
0$ for all ${\boldsymbol{v}} \in \mathbb{Z}^n\setminus \{{\boldsymbol{0}}\}$. Then
\begin{equation*}
\operatorname{Res}_{{\boldsymbol{\mathcal A}}}(f_{0},f_{1},\dots,f_{n})= \pm
\bigg(\prod_{{\boldsymbol{v}} }\operatorname{Res}_{{\mathcal A}_{1}^{{\boldsymbol{v}}
},\dots,{\mathcal A}_{n}^{{\boldsymbol{v}} }}(f_{1}^{{\boldsymbol{v}} },\dots,f_{n}^{{\boldsymbol{v}}
})^{-h_{{\mathcal A}_{0}}({\boldsymbol{v}} )} \bigg) f_{0}(Z(f_{1},\dots,f_{n})) ,
\end{equation*}
the product being over all primitive elements ${\boldsymbol{v}} \in\mathbb{Z}^n$.
\end{proposition}
\begin{proof}
This formula has been obtained, under some restrictions on the
supports, by Pedersen and Sturmfels in \cite[Theorem 1.1]{PS93}. The
general case can be found in~\cite[Theorem 1.1]{DS:srmet}.
\end{proof}
From now on, we fix a family of non-empty finite subsets ${\mathcal A}_{1},\dots,
{\mathcal A}_{n}$ of $\mathbb{Z}^{n}$ such that
$\operatorname{MV}_{\mathbb{R}^{n}}(Q_{1},\dots,Q_{n})\ge1$, where $Q_{i}=\operatorname{conv}({\mathcal A}_{i})$. We consider also a family of Laurent
polynomials $f_{1},\dots, f_{n}\in \mathbb{C}[{\boldsymbol{x}}^{\pm1}]$ with
$\operatorname{supp}(f_{i})\subset {\mathcal A}_{i}$ and
$\operatorname{Res}_{{\mathcal A}_{1}^{{\boldsymbol{v}}},\dots,{\mathcal A}_{n}^{{\boldsymbol{v}}}}(f_{1}^{{\boldsymbol{v}}},\dots,f_{n}^{{\boldsymbol{v}}})\ne
0$ for all ${\boldsymbol{v}}\in \mathbb{Z}^{n}\setminus \{0\}$. By
Bernstein's theorem \cite[Theorem~B]{Ber75},
\begin{equation*}
\deg(Z({\boldsymbol{f}}))= \operatorname{MV}_{\mathbb{R}^{n}}(Q_{1},\dots, Q_{n})\ge 1.
\end{equation*}
Consider the projection $ \pi_{{\boldsymbol{e}}_{1}}\colon \mathbb{R}^{n}\to \mathbb{R}^{n-1}$
given by $\pi_{{\boldsymbol{e}}_{1}}(x_1,x_{2},\dots, x_n)=(x_2,\dots,x_{n})$. If we
regard each $f_i$ as a Laurent polynomial in the group of variables ${\boldsymbol{x}}' :=
(x_2,\dots,x_{n}) $ with coefficients in the ring $\mathbb{C}[{x_1}^{\pm1}]$,
its support with respect to ${\boldsymbol{x}}'$ is contained in the
finite subset $\pi_{{\boldsymbol{e}}_1}({\mathcal A}_i)$ of $\mathbb{R}^{n-1}$. Set then
$$
R({\boldsymbol{f}})= \operatorname{Res}_{\pi_{e_1}({\mathcal A}_1),\dots,\pi_{e_1}({\mathcal A}_{n})}
\big(f_1(x_1,{\boldsymbol{x}}') ,\dots, f_n(x_1,{\boldsymbol{x}}')\big) \in \mathbb{C}[x_1^{\pm1}]
$$
for the evaluation of the resultant at these coefficients in
$\mathbb{C}[x_{1}^{\pm1}]$.
Recall that the {sup-norm} of a Laurent polynomial $f\in
\mathbb{C}[{\boldsymbol{x}}^{\pm1}]$ is defined as
\begin{math}
\|f\|_{\sup}=\sup_{{\boldsymbol{w}}\in (S^1)^n}|f({\boldsymbol{w}})|.
\end{math}
In general, it holds that
\begin{equation} \label{eq:17}
\operatorname{h}(f)\le \log \|f\|_{\sup}
\end{equation}
This is a consequence of Cauchy's formula for the coefficients of the
Laurent expansion of a holomorphic function on $(\mathbb{C}^{\times})^{N}$ (see for
instance \cite[page 1255]{Som04}).
The following result gives a bound for the sup-norm of $R({\boldsymbol{f}})$. Its
proof is a variant of that for \cite[Lemma~1.3]{Som04}.
\begin{lemma}\label{bound}
Let notation be as above. Then
$$
\log\|R({\boldsymbol{f}})\|_{\sup}\le \sum_{j=1}^n\operatorname{MV}_{{\boldsymbol{e}}_1^\bot}(\{\pi_{{\boldsymbol{e}}_1}(Q_\ell)\}_{\ell\ne j})
\log\|f_j\|_{\sup}.
$$
\end{lemma}
\begin{proof}
Let $k\ge 1$. Proposition \ref{additivity} implies that
\begin{equation}\label{eq:19}
R({\boldsymbol{f}})^{k^{n}}=\operatorname{Res}_{k\pi_{{\boldsymbol{e}}_1}({\mathcal A}_1),\dots,k\pi_{{\boldsymbol{e}}_1}({\mathcal A}_{n})}
(f_1(x_1,{\boldsymbol{x}}')^k ,\dots, f_n(x_1,{\boldsymbol{x}}')^k),
\end{equation}
where $k\pi_{{\boldsymbol{e}}_1}({\mathcal A}_j)$ denotes the pointwise sum of $k$ copies
of $\pi_{{\boldsymbol{e}}_1}({\mathcal A}_j)$. For short, set
$R_{k}=\operatorname{Res}_{k\pi_{{\boldsymbol{e}}_1}({\mathcal A}_1),\dots,k\pi_{{\boldsymbol{e}}_1}({\mathcal A}_{n})}$ and
${\boldsymbol{f}}^{k}=(f_{1}^{k},\dots,f_{n}^{k})$, so that the identity above can
be rewritten as $ R({\boldsymbol{f}})^{k^{n}}=R_{k}({\boldsymbol{f}}^{k})$. By Proposition
\ref{prop:6}, the partial degrees of this
resultant are given by
\begin{displaymath}
\deg_{{\boldsymbol{u}}_{j}}(R_{k})=\operatorname{MV}_{{\boldsymbol{e}}_1^\bot}(\{k\pi_{{\boldsymbol{e}}_1}(Q_\ell)\}_{\ell\ne
j})=k^{n-1}\operatorname{MV}_{{\boldsymbol{e}}_1^\bot}(\{\pi_{{\boldsymbol{e}}_1}(Q_\ell)\}_{\ell\ne j}),
\end{displaymath}
where ${\boldsymbol{u}}_{j}$ is a group of $\#k{\mathcal A}_{j}$ variables, for $j=1,\dots,
n$. In particular, the logarithm of its number of monomials is bounded
from above as
\begin{multline*}
\log(\#\operatorname{supp}(R_{k}))\le\log\bigg( \prod_{j=1}^{n}{\#k{\mathcal A}_{j}+
\deg_{{\boldsymbol{u}}_{j}}(R_{k})\choose \#k{\mathcal A}_{j}}\bigg) \\ \le
\sum_{j=1}^{n}\deg_{{\boldsymbol{u}}_{j}}(R_{k})\log(\#k{\mathcal A}_{j}+1)= O(k^{n-1}\log(k+1)),
\end{multline*}
since $\#k{\mathcal A}_{j}\le (k+1)^{c_{1}n}$ for a constant $c_{1}$
independent of $k$. By Proposition \ref{prop:6}, the height of this
resultant is bounded from above by
\begin{displaymath}
\operatorname{h}(R_{k})\le \sum_{j=1}^{n}\deg_{{\boldsymbol{u}}_{j}}(R_{k})\log(\#k{\mathcal A}_{j})= O(k^{n-1}\log(k+1)).
\end{displaymath}
Let $w_{1}\in S^{1}$. Using~\eqref{eq:19}, \eqref{eq:17}, the
previous bounds, and the fact that $\|f_{j}^{k}\|_{\sup}=\|f_{j}\|^{k}_{\sup}$, we deduce that
\begin{align*}
{k^{n}}\log\|R({\boldsymbol{f}})\|_{\sup}&= \log\|R_{k}({\boldsymbol{f}}^{k}) \|_{\sup}\\
&\le
\operatorname{h}(R_{k}) + \sum_{j=1}^{n}\deg_{{\boldsymbol{u}}_{j}}(R_{j})
\log\|f_{j}^{k}\|_{\sup} + \log(\#\operatorname{supp}(R_{k})) \\
&= k^{n} \Big(\sum_{j=1}^n\operatorname{MV}_{{\boldsymbol{e}}_1^\bot}(\{\pi_{{\boldsymbol{e}}_1}(Q_\ell)\}_{\ell\ne j})
\log\|f_j\|_{\sup}\Big) + O(k^{n-1}\log(k+1)).
\end{align*}
The result then follows by dividing both sides of this inequality by
$k^n$ and letting $k\to\infty$.
\end{proof}
For ${\boldsymbol{a}}\in\mathbb{Z}^n\setminus\{0\}$ and $z$ an additional variable, set
$$
E_{{\boldsymbol{a}}}({\boldsymbol{f}})= \operatorname{Res}_{\{{\boldsymbol{0}}, {\boldsymbol{a}}\}, {\mathcal A}_{1}, \dots,
{\mathcal A}_{n}}(z-{\boldsymbol{x}}^{\boldsymbol{a}},f_1,\ldots,f_n) \in \mathbb{C}[z].
$$
Due to the Poisson formula for sparse resultants given in Proposition
\ref{prop:5}, we have that $Z(E_{{\boldsymbol{a}}}({\boldsymbol{f}}))=\chi_*^{\boldsymbol{a}}(Z({\boldsymbol{f}})),$
and so $E_{{\boldsymbol{a}}}({\boldsymbol{f}})$ can be regarded as an elimination polynomial
for the cycle $Z({\boldsymbol{f}})$ with respect to the monomial projection
$\chi^{{\boldsymbol{a}}}$.
By \cite[Theorem 1.4]{DS:srmet}, there exists $m\in \mathbb{Z}$ such that
\begin{equation} \label{eq:6}
E_{{\boldsymbol{e}}_1}({\boldsymbol{f}})(x_1) =x_{1}^{m}R({\boldsymbol{f}}).
\end{equation}
Hence, Lemma \ref{bound}
can be regarded as a bound for the sup-norm of the elimination
polynomial $E_{{\boldsymbol{e}}_{1}}({\boldsymbol{f}})$. Our next step is to extend this
result to an arbitrary ${\boldsymbol{a}}$. Recall that $\pi_{\boldsymbol{a}}\colon \mathbb{R}^n\to
{\boldsymbol{a}}^\bot$ denotes the orthogonal projection onto the hyperplane
${\boldsymbol{a}}^{\bot}\subset\mathbb{R}^{n}$.
\begin{lemma}\label{prop:1}
Following the notation above,
$$
\log\|E_{{\boldsymbol{a}}}({\boldsymbol{f}})\|_{\sup}\le \|{\boldsymbol{a}}\|_{2} \sum_{j=1}^n
\operatorname{MV}_{{\boldsymbol{a}}^\bot}(\{\pi_{\boldsymbol{a}}(Q_\ell)\}_{\ell\ne j})
\log\|f_j\|_{\sup}.
$$
\end{lemma}
\begin{proof}
Consider first the case where ${\boldsymbol{a}}\in\mathbb{Z}^n$ is primitive. The
quotient $\mathbb{Z}^n/ {\boldsymbol{a}}\mathbb{Z}$ is torsion-free and so ${\boldsymbol{a}}$ can be
completed to a basis of $\mathbb{Z}^n$. Equivalently, there is an invertible
matrix $A\in \operatorname{SL}_n(\mathbb{Z})$ with first row ${\boldsymbol{a}}$. Set ${\boldsymbol{a}}, {\boldsymbol{a}}_2,
\dots, {\boldsymbol{a}}_n$ and ${\boldsymbol{b}}_1,{\boldsymbol{b}}_2, \dots, {\boldsymbol{b}}_n$ for the rows of
$A$ and of $A^{-1}$, respectively. There is a commutative diagram
\begin{displaymath}
\xymatrix{
(\mathbb{C}^\times)^n
\ar[r]^{\chi^{\boldsymbol{a}}} \ar[d]^{\varphi_A} & \mathbb{C}^\times \\
(\mathbb{C}^\times)^n \ar@/^1pc/[u]^{\varphi_{A^{-1}}} \ar[ur]_{\chi^{{\boldsymbol{e}}_{1}}} &}
\end{displaymath}
where $\varphi_A$ and $\varphi_{A^{-1}}$ are the monomial
isomorphisms given by ${\boldsymbol{x}}\mapsto {\boldsymbol{x}}^A= ({\boldsymbol{x}}^{{\boldsymbol{a}}_1},\dots, {\boldsymbol{x}}^{{\boldsymbol{a}}_n})$
and ${\boldsymbol{x}}\mapsto {\boldsymbol{x}}^{A^{-1}}= ({\boldsymbol{x}}^{{\boldsymbol{b}}_1},\dots, {\boldsymbol{x}}^{{\boldsymbol{b}}_n})$,
respectively.
Let ${\boldsymbol{y}}=(y_1,\dots, y_n)$ denote the coordinates of the algebraic torus below. For $\ell=1,\ldots, n,$ set
$$
f_\ell^A({\boldsymbol{y}})= \varphi_{A^{-1}}^{*}f_\ell=f_\ell({\boldsymbol{y}}^{{\boldsymbol{b}}_1},\dots,
{\boldsymbol{y}}^{{\boldsymbol{b}}_n}) \in \mathbb{C}[{\boldsymbol{y}}^{\pm1}],
$$
so that $(\varphi_A)_{*}Z({\boldsymbol{f}})=Z({\boldsymbol{f}}^A)$. Hence,
$E_{{\boldsymbol{a}}}({\boldsymbol{f}})=E_{{\boldsymbol{f}}^A, {\boldsymbol{e}}_1}$ and so Lemma~\ref{bound} combined
with \eqref{eq:6} implies that
\begin{equation}\label{eq:10}
\log\|E_{{\boldsymbol{a}}}({\boldsymbol{f}})\|_{\sup}\le \sum_{j=1}^n \operatorname{MV}_{{\boldsymbol{e}}_1^\bot}(\{\pi_{{\boldsymbol{e}}_{1}}(\operatorname{N}(f_\ell^A))\}_{\ell\ne j})
\log\|f_j^A\|_{\sup},
\end{equation}
where $\operatorname{N}(f_\ell^A)$ is the Newton polytope of $f_{\ell^{A}}$. We
have
\begin{math}
\pi_{{\boldsymbol{e}}_{1}}(\operatorname{N}(f_\ell^A))= \wt B (\operatorname{N}(f_\ell)) \subset \wt B(Q_{\ell})
\end{math}
for the linear map $\wt B\colon \mathbb{R}^n\to \mathbb{R}^{n-1}$ given by $\wt
B({\boldsymbol{x}})=(\langle {\boldsymbol{x}},{\boldsymbol{b}}_2\rangle, \dots, \langle
{\boldsymbol{x}},{\boldsymbol{b}}_n\rangle)$. Let $\{{\boldsymbol{v}}_2,\ldots,{\boldsymbol{v}}_n\}$ be an orthonormal
basis of ${\boldsymbol{a}}^\bot$ and consider a second commutative diagram
\begin{displaymath}
\xymatrix{
\mathbb{R}^n
\ar[r]^{\wt B} \ar[d]_{\pi_{\boldsymbol{a}}} & \mathbb{R}^{n-1}\ar[d]^C \\
{\boldsymbol{a}}^\bot \ar[r]_U & {\boldsymbol{a}}^\bot}
\end{displaymath}
where $C$ is the linear map defined by $ C (y_2,\dots, y_n)=y_2{\boldsymbol{v}}_2
+ \cdots + y_n {\boldsymbol{v}}_n$. It is easy to verify that $U\in
\operatorname{GL}({\boldsymbol{a}}^\bot)$ is uniquely determined by $\pi_{\boldsymbol{a}}$, $\wt{B}$ and
$C$. Since $C$ maps the canonical basis of $\mathbb{R}^{n-1}$
into an orthonormal basis of ${\boldsymbol{a}}^\bot$, it is an isometry between
these two spaces. On the other hand, a straightforward computation
shows that
$$U({\boldsymbol{v}}_j)=\sum_{k=2}^n\langle {\boldsymbol{v}}_j,{\boldsymbol{b}}_k\rangle {\boldsymbol{v}}_k={\boldsymbol{b}}_j, \ \quad j=2,\ldots, n.
$$
We note that ${\boldsymbol{b}}_2,\dots, {\boldsymbol{b}}_n$ is a basis of the $\mathbb{Z}$-module
${\boldsymbol{a}}^\bot\cap \mathbb{Z}^n$. The Brill-Gordan formula~\cite[Chapitre~3,
\S~11, Proposition~15]{Bourbaki:ema} implies that $\operatorname{vol}(
{\boldsymbol{a}}^\bot/({\boldsymbol{a}}^\bot \cap \mathbb{Z}^n))=\|{\boldsymbol{a}}\|_{2}$. Hence,
\begin{equation}
\label{eq:16}
|\det(U)|=\operatorname{vol}( {\boldsymbol{a}}^\bot/({\boldsymbol{a}}^\bot \cap \mathbb{Z}^n))=\|{\boldsymbol{a}}\|_{2}.
\end{equation}
Since $C$ is an isometry, \cite[Theorem 4.12(a)]{CLO05} implies that, for $j=1,\dots,n$,
\begin{equation*}
\operatorname{MV}_{{\boldsymbol{e}}_1^\bot}(\{\wt B(Q_\ell)\}_{\ell\ne j}) =
\operatorname{MV}_{{\boldsymbol{a}}^\bot}(\{C\circ \wt B(Q_\ell)\}_{\ell\ne j})
= \operatorname{MV}_{{\boldsymbol{a}}^\bot}(\{U\circ\pi_{\boldsymbol{a}} (Q_\ell)\}_{\ell\ne j}).
\end{equation*}
By \eqref{eq:16}, $\|{\boldsymbol{a}}\|^{-1/(n-1)}U$ is a volume preserving
map. Applying \cite[Theorem~4.12(a,b)]{CLO05}, we deduce that
\begin{equation*}
\operatorname{MV}_{{\boldsymbol{a}}^\bot}(\{U\circ\pi_{\boldsymbol{a}} (Q_\ell)\}_{\ell\ne j})
=\|{\boldsymbol{a}}\|_{2} \operatorname{MV}_{{\boldsymbol{a}}^\bot}(\{\pi_{\boldsymbol{a}} (Q_\ell)\}_{\ell\ne j}).
\end{equation*}
In addition, $\varphi_A$ gives an automorphism
of $(S^1)^n$ and so $\|f_\ell^A\|_{\sup}=\|f_\ell\|_{\sup}$. We
conclude that, when ${\boldsymbol{a}}$ is primitive,
\begin{equation}\label{eq:11}
\log\|E_{{\boldsymbol{a}}}({\boldsymbol{f}})\|_{\sup}\le \|{\boldsymbol{a}}\|_{2} \sum_{j=1}^n
\operatorname{MV}_{{\boldsymbol{a}}^\bot}(\{\pi_{\boldsymbol{a}}(Q_\ell)\}_{\ell\ne j})
\log\|f_j\|_{\sup}.
\end{equation}
Now let ${\boldsymbol{a}}\in \mathbb{Z}^n\setminus \{0\}$ be any vector. Choose
a primitive ${\boldsymbol{a}}'\in \mathbb{Z}^{n}$ and $m\in \mathbb{Z}_{\ge1} $ such that ${\boldsymbol{a}}= m {\boldsymbol{a}}'$.
Using Proposition \ref{additivity}, we deduce that
$$
E_{{\boldsymbol{a}}}({\boldsymbol{f}}) (z)=\pm \prod_{\omega\in \mu_m} E_{{\boldsymbol{a}}'}({\boldsymbol{f}})(\omega z)
$$
where $\mu_m$ denotes the set of $m$-th roots of $1$. Hence
$\|{\boldsymbol{a}}\|_{2}=m\|{\boldsymbol{a}}'\|_{2} $, $\pi_{\boldsymbol{a}}=\pi_{{\boldsymbol{a}}'}$ and
$\log\|E_{{\boldsymbol{a}}}({\boldsymbol{f}})\|_{\sup}\le m \log\|E_{{\boldsymbol{a}}'}({\boldsymbol{f}})\|_{\sup}$.
The result follows from the bound~(\ref{eq:11}) applied to ${\boldsymbol{a}}'$.
\end{proof}
\begin{proof}[Proof of Theorem \ref{thm:1}]
Let ${\boldsymbol{a}}\in \mathbb{Z}^{n}\setminus \{{\boldsymbol{0}}\}$. Applying
Proposition \ref{prop:5} with $f_0=z-{\boldsymbol{x}}^{\boldsymbol{a}}$ and
$f_0=z-{\boldsymbol{x}}^{-{\boldsymbol{a}}},$ we get that the product of the leading and the
constant coefficients of $E_{{\boldsymbol{a}}}({\boldsymbol{f}})$ is equal to
$$\pm \prod_{{\boldsymbol{v}}}\operatorname{Res}_{{\mathcal A}_{1}^{{\boldsymbol{v}}},\dots,{\mathcal A}_{n}^{{\boldsymbol{v}}}}(f_{1}^{{\boldsymbol{v}}},\dots,f_{n}^{{\boldsymbol{v}}})^{|\langle
{\boldsymbol{v}},{\boldsymbol{a}}\rangle|}.
$$
Recall also that $Z(E_{{\boldsymbol{a}}}({\boldsymbol{f}})(z))={\boldsymbol{\chi}}_*^{\boldsymbol{a}}\big(Z({\boldsymbol{f}})\big).$
The Erd\"os-Tur\'an's theorem (Theorem \ref{mtunivariate}) then implies that
$$\Delta_{\rm ang}\big({\boldsymbol{\chi}}_*^{\boldsymbol{a}}(Z({\boldsymbol{f}}))\big)\leq c\, \sqrt{\frac{1}{D}\log\Bigg(\frac{\|E_{{\boldsymbol{a}}}({\boldsymbol{f}})(z)\|_{\sup}}
{\prod_{{\boldsymbol{v}}}|\operatorname{Res}_{{\mathcal A}_{1}^{{\boldsymbol{v}}},\dots,{\mathcal A}_{n}^{{\boldsymbol{v}}}}(f_{1}^{{\boldsymbol{v}}},\dots,f_{n}^{{\boldsymbol{v}}})|^{\frac{|\langle{\boldsymbol{v}},{\boldsymbol{a}}\rangle |}{2}}}\Bigg)},
$$
with $c=2.5619\dots$
Lemma \ref{prop:1} implies that $ \log\bigg(\frac{\|E_{{\boldsymbol{a}}}({\boldsymbol{f}})(z)\|_{\sup}}
{\prod_{{\boldsymbol{v}}}|\operatorname{Res}_{{\mathcal A}_{1}^{{\boldsymbol{v}}},\dots,{\mathcal A}_{n}^{{\boldsymbol{v}}}}(f_{1}^{{\boldsymbol{v}}},\dots,f_{n}^{{\boldsymbol{v}}})|^{\frac{|\langle{\boldsymbol{v}},{\boldsymbol{a}}\rangle
|}{2}}}\bigg)$ is bounded from above by the quantity
\begin{displaymath}
{\|{\boldsymbol{a}}\|_{2}} \log\Bigg(\frac{\prod_{i=1}^{n}\|f_i\|_{\sup}^{D_{{\boldsymbol{w}},i}}}
{ \prod_{{\boldsymbol{v}}}|\operatorname{Res}_{{\mathcal A}_{1}^{{\boldsymbol{v}}},\dots,{\mathcal A}_{n}^{{\boldsymbol{v}}}}(f_{1}^{{\boldsymbol{v}}},\dots,f_{n}^{{\boldsymbol{v}}})|^{\frac{|\langle
{\boldsymbol{v}},{\boldsymbol{w}} \rangle|}{2}}}\Bigg),
\end{displaymath}
for ${\boldsymbol{w}}=\frac{{\boldsymbol{a}}}{\|{\boldsymbol{a}}\|_{2}}\in S^{n-1}$.
From the definitions of $\theta(Z({\boldsymbol{f}}))$ and of the Erd\"os-Tur\'an
size~$\eta({\boldsymbol{f}})$ given in \eqref{theta} and \eqref{eq:1}, respectively, we get
$$\theta(Z({\boldsymbol{f}}))\leq \min\{1,c\, \sqrt{\eta({\boldsymbol{f}})}\}.
$$
Applying Theorem \ref{thm:2} and the fact that the function
$ t^{\frac23}(9-\log(t))$ is monotonically
increasing in the interval $(0,1]$, we deduce that
\begin{align*}
\Delta_{\rm ang}(Z({\boldsymbol{f}}))&\le 22 n\bigg(\frac83\bigg)^{n} \big(9-\log(\min\{1,c\,
\sqrt{\eta({\boldsymbol{f}})}\})\big)^{\frac23(n-1)} \min\{1,c\,
\sqrt{\eta({\boldsymbol{f}})}\}^{\frac23}\\
&\le 22 n\bigg(\frac83\bigg)^{n}
2^{-\frac23(n-1)}(18+\log^{+}({\eta({\boldsymbol{f}})}^{-1}))^{\frac23(n-1)}
c^{\frac23} {\eta({\boldsymbol{f}})}^{\frac13}\\
&\le
66\,n\, 2^{n} (18+\log^{+}({\eta({\boldsymbol{f}})}^{-1}))^{\frac23(n-1)}
{\eta({\boldsymbol{f}})}^{\frac13},
\end{align*}
which gives the bound for the angle discrepancy. For the radius
discrepancy, we use the bounds given in Theorem \ref{mtunivariate},
\eqref{theta}, and Theorem \ref{thm:2} to get, for $0<\varepsilon<1,$
\begin{displaymath}
\Delta_{\rm rad}({\boldsymbol{f}},\varepsilon)\leq\sum_{j=1}^n\Delta_{\rm rad}({\boldsymbol{\chi}}_*^{{\boldsymbol{e}}_j}(Z({\boldsymbol{f}})),\varepsilon)\\
\leq \frac{2n}{\varepsilon}\,\eta({\boldsymbol{f}}).
\end{displaymath}
This concludes with the proof of the theorem.
\end{proof}
We next study a number of basic properties of the Erd\"os-Tur\'an
size. The following proposition shows that this notion generalizes the
measure of polynomials that appears in the statement of Theorem
\ref{mtunivariate}.
\begin{proposition} \label{prop:2}
Let $d\ge1$ and
$f= a_0+\cdots+a_dx^d \in \mathbb{C}[x] $
with $a_0a_d \ne 0$. Then
$$
\eta(f)=\frac{1}{d}\log\bigg(\frac{\|f\|_{\sup}}{\sqrt{|a_0a_d|}}\bigg).$$
\end{proposition}
\begin{proof}
The directional resultants of $f$ are
\begin{displaymath}
\operatorname{Res}_{{\boldsymbol{v}}}(f^{{\boldsymbol{v}}})
\begin{cases}
\pm a_{0}& \text{ for } {\boldsymbol{v}}=1,\\
\pm a_{d}& \text{ for } {\boldsymbol{v}}=-1.
\end{cases}
\end{displaymath}
Moreover, $D=\operatorname{MV}_{\mathbb{R}}([0,d])=d$, $D_{{\boldsymbol{w}},1}=1$ for ${\boldsymbol{w}}\in
S^{0}=\{\pm1\}$, and $|\langle {\boldsymbol{v}},{\boldsymbol{w}}\rangle|=1$ for all
${\boldsymbol{v}},{\boldsymbol{w}}\in \{\pm1\}$. The formula for $\eta(f)$ then boils down to
$\frac{1}{d}\log\Big(\frac{\|f\|_{\sup}}{\sqrt{|a_0a_d|}}\Big).$
\end{proof}
We denote by $\Delta^{n}=\operatorname{conv}({\boldsymbol{0}}, {\boldsymbol{e}}_1,\ldots,{\boldsymbol{e}}_n)\subset
\mathbb{R}^{n}$ the
standard $n$-simplex.
\begin{proposition} \label{prop:3} Let ${\mathcal A}_{1},\dots,{\mathcal A}_{n}$ be a
family of non-empty finite subsets of $ \mathbb{Z}^{n} $ such that
$\operatorname{MV}_{\mathbb{R}^{n}}(Q_{1},\dots,Q_{n})\ge1$ with $Q_{i}=\operatorname{conv}({\mathcal A}_{i})$.
Let $f_{1},\dots, f_{n}\in \mathbb{C}[x_{1}^{\pm1},\dots, x_{n}^{\pm1}]$
with $\operatorname{supp}(f_{i})\subset {\mathcal A}_{i}$ and such that
$\operatorname{Res}_{{\mathcal A}_{1}^{{\boldsymbol{v}}},\dots,{\mathcal A}_{n}^{{\boldsymbol{v}}}}(f_{1}^{{\boldsymbol{v}}},\dots,f_{n}^{{\boldsymbol{v}}})\ne
0$ for all ${\boldsymbol{v}}\in \mathbb{Z}^{n}\setminus \{0\}$.
\begin{enumerate}
\item \label{item:6} $\eta({\boldsymbol{f}})<\infty$.
\item \label{item:7} Let $\gamma_{1},\dots, \gamma_{n}\in
\mathbb{C}^{\times}$. Then $\eta(\gamma_{1} f_{1},\dots,\gamma_{n}f_{n})=\eta(f_{1},\dots,f_{n})$.
\item \label{item:8} Let $d_j\in\mathbb{Z}_{\ge1}$ and ${\boldsymbol{b}}_j\in\mathbb{Z}^n$ such that
$Q_{j}\subset d_j\Delta^{n}+{\boldsymbol{b}}_j$, $j=1,\dots,n$. Then
\begin{multline*}
\eta({\boldsymbol{f}})\le
\frac{1}{\operatorname{MV}_{\mathbb{R}^{n}}(Q_{1},\dots,Q_{n})}\bigg((n+\sqrt{n})\bigg(\prod_{j=1}^{n}d_j\bigg)\sum_{j=1}^n\frac{\log\|f_j\|_{\sup}}{d_j}\\
+ \sum_{{\boldsymbol{v}}}\frac{{\|{\boldsymbol{v}}\|_{2}}}{2}\log^+|\operatorname{Res}_{{\mathcal A}_{1}^{{\boldsymbol{v}}},\dots,{\mathcal A}_{n}^{{\boldsymbol{v}}}}(f_{1}^{{\boldsymbol{v}}},\dots,f_{n}^{{\boldsymbol{v}}})^{-1}|\bigg),
\end{multline*}
the second sum being taken over all primitive vectors
${\boldsymbol{v}}\in\mathbb{Z}^n$. Moreover, if $f_{1},\dots, f_{n}\in
\mathbb{Z}[x_{1}^{\pm1},\dots, x_{n}^{\pm1}]$, then
\begin{displaymath}
\eta({\boldsymbol{f}})\le
\frac{(n+\sqrt{n})\big(\prod_{j=1}^{n}d_j\big)}{\operatorname{MV}_{\mathbb{R}^{n}}(Q_{1},\dots,Q_{n})} \sum_{j=1}^n\frac{\log\|f_j\|_{\sup}}{d_j}.
\end{displaymath}
\end{enumerate}
\end{proposition}
\begin{proof}
The statement of \eqref{item:6} is clear, since $\eta({\boldsymbol{f}})$ is defined as the
supremum of a continuous function over the compact set~$S^{n-1}$.
For \eqref{item:7}, let ${\boldsymbol{a}}\in \mathbb{Z}^{n}\setminus \{{\boldsymbol{0}}\}$.
As explained in the proof of Theorem \ref{thm:1}, the product of the leading and the
constant coefficients of $E_{{\boldsymbol{a}}}({\boldsymbol{f}})$ is equal to
$$\pm \prod_{{\boldsymbol{v}}}\operatorname{Res}_{{\mathcal A}_{1}^{{\boldsymbol{v}}},\dots,{\mathcal A}_{n}^{{\boldsymbol{v}}}}(f_{1}^{{\boldsymbol{v}}},\dots,f_{n}^{{\boldsymbol{v}}})^{|\langle
{\boldsymbol{v}},{\boldsymbol{a}}\rangle|}.
$$
Hence, the denominator in the definition of $\eta({\boldsymbol{f}})$ is
multihomogeneous in the coefficients of each $f_{j}$, of partial
degrees equal to ${\|{\boldsymbol{a}}\|_{2}}^{-1}$ times those of
$E_{{\boldsymbol{a}}}({\boldsymbol{f}})$.
Hence,
\begin{displaymath}
\frac{1}{\|{\boldsymbol{a}}\|_{2}}\deg_{f_{j}}(E_{{\boldsymbol{a}}}({\boldsymbol{f}}))=
\operatorname{MV}_{{\boldsymbol{a}}^\bot}\big(\pi_{\boldsymbol{a}}(Q_1),\dots,
\pi_{\boldsymbol{w}}(Q_{j-1}),\pi_{\boldsymbol{a}}(Q_{j+1}),\dots,\pi_{\boldsymbol{a}}(Q_n))=D_{{\boldsymbol{w}},j}
\end{displaymath}
for ${\boldsymbol{w}}=\frac{{\boldsymbol{a}}}{\|{\boldsymbol{a}}\|_2}$, which implies the statement.
For \eqref{item:8}, let ${\boldsymbol{w}}\in S^{n-1}$. Then
$\pi_{\boldsymbol{w}}(Q_j)\subset\pi_{\boldsymbol{w}}(d_j \Delta^{n}+{\boldsymbol{b}}_j)$. Due to
the monotonicity of the mixed volume with respect to the inclusion,
plus its properties of homogeneity and invariance under translation,
we deduce that, for $j=1,\dots,n$,
\begin{multline}\label{huno}
\operatorname{MV}_{{\boldsymbol{w}}^{\bot}}(\{\pi_{\boldsymbol{w}}(Q_\ell)\}_{\ell\ne
j})\leq\operatorname{MV}_{{\boldsymbol{w}}^{\bot}}(\{\pi_{\boldsymbol{w}}(d_\ell\Delta^{n}+{\boldsymbol{b}}_\ell)\}_{\ell\ne
j})\\ \leq(n-1)!\Big(\prod_{\ell\ne j} d_{\ell}\Big)\operatorname{vol}_{{\boldsymbol{w}}^{\bot}}(\pi_{\boldsymbol{w}}(\Delta^{n})) .
\end{multline}
The projected simplex $\pi_{\boldsymbol{w}}(\Delta^{n})$ can be covered by the union
of the projection of its facets. One of the facets of $\Delta^{n}$ has
$(n-1)$-dimensional volume equal to $\frac{\sqrt{n}}{(n-1)!}$, while
the other $n$ facets have $(n-1)$-dimensional volume equal to
$\frac{1}{(n-1)!}.$ Since the volume cannot increase under orthogonal
projections, we have that
\begin{equation}\label{hdos}
\operatorname{vol}_{{\boldsymbol{w}}^{\bot}}(\pi_{\boldsymbol{w}}(\Delta^{n}))\leq \frac{\sqrt{n}+n}{(n-1)!}.
\end{equation}
In addition, $|\langle {\boldsymbol{v}},{\boldsymbol{w}}\rangle|\le \|{\boldsymbol{v}}\|_{2}$ since
${\boldsymbol{w}}\in S^{n-1}$. Then, the first part of the statement follows from
\eqref{huno},\,\eqref{hdos} and the definition of $\eta({\boldsymbol{f}})$. The
second part follows from the fact that, if the coefficients of the
$f_{i}$'s are integers, then the relevant directional resultants are
nonzero integers and so their absolute values are at least 1.
\end{proof}
Let us consider the statement of Proposition \ref{prop:3}\eqref{item:8},
in the classical dense case $Q_j=d_j\Delta^{n}$ for all $j$. In this situation, the
only primitive vectors ${\boldsymbol{v}}$ to consider are ${\boldsymbol{v}}={\boldsymbol{e}}_{i}$,
$i=1,\dots,n$, and ${\boldsymbol{v}}={\boldsymbol{e}}_{0}:=- \sum_{i=1}^{n}{\boldsymbol{e}}_{i}.$ Given
$d_{j}\ge1$, $j=1,\dots,n$, we write $ \operatorname{Res}_{d_{1},\dots,d_{n}} $ for
the resultant of $n$ homogeneous polynomials in $n$ variables of respective degrees
$d_{1},\dots,d_{n},$ as defined in \cite{CLO05}. Given a system of polynomials $f_1,\ldots,
f_n\in\mathbb{C}[x_1,\ldots, x_n]$ with $\deg(f_{j})\le d_{j}$ and
$i=0,\dots,n$, the initial polynomials
$f_{1}^{{\boldsymbol{e}}_i},\dots,f_{n}^{{\boldsymbol{e}}_i}$ form a system of $n$ polynomials
of degrees $d_{1},\dots,d_{n}$. In particular, we can evaluate
$\operatorname{Res}_{d_{1},\dots,d_{n}}$ at these polynomials. If we assume that
these resultants are nonzero, we obtain
\begin{multline*}
\eta({\boldsymbol{f}})\leq(n+\sqrt{n})\sum_{j=1}^n\frac{\log\|f_j\|_{\sup}}{d_j}
+\frac{1}{2\prod_{j=1}^{n}d_j}\bigg(
\sqrt{n}\log^+|\operatorname{Res}_{d_{1},\dots,d_{n}}(f_{1}^{{\boldsymbol{e}}_{0}},\dots,f_{n}^{{\boldsymbol{e}}_{0}})^{-1}|
\\+
\sum_{i=1}^n\log^+|\operatorname{Res}_{d_{1},\dots,d_{n}}(f_{1}^{{\boldsymbol{e}}_i},\dots,f_{n}^{{\boldsymbol{e}}_i})^{-1}|\bigg).
\end{multline*}
In particular, if $f_1,\ldots, f_n\in\mathbb{Z}[x_1,\ldots, x_n]$, then
\begin{equation*}
\eta({\boldsymbol{f}})\le (n+\sqrt{n})\sum_{j=1}^n\frac{\log\|f_j\|_{\sup}}{d_j}.
\end{equation*}
\section{Asymptotic equidistribution} \label{sec:equid-prob}
We will apply here the results in the previous sections to study the
asymptotic distribution of the roots of a sequence of systems of
Laurent polynomials over $\mathbb{Z}$ and of random systems of Laurent
polynomials over~$\mathbb{C}$.
First, we will consider polynomials over $\mathbb{Z}$.
Let $Q_i,\dots, Q_{n}\subset \mathbb{R}^{n}$ be a family of lattice polytopes
such that $\operatorname{MV}_{\mathbb{R}^{n}}(Q_1,\ldots, Q_n)\ge1$. For each integer $\kappa\geq1$ and
$i=1,\dots, n$, consider the finite subset of $\mathbb{Z}^{n}$ given by
\begin{equation}
\label{eq:4}
{\mathcal A}_{\kappa,i}=\kappa Q_i\cap\mathbb{Z}^n .
\end{equation}
\begin{proposition} \label{prop:7} For $\kappa\ge1 $ let
${\boldsymbol{f}}_{\kappa}=(f_{\kappa,1},\dots, f_{\kappa,n}) $ be a family of Laurent
polynomials in $\mathbb{Z}[x_1^{\pm1},\dots, x_n^{\pm1}] $ such that $
\operatorname{supp}(f_{\kappa,i})\subset \kappa Q_{i}$ and
$\operatorname{Res}_{{\mathcal A}_{\kappa,1}^{{\boldsymbol{v}}},\dots,{\mathcal A}_{\kappa,n}^{{\boldsymbol{v}}}}(f_{\kappa,1}^{{\boldsymbol{v}}},\ldots,f_{\kappa,n}^{{\boldsymbol{v}}})\ne0$
for all ${\boldsymbol{v}}\in \mathbb{Z}^{n}\setminus \{{\boldsymbol{0}}\}$. Then there is a
constant $c_{1} >0$ which does not depend on $\kappa$ such that
$$
\Delta_{\rm ang}(Z({\boldsymbol{f}}_\kappa))\le c_{1} \,
\bigg(\frac{\sum_{i=1}^{n}\log\|f_{\kappa,i}\|_{\sup}}{\kappa}\bigg)^{\frac13}
\bigg(1+\log^{+}\bigg(\frac{\kappa}{\sum_{i=1}^{n}\log\|f_{\kappa,i}\|_{\sup}}\bigg)\bigg)^{\frac{2}{3}(n-1)}
$$
and, for any $0<\varepsilon<1$,
$$
\Delta_{\rm rad}(Z({\boldsymbol{f}}_\kappa),\varepsilon)\le c_{1} \,
\frac{\sum_{i=1}^{n}\log\|f_{\kappa,i}\|_{\sup}}{\varepsilon \kappa}.
$$
\end{proposition}
\begin{proof}
This follows easily from Theorem \ref{thm:1} and Proposition \ref{prop:3}\eqref{item:8}.
\end{proof}
\begin{proof}[Proof of Theorem \ref{cor:2}]
Following the notation in the statement of Theorem \ref{cor:2},
$\nu_{Z({\boldsymbol{f}}_{\kappa})}$ is the discrete measure associated to
$Z({\boldsymbol{f}}_{\kappa})$ and $\nu_{\text{\rm Haar}}$ is the measure
on~$(\mathbb{C}^{\times})^{n}$ induced by the Haar probability measure on
$(S^{1})^{n}$.
We have to show that, for every continuous function with compact
support $h$,
\begin{displaymath}
\lim_{\kappa\to\infty}\int_{(\mathbb{C}^{\times})^{n}}h \, \d\nu_{Z({\boldsymbol{f}}_{\kappa})}=\int_{(\mathbb{C}^{\times})^{n}}h \, \d\nu_{\text{\rm Haar}}.
\end{displaymath}
It is enough to prove the statement for the characteristic function $h_{U}$
of the open sets of the form
\begin{equation} \label{eq:13}
U= \{(z_{1},\dots, z_{n})\in(\mathbb{C}^\times)^n \mid
r_{1,j}<|z_j|<r_{2,j},\,\alpha_j<\arg(z_j)<\beta_j \text{ for all }
j\},
\end{equation}
with $0\leq r_{1,j}<r_{2,j}\le \infty$, $ r_{i,j}\ne 1$ and
$-\pi<\alpha_j<\beta_j\leq\pi$, since any continuous function with
compact support can be uniformly approximated by a linear combinations
of the aforementioned characteristic functions.
Consider first the case where $U\cap(S^1)^n=\emptyset.$ Due to the
conditions imposed on the numbers $r_{i,j},$ there exists
$\varepsilon>0$ such that $U$ is disjoint with the set
\begin{displaymath}
\{{\boldsymbol{\xi}}\in\mathbb{C}^n
\mid \,1-\varepsilon<|\xi_j|<(1-\varepsilon)^{-1} \text{ for all } j\}.
\end{displaymath}
Hence,
$$
\int_{(\mathbb{C}^\times)^n}h_U\,\d\delta_{Z({\boldsymbol{f}}_\kappa)}=\frac{\deg(Z({\boldsymbol{f}}_\kappa)|_{U})}{\kappa^{n}\operatorname{MV}({\boldsymbol{Q}}) }\le
\Delta_{\rm rad}({\boldsymbol{f}}_{\kappa}, {\varepsilon}),
$$
where $\operatorname{MV}({\boldsymbol{Q}})$ denotes the mixed volume of the polytopes $Q_{1},\dots, Q_{n}\subset\mathbb{R}^{n}$ and
$Z({\boldsymbol{f}}_\kappa)|_{U}=\sum_{{\boldsymbol{\xi}}\in|Z({\boldsymbol{f}}_\kappa)|\cap U}m_{\boldsymbol{\xi}}[{\boldsymbol{\xi}}]$. Proposition \ref{prop:7} implies
that this integral goes to $0$ for $\kappa\to\infty$, which proves the statement in this case, since
$\int_{\mathbb{C}^n}h_U\,\d\nu_{\text{\rm Haar}}=0$.
Consider now the case where $U\cap(S^1)^n\neq\emptyset$.
Set
\begin{math}
\ov U= \{{\boldsymbol{z}} \mid \alpha_j\le \arg(z_j)\le \beta_j \text{ for all }
j\}.
\end{math}
Then
\begin{multline*}
\int_{(\mathbb{C}^\times)^n}h_U\,\d\delta_{Z({\boldsymbol{f}}_\kappa)}-\int_{(\mathbb{C}^\times)^n}h_U\,\d\nu_{\text{\rm Haar}}=
\int_{(\mathbb{C}^\times)^n}h_U\,\d\delta_{Z({\boldsymbol{f}}_\kappa)}- \prod_{j=1}^{n} \frac{\beta_{j}-\alpha_{j}}{2\pi}\\
=\int_{(\mathbb{C}^\times)^n}\bigg(h_{\overline{U}}-\prod_{j=1}^{n}
\frac{\beta_{j}-\alpha_{j}}{2\pi}\bigg)\d\delta_{Z({\boldsymbol{f}}_\kappa)}-\int_{(\mathbb{C}^\times)^n}h_{\overline{U}\setminus
U}\,\d\delta_{Z({\boldsymbol{f}}_\kappa)}.
\end{multline*}
We have
$$
\int_{(\mathbb{C}^\times)^n}\bigg|h_{\overline{U}}-\prod_{j=1}^{n} \frac{\beta_{j}-\alpha_{j}}{2\pi}\bigg|\d\delta_{Z({\boldsymbol{f}}_\kappa)}\leq
\bigg|\frac{\deg(Z({\boldsymbol{f}}_\kappa)_{{\boldsymbol{\alpha}},{\boldsymbol{\beta}}})}{\kappa^{n}\operatorname{MV}({\boldsymbol{Q}})} - \prod_{j=1}^{n}
\frac{\beta_{j}-\alpha_{j}}{2\pi} \bigg| \leq \Delta_{\rm ang}({\boldsymbol{f}}_\kappa).
$$
Again, Proposition \ref{prop:7} implies
that this integral goes to $0$ for $\kappa\to\infty$.
On the other hand, $\ov
U\setminus U$ is a union of a finite number of subsets $U_{l}$ of the
form \eqref{eq:13} such that $U_{l}\cap(S^1)^n=\emptyset$ for all $l$.
By the previous considerations, $\int_{(\mathbb{C}^\times)^n}h_{U_l}\d\delta_{Z({\boldsymbol{f}}_\kappa)}\to_{\kappa} 0$ and so
$\int_{(\mathbb{C}^\times)^n}h_{\ov U\setminus U}\, \d\delta_{Z({\boldsymbol{f}}_\kappa)}\to_{\kappa}0.$
Hence
\begin{displaymath}
\lim_{\kappa\to\infty}\int_{(\mathbb{C}^\times)^n}h_{{U}}\,
\d\delta_{Z({\boldsymbol{f}}_\kappa)} = \prod_{j=1}^{n}
\frac{\beta_{j}-\alpha_{j}}{2\pi} =\int_{\mathbb{C}^n}h_U\,\d\nu_{\text{\rm Haar}}=0,
\end{displaymath}
which concludes the proof.
\end{proof}
We will now consider random systems of Laurent polynomials with complex coefficients.
To explain and prove our results, we have to consider metrics and
measures on projective spaces over $\mathbb{C}$. Let $N\ge 1$
and consider the standard Riemannian structure on $\mathbb{C}^{N}$ induced by
the Euclidean norm $\|\cdot\|_{2}$. Let $S^{2N-1}=\{{\boldsymbol{z}}\in \mathbb{C}^{N}
\mid \|{\boldsymbol{z}}\|_{2} =1\}$ be the unit sphere with the induced Riemannian
structure. The map $S^{2N-1}\to \P(\mathbb{C}^{N})$ given by $(z_{0},\dots,
z_{N-1})\mapsto (z_{0}:\dots:z_{N-1})$ gives a principal bundle with
fiber $S^{1}$. The Fubini-Study metric on $\P(\mathbb{C}^{N})$ is defined as
the unique Riemannian structure such that this map is a Riemannian
submersion, see \cite{KN:fdgII} for details.
The geodesics of $\P(\mathbb{C}^{N})$ coincide with lines. Hence, we can
define a distance between two points ${\boldsymbol{z}}_{1}$ and ${\boldsymbol{z}}_{2}$ as the
length of the line segment joining them, and we will denote it by
$\operatorname{dist}_{\operatorname{FS}}({\boldsymbol{z}}_{1},{\boldsymbol{z}}_{2})$. However, it will be more convenient
to consider the distance function $\operatorname{dist}:=\sin(\operatorname{dist}_{\operatorname{FS}})$. This
function can be computed with the formula
\begin{equation} \label{eq:14}
\operatorname{dist}({\boldsymbol{z}}_{1},{\boldsymbol{z}}_{2})= \sqrt{1-\Big
(\frac{|\langle
\wt {\boldsymbol{z}}_{1},\wt {\boldsymbol{z}}_{2}\rangle|}{\|\wt {\boldsymbol{z}}_{1}\|_{2}\|\wt {\boldsymbol{z}}_{2}\|_{2}}\Big)^{2}}
\end{equation}
for any choice of representatives $\wt {\boldsymbol{z}}_{i}\in\mathbb{C}^{N}\setminus \{{\boldsymbol{0}}\}$, $i=1,2$.
\begin{lemma} \label{lemm:3} Let ${\boldsymbol{z}}_{1},{\boldsymbol{z}}_{2}\in \P(\mathbb{C}^{N})$ with $\wt {\boldsymbol{z}}_{i}\in \mathbb{C}^{N}$, $i=1,2$, representatives of these
points such that $\|\wt{\boldsymbol{z}}_{2}\|_2=1$. Then $\operatorname{dist}({\boldsymbol{z}}_{1},{\boldsymbol{z}}_{2})\le
\|\wt{\boldsymbol{z}}_{2}-\wt{\boldsymbol{z}}_{1}\|_{2}$.
\end{lemma}
\begin{proof}
We have that
\begin{displaymath}
\|\wt{\boldsymbol{z}}_{2}-\wt{\boldsymbol{z}}_{1}\|^{2}_{2}= \langle
\wt{\boldsymbol{z}}_{2}-\wt{\boldsymbol{z}}_{1},\wt{\boldsymbol{z}}_{2}-\wt{\boldsymbol{z}}_{1}\rangle=1+\|\wt{\boldsymbol{z}}_{1}\|_{2}^{2}-2\operatorname{Re} (\langle
\wt {\boldsymbol{z}}_{1},\wt {\boldsymbol{z}}_{2}\rangle)\ge 1+\|\wt{\boldsymbol{z}}_{1}\|_{2}^{2}-2|\langle
\wt {\boldsymbol{z}}_{1},\wt {\boldsymbol{z}}_{2}\rangle|.
\end{displaymath}
On the other hand, the
formula \eqref{eq:14} gives $ \operatorname{dist}({\boldsymbol{z}}_{1},{\boldsymbol{z}}_{2})^{2}= 1-\big(\frac{|\langle
\wt {\boldsymbol{z}}_{1},\wt
{\boldsymbol{z}}_{2}\rangle|}{\|\wt{\boldsymbol{z}}_{1}\|_{2}}\big)^{2}$. Hence,
\begin{displaymath}
\|\wt{\boldsymbol{z}}_{2}-\wt{\boldsymbol{z}}_{1}\|^{2}_{2}-
\operatorname{dist}({\boldsymbol{z}}_{1},{\boldsymbol{z}}_{2})^{2}= \Big( \|\wt{\boldsymbol{z}}_{1}\|_{2}-\frac{|\langle
\wt {\boldsymbol{z}}_{1},\wt
{\boldsymbol{z}}_{2}\rangle|}{\|\wt{\boldsymbol{z}}_{1}\|_{2}}\Big)^{2}\ge0,
\end{displaymath}
which proves the statement.
\end{proof}
We will need the following \L ojasiewicz inequality for a
hypersurface of a complex projective space. For a homogeneous
polynomial $f\in \mathbb{C}[x_{0},\dots, x_{N-1}]$ of degree $d,$ and a point
${\boldsymbol{z}}\in \P(\mathbb{C}^{N})$, the value
\begin{displaymath}
\frac{|f({\boldsymbol{z}})|}{\|{\boldsymbol{z}}\|_{2}^{d}}
\end{displaymath}
is well-defined. For a subset $E\subset \P(\mathbb{C}^{N})$, we write
$\operatorname{dist}({\boldsymbol{z}}, E)$ for the distance between~${\boldsymbol{z}}$ and $E$.
\begin{lemma} \label{lemm:2}
Let $f\in \mathbb{C}[x_{0},\dots,x_{N-1}]$ be a homogeneous polynomial of degree $d\ge 0$ and
${\boldsymbol{z}}\in \P(\mathbb{C}^{N})$. Then
\begin{displaymath}
\frac{|f({\boldsymbol{z}})|}{\|{\boldsymbol{z}}\|_{2}^{d}}\ge \bigg(\sup_{{\boldsymbol{x}}\in \P(\mathbb{C}^{N})}
\frac{|f({\boldsymbol{x}})|}{\|{\boldsymbol{x}}\|_{2}^{d}} \bigg) \operatorname{dist}({\boldsymbol{z}},V(f))^{d}.
\end{displaymath}
\end{lemma}
\begin{proof}
Let ${\boldsymbol{z}}, {\boldsymbol{x}}\in \P(\mathbb{C}^{N})$ such that ${\boldsymbol{x}}\notin V(f)$.
Let $\wt{\boldsymbol{z}}, \wt{\boldsymbol{x}}$ be representatives
of these points in the sphere $S^{2N-1}$ and set
\begin{math}
f_{\wt {\boldsymbol{x}}}(t)=f(\wt {\boldsymbol{z}}+ t \wt {\boldsymbol{x}}) \in \mathbb{C}[t].
\end{math}
This is a univariate polynomial of degree $d$ with leading coefficient
$f(\wt{\boldsymbol{x}})$. Then, there exist $\xi_{j}\in \mathbb{C}$, $j=1,\dots, d$, such
that
\begin{math}
f_{\wt
{\boldsymbol{x}}}=f(\wt{\boldsymbol{x}})\prod_{j}(t-\xi_{j})
\end{math}
and so
\begin{displaymath}
|f(\wt{\boldsymbol{z}})|= |f_{\wt {\boldsymbol{x}}}(0)|=|f(\wt{\boldsymbol{x}})| \prod_{j}|\xi_{j}| .
\end{displaymath}
For each $j$, we have that $\wt {\boldsymbol{z}} + \xi_{j}\wt{\boldsymbol{x}}\in V(f)$. Using Lemma \ref{lemm:3}, we deduce that
\begin{displaymath}
|\xi_{j}|=\|(\wt {\boldsymbol{z}} + \xi_{j}\wt{\boldsymbol{x}})-\wt{\boldsymbol{z}}\|_{2} \ge \operatorname{dist}({\boldsymbol{z}}, \wt {\boldsymbol{z}} + \xi_{j}\wt{\boldsymbol{x}}) \ge \operatorname{dist}({\boldsymbol{z}}, V(f)).
\end{displaymath}
We deduce that
\begin{displaymath}
\frac{|f({\boldsymbol{z}})|}{\|{\boldsymbol{z}}\|_{2}^{d}}= |f(\wt{\boldsymbol{z}})|\ge |f(\wt{\boldsymbol{x}})| \operatorname{dist}({\boldsymbol{z}}, V(f))^{d}
=\frac{|f({\boldsymbol{x}})|}{\|{\boldsymbol{x}}\|_{2}^{d}} \operatorname{dist}({\boldsymbol{z}}, V(f))^{d}.
\end{displaymath}
Since this holds for all ${\boldsymbol{x}}\notin V(f)$, the result follows.
\end{proof}
Let $\mu_{\operatorname{FS}}$ denote the measure on $\P(\mathbb{C}^{N})$ induced by the
Fubini-Study metric. Then $\mu_{\operatorname{FS}}(\P(\mathbb{C}^{N}))= \frac{\pi^{N-1}}{(N-1)!}$. We
will consider the normalized measure given by
\begin{displaymath}
\mu= \frac{(N-1)!}{\pi^{N-1}} \mu_{\operatorname{FS}}.
\end{displaymath}
A result of Beltr\'an and Pardo \cite[Theorem 1]{BP:edcnsm} shows
that, for a hypersurface $H\subset \P(\mathbb{C}^{N})$ of degree $d$,
the normalized measure of the tube
around $H$ of radius $\rho\ge0$ is bounded from above by
\begin{equation*}
15 d (N-1)^{2}\rho^{2}.
\end{equation*}
Applying this result, we deduce the following bound for the volume of
the set where a polynomial can take small values. For $\delta > 0$ and a homogeneous
polynomial $f\in \mathbb{C}[{\boldsymbol{x}}]$, we consider the
subset of $\P(\mathbb{C}^{N})$ given by
\begin{equation} \label{eq:8}
V(f)_{\delta} = \Big\{{\boldsymbol{z}} \in \P(\mathbb{C}^{N}) \Big|\
\frac{|f({\boldsymbol{z}})|}{\|{\boldsymbol{z}}\|_2^{d}}<\delta \Big\}.
\end{equation}
\begin{proposition} \label{prop:4} Let $\delta > 0$ and $f\in
\mathbb{C}[x_{0},\dots,x_{N-1}]$ a homogeneous polynomial of degree $d\ge
1$. Then
\begin{displaymath}
\mu(V(f)_{\delta}) \le 15
d N^{3} \Big( \frac{\delta}{\|f\|_{\sup}}\Big) ^{\frac2d}.
\end{displaymath}
In particular, if $f\in \mathbb{Z}[x_{0},\dots,x_{N-1}]$, then $
\mu(V(f)_{\delta} ) \le 15 d N^{3}
{\delta}^{\frac2d}$.
\end{proposition}
\begin{proof}
Let ${\boldsymbol{z}}\in V(f)_{\delta}$. Using Lemma \ref{lemm:2}, we deduce that
\begin{displaymath}
\delta> \bigg(\sup_{{\boldsymbol{x}}}
\frac{|f({\boldsymbol{x}})|}{\|{\boldsymbol{x}}\|_{2}^{d}} \bigg) \operatorname{dist}({\boldsymbol{z}},V(f))^{d}
\ge \frac{\|f\|_{\sup}}{N^{\frac{d}{2}}} \operatorname{dist}({\boldsymbol{z}},V(f))^{d}.
\end{displaymath}
Hence,
\begin{displaymath}
\operatorname{dist}({\boldsymbol{z}},V(f))<
N^{\frac{1}{2}}\bigg(\frac{\delta}{\|f\|_{\sup}}
\bigg)^{\frac1d}
\end{displaymath}
and so $V(f)_{\delta}$ is contained in the tube around $V(f)$ of
radius $N^{\frac{1}{2}}\big(\frac{\delta}{\|f\|_{\sup}}
\big)^{\frac1d}$. The first part of the result follows then from the
Beltr\'an--Pardo bound for the volume of this tube. The second part
follows from the fact that $ \|f\|_{\sup}\ge |f|\ge 1$, because of the
inequality \eqref{eq:17} and the fact
that the coefficients of $f$ are integer numbers.
\end{proof}
Let us keep the preceding notation and set ${\boldsymbol{\mathcal A}}_{\kappa}=({\mathcal A}_{\kappa,1},\dots,
{\mathcal A}_{\kappa,n})$ with ${\mathcal A}_{\kappa,i}$ as in~\eqref{eq:4}.
Each point of the
projective space $\P(\mathbb{C}^{{\boldsymbol{\mathcal A}}_{\kappa}})$ can be identified with a system
${\boldsymbol{f}}_\kappa=(f_{\kappa,1},\dots,f_{\kappa,n})$ of Laurent polynomials such that
$\operatorname{supp}(f_{\kappa,i})\subset \kappa Q_{i}$, $i=1,\dots, n$, modulo a multiplicative
scalar. The associated cycle $Z({\boldsymbol{f}}_\kappa)$ is well-defined,
since it does not depend on this multiplicative scalar.
Set $\mu_{\kappa}$ for the normalized Fubini-Study measure on $\P(\mathbb{C}^{{\boldsymbol{\mathcal A}}_{\kappa}})$ and
let $\lambda_{\kappa}\colon \P(\mathbb{C}^{{\boldsymbol{\mathcal A}}_{\kappa}}) \to \mathbb{R}_{\ge0}$ be a probability density
function, that is, a $\mu_{\kappa}$-measurable function with
\begin{displaymath}
\int_{\P(\mathbb{C}^{{\boldsymbol{\mathcal A}}_{\kappa}})}\lambda_{\kappa}\, \d\mu_{\kappa}=1.
\end{displaymath}
Let ${\boldsymbol{f}}_\kappa$ be a random system of Laurent polynomials with
$\operatorname{supp}(f_{\kappa,i})\subset \kappa Q_{i}$, $i=1,\dots, n$, distributed according
to the probability law given by $\lambda_{\kappa}$ with respect to $\mu_{\kappa}$. We
can then consider the angle discrepancy of $Z({\boldsymbol{f}}_\kappa)$ and, for
$0<\varepsilon< 1$, the radius discrepancy of $Z({\boldsymbol{f}}_\kappa)$ with respect to
$\varepsilon$, as random variables on $\P(\mathbb{C}^{{\boldsymbol{\mathcal A}}_{\kappa}})$. We denote
by $ \mathbb{E}(\Delta_{\rm ang}(Z({\boldsymbol{f}}_\kappa));{\lambda_{\kappa}})$ and
$\mathbb{E}(\Delta_{\rm rad}(Z({\boldsymbol{f}}_\kappa),\varepsilon);{\lambda_{\kappa}})$ the expected value of
these random variables.
\begin{theorem}\label{eqprob0}
For $\kappa\ge1$, let $\lambda_{\kappa}$ be a probability density function
on $\P(\mathbb{C}^{{\boldsymbol{\mathcal A}}_{\kappa}})$ and
${\boldsymbol{f}}_\kappa=(f_{\kappa,1},\dots,f_{\kappa,n})$ a random system of
Laurent polynomials with $\operatorname{supp}(f_{\kappa,i})\subset \kappa Q_{i}$,
$i=1,\dots, n$, distributed according to the probability law given
by $\lambda_{\kappa}$ with respect to~$\mu_{\kappa}$. Assume that the
sequence $(\lambda_{\kappa})_{\kappa\ge1}$ is uniformly bounded. Then
there is a constant $c_{2} >0$ which does not depend on $\kappa$
such that
$$
\mathbb{E}(\Delta_{\rm ang}(Z({\boldsymbol{f}}_\kappa));{\lambda_{\kappa}})\le c_{2} \, \frac{\log(\kappa+1)^{\frac{2}{3}n-\frac13}}{\kappa^{\frac13}}
$$
and, for any $0<\varepsilon<1$,
$$
\mathbb{E}(\Delta_{\rm rad}(Z({\boldsymbol{f}}_\kappa),\varepsilon);{\lambda_{\kappa}})\le c_{2} \,
\frac{\log(\kappa+1)}{\varepsilon \kappa}.
$$
In particular,
\begin{displaymath}
\lim_{\kappa\to\infty}\mathbb{E}(\Delta_{\rm ang}(Z({\boldsymbol{f}}_\kappa));{\lambda_{\kappa}})=0
\quad\text{and}\quad \lim_{\kappa\to\infty}\mathbb{E}(\Delta_{\rm rad}(Z({\boldsymbol{f}}_\kappa),\varepsilon);{\lambda_{\kappa}})=0.
\end{displaymath}
\end{theorem}
\begin{proof}
We first estimate the expected value of the angle discrepancy,
which is given by the formula
\begin{displaymath}
\mathbb{E}(\Delta_{\rm ang}(Z({\boldsymbol{f}}_\kappa));\lambda_{\kappa})=\int_{\P(\mathbb{C}^{{\boldsymbol{\mathcal A}}_\kappa})}\Delta_{\rm ang}(Z({\boldsymbol{f}}_\kappa))\lambda_\kappa({\boldsymbol{f}})
\, \d\mu_{\kappa}.
\end{displaymath}
Consider the Minkowski sum $Q=\sum_{i=1}^{n}Q_{i}$, which is a
lattice polytope on $\mathbb{R}^{n}$ of dimension $n$ because of the
assumption that the mixed volume of $Q_{1},\dots,Q_{n}$ is positive.
For each primitive vector ${\boldsymbol{v}}\in \mathbb{Z}^{n}$ which is an inner normal
to a facet of~$Q$, consider the directional resultant
\begin{displaymath}
R_{\kappa,{\boldsymbol{v}}}=\operatorname{Res}_{{\mathcal A}_{\kappa,1}^{{\boldsymbol{v}}},\dots,{\mathcal A}_{\kappa,n}^{{\boldsymbol{v}}}}\in \mathbb{Z}[{\boldsymbol{u}}_{1},\dots, {\boldsymbol{u}}_{n}],
\end{displaymath}
where ${\boldsymbol{u}}_{i}$ is a group of $\#{\mathcal A}_{\kappa,i}$
variables. Proposition \ref{prop:3}\eqref{item:7} implies that its
total degree is bounded by $\deg(R_{\kappa,{\boldsymbol{v}}})=
c_{{\boldsymbol{v}}}\kappa^{n-1}$ for a constant $c_{{\boldsymbol{v}}}$ independent of
$\kappa$. Its total number of variables is $\# {\boldsymbol{\mathcal A}}_{\kappa} =
\sum_{i}{\mathcal A}_{\kappa,i}= \sum_{i}\kappa Q_{i}\cap\mathbb{Z}^{n}$. This number
can be bounded by $c_{3}\kappa^{n}$ for a constant $c_{3}$ independent
of~$\kappa$.
Set $\delta_{\kappa,{\boldsymbol{v}}}=\kappa^{-2n \deg(R_{\kappa,{\boldsymbol{v}}})}$. Consider the
subset $ V(R_{\kappa,{\boldsymbol{v}}})_{\delta_{\kappa,{\boldsymbol{v}}}}\subset \P(\mathbb{C}^{{\boldsymbol{\mathcal A}}_{\kappa}}) $ as
defined in~\eqref{eq:8} and put
\begin{displaymath}
U_{\kappa}=\bigcup_{{\boldsymbol{v}}}V(R_{\kappa,{\boldsymbol{v}}})_{\delta_{\kappa,{\boldsymbol{v}}}},
\end{displaymath}
the union being over all primitive inner normal vectors to facets
of $Q$. Using the fact that $0\leq \Delta_{\rm ang}(Z({\boldsymbol{f}}_\kappa))\leq1 $ and the hypothesis that
the functions $\lambda_{\kappa}$ are uniformly bounded, we deduce that
\begin{displaymath}
0\le \int_{U_{\kappa}}\Delta_{\rm ang}(Z({\boldsymbol{f}}_\kappa))\lambda_\kappa({\boldsymbol{f}})
\, \d\mu_{\kappa} \le \Big( \sup_{{\boldsymbol{f}}_\kappa} \lambda_{\kappa}({\boldsymbol{f}}_\kappa) \Big)
\mu_{d}(U_{\kappa})\le c_{4} \mu_{\kappa}(U_{\kappa})
\end{displaymath}
for a constant $c_{4}$ independent of $\kappa$. By Proposition~\ref{prop:4},
\begin{multline} \label{eq:5}
\mu_{\kappa}(U_{\kappa}) \le\sum_{{\boldsymbol{v}}}
\mu_{\kappa}(V(R_{\kappa,{\boldsymbol{v}}})_{\delta_{\kappa,{\boldsymbol{v}}}}) \le \sum_{{\boldsymbol{v}} }15 \deg(R_{\kappa,{\boldsymbol{v}}}) \big(\#
{\boldsymbol{\mathcal A}}_{\kappa}\big)^{3} \delta_{\kappa,{\boldsymbol{v}}}^{\frac{2}{\deg(R_{\kappa,{\boldsymbol{v}}})}}\\
\le 15 \Big(\sum_{{\boldsymbol{v}}}c_{{\boldsymbol{v}}} \kappa^{n-1}\Big) (c_{3} \kappa^{n})^{3}
\kappa^{-4n} = c_{5} \kappa^{-1},
\end{multline}
with $c_{5} = 15 \big(\sum_{{\boldsymbol{v}}}c_{{\boldsymbol{v}}} \big) c_{3}^{3}$. Hence,
$\mu_\kappa(U_{\kappa})\to_{\kappa}0$ and so
$\int_{U_{\kappa}}\Delta_{\rm ang}(Z({\boldsymbol{f}}_\kappa))\lambda_\kappa({\boldsymbol{f}}_\kappa) \, \d\mu_{\kappa}\to_{\kappa}0 $ as
well.
Let ${\boldsymbol{f}}_\kappa\in \P(\mathbb{C}^{{\boldsymbol{\mathcal A}}_{\kappa}})\setminus U_{\kappa}$ and choose a
representative $\wt {\boldsymbol{f}}_\kappa=(\wt f_{\kappa,1},\dots, \wt f_{\kappa,n})\in
\mathbb{C}^{{\boldsymbol{\mathcal A}}_{\kappa}}\setminus \{{\boldsymbol{0}}\}$ with $\|\wt {\boldsymbol{f}}_\kappa\|_{2}=1$. By
Proposition \ref{prop:3}\eqref{item:7}, $\eta({\boldsymbol{f}}_\kappa)=\eta(\wt
{\boldsymbol{f}}_\kappa)$. Note that the Minkowski sum $\sum_{i}\kappa Q_{i}$ coincides with
$\kappa Q$. Hence, the only non trivial directional resultants of the family
of finite sets ${\mathcal A}_{\kappa,1},\dots, {\mathcal A}_{\kappa,n}$ are those of the form
$R_{\kappa, {\boldsymbol{v}}}$ as considered above. As before, let $\Delta^{n}\subset\mathbb{R}^{n}$
be the standard $n$-simplex. Choose $e\ge1$ and ${\boldsymbol{b}}_{i}\in
\mathbb{Z}^{n}$ such that $Q_{i}\subset e\Delta^{n}+{\boldsymbol{b}}_{i}$ for all
$i$. Hence, $\kappa{\mathcal A}_{\kappa,i}\subset \kappa e Q_{i}+\kappa{\boldsymbol{b}}_{i}$ for
all~$i$. Proposition~\ref{prop:3}\eqref{item:8} implies that there is
a constant $c_{6}$ independent of $\kappa$ such that
\begin{align*}
\eta({\boldsymbol{f}}_\kappa)& \le \frac{1}{\kappa^{n}\operatorname{MV}({\boldsymbol{Q}})}\Big((\kappa e)^{n-1}
(n+\sqrt{n})\sum_{i=1}^{n}\log \|\wt f_{\kappa,i}\|_{\sup} + \frac12 \sum_{{\boldsymbol{v}}}
\|{\boldsymbol{v}}\|_{2} \log^{+}|R_{\kappa,{\boldsymbol{v}}}({\boldsymbol{f}}^{{\boldsymbol{v}}}_\kappa)^{-1}|\Big)\\
& \le \frac{1}{\kappa^{n}\operatorname{MV}({\boldsymbol{Q}})}\Big((\kappa e)^{n-1}
(n+\sqrt{n})\sum_{i=1}^{n}\log (\#{\mathcal A}_{\kappa,i}) + n \sum_{{\boldsymbol{v}}}
\|{\boldsymbol{v}}\|_{2} \deg(R_{\kappa,{\boldsymbol{v}}})\log(\kappa) \Big)\\
& \le c_{6}\frac{\log(\kappa+1)}{\kappa},
\end{align*}
the second and fourth sums being over the primitive inner normals
${\boldsymbol{v}}$ to the facets of~$Q$. Here, we used the fact that $\|\wt
f_{\kappa, i}\|_{\sup}\le \#{\mathcal A}_{\kappa,i}$ for $\wt {\boldsymbol{f}}_\kappa$ in the unit sphere of
$\mathbb{C}^{{\boldsymbol{\mathcal A}}_{\kappa}}$, the definition of the set $U_{\kappa}$, the bound $
\#{\mathcal A}_{\kappa,i}\le \#{\boldsymbol{\mathcal A}}_{\kappa}\le c_{3}\kappa^{n}$ and the inequality
$\deg(R_{\kappa,{\boldsymbol{v}}})\le c_{{\boldsymbol{v}}}\kappa^{n-1}$ that we explained before.
Using Theorem \ref{thm:1} and the fact that the function
$t^\frac13\log\left(\frac{\alpha}{t}\right)^{\frac{n-1}{3}}$ is
increasing for small values of $t>0,$ we deduce that, for ${\boldsymbol{f}}_\kappa\in
\P(\mathbb{C}^{{\boldsymbol{\mathcal A}}_{\kappa}})\setminus U_{\kappa}$,
\begin{multline*}
\Delta_{\rm ang}(Z({\boldsymbol{f}}_\kappa))\leq c_{7} \eta({\boldsymbol{f}}_\kappa)^{\frac13}\log\Big(\frac{c_{8}}{\eta({\boldsymbol{f}}_\kappa)}\Big)^{\frac{2}{3}(n-1)}
\\ \leq c_{9} \Big(\frac{\log(\kappa+1)}{\kappa}\Big)^{\frac13} \log \Big(\frac{\kappa}{\log(\kappa+1)}\Big)^{\frac{2}{3}(n-1)}
\leq c_{10} \, \frac{\log(\kappa+1)^{\frac23 n-\frac13}}{\kappa^{\frac13}}
\end{multline*}
for suitable constants $c_7$, $c_{8}$, $c_{9}$ and $c_{10}$. This
proves the first part of the statement. For the radius discrepancy,
we proceed in a similar way: given $\varepsilon>0$, we write
$\mathbb{E}(\Delta_{\rm rad}({\boldsymbol{f}}_\kappa,\varepsilon);\lambda_{\kappa})$ as an integral, which we split
into two parts. We bound the first using that
$0\leq\Delta_{\rm rad}({\boldsymbol{f}}_\kappa,\varepsilon)\leq1$ and the estimate~\eqref{eq:5},
while the second integral can be bounded by applying Theorem
\ref{thm:1}.
\end{proof}
\begin{proof}[Proof of Theorem \ref{thm:3}]
The proof is similar to the one given for Theorem \ref{cor:2}. Write
$\nu_{\kappa}=\frac{\mathbb{E}(Z({\boldsymbol{f}}_\kappa);{\lambda_{\kappa}})}{\kappa^{n}\operatorname{MV}({\boldsymbol{Q}})}$ for short,
where $\mathbb{E}(Z({\boldsymbol{f}}_\kappa);{\lambda_{\kappa}})$ is the expected zero density measure
of~${\boldsymbol{f}}_{\kappa}$ .
To prove the statement, it is enough to show that, for all subsets
$U$ as in~\eqref{eq:13},
\begin{displaymath}
\lim_{\kappa\to \infty}\nu_{\kappa}(U)= \nu_{\text{\rm Haar}}(U\cap (S^{1})^{n})=
\begin{cases}
\prod_{j=1}^{n} \frac{\beta_{j}-\alpha_{j}}{2\pi} & \text{ if } U\cap
(S^{1})^{n}\ne\emptyset, \\
0 & \text{ otherwise.}
\end{cases}
\end{displaymath}
If $U\cap(S^1)^n=\emptyset,$ then there exists
$\varepsilon>0$ such that
\begin{displaymath}
\deg(Z({\boldsymbol{f}}_\kappa)|_{U})\le
\deg(Z({\boldsymbol{f}}_\kappa)) \Delta_{\rm rad}({\boldsymbol{f}}_\kappa, {\varepsilon})\le
\kappa^{n}\operatorname{MV}({\boldsymbol{Q}}) \Delta_{\rm rad}({\boldsymbol{f}}_\kappa, {\varepsilon})
\end{displaymath}
and so
\begin{math}
\nu_{\kappa}(U)\le \mathbb{E}(\Delta_{\rm rad}({\boldsymbol{f}}_\kappa,\varepsilon);\lambda_{\kappa}).
\end{math}
Theorem \ref{eqprob0} then implies that
\begin{math}
\lim_{\kappa\to \infty}\nu_{\kappa}(U)=
0= \nu_{\text{\rm Haar}}(U).
\end{math}
If $U\cap(S^1)^n\neq\emptyset$, we set
\begin{math}
\ov U= \{{\boldsymbol{z}} \mid \alpha_j\le \arg(z_j)\le \beta_j \text{ for all }
j\},
\end{math}
and then we have
\begin{displaymath}
\nu_{\kappa}(U) - \prod_{j=1}^{n} \frac{\beta_{j}-\alpha_{j}}{2\pi}
= \Big( \nu_{\kappa}(\ov U) - \prod_{j=1}^{n}
\frac{\beta_{j}-\alpha_{j}}{2\pi}\Big) - \nu_{\kappa}(\ov U\setminus U).
\end{displaymath}
Set $R_{\kappa}=\prod_{{\boldsymbol{v}}}R_{\kappa,{\boldsymbol{v}}}$ for the product of the
directional resultants of
${\mathcal A}_{1},\dots,{\mathcal A}_{n}$. Then
\begin{align*}
\Big| \nu_{\kappa}(\ov U) - \prod_{j=1}^{n}
\frac{\beta_{j}-\alpha_{j}}{2\pi} \Big| & =
\int_{\P(\mathbb{C}^{{\boldsymbol{\mathcal A}}_\kappa})\setminus V(R_{\kappa})}\Big|\frac{\deg(Z({\boldsymbol{f}}_\kappa)_{{\boldsymbol{\alpha}},{\boldsymbol{\beta}}})}{\kappa^{n}\operatorname{MV}({\boldsymbol{Q}})} - \prod_{j=1}^{n}
\frac{\beta_{j}-\alpha_{j}}{2\pi} \Big| \lambda_{\kappa}({\boldsymbol{f}}_\kappa)\,\d\mu_\kappa \\
& \le
\int_{\P(\mathbb{C}^{{\boldsymbol{\mathcal A}}_\kappa})}\Delta_{\rm ang}({\boldsymbol{f}}_\kappa) \lambda_{\kappa}({\boldsymbol{f}}_\kappa)\,\d\mu_\kappa.
\end{align*}
We have that $\ov
U\setminus U$ is a union of a finite number of subsets $U_{l}$ of the
form \eqref{eq:13} such that $U_{l}\cap(S^1)^n=\emptyset$ for all $l$.
By the previous considerations, $\lim_{\kappa\to\infty}\nu_{\kappa}(U_{l})=0$ and so
$\lim_{\kappa\to\infty}\nu_{d}(\ov U\setminus U)= 0$. Theorem \ref{eqprob0} then
implies that
\begin{displaymath}
\lim_{\kappa\to \infty}\nu_{\kappa}(U)= \lim_{\kappa\to \infty}\nu_{\kappa}(\ov U)=
\prod_{j=1}^{n} \frac{\beta_{j}-\alpha_{j}}{2\pi} = \nu_{\text{\rm Haar}}
(U).
\end{displaymath}
\end{proof}
\begin{remark} \label{rem:3} It is not clear to us whether the upper
bound in Proposition \ref{prop:4} for the volume of the set
$V(f)_{\delta}$ is sharp or not. It would
be interesting to clarify this point, as a qualititive improvement
on this bound might enlarge the range of applicability of theorems
\ref{eqprob0} and \ref{thm:3}.
\end{remark}
\begin{remark} \label{rem:1} In some situations, it might be
interesting to consider probability distributions on the complex
linear space $\mathbb{C}^{{\boldsymbol{\mathcal A}}_{\kappa}}$ rather than on
$\P(\mathbb{C}^{{\boldsymbol{\mathcal A}}_{\kappa}})$. For a point ${\boldsymbol{f}}_\kappa\in \P(\mathbb{C}^{{\boldsymbol{\mathcal A}}_{\kappa}})$, the
associated cycle $Z({\boldsymbol{f}}_\kappa)$ does not depend on the choice of a
representative in $\mathbb{C}^{{\boldsymbol{\mathcal A}}_{\kappa}}$ for this point and, \emph{a
fortiori}, the same holds for the angle and radius discrepancies of
$Z({\boldsymbol{f}}_\kappa)$. Hence, one might consider random variables on
$\mathbb{C}^{{\boldsymbol{\mathcal A}}_{\kappa}}$ arising from this cycle as random variables on
$\P(\mathbb{C}^{{\boldsymbol{\mathcal A}}_{\kappa}})$, by applying Federer's coarea formula (see for
instance~\cite[Theorem 20]{BP:edcnsm}).
In precise terms, the normal Jacobian of the map
$\varpi\colon \mathbb{C}^{{\boldsymbol{\mathcal A}}_{\kappa}}\setminus \{{\boldsymbol{0}}\}\to \P(\mathbb{C}^{{\boldsymbol{\mathcal A}}_{\kappa}})$
with respect to the Euclidean structure on
$\mathbb{C}^{{\boldsymbol{\mathcal A}}_{\kappa}}\setminus \{{\boldsymbol{0}}\}$ and the Fubini-Study one on
$\P(\mathbb{C}^{{\boldsymbol{\mathcal A}}_{\kappa}})$ is given, for ${\boldsymbol{g}}\in \mathbb{C}^{{\boldsymbol{\mathcal A}}_{\kappa}}\setminus
\{{\boldsymbol{0}}\}$, by
\begin{displaymath}
\operatorname{NJ}_{{\boldsymbol{g}}}\varpi= \|{\boldsymbol{g}}\|^{-2N_{\kappa}}
\end{displaymath}
with $N_{\kappa}=\#{\boldsymbol{\mathcal A}}_{\kappa}-1$. Given a probability
density function $\lambda_{\kappa}\colon \mathbb{C}^{{\boldsymbol{\mathcal A}}_{\kappa}}\to \mathbb{R}$, one might derive a corresponding
probability density function on $\P(\mathbb{C}^{{\boldsymbol{\mathcal A}}_{\kappa}})$ by integrating along
the fibers of $\varpi$ as
\begin{equation} \label{eq:9}
\lambda_{\kappa}({\boldsymbol{f}}_\kappa)= \frac{\pi^{N_{\kappa}}}{N_{\kappa}!}
\int_{\varpi^{-1}({\boldsymbol{f}}_\kappa)}{\Lambda_{\kappa}({\boldsymbol{g}})}{\|{\boldsymbol{g}}\|_{2}^{2N_{\kappa}}} \,
\d \varpi^{-1}({\boldsymbol{f}}_\kappa),
\end{equation}
where $ \d \varpi^{-1}({\boldsymbol{f}}_\kappa)$ is the volume form of the fiber
$\varpi^{-1}({\boldsymbol{f}}_\kappa)$. The probability distribution given by $\Lambda_{\kappa}$ of,
for instance, the angle discrepancy, can then be computed, for any
Borel subset $I\subset [0,1]$, as
\begin{displaymath}
\operatorname{Prob}(\Delta_{\rm ang}(Z({\boldsymbol{f}}_\kappa)) \in I;\Lambda_{\kappa})=\int_{\Delta_{\rm ang}^{-1}(I)}
\lambda_{\kappa}({\boldsymbol{f}}_\kappa)\, \d \mu_{\kappa},
\end{displaymath}
with $\Delta_{\rm ang}^{-1}(I)= \{{\boldsymbol{f}}_\kappa\in \P(\mathbb{C}^{{\boldsymbol{\mathcal A}}_{\kappa}})\mid \,
\Delta_{\rm ang}(Z({\boldsymbol{f}}_\kappa))\in I\}$. This is a consequence of the coarea
formula.
\end{remark}
\begin{example} \label{exm:1} Let ${\boldsymbol{f}}_\kappa=(f_{\kappa,1},\dots, f_{\kappa,n})$ be
a random system of Laurent polynomials with $\operatorname{supp}(f_{\kappa,i})\subset
\kappa Q_{i}$ whose coefficients $\{f_{\kappa,i,{\boldsymbol{a}}}\}_{{\boldsymbol{a}}\in{\mathcal A}_{\kappa,i}}$ are
independent complex Gaussian random variables with mean 0 and
variance 1. This is a probability distribution on~$\mathbb{C}^{{\boldsymbol{\mathcal A}}_{\kappa}}$
whose density function is defined, for ${\boldsymbol{f}}_\kappa\in \mathbb{C}^{{\boldsymbol{\mathcal A}}_{\kappa}}$,
as
\begin{displaymath} \Lambda_{\kappa}({\boldsymbol{f}}_\kappa)= \prod_{i=1}^{n}
\prod_{{\boldsymbol{a}}\in{\mathcal A}_{\kappa,i}}\frac{1}{\pi}\operatorname{e}^{-|f_{\kappa,i,{\boldsymbol{a}}}|^{2}}=\frac{1}{\pi^{\#{\boldsymbol{\mathcal A}}_{\kappa}}}\operatorname{e}^{-\|{\boldsymbol{f}}_\kappa\|_{2}^{2}}.
\end{displaymath}
The random cycle $Z({\boldsymbol{f}}_\kappa)$ might be described by a probability
distribution on $\P(\mathbb{C}^{{\boldsymbol{\mathcal A}}_{\kappa-}})$. The corresponding density function
is the constant function $\lambda_{\kappa}=1$. This can be seen by computing the
integral along the fibers \eqref{eq:9}, or simply by observing that
$\Lambda_{\kappa}$ is a function of the radius $\|{\boldsymbol{f}}_\kappa\|_{2}$.
Theorem \ref{thm:3} implies then that the sequence of roots of
${\boldsymbol{f}}_\kappa$ converge weakly to the equidistribution on $(S^{1})^{n}$
when $\kappa\to \infty$. In this way, we recover a result of Bloom and
Shiffman \cite[Example 3.5]{BS07}.
\end{example}
\bibliographystyle{amsalpha}
|
{
"timestamp": "2014-08-07T02:10:02",
"yymm": "1203",
"arxiv_id": "1203.1843",
"language": "en",
"url": "https://arxiv.org/abs/1203.1843"
}
|
\section{Introduction}
\label{sec1}
The study of interacting bosonic atoms in a disordered potential
landscape, called in the literature as ``dirty boson
problem``~\ci{Fisher}, has originally been introduced in the context
of the motion of superfluid helium in porous Vycor glass~\ci{Chan}.
Due to the frozen environment, disorder ensembles averages of
physical observables have to be determined, which depend on many
system parameters as, for instance, the strength of a repulsive
interaction between two particles of the Bose gas as well as the
strength and the correlation length which characterize the disorder
potential. The main and intriguing part of the problem is the
competition between the repulsive two-particle interaction and the
localization property of disorder. From a theoretical point of view,
the disorder potential was introduced by investigating the Anderson
localization phenomenon for fermions~\ci{Anderson}. Much attention
has recently been paid for the Anderson localization and the
propagation of bosonic matter waves in random external
potentials~\ci{LSP}. Experimentally, the bosonic matter waves have
been studied in the random potential produced either by laser
speckles~\ci{Billy} or by an incommensurable optical
lattice~\ci{Roati}. Whereas the laser speckle disorder potential is
created by a laser beam scattered from a diffusive glass
plate~\ci{Goodman1}, the incommensurable optical lattice is produced
through two interfering laser beams with incommensurable
wavelengths. However, one needs to remark that
such lattices exhibit certain
pathological features, which do not occur in genuinely random
lattices, such as a transition between localized and delocalized
states, even in one spatial dimension~\ci{Boers}. In that sense the
quasi-periodic lattices should be considered as to be quasi-random
ones. Recent progress in different experimental realizations of
laser speckle disorder is reported in Refs.~\ci{Kondov,Pezze}.
According to the laser speckle theory described in the seminal
work of Goodman~\ci{Goodman1,Goodman2}, the monochromatic light
reflected from a rough surface on the scale of an optical wavelength
yields many independent dephased but coherent wavelets
which interfere at a distance, which is essentially larger than the
wavelength. This results in a granular pattern of intensity that is
called Gaussian speckle as the real and imaginary parts of the field
amplitude form a circular complex Gaussian distribution at any fixed
spatial point. Details of the speckle formation will be considered
in the next section of the paper. Here, we note that this
distribution consists of the first-order statistics of the speckle
disorder, while the second-order statistics of disorder is
represented by its autocorrelation function.
In order
to understand the underlying physics of laser speckles, let us
briefly describe their formation in $2d$. Object waves are fields,
which are a result of the incident polarized monochromatic field
reflection from a rough surface, and they are described in a plane
$\alpha,\beta$ immediately adjacent to the surface in terms of a complex function
$a(\alpha,\beta)$~\ci{Dainty}. The Huygens-Fresnel principle
establishes in the Fresnel approximation a relation between these object
waves $a(\alpha,\beta)$ and the complex waves $A(x,y)$ in the
observation plane $x,y$ through an integral which resembles a
Fourier transformation. Hence, the wave $A(x,y)$ is a result of the
interference of all object waves in the
$x,y$ plane. As in the Fresnel approximation one assumes the
condition $z \gg (\alpha^2+\beta^2)_{\rm max}/\lambda$, where $z$
denotes the distance between the object wave $\alpha,\beta$ plane as well as
the observation wave $x,y$ plane and $\lambda$ denotes the light wavelength,
the waves $A(x,y)$ are called to be in far field~\ci{Dainty}.
In the Fourier mapping of object waves for the formation of
far fields both the form and the finite size of
the diffraction aperture ${\cal A}$ in the $\alpha,\beta$ plane plays a central role. It
determines the form of the
autocorrelation function as well as its correlation length,
which characterizes the average size of the speckle, i.e.~a grain
of the above mentioned intensity pattern. Typically, the expression
for the autocorrelation function consists of a constant and a
spatially varying part. The latter, which is of interest for various speckle
applications, has one central maximum and
a set of side maxima of decaying height, which are separated from each
other by zeros. This analytical structure is principal in the
theory of laser speckles, since it is the result of the Fourier
transformation of the finite-size diffraction aperture ${\cal A}$.
Due to the existence of zeros, it can qualitatively not be approximated
by a function of a Gaussian form as was assumed and even
numerically derived in Refs.~\ci{Pilati1,Pilati2,Piraud}. It is
interesting that the experiment demonstrates an ambiguity in the following
respect: whereas the function with zeros is exploited in the
papers~\ci{Billy,Clement1a,Clement1b}, the spatial autocorrelation function is fitted
by a Gaussian in Refs.~\ci{Kondov,Pezze,Chen1,Chen2,Chen3,Chen4}. Calculating a standard
deviation of the second-order moment of the random intensity, it was
shown in Ref.~\ci{Clement2} that for $1d$ the autocorrelation
function derived in Ref.~\ci{Goodman1,Goodman2} can be well approximated
by a Gaussian form. However, a Fourier transform of this
autocorrelation function, the power spectral density, which is
essential for the theory of a Bose-Einstein condensate (BEC) in an
external disorder potential, behaves, unlike the Gaussian function,
as the triangle function ${\rm tri}(x)=1-|x|$ for $|x|\leq 1$ and
otherwise zero for any dimensionality. For $1d$ and $2d$ this was
shown by Goodman in Refs.~\ci{Goodman1,Goodman2}, the corresponding $3d$ case is dealt with
below in the text. This triangle function makes the upper limit of
the integration in momentum space finite. For those reasons
the recently proposed Gaussian autocorrelation function for the laser
speckle is not suitable for a comprehensive description of a BEC
in laser speckle disorder.
The present paper is organized as follows. We start with describing
the basic principles of the laser speckle theory in Sec.~\ref{sec2}.
Following a scheme described in Refs.~\ci{Goodman1,Goodman2}, we
will then derive in Sec.~\ref{sec3} the expressions for the
autocorrelation function of laser speckles and their Fourier
transforms ranging from $1d$ to $3d$ with special emphasize on
discussing both isotropic and anisotropic cases. The scheme of the
possible experimental realization of the $3d$ isotropic speckle will
be outlined in Sec.~\ref{sec4}.
Note, however, that we consider in our paper a true $3d$ speckle pattern,
not a quasi-three dimensional one of a transverse $2d$ speckle with
a longitudinal depth in the autocorrelation function as described in
Ref.~\ci{Leushacke} and section 4.4.3 of the Goodman book~\ci{Goodman2},
which has been applied in many experiments (see, for instance,
Ref.~\ci{Clement2}). This depth autocorrelation function concept
assumes the existence of an additional spatial direction for the
relevant speckle and can only be valid for $1d$ or $2d$ speckles. As
is further discussed in Refs.~\ci{Leushacke,Goodman2}, the depth size is essentially
larger than ones in other dimensions. Here we consider a $3d$ volume
speckle with compatible speckle grain sizes in all spatial
directions, which was already simulated in
Refs.~\ci{Pilati1,Pilati2}. Since the existing speckle patterns are
experimentally produced mainly in a $2d$ geometry, we will propose a
special scheme for its possible realization in a $3d$ volume. In the
subsequent Sec.~\ref{sec5} the effect of a weak $3d$ isotropic
speckle on various thermodynamic properties of a dilute Bose gas
will be considered at zero temperature. To this end we calculate
both condensate depletion and sound velocity of a BEC within a
perturbative solution of the Gross-Pitaevskii equation. Afterwards,
in Sec.~\ref{sec6}, we reproduce the expression of the normalfluid
density of a BEC in an external disorder potential obtained earlier
within the treatment of Landau. From this rederivation we realize
that condensate particles, which are scattered by a disorder
potential, form a gas of quasiparticles, which is responsible for
the normalfluid component. Finally, we summarize and analyze the
results obtained in the paper in Sec.~\ref{sec7}.
\section{Fundamentals of laser speckle theory}
\label{sec2}
According to Refs.~\ci{Goodman1,Goodman2, Dainty} the circular
Gaussian probability density function
\begin{equation} p(A_R,A_I)=\displaystyle\frac{1}{2\pi \eta^2}\exp\left(-\displaystyle\frac{A_R^2+A_I^2}{2\eta^2} \right)\,,
\lab{lspeck1}
\end{equation}
for the real $A_R$ and imaginary $A_I$ parts of a far-field $A(x,y)$
at each point $x,y$ with the variance $\eta=\sqrt{\langle
|A|^2\rangle}$ represents the background of the theory of laser
speckles. Another basis of the theory is the $M$-fold joint Gaussian
probability density function
\begin{equation} p([A])=\displaystyle\frac{1}{(2\pi)^M |C_A| }\exp\left(-\displaystyle\frac{[A^*][A]}{[C_A]}
\right)\, \lab{lspeck2} \end{equation}
for far-fields $A(x,y)$ at different points $x,y$. Here $[C_A]$ is a
Hermitian symmetric matrix with determinant $|C_A|$, whose elements
are given by $(C_A)_{i,j}=\langle A^*(x_i,y_i) A(x_j,y_j)\rangle$
for a set of far-fields $[A] \equiv \{A(x_1,y_1),A(x_2,y_2),\ldots
A(x_M,y_M)\}$ at $M$ points of the $x,y$ plane. Note that the
notation $\langle \cdots \rangle$ in the expressions for $\eta^2$
and $(C_A)_{i,j}$
and throughout below in the text means the disorder ensemble average.
Furthermore, one assumes that the indices $i,j$ at $(C_A)_{i,j}$ are taken
for adjacent spatial positions.
Expressions~\re{lspeck1} and~\re{lspeck2} are the result of the
central limit theorem of probability theory~\ci{Middleton}, which
claims the following: if complex random variables are the sum of
other independent complex random variables then, at the increase of
the number of second ones, the first ones are distributed according to
the Gaussian law. Applying the theorem for our case we have
far fields $A(x,y)$ as result of the interference of independent
object waves at all positions of $x,y$ plane. As we will see below, there are mainly two
physical conditions for the validity of the central limit
theorem. They are related to the physics of providing independence of the
object waves and to the method of their summation within
the interference process. We will now describe both of them in more detail.
A requirement for the object wave $a(\alpha,\beta)$ to be
independent leads to some limitations for its statistical
properties~\ci{Dainty}. First of all, formed after the reflection of
monochromatic light from the rough surface, the individual wavelet
$a(\alpha,\beta)$ should be completely polarized. Second, the
first-order probability density of its phase should be uniform in
the interval $-\pi$ to $\pi$. And at last, the object wave
$a(\alpha,\beta)$ should be quasi-homogeneous, which means that its
autocorrelation function $C_a$ consists of a slowly-varying
intensity $I_a$ envelope and a short-range normalized correlation
function $C_a'$:
\begin{eqnarray}
C_a(\alpha_1 ,\beta_1 ;\alpha_2 ,\beta_2) \equiv \langle
a^*(\alpha_1 ,\beta_1) a(\alpha_2 ,\beta_2)\rangle
\nonumber \\
=I_a\left(
\displaystyle\frac{\alpha_1+\alpha_2}{2},\displaystyle\frac{\beta_1+\beta_2}{2}\right)
C_a'(\alpha_2-\alpha_1,\beta_2-\beta_1)\,.
\lab{lspeck3}
\end{eqnarray}
If we increase in this expression the range of variation of the
correlation part, i.e.~the correlation length of the object wave,
the changing range of the intensity becomes smaller. However, in
order to entirely satisfy the independence condition of the object
waves, their correlation length in Eq.~\re{lspeck3} should be as
short as possible, which means that $C_a'(\alpha_,\beta)$ has to be
delta correlated. The latter introduces some demands upon the
properties of the random light scatterer, which is called in the
literature as the diffusor. Typically, a diffusor is an optically
homogeneous transparent glass plate with no reflection centers for
light in the volume and a geometrically inhomogeneous distribution
of reflection centers with random heights on its surface. As
mentioned in the introductory section of the paper, the scattering
rough surface generates object waves in a plane $\alpha,\beta$,
which is closely situated at the surface, when the monochromatic
polarized incident light transmits through the plate.
Another realization of object waves is considered in
Refs.~\ci{Goodman1,Goodman2}, where the lateral monochromatic light
was directly incident on the rough surface. For our purpose to
calculate the normalized speckle autocorrelation function, the
optical property of the medium, from which light falls on the
rough surface, is merely dropped from the consideration.
Assuming a Gaussian probability density of the surface height
$h(\alpha,\beta)$ with the autocorrelation function
$C_h(\alpha,\beta)$ and a variance $\eta_h^2$ and assuming also a
Gaussian probability density of object wave phases with a variance
$\eta_\phi^2$, Goodman derived the relation~\ci{Goodman1,Goodman2}
\begin{equation}
C_a'(\alpha_,\beta)=\exp \left( -\eta_\phi^2
[1-C_h(\alpha,\beta)] \right)\,,
\lab{lspeck4}
\end{equation}
where $\eta_\phi= 2\pi \eta_h /\lambda$. This function can be
approximated by a delta function $\delta (\alpha_,\beta)$ when
$\eta_\phi>1$ and thus $\eta_h>\lambda/2$ and the mean distance
between two inhomogeneous $h(\alpha,\beta)$ is larger than
$\lambda$. The delta functional autocorrelation of object waves
provides their independence from each other. On the other hand, the
Gaussian probability density of phases reduces to a uniform one for
$\eta_\phi>1$ supporting the second requirement for the object waves
outlined above. Therefore, the requirements for object waves
described in the previous paragraph can be experimentally realized
if the size of the surface inhomogeneities and the distance between
them are larger than the wave length of the light. As a next
implication of the presented analysis we can suppose that for the
outlined system parameters the object wave probability density
itself may have a circular Gaussian form for the real and imaginary
components of the wavelet $a(\alpha,\beta)$.
The next important step of the theory of Gaussian laser speckles is
the formation of far fields for a given set of object waves. It is
based on the Huygens-Fresnel principle of optics, which preserves
the individual wavelet picture, i.e., works in the limit of optics,
where deviations from geometrical optics are small. The interference
of the object waves yields an amplitude $A(x,y)$, which reads in the
above mentioned far field Fresnel approximation as follows
\begin{equation}
A(x,y)=\int_{\cal A}
a(\alpha,\beta)\exp \left[ -\displaystyle\frac{2\pi i}{\lambda z} (x\alpha +
y\beta)\right] d\alpha d\beta \,,
\lab{lspeck5}
\end{equation}
where we have
omitted unimportant multipliers in front and inside of the integral.
This expression resembles, indeed, a Fourier
transformation and, thus, conserves the
principle that each object wave contributes individually to the
interference.
Using Eqs.~\re{lspeck3} and~\re{lspeck5} it is straight-forward to derive the
expression for the autocorrelation function
\begin{equation}
C_A(x_1 ,y_1 ;x_2,y_2) = \langle A^*(x_1 ,y_1) A(x_2 ,y_2)\rangle \,.
\lab{lspeck6}
\end{equation}
It turns out to be given by
\begin{equation}
C_A(x_1 ,y_1 ;x_2 ,y_2) = I_A\left(
\displaystyle\frac{x_1+x_2}{2},\displaystyle\frac{y_1+y_2}{2}\right) C_A'(\Delta x,\Delta y)
\lab{lspeck7}
\end{equation}
for $\Delta x=x_2-x_1$ and $\Delta y=y_2-y_1$ with
\begin{equation}
I_A(x,y)=\int_{\cal A} C_a'(\alpha'',\beta'')\exp \left[
-\displaystyle\frac{2\pi i}{\lambda z} (x\alpha'' + y\beta'')\right] d\alpha''
d\beta''
\lab{lspeck8}
\end{equation}
and
\begin{equation}
C_A'(x,y)=\int_{\cal A}
I_a(\alpha',\beta')\exp \left[ -\displaystyle\frac{2\pi i}{\lambda z} (x\alpha' +
y\beta')\right] d\alpha' d\beta'
\lab{lspeck9}
\end{equation}
for the novel variables $\alpha'=(\alpha_1+\alpha_2)/2$,
$\beta'=(\beta_1+\beta_2)/2$ and $\alpha''=\alpha_2-\alpha_1$,
$\beta''=\beta_2-\beta_1$. Eq.~\re{lspeck7} shows that the far
fields $A(x,y)$ are
quasi-homogeneous like the object waves $a(\alpha,\beta)$.
This is a direct result of the Fourier transformation
(\ref{lspeck5}). Another important result of this linear
transformation is the implicit proof of the above made supposition
that object waves are circularly Gaussian distributed. Indeed, only
Gaussian distributed object waves can contribute through the linear
mapping to Gaussian far fields. As we will see below the role of the
so far uninvestigated parameter, the aperture $\cal A$, will lead to
the formation of a correlation length of the correlation function
$C_A'(x,y)$.
The autocorrelation function $C_I(x,y)$ of the far-field intensity
$I(x,y)=|A(x,y)|^2$ can be calculated using the Wick theorem for
variables distributed according to the Gaussian law. A simple
calculation gives the expression
\begin{equation}
C_I(x,y)=\langle I \rangle^2 \left[1+|C_A'(x,y)|^2\right]\,,
\lab{lspeck10}
\end{equation}
where the
normalized autocorrelation function for the far-field $C_A'(x,y)$ is
defined as $C_A'(x,y)/C_A'(0,0)$, where $C_A'(0,0)=\langle I \rangle$.
As already mentioned above, if in Eq.~\re{lspeck3} the autocorrelation
function of object waves $C_a'(\alpha_,\beta)$ is delta correlated,
then the intensity function of these waves $I_a(\alpha,\beta)$ can
be approximated as a constant.
Assuming that the $\alpha,\beta$ plane is close to the
rough surface, one can write $|a(\alpha,\beta)|=\kappa
|P(\alpha,\beta)|$, where $P(\alpha,\beta)$ is the incident to the
glass plate light wave and $\kappa$ is the average reflectivity of
surface, for each position $\alpha,\beta$. Then the intensity of the
object waves at $\alpha,\beta$ is determined by the relation
$I_a(\alpha,\beta)=\kappa^2 |P(\alpha,\beta)|^2$. Therefore, the
expression for the normalized far-field autocorrelation function
reads
\begin{equation}
C_A'(x,y)=\displaystyle\frac{\int_{\cal A} |P(\alpha,\beta)|^2 \exp
\left[ -\displaystyle\frac{2\pi i}{\lambda z} (x\alpha + y\beta)\right] d\alpha
d\beta}{\int_{\cal A} |P(\alpha,\beta)|^2 d\alpha d\beta}\,.
\lab{lspeck11}
\end{equation}
\section{Speckle autocorrelation function for apertures in $1d$
to $3d$ dimensions} \label{sec3}
As already mentioned in the introductory section, the investigation
of a BEC in the laser speckle disorder has found much attention from
both a theoretical and an experimental point of view. In particular,
a variety of isotropic and anisotropic speckles have been the
subject of these works.
Motivated by this interest, we will describe in the present section
the derivation of the speckle autocorrelation function for different
apertures ranging from one to three dimensions by generalizing
the appropriate expressions from the previous section to these dimensions.
\subsection{Real space}
Due to the analytic form of Eq.~\re{lspeck11}, we can take the
intensity of the incident wave $|P(\alpha,\beta)|^2$ to be unity
over the whole aperture region of the $\alpha,\beta$ plane. Writing
the function $|P|^2$ in the form $|P|_{d,{\cal A}}^2$, where $d$ is
the space of dimensionality and ${\cal A}$ is the form of the
aperture, we have the following expressions:
\begin{equation}
|P(\alpha,\beta)|_{2d,{\rm rct}}^2={\rm rect}
\left( \displaystyle\frac{\alpha}{L_{\alpha}}\right) {\rm rect} \left(
\displaystyle\frac{\beta}{L_{\beta}}\right)
\lab{lspeck14}
\end{equation}
for the $2d$ anisotropic rectangular aperture with sizes $L_{\alpha}$
and $L_{\beta}$, where the function ${\rm rect} (x)=1$ for $|x|\leq
1/2$ and zero otherwise;
retaining in Eqs.~\re{lspeck14} only the first ${\rm rect} (x)$
function and equating $L_{\alpha}=L$ one obtains the expression of
$|P(\alpha)|_{1d,{\rm inv}}^2$ for the $1d$ interval aperture of the
size $L$; the analytic form of $|P(\alpha,\beta)|_{2d,{\rm qdt}}^2$
for the $2d$ quadratic aperture of the size $L$ is obtained from
Eqs.~\re{lspeck14} if we specialize this equation according to
$L_{\alpha}=L_{\beta}=L$;
\begin{equation} |P(\alpha,\beta)|_{2d,{\rm crc}}^2={\rm circ} \left(
\displaystyle\frac{2 r}{D}\right)
\lab{lspeck15}
\end{equation}
for the $2d$ isotropic circular
aperture with the diameter $D$ and $r=\sqrt{\alpha^2+\beta^2}$, where the
function ${\rm circ} (x)=1$ for $|x|\leq 1$ and zero otherwise;
\begin{equation}
|P(\alpha,\beta,\gamma)|_{3d,{\rm rcpl}}^2={\rm rect}
\left( \displaystyle\frac{\alpha}{L_{\alpha}}\right) {\rm rect} \left(
\displaystyle\frac{\beta}{L_{\beta}}\right){\rm rect} \left(
\displaystyle\frac{\gamma}{L_{\gamma}}\right)
\lab{lspeck17}
\end{equation}
for the $3d$
anisotropic rectangular parallelepiped aperture of sizes
$L_{\alpha}$, $L_{\beta}$ and $L_{\gamma}$;
the expression $|P(\alpha,\beta,\gamma)|_{3d,{\rm cub}}^2$ of the
$3d$ cubic aperture of the size $L$ is obtained from
Eq.~\re{lspeck17} by setting $L_{\alpha}=L_{\beta}=L_{\gamma}=L$;
\begin{equation}
|P(\alpha,\beta,\gamma)|_{3d,{\rm sph}}^2={\rm circ} \left( \displaystyle\frac{2 r}{D}\right)
\lab{lspeck18}
\end{equation}
for the $3d$ isotropic sphere aperture with the
diameter $D$ and $r=\sqrt{\alpha^2+\beta^2+\gamma^2}$;
\begin{equation}
|P(r,\gamma)|_{3d,{\rm cyl}}^2={\rm circ} \left( \displaystyle\frac{2 r}{D}\right){\rm rect} \left(
\displaystyle\frac{\gamma}{L_{\gamma}}\right)
\lab{lspeck19}
\end{equation}
for the $3d$ anisotropic cylinder aperture with the diameter of
circle $D$, $r=\sqrt{\alpha^2+\beta^2}$ and size $L_{\gamma}$ along
the $\gamma$ axis.
Substituting the expressions~\re{lspeck14}--\re{lspeck19} of the
$|P|_{d,{\cal A}}^2$ function in Eq.~\re{lspeck11} and calculating
the respective integrals, we obtain the corresponding expressions
for the correlation function $|C_A'|^2$:
\begin{equation} |C_A'(\Delta x,\Delta y)|_{2d,{\rm rct}}^2= {\rm sinc}^2
\left( \displaystyle\frac{L_{\alpha}\Delta x}{\lambda z}\right){\rm sinc}^2 \left(
\displaystyle\frac{L_{\beta}\Delta y}{\lambda z}\right)
\lab{lspeck22}
\end{equation}
where ${\rm sinc}(y)=\sin(\pi y)/(\pi y)$, for the $2d$ anisotropic
rectangular aperture with $z$ being the distance between object wave
and far field planes; retaining in this equation only first ${\rm
sinc}^2 (y)$ function, the dependence on $\Delta x$ and assuming
$L_{\alpha}=L$ one obtains the expression $|C_A'(\Delta x)|_{1d,
{\rm inv}}^2$ for the $1d$ interval aperture with $z$ being the
distance between object wave and far field intervals; the expression
$|C_A'(\Delta x,\Delta y)|_{2d,{\rm qdt}}^2$ for the $2d$ quadratic
aperture one can derive from Eq.~\re{lspeck22} for
$L_{\alpha}=L_{\beta}=L$;
\begin{equation}
|C_A'(r)|_{2d,{\rm crc}}^2=\left|
2\displaystyle\frac{J_1\left( \displaystyle\frac{\pi Dr}{\lambda z}\right)}{\displaystyle\frac{\pi Dr}{\lambda
z}} \right|^2\,,
\lab{lspeck23}
\end{equation}
where $J_1(x)$ is a Bessel
function of the first kind and of the first order, for the $2d$
isotropic circular aperture with $r=\sqrt{(\Delta x)^2+(\Delta y)^2}$;
\begin{eqnarray}
&& \hspace*{0.5cm} |C_A'(\Delta x,\Delta y,\Delta z)|_{3d,{\rm rcpl}}^2= \nonumber\\
&&{\rm sinc}^2 \left( \displaystyle\frac{L_{\alpha}\Delta x}{\lambda z}\right){\rm sinc}^2
\left( \displaystyle\frac{L_{\beta}\Delta y}{\lambda z}\right){\rm sinc}^2 \left(
\displaystyle\frac{L_{\gamma}\Delta z}{\lambda z}\right)
\lab{lspeck25}\end{eqnarray}
for the $3d$ anisotropic rectangular parallelepiped aperture with
$z$ as the distance between the object wave and the far field
volumes; the expression $|C_A'(\Delta x,\Delta y,\Delta z)|_{3d,{\rm
cub}}^2$ for the $3d$ cubic aperture is obtained from
Eq.~\re{lspeck25} by assuming
$L_{\alpha}=L_{\beta}=L_{\gamma}=L$;
\begin{eqnarray}
&&\hspace*{1cm}|C_A'(r)|_{3d,{\rm sph}}^2= \lab{lspeck26} \\
&&\left| 3\left( \displaystyle\frac{\lambda z}{\pi
Dr}\right)^3\left[ \sin \left( \displaystyle\frac{\pi Dr}{\lambda z}\right)-\left(
\displaystyle\frac{\pi Dr}{\lambda z}\right) \cos \left( \displaystyle\frac{\pi Dr}{\lambda
z}\right)\right] \right|^2
\nonumber
\end{eqnarray}
for the $3d$
isotropic sphere aperture with $r=\sqrt{(\Delta x)^2+(\Delta
y)^2+(\Delta z)^2}$;
\begin{equation}
|C_A'(r,\Delta z)|_{3d,{\rm cyl}}^2=\left|
2\displaystyle\frac{J_1\left( \displaystyle\frac{\pi Dr}{\lambda z}\right)}{\displaystyle\frac{\pi Dr}{\lambda
z}} \right|^2 {\rm sinc}^2 \left( \displaystyle\frac{L_{\gamma}\Delta z}{\lambda
z}\right)
\lab{lspeck27}
\end{equation}
for the $3d$ anisotropic cylinder aperture with $r=\sqrt{(\Delta
x)^2+(\Delta y)^2}$.
Expressions of the autocorrelation function for a $2d$ quadratic
aperture
$|C_A'(\Delta x,\Delta y)|_{2d,{\rm qdt}}^2$ and for a
$2d$ circular aperture in Eq.~\re{lspeck23} are derived by Goodman
in Refs.~\ci{Goodman1,Goodman2}. As can be seen from the formulas
for other cases of the aperture, they are closely related to both of
these Goodman cases of the aperture. However, the derivation of the $3d$
isotropic sphere autocorrelation function (\ref{lspeck26}), which is
a result of the present paper, required some additional effort.
The analytical forms of the autocorrelation functions $|C_A'|^2$ are
similar in every spatial direction. They have one central maximum and a set
of side maxima of decaying height, which are separated from each other by
zeros. As was pointed out in the introductory
section, it is obvious that
these forms can not be fitted by a Gaussian. The
argument of the autocorrelation function, which corresponds to its first zero, provides the
correlation length of the disorder, i.e.~the average size of the speckle grain,
for the appropriate spatial direction. Denoting it by
$\delta x$ we have, for instance, for $1d$ speckle
\begin{equation}
\delta x=\displaystyle\frac{\lambda z}{L}\,.
\lab{lspeck27a}
\end{equation}
The main interest of the present paper is the $3d$ spherical
aperture of Eq.~(\ref{lspeck26})
since we will carry out the calculation of BEC properties
for this particular case of laser speckles. Numerically solving the equation
$\sin(x)-x \cos(x)=0$ we find first its solution to be at $x_c=4.493$, thus
the disorder correlation length is given by $r_c=1.4302~\lambda z/D$.
In order
to establish a physical meaning of $\delta x$ we introduce the
"wave number" $k_{\rm eff}$, which is related to the vector
$\alpha,\beta$ in the above Fourier transform formulas, by the relation
$k_{\rm eff}=2\pi x/(\lambda z)$. If we substitute in it $\delta x$ from
Eq.~\re{lspeck27a} then we obtain $k_{\rm eff}=2\pi/L$. For a circular and
a spherical aperture the "wave number" is $k_{\rm eff}=2\pi/D$. However, the
sense of $k_{\rm eff}$ is in an uncertainty of the wave vector when the
problem of wave propagation is solved in the restricted area. It is
well known that in this area the wave vector is determined within the
resolution $k_{\rm eff}$. Therefore, we can say that the origin of a
speckle grain with a correlation length $\delta x$ as its size represents the
spatial uncertainty in the determination of far fields, which is introduced by the
finite size of the aperture.
\subsection{Fourier space}
For many applications the Fourier transform of the far-field intensity
autocorrelation function, or the power spectral density, of the
speckle is of considerable interest. In the literature on laser
speckle theory~\ci{Goodman1,Goodman2} it is defined according to
\begin{equation}
C_I({\bf k})=\int C_I({\bf x})e^{-i2\pi {\bf k}{\bf x}} d^d x\,.
\lab{lspeck28}
\end{equation}
Substituting
in it Eq.~\re{lspeck10} for $C_I({\bf x})$ one obtains
\begin{equation}
C_I({\bf k})= \langle I \rangle^2 \left[\delta ({\bf k})+|C_A'({\bf k})|^2\right]\,.
\lab{lspeck29}
\end{equation}
In the perturbative considerations of BEC in the speckle potential
the Fourier transform $|C_A'({\bf k})|^2$ plays the central role. It
has the following expressions for the real space autocorrelation
functions taken from Eqs.~\re{lspeck22}--\re{lspeck27}:
\begin{equation}
|C_A'({\bf k})|_{2d,{\rm rct}}^2= \displaystyle\frac{(\lambda z)^2}{L_{\alpha} L_{\beta}} {\rm tri}
\left( \displaystyle\frac{k_x\lambda z}{L_{\alpha}}\right){\rm tri} \left(
\displaystyle\frac{k_y\lambda z}{L_{\beta}}\right)
\lab{lspeck32}
\end{equation}
where the triangle function is defined as ${\rm tri}(x)=1-|x|$ for
$|x|\leq 1$ and zero otherwise, for the $2d$ anisotropic rectangular
aperture; the expression $|C_A'({\bf k})|_{1d,{\rm inv}}^2$ for the
$1d$ interval aperture one can get from Eq.~\re{lspeck32} by
assuming $L_{\alpha}= L_{\beta}=L$, $k_x=k_y=k$ and taking the
square root of its right-hand side; the expression
$|C_A'({\bf k})|_{2d,{\rm qdt}}^2$ for the $2d$ quadratic aperture is
obtained from Eq.~\re{lspeck32} with the assumption $L_{\alpha}=
L_{\beta}=L$;
\begin{eqnarray}
&& |C_A'({\bf k})|_{2d,{\rm crc}}^2= 2\left(\displaystyle\frac{2\lambda z}{\pi D}\right)^2
\lab{lspeck33} \\
&&\times \left[ \cos^{-1}\left(
\displaystyle\frac{k\lambda z}{D}\right)-\displaystyle\frac{k\lambda z}{D}\sqrt{1-\left(
\displaystyle\frac{k\lambda z}{D}\right)^2}\,\right] \nonumber
\end{eqnarray}
for the
$2d$ isotropic circular aperture with $k=\sqrt{k_x^2+k_y^2}$;
\begin{eqnarray}
&&|C_A'({\bf k})|_{3d,{\rm rcpl}}^2=
\displaystyle\frac{(\lambda z)^3}{L_{\alpha} L_{\beta} L_{\gamma}}
\lab{lspeck35} \\
&&\times {\rm tri} \left( \displaystyle\frac{k_x\lambda z}{L_{\alpha}}\right) {\rm tri} \left(
\displaystyle\frac{k_y\lambda z}{L_{\beta}}\right) {\rm tri} \left( \displaystyle\frac{k_z\lambda
z}{L_{\gamma}}\right)\nonumber
\end{eqnarray}
for the $3d$
anisotropic rectangular parallelepiped aperture;
the expression $|C_A'({\bf k})|_{3d,{\rm cub}}^2$ for the $3d$ cubic
aperture is obtained from Eq.~\re{lspeck35} by specializing $L_{\alpha}=
L_{\beta}= L_{\gamma}=L$;
\begin{equation}
|C_A'({\bf k})|_{3d,{\rm sph}}^2=\displaystyle\frac{3}{\pi}\left(\displaystyle\frac{2\lambda
z}{4D}\right)^3(b^3-12b+16)
\lab{lspeck36}
\end{equation}
for the $3d$
isotropic sphere aperture with $b=2k\lambda z/D$ and
$k=\sqrt{k_x^2+k_y^2+k_z^2}$;
\begin{eqnarray}
&&|C_A'({\bf k})|_{3d,{\rm cyl}}^2= \left(\displaystyle\frac{\lambda z}{L_{\gamma}}\right) {\rm tri} \left( \displaystyle\frac{k_z\lambda
z}{L_{\gamma}}\right) \lab{lspeck37} \\
&&\times 2\left(\displaystyle\frac{2\lambda z}{\pi D}\right)^2 \left[ \cos^{-1}\left(
\displaystyle\frac{k\lambda z}{D}\right)-\displaystyle\frac{k\lambda z}{D}\sqrt{1-\left(
\displaystyle\frac{k\lambda z}{D}\right)^2}\,\right]
\nonumber
\end{eqnarray}
for the $3d$
anisotropic cylinder aperture with $k=\sqrt{k_x^2+k_y^2}$.
The equation for the quadratic aperture $|C_A'({\bf k})|_{2d,{\rm
qdt}}^2$ and Eq.~\re{lspeck33} have been derived by Goodman in
Refs.~\ci{Goodman1,Goodman2}. Other expressions of $|C_A'({\bf
k})|^2$, except for the $3d$ isotropic sphere aperture case, can be
obtained by using these formulas. Eq.~\re{lspeck36} is a result of
this paper. In all our formulas for the anisotropic aperture we have
assumed that the size deviation of the aperture with respect to its
average isotropic size is essentially less than the distance $z$.
As is seen from the formulas of $|C_A'({\bf k})|^2$ expressed through the
triangle function their value becomes zero when their argument is
unity. For the $2d$ circle and the $3d$ sphere apertures $|C_A'({\bf k})|^2$
is zero for $k\lambda z/D=1$. Hence, the wave vector of the Fourier
transform autocorrelation function only varies in a finite interval
from zero, in contrast to the case for a Gaussian
function. This fact is another reason why the speckle autocorrelation
function can not be approximated by a Gaussian form.
It is worth to discuss the expression $|C_A'(r)|^2={\rm sinc}^2(k_L
r)$ with $k_L= D/\lambda z$ for the autocorrelation function used
in Ref.~\ci{Kuhn} for the $3d$ isotropic aperture. It is similar to our correlation function
$|C_A'(\Delta x)|_{1d,{\rm inv}}^2$ for the $1d$ interval aperture.
The authors of Ref.~\ci{Kuhn} claim that this expression is valid
for $z \sim (\alpha^2+\beta^2)_{\rm max}/\lambda$, which is outside
of the far-field limit. However, that limit destroys the
fundamentals of the Gaussian speckle theory as they are described in
Sec.~\ref{sec2}. Therefore, it is unclear whether $|C_A'(r)|^2$ of
Ref.~\ci{Kuhn} is related to laser speckles or not.
Furthermore, we discuss the definition of the speckle correlation
length to be the width at the half value of the maximum of
$|C_A'(r)|^2$ for $r=0$, when the last one is approximated by a
Gaussian function.
Probably, this definition was introduced first by Modugno in
Ref.~\ci{Modugno}, when he considered $|C_A'(\Delta x)|_{1d,{\rm
inv}}^2$. It was found in Ref.~\ci{Modugno} that the correlation
length is given by $\delta x=0.88\lambda z/L$, while from
Eq.~\re{lspeck27a} the exact value turns out to be $\delta x=\lambda
z/L$. It is interesting that the Gaussian $|C_A'(r)|^2$ has been
obtained in the numerical simulation of the $3d$ isotropic laser
speckle in Refs.~\ci{Pilati1,Pilati2} which should be compared with
the exact $|C_A'(r)|_{3d,{\rm sph}}^2$ in Eq.~\re{lspeck26}, with
the correlation length $r_c=1.1\lambda z/D$, however, the exact one
is $r_c=1.4302 \lambda z/D$, see the discussion after
Eq.~\re{lspeck27a}. It seems that we can explain the reason why the
authors of Refs.~\ci{Pilati1,Pilati2} obtained the Gaussian form of
$|C_A'(r)|^2$. They used the speckle simulation method proposed by
Huntley in Ref.~\ci{Huntley} which we briefly review for the $2d$
case. Let us consider to this end two square planes $\alpha,\beta$
and $x,y$ with the same size $L$. According to the Huntley method
one uses Eq.~\re{lspeck5} in order to perform a double Fourier
transformation. In the first inverse Fourier transformation the
complex object waves $a(\alpha,\beta)$ on the mesh points in the
$\alpha,\beta$ plane are simulated through the given Gaussian
distributed complex random waves $A(x,y)$ on the mesh points in the
$x,y$ plane. Afterwards, one cuts by a circle with radius $D/2$ the
$\alpha,\beta$ region of the obtained $a(\alpha,\beta)$ such that it
vanishes outside of this region. In the second direct Fourier
transformation the derived complex waves $a(\alpha,\beta)$ form the
final complex far-fields $A(x,y)$. Huntley has investigated in
Ref.~\ci{Huntley} only the first-order statistical property of the
simulated pattern, i.e.~the probability density of the intensity,
and showed that it corresponds to the theoretical laser speckles of
Ref.~\ci{Goodman1}. However, the proposed simulation method can
drastically deviate in the second-order statistical property of a
speckle, i.e.~its autocorrelation function, from the theoretical
one.
Indeed, in accordance with the theory of a speckle autocorrelation
function as presented in this section, after the first Fourier mapping the
object waves $a(\alpha,\beta)$ acquire a correlation with the
correlation length $\delta \alpha=\delta \beta=\lambda z/L$, where
$L$ is size of the square $x,y$ plane. More precisely, now the
function $C_a'(\alpha,\beta)$ is not delta correlated. However,
according to Eq.~\re{lspeck3}, the broadening of the
$C_a'(\alpha,\beta)$ function reduces to a changing of a constant
character of the $I_a(\alpha,\beta)$ function to one of varying in space in the
$\alpha_,\beta$ plane. Substituting this function of
$I_a(\alpha,\beta)$ in Eq.~\re{lspeck9} and integrating over
$\alpha$ and $\beta$ gives the function $C_A'(x,y)$ which may
qualitatively be different from the one discussed in this section.
A simulation method, which is consistent with the above laser speckle theory,
is described in the book of Goodman~\ci{Goodman2}. There are other
numerical methods in Refs.~\ci{Makse1,Makse2}, in which the exact form of the real
space autocorrelation function is used to generate the speckle
pattern. In particular, one of such methods was exploited for the
simulation of $1d$ speckle in Ref.~\ci{Sucu}.
\section{Experimental realization of $3d$ isotropic speckle}
\label{sec4}
As was already mentioned in the introductory section, we consider here a true
$3d$ speckle, not the quasi-three dimensional one consisting of a
transverse $2d$ speckle with a longitudinal depth in the
autocorrelation function as described in details in
Ref.~\ci{Leushacke} and section 4.4.3 of the Goodman
book~\ci{Goodman2} and applied in many experiments. At a first
glance, it seems exotic and unrealistic to experimentally realize
such a $3d$ volume speckle pattern. However, in the present section
we will describe the physical principle how it can be generated.
In the typical $2d$ geometry of the experimental realization of a
speckle a lens, which collects the incident light, is installed close to the
glass plate such that its focal plane coincides with the far-field
plane~\ci{Clement2}. This idea of a speckle formation in the focal
plane can be generalized to a full $3d$ geometry, when the speckle is formed in
the focal point, i.e.~the focus, of an empty ellipsoidal optic cavity according to
the scheme displayed in Fig.~1.
\begin{figure}[t]
\begin{center}
\includegraphics[scale=0.5]{figure1.eps}
\end{center}
\caption{Cross section of the ellipsoidal reflective cavity with spheres
$A$ and $B$ in its focuses. Focus $A$ contains a small size absolute
spherical light reflector in the center and volume optic
inhomogeneities. Incident laser beams (thick yellow arrows), after
reflection from the reflector, scatter additionally from inhomogeneities
producing individual wavelets (thin yellow lines), which are
collected in focus $B$, where a BEC is deposited.}
\lab{fig1}
\end{figure}
Let us consider that cavity, whose inside surface
reflects absolutely the light emitted from one of its focus (point $A$) and
collects it at the second focus (point $B$). Two laser beams (thick
yellow arrows) are incident through holes in the cavity surface into
the small metallic sphere, i.e.~the reflector, with absolute light
reflection, located in the center of the glass sphere $A$. It is assumed
that laser beams cover the entire surface of this reflector and a BEC
is deposited in the sphere $B$, which is located at the second focus
of the ellipsoid.
The glass sphere $A$ additionally contains the located randomly
light scattering centers, for instance, absolutely light reflective
metallic polyhedrons with a random average size of each facet. The
theory how to derive the $2d$ object wave autocorrelation function
$C_a'(\alpha,\beta)$, described in Refs.~\ci{Goodman1,Goodman2}, can
be easily generalized to the derivation of
$C_a'(\alpha,\beta,\gamma)$ for such a $3d$ case with the same
expression~\re{lspeck4}. However, now this expression is a function
of $3d$ other quantities. The condition, at which
$C_a'(\alpha,\beta,\gamma)$ becomes delta correlated and the object
waves are independent, is the same as for $C_a'(\alpha,\beta)$.
Therefore, if the mean distance between these light scattering
centers and the average size of polyhedrons are larger than the
light wavelength, then $C_a'(\alpha,\beta,\gamma)$ will be delta
correlated. On the other hand, each sphere with radius
$(\alpha^2+\beta^2+\gamma^2)^{1/2}$ and with the same center as the
sphere $A$ will be the object wave volume, whereas for comparison
for $2d$ we had a $\alpha,\beta$ object wave plane.
Incident laser beams, after reflection from the reflector, scatter
additionally from scattering centers and produce individual and
independent wavelets, the object waves $a(\alpha,\beta,\gamma)$, indicated via thin
yellow lines in Fig.~\ref{fig1}, which are collected in the sphere $B$, where a BEC
is deposited. For the presented geometry the far-field condition is
satisfied, since a distance $z$ between the object wave and the far-field
volumes, i.e.~the length of each wavelet trajectory between two focuses
of the ellipsoid, is larger than the size of the object wave sphere
$A$.
The described scheme can be generalized for the experimental
realization of any $3d$ anisotropic speckle. To this end one only needs to
change the form of the spherical aperture $A$, which contains
the glass and the light scattering centers, into a suitable one listed in
the previous section. The spherical form of the
metallic reflector retains unchanged.
At the end of this section, it is worthwhile to discuss the possible
realization of a $3d$ volume speckle pattern using $2d$ plane
speckles. Such a scenario presumes a $3d$ speckle as a result of the
sum or, more clearly, as a linear interference of two and more $2d$
speckles. While theoretically this scenario is discussed by Pilati
{\it at} {\it al.} in Ref.~\ci{Pilati2}, the experiment, in which
two perpendicular $2d$ speckle planes form a $3d$ speckle pattern,
was realized in Ref.~\ci{Jendrzejewski} by Jendrzejewski {\it et}
{\it al.} Instantly the question arises whether the random pattern
realized in such a way belongs to the class of speckles or not. In
spite of an additional theoretical analysis, which is required to
answer that question in detail, the following argument shows that
the possible conclusion is negative. Indeed, according to the
fundamentals of the laser speckle theory of Goodman,
Refs.~\ci{Goodman1,Goodman2}, and Dainty, Ref.~\ci{Dainty}, see also
Secs.~\ref{sec2} and~\ref{sec3} of this paper, the correlated
speckle pattern in any dimension,
except the one described in Ref.~\ci{Leushacke} and its analogue for
$1d$ (see next paragraph), is a result of the Fourier transform over
the restricted aperture object wave region in the same dimension.
This means that a $3d$ volume speckle can be obtained only by a $3d$
object wave volume. Physically it means that the single connected
spatial domain of each $3d$ speckle grain, which is a result of $3d$
correlations, can not be obtained by a linear combination of randomly
sized and independent $2d$ speckle grains.
By that reason, the true $3d$ speckle cannot be obtained even by a
combination of quasi-three dimensional speckles, which we
discussed at the beginning of this section.
\section{BEC depletion and sound velocity in weak $3d$ isotropic speckle}
\label{sec5}
The interaction potential of light with an atom at position ${\bf r}$ is determined by the far-field intensity $I({\bf r})=
|A({\bf r})^2|$ and has the form $V({\bf r})=t I({\bf r})$, see for instance
Refs.~\ci{Clement2,Kuhn}, where the constant $t$ is a function of
the atomic and light characteristics. At the derivation of
$V({\bf r})$ it was assumed that the incident laser wave does not induce an
atomic electron interlevel transition, but merely deforms the atomic
ground state.
It is convenient to define the interaction potential as $V({\bf r})=V_0+\Delta V({\bf r})$,
where $\Delta V({\bf r})=V({\bf r})-V_0$ and
$V_0=\langle I \rangle$. Using the obvious property $\langle \Delta V({\bf r})\rangle =0$, a simple
calculation shows that
\begin{equation} \langle V({\bf r}')V({\bf r}'+{\bf r}) \rangle =
V_0^2 \left[ 1+\displaystyle\frac{\langle \Delta V({\bf r}')
\Delta V({\bf r}'+{\bf r}) \rangle}{V_0^2}\right]
\lab{lspeck38}
\end{equation}
and, therefore, we have the following relationships between the laser
speckle autocorrelation and the disorder potential correlation
functions:
\begin{eqnarray}
|C_I({\bf r})|^2&=& \langle V({\bf r}') V({\bf r}'+{\bf r}) \rangle \, , \nonumber \\
|C_A'({\bf r})|^2 &=&\displaystyle\frac{\langle \Delta V({\bf r}') \Delta V({\bf r}'+{\bf r}) \rangle}{V_0^2} \, .
\lab{lspeck39}
\end{eqnarray}
Our interest is a Bose gas with a contact interaction. Taking
into account that, according to the novel definition of $V({\bf r})$, the
chemical potential for the ground state of BEC will be renormalized according to
$\mu \rightarrow \mu-V_0$, the Gross-Pitaevskii equation (GPE) reads
\begin{equation}
\left[-\displaystyle\frac{\hbar^2}{2m}{\mbox{\boldmath $\nabla$}}^2 + \Delta V({\bf r})
+g|\Psi({\bf r})|^2 - \mu \right] \Psi({\bf r})=0 \,.
\lab{lspeck40}
\end{equation}
Here $g=4\pi \hbar^2 a/m$ denotes the strength of the contact interaction
with the scattering length $a$.
Under the assumption that the disorder potential is weak, one can
expand the solution
\begin{equation}
\Psi({\bf r})=\psi_0+\psi_1({\bf r})+\psi_2({\bf r})+\cdots
\lab{lspeck41}
\end{equation}
and solve the GPE (\ref{lspeck40}) perturbatively in the respective order of
$\Delta V({\bf r})$~\ci{Krumnow}. For the ground state all functions
of the expansion as well as $\Psi({\bf r})$ are real. In this way the problem is
reduced to find the total particle density $n=\langle \Psi({\bf r})^2\rangle$
and the condensate density $n_0=\langle \Psi({\bf r}) \rangle^2$. In
particular, the lowest order expression for
the condensate depletion reads
\begin{equation}
n-n_0=n_0\, \int \, \displaystyle\frac{d^3 k}{(2\pi)^3} \displaystyle\frac{R({\bf k})}{[\hbar^2 {\bf k}^2/2m + 2ng]^2} +\cdots\,,
\lab{lspeck42}
\end{equation}
where we introduced the literature notation $R({\bf k})= R|C_A'({\bf k})|^2$ with
$R=V_0^2$.
In order
to further apply our formula Eq.~\re{lspeck36} for the $3d$
isotropic autocorrelation function $|C_A'({\bf k})|_{3d,{\rm sph}}^2$, one
needs to make a remark. According to the definition in Eq.~\re{lspeck28},
the Fourier transforms of autocorrelation functions carry a physical dimension.
In particular, the correlation function
$|C_A'({\bf k})|_{3d,{\rm sph}}^2$, calculated with Eq.~\re{lspeck28}, is proportional to the inverse volume of the
$3d$ isotropic aperture $3/(4\pi)(2/D)^3$ times $(\lambda z)^3$.
If we introduce the correlation length as $\sigma=\lambda z/D$, then
the proportionality factor is $3(2\sigma)^3/(4\pi)$. In the following, we assume that
$|C_A'({\bf k})|_{3d,{\rm sph}}^2$ is already normalized by that factor.
It is convenient to introduce also the BEC coherence length according
to $\xi=[\hbar^2/(2mng)]^{1/2}=1/\sqrt{8\pi na}$.
Substituting the normalized correlation function $R|C_A'({\bf k})|_{3d,{\rm sph}}^2$
from Eq.~\re{lspeck36} in Eq.~\re{lspeck42} and performing the
integration, we get the expression $n-n_0=n_{\rm HM}f(\sigma/\xi)$,
where the depletion $n_{\rm HM}=[m^2R/(8\pi^{3/2}
\hbar^4)]\sqrt{n/a}$ was obtained by Huang and Meng in Ref.~\ci{Huang} (see also Ref.~\cite{Falco}) for
delta correlated disorder $R({\bf r})$ and the condensate depletion function is defined via
\begin{eqnarray}
f\left(\displaystyle\frac{\sigma}{\xi}\right)&=&\displaystyle\frac{1}{\sqrt{2}\pi}\displaystyle\frac{\sigma}{\xi}
\left[4-\left(\displaystyle\frac{8\sigma^2}{\xi^2}+6\right) \ln
\left(1+\displaystyle\frac{\xi^2}{2\sigma^2}\right)\right. \nonumber \\
&& \left. + \displaystyle\frac{4}{\sqrt{2}}\displaystyle\frac{\xi}{\sigma} {\rm arctan}
\left(\displaystyle\frac{\xi}{\sqrt{2}\sigma}\right)\right]\, .
\lab{lspeck43}
\end{eqnarray}
The function $f(\sigma/\xi)$, which is depicted in Fig.~\ref{fig2},
has the following asymptotics for small $\sigma/\xi$
\begin{equation}
f\left(\displaystyle\frac{\sigma}{\xi}\right)\approx
1-\displaystyle\frac{14\sqrt{2}}{3\pi}\left(\displaystyle\frac{\sigma}{\xi}\right)^3
-\displaystyle\frac{18\sqrt{2}}{5\pi}\left(\displaystyle\frac{\sigma}{\xi}\right)^5+\cdots
\lab{lspeck44}
\end{equation}
and, correspondingly, for large $\sigma/\xi$
\begin{equation}
f\left(\displaystyle\frac{\sigma}{\xi}\right)\approx
\displaystyle\frac{1}{2^{5/2}\pi}\left[\displaystyle\frac{1}{3}
\left(\displaystyle\frac{\xi}{\sigma}\right)^3-\displaystyle\frac{1}{10}
\left(\displaystyle\frac{\xi}{\sigma}\right)^5 \right]+\cdots
\lab{lspeck45}
\end{equation}
Introducing the appropriate correlation length for each aperture,
as described in Sec.~\ref{sec3}, one can show that, when this
correlation length tends to zero, then the corresponding correlation
function $|C_A'({\bf r})|^2$ tends to the delta function. The same
behavior has our function $|C_A'({\bf r})|_{3d,{\rm sph}}^2$ in the limit $\sigma
\rightarrow 0$. Therefore, we should reproduce the Huang and Meng
result $n_{\rm HM}$ for the condensate depletion in this limit.
Indeed, when $\sigma/\xi \rightarrow 0$ we read off from Eq.~\re{lspeck44} that one
obtains $f(\sigma/\xi)\rightarrow 1$.
\begin{figure}[t]
\begin{center}
\includegraphics[width=8cm,scale=1]{figure2.eps}
\end{center}
\caption{Condensate depletion function $f(\sigma/\xi)$ from Eq.~(\ref{lspeck43}).}
\lab{fig2}
\end{figure}
For the $3d$ isotropic Bose gas with contact interaction the
normalfluid density $n_N$ is determined by the equation
$n_N=4(n-n_0)/3$ (see Sec.~\ref{sec6} below and Ref.~\ci{Krumnow} as
well as the references therein), from which $n_N$ is proportional to
the function of $f(\sigma/\xi)$.
In Ref.~\ci{Krumnow} the
sound velocity of a dipolar BEC in a weak external disorder potential is calculated
within a hydrodynamic approach. To this end a general derivation was performed
which is applicable for an arbitrary interaction potential.
For an isotropic $3d$ system with contact
interaction it has the form:
\begin{eqnarray}
\displaystyle\frac{c}{c_0}&=&1+ \, \int \,
\displaystyle\frac{d^3 k}{(2\pi)^3} \displaystyle\frac{R({\bf k})}{(\hbar^2 {\bf k}^2/2m + 2ng)^2} \nonumber \\
&& \times \left\{ \displaystyle\frac{\hbar^2 {\bf k}^2/2m}{(\hbar^2 {\bf k}^2/2m + 2ng)}-(\hat{\bf q}\hat{\bf k})^2 \right\} +\cdots\,,
\lab{lspeck46}
\end{eqnarray}
where $c_0=(ng/m)^{1/2}$ is the sound velocity in a system without
disorder and the scalar product between the sound direction $\hat{\bf q}$ and the direction of
wave propagation $\hat {\bf k}$ has the form $\hat{\bf q}\hat{\bf k}=\cos \vartheta$ for an
isotropic system.
Calculating the integral in Eq.~\re{lspeck46}, we obtain
$c/c_0=1+n_{\rm HM}s(\sigma/\xi)/(2n)$, where the sound velocity function reads
\begin{eqnarray}
s\left(\displaystyle\frac{\sigma}{\xi}\right)&=&\displaystyle\frac{2^{3/2}}{\pi}\displaystyle\frac{\sigma}{\xi}
\left[\displaystyle\frac{14}{3}-\left(\displaystyle\frac{28\sigma^2}{3\xi^2}+4\right) {\rm ln}
\left(1+\displaystyle\frac{\xi^2}{2\sigma^2}\right)\right. \nonumber \\
&& + \left. \displaystyle\frac{5}{3\sqrt{2}}\displaystyle\frac{\xi}{\sigma} \arctan
\left(\displaystyle\frac{\xi}{\sqrt{2}\sigma}\right)\right]
\lab{lspeck47}
\end{eqnarray}
It is depicted in Fig.~\ref{fig3} and
has the following asymptotics for small $\sigma/\xi$
\begin{equation}
s\left(\displaystyle\frac{\sigma}{\xi}\right)\approx
\displaystyle\frac{5}{3}+\displaystyle\frac{2^{3/2} 3}{\pi}\left(\displaystyle\frac{\sigma}{\xi}\right)
-\displaystyle\frac{2^{3/2} 62}{9\pi}\left(\displaystyle\frac{\sigma}{\xi}\right)^3+\cdots
\lab{lspeck48}
\end{equation}
and for large $\sigma/\xi$
\begin{equation}
s\left(\displaystyle\frac{\sigma}{\xi}\right)\approx
\displaystyle\frac{2^{1/2}}{\pi}\left[-\displaystyle\frac{7}{3}
\left(\displaystyle\frac{\xi}{\sigma}\right)+\displaystyle\frac{13}{18}
\left(\displaystyle\frac{\xi}{\sigma}\right)^3 \right]+\cdots \, ,
\lab{lspeck49}
\end{equation}
respectively.
\begin{figure}
\begin{center}
\includegraphics[width=8cm,scale=1]{figure3.eps}
\end{center}
\caption{Sound velocity function $s(\sigma/\xi)$ from Eq.~(\ref{lspeck47}).}
\lab{fig3}
\end{figure}
Again, when the correlation length $\sigma \rightarrow 0$ and thus
the correlation function $|C_A'({\bf r})|_{3d,{\rm sph}}^2$ is delta
correlated, we reproduce the result $s(\sigma/\xi)\approx 5/3$ of
Ref.~\ci{Giorgini}, obtained for delta correlated $R({\bf r})$.
As shown in this section, the finite range of integration for the
vector $\bf k$, when a Fourier transform of a speckle correlation
function is being applied, essentially simplifies the analytic
calculation of the BEC properties. This is an essential advantage of
applying the laser speckle theory to the BEC investigation. Conversely,
due to the infinite limit of integration on $\bf k$,
the Gaussian disorder correlation function, which is often used in
the literature, introduces some difficulties in its application to
the BEC theory.
\section{Landau derivation of normalfluid density}
\label{sec6}
Let $K_0$ be a reference frame, and $K$ a second frame with relative
velocity $- {\bf v}$ with respect to $K_0$. According to the
Galilean transformation in classical mechanics, the energy $E_0$ of
a system in the frame $K_0$ and its energy $E$ in the frame $K$ are
related to each other by:
\begin{equation} E=E_0-{\bf P}_0 {\bf v} + \frac{M}{2} {\bf v}^2 \,,
\lab{lspeck50} \end{equation}
where ${\bf P}_0$ and $M$ are the total momentum and the mass of the
system, respectively.
Following to Refs.~\ci{Landau1a,Landau1b} let us assume that, at
temperature $T=0$, the condensate is in rest, i.e., in the frame
$K_0$, and its energy is $E_0=0$ with momentum ${\bf P}_0=0$. If one
quasiparticle with mass $m$ appears in the condensate with energy
$\varepsilon ({\bf p})$, where ${\bf p}$ is a momentum of the
quasiparticle, then in the frame $K_0$ energy and momentum now
become $E_0=\varepsilon ({\bf p})$ and ${\bf P}_0={\bf p}$. Hence,
from Eq.~\re{lspeck50} the energy $E$ in frame $K$ will be
$E=\varepsilon ({\bf p})-{\bf p}{\bf v}+M {\bf v}^2/2$ and the
energy of the quasiparticle in frame $K$ after a Galilean
transformation has a form $\varepsilon ({\bf p})-{\bf p}{\bf v}$.
According to the Landau two-fluid theory~\ci{Landau1a,Landau1b} of
liquid helium II, a gas of quasiparticles, for instance phonons,
constitutes the normalfluid density at low temperatures. For $T=0$
no quasiparticles exist, thus the helium is entirely superfluid. If
a gas of quasiparticles appears in the system for finite but low
temperatures, which has zero center mass velocity in the frame $K_0$
and moves with constant velocity $-{\bf v}$ with respect to the
frame $K$,
in which the helium liquid is in the rest, then the total momentum
of the gas per volume in the frame $K$ is given by
\begin{equation}
\displaystyle\frac{\bf P}{V}= \int {\bf p} \, N(\varepsilon ({\bf p})-{\bf p}{\bf v}) \displaystyle\frac{d^3
p}{(2\pi \hbar)^3}\,,
\lab{lspeck51}
\end{equation}
where $N(\varepsilon ({\bf p}))$ is the average occupation number of
states by phonons with energy $\varepsilon ({\bf p})$.
Eq.~\re{lspeck51} describes the thermodynamic property of a gas of
phonons. However, it can be generalized to our BEC system in the
external disorder potential at $T=0$, if we assume that, after
scattering with the disorder, particles of the condensate become the
quasiparticles of the normalfluid density.
It is clear that it occurs when the disorder is attached to the
frame $K_0$. To this end we replace in Eq.~\re{lspeck51} the
thermodynamic quantity $N(\varepsilon ({\bf p})-{\bf p}{\bf v})$ by
the quantum one $|\Psi ({\bf p}-m {\bf v})|^2$, where the wave
function is a solution of the GPE with disorder and written in
momentum representation. After that we average both sides of
Eq.~\re{lspeck51} over the disorder ensemble. The obtained mean
square of the modulo of the wave function is now homogeneous in
space, so it can be expressed in terms of the energy of a
quasiparticle, as the Hamiltonian is commutative with the momentum
operator and thus the eigenfunction of the latter can be taken as
the eigenfunction of the former~\ci{Landau2}. Recalling that the
expression for the total density is $n=\langle \Psi^2 \rangle$, we
obtain
\begin{equation}
\displaystyle\frac{\langle {\bf P} \rangle }{V}= \int
{\bf p} \, n(\varepsilon ({\bf p})-{\bf p}{\bf v}) \displaystyle\frac{d^3 p}{(2\pi
\hbar)^3}\,.
\lab{lspeck52}
\end{equation}
In order to derive the expression for the normalfluid density we
expand the integrand of Eq.~\re{lspeck52} in power of ${\bf p}{\bf
v}$ and, in the limit ${\bf v} \rightarrow {\bf 0}$, retain only
its first two terms. After integrating over the directions of the
vector ${\bf p}$ the zeroth order term of this expansion disappears.
Thus one obtains
\begin{equation} \displaystyle\frac{\langle{\bf P}\rangle}{V}= -\int {\bf p} \,({\bf p}{\bf v})
\displaystyle\frac{d n(\varepsilon ({\bf p}))}{d\varepsilon ({\bf p})} \displaystyle\frac{d^3
p}{(2\pi \hbar)^3}\,. \lab{lspeck53} \end{equation}
This expression is the main result of the normalfluid density Landau
theory, when the two replacements $\langle {\bf P}\rangle$ by ${\bf
P}$ and $n(\varepsilon ({\bf p}))$ by $N(\varepsilon ({\bf p}))$ are
performed.
Taking into account that ${\bf p} ({\bf p}{\bf v})= p_z^2 {\bf v}$,
the expression for the normalfluid density reduces to
\begin{equation} \rho_n= -\int p_z^2 \, \displaystyle\frac{d n(\varepsilon ({\bf
p}))}{d\varepsilon ({\bf p})} \displaystyle\frac{d^3 p}{(2\pi \hbar)^3}\,.
\lab{lspeck54} \end{equation}
From Eq.~\re{lspeck42} we have the expression of
the total density Fourier transform
\begin{equation} n(\varepsilon ({\bf p}))=(2\pi)^3 n_0 \delta ({\bf k})+\displaystyle\frac{n_0
R(\bf k)}{(\hbar^2 {\bf k}^2/2m + 2ng)^2}\,, \lab{lspeck55} \end{equation}
in first order of $R(\bf k)$, from which the energy of the
quasiparticles follows to be $\varepsilon ({\bf p})={\bf p}^2/2m +
2ng$, where ${\bf p}=\hbar {\bf k}$. Substituting $\varepsilon ({\bf
p})$ in Eq.~\re{lspeck54} and performing its integral by parts and
using in the obtained expression $n(\varepsilon ({\bf p}))$ from
Eq.~\re{lspeck55}, one gets
\begin{equation}
\rho_n=\rho_0\, \int \, \displaystyle\frac{d^3 k}{(2\pi)^3} \displaystyle\frac{p_z^2
R(\bf k)}{{\bf p}^2 (\hbar^2 {\bf k}^2/2m + 2ng)^2}\,,
\lab{lspeck56}
\end{equation}
where $\rho_0=mn_0$.
It is interesting that there is the relationship
$\varepsilon_{\rm B} ({\bf p})=\varepsilon^{1/2} ({\bf p}){\bf p}/(2m)^{1/2}$ between our $\varepsilon ({\bf p})$
and the Bogoliubov quasiparticle energy $\varepsilon_B ({\bf p})$.
If we use this relation, then we obtain
\begin{equation}
\rho_n=\displaystyle\frac{\rho_0}{4}\,
\int \, \displaystyle\frac{d^3 k}{(2\pi)^3} \displaystyle\frac{{\bf p}^2\, p_z^2 R(\bf k)}{m^2
\varepsilon_B^4 ({\bf p})}\,.
\lab{lspeck57}
\end{equation}
This expression without the prefactor $1/4$ coincides with Eq.~(19)
of Ref.~\ci{Giorgini} for the normalfluid density $\rho_{n,LR}$,
obtained within the linear response approach, if we replace $V\int
d^3 k/(2\pi)^3$ by $\sum_{\bf k}$. The prefactor $1/4$ appears from
the relation between $\varepsilon ({\bf p})$ and $\varepsilon_{\rm
B} ({\bf p})$. For a $3d$ isotropic BEC system we have
$p_x^2=p_y^2=p_z^2$ and ${\bf p}^2=3p_z^2$. Multiplying the
right-hand side of Eq.~\re{lspeck56} with 3 and canceling $3p_z^2$
and ${\bf p}^2$ in the numerator and the denominator, we obtain
Eq.~\re{lspeck42}, therefore,
$n-n_0=3\rho_{n,LR}/(4m)$~\ci{Giorgini}.
It is worth to discuss the validity to use the Landau approach for
BEC with the disorder. According to a remark in the text
book~\ci{Pitaevskii} the Landau approach should not be applicable
for such a system. Indeed, the applied Landau derivation of the
normalfluid density presumes the validity of the quasiparticle
concept (see, for instance, Refs.~\ci{Landau1a,Landau1b}), in which
there are no collisions not only between quasiparticles but also of
last ones with the external disorder potential. More exactly,
according to this concept quasiparticles should be well defined and
their gas should be ideal.
In our case, effective quasiparticles with the mean-field energy
$\varepsilon ({\bf p})$ and the quantum state distribution at
temperature $T=0$, represented by the total density $n(\varepsilon
({\bf p}))$, appear in the system after the disorder ensemble
average. However, after this averaging the real space is homogeneous
and there is no reason for the gas of effective quasiparticles to be
not ideal. Hence, if for the conventional quasiparticles the source
of their appearance is the low temperature, here it is the
scattering of the condensate particles with the disorder and then
their excitation and departure from the condensate. This physical
conclusion naturally arises from the Landau derivation of the
normalfluid density.
\section{Summary and conclusion}
\label{sec7}
At first, we have summarized the derivation of the
autocorrelation function of the laser speckle in $1d$ and $2d$
following the seminal work of Goodman. We showed that a Gaussian
approximation of this function, proposed in some recent papers, is
inconsistent with the background of laser speckle theory. Then
we have proposed a possible experimental realization for an
isotropic $3d$ laser speckle potential and derived its corresponding
autocorrelation function. Using a Fourier transform of that
function, we calculated both condensate depletion and sound velocity
of a BEC in a weak speckle disorder within a perturbative solution
of the Gross-Pitaevskii equation. At the end, we reproduced the
expression of the normalfluid density obtained earlier within the
treatment of Landau. This physically transparent derivation showed
that condensate particles, which are scattered by disorder, form a
gas of quasiparticles which is responsible for the normalfluid
component. We have justified the validity of the Landau approach to
our BEC system with disorder.
\section{Acknowledgements}
\label{sec8}
One of the authors, B. A., thanks the Volkswagen Foundation for
partial support of the work. B. A. is also grateful to Center for International Cooperation
at the Freie Universit\"at
Berlin for its hospitality. Both authors appreciate Hagen
Kleinert and the members of his group for many discussions.
|
{
"timestamp": "2013-01-01T02:00:47",
"yymm": "1203",
"arxiv_id": "1203.1698",
"language": "en",
"url": "https://arxiv.org/abs/1203.1698"
}
|
\section{Introduction}
The $k$-core of a graph is its maximal subgraph with minimum degree at
least~$k$. The $k$-core of a graph is unique
and it can be obtained by iteratively deleting vertices of degree
smaller than $k$. The $k$-core of a graph that already has minimum
degree at least~$k$ is the graph itself. So we also say that graphs
(and multigraphs) with minimum degree at least~$k$ are $k$-cores.
The investigation of $k$-cores in random graphs was started by
Bollob{\'a}s~\cite{Bollobas84} in 1984 in connection with $k$-connected
subgraphs in random graphs. There has been much success in the use of
$k$-cores due to their amenability to analysis. For some earlier
results on the $k$-cores of random graphs,
see~\cite{Luczak91,Luczak92,Molloy92}.
A seminal result in this area was proved by Pittel, Spencer and
Wormald~\cite{PittelSpencerWormald96}: they determined the threshold
$c_k$ for the emergence of a giant $k$-core in $G(n,m)$. Roughly
speaking, if the average degree is below this threshold, the $k$-core
of $G(n,m)$ is empty with probability going to $1$ as $n\to\infty$,
and above the threshold the $k$-core has a linear number of vertices
with probability going to $1$. After this result, many proofs using a
variety of techniques were given for the emergence of a giant $k$-core
in graphs and hypergraphs;
see~\cite{Cooper04,FernholzRamachandaram04,CainWormald06,Kim06,
JansonLuczak07,JansonLuczak08,Riordan08}.
We are interested in finding how robust this giant $k$-core of
$G(n,m)$ is as a $k$-core. More precisely, if we delete a random edge
in the $k$-core of $G(n,m)$ and obtain its new $k$-core, is the new
$k$-core much smaller than the original one? This can be seen as a
measure of the robustness of the giant $k$-core. We do not restrict
ourselves to the $k$\nobreakdash-core of $G(n,m)$: we consider a $k$-core chosen
uniformly at random with given number of vertices and edges, then we
delete an edge from it uniformly at random and obtain the new
$k$-core.
We define a constant $c_k'$ and analyse the behaviour of the random
$k$-cores with average degree below and above $c_k'$. We work with
multigraphs with given degree sequence and then we deduce the desired
results for simple graphs. Throughout the paper we use a simple
deletion algorithm (and some variants) to find the $k$-core of a
graph: the algorithm iteratively removes vertices of degree less than
$k$ until all remaining vertices have degree at least $k$. We couple
this deletion algorithm with a random walk. For the case with
bounded average degree $c > c_k'+\psi(n)$ with $\psi(n) =
\omega(n^{-1/4})$ and $\psi(n)>0$, this strategy works quite well: we
prove that the deletion algorithm and the random walk both
terminate/die in less than $t(n)$ steps with probability going to $1$,
for every $t(n) = \omega(\psi(n)^{-1})$. This also implies that, when
$2m/n = c_k+\phi(n) > c_k+n^{-\delta}$, where $\delta \in(0,1/4)$
is a constant, the probability of deleting $\omega(\psi(n)^{-1})$
vertices of the $k$-core of $G(n,m)$ to find its new $k$-core after
deleting a single random edge goes to zero.
For the case with average degree $c\leq c_k'-\varepsilon$ where $\varepsilon$ is a
positive constant, we use the random walk to show that, for any
$h(n)\to\infty$, with probability going to $1$, the deletion algorithm
deletes $\Theta(n)$ vertices or at most $h(n)$ vertices. When $c\to
k$, the probability of deleting $\Theta(n)$ vertices goes to~$1$. Then
we use the differential equation method as described
in~\cite{Wormald99} to show that, if $\Theta(n)$ vertices are deleted,
then the deletion algorithm will not stop until the $k$-core has less
than $\gamma n$ vertices a.a.s.\ (where we can choose $\gamma$ as small
as we want). Using a result in~\cite{JansonLuczak07}, we prove that
in this case the $k$-core must be empty a.a.s.{} This finishes the proof
that, for $k+\varepsilon \leq c\leq c_k'+\varepsilon$ and any $h(n)\to\infty$, the
deletion algorithm deletes $n$ vertices or at most $h(n)$ vertices
a.a.s.; and that for $c\to k$, we delete $n$ vertices a.a.s.{} Proving that
the probability of deleting all vertices in the case $k+\varepsilon \leq
c\leq c_k'-\varepsilon$ is bounded away from zero require some more work: we
couple the deletion algorithm for multigraphs and simple graphs for
$t(n)\to\infty$ steps. This will then imply that the probability of
deleting $h(n)$ vertices for some $h(n)\to\infty$ is bounded away from
zero and so we must delete all vertices with probability bounded away
from zero.
\section{Main results}
Let $\ensuremath{\mathop{G}} = \simpleknm$ be a graph sampled uniformly at random from
the (simple) $k$-cores with vertex set $[n]$ and $m = m(n)$ edges. For
any graph $H$, let $K(H)$ denote the $k$-core of $H$ and let $W(H)$ be
$\card{V(H)} - \card{V(K( H - e))}$, where $e$ is an edge chosen
uniformly at random from the edges of~$H$. That is, $W(H)$ is the
number of vertices we delete from $H-e$ to obtain its $k$-core.
For
every $k\geq 0$, let
\begin{equation*}
f_k(\lambda) = e^\lambda - \sum_{i=0}^{k-1}
\frac{\lambda^i}{i!}
\quad\text{and}\quad
h_k(\mu) = \frac{e^\mu \mu}{f_{k-1}(\mu)}.
\end{equation*}
For
$k\geq 3$, let $c_k = \inf\set{h_k(\mu) \colon \mu > 0}$ and let
$\mu_{k,c_k}$ be such that $c_k = h_k(\mu_{k,c_k})$. We discuss the
existence of $c_k$ and $\mu_{k,c_k}$ later. Let
\begin{equation*}
c_k' = \frac{\mu_{k, c_k}
f_{k-1}(\mu_{k, c_k})}{f_k(\mu_{k, c_k})}.
\end{equation*}
Throughout the text, let $c = 2m/n$. The asymptotics will always be
with respect to $n\to \infty$. For a sequence of probability spaces
$(\Omega_n, \mathbb{P}_n)_{n\in\ensuremath{\mathbb{N}}}$, we say that a sequence of
events $(E_n)_{n\in\ensuremath{\mathbb{N}}}$ holds asymptotically almost surely (a.a.s.)
if $\mathbb{P}_n(E_n) \to 1$ as $n\to\infty$.
\begin{thm}
\label{thm:main}
Let $k\geq 3$ be a fixed integer. Let $m = m(n)$ and $c = 2m/n$. Then
the following hold.
\begin{itemize}
\item[(i)] If $c\geq k$ and $c\to k$, then $W(\simpleknm) = n$ a.a.s.
\item[(ii)] Let $\varepsilon > 0$ be a fixed real. Suppose that $k+\varepsilon \leq
c\leq c_k'-\varepsilon$. For any function $h(n)\to \infty$, we have that
a.a.s.\ $W(\simpleknm) \leq h(n)$ or $W(\simpleknm) = n$. Moreover,
$W(\simpleknm) = n$ with probability bounded away from zero.
\item[(iii)] Let $\psi(n) = \omega(n^{-1/4})$ be a positive function
and let $C_0$ be a constant. Suppose that $c_k'+\psi(n)\leq c\leq
C_0$. For every $h(n) = \omega(\psi(n)^{-1})$, we have that
$\prob{W(\simpleknm)\geq h(n)}\to 0$.
\end{itemize}
\end{thm}
We apply Theorem~\ref{thm:main} to study the robustness of the
$k$-core of $G(n,m)$, the random graph chosen uniformly at random
from all graphs on $[n]$ with $m$ edges.
\begin{cor}
\label{cor:Gnm}
Let $k\geq 3$ be a fixed integer. Let $m = m(n)$ and suppose that
$c = 2m/n = c_k+\psi(n) \geq c_k+ n^{-\delta}$ and $c\leq C_0$,
where $\delta$ is a constant in $(0,1/4)$ and $C_0$ is a
constant. Then, for every $h(n) = \omega(\psi(n)^{-1})$, we have
that $\prob{W\paren[\big]{K\paren[\big]{G(n,m)}} \geq h(n)}\to 0$.
\end{cor}
We remark that there are some known results about the $k$-core of
random graphs with given degree sequence under some constraints on the
degree sequences
(see~\cite{JansonLuczak07,Cooper04,FernholzRamachandaram04}). Since
the degree sequence of a graph $G$ and the degree sequence of $G-e$
for some edge $e\in E(G)$ are very similar, it is intuitive that one
can draw some conclusions about $W(G(k, n, m))$. Indeed, in the case
$c\in [c_k+\varepsilon, C_0]$ one can use~\cite{JansonLuczak07} to conclude
that $W(G(k,n,m)) = o(n)$ a.a.s.{} We were not able to derive results for
the cases (i) and (ii) directly from known results.
\subsection{Models of random multigraphs}
We use the allocation model restricted to $k$-cores (here we allow
multigraphs): let $a: [2m] \to [n]$ be chosen uniformly at random
among the functions such that $|a^{-1}(v)| \geq k$ for any $v\in [n]$;
let $\ensuremath{\mathop{G_{\multisub}}} = \multiknm$ be the multigraph on $[n]$ obtained by adding
an edge joining $a(i)$ and $a(m+i)$ for every $i\in [m]$. Then every
simple $k$-core with $n$ vertices and $m$ edges is generated by $m!
2^m$ allocations. This implies that $\multiknm$ conditioned upon
simple graphs is a uniform probability space on $k$-cores with vertex
set $[n]$ and $m$ edges. Multigraphs do not necessarily have the same
probability in $\multiknm$.
Let $\ensuremath{\Dcal_k(n,m)}$ be the set of $\mathbf{d}\in\ensuremath{\mathbb{N}}^n$ with $\sum_{i=1}^n d_i = 2m$
and $\min_i d_i \geq k$. For every multigraph $H$ with vertex set
$[n']$, let $\mathbf{d}(H)$ denote the degree sequence of $H$, that is,
$(\mathbf{d}(H))_i$ is the degree of vertex $i$. For any $\mathbf{d}=(d_1,\dotsc,
d_n)\in \ensuremath{\mathbb{N}}^n$, let $D_j(\mathbf{d})$ be the number of occurrences of $j$
in $\mathbf{d}$ and let $\eta(\mathbf{d}) = \sum_{i=1}^n \binom{d_i}{2}/m$. We will
work with $k$-cores generated using the pairing model with degree
sequences in $\ensuremath{\Dcal_k(n,m)}$. Given a degree sequence $\mathbf{d}$, let $\ensuremath{\mathop{G_{\multisub}}}(\mathbf{d})$
denote the graph generated using the pairing model: arbitrarily choose
a partition of $[2m]$ into sets $S_1,\dotsc, S_n$ (which we call bins)
such that $|S_i| =d_i$ for very $i$, add a perfect matching uniformly
at random on $[2m]$ and contract each $S_i$ to obtain a
multigraph. Then $\multiknm$ conditioned upon $\mathbf{d}(\multiknm)=\mathbf{d}$ has
the same distribution as $\ensuremath{\mathop{G_{\multisub}}}(\mathbf{d})$.
It is clear that $\mathbf{d}(\ensuremath{\mathop{G_{\multisub}}})$ has multinomial distribution
conditioned upon each coordinate being at least $k$, which we denote
by $\multinomialknm$. We say that a variable $Y$ taking integer values
has truncated Poisson distribution with parameters $(k,\lambda)$
(which we denote by $\tpoisson{k}{\lambda}$) if, for every integer
$j$,
\begin{equation*}
\prob{Y = j}
=
\begin{cases}
{\displaystyle \frac{\lambda^j}{j!f_k(\lambda)}},& \text{ if }j\geq k;\\
0,&\text{ otherwise.}
\end{cases}
\end{equation*}
By straightforward computations, one can show that $\multinomialknm$
has the same distribution as $\mathbf{Y} = (Y_1,\dotsc, Y_n)$ where the
$Y_i$'s are independent truncated Poisson variables with parameters
$(k, \lambda)$ conditioned upon the event $\Sigma$ that $\sum_{i=1}^n Y_i =
2m$.
\section{Random walks and a deletion procedure}
\subsection{A deletion procedure}
\label{sec:exploration_multi_super}
We are given a degree sequence $\mathbf{d}\in\ensuremath{\Dcal_k(n,m)}$. Here we describe a
procedure for finding the $k$-core of $\ensuremath{\mathop{G_{\multisub}}}(\mathbf{d})-e$, where $e$ is a
random edge in $\ensuremath{\mathop{G_{\multisub}}}(\mathbf{d})$. We will sample $\ensuremath{\mathop{G_{\multisub}}}(\mathbf{d})$ using the
pairing model by discovering one edge at a time. We start by choosing
$e$ by picking two points uniformly at random from the set of all
points.
\medskip
\noindent\textbf{Deletion procedure $(\mathbf{d})$}
\begin{itemize}
\item Partition $[2m]$ into $n$ bins $S_1,\dotsc, S_n$ such that
$|S_i|=d_i$ for every $1\leq i\leq n$.
\item Iteration $0$: Choose $e$ by picking distinct points $u$ and
$v$ uniformly at random from $[2m]$. Delete $u$ and $v$ and
\textbf{mark} all points in bins of size less than $k$.
\item Loop: While there is a marked undeleted point, choose one such point
$u$ and find the other end $v$ of the edge incident to $u$. Delete
$u$ and $v$. If $v$ was in a bin of size exactly $k$ (now of size
$k-1$ because we deleted $v$), mark all the
other points in this bin.
\end{itemize}
After the deletion procedure is over, the $k$-core can be obtained
by adding a random matching uniformly at random on the surviving
points. Let $Z_0(\mathbf{d})$ denote the number of marked points after the
deletion of the edge $e$ chosen in Iteration~$0$. Note that
$Z_0(\mathbf{d})\in\set{0, k-2,k-1,2(k-1)}$.
Let $Y_j(\mathbf{d})$ be the number of undeleted marked points after the
$j$-th iteration of the loop (and $Y_0(\mathbf{d}) := Z_0(\mathbf{d})$). The
procedure stops when $Y_j(\mathbf{d}) = 0$. Let $Z_j(\mathbf{d})$ be the number of
points that are marked in the $j$-th iteration of the loop. Let
$W(\mathbf{d}) = \sum_{j}\ceil[\big]{\frac{Z_j(\mathbf{d})}{k-1}}$. Note that
$W(\mathbf{d}) = W(\ensuremath{\mathop{G_{\multisub}}}(\mathbf{d}))$.
We mark new points in an iteration of the loop if $v$ lies in a bin of
(current) size~$k$. The probability that this happens (denoted by
$p_j(\mathbf{d})$) is the ratio of the number of unmarked points in bins of
(current) size $k$ and the number of undeleted points other than the
one we are exploring. If $v$ is also a marked point, then no new
points will be marked and $v$ is deleted. In this case, $Z_j(\mathbf{d}) =
-1$ and the probability that this happens (denoted by $p_j'(\mathbf{d})$) is
the ratio of the marked undeleted points other than $u$ and the number
of undeleted points other than $u$. Thus, in the $j$-th iteration of
the loop,
\begin{equation*}
Z_j(\mathbf{d})
=
\begin{cases}
k-1,&\text{ with probability } p_j(\mathbf{d});\\
-1,& \text{ with probability } p_j'(\mathbf{d})\\
0,&\text{otherwise}.
\end{cases}
\end{equation*}
The probabilities of $p_j(\mathbf{d})$ and $p_j'(\mathbf{d})$ are analyzed later.
\subsection{Random walks}
Given $c$ and $k$, we will define random walks in $\ensuremath{\mathbb{Z}}$ that will
help us to study the behaviour of the deletion procedure. Let
$\lambda_{k,c}$ be the (unique) positive root of $\lambda
f_{k-1}(\lambda) /f_{k}(\lambda) = c$. Such root always exists for $c
> k$. For more properties of $\lambda_{k, c}$,
see~\cite{PittelWormald03}. Let
\begin{equation*} q_{k,c} =
\frac{\lambda_{k,c}^{k-1}}{(k-1)!f_{k-1}(\lambda_{k,c})}.
\end{equation*}
Let $Z(k,c)$ be a random variable such that
\begin{align*}
Z(k,c)
&=
\begin{cases}
k-1,&\text{with probability } q_{k,c};\\
0,&\text{otherwise.}
\end{cases}
\end{align*}
Let $Y_0 = Z_0(\mathbf{d})$. For $j>0$, let $Y_j = Y_{j-1}+ Z_j-1$ where
$Z_j$ has same distribution as $Z(k,c)$ and the variable $Z_j$ is
independent from $Z_1,Z_2,\dotsc, Z_{j-1}$. Thus, we defined a random
walk such that the position in iteration $j$ is $Y_j$ and the drift is
given by $Z_j-1$. Similarly, for $\xi =\xi(n) \geq 0$ and $\xi \leq
1-q_{k,c}$, define the random variable $Z^+(k,c,\xi)$ by
\begin{align*}
Z^+(k,c)
&=
\begin{cases}
k-1,&\text{with probability } q_{k,c}+\xi;\\
0,&\text{otherwise.}
\end{cases}
\end{align*}
Let $Y_0^+ = Z_0(\mathbf{d})$. For $j>0$, let $Y_j^+ = Y_{j-1}^+ +
Z_j^+-1$ where $Z_j^+$ has same distribution as $Z^+(k,c,\xi)$ and the
variable $Z_j^+$ is independent from $Z_1^+,Z_2^+,\dotsc, Z_{j-1}^+$.
Note that $(Y_j)_{j\in\ensuremath{\mathbb{N}}}$ and $(Y_j^+)_{j\in\ensuremath{\mathbb{N}}}$ are actually
branching processes.
For $\xi =\xi(n) \geq 0$ and $\xi \leq q_{k,c}$, define
the random variable and $Z^-(k,c,\xi)$ by
\begin{align*}
Z^-(k,c)
&=
\begin{cases}
k-1,&\text{with probability } q_{k,c}-\xi;\\
-1, &\text{with probability } \xi;\\
0,&\text{otherwise.}
\end{cases}
\end{align*}
Let $Y_0^- = Z_0(\mathbf{d})$. For $j>0$, let $Y_j^- = Y_{j-1}^- +
Z_j^- -1$ where $Z_j^-$ has same distribution as $Z^-(k,c,\xi)$ and
the variable $Z_j^-$ is independent from $Z_1^-,Z_2^-,\dotsc,
Z_{j-1}^-$.
We say that $Y_j$ is the number of particles alive in iteration $j$
and that $Z_j$ is the number of particles born in iteration $j$ (and
similarly for $Y_j^+$, $Z_j^+$, and $Y_j^-$, $Z_j^-$).
The random walk given by $Z^+(k,c,\xi)$ is going to be used to bound
the number of marked points in the deletion process by above, while
the random walk given by $Z^-(k,c,\xi)$ will bound it from below.
Here we will prove some properties of these random walks.
Recall that $h_k(\mu) = \mu e^\mu/f_{k-1}(\mu)$ and $c_k =
\inf\set{h_k(\mu):\mu > 0)} = h_k(\mu_{k,c_k})$. Here we justify why
the infimum is reached and why it is reached by a unique $\mu$. It is
easy to see that $h_k$ is differentiable. Moreover, $h_k(\mu)\to\infty$ when
$\mu \to 0$ and when $\mu \to \infty$. The first derivative of
$h_k(\mu)$ is
\begin{equation*}
\frac{e^\mu}{f_{k-1}(\mu)}\paren[\Big]{1+\mu-
\mu\frac{f_{k-2}(\mu)}{f_{k-1}(\mu)}}
\end{equation*}
Using the fact that $f_{k-2}(\mu) = f_{k-1}(\mu) + \mu^{k-2}/(k-2)!$,
it is clear that this derivative is at least $0$ iff
\begin{equation}
\label{eq:min_req}
\frac{\mu^{k-1}}{(k-2)!} \leq f_{k-1}(\mu)
\end{equation}
and the functions on both sides are convex and increasing for $\mu >
0$. Thus, the function $h_k(\mu)$ must reaches its infimum in a
unique point $\mu_{k,c_k}$ and the equation $h_k(\mu) = c$ has exactly
two roots when $c > c_k$. Let $\mu_{k,c}$ denote the largest root of
the equation $h_k(\mu) = c$. Recall that $c_k' = h_k(\mu_{k,c_k})$.
\begin{prop}
\label{prop:mean}
The following hold:
\begin{itemize}
\item[(i)] $\esp{Z(k,c)}$ is a strictly decreasing function of $c$
for $c > k$ and $\esp{Z(k, c_k')} = 1$.
\item[(ii)] For any $\varepsilon > 0$ with $c_{k}'-\varepsilon > k$, there exists
a positive constant $\alpha$ such that $\esp{Z(k,c_k'-\varepsilon)} >
1+\alpha$.
\item[(iii)] Let $\psi(n)$
be a nonnegative function with $\psi(n) \leq C_0$, where $C_0$ is
constant. There exists a positive constant $\beta$ such that
$\esp{Z(k,c_k'+\psi(n))} \leq 1-\beta\psi(n)$.
\end{itemize}
\end{prop}
\begin{proof}
Let $g(c) = \esp{Z(k,c)}$. Note that $g(c) = (k-1) q_{k,c}$. By the
definition of $c_k'$, we have that~\eqref{eq:min_req} holds with
equality for $\mu = \mu_{k, c_k}$. This clearly implies $g(c_k') =
1$. We have that $\lambda_{k,c}$ is a strictly increasing function of
$c$ and vice-versa (see the derivative computation in~\cite[Lemma
1]{PittelWormald03}). If $c > k$, then $\lambda_{k,c} > 0$. Thus, by
considering $c = c(\lambda) = \lambda
f_{k-1}(\lambda)/f_{k}(\lambda)$ and differentiating with respect
to $\lambda$, we get
\begin{equation*}
\begin{split}
\frac{d}{d\lambda}q_{k, c}
&=
\frac{\lambda^{k-2}
\left(
k-1
-
\esp{\tpoisson{k-1}{\lambda}}
\right)}{(k-2)!f_{k-1}(\lambda)}
< 0
\end{split}
\end{equation*}
since $\esp{\tpoisson{k-1}{\lambda_{k,c}}} > k-1$. Thus, $g(c)$ is
strictly decreasing for $c > k$.
It is easy to see that $c(\lambda)$ is a smooth function on $\lambda
\in [\varepsilon',\infty)$ for any $\varepsilon' >0$. By the Inverse Function
Theorem, this implies that $\lambda_{k,c}$ is a smooth function on
$c\in[c(\varepsilon'),C_0]$ and so $g(c)$ is a smooth function on
$c$. Thus, the supremum $\sup\set{g'(c): c_k'\leq c\leq C_0}$ and
the infimum $\inf\set{g'(c): c_k'\leq c\leq C_0}$ are both achieved
and are both negative constants since $g(c)$ is strictly decreasing. By
the Mean Value Theorem, there are positive constants $\alpha$ and
$\beta$ such that $g(c) \geq 1 + \alpha|c-c_k'|$ for $c(\varepsilon') < c < c_k'$ and
$g(c) \leq 1 -\beta|c-c_k'|$ for $c_k' < c < C_0$.
\end{proof}
\begin{prop}
\label{prop:meangreaterthan1}
Let $k,c,\xi$ be such that $\esp{Z^- (k,c,\xi)} > 1+ \varepsilon$, for some
constant $\varepsilon > 0$. Then $\prob{Y_j^- > 0,\ \forall j\geq 0}$ is
bounded away from $0$ and, for any function $h(n)\to \infty$,
\begin{equation*}
\prob[\Big]{Y_j^- > 0,\ \forall j\geq h(n)}
= 1 + o(1).
\end{equation*}
\end{prop}
\begin{proof}
The first part follows from the fact that $(Y_j^-)_{j\geq 0}$ is a
random walk in $\ensuremath{\mathbb{R}}$ with positive expected drift (see
e.g.~\cite[p.~366]{Feller1}). The second part is a straightforward
application of the method of bounded differences since the variables
$Z_j^-$ are independent random variables with range
$[-1,k-1]$ (see~\cite{McDiarmid89}).
\end{proof}
\section{The case $c > c_k'+\omega(n^{-1/4})$}
Here we prove Theorem~\ref{thm:main}(iii). We start by proving a version
of Theorem~\ref{thm:main}(iii) for random multigraphs with given
degree sequence.
\begin{thm}
\label{thm:c_greater_ck_prime_degrees}
Let $\psi(n) = \omega(n^{-1/4})$ be a positive function and let
$C_0$ be a constant. Suppose that $m = m(n)$ is such that $c =
2m/n$ satisfies $c_k' + \psi(n)\leq c \leq C_0$. Let $\mathbf{d}
\in\ensuremath{\Dcal_k(n,m)}$ be such that $|D_k(\mathbf{d}) - \esp{D_k(\mathbf{Y})}| \leq n \phi(n)$
for $\phi(n) = o(\psi(n))$, where $\mathbf{Y} = (Y_1,\dotsc, Y_n)$ and the
$Y_i$'s are independent truncated Poisson variables with parameters
$(k, \lambda_{k,c})$. For every $h(n) = \omega(\psi(n)^{-1})$, we have
that $\prob{W(\ensuremath{\mathop{G_{\multisub}}}(\mathbf{d})) \geq h(n)} = o(1)$.
\end{thm}
Using Theorem~\ref{thm:c_greater_ck_prime_degrees}, we can deduce a
result about multigraphs with given number of vertices and edges,
which is then used to prove Theorem~\ref{thm:main}(iii).
\begin{cor}
\label{cor:coupling_multi}
Let $\psi(n) = \omega(n^{-1/4})$ be a positive function and let
$C_0$ be a constant. Suppose that $c = 2m/n$ is such that $c_k' +
\psi(n)\leq c \leq C_0$. For every $h(n) = \omega(\psi(n)^{-1})$,
we have that $\prob{W(\multiknm) \geq h(n)} = o(1)$.
\end{cor}
Now we prove Theorem~\ref{thm:c_greater_ck_prime_degrees}. We will
choose $\xi$ big enough so that $Z_j(\mathbf{d})$ is stochastically bounded
from above by $Z_j^+$ for $j\leq t(n)$ steps, where $Z_j^+$ has the same distribution as $Z^+(k,c, \xi)$. Recall we start the
deletion process with $n$ bins with $d_i$ points inside each bin
$i$. Let $p$ denote the initial ratio between the number of points in
bins of size $k$ and the total number of points. Note that $p =
kD_k(\mathbf{d})/2m = q_{k,c}(1+\phi_1(n))$, for some function $\phi_1(n)$
such that $\phi_1(n) = O(\phi(n))$. Choose $t(n) = \psi(n)^{-1}
n^\alpha$, where $\alpha$ is constant in $(0,1/2)$. Then, for $1\leq
j\leq t(n)$,
\begin{equation*}
\frac{kD_k(\mathbf{d})-(j+2)(k-1)}{2m-2j-2} \leq p_j(\mathbf{d})\leq\frac{kD_k(\mathbf{d})}{2m-2j-2}
\end{equation*}
and so $p_j(\mathbf{d}) = p + O(t(n)/n)$. We can assume $h(n) \leq
t(n)$. Proposition~\ref{prop:mean} implies that $q_{k,c}\leq 1/(k-1)$.
Since $t(n)/n = o(\psi(n))$, we can choose $\xi > 0$ such that $\xi =
o(\psi(n))$ and $\xi < 1-q_{k,c}$ and $Z^+_j \geq Z_j$ for all $j\leq
t$. Now $\esp{Z^+(k,c,\xi)} \leq 1 - \beta\psi(n) + (k-1)\xi$ according
to Proposition~\ref{prop:mean} for some positive constant
$\beta$. Since $\xi = o(\psi)$, we have $\esp{Z^+(k,c,\xi)} \leq 1 -
\beta'\psi(n)$ for some positive constant $\beta'$. Thus, we have that
$\esp{Y_{t}^+} = O((1-\beta'\psi(n))^{t(n)}) =
O(\exp(-t(n)\beta'\psi(n) )) = o(1)$ because $t(n) =
n^{\alpha}/\psi(n)$ with $\alpha > 0$. This implies that the deletion
procedure stops before $t(n)$ steps a.a.s., which proves
Lemma~\ref{thm:c_greater_ck_prime_degrees}.
\subsection{Proof of Corollary~\ref{cor:coupling_multi} and
Theorem~\ref{thm:main}(iii)}
\label{sec:proofcor_super}
Let $h(n)=\omega(\psi(n)^{-1})$. Choose $\phi(n)$ such that $\phi(n) =
o(\psi(n))$ and $\phi(n) = \omega(n^{-1/4})$. First we will prove
Corollary~\ref{cor:coupling_multi}. We will show that the degree
sequences that satisfy the hypotheses in
Lemma~\ref{thm:c_greater_ck_prime_degrees} are the `typical' degree
sequences for $\multiknm$. Let $\ensuremath{\tilde\Dcal_k(n,m)}$ be the set of degree sequences
$\mathbf{d}$ satisfying $|D_k(\mathbf{d}) - \esp{D_k(\mathbf{Y})}|\leq n\phi(n)$. Recall that
$\mathbf{d}(\multiknm)$ has the same distribution as $\mathbf{Y} = (Y_1,\dotsc,
Y_n)$ such the $Y_i$'s are independent truncated Poisson variables
with parameters $(k, \lambda_{k,c})$ and conditioned to the event
$\Sigma$ that $\sum_i Y_i = 2m$. Using Chebyshev's inequality,
\begin{equation*}
\prob{|D_k(\mathbf{Y}) - \esp{D_k(\mathbf{Y})}| \geq \phi(n) n}
\leq
\frac{n}{n^2\phi(n)^2}.
\end{equation*}
By~\cite[Theorem~4(a)]{PittelWormald03}, it is easy to see that the
probability of $\Sigma$ is $\Omega(1/\sqrt{n})$. Thus,
\begin{equation}
\label{eq:probtypicaldegree}
\prob{\mathbf{d}(\multiknm) \not\in \ensuremath{\tilde\Dcal_k(n,m)}}
\leq
\frac{\prob{\mathbf{Y} \not\in\ensuremath{\tilde\Dcal_k(n,m)}}}{\prob{\Sigma}}
=
O\left(\frac{n\sqrt{n}}{n^2 \phi(n)^2}\right)
=o(1).
\end{equation}
For every $n\in\ensuremath{\mathbb{N}}$, since the set $\ensuremath{\tilde\Dcal_k(n,m)}$ is finite, there exists
a degree sequence $\mathbf{d}^*(n)$ such that
$\prob[\big]{W(\ensuremath{\mathop{G_{\multisub}}}(\mathbf{d}^*(n))) \geq h(n)}=
\max\set[\big]{\prob[\big]{W(\ensuremath{\mathop{G_{\multisub}}}(\mathbf{d})) \geq h(n)}: \mathbf{d} \in
\ensuremath{\tilde\Dcal_k(n,m)}}$. Set $r(n) =\prob[\big]{W(\ensuremath{\mathop{G_{\multisub}}}(\mathbf{d}^*(n))) \geq
h(n)}$. Theorem~\ref{thm:c_greater_ck_prime_degrees} implies that
$r(n) = o(1)$. Thus, for any sequence $(\mathbf{d}(n))_{n\in\ensuremath{\mathbb{N}}}$ such that
$\mathbf{d}(n)\in\ensuremath{\tilde\Dcal_k(n,m)}$ for every $n\in\ensuremath{\mathbb{N}}$, we have that
$\prob{W(\ensuremath{\mathop{G_{\multisub}}}(\mathbf{d}(n))) \geq h(n)}\leq r(n) = o(1)$. This is usually
expressed by saying that $\prob{W(\ensuremath{\mathop{G_{\multisub}}}(\mathbf{d}(n))) \geq h(n)}\to 0$
uniformly for $\mathbf{d}\in\ensuremath{\tilde\Dcal_k(n,m)}$. Together
with~\eqref{eq:probtypicaldegree}, this implies that
$\prob{W(\multiknm)\geq h(n)} = o(1)$, proving
Corollary~\ref{cor:coupling_multi}.
We will now prove Theorem~\ref{thm:main}(iii). To deduce the result for
simple graphs, we impose further conditions on the degree sequences:
let $\ensuremath{\hat\Dcal_k(n,m)}$ be the set of degree sequences in $\ensuremath{\tilde\Dcal_k(n,m)}$ that satisfy
the conditions that $\max_i d_i \leq n^{\varepsilon}$ for some $\varepsilon
\in(0,0.25)$ and that $|\eta(\mathbf{d}) - \esp{\eta(\mathbf{Y})}| \leq
\phi(n)$. One can easily prove that $\var{Y_i(Y_i-1)} = O(1)$ and so
uniformly for $n$ and~$m$ with $c < C_0$, by Chebyshev's inequality,
\begin{equation*}
\begin{split}
\prob[\Big]{|\eta(\mathbf{Y}) - \esp{\eta(\mathbf{Y})}|
\geq \phi(n)}
=
O\left(
\frac{1}{n\phi(n)^2}
\right).
\end{split}
\end{equation*}
For $j_0 > 2e\lambda_{k,c}$, we have that $\prob{Y_1 > j_0} =
O(\exp(-j_0/2))$. This holds because the ratio
$\prob{Y_1=j+1}/\prob{Y_1=j}$ is less than $1/e$ for $j \geq j_0/2$
(This is the same equation as~\cite[Equation
(27)]{PittelWormald03}). Thus, $\prob{\max_j Y_j \geq n^{\varepsilon}} =
O(n\exp(- n^{\varepsilon}/2))$. This implies that $\prob{\mathbf{d}(\multiknm)\in
\ensuremath{\hat\Dcal_k(n,m)}}$ is also $1+o(1)$. For $\mathbf{d}\in\ensuremath{\hat\Dcal_k(n,m)}$, the probability of
that $\ensuremath{\mathop{G_{\multisub}}}(\mathbf{d})$ is simple is already known (see~\cite{McKay85,
McKayWormald91}):
\begin{equation*}
\begin{split}
\prob{\ensuremath{\mathop{G_{\multisub}}}(\mathbf{d})\text{ simple}}
&=
\exp
\left(
-\frac{\eta(\mathbf{d})}{2}
-\frac{\eta(\mathbf{d})^2}{4}
+
O\left(\frac{\max_i d_i^4}{n}\right)
\right)
\\
&\sim
\exp
\left(
-\frac{\bar\eta_{c}}{2}
-\frac{(\bar\eta_{c})^2}{4}
+
O\left(\frac{\max_i d_i^4}{n}\right)
\right),
\end{split}
\end{equation*}
where $\bar \eta_c := \lambda_{k,c} f_{k-2}(\lambda_{k,c})/
f_{k-1}(\lambda_{k,c})$. We can apply the same argument on the
uniformity of the bound for $\prob{W(\ensuremath{\mathop{G_{\multisub}}}(\mathbf{d}))\geq h(n)} = o(1)$ as
above to $\prob{\ensuremath{\mathop{G_{\multisub}}}(\mathbf{d})\text{ is simple}} - \exp\left(-\eta_{c}/2
-\eta_{c}^2/4\right)$ and conclude that $$\prob{\multiknm \text{ is
simple}} = \exp\left(-\eta_{c}/2 -\eta_{c}^2/4\right) +o(1) =
\Omega(1)$$ and so
\begin{equation*}
\begin{split}
&\prob[\Big]{W(\simpleknm)\geq h(n)}
\\
&=
\probcond[\Big]{W(\multiknm)\geq h(n)}{\multiknm\text{ is simple}}
\\
&\leq
\frac{\prob{W(\multiknm)\geq h(n)}}
{\prob{\multiknm\text{ is simple}}}
\\
&= o(1).
\end{split}
\end{equation*}
This finishes the proof of Theorem~\ref{thm:main}(iii).
\subsection{The $k$-core of $G(n,m)$}
In this section we prove Corollary~\ref{cor:Gnm}. We will
use~\cite[Theorem 2]{PittelSpencerWormald96}. Although this result
does not state the number of edges in the $k$-core, it can be obtained
from its proof with the main steps in~\cite[Equations
(6.18),(6.34)]{PittelSpencerWormald96} and~\cite[Corollary
1]{PittelSpencerWormald96} applied to $J_1$. We
restate~\cite[Theorem~2]{PittelSpencerWormald96} with the number of
edges here:
\begin{thm}[\protect{\cite[Theorem~2]{PittelSpencerWormald96}}]
Suppose $c > c_k + n^{-\delta}$, $\delta\in (0,1/2)$ being
fixed. Fix $\sigma \in (3/4,1-\delta/2)$ and $\bar \zeta =
\min\set{2\sigma -3/2, 1/6}$. Then with probability $\geq 1 +
O(\exp(-n^\zeta))$ ($\forall \zeta < \bar\zeta$), the random graph
$G(n,m=cn/2)$ contains a giant $k$-core with $e^{-\mu_{k,c}}
f_k(mu_{k,c}) n + O(n^{\sigma})$ vertices and $(1/2) \mu_{k,c}
e^{-\mu_{k,c}} f_{k-1}(\mu_{k,c})n + O(n^{\sigma})$ edges.
\end{thm}
We are now ready to prove Corollary~\ref{cor:Gnm}. Recall that $c
\geq c_k + n^{-\delta}$, where $\delta \in (0, 1/4)$. So $\delta =
1/4-\varepsilon$, where $\varepsilon$ is a constant in $(0,1/4)$. Let $\varepsilon' <
\varepsilon$ be a constant such that $\varepsilon'< 1/4-\delta/2$. Fix $\sigma =
3/4+\varepsilon'$. Thus, the average degree of the $k$-core is
\begin{equation*}
\frac{\mu_{k,c}
f_{k-1}(\mu_{k,c})}{f_k(\mu_{k,c})}(1+O(n^{-1/4+\varepsilon'}).
\end{equation*}
Recall that $h'(\mu_{k,c_k})=0$ and $h'(\mu) > 0$ for $\mu
>\mu_{k,c_k}$. This implies that $\mu_{k, c} = \mu_{k, c_k} +
\Omega(c-c_k)$. Moreover, the function $x\mapsto xf_{k-1}(x)/ f_k(x)$
is smooth. Thus, the average degree of the $k$-core of $G(n,m)$ is
$(c_k' + \Theta(c-c_k))(1+O(n^{-1/4+\varepsilon'}))$. Since $c-c_k' >
n^{-\delta} = n^{-1/4+\varepsilon}$ with $\varepsilon > \varepsilon'$, the average degree
of the $k$-core is $c_k'+\Omega(c-c_k)$. We can now apply
Theorem~\ref{thm:main}(iii) to obtain the desired result.
\section{The case $k\leq c \leq c_k'-\varepsilon$: deleting $\Theta(n)$
vertices}
\label{sec:inter}
The following result is an intermediate step for the proof of
Theorem~\ref{thm:main}(i) and (ii) .
\begin{thm}
\label{thm:inter_degrees}
Let $\varepsilon > 0$ be a fixed real. Suppose that $k\leq c \leq
c_k'-\varepsilon$. Let $\phi(n) = o(1)$. Let $\mathbf{d}$ be such that $D_k(\mathbf{d})
\geq \esp{D_k(\mathbf{Y})}(1 - \phi(n))$, where $\mathbf{Y} = (Y_1,\dotsc, Y_n)$
and the $Y_i$'s are independent truncated Poisson variables with parameters
$(k, \lambda_{k,c})$. Then there exists a constant $\varepsilon' > 0$
(depending on $\varepsilon$) such that, for every function $h(n)\to
\infty$, we have that a.a.s.\ $W(\ensuremath{\mathop{G_{\multisub}}}(\mathbf{d})) \leq h(n)$ or $W(\ensuremath{\mathop{G_{\multisub}}}(\mathbf{d}))
\geq \varepsilon' n$. Moreover, $W(\ensuremath{\mathop{G_{\multisub}}}(\mathbf{d})) \geq \varepsilon' n$ with
probability bounded away from zero.
\end{thm}
The proof of the following corollary is very similar to the proof
in Section~\ref{sec:proofcor_super} and so we omit it.
\begin{cor}
\label{cor:multi_inter}
Let $\varepsilon > 0$ be a fixed real. Suppose that $k\leq c \leq
c_k'-\varepsilon$. Then there exists a constant $\varepsilon' > 0$ (depending on
$\varepsilon$) such that, for every function $h(n)\to \infty$, we have that
a.a.s.\ $W(\multiknm) \leq h(n)$ or $W(\multiknm) \geq \varepsilon' n$.
Moreover, $W(\multiknm)) \geq \varepsilon' n$ with probability bounded away
from zero.
\end{cor}
For the case $c\to k$, Theorem~\ref{thm:inter_degrees} implies a
stronger result because there is a function $h(n)\to \infty$ such that
$W(\multiknm)\geq h(n)$ steps a.a.s.\ From this
one can deduce the following result.
\begin{cor}
\label{cor:goingtok}
If $c\geq k$ and $c\to k$, then there exists a constant $\varepsilon'> 0$
such that $W(\multiknm)) \geq \varepsilon' n$ a.a.s.
\end{cor}
Now we prove Theorem~\ref{thm:inter_degrees}. We will choose $\xi$ so
that $Z_j(\mathbf{d})$ is stochastically bounded from below by $Z_j^-$, where
$Z_j^-$ has the same distribution as $Z^-(k,c,\xi)$, and so that
$\esp{Z^-(k,c,\xi)}$ is bounded away from $1$ from above. For $j\geq
1$, we have already seen that $p_j(\mathbf{d}) =
(1+O(j/n)+O(\phi(n)))q_{k,c}$. Moreover, for $j\geq 1$, the
probability that $Z_{j}(\mathbf{d})=-1$ is at most $(k-1)(j+2) /(2m-2j-2)$.
Proposition~\ref{prop:mean} implies that $q_{k,c} > 1/(k-1)$ and that
$\esp{Z^-(k,c,\xi)}\geq 1 + \alpha' -(k-1)\xi$ for some constant
$\alpha'>0$. Choose $\xi \in (0,\alpha'/(k-1))$. Thus, we have
$\esp{Z^-(k,c,\xi)}\geq 1 +\alpha$ for some $\alpha > 0$. We can now
choose $\varepsilon'' > 0$ small enough so that $p_j(\mathbf{d}) \geq q_{k,c} -\xi$
and $\prob{Z_j(\mathbf{d})=-1}\leq \xi$ for all $j\leq \varepsilon''n$. Thus, we can
couple the processes for at least $t(n) = \varepsilon''n$ steps.
By Proposition~\ref{prop:meangreaterthan1}, a.a.s.\ either $Y_j^- \leq
0$ for some $j\leq h(n)$ or $Y_{j}^- > 0$ for all~$j$. Moreover,
the latter occurs with probability bounded away from zero. Since the
coupling holds for $t(n)$ steps with $Z_j^-\leq Z_{j}(\mathbf{d})$, a.a.s.\
either $Y_j(\mathbf{d}) = 0$ for some $j\leq h(n)$ or $Y_{j}(\mathbf{d}) > 0$ for
$1\leq j\leq \varepsilon''n$. Thus, a.a.s.\ either $W(\ensuremath{\mathop{G_{\multisub}}}(\mathbf{d}))\leq
h(n)+2$ or $W(\multiknm) \geq \varepsilon''n/(k-1)$. This completes the proof
of Theorem~\ref{thm:inter_degrees}.
\section{The case $k\leq c \leq c_k'-\varepsilon$}
In this section we will prove Theorem~\ref{thm:main}(i). We will also
prove Theorem~\ref{thm:main}(ii) except for the claim that
$W(\simpleknm) = n$ with probability bounded away from zero, which is
handled in Section~\ref{sec:simple}. We use
the differential equation method as described in~\cite[Theorem
6.1]{Wormald99} with stopping times. We will also use some results
from~\cite{CainWormald06}.
We will use the pairing-allocation model $\ensuremath{\mathcal{P}}(M,L, V, k)$ as
described in~\cite{CainWormald06}: given a set $M$ of points together
with a perfect matching $E_M$ on $M$ and two disjoint set $L, V$ let
$h$ be chosen uniformly at random from the functions mapping $M$ to
$L\cup V$ such that $|h^{-1}(v)| \geq k$ for all $v\in V$ and
$|h^{-1}(v)|=1$ for all $v\in L$. Let
$G_{\ensuremath{\mathcal{P}}} = G_{\ensuremath{\mathcal{P}}}(M,L,V,k)$ be the multigraph obtained by adding
edges joining $h(a)$ and $h(b)$ for every $ab\in E_M$ and $h(a),
h(b)\in V$. Note that $\multiknm = G_{\ensuremath{\mathcal{P}}}([2m],\varnothing, [n],k)$
with $E_M = \set{\set{i, m+i}: i\in[m]}$.
We say that the vertices in $V$ are heavy vertices and the vertices in
$L$ are light vertices. We will also say that point $i\in M$ is in $v$
if $h(i) = v$.
Cain and Wormald~\cite{CainWormald06} analyse a deletion procedure for
obtaining the $k$-core. Here we will use a similar procedure with the
only modifications being in the first step. The procedure receives as
input $h:[2m]\to [n]$ such that $|h^{-1}(v)| \geq k$ for all $v\in
[n]$.
\medskip
\noindent \textbf{Deletion procedure -- pairing--allocation $(h)$:}
\begin{itemize}
\item Let $M = [2m]$, $L = \varnothing$ and $V = [n]$.
\item Iteration 0: Choose $i\in [m]$ uniformly at random. Find $v =
h(i)$ and $u = h(m+i)$. Delete $i$ and $m+i$ from $M$. If $u\neq v$
and $|h^{-1}(v)|= k$, then delete $v$ from~$V$, add $k-1$ new
elements to $L$ and redefine the action of $h$ on
$h^{-1}(v)\setminus\set{i}$ as a bijection to the new
elements. Similarly to~$u$, if $u\neq v$ and $|h^{-1}(u)|= k$, then
delete $u$ from~$V$, add $k-1$ new elements to $L$ and redefine the
action of $h$ on $h^{-1}(u)\setminus\set{m+i}$ as a bijection to the
new elements. If $u=v$ and $|h^{-1}(v)|\leq k+1$, then delete $v$
from $V$, add $|h^{-1}(v)|-2$ new elements to $L$ and redefine the
action of $h$ on $h^{-1}(v)\setminus\set{i, m+i}$ as a bijection to
the new elements.
\item Loop: While $L\neq\varnothing$, choose $j\in h^{-1}(L)$ uniformly at
random. Delete $j$ and $m+j$ of $M$ and delete $h(j)$ from $L$. Find
$v= h(m+j)$. If $v \in L$, delete $v$ from $L$. If $v\in V$ and
$|h^{-1}(v)|=k$, then delete $v$ from $V$, add $k-1$ new elements to
$L$ and redefine the action of $h$ on $h^{-1}(v)\setminus\set{i}$ as
a bijection to the new elements.
\end{itemize}
Let $h_0, M_0. L_0, V_0$ be the values of $h, M, L, V$, resp., after
Iteration $0$. Let $h_i, M_i, L_i, V_i$ be the values of $h, M, L, V$,
resp., after the $i$-th iteration of the loop. Then the proof
of~\cite[Lemma~6]{CainWormald06} gives us the same conclusion
as~\cite[Lemma 6]{CainWormald06}:
\begin{lem}
Starting with $h = \ensuremath{\mathcal{P}}([2m], \varnothing, [n], k)$ and
conditioning upon the values of $M_i, L_i$ and $V_i$, we have that
$h_i$ has same distribution as $\ensuremath{\mathcal{P}}(M_i, V_i, L_i,k)$.
\end{lem}
\subsection{The case $c\to k$}
Here we prove Theorem~\ref{thm:main}(i). We assume that $c = 2m/n = k+
\phi(n)$, where $\phi(n) = o(1)$ and $\phi(n) \geq 0$. Let $S_i$
denote the number of points in heavy vertices just after the $i$-th
iteration of the loop. Let $S_0$ denote the number of points in heavy
vertices after Iteration~$0$. We will use $x$ as $i/n$ and $y(i/n)$
to approximate $S_i/n$.
Define $D_\gamma = \set{(x,y)\colon -\gamma < 2x < k -\gamma,\ \gamma < y
< k + \gamma}.$ Note that $D_\gamma$ is bounded, connected and
open. We choose $\gamma < \min\set{\gamma_0/3, k}$ so that the
$k$-core cannot not empty and smaller than $\gamma_0 n$ a.a.s.
($\gamma_0$ is given by Lemma~\ref{lem:small}). Moreover, we work with
$n$ big enough so that $\phi(n) < \gamma$. After the first step there
are at most $2(k-1)$ points in $L_0$ and all the other vertices in
$V_0$ and so $S_0 \geq 2m-2(k-1)$. Then it is clear
that $S_0/n \leq k + \phi < k + \gamma$ and $S_0/n > \gamma$.
Let $T_D = \min\set{i: (i/n, S_i/n)\not\in D}$. Let $W_i$ denote the
number of light vertices after iteration $i$ is performed. We also use
the stopping time $T = \min\set{i: W_i = 0}$. That is, there are no
light vertices to be deleted and the deletion process has actually
ended. We need to check the boundedness hypothesis, trend hypothesis
and Lipschitz hypothesis (see~\cite{Wormald99} for more details). The
boundedness hypothesis is trivially true: $|S_i - S_{i+1}|\leq k$
always.
Now we check the trend hypothesis. Let $f(x,y) = -ky/(k-2x)$. Let
$H_i$ denote the history of the process at iteration $i\geq 1$. We
need to show that $\xi_1 :=| \esp{S_{i+1}-S_i | H_{i}}-f(i/n,S_i/n)| =
o(1)$ while the $i < T$ and $i < T_D$. We have that $S_{i+1}-S_i$ is
zero if $j$ is matched to a light vertex, is $-1$ if $j$ is matched to
a point in a heavy vertex with degree $> k$ and is $-k$ if $j$ is
matched to a point in a heavy vertex with degree exactly $k$. The
probability that $j$ is matched to a point in a heavy vertex is $S_i/
(2m-2i-2)$. The probability that such a heavy vertex has degree $k$ is
at least $1 - \sum\set{d_i: d_i > k}/ S_i$ where $\mathbf{d}$ is the degree
sequence (we do not sample the degree sequence, we just
decide if the vertex had degree $k$ or not). But for every possible
degree sequence $1 - \sum\set{d_i: d_i > k}/ S_i \geq 1 + n\phi(n)/S_i
= 1+o(1)$ whenever $S_i \geq \gamma n$.Then
\begin{equation*}
\esp{S_{i+1}-S_i | H_{i}}
=
\frac{-k |S_i|}{2m-2i-2}+o(1),
\end{equation*}
and so the trend hypothesis holds. It is easy to see that the
Lipschitz hypothesis also holds in $D_\gamma \cap \set{(x, y): x\geq 0}$.
According to~\cite[Theorem
6.1]{Wormald99}, the $y'(x) = f(x,y)$ has a unique solution in
$D_\gamma$, say $y^*$, with $y(0)=k$ and a unique solution in
$D_\gamma$, say $y^{**}$, with $y(0) = S_0/n$. Note that $y^*$ is a
fixed function while $y^{**}$ is a random variable because $S_0$ is a
random variable. The Lipschitz condition implies that, for any $x$
with both $(x, y^*(x))$ and $(x, y^{**}(x))$ in $D_\gamma$, we have
that $|y^*(x)-y^{**}(x)| = x |k-S_0/n| R =: \xi_3$, where $R$ is some
big constant and so $\xi_3=o(1)$. Let $\xi_2 = o(1)$ and $\xi_2 >
\xi_1$ and $\xi_2 > \xi_3$. By~\cite[Theorem
6.1]{Wormald99}, there is a constant $C$ and a function $\xi \to 0$,
such that, a.a.s.\ at each step $i < \min\set{T,n\sigma}$ we have that
\begin{equation}
\label{eq:formula}
|S_i - ny^{*}(i/n)| \leq \xi n,
\end{equation}
where $\sigma$ denotes the supremum of $x$ such that $(x', y^{*}(x'))$
and $(x', y^{**}(x'))$ are at $\ell^{\infty}$-distance at least
$C\xi_2$ of the boundary of $D_{\gamma}$ for all $0\leq x'\leq x$.
Let $\varepsilon'$ be given by Corollary~\ref{cor:goingtok}. For $\varepsilon' < x <
(k-\gamma)/2$, we have that
\begin{equation*}
(k-2x) - y^*(x)
=
(k-2x)
\left( 1 - \left(\frac{k-2x}{k}\right)^{k/2-1}\right)
\geq
\frac{2\gamma\varepsilon'}{k}.
\end{equation*}
This implies that, if~\eqref{eq:formula} holds at $i$ where $\varepsilon'n
<2i< (k-\gamma)n$, then $W_i = 2m-2i-2-S_i = \Omega(n)$. Thus,
if~\eqref{eq:formula} holds for some step $i\in (\varepsilon'n, \sigma n]$
with $T >i$, then $T > i+1$ because there are still $\Omega(n)$ points
to be deleted. This implies that, conditioning upon $T > \varepsilon'n$, we
have that $T > \sigma n$ a.a.s.
For any constant $\alpha \in (0,\gamma)$, using the fact that $\xi_3=
o(1)$, there exists $x$ such that $x\leq \sigma n$ and $(x,y^*(x))$
and $(x,y^{**}(x))$ are at $\ell^{\infty}$-distance $(C\xi_2, \alpha)$
of the boundary of $D_\gamma$. For such an $x$ we have $T > x$ a.a.s.\
because $T > \sigma n$ a.a.s.{} Thus, \eqref{eq:formula}~holds a.a.s.{}
Since $x$ is at $\ell^\infty$-distance at most $\alpha$ of the
boundary of $D_\gamma$, either $2x \geq k - \gamma - \alpha$ or
$y^*(x) \leq \gamma+\alpha$. We excluded $y^*(x) \geq k+\gamma-\alpha$
because $y^*(0) = k$ and $y^*$ decreases as $x$ increases. For $n$
sufficiently large so that $|\xi(n)| < \gamma$, the
equation~\eqref{eq:formula} for at either $2x \geq k - \gamma - \alpha$ or
$y^*(x) \leq \gamma+\alpha$ shows that $S_i\leq n\gamma_0$ a.a.s.
Since $T >\varepsilon'n$ a.a.s.\ by Corollary~\ref{cor:goingtok}, the $k$-core
would have to be smaller than $\gamma_0 n$ and so it must be empty
a.a.s.\ (see Section~\ref{sec:small}). We conclude that $W(\multiknm) =
n$ a.a.s.{} Since the probability that $\multiknm$ is simple is
$\Omega(1)$ (see~\cite{McKay85, McKayWormald91}), we have that
$W(\simpleknm) = n$ a.a.s.
\subsection{The case $c\in [k+\varepsilon, c+k' - \varepsilon]$}
\label{sec:de-inter}
We prove Theorem~\ref{thm:main}(ii) except for the claim that
$W(\simpleknm) = n$ with probability bounded away from zero, which is
addressed in Section~\ref{sec:simple}. Let $h(n)\to \infty$. Let
$\varepsilon'$ be given by Corollary~\ref{cor:multi_inter}. Assume that $c\to
C \in [k+\varepsilon, c+k' -\varepsilon]$, where $C$ is a constant. We will explain
later how to drop this constraint.
Again we use the differential equation method as in~\cite{Wormald99}
and the deletion procedure described in the beginning of the section.
For each~$i$, let $S_i$ denote the number of points in heavy vertices
just after iteration $i$, let $T_i$ denote the number of heavy
vertices just after iteration $i$ and let $W_i$ denote the number of
points in light vertices just after iteration~$i$. We will use the
differential equation method to approximate $S_i$ and $T_i$. Note that
$W_i = 2m - 2i - 2 - S_i$. We will use $y(i/n)$ to approximate $S_i/n$
and $z(i/n)$ to approximate $T_i/n$.
Let $\gamma$ be a positive constant with $\gamma < \min\set{1, C-k}$
to be chosen later. Define
\begin{equation*}
D_\gamma = \set{(x,y,z)\colon \gamma < z < 1 + \gamma,\
-\gamma < x < C-\gamma,\
\gamma < y < C+\gamma, y > (k+ \gamma)z}.
\end{equation*}
Then $D_\gamma$ is bounded, connected and open. We have $T_0\in
\set{n, n-1, n-2}$ and $S_0 \in[2m-2-2(k-1), 2m-2]$. Thus, $T_0/n = 1 + o(1/n)$ and $S_0/n = C+o(1)$.
Then $D_\gamma$ contains the closure of the points $(0, y,z)$ such
that $\prob{S_i = yn \text{ and }T_i=zn}\neq 0$ for some~$n$.
We use the stopping time $T = \min\set{i: W_i = 0}$ again. We have to
check the boundedness hypothesis, the trend hypothesis and the
Lipschitz hypothesis. The boundedness hypothesis is again easy:
$|S_{i+1}-S_i|\leq k$ and $|T_{i+1}-T_i|\leq 1$ always.
The trend hypothesis is exactly like in~\cite{CainWormald06} with
$|\esp{T_{i+1}-T_i| H_i} -f_z(i/n)| = \xi_z = o(1)$ and
$|\esp{S_{i+1}-S_i| H_i} -f_y(i/n)| = \xi_y = o(1)$ with
\begin{align*}
f_z(x)
&=
- \frac{y}{C-2x}
\left(1-\frac{\mu z}{y}\right)
\quad\text{and}\quad f_y(x)
=
- \frac{y}{C-2x}
\left(k-(k-1)\frac{\mu z}{y}\right),
\end{align*}
where $\mu = \lambda_{k, y/z}$. The Lipschitz hypothesis is
straightforward to check.
According to~\cite[Theorem 6.1]{Wormald99}, the $y'(x) = f_y(x)$ and
$z'(x) = f_z(x)$ has unique solutions $(y^*, z^*)$ and $(y^{**},
z^{**})$, with initial conditions $y(0)=C$ and $z(0)=1$, and $y(0) =
S_0/n$ and $z(0) = T_0$, resp. The Lipschitz hypothesis implies that,
there exists a constant $R$ such that, for any $x$ with both $(x,
y^*(x),z^*(x))$ and $(x, y^{**}(x), z^{**}(x))$ in $D_\gamma$, we have
$\max\set{|y^*(x)-y^{**}(x)|,|z^*(x)-z^{**}(x)|} \leq x
|k-S_0/n|R =: \xi_3$. Note that $\xi_3=o(1)$. Let $\xi_2 > \xi_z$,
$\xi_2 > \xi_y$, $\xi_2>\xi_3$, $\xi_2 = o(1)$. Thus, by~\cite[Theorem
6.1]{Wormald99}, there is a constant $C_0$ and a function $\xi \to 0$,
such that, a.a.s.\ at each step $i < \min\set{T,n\sigma}$ we have that
\begin{equation}
\label{eq:formula2}
\max\set{|S_i - ny^{*}(i/n)|, |T_i - n z^*(i/n)|} \leq \xi n,
\end{equation}
where $\sigma$ denotes the supremum of $x$ such that, for all $0\leq
x'\leq x$, we have $(x' ,
y^{*}(x'),z^*(x'))$ and $(x', y^{**}(x'),z^{**}(x'))$ are at
$\ell^{\infty}$-distance at least $C_0\xi_2$ of the boundary of
$D_{\gamma}$.
According to~\cite{CainWormald06}, we have that $\mu^2/(C-2x)$ and
$(ze^{\mu})/f_k(\mu)$ are constants. With initial conditions $y(0) =
C$ and $z(0) = 1$, we get $\mu^2/(C-2x)=\lambda_{k,C}^2/C$ and
$ze^{\mu}/f_k(\mu) = e^{\lambda_{k,C}}/f_k(\lambda_{k,C})$, which can be used
to deduce that
\begin{equation}
\label{eq:diffeq}
y^* = (C-2x)\frac{h_k(\lambda_{k,C})}{h_k(\mu)}.
\end{equation}
For $x\geq \varepsilon' /2$, we must have $\mu(x) \leq \lambda_{k,C}
\sqrt{1-\varepsilon'/C}$ and so $h_k(\mu)\geq (1+\varepsilon'')h_k(\lambda_{k,C})$,
for some $\varepsilon''>0$. Thus, for every $x$ such that $\varepsilon' \leq 2x \leq
C-\gamma$ using~\eqref{eq:diffeq},
\begin{equation*}
C-2x - (C-2x)\frac{h_k(\lambda_{k,C})}{h_k(\mu)} \geq \gamma\varepsilon''.
\end{equation*}
This implies that, if~\eqref{eq:formula} holds at $i$ with $\varepsilon'n
<2i<(C-\gamma)n$, then $W_i = 2m-2i-2-S_i = \Omega(n)$. Thus,
if~\eqref{eq:formula} holds for some step $i\in (\varepsilon'n, \sigma n]$
with $T >i$, then $T > i+1$ because there are still $\Omega(n)$ points
to be deleted. This implies that, conditioning upon $T > \varepsilon'n$, we
have that $T > \sigma n$ a.a.s.
For any constant $\alpha \in (0,\gamma)$, using the fact that $\xi_3=
o(1)$, there exists $x$ such that $x\leq \sigma n$ and $(x,y^*(x),
z^*(x))$ and $(x,y^{**}(x), z^{**}(x))$ are at
$\ell^{\infty}$-distance $(C_0\xi_2, \alpha)$ of the boundary of
$D_\gamma$. For such an $x$ we have $T > x$ a.a.s.\ because $T > \sigma
n$ a.a.s.{} Thus, \eqref{eq:formula2}~holds a.a.s.{} Since $x$ is at
$\ell^\infty$-distance at most $\alpha$ of the boundary of $D_\gamma$,
either $z^*(x)\leq \gamma+\alpha$ or $2x\geq C-\gamma-\alpha$ or
$y^*(x) \leq \gamma +\alpha$ or $y^*(x) / z^*(x) \leq k + \gamma
+\alpha$. We excluded $y^*(x) \geq C+\gamma-\alpha$ and $z^*(x)\geq
1+\gamma-\alpha$ because $y^*(0) = C$ and $z^*(0)=1$ and $f_y$ and
$f_z$ are decreasing.
If $2x \geq C - \gamma - \alpha$, then, using that $\mu^2/(C-2x)$
remains constant and $\mu(0) = \lambda_{k,C}$ with $C < c_k' $, we have
that $h_k(\mu)\geq h_k(\lambda_{k,C})$ and so $y^*(x) \leq C-2x\leq
\gamma+\alpha$. For $n$ sufficiently large so that $|\xi(n)| < \gamma$, it is easy to see that the
equation~\eqref{eq:formula} for at $y^*(x) \leq \gamma+\alpha$ or
$z^*(x)\leq \gamma+\alpha$ implies that $S_i\leq 3\gamma n$ a.a.s.{} We
still have to check what happens when $y^*(x) / z^*(x) \leq k + \gamma
+\alpha$. In this case, $\mu = O(\gamma+\alpha)$. This holds because
the function $g_k(x) = x f_{k-1}(x)/f_k(x)$ with domain $(0, 2\gamma)$
is strictly increasing and the limit of its one-sided derivative with
$x\to 0$ is $1/(k+1)$. This limit can be computed by the derivative of
this function which is $(1/x)g_k(x) (1+g_{k-1}(x)-g_k(x))$ and then
using Taylor's approximation for $g_k(x)$ and $g_{k-1}(x)$ around
$x\to 0$. For more details on this computation see~\cite[Lemma
1]{PittelWormald03} and its proof. Using that $\mu^2/(C-2x)$ during
the process, we then have $C-2x = O(\gamma^2)$, we can then conclude
that the $S_i = O(\gamma n)$ a.a.s.
Thus, conditioned upon $T > \varepsilon'n$ the $k$-core has at most $O(\gamma
n)$ vertices a.a.s.{} Let $\gamma_0$ be the constant given by
Lemma~\ref{lem:small}. By choosing $\gamma$ small enough, we can
conclude that, conditioned upon $T > \varepsilon'n$, the $k$-core has less than
$\gamma_0 n$ vertices a.a.s.\ and which implies, by
Lemma~\ref{lem:small}, that the $k$-core must be empty a.a.s.{} By
Corollary~\ref{cor:multi_inter}, we have that $W(\multiknm)\leq h(n)$
or $W(\multiknm)=n$ with probability $1+o(1)$ conditioned upon $T >
\varepsilon'n$ (where the convergence depends on $c$).
Recall that we assumed $c\to C$. We show how to drop this assumption
here. Let $(c_i)_{i\in \ensuremath{\mathbb{N}}}$ such that every $c_i\in[k+\varepsilon, c+k'
-\varepsilon]$. Let $r(n)$ be the probability that neither
$W(\multiknm)\leq h(n)$ nor $W(\multiknm)=n$. Then every subsequence
of $(c_i)_{i\in \ensuremath{\mathbb{N}}}$ has a subsequence that converges to some
constant $C_0$ and in that subsequence $r(n)\to 0$. So by the
subsubsequence principle $r(n)\to 0$. Since the probability that
$\multiknm$ is simple is $\Omega(1)$, we have that $W(\simpleknm)\leq
h(n)$ or $W(\simpleknm)=n$ a.a.s.
\subsection{No small $k$-cores}
\label{sec:small}
\begin{lem}
\label{lem:small}
Let $C_0$ be a constant. Suppose that $m = m(n)$ satisfies $kn \leq
2m\leq C_0 n$. Then there exists a constant $\gamma$ such that
a.a.s.\ the graph obtained from $\multiknm$ by deleting an edge chosen
uniformly at random either has a $k$-core of size at least
$\gamma n$ or its $k$-core is empty.
\end{lem}
\begin{proof}
This is an application of a result by Luczak and Janson~\cite[Lemma
5.1]{JansonLuczak07}: if a degree sequence $(\mathbf{d}_n)_{n\in \ensuremath{\mathbb{N}}}$
satisfies $\sum_{i} e^{\alpha d_i} \leq Rn$ for constants $\alpha$
and $R$, then there is a constant~$\gamma$ such that a.a.s.\ no
subgraph of $\ensuremath{\mathop{G_{\multisub}}}(\mathbf{d})$ with less than $\gamma n$ vertices has
average degree at least~$k$. We set $\alpha < 1/3$ and we will
choose $R$ later. Let $\ensuremath{\check\Dcal_k(n,m)} \subseteq \ensuremath{\Dcal_k(n,m)}$ be the set of degree
sequences $\mathbf{d}$ such that $\sum_{i} e^{\alpha d_i} \leq Rn$. It
suffices to show that the degree sequence $\mathbf{d} = \mathbf{d}(\multiknm)$ is
in $\ensuremath{\check\Dcal_k(n,m)}$ a.a.s.
Let $\mathbf{Y} = (Y_1,\dotsc, Y_n)$ be such that the $Y_i$'s are
independent random variables with distribution
$\tpoisson{k}{\lambda_{k,c}}$. As already mentioned before, $\mathbf{d}$
has the same distribution of $\mathbf{Y}$ conditioned upon the event that
$\sum_i Y_i = 2m$. Using~\cite[Theorem 4]{PittelWormald03}, one can
prove that $\prob{\sum_{i} Y_i = 2m} = \Omega(1/ \sqrt{n})$.
For $J_0$ big enough (depending only on~$C_0$), we have that
$\lambda_{k,C_0}/J_0\leq e^{-1}$, which implies
$\lambda_{k,c}/J_0\leq e^{-1}$. Clearly, $\sum_{j\leq J_0} e^{\alpha
j}D_j(\mathbf{Y}) \leq e^{\alpha J_0} n$. Let $J_1 = J_0+ (1+\beta) \log n$
with $\beta \in (1/2,(2\alpha)^{-1}-1)$. Let $p =
\lambda_{k,c}^{J_0-1}/((J_0-1)!f_k(\lambda_{k,c}))$. Then
\begin{equation*}
\begin{split}
\prob{\exists j > J_1 \text{ with }D_j(\mathbf{Y})>0}
&\leq
n p\sum_{i\geq 0}\frac{1}{e^{(1+\beta)\log n+i}}
\leq
\frac{p n^{-\beta}}{1-e^{-1}}
= O(n^{-\beta}).
\end{split}
\end{equation*}
Using the fact that $\prob{\sum_i Y_i = 2m} = \Omega(1 / \sqrt{n})$
and $\beta \in (1/2,(2\alpha)^{-1}-1)$, we conclude that $\prob{\max_i
d_i > J_1} = o(1)$. And so $\sum_{j>J_1} e^{\alpha j}D_j(\mathbf{Y}) = 0$
a.a.s.
Now we consider $j\in (J_0, J_1]$. By Hoeffding's inequality and using
the fact that $\prob{\sum_i Y_i =2m} = \Omega(1/ \sqrt{n})$, we have
that $\prob{|D_j(\mathbf{Y}) -p^{(j)}n|\geq a \sqrt{n}} = O(\sqrt n) e^{-a^2}
$, where $p^{(j)} = \lambda_{k,c}^j/(j!f_{k}(\lambda_{k,c}))$. Thus,
\begin{equation*}
\begin{split}
\prob[\Big]{|D_j(\mathbf{Y}) -p^{(j)}n|\geq a \sqrt{n} \text{ for some }j\in(J_0,J_1]}
&=
(1+\beta) \log n O(\sqrt n) e^{-a^2}
\\
&=
O(n^{-\beta'}),
\end{split}
\end{equation*}
for $a = \sqrt{(1+\beta')\log n}$ with $\beta'>0$. Thus, a.a.s.
\begin{equation*}
\begin{split}
\sum_{j=J_0+1}^{J_1}
&e^{\alpha j} D_j(\mathbf{Y})
\leq
e^{\alpha J_0}
\sum_{j=1}^{(1+\beta)\log n}
e^{\alpha j} \left(p\frac{1}{e^j}n + a\sqrt{n}\right)
\\
&\leq
e^{\alpha J_0}
\left(
n p \sum_{j=1}^{(1+\beta)\log n} e^{-2j/3}
+
e^{\alpha (J_1-J_0)}(J_1-J_0) a\sqrt{n}\right)
\\
&\leq
e^{\alpha J_0}\left(
\frac{p}{e^{2/3}(1-e^{-2/3})}
+ \frac{n^{(1+\beta)\alpha + 1/2}(\log n)^{3/2}}{n}\sqrt{1+\beta'}(1+\beta)
\right) n.
\end{split}
\end{equation*}
Using that $1+\beta < (2\alpha)^{-1}$, we can set $R = e^{\alpha
J_0}(1 + p/(1-e^{-2/3})+\sqrt{1+\beta'}(1+\beta))$.
\end{proof}
\section{Working with simple graphs}
\label{sec:simple}
\begin{lem}
\label{lem:simple_infty}
Let $\varepsilon > 0$ be a fixed real. Suppose that $c = 2m/n$ satisfies
$k+\varepsilon \leq c\leq c_k'-\varepsilon$. Then there exists a function
$h(n)\to\infty$ such that $\prob{W(\simpleknm) \geq h(n)} = \Omega(1)$.
\end{lem}
Together with Section~\ref{sec:de-inter}, this lemma implies that
$\prob{W(\simpleknm) = n} = \Omega(1)$, which completes the proof of
Theorem~\ref{thm:main}(ii).
We now prove Lemma~\ref{lem:simple_infty}. We will work with degree
sequences. Let $\ensuremath{\mathop{G}}(\mathbf{d})$ be chosen uniformly at random from all
(simple) $k$-cores with degree sequence~$\mathbf{d}$. Let $\phi(n) = o(1)$
with $\phi(n) = \omega(n^{-1/4})$. Let $\mathbf{Y} = (Y_1,\dotsc, Y_n)$ be
such that the $Y_i$'s are independent truncated Poisson variables with
parameters $(k, \lambda_{k, c})$. Let $\ensuremath{\tilde\Dcal_k(n,m)}$ be the degree
sequences $\mathbf{d}$ such that $|D_k(\mathbf{d})-\esp{D_k(\mathbf{Y})}|\leq n\phi(n)$ and
$\max_i d_i \leq n^{\beta}$ for some $\beta \in(0,0.25)$ and
$|\eta(\mathbf{d}) - \esp{\eta(\mathbf{Y})}| \leq \phi(n)$. Similarly to the proof
in Section~\ref{sec:proofcor_super}, one can prove that
$\mathbf{d}(\simpleknm)\in\ensuremath{\tilde\Dcal_k(n,m)}$ a.a.s.{} Thus, it suffices to show that,
there exists $h(n)\to\infty$ such that
\begin{equation*}
\prob{W(\ensuremath{\mathop{G}}(\mathbf{d}))\geq h(n)} = \Omega(1),
\end{equation*}
for $\mathbf{d}\in\ensuremath{\tilde\Dcal_k(n,m)}$.
For $\mathbf{d}\in\ensuremath{\tilde\Dcal_k(n,m)}$, we will couple deletion algorithms for $\ensuremath{\mathop{G}}(\mathbf{d})$
and $\ensuremath{\mathop{G_{\multisub}}}(\mathbf{d})$ so that they coincide for $t(n) \to \infty$ steps.
We use a deletion algorithm that is essentially the same as the one we
used in the other sections. The only difference is that we explore a
whole vertex at a time (instead of an edge at a time) and mark the
vertices that have to be deleted.
\noindent \textbf{Deletion procedure by vertex:}
\begin{itemize}
\item Iteration 0: Choose an edge $uv$ uniformly at random, delete
$uv$ and mark the vertices with degree less than $k$.
\item Loop: While there is an undeleted marked vertex, say $w$, find
its neighbours, delete $w$ and the edges incident to it, and then
mark all neighbours of $w$ that now have degree less than $k$.
\end{itemize}
If we can do such a coupling for $t(n)\to\infty$ iterations of the loop,
then we can choose $h(n)\to\infty$ such $h(n)\leq
\min\set{t(n),\varepsilon'n}$ with $\varepsilon'$ as in
Theorem~\ref{thm:inter_degrees} so that $\prob{W(\ensuremath{\mathop{G_{\multisub}}}(\mathbf{d}))\geq
h(n)} = \Omega(1)$. This would imply that the deletion algorithm did not
stop for at least $h(n)$ steps and so $\prob{W(\ensuremath{\mathop{G}}(\mathbf{d}))\geq h(n)}
\geq \Omega(1)$.
In the rest of this section, we show that there exists $t(n)\to\infty$
such that we can couple the deletion algorithms for $\ensuremath{\mathop{G}}(\mathbf{d})$ and
$\ensuremath{\mathop{G_{\multisub}}}(\mathbf{d})$ so that they coincide for $t(n)$ iterations of the
loop. For now assume that $t(n)\to\infty$ with $t(n)\leq \log
n$. Later we add more restrictions on the growth of $t(n)$. We show
that the probabilities that a certain edge $uv$ is chosen in the first
step are asymptotically equivalent for $\ensuremath{\mathop{G}}(\mathbf{d})$ and
$\ensuremath{\mathop{G_{\multisub}}}(\mathbf{d})$ and so the first step can be coupled. For the other
steps $i\leq t(n)$, we show that the probabilities that the set of
neighbours of the vertex $w$ is some specific set are again
asymptotically equivalent for $\ensuremath{\mathop{G}}(\mathbf{d})$ and $\ensuremath{\mathop{G_{\multisub}}}(\mathbf{d})$ with
some error $\xi(n) = o(1)$. So we can couple the deletion algorithms
for $t(n)$ steps, where $t(n)$ will depend on $\xi(n)$. In the
computations in this section, we will use $\funcaoprob_{\probmultisub}$ to denote the
probabilities in the deletion procedure for $\ensuremath{\mathop{G_{\multisub}}}(\mathbf{d})$ and we will
use $\funcaoprob$ to denote the probabilities in the deletion
procedure for $G(\mathbf{d})$. First we analyse the procedure for
multigraphs. Let $uv \in \binom{V}{2}$. Then
\begin{equation*}
\begin{split}
\probmulti{uv\text{ is chosen in the first step}}
&=
\frac{\probmulti{uv\in E(\ensuremath{\mathop{G_{\multisub}}}(\mathbf{d}))}}{m}
\\
&=
\frac{d_ud_v}{m}\frac{(2m -2)!2^{ m} m!}{(2 m)! 2^{ m-1}(m-1)!}
\\
&=
\frac{d_ud_v}{m(2m-1)}
=
\frac{d_ud_v}{m(2m)}(1+\xi_1(n)),
\end{split}
\end{equation*}
where $\xi_1(n) = o(1)$.
In $i$-th iteration of the loop, we delete a vertex $w$ and find its
set of neighbours $U$. Let $\ell$ be the current degree of $w$ and let
$\set{u_1,\dotsc, u_\ell}$ be a subset of $\ell$ undeleted
vertices. Let $x_1,\dotsc, x_\ell$ be an enumeration of the points
inside $w$. Let $y_1,\dotsc, y_\ell$ be the points matched to
$x_1\dotsc, x_\ell$. Let $\check m$ be the number of undeleted edges
at the beginning of the $i$-th iteration of the loop and let
$\check\mathbf{d}$ be the degree sequence of the current graph. Using $[x]_j := (x)(x-1)\dotsc(x-j+1)$, we have
\begin{equation*}
\prob{U = \set{u_1,\dotsc, u_\ell}}
=
\ell!
\prob{y_i \in u_i \ \forall i}
=
\frac{\ell!\prod_{i=1}^\ell \check d_{u_i}}{2^\ell[\check m]_\ell}(1+\xi_2(n)),
\end{equation*}
where $\xi_2(n)= o(1)$ because $\check m\geq m-kt(n)\geq m-k\log n$.
Now we have to compute estimates for the probabilities in the
deletion algorithm for simple graphs. The following lemma is an
application of~\cite[Theorem 10]{McKay81}.
\begin{lem}
\label{lem:mckay_main}
Let $\mathbf{d}\in\ensuremath{\Dcal_k(n,m)}$ be such that $\max_i d_i\leq n^{0.25}$.
Let $H$ be a graph on $[n]$ with at most $k t(n)$ edges. Let $L$
be a supergraph of $H$ with at most $k$ edges more than $H$ such that
there is a simple graph $G$ with degree sequence $\mathbf{d}$ such that
$G\cap L = H$. Then
\begin{equation*}
\probcond{L\subseteq \ensuremath{\mathop{G}}(\mathbf{d})}{H\subseteq \ensuremath{\mathop{G}}(\mathbf{d})}
=
\frac{\prod_{v=1}^{n}[d_i - h_i]_{j_i}}
{2^{\card{E(J)}}[m]_{\card{E(J)}}} (1+ \nu(n)),
\end{equation*}
where $\mathbf{h}$ is the degree sequence of $H$, $J = L-E(H)$, $\mathbf{j}$ is
the degree sequence of $J$, and $\nu(n)=o(1)$.
\end{lem}
Notice that to use this lemma one has to check the existence of a
simple graph $G$ with certain properties. In our case, Erd\H os-Gallai
Theorem will be enough to ensure such simple graph exists.
\begin{lem}
\label{lem:exists_simplegraph}
Let $n$ be sufficiently large so that $n - n^{0.25} - k\log n >
\sqrt{n}$. Let $n'\geq n - \log n$. Let $\mathbf{g}$ be a sequence on
$[n']$ such that $g_1 \geq g_2\geq \dotsm\geq g_{n'}$, $\sum_i
g_i$ is even, $g_1 \leq n^{0.25}$, $\card{\set{j\colon g_j =
0}}\leq k\log n$. Then there exists a simple graph with degree
sequence $\mathbf{g}$.
\end{lem}
The proofs for these lemmas are presented in
Section~\ref{sec:proofsimple}. Now we can analyse the deletion
algorithm for simple graphs. Let $uv \in \binom{V}{2}$. Then
\begin{equation*}
\probsimple{uv\text{ is chosen in the first step}}
=
\probsimple{uv\in E(\ensuremath{\mathop{G}}(\mathbf{d}))}\frac{1}{m}.
\end{equation*}
We need to compute $\probsimple{uv\in E(\ensuremath{\mathop{G}}(\mathbf{d}))}$. Note that this is
the same as $\probcond{L\subseteq G(\mathbf{d})}{H\subseteq G(\mathbf{d})}$ with $L
= ([n],\set{uv})$ and $H = ([n], \varnothing)$. In order to use
Lemma~\ref{lem:mckay_main}, we need to check if there is a simple
graph $G$ with $G\cap L = H$ with degree sequence $\mathbf{d}$. This is the
same as saying that there exists a simple graph $G$ with degree
sequence $\mathbf{d}$ such that $uv\not\in E(G)$. It suffices to show that,
for every set of vertices $S\subseteq [n]\setminus\set{u,v}$ of size
$d_v$, there is a simple graph with degree sequence $\mathbf{d}'$, where
$\mathbf{d}'$ is obtained from $\mathbf{d}$ be deleting $v$ and decreasing the
degree of every vertex in $S$ by $1$ (that is, $S$ can be the set of
neighbours of $v$ and it does not include $u$). Note that $\sum_i
d_i'$ is even because $\sum_j d_j$ is even. Moreover, $n-1 \geq
n-\log n$ and $\max_i d_i'\leq n^{0.25}$ and $\mathbf{d}'$ has no
zeroes. By Lemma~\ref{lem:exists_simplegraph}, there is a simple graph
with degree sequence $\mathbf{d}'$ and so we can use
Lemma~\ref{lem:mckay_main} to show that
\begin{equation*}
\prob{uv\in \ensuremath{\mathop{G}}} = \frac{d_ud_v}{2 m} (1+\xi_3(n)),
\end{equation*}
where $\xi_3(n) = o(1)$. Now suppose we are in the $i$-th iteration of
the loop and deleting a vertex $w$. Let $\check n$ be the number of
undeleted vertices in the beginning of iteration $i$ and let $\check
m$ be the number of undeleted edges at the beginning of iteration
$i$. Let $\check\mathbf{d}$ be the current degree sequence (that is, $\check
d_u$ is the number neighbours $u$ has among the undeleted vertices).
At each iteration we delete at most $k$ edges and only one vertex. So
$\check n \geq n - t(n)$ and $\check m \geq m - kt(n)$. Let
$\ell:=\check d_w$ and $\set{u_1,\dotsc, u_\ell}$ be a set with $\ell$
(undeleted) vertices. Let $U$ be the neighbours of $w$ discovered in
iteration $i$. We want to compute the probability that
$U=\set{u_1,\dotsc, u_\ell}$. In order to use
Lemma~\ref{lem:mckay_main}, we have to check if there exists a simple
graph $G$ with degree sequence $\check\mathbf{d}$ such that $G\cap L = H$,
where $H$ is the graph discovered so far (which includes the deleted
vertices) and $L = ([n],E(H)\cup\set{wu_1,\dotsc,wu_\ell})$, which is
the same as checking if it is possible to get a simple graph such that
$w$ as no neighbours in $\set{u_1,\dots, u_\ell}$. Let $U'$ be a
set of $\ell$ undeleted vertices such that $U'\cap \set{u_1,\dotsc,
u_\ell} = \varnothing$. There are plenty of choices for $U'$ since
$t(n)\leq \log(n)$. Let $\mathbf{d}'$ be the degree sequence on $\check n -1$
obtained from $\check\mathbf{d}$ by deleting $w$ and decreasing the degree of
each vertex in $U'$ by $1$. Then $\check n \geq n -\log n$, $\max_i
d_i'\leq n^{0.25}$ and $\card{\set{j\colon d_j' = 0}}\leq kt(n)\leq k\log
n$. Using Lemma~\ref{lem:exists_simplegraph}, there is a simple graph
with degree sequence $\mathbf{d}'$ and so, by Lemma~\ref{lem:mckay_main},
\begin{equation*}
\prob{U = \set{u_1,\dotsc, u_\ell}}
=
\frac{\ell!\prod_{i=1}^\ell \check d_{u_i}}{2^\ell[\check m]_\ell}(1+\xi_4(n)),
\end{equation*}
where $\xi_4(n) = o(1)$. Thus, there exists a function $\xi(n)=o(1)$ such
that
\begin{equation*}
\prob{U = \set{u_1,\dotsc, u_\ell}} = \probmulti{U =
\set{u_1,\dotsc, u_\ell}}(1+\xi(n))
\end{equation*}
and, for every $uv\in \binom{V}{2}$,
\begin{equation*}
\probsimple{uv\text{ is chosen in the first step}}
\sim
\probmulti{uv\text{ is chosen in the first step}}.
\end{equation*}
We conclude that the deletion algorithms can be
coupled for $t(n)$ steps as long as $(1+\xi)^t=1+o(1)$. Thus, it
suffices to choose $t = o(1/\xi)$.
\subsection{Proofs of Lemma~\ref{lem:mckay_main} and
Lemma~\ref{lem:exists_simplegraph}}
\label{sec:proofsimple}
\begin{proof}[Proof of Lemma~\ref{lem:mckay_main}]
Let $\Delta_L$ be the maximum degree in $L$ and let $\Delta$ be the
maximum degree in $D$. Note that $\Delta_L \leq |E(H)|+k \leq k\log
n + k$ and $\Delta \leq n^{0.25}$. Then
\begin{equation*}
\begin{split}
\card{E(\ensuremath{\mathop{G}}(\mathbf{d}))}-\card{E(H)}-\card{E(J)}
&\geq m - kt(n)-k \geq
n^{0.25}(2n^{0.25})
\\
&\geq \Delta (\Delta+\Delta_L) =: D.
\end{split}
\end{equation*}
So we can use part (a) of~\cite[Theorem 2.10]{McKay81} to obtain that
\begin{equation*}
\begin{split}
\probcond{L\subseteq \ensuremath{\mathop{G}}(\mathbf{d})}{H\subseteq \ensuremath{\mathop{G}}(\mathbf{d})}
&
\leq
\frac{\prod_{v=1}^{ n}[d_i - h_i]_{j_i}}
{2^{\card{E(J)}}[ m - \card{E(H)}-D]_{\card{E(J)}}}
\\&=
\frac{\prod_{v=1}^{ n}[d_i - h_i]_{j_i}}
{2^{\card{E(J)}}[ m]_{\card{E(J)}}}(1+\nu_1(n))
\end{split}
\end{equation*}
with $\nu_1(n) = o(1)$ because $|E(J)|\leq k$ and $ m -
\card{E(H)}-D \geq m - k\log n- 2\sqrt{n}$. Now we will use part (b)
of Theorem 2.10 from~\cite{McKay81}. We have to check the conditions
for (b):
\begin{equation*}
\begin{split}
\card{E(\ensuremath{\mathop{G}}(\mathbf{d}))}-\card{E(H)}-\card{E(J)}
&\geq
m -
kt(n)-k \geq
k\geq n^{0.25}(n^{0.25}+1)
\\
&\geq
\Delta(\Delta+\Delta_L+2)
+\Delta (\Delta_L+1),
\end{split}
\end{equation*}
so we can apply~\cite{McKay81}[Theorem 2.10(b)]. We have to bound some
errors given by~\cite{McKay81}[Theorem 2.10(b)]. We have that
\begin{equation*}
\begin{split}
0&\leq \frac{\Delta (\Delta_L+1)}
{ m-\card{E(H)}-\card{E(J)} -
\Delta (\Delta+\Delta_L+2)}
\\
&\leq
\frac{n^{0.25}(n^{0.25}+1)}
{n - k\log n - n^{0.25}(2n^{0.25}+2)}
=:
\nu_2(n),
\end{split}
\end{equation*}
with $\nu_2(n) = o(1)$, and
\begin{equation*}
\begin{split}
0&\leq
\frac{\Delta^2}
{2(\card{E(G)}-\card{E(H)}-D-(1-1/e)\card{E(J)})}
\\
&\leq
\frac{\sqrt{n}}
{2(n -k\log n- n^{0.25}(2n^{0.25}) -(1-1/e)k)}
=: \nu_3(n),
\end{split}
\end{equation*}
with $\nu_3(n) = o(1)$ Then~\cite{McKay81}[Theorem 2.10(b)] implies that
\begin{equation*}
\begin{split}
\probcond{L\subseteq \ensuremath{\mathop{G}}(\mathbf{d})}{H\subseteq \ensuremath{\mathop{G}}(\mathbf{d})}
&\geq
\frac{\prod_{v=1}^{ n}[d_i - h_i]_{j_i}}
{2^{\card{E(J)}}[ m]_{\card{E(J)}}}
(1+\nu_4(n))\left(\frac{1+\nu_2(n)}{1+\nu_3(n)}\right)^{E(J)}.
\end{split}
\end{equation*}
with $\nu_4(n) = o(1)$. Since $\nu_i(n)=o(1)$ for $i=1,2,3,4$, we
can conclude that
\begin{equation*}
\probcond{L\subseteq \ensuremath{\mathop{G}}(\mathbf{d})}{H\subseteq \ensuremath{\mathop{G}}(\mathbf{d})}
=
\frac{\prod_{v=1}^{ n}[d_i - h_i]_{j_i}}
{2^{\card{E(J)}}[ m]_{\card{E(J)}}}
(1+\nu(n)),
\end{equation*}
where $\nu = o(1)$.
\end{proof}
\begin{proof}[Proof of Lemma~\ref{lem:exists_simplegraph}]
We will use Erd\H os-Gallai Theorem: $\mathbf{g}$ is the degree sequence of
a simple graph iff, for every $1\leq \ell \leq n'$,
\begin{equation*}
\sum_{i=1}^{\ell} g_i
\leq
\ell(\ell-1)
+
\sum_{j=\ell+1}^{n'}\min\set{\ell, g_j}.
\end{equation*}
If $\ell \geq n^{0.25}+1$, then $\sum_{i=1}^{\ell} g_i \leq \ell
g_1 \leq \ell(\ell-1)$. If $\ell \leq n^{0.25}$,
\begin{equation*}
\begin{split}
\sum_{i=1}^{\ell} g_i
&\leq
\ell n^{0.25}
\leq
\sqrt{n}
\leq
n - n^{0.25} - k\log n
\leq
n' - \ell - \card{\set{j\colon g_j = 0}}
\\
&=
\sum_{j= \ell+1}^{n'} 1 - \card{\set{j\colon g_j = 0}}
\leq
\sum_{j= \ell+1}^{n'}\min\set{\ell, g_j}.
\end{split}
\end{equation*}
\end{proof}
\section*{Acknowledgments}
I would like to thank my advisor Nicholas Wormald for his supervision
during this project.
\bibliographystyle{plain}
|
{
"timestamp": "2012-09-12T02:08:13",
"yymm": "1203",
"arxiv_id": "1203.2209",
"language": "en",
"url": "https://arxiv.org/abs/1203.2209"
}
|
\section{Introduction}
The study of globular clusters systems (GCSs) is motivated by the hope of finding clues to the formation
and history of their host galaxies (for reviews see \citealt{brodie06,harris10}). Globular cluster (GC) formation
occurs in a large variety of star forming environments and at all epochs.
The formation of massive clusters is apparently favored in star bursts as they occur in
merger events (e.g. \citealt{whitmore95,degrijs03a}), but one finds GCs also in star-forming disks of normal spiral galaxies where enhanced star
formation rates again seem to be related to an enhanced efficiency of massive cluster formation \citep{larsen99,larsen00}.
Against traditional wisdom, intermediate-age globular clusters exist even in the Milky Way \citep{davies11}.
The old globular cluster systems of elliptical galaxies exhibit characteristic properties. A striking pattern is the color-bimodality, featuring
a blue (bona fide metal-poor) peak and a red (bona fide metal-rich) peak (e.g. \citealt{larsen01, kundu01}).
Since early merger events are believed to be the driver for the formation of elliptical galaxies, the properties of GCSs of known
merger remnants may provide more insight into the formation mechanisms.
The nearest merger remnant is NGC 5128 (Cen A) at a distance of 3.8 Mpc. Its stellar population is best described by
the bulk of stars (about 80\%) being very old, while a younger component (2-4 Gyr) contributes 20\%-30\% \citep{rejkuba11}. The GCS is well investigated regarding kinematics, ages, and abundances \citep{peng04a,peng04b, woodley10a,woodley10b}.
Most GCs are old, but there are also intermediate-age and younger clusters. The kinematics of both planetary
nebulae and GCs indicate the existence of a massive dark halo \citep{woodley10a}.
Cen A is a ``double-double" radio galaxy (e.g. \citealt{saikia09}) with probably recurrent nuclear activity.
Next in distance to Cen A is NGC 1316 (Fornax A) in the outskirts of the Fornax cluster. There is a vast quantity of literature on NGC 1316 with studies in many
wavelength bands. We give a representative overview.
The morphology of NGC 1316 is very different from Cen A and is characterized by an
inner elliptical body with a lot of fine dust structure in its central region, best admired in HST images, and an extended elongated
structure with loops, tails, and tidal arms which cover almost an area on the sky comparable to the full moon. These were first described by \citet{schweizer80} in
a wide-field photographic study. \citet{schweizer81} also noted the high central surface brightness. Furthermore, he performed kinematical measurements and found an abnormally low $M/L_B = 1.8$ which in part is a consequence of his large adopted distance of 32.7 Mpc.
More modern values are higher, e.g. $M/L_V = 2.5$ \citep{shaya96} and $M/L_{K_s} = 0.65$ \citep{nowak08}, but still clearly indicate an intermediate-age
population. \citet{schweizer80}, moreover, detected an inner ionizing rotating disk and a giant HII-region south of the center. He
suggested a merger which occurred between 0.5 and 2 Gyr ago.
\citet{mackie98}, on the base of deep B-band imaging and imaging in H$_\alpha$ and NII, revisited the morphology and studied the
distribution of gaseous line emission. They detected faint emission to the North, between the nucleus and the companion galaxy
NGC 1317. They also detected the interesting ``Extended Emission Line Region (EELR)", an elongated feature of length 1.5\arcmin
South-West to the nucleus at a distance of about 6\arcmin, in a region without any signature of ongoing star formation.
Using ROSAT data, they detected hot gas apparently associated with Schweizer's tidal features L1 and L2 (see Fig.2 of
\citealt{schweizer80}). Their general
assessment of NGC 1316's history is a disk-disk or disk-E merger older than 1 Gyr and a smaller merger about 0.5 Gyr ago.
\citet{horellou01} studied the content of atomic and molecular gas. Molecules are abundant in the central region, but HI has been detected only in some single spots, among them the EELR and the southern HII-region.
NGC 1316 has been observed in the X-rays with practically all X-ray satellites (Einstein: \citealt{fabbiano92}; ROSAT:
\citealt{feigelson95}, \citealt{kim98}; ASCA: \citealt{iyomoto98}). More recently, \citet{kim03} used Chandra to constrain the
temperature of the hot ISM ($\approx$0.6 keV), to detect a low-luminosity AGN, and to detect 96 X-ray point sources. They
quote a range 0.25 Z$_\odot$ $<$ Z $<$ 1.3 Z$_\odot$ for
the metal-abundance of the ISM. The values derived from Suzaku-data are lower than solar \citep{konami10} (e.g. Fe is only 0.44 solar).
NGC 1316 also is among the sample of elliptical galaxies by \citet{fukuzawa06} who quote X-ray based dynamical masses.
\citet{nagino09} used XMM-Newton to constrain the gravitational potential.
Kinematics of the stellar population and abundances have been studied by \citet{kuntschner00}, \citet{thomas05}, and \citet{bedregal06}. The luminosity-weighted abundance is higher than
solar (Z $\approx$ 0.03).
\citet{arnaboldi98} presented kinematics of 43 planetary nebulae and long-slit spectroscopy along the major and the minor axis. They analyzed the velocity field
and gave some dynamical considerations, resulting in a total mass of $2.9~10^{11} M_\odot$
and a mass-to-light ratio in the B-band of 8 within 16 kpc.
\citet{nowak08} investigated the very inner kinematics in order to constrain the mass of the supermassive black hole. They found
a mass of about $1.5~10^8~M_\odot$ marginally consistent with the mass-sigma relation from \citet{tremaine02}.
Noteworthy is the double giant radio lobe with centers in projected galactocentric distances of about 100 kpc (e.g. see Fig.1 of \citealt{horellou01}).
\citet{lanz10} used mid-infrared and X-ray data to develop a scenario in which the lobes have been created by a nuclear outburst
about 0.5 Gyr ago.
Very recently, \citet{mcneil12} presented radial velocities of almost 800 planetary nebulae in NGC 1316, one of the largest samples so far. A spherical Jeans model indicates
a high dark matter content, characterized by a dark halo with a large core.
The GCS has been investigated several times with various intentions. Apparently, the first observations devoted
to GCs were done using HST (WF/PC-I) \citep{shaya96}. These authors list 20 clusters in the very central region.
One object has $M_V = -12.7$ which, if old, today would be called an Ultracompact Dwarf (e.g. \citealt{mieske08}).
More work on GCs, using the HST and the NTT, was presented by \citet{goudfrooij01a,goudfrooij01b,goudfrooij04}. Based
on infrared colors, they found ages for the brightest clusters consistent with 2-3 Gyrs. The luminosity function (LF)
of red GCs was found to be a power law with the exponent -1.2, flatter than that of normal ellipticals, while
the blue clusters exhibit the normal LF with a turn-over at about $M_V \approx -7.2$ (but see our remarks in
Sec.\ref{sec:discussion_color}).
At the same time, \citet{gomez01} studied the GCS photometrically, using images obtained with the
3.6m telescope on La Silla, ESO, in B,V,I. They did not detect a clear bimodality but a difference in the azimuthal
distribution of blue and red clusters in the sense that the red clusters follow the ellipticity of the galaxy's bulge, while
the blue clusters are more circularly distributed. Moreover, they confirmed the very low specific frequency previously
found by \citet{grillmair99} of $S_N= 0.4$.
Our intention is to study the GCS on a larger field than has been done before. Moreover, our Washington photometry permits a useful comparison
to the Washington photometry of elliptical galaxies, obtained with the same instrumentation
\citep{dirsch03a,dirsch03b,dirsch05,bassino06a}.
We adopt a distance of 17.8 Mpc , quoted by \citet{stritzinger10} using the four type Ia supernovae which appeared so far in NGC 1316.
The high supernova rate is a further indicator for an intermediate-age population.
The surface brightness fluctuation distance \citep{blakeslee09} is $21\pm0.6$ Mpc which may indicate a still unsolved zero-point problem.
This paper is the first in a series devoted to NGC 1316 and its globular cluster system. Future papers
will treat the kinematics
and dynamics of the GCS as well as
SH2, the HII-region
detected by \citet{schweizer80}.
\section{Observations and reductions}
\subsection{Data}
Since the present data have been taken and reduced together with the data leading to the
Washington photometry of NGC 1399 \citep{dirsch03b}, all relevant information regarding the
reduction technique can be found there. We thus give the most basic information only.
The data
set consists of Washington wide-field images of NGC 1316 taken with
the MOSAIC camera mounted at the prime focus of the CTIO 4m Blanco telescope during the night 20/21 November 2001
(the entire run had three nights). We used the Kron-Cousins R and Washington C filters. Although the genuine Washington
system uses T1 instead of R, \citet{geisler96} has shown that the Kron-Cousins R filter
is more efficient than T1, due to its larger bandwidth and higher throughput, and that R and
T1 magnitudes are closely related, with only a very small color term and a zero-point difference of 0.02 mag.
The MOSAIC wide-field camera images a field of 36\arcmin $\times$ 36\arcmin, with a pixel scale of
0.27\arcsec /pixel. The observations were performed in the 16 channel read-out mode. We obtained three R exposures
with an exposure time of 600 sec each, and three C-exposures with an exposure time of 1200 sec each. Additionally,
we observed standard stars for the photometric calibration. In the following, we refer to colors as C-R and not as C-T1,
although the calibration provides Washington colors. The C-images have not been dithered.
The seeing was about 1\arcsec\ in the R-images and 1.5\arcsec\ in the C-images.
\subsection{MOSAIC reduction and photometry}
The MOSAIC data were reduced using the mscred package within IRAF. In particular this
software is able to correct for the variable pixel scale across the CCD which would cause otherwise a
4 percent variability of the brightness of stellar-like objects from the center to the corners. The flatfielding
resulted in images that had remaining sensitivity variations of the order of 1.5\%.
In particular Chip 4 and
Chip 5 showed discernible remaining flatfield structure (but within the given deviation).
The actual photometry has been performed by using DAOPHOT II.
\subsection{Photometric calibration}
Standard fields for the photometric calibration have been observed in each
of the 3 nights. The weather conditions were photometric. We observed 4-5 fields, each containing about 10 standard stars from
the list of \citet{geisler96}, with a large coverage of airmasses (typically from 1.0 to 1.9). It was
possible to use a single transformation for all three nights, since the coefficients derived for the
different nights were indistinguishable within the errors.
We derived the following relations between instrumental and standard magnitudes:
\begin{eqnarray}
\nonumber
\mathrm{R} =
&\mathrm{R}_\mathrm{inst}+(0.72\pm0.01)-(0.08\pm0.01)X_\mathrm{R}\\
\nonumber
&+(0.021\pm0.004)(\mathrm{C}-\mathrm{R})\\
\nonumber
\mathrm{C} = &\mathrm{C}_\mathrm{inst}+(0.06\pm0.02)-(0.30\pm0.01)X_\mathrm{C}\\
\nonumber
&+(0.074\pm0.004)(\mathrm{C}-\mathrm{R})
\end{eqnarray}
The standard deviation of the difference between our calibrated and tabulated
magnitudes is 0.018\,mag in R and 0.027\,mag in C.
To calibrate the NGC\,1316 field we identified isolated stars which
were used to determine the zero points. The scatter between the zero points determined from
individual stars is 0.03\,mag and most probably due to flat field
uncertainties.
The final uncertainties of the zero points are 0.03\,mag and 0.04\,mag for R
and C, respectively. This results in an absolute calibration uncertainty in C-T1 of
0.05\,mag (the uncertainty in the color term can be neglected). See, however, Fig. \ref{fig:comparison}
which indicates that the calibration is quite precise in the interval 0.5 $< $C-R $<$ 2.5.
The foreground reddening towards NGC\,1316 according to
\citet{schlegel98} is E$_{\mathrm{B-V}}=0.02$. Using
E$_{\mathrm{C-T1}}=1.97$\,E$_{\mathrm{B-V}}$
\citep{harris77} we had to correct C-R by 0.04\,mag. In the following we neglect the foreground reddening, since
no conclusion depends strongly on an absolute precision of this order.
\subsection{Photometric uncertainties and selection of point sources}
\label{sec:selection}
Fig. \ref{fig:errors} shows the photometric errors as given by DAOPHOT for all sources. The upper panel
shows the uncertainties in R, the lower panel in C-R.
The point sources and resolved sources are clearly separated for magnitudes fainter than R=18. Brighter point
sources are saturated. The uncertainties in color grow quickly for R$ >$ 24 mag. However, we want to select GC candidates not only on account of photometric uncertainties, but by goodness
of fit and degree of resolution through the ALLSTAR parameters chi and sharp.
These parameters are plotted in Fig. \ref{fig:chisharp} with sharp in the upper panel and chi in the lower panel. Positive
values of sharp indicate resolved objects, negative values deficiencies in the photometry, mainly for the faintest objects.
The unsaturated resolved objects are of course galaxies, but we cannot exclude that some extended GCs are among
them (see the remark in Section \ref{sec:previousphot}).
Guided by Fig. \ref{fig:chisharp}, we select as point sources objects within the sharp-parameter interval $-0.5 < sharp < 0.8$
and chi less than 4.
We use the R-image due to its better seeing. Furthermore, we select the magnitude interval 18 $<$ R $<$ 24 and the
uncertainties in R and C-R to be less than 0.1 mag and 0.2 mag, respectively. This selection reduces the full sample
of about 22000 objects found in the entire field to 4675.
The resulting photometric errors for the so selected point sources are shown in Fig.\ref{fig:resultingerrors}.
\begin{figure}[h]
\begin{center}
\includegraphics[width=0.35\textwidth]{errors.jpg}
\caption{Photometric uncertainties as given by DAOPHOT in R (upper panel) and C-R (lower panel). }
\label{fig:errors}
\end{center}
\end{figure}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.35\textwidth]{chisharp.jpg}
\caption{The chi- and sharp parameters of DAOPHOT. We select as point sources objects within the sharp-parameter interval -0.5 $<$ sharp $<$ 0.8 and chi $<$ 4. }
\label{fig:chisharp}
\end{center}
\end{figure}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.35\textwidth]{resultingerrors.pdf}
\caption{Photometric uncertainties of selected point sources. Since the C-images are not dithered the larger errors at a given mag
are explainable by sources lying close to the CCD-gaps.}
\label{fig:resultingerrors}
\end{center}
\end{figure}
\subsection{Comparison with previous photometry}
\label{sec:previousphot}
We now compare for confirmed clusters in NGC 1316 our photometry with the B-I photometry of \citet{goudfrooij01a}. There are 17 clusters and 10 stars for which B-I and C-R colors
exist. Our point source selection excludes the object 211 of Goudfrooij et al. which seems to be very extended.
Fig.\ref{fig:comparison} shows the comparison. Star symbols denote stars, circles denote clusters. For C-R colors bluer than 2, the agreement is very satisfactory with the exception of one cluster (the object 115 in Goudfrooij at al.'s list). This cluster is projected onto the immediate vicinity of a dust patch which may have influenced its
photometry. The stars redder than C-R=2.5 deviate significantly which plausibly shows a deficiency of the photometric calibration which may be not valid for such red
objects. The dotted line represents the theoretical integrated single stellar population colors for solar metallicity, taken from \citet{marigo08}. Again, the agreement is very satisfactory.
All clusters (except 115 of Goudfrooij et al. ) are located outside of the inner dust structures. However, only a strong reddening would
be detectable. The reddening vector in Fig.\ref{fig:comparison} is E(B-I) = 1.18~E(C-R), adopting the reddening law of \citet{harris77}, thus almost
parallel to the (B-I)-(C-R)-relation.
\begin{figure}[h]
\begin{center}
\includegraphics[width=0.35\textwidth]{comp_goud.pdf}
\caption{Comparison of our C-R colors with the B-I colors of \citet{goudfrooij01a} for stars and confirmed globular clusters in NGC 1316. Star symbols denote stars,
circles denote clusters. The dotted line delineates theoretical integrated single stellar population colors for solar metallicity, taken from \citet{marigo08}.}
\label{fig:comparison}
\end{center}
\end{figure}
\section{The CMD of point sources}
The CMD of the selected 4675 point sources is shown in Fig.\ref{fig:CMDbig} (left panel) together with their location
(in pixels; right panel). Visible are the vertical stellar sequences at C-R=0.7 and C-R=3. The bulk of GC candidates
are found between C-R=1 and C-R=1.8 for R-magnitudes fainter than 20. There may be also GCs brighter than
R=20. The bulge of NGC 1316 is striking. We recall that the optically visible full diameter of NGC 1316 is about 27\arcmin, corresponding
to 6000 pixels. At first glance there is no bimodality. A more detailed look into color intervals will modify this impression.
R=18 mag is approximately the saturation limit for point sources. A comparison with the field around NGC 1399 \citep{dirsch03b}
(their Fig.3) shows that the present data are considerably deeper. At a magnitude level of R=24 mag, the incompleteness
due to faint sources in C starts at C-R=1.3, while around NGC 1316, it starts at C-R=2.3.
The inner blank region of NGC 1316 shows the incompleteness where the galaxy light gets too bright. To avoid this
incompleteness region, we shall consider only GC candidates with distances larger than 2\arcmin.
\begin{figure*}[t]
\begin{center}
\includegraphics[width=0.3\textwidth,angle=-90]{CMD_big.pdf}
\caption{Left panel: The CMD for 4675 point sources in the MOSAIC field around NGC 1316. The bulk of the GCs are found in the color interval 1$<$C-R$<$2, but there are also bluer GCs. However,
most point sources bluer than C-R=1 and
23 $<$ R $<$ 24 are unresolved background galaxies. Right panel: distribution of point sources in our MOSAIC field. The inner galaxy bulge is strikingly
visible. The size is 34\arcmin $\times$ 34\arcmin. North is up, East to the left.}
\label{fig:CMDbig}
\end{center}
\end{figure*}
Fig. \ref{fig:CMD_radius_selected} shows CMDs with three different radius selections (see figure caption). In the inner region
(which in fact is blank within the inner 30\arcsec; here we consider also clusters closer to the center than 2\arcmin) the vertical sequence of bright objects at C-R=1.5 is striking. Some
of these GC candidates are already confirmed clusters from the radial velocity sample of \citet{goudfrooij01b}.
Very interesting is the blue object with C-R=0.64 (object 119 in Goudfrooij et al.'s list) which must be quite young, about 0.5 Gyr (see Sec. \ref{sec:discussion_color}),
assuming solar metallicity.
Two more objects with very red colors, which appear
as GCs in the list of Goudfrooij et al. (numbers 122 and 211), are not marked. With R=20.37, C-R = 3.26, and R= 20.77, C-R=3.57,
they fall drastically outside the color range of GCs. They are not obviously associated with dust. On our FORS2/VLT pre-images (seeing about 0.5\arcsec), used for
the spectroscopic campaign, they are
clearly resolved and might be background galaxies, in which case the velocities are incorrect.
However, if one bright young cluster exists, then we expect also fainter ones of similar age. We shall show that statistically.
The middle panel demonstrates that the occurrence of these bright clusters is restricted to the inner region, but there are still
overdensities of GC candidates. The right panel shows almost exclusively foreground stars and background galaxies.
We now have a closer look at the distribution of colors.
\begin{figure*}[t]
\begin{center}
\includegraphics[width=0.8\textwidth,angle=0]{CMD_radius_selected.pdf}
\caption{The three CMDs refer to different radius selections. The left panel shows GC candidates within the inner 4\arcmin, the middle
panel between 4\arcmin\ and 10\arcmin, and the outer panel at radii larger than 10\arcmin. The respective area sizes then are related
as 1:5.2:25.3. The inner CMD is thus dominated by genuine GCs. In the left panel, spectroscopically confirmed GCs \citep{goudfrooij01b}
are marked. Note the blue object at C-R = 0.64 which must be a quite young cluster.}
\label{fig:CMD_radius_selected}
\end{center}
\end{figure*}
\section{Color distribution}
\label{sec:color}
Fig. \ref{fig:colordistribution} displays the color distribution in different radial regimes (left: radius $<$ 10\arcmin; middle: 4\arcmin $<$ radius $<$ 10\arcmin; right: 2 $<$ radius $<$ 4\arcmin) for all magnitudes (R $<$ 24 mag)(lower row) , for bright magnitudes (R $<$ 22) (upper row),
and for an intermediate sample magnitudes (R $<$ 23) (middle row). A radius of 4\arcmin\ approximately marks the extension of the bulge. The ordinates are given in numbers per arcmin$^2$.
The corresponding values and their uncertainties are given in Table \ref{tab:color24} for the case R $<$ 24 mag and in Table \ref{tab:color23}
for the case R$<$ 23 mag.
The counts are background corrected, the background being evaluated outside 13\arcmin.
\begin{figure*}[]
\begin{center}
\includegraphics[width=0.6\textwidth]{colordistri.pdf}
\caption{Color histograms, corrected for background counts, for three radial regimes and for different magnitude ranges. Left panel: 0-10 arcmin, middle panel:
4-10 arcmin, right panel: 2-4 arcmin. The upper row displays magnitudes brighter than R=22, the middle row objects brighter than R=23, the lower row
objects brighter than R=24. The respective backgrounds are indicated by the dotted histograms.
The vertical dashed line indicates the color of NGC 1316 at 2\arcmin, outside the dusty region.
}
\label{fig:colordistribution}
\end{center}
\end{figure*}
The main observations from Fig.\ref{fig:colordistribution} are:\\
\noindent
-- the entire system shows a continuous distribution of colors with no sign of bimodality\\
-- the peak of the color distribution is displaced bluewards with respect to the galaxy color, more so for the outer clusters\\
-- the inner bright clusters show a pronounced bimodality with peaks at C-R=1.4 and C-R=1.1. These peaks are already indicated
in the small, but clean, spectroscopic sample of \citet{goudfrooij01a}\\
-- the outer clusters show only a peak a C-R=1.1\\
-- there is evidence for a small GC population bluer than C-R=1.0 which is approximately the metal-poor limit for old clusters (see Fig.\ref{fig:washington}).\\
The color distribution of GCs in a typical elliptical galaxy in our photometric system is bimodal with two peaks at C-R=1.35
and C-R=1.75 \citep{bassino06b} with little scatter. A Gauss fit to the entire sample in the
color range 0.8-2 gives a peak color
of $1.33 \pm 0.01$ and a sigma of $0.3\pm 0.01$ which resembles indeed the position of the ``universal'' blue peak color in the GCS of elliptical galaxies
while the distribution is broader.
On first sight, this is somewhat surprising since we expect
a quite different composition of GCs: there should be a mixture of old GCs from the pre-merger components, and an unknown fraction of GCs, presumably a large one,
which stem from one or more star-bursts. Later star forming-events with GC production might also have occurred.
The facts that the already confirmed clusters define such a sharp color peak (Fig. \ref{fig:CMD_radius_selected}) and that the existence of intermediate-age clusters has already
been shown, strongly suggest that among the bright clusters, intermediate-age objects are
dominating. Older clusters are then cumulatively mixed in with decreasing brightness.
The peak color for the outer sample is $1.23\pm0.037$ and for the inner sample $1.37\pm0.01$.
This difference must, in part, be due to a different composition of GC populations since the density of red clusters shows a steeper decline, but we
cannot exclude a contribution from differential reddening. However, this must be small since there are hardly any cluster candidates
redder than C-R=1.9. Moreover, the excellent reproduction of the theoretical color-color relations in Fig. \ref{fig:comparison}
indicates the absence of {\bf strong} reddening for this inner sample where we would expect to see reddening effects first.
\begin{table}[h!]
\caption{Background subtracted numbers per arcmin$^2$ of GC candidates brighter than R=24 mag in color bins and for different radial selections.Bin widths are 0.1 mag. The background is defined by r$>$13\arcmin. The uncertainties are based on the
square root of the raw counts. Negative values are kept for formal correctness.}
\begin{center}
\resizebox{9cm}{!}{
\begin{tabular}{ccccc}
Color & background & 2$<$r$<$10 & 4$<$r$<$10 & 2$<$r$<$4\\
\hline
0.650 & 0.144 $\pm$ 0.014 & 0.041 $\pm$ 0.028 & 0.026 $\pm$ 0.029 & 0.148 $\pm$ 0.089\\
0.750 & 0.167 $\pm$ 0.015 & -0.01 $\pm$ 0.026 & -0.01 $\pm$ 0.028 & -0.03 $\pm$ 0.061\\
0.850 & 0.155 $\pm$ 0.014 & 0.033 $\pm$ 0.028 & 0.023 $\pm$ 0.030 & 0.111 $\pm$ 0.085\\
0.950 & 0.167 $\pm$ 0.015 & 0.085 $\pm$ 0.032 & 0.064 $\pm$ 0.033 & 0.231 $\pm$ 0.104\\
1.050 & 0.159 $\pm$ 0.015 & 0.141 $\pm$ 0.034 & 0.083 $\pm$ 0.034 & 0.398 $\pm$ 0.122\\
1.150 & 0.163 $\pm$ 0.015 & 0.216 $\pm$ 0.038 & 0.113 $\pm$ 0.035 & 0.766 $\pm$ 0.158\\
1.250 & 0.156 $\pm$ 0.014 & 0.214 $\pm$ 0.037 & 0.089 $\pm$ 0.034 & 0.693 $\pm$ 0.151\\
1.350 & 0.138 $\pm$ 0.014 & 0.225 $\pm$ 0.037 & 0.084 $\pm$ 0.032 & 0.896 $\pm$ 0.166\\
1.450 & 0.121 $\pm$ 0.013 & 0.226 $\pm$ 0.036 & 0.087 $\pm$ 0.031 & 0.860 $\pm$ 0.162\\
1.550 & 0.098 $\pm$ 0.011 & 0.176 $\pm$ 0.032 & 0.053 $\pm$ 0.026 & 0.777 $\pm$ 0.153\\
1.650 & 0.077 $\pm$ 0.010 & 0.137 $\pm$ 0.028 & 0.063 $\pm$ 0.025 & 0.613 $\pm$ 0.136\\
1.750 & 0.058 $\pm$ 0.009 & 0.076 $\pm$ 0.022 & 0.025 $\pm$ 0.020 & 0.393 $\pm$ 0.110\\
1.850 & 0.047 $\pm$ 0.008 & 0.077 $\pm$ 0.021 & 0.055 $\pm$ 0.021 & 0.165 $\pm$ 0.075\\
1.950 & 0.036 $\pm$ 0.007 & 0.015 $\pm$ 0.015 & 0.005 $\pm$ 0.014 & 0.070 $\pm$ 0.054\\
2.050 & 0.054 $\pm$ 0.009 & -0.01 $\pm$ 0.014 & -0.00 $\pm$ 0.016 & -0.02 $\pm$ 0.028\\
2.150 & 0.036 $\pm$ 0.007 & 0.021 $\pm$ 0.015 & 0.009 $\pm$ 0.015 & 0.096 $\pm$ 0.060\\
\hline
\end{tabular}
}
\end{center}
\label{tab:color24}
\end{table}%
\begin{table}[h!]
\caption{Background subtracted numbers per arcmin$^2$ of GC candidates brighter than R=23 mag in color bins and for different radial selections. Bin widths are 0.1 mag. The background is defined by r$>$13\arcmin. The uncertainties are based on the
square root of the raw counts. Negative values are kept for formal correctness.}
\begin{center}
\resizebox{9cm}{!}{
\begin{tabular}{ccccc}
Color & background & 2$<$r$<$10 & 4$<$r$<$10 & 2$<$r$<$4\\
\hline
0.650 & 0.042 $\pm$ 0.007 & 0.032 $\pm$ 0.017 & 0.026 $\pm$ 0.018 & 0.064 $\pm$ 0.054\\
0.750 & 0.048 $\pm$ 0.008 & 0.003 $\pm$ 0.015 & 0.004 $\pm$ 0.016 & -0.02 $\pm$ 0.028\\
0.850 & 0.039 $\pm$ 0.007 & 0.018 $\pm$ 0.015 & 0.018 $\pm$ 0.016 & 0.014 $\pm$ 0.038\\
0.950 & 0.034 $\pm$ 0.007 & 0.056 $\pm$ 0.018 & 0.042 $\pm$ 0.018 & 0.152 $\pm$ 0.071\\
1.050 & 0.047 $\pm$ 0.008 & 0.099 $\pm$ 0.023 & 0.066 $\pm$ 0.022 & 0.165 $\pm$ 0.075\\
1.150 & 0.056 $\pm$ 0.009 & 0.147 $\pm$ 0.027 & 0.068 $\pm$ 0.023 & 0.554 $\pm$ 0.128\\
1.250 & 0.065 $\pm$ 0.009 & 0.101 $\pm$ 0.025 & 0.041 $\pm$ 0.022 & 0.280 $\pm$ 0.096\\
1.350 & 0.047 $\pm$ 0.008 & 0.096 $\pm$ 0.023 & 0.013 $\pm$ 0.017 & 0.377 $\pm$ 0.106\\
1.450 & 0.046 $\pm$ 0.008 & 0.142 $\pm$ 0.026 & 0.034 $\pm$ 0.019 & 0.591 $\pm$ 0.130\\
1.550 & 0.030 $\pm$ 0.006 & 0.098 $\pm$ 0.021 & 0.020 $\pm$ 0.015 & 0.448 $\pm$ 0.113\\
1.650 & 0.032 $\pm$ 0.007 & 0.066 $\pm$ 0.019 & 0.032 $\pm$ 0.017 & 0.233 $\pm$ 0.084\\
1.750 & 0.023 $\pm$ 0.006 & 0.038 $\pm$ 0.015 & 0.007 $\pm$ 0.012 & 0.189 $\pm$ 0.075\\
1.850 & 0.013 $\pm$ 0.004 & 0.044 $\pm$ 0.014 & 0.039 $\pm$ 0.015 & 0.040 $\pm$ 0.038\\
1.950 & 0.013 $\pm$ 0.004 & 0.002 $\pm$ 0.008 & 0.002 $\pm$ 0.009 & 0.013 $\pm$ 0.027\\
2.050 & 0.030 $\pm$ 0.006 & -0.01 $\pm$ 0.010 & -0.01 $\pm$ 0.011 & -0.03 $\pm$ 0.006\\
2.150 & 0.020 $\pm$ 0.005 & 0.002 $\pm$ 0.010 & -0.00 $\pm$ 0.010 & 0.006 $\pm$ 0.027\\
\hline
\end{tabular}
}
\end{center}
\label{tab:color23}
\end{table}%
\section{Spatial distribution of clusters}
\subsection{Density profile and total numbers}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.4\textwidth]{profil_all.pdf}
\caption{Surface densities for selected point sources. The dashed horizontal line indicates the background. Inside 2\arcmin, the counts are severely affected by incompleteness. The profile
seems to be inflected at a radius of 4\arcmin. We define the background by counts outside 13\arcmin.}
\label{fig:profile_all}
\end{center}
\end{figure}
Given the complicated morphology and a probable mixture of bulge and disk symmetry, a surface density profile for GCs in NGC 1316 will not have the same meaning
as in spherical elliptical galaxies, where deprojection is possible. We rather use the density profile to
evaluate the total extent of the cluster system and possible features in the profile. For convenience we
count in circular annuli and select point sources in the color range 0.9-2 according to the above selection.
The full annuli become sectors for distances larger than 10\arcmin, when the image border is reached. We correct for this geometrical incompleteness.
First we show the general density profile, then the two-dimensional distribution in various color intervals, and finally the
radial density profiles in these intervals.
Fig. \ref{fig:profile_all} shows the resulting surface densities. Inside 2\arcmin\ the counts are severely affected by incompleteness
due to the galaxy light and possible extinction by dust. Outside 13\arcmin\ the counts are not distinguishable anymore from the background, for which we
determine 0.946 objects/arcmin$^2$.
Summing up the bins in Fig.\ref{fig:profile_all}, we find 636 $\pm$ 35 as the total number of GC candidates down to our magnitude limit of R=24 and in the color range 0.9 $<$ C-R $<$ 2.0 inside 13\arcmin. Correcting for the
inner incompleteness would not enhance this number considerably. In case of an old GCS of a normal elliptical galaxy with a
Gaussian-like luminosity function and a turn-over magnitude (TOM) corresponding to its distance, one would roughly double this number to have a fair estimation of the total number.
Since a single TOM, valid for the entire cluster system, does not exist, one would underestimate
this number. About 1400-1500 clusters seems to be a good guess, somewhat larger than the number given by \citet{gomez01}.
In any case, it is a relatively poor cluster system. We come back to this point in the discussion.
This radius of 13\arcmin\ agrees well with the extension of NGC 1316 seen in Fig.\ref{fig:picture} which we expect if the extension of
NGC 1316 is determined by the dynamical processes during the merger event(s). One notes an inflection point in Fig.\ref{fig:profile_all}
at about 4\arcmin. Its nature becomes clearer if we subtract the background and plot surface densities for different
color intervals. This is done in Fig.\ref{fig:profile_density_linear} in a linear display, including the interval 0.8$<$C-R$<$1.1,
and in Fig.\ref{fig:profile_colors} in double-logarithmic display. The existence of a blue population is statistically well visible, although
poor. GC candidates in the color interval 1.3-1.6 show a sharp decline at 4\arcmin, which defines the bulge. The
outer clusters in this sample are mainly located to the South.
\subsection{The two-dimensional distribution of clusters}
\label{sec:2dim}
\begin{figure*}[ht]
\begin{center}
\includegraphics[angle=-90,width=0.7\textwidth]{xyplot_color_richtig.pdf}
\caption{This plot shows the 2-dimensional distribution of GC candidates in six C-R color intervals. North is up, East to the left. The origin
is NGC 1316. See text for details. The most
important observations are: there are clusters bluer than C-R=1.1, clusters in the range 1.1-1.3 show a symmetry with a shifted major axis,
clusters of intermediate color define mostly the bulge but there is also an overdensity of clusters related to outer morphological features to the
South. }
\label{fig:xyplot_color}
\end{center}
\end{figure*}
NGC 1316 exhibits an elliptical symmetry only in its inner parts. At larger radii,
the overall morphology is characterized by tidal(?) tails and loops.
It is now interesting to
ask whether morphological features of NGC 1316 can be found back in the distribution of GCs. Naturally, one does that
in color intervals to suppress the background and (hopefully) get insight into the nature of the cluster candidates.
We plausibly assume that intermediate-age and younger clusters have at least the metallicity of the stellar bulge, i.e. at
least solar metallicity. Refer to Fig.\ref{fig:washington} for a theoretical relation between age and Washington colors.
Fig.\ref{fig:xyplot_color} shows the 2-dimesional distribution of objects in six different color-intervals.
Interval 0.4$<$ C-R $<$0.8:\\
Here we expect younger clusters with ages about 0.5 Gyr or somewhat younger. NGC 1316 is statistically not visible but note the cluster with C-R = 0.6 in Fig.\ref{fig:CMD_radius_selected}. We
anticipate (Richtler et al., in prep.) that we identified another object with C-R=0.4 by its radial velocity. Certainly there are not
many bright clusters within this age range and they do not seem to belong to a particularly strong star formation epoch.
Interval 0.8$<$ C-R $<$1.1\\
Now the field of NGC 1316 becomes recognizable. With solar metallicity or higher, ages are between 0.6 and 1 Gyr. Noteworthy are the clumps roughly 1.5\arcmin\ to the NE
and a slight overdensity which is projected roughly onto the L1-feature.
Interval 1.1$<$ C-R $<$1.3:\\
In this interval we expect old, metal-poor clusters and younger clusters between 1 and 1.5 Gyr. NGC 1316 is now clearly
visible. The major axis of the distribution seems to show a position angle of about 90$^\circ$, different from the main
body. However, the more quantitative source counts in dependence on azimuth still show an overdensity at the
position angle of the major axis of NGC 1316 towards the North-East, while the peak towards the South-West is less
pronounced (Fig. \ref{fig:azimut}).
The fraction of intermediate-age clusters is unknown. We shall discuss this issue further in Sec. \ref{sec:subpop}, where we
argue that younger clusters actually provide the dominating population.
Interval 1.3$<$ C-R $<$1.6\\
This interval samples the maximum of the color distribution. One notes the sharp definition of the galaxy's luminous body. Perpendicular to
the major axis the cluster reaches the background at about 2.5\arcmin. A gap is visible at a distance of approximately
4\arcmin\ to the South. Even further to the South, the cluster candidates populate a region roughly defined
by Schweizer's L1-structure.
Note also the
horizontal boundary of cluster candidates at a distance of 8\arcmin\ which delineates nicely the sharp ridge in NGC 1316.
Interval 1.6$<$ C-R $<$1.9\\
Here we expect the majority of the metal-rich GCs of the pre-merger components. The fact that NGC 1316 is still well visible,
indicates a similar concentration as that of the metal-poor clusters. The overall symmetry follows the main body of NGC 1316,
which indicates the presence of an old population. In fact, as we argue in Section \ref{sec:scenario}, this population dominates
the bulge mass.
Interval 1.9$<$ C-R $<$2.2\\
One would expect old, metal-rich, and reddened clusters in this interval. However, NGC 1316 is not longer visible.
\subsection{Density profiles in various color intervals}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.4\textwidth]{density_profile_linear.pdf}
\caption{Background subtracted surface densities for selected point sources in different C-R color intervals and linearly displayed.
See Sec.\ref{sec:2dim} for the significance of these intervals and Fig.\ref{fig:profile_colors} for a logarithmic display. Although
only a poor population, clusters bluer than 1.1 are statistically well visible. }
\label{fig:profile_density_linear}
\end{center}
\end{figure}
\begin{figure*}[t]
\begin{center}
\includegraphics[width=0.8\textwidth,angle=0]{profile_colors.pdf}
\caption{Surface densities in double-logarithmic display for point sources with the background subtracted and for three different C-R color intervals:0.9-1.3 (left panel),
1.3-1.6 (middle panel),1.6-1.9(right panel). For the blue and the red samples, the power-law indices, valid for
the radial range between log(r)=0.4 and log(r)=0.9 are
indicated. There is no uniform power-law for the intermediate sample. The dashed horizontal lines indicate the backgrounds. The vertical line is an excessively large uncertainty in one bin. It becomes
clear that the inflection point in Fig. \ref{fig:profile_all} is caused by the intermediate sample, which sharply falls off at a distance
of 4.5\arcmin. We identify this sample with the bulk
of intermediate-age clusters. }
\label{fig:profile_colors}
\end{center}
\end{figure*}
We now consider the radial distributions in different color intervals.
This is done in Figs. \ref{fig:profile_density_linear} and \ref{fig:profile_colors}. Fig. \ref{fig:profile_density_linear} plots
the surface densities linearly and includes the color interval 0.8 $<$ C-R $<$ 1.1 where the clusters should be younger than
1 Gyr. This population is well visible in Fig. \ref{fig:profile_density_linear} at radii corresponding to the bulge region, and less conspicous at larger
radii (due to the poor statistics we choose a larger bin in Fig.\ref{fig:profile_colors}).
In Fig. \ref{fig:profile_colors}, the left panel shows a blue sample (0.9 $<$ C-R $<$1.3), the middle panel an intermediate sample
(1.3 $<$ C-R $<$1.6), and the right panel (1.6$<$ C-R $<$1.9) the red sample. It becomes clear that the inflection point in Fig.\ref{fig:profile_all} is caused by the intermediate sample
whose density shows a rapid decline at 4.0\arcmin\ and remains more or less constant for larger radii.
Since we are azimuthally averaging along structures which are azimuthally strongly inhomogeneous, this constancy
does not reflect a true radial constancy.
Interestingly, the inflection point is not visible in the luminosity profile of NGC 1316.
Given all evidence, we identify this
sample with the bulk of intermediate-age clusters.
For the blue and the red sample, the indicated power-law indices refer to the
radial interval 0.4 $<$ log(r) $<$ 0.9. The red GC candidates show a somewhat steeper decline. They perhaps preserved their initially
more concentrated profile.
Fig.\ref{fig:xyplot_color} shows that the distribution of the intermediate sample at larger radii is concentrated towards the south,
roughly delineating Schweizer's L1-structure.
\section{Azimuthal distribution}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.5\textwidth]{pos30.pdf}
\caption{The distribution of position angles in the radial ranges 2\arcmin\ - 4\arcmin\ (upper row) and 4\arcmin\- 10\arcmin\
(lower row) for three color intervals. The major axis is indicated by the vertical dotted lines. Sphericity is detected nowhere.}
\label{fig:azimut}
\end{center}
\end{figure}
Fig. \ref{fig:azimut} shows the azimuthal distribution for three color-subsamples of GC candidates for the radial
range 2\arcmin\ - 4\arcmin\ (inner sample) and 4\arcmin\ - 10\arcmin\ (outer sample). The bin width
is 30$^\circ$ and the counts have been performed in circular annuli. The abscissas are the position angles measured
counterclockwise with zero in the North. The ordinates are background-subtracted number densities per arcmin$^2$.
The first observation is that no subsample shows a spherical distribution, This stands in some contradiction to
\citet{gomez01} who found, using BRI colors, for their blue sample a spherical distribution, while their red sample
followed the galaxy's ellipticity. We suspect that this is due to our more precise background subtraction and, because of the C filter inclusion,
a more precise color subsampling. The highest degree of symmetry is found for the inner intermediate-color sample, once
more demonstrating the strong confinement of this population to the bulge. The outer intermediate sample shows the
strong dominance of the southern region. Most of these clusters might be attributed to the L1 feature.
The red sample, in which we expect old metal-rich clusters, also shows the bulge symmetry.
\section{Luminosity function}
The luminosity function (LF) of the GCS has been derived to very faint magnitudes by HST \citep{goudfrooij01b}, although
only for the central parts. The main result is that the LF misses a well-defined turn-over magnitude (TOM) , which for ``normal''
elliptical galaxies is an excellent distance indicator \citep{villegas10,richtler03}.
This can be understood by the relative youth of the cluster system which contains still unevolved clusters, an effect
already noted by \citet{richtler03} in galaxies with younger populations.
Our data cannot compete in depth with the HST observations, but we can probe the outer parts of the GCS, where
our data are complete down to R=24 mag, in order to see whether a TOM is visible. With a distance modulus of
31.25 and a TOM of $\mathrm M_V = -7.4$ we expect it at about V= 23.8 and with a mean V-R=0.5. At R=23.3, a TOM-brightness
outside the main body of NGC 1316 should be clearly visible.
As the radial interval we choose 4-9 arcmin\ in order to avoid the galaxy light, have reasonable number counts
and avoid the very outskirts. Fig. \ref{fig:LKF} shows the background-corrected LF in this radial range, where the background counts
refer to the complete field outside 12\arcmin. Surface densities are numbers/arcmin$^2$. The uncertainties are calculated as the square root of background
corrected counts. The dotted histogram shows the background counts.
The expected turn-over at R$\approx$23.3 mag is not visible, resembling the HST results of \citet{goudfrooij04} for the central part and for the
red clusters.
The conclusion is that in this outer region the LF of intermediate-age clusters contribute significantly.
The data of \citet{goudfrooij04} indicate a turn-over for their blue sample, but see our remarks in Sect.\ref{sec:discussion_color}
for counterarguments.
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.4\textwidth]{LKF_4_9.pdf}
\caption{The luminosity function in R inside a circular annulus between 4\arcmin\ and 9\arcmin\ radius. The surface
density is counted in numbers/arcmin**2. The background counts have been performed outside 12\arcmin. We do not see the expected turn-over at R$\approx$ 23.3. One concludes
that also outside the main body of NGC 1316 younger clusters dominate the LF.}
\label{fig:LKF}
\end{center}
\end{figure}
\section{Discussion}
\subsection{External and internal reddening}
\label{sec:reddening}
The foreground reddening toward NGC 1316 is low. The all-sky maps of
\citet{burstein82} and \citep{schlegel98} give E(B-V)=0 and
E(B-V)=0.02, respectively. This difference is the zero-point difference between the two
reddening laws.
According to Schlegel et al., the accuracy
of both values is comparable below a reddening of E(B-V)=0.1. Therefore the
reddening in C-R is E(C-R)= 0.04 or probably smaller, comparable with
the uncertainty of the photometric calibration. We thus prefer not to correct
for foreground reddening, also given that no conclusion depends on such
a precision. The reddening by dust within NGC 1316 outside a radius of 1\arcmin\ seems to be low as well.
We point to the fact that our reddest GC candidates have C-R$\approx$1.9, which the appropriate color for old, metal-rich
GCs.
\subsection{Theoretical Washington colors}
We use theoretical isochrones for relating cluster ages to Washington colors.
Fig. \ref{fig:washington} compares the color distribution of GC candidates with theoretical models of single stellar populations from \citet{marigo08}, using their web-based tool (http://stev.oapd.inaf.it/cgi-bin/cmd). Models are shown for five different metallicities. The
highest metallicity refers to the stellar population (e.g. \citealt{kuntschner00})
Overplotted (in arbitrary units) is a Gaussian representing the color distribution for GC candidates between 2\arcmin\ and 10\arcmin\ radius.
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.5\textwidth]{marigo_extended.pdf}
\caption{This graph shows theoretical single stellar population models from \citet{marigo08} for five different metallicities. Overplotted (in arbitrary units) is a Gaussian representing the color distribution for GC candidates between 2 and 10 arcmin radius. If the cluster sample within $1.3 < C-R < 1.6$ is dominated by objects with at least solar metallicity, many clusters are younger than 2 Gyr. }
\label{fig:washington}
\end{center}
\end{figure}
\subsection{Color distribution and ages}
\label{sec:discussion_color}
What are the implications of the color distribution in Fig. \ref{fig:colordistribution} for possible scenarios of the cluster formation history
of NGC 1316? First we compare our C-R photometry with the HST-photometry in B,V,I of the cluster sample of \citet{goudfrooij01a}.
These authors find two peaks at B-I=1.5 and B-I=1.8. Translating these values into Washington colors with the help of Fig. \ref{fig:comparison}, one estimates for these peaks C-R=1.1 and C-R=1.4, respectively. This is indeed consistent with what is
seen in Fig. \ref{fig:colordistribution}. Already \citet{goudfrooij01b} interpreted the peak at B-I=1.8 as the signature of intermediate-age
clusters with ages around 3 Gyr which was backed up by the spectroscopic ages and metallicities derived for a few bright objects \citep{goudfrooij01a}.
From Fig.\ref{fig:washington} one would rather assign an age of somewhat less than 2 Gyr.
If the bluer peak indicates old, metal-poor clusters of the pre-merger components, the metallicity
distribution of old clusters must be strongly biased to very metal-poor objects and thus be very different from that in ``normal'' globular cluster systems of ellipticals (and also spirals) which show
a more or less universal peak at C-R = 1.35 \citep{bassino06b}.
Since we can identify clusters with ages even younger than what would correspond to C-R=1.1, that means below 1 Gyr, one rather expects a mixture of old and
intermediate-age clusters of unknown proportions. The hypothesis that the blue peak does not represent an overabundance
of very metal-poor clusters, but marks a second burst of star formation with an age of about 1 Gyr, is interesting. A counterargument
could be that the luminosity function in a V,V-I diagram seems to show a turn-over at about V= 24.6 mag \citep{goudfrooij04}. However,
a number of points related to Fig.3 of Goudfrooij et al. create doubts whether this turn-over really corresponds to the universal turn-over magnitude in GCS. Firstly, \citet{goudfrooij04} adopted a distance of 22.9 Mpc which means a brightening of 0.5 mag in absolute
magnitudes with respect to the supernova distance of 17.8 Mpc, we adopted. So the turn-over would be at $M_V = - 6.7$, i.e. almost 0.7 mag
fainter than the ''normal'' turn-over, while one would expect the turn-over to be brighter due to the bias to low metallicities. Secondly, the turn-over itself is not well defined, the highest number count being even fainter
than the formal turn-over from the fit, leaving the question open whether there is a decline at fainter magnitudes or not. Thirdly, their color interval which defines "old clusters" is $0.55 < V-I < 0.97$, a sample which
clearly includes intermediate-age clusters (here we also point to the superiority of the Washington system with respect to V-I and
refer the reader to Fig. 7 of \citealt{dirsch03b} ).
Evidence from our photometry comes from the fact that the red peak is strongly constrained to the bulge (Figs. \ref{fig:profile_colors}
and \ref{fig:profile_density_linear}). The blue clusters, on the other hand, are abundant still in the outskirts of NGC 1316. If this outer cluster
population would consist exclusively of old clusters, then a turn-over magnitude should be visible in our luminosity function
which is not even corrected for incompleteness.
The absence of the striking red peak in the outer region shows that these stellar populations do not stem from the first star-burst.
The dynamical age of these regions is unknown, but plausibly younger than the inner star burst.
The light of the inner bulge is redder than the peak color of the GCS by about 0.15 mag in C-R and shows only a small
radial transition to bluer colors. \footnote{Strictly speaking, this difference is not exactly valid,
because we measure the color of the stellar population by using the total projected light and calculate
a color for the GCs by averaging over magnitudes. The difference, however, is negligible.}
A differential reddening between clusters and stellar population is not probable. The selection of cluster candidates between
1\arcmin\ and 3 \arcmin\ (236 objects), where the reddening should be strongest, still gives a peak color of C-R=1.38, while the galaxy has C-R=1.57. An explanation for this difference may be sought in the fact that the clusters are (more or less) single stellar populations of intermediate-age, while the total bulge light has a contribution from old
populations. We shall argue that this contribution is significant.
We find statistical evidence for clusters between $0.8 < C-R < 1.1$ which are neither old and metal-poor nor
can be related to a starburst 2 Gyr ago, but must be younger even if their metallicities are lower than solar.
Plausibly, these clusters should have a metallicity at least as high as the
intermediate-age stellar population, if they formed in NGC 1316.
However, there is still the possibility that infalling dwarf galaxies provided clusters with low metallicities
\subsection{Comparison with spectroscopic ages}
\citet{goudfrooij01b} provide spectroscopic ages and metallicities for three clusters, their objects No.103, No.114, and No. 210. There exist C-R measurements for 103 and 210 (114 is too close to the center). These clusters have solar abundance and a common
age (within the uncertainties) of 3 Gyr. The C-R colors for 103 and 210 are 1.48 and 1.51, respectively. A comparison with Fig.\ref{fig:washington} shows that these colors are a bit too blue for full consistency with the model colors for 3 Gyr and solar
abundance. A shift to the red of 0.1 mag in C-R would be necessary to achieve perfect agreement, but given that various models
with their respective uncertainties and the uncertainty of the absolute photometric calibration are involved, one can consider
the agreement to be satisfactory. In any case, these two clusters do not seem to be reddened.
\subsection{Subpopulations}
\label{sec:subpop}
A bimodal color distribution is not exclusively a feature of elliptical galaxies, as the example of the Milky Way shows.
As a working hypothesis, we assume that the color histogram of old GCs of NGC 1316 also has a bimodal appearance. Moreover, we
assume that clusters redder than the galaxy color belong to the old, metal-rich subpopulation.
\citet{bassino06b} compiles values for the blue and the red peak of early-type galaxies in the Washington system. Adopting these
values, we fix the Gaussian of the red peak with a maximum at C-R=1.75 and a dispersion of 0.15. We cannot distinguish
old, metal-poor clusters from younger, metal-rich clusters that populate the same color range, but may assume that their
color distribution follows the almost universal distribution known from other galaxies, for which \citet{bassino06b}
quotes C-R=1.32 for the peak color and 0.15 mag for the dispersion.
In order to get an impression of what the pre-merger GCSs could look like, we fit the corresponding Gaussian to the color bins redder than C-R=1.6, assuming that
all clusters redder than the galaxy light are old. The amount of old, metal-poor clusters is unknown, but we assume two cases: equal to
and twice the metal-rich population. The subtraction of these two Gaussians from the Gaussian describing the entire
sample down to R=24 mag in Sect. \ref{sec:color}
reveals the GC population which has been formed in the merger and and perhaps in following star-forming events.
Fig. \ref{fig:subpop} shows these two cases. In the left panel (case 1) the dashed line results from the subtraction of the assumed
bimodal color distribution of old clusters. In this case, intermediate-age clusters are dominating the color interval 1.3-1.6. In the
the right panel (case 2), old clusters dominate which contradicts the behaviour of clusters in this color range. Therefore, case 1
may be closer to the true situation.
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.5\textwidth]{subpops.pdf}
\caption{In both panels, the broad Gaussian (solid line) represents the color distribution of the entire GCS. The two narrower Gaussians
represent the assumed bimodality of old metal-poor and metal-rich clusters. The dashed line results from subtracting the bimodal
color distribution. In the left panel (case 1), intermediate-age clusters dominate, in the right panel (case 2), old clusters dominate.
Case 2 is less consistent with the properties of the GC population. }
\label{fig:subpop}
\end{center}
\end{figure}
\subsection{A star formation history}
\label{sec:scenario}
The star formation history of NGC 1316 was probably complex. A merger event is not a simple infall, but is preceded by close encounters which can trigger
starbursts, in time separated by the orbital period. See, for example, \citet{teyssier10} and \citet{dimatteo08} for modern simulations of mergers and starbursts.
Teyssier et al. resolve the multi-phase ISM and find that the main trigger for starbursts is not a large-scale gas flow, but a fragmentation of gas into cold and dense
clumps, favoring star cluster formation. A realistic scenario would probably involve many individual starbursts which perhaps can be identified in future high S/N spectra
of globular clusters in NGC 1316. Our intention in simplifying the star formation history is to show that in order to bring photometric properties and dynamical determined
mass-to-light ratios into agreement, the properties of the pre-merger population rule out an elliptical galaxy. Moreover, a contribution of a population significantly
younger than 2 Gyr is needed.
We therefore ask whether it is possible to find a simple model, inspired by the color distribution of GCs, which is consistent
with the color and the mass-to-light ratio of the bulge stellar population.
We adopt three populations: a
pre-merger population (pop.1) with
an a-priori unknown age, and two populations representing starbursts with ages 2 Gyr (pop. 2) and 0.8 Gyr (pop. 3).
Since the luminosity weighted metallicity of the bulge is at least solar \citep{kuntschner00}, we adopt solar metallicity for each of them.
The color of NGC 1316 then results from a composite of these populations to which we assign
the integrated color C-R=1.55 (see Fig.\ref{fig:colorprofile}).
Mass-to-light ratios for the inner bulge have been quoted by several authors in various photometric bands. To transform an M/L-value from
one band to another, is in the case of NGC 1316 not completely trivial, since published aperture photometries \citep{prugniel98} measure
a color which is reddened by the dust within about 1\arcmin\ radius. We therefore adopt theoretical colors (including Washington colors) from
the Padova models (http://stev.oapd.inaf.it/cgi-bin/cmd; \citealt{marigo08}).
The relation between M/L-ratios (denoted by $\Gamma$) in filters i and j is
$$\frac{\Gamma_i}{\Gamma_j} = \frac{L_j}{L_i} \frac{L_{i,\odot}}{L_{j,\odot}} $$
C-R=1.55 corresponds to a good approximation to a single stellar population with solar metallicity and an age of 2.5 Gyr. For that population we find B-V=0.81, B-R=1.34,
and R-I=0.56. Furthermore, we adopt the solar absolute magnitudes of \citet{binney98} (p.53) and thus can transform M/L-values into the R-band.
\citet{schweizer81} gives $\mathrm M/L_V = 1.3 \pm 0.2$ for a distance
of 32.7 Mpc, corresponding to $\mathrm M/L_R = 2.1$ for our distance (M/L is inversely proportional to the distance).
\citet{shaya96} give $M/L_V \approx 2.2 \pm 0.2$, corresponding to $M/L_R \approx 1.9$ (distance 16.9 Mpc).
\citet{arnaboldi98}, by using dynamics of planetary nebulae, quote $\mathrm M/L_B \approx 7.7$ inside a radius of
200\arcsec\ (distance 16.9 Mpc).
However, they find lower values for smaller radii, down to $\mathrm M/L_B \approx 4.3$ at a radius
of 45\arcsec\ which corresponds to $\mathrm M/L_R \approx 3.1$.
This early result, based on only a few planetary nebulae, is superseded by the recent work of \citet{mcneil12}, who quote $\mathrm M/L_B = 2.8$ (distance 21.5 Mpc)
for the stellar population, which transforms into $\mathrm M/L_R \approx 2.6$.
Due to the large database of their dynamical model, this value certainly has a high weight. However, it is based on spherical modeling and the authors caution that
it may change with more realistic models.
\citet{nowak08}, modeling the central kinematics with a black hole, have found for the central region a value in the $K_s$-band between
0.7 and 0.8 (distance 18.6 Mpc), corresponding to a value between 2.8 and 3.2 in the R-band (setting $M_Ks/M_{Ks,\odot}$ = $M_K/M_{K,\odot}$ and adopting
R-K=2.60 for the stellar population).
Future work will probably lower the uncertainties. For the moment, $M/L_R \approx 2.5$ seems to be a good value to adopt.
Perhaps the best empirical determinations of M/L-values of
stellar populations of early-type galaxies come from the SAURON-group \citep{cappellari06}. The highest values, they
quote in the I-band inside one effective radius, are around 5 (their "Jeans"-values). Considering that a dark halo can contribute
up to 8\% of the total mass, values around 4 for metal-rich, old populations seem reasonable. The Padova models give $M/L_I$=4.1 for a 12 Gyr old
population of solar metallicity (with $M_{I,\odot}$=+4.10), assuming a log-normal IMF \citep{chabrier01}, reproducing the empirical values quite well.
Adopting their M/L-ratios also for younger populations, we now can assign colors and M/L-values to our model populations
and estimate under simplifying assumptions the properties of the pre-merger population.
Table \ref{tab:pops} lists the models, ages, metallicities, Washington colors, and R-band mass-to-light-ratios (columns 1-5).
We assume two different pre-merger populations with ages of 12 Gyr (pop.1a) and 7 Gyr (pop.1b).
The equation to be solved is then
$$\mathrm (C-R)_{integrated} = 1.55 = -2.5 \cdot log \left (\frac{(L1+L2+L3)_C}{(L1+L2+L3)_R} \right ) $$
where L1, L2, L3 are the respective luminosities in C and R. If we use the R-luminosity of the bulge as unit and
assume a certain mass proportion of pop.2 and pop.3, one can solve for L2 and L3. Since in case 1 of Fig. \ref{fig:subpop} the young clusters
are of comparable number, we assume pop.2 and pop.3 to contribute with
equal stellar mass. L2 and L3 (R-band) are given in column 6. Column 7 gives the resulting stellar masses for the bulge
(inside a radius of 240\arcsec\ ) using the bulge luminosity of Table \ref{tab:photmodel}. These values are upper limits since
in reality one must use the deprojected bulge luminosities, but the global symmetry of NGC 1316 is not known.
Having these values, one can
calculate the resulting M/L of the composite population.
\begin{table}[h!]
\caption{Properties of three adopted single stellar populations and their composite}
\begin{center}
\resizebox{9cm}{!}{
\begin{tabular}{ccccccc}
\hline
Pop. & Z & age[Gyr] & C-R & $M/L_R$ &$\mathrm L_R$& $\mathrm M[M_\odot] $ \\
(1) & (2) & (3) & (4) & (5) & (6) & (7) \\
\hline
1a & 0.019& 12 &1.99 & 5.2 & 0.60 & $\mathrm 3.18 \times 10^{11}$ \\
2 & 0.019 &2 &1.49 & 1.1 & 0.13 & $\mathrm 1.55 \times 10^{10}$\\
3 & 0.019 & 0.8 & 0.96 & 0.57 & 0.27 & $\mathrm 1.53 \times10^{10}$\\
\hline
composite & & &1.55 &3.4& 1 & $\mathrm 3.48 \times 10^{11}$ \\
\hline
\hline
1b & 0.019 & 7 & 1.84 & 3.4 & 0.67 & $\mathrm 2.36 \times 10^{11}$\\
2 & 0.019 &2 &1.49 & 1.1 & 0.11 & $\mathrm 1.22 \times 10^{10}$\\
3 & 0.019 & 0.8 & 0.96 & 0.57 & 0.22 & $\mathrm 1.26 \times 10^{10}$\\
\hline
composite & & & 1.55 & 2.55 &1 & $\mathrm 2.60 \times 10^{11}$\\
\hline
\hline
\end{tabular}
}
\end{center}
\label{tab:pops}
\end{table}
What one learns from this exercise is that a pre-merger
population of 12 Gyr with solar metallicity, representative of an elliptical galaxy, does not agree well with a composite $M/L_R$=2.5.
In contrast, a 7 Gyr old population fits well, which one would expect if the pre-merger was one or two spiral galaxies with an already
existing mix of old to intermediate-age populations.
Moreover, a population significantly younger than 2 Gyr is necessary in case of $M/L_R$=2.5. If only the 2 Gyr population would exist, it would be so dominant ($\mathrm L2_R=0.85$
in the case of a 12 Gyr pre-merger population) that $\mathrm M/L_R=1.7$. This value would even be lower in the case of a younger pre-merger population.
These considerations are viable only for the bulge. In the outer parts, the galaxy gets bluer, but we have no knowledge of
the metallicities or ages. However, one might suspect that younger ages dominate over decreasing metallicity, since the
Southern L1-structure is traced by GCs of probable intermediate age.
We thus have a total stellar bulge mass of about $2.6\times10^{11}$ $M_\odot$ of which only
10\% have
been produced in both starbursts. These 10\% are, however, responsible for
the majority of bright (!) GCs, emphasizing once more that the efficiency of
GC formation in phases of high star formation rates is greatly enhanced \citep{whitmore95,larsen00,degrijs03a,degrijs03b,kravtsov05}.
One has to account for a gas mass of $2.6\times10^{10}$ $M_\odot$ which has been transformed into
stars. Since we must assume a strong galactic wind with heavy mass loss during the burst phases, the total gas mass needed was probably higher. A gas mass of this order is easily provided by one or two
spiral galaxies (e.g. \citealt{mcgaugh97}).
\section{Conclusions}
We present wide-field photometry (36 $\times$ 36 arcmin$^2$) in Washington C and Kron-Cousins R around the
merger remnant NGC 1316 (Fornax A) in order to investigate the globular cluster system of NGC 1316 on a larger scale
than has been done before. The data consist of MOSAIC images obtained with the 4-m Blanco telescope at
CTIO. The Washington system is particularly interesting due to its good metallicity resolution in GCSs containing
old clusters and for the comparison with the well-established bimodal color distribution of elliptical galaxies.
Our main findings are:\\
The GC candidates are well confined to the region of the optical extension of NGC 1316. Outside a radius of
12\arcmin\ (corresponding to 62 kpc), a cluster population is statistically not detectable.\\
The entire system down to R = 24 mag shows a broad
Gaussian-like distribution with a peak at C-R = 1.37, somewhat bluer than the color of the inner bulge. A selection of brighter GC candidates shows a bimodal color distribution, best visible for bulge objects, with peaks
at C-R=1.1 and C-R=1.4. We find a small population of GC candidates bluer
than C-R=1.0, which is the limit for old metal-poor clusters. Clusters bluer than C-R=0.8 are statistically not
detectable, but casual confirmations through radial velocities show that there are sporadically clusters
as blue as C-R=0.4. The brighter population of cluster candidates therefore consists of intermediate-age clusters,
while older clusters of the pre-merger galaxy are progressively mixed in at fainter magnitudes.
Assuming that younger cluster have at least the metallicity of the bulge population, one can assign ages, using
theoretical Washington isochrones and integrated colors. The peak at C-R=1.4 then corresponds to an age of
about 1.8 Gyr, the peak at C-R=1.1 to an age of 0.8 Gyr, and the bluest colors to less than 0.5 Gyr.
The peaks plausibly stem from epochs of very high star formation rates, connected to one or several merger events.
The bluest clusters could indicate ongoing star formation younger than 0.8 Gyr. Alternatively, they could be
metal-poorer clusters of somewhat older age from infalling dwarf galaxies or could be formed in one of the star bursts from metal-poor gas of one of the pre-merger components.
The intermediate color interval $1.3 < C-R < 1.6$ shows a radial surface density profile very different from the other color regimes. Here we
find the bulk of intermediate-age clusters. These objects are strongly confined to the inner bulge, showing an inflection at about 4\arcmin.
For radii larger than 4\arcmin, their distribution is azimuthally inhomogeneous with a
pronounced concentration in the area of Schweizer's (1980) L1-structure.
The blue clusters ($0.9 < C-R < 1.3$)
fall off without showing an inflection with a power-law exponent of -1.6. The red clusters ($1.6 < C-R < 2$) follow a somewhat steeper
power-law with an exponent of -2.1.
Outside the bulge, the luminosity function of cluster candidates does not show the turn-over expected for an old
cluster system, indicating that also at larger radii, younger clusters contribute significantly.
Guided by the color distribution of GCs, we present a simple model, which as the main ingredient adopts two burst of star formation
with ages 2 Gyr and 0.8 Gyr added to an older population of solar metallicity. In order to reproduce the color of the bulge of NGC 1316, and the stellar M/L-value from dynamical estimates, this older population has an age of about 7 Gyr as a single stellar population. In reality, this population is expected to be a mix of populations and thus the
age indicates spiral galaxies as merger components rather than an old elliptical galaxy. The existence of a population significantly younger than 2 Gyr
is necessary in order to avoid the dominance of this population which would result in too low M/L-values.
The stellar masses of the younger populations account for only 10\% of the total stellar mass, while their luminosity contributes 30\% of the total luminosity.
As an appendix, we present our image and add morphological remarks.
The wide-field morphology of NGC 1316 does not reveal new features with respect to the photographic morphological study
of \citet{schweizer80} with the exception of a faint arc which morphologically could be a continuation of the prominent L1-feature.
We also present a color map of the inner regions showing details, which to our knowledge have not been mentioned in the literature, among them a curved feature which may be the relic of a spiral arm.
The color gradient is very shallow out to 6\arcmin\
pointing to a well mixed stellar population.
We construct a new spherical model of the R-brightness profile and give its main characteristics.
\begin{acknowledgements}
We thank an anonymous referee for valuable remarks and constructive criticism.
TR acknowledges financial support from the Chilean Center for Astrophysics,
FONDAP Nr. 15010003, from FONDECYT project Nr. 1100620, and
from the BASAL Centro de Astrofisica y Tecnologias
Afines (CATA) PFB-06/2007. He also thanks the Arryabhatta Institute for Observational Sciences, Nainital, for warm hospitality
and financial support. LPB gratefully acknowledges support
by grants from Consejo Nacional de Investigaciones Cient\'ificas
y T\'ecnicas and Universidad Nacional de La Plata (Argentina).
We thank Richard Lane for a careful reading of a draft version.
\end{acknowledgements}
|
{
"timestamp": "2012-03-09T02:04:18",
"yymm": "1203",
"arxiv_id": "1203.1879",
"language": "en",
"url": "https://arxiv.org/abs/1203.1879"
}
|
\section{Introduction}
The link scheduling problem for wireless networks has received
considerable attention in the recent past. In a wireless network
with shared spectrum, interference from neighboring nodes prevents
all nodes in the network from transmitting simultaneously at full
interference free rate. A link scheduler chooses a set of links to
deactivate at every time instant to eliminate their interference on
other links and only active links transmit data. An important
performance objective of a scheduler is throughput optimality,
\emph{i.e.,} for any given network, the scheduler should keep all
the queues in the network stable for the largest set of arrival
rates that are stabilizable for that network.\par For wireless
networks in which a set of link activation vectors are defined
according to a general binary interference model, the
Maxweight policy or the dynamic back-pressure policy is known to be
throughput optimal \cite{Tass}. Maxweight type policies have also been
shown to be throughput optimal for wireless networks with fading
channels, where the link rates vary over time \cite{Stolyar,Eryilmaz}.
However, the Maxweight policy suffers from high computational
complexity (NP-hard in many cases, including $k$-hop interference
models, k$>$1) \cite{sharma}, and has therefore motivated the study
of schedulers that have low complexity, are amenable to distributed
implementation and also offer provable performance guarantees.
Examples of such schedulers include Greedy Maximal Scheduling (GMS)\cite{Linshroff}
and Maximal Scheduling\cite{Chaporkar}, which have been widely studied for wireless
networks with static channels. \par
There has been a number of studies that analyze the performance of GMS as a function of the network topology. The main parameter of focus has been efficiency, which is defined as the largest fraction of the network capacity region guaranteed to be stable under GMS. In \cite{Linshroff}, efficiency has been evaluated as a function of the local pooling factor of a network graph (LPF),
which depends on the network topology and interference constraints.
Later, using the LPF, GMS has been shown to be throughput optimal for a
wide class of network graphs under the node exclusive interference
model \cite{Birand, Zussman}.\par The performance analysis of the
aforementioned low complexity schedulers does not however, carry
over to the scenario with fading, in which link rates are
time-varying. For instance, unlike a static network, one cannot
conclude in a network with time-varying links that satisfying local
pooling under GMS implies throughput optimality. It is only known
that in the case of the node-exclusive interference model, GMS can
achieve at least half the network stability region. Thus, it is of interest to investigate if for networks with time varying link rates, GMS performs as well as it does in networks with fixed link rates.
\cite{Linimperfect}. \par In this paper, we develop a greedy link
scheduler, GFS, for wireless networks with fading channels, which,
although not throughput optimal, has low computational complexity
and offers provably good performance guarantees. We show that the
performance of our greedy scheduler can be related to the LPF of a
network graph. We then conjecture that the performance of GFS is a lower bound on the performance of GMS for wireless networks with time-varying link rates.
\vspace{-0.05in}
\section{System Model}
\begin{figure}
\psfrag{a1}{$S_1$\vspace{10mm}} \psfrag{a2}{\vspace{4mm} $S_2
\newline$} \psfrag{c11}{\hspace{-3mm} $c_{1}^{1}$}
\psfrag{c21}{\vspace{10mm}$c_{2}^{1}$} \psfrag{c12}{$c_{1}^{2}$}
\psfrag{c22}{$c_{2}^{2}$} \psfrag{c1b}{$\overline{c}_{1}$}
\psfrag{c2b}{$\overline{c}_{2}$} \psfrag{L}{$\Lambda$}
\psfrag{l1}{$l_1$} \psfrag{l2}{$l_2$}
\begin{center}
\includegraphics[bb=0bp 50bp 1165bp 380bp,clip,scale=0.18]{twolinkfading}
\par
\end{center}
\caption{Figure shows an example of two interfering links with two fading states $S_1$ and $S_2$, occurring with probability $\pi^1$ and $\pi^2$. The network stability region region, $\Lambda$ is the interior of the region enclosed by the solid lines.}
\label{fig:capacity-region}
\vspace{-0.2in}
\end{figure}
We consider a
wireless network modeled as a graph
$\mathcal{G}=(\mathcal{V},\mathcal{E})$ with edges representing
links. We assume a single hop traffic model where each edge
represents a source-destination pair. Time is divided into slots and
packets arrive at the source node following an i.i.d. process with
a finite mean at the start of each time slot. Let $A_l(t)$ denote the number of packets arriving during time slot $t$. $A_l(t)$ has a mean $\lambda_l$. The vector of channel
states across all links in the network is assumed to be fixed over
the duration of a time slot but changing after every time slot. The
set of channels in the network can assume a state $j \in \{1,\ldots
,J \}$ according to stationary probability $\pi^j$. In each time
slot $t$, the achievable rate of link $l\in\mathcal{E}$, denoted by
$c_l(t)$, assumes value $c_{l}^{j}$ if the network is in fading state
$j$ at time slot $t$. The expected rate of a link, denoted by
$\overline{c}_{l}$ is given by $\overline{c}_l=\sum_{j=1}^{J}\pi^j
c_{l}^{j}$. We assume a generalized binary interference model, in
which each link $l$ is associated with an interference set, denoted
by $\mathcal{I}_l\subset\mathcal{E}$. Set $\mathcal{I}_l$ consists
of the set of links that cannot be active whenever link $l$ is
active.\par
Let $\vec{r}^{j}$ denote a $1\times|\mathcal{E}|$
rate allocation vector for a network that is in channel state $j$,
where $r^{j}_{i}$ is the rate allocated to link $i\in\mathcal{E}$.
Any rate allocation vector $\vec{r}^{j}$ must satisfy the following
properties:
\begin{itemize}
\item[{(a)}]$r^{j}_{l}>0$ implies $r^{j}_{k}=0$, for all
$k\in\mathcal{I}_l$, where $k\neq l$
\item[{(b)}]There exists no link $k\in \mathcal{E}$ such that
$r^{j}_{k}\neq c^{j}_{k}$ and $k\notin\mathcal{I}_l$ for all $l$
satisfying $r^{j}_{l}>0$. In other words there exists no link that
does not interfere with any other active link and is yet not
scheduled.
\end{itemize}
Let $\mathcal{R}^j$ denote the set of all feasible rate allocation
vectors for a wireless network graph when the network is in channel
state $j$. Similarly, $\mathcal{R}^j_{\mathcal{L}}$ is the set of all
feasible rate allocation vectors on the subgraph
$\mathcal{L}\subset\mathcal{G}$. The stability region of the
network~\cite{Tass}\cite{NeelyTass} is then given by the interior of the set
$\Lambda=\{\vec{\lambda}:\vec{\lambda}\preceq\vec{\phi},\ \text{for
some}\ \vec{\phi}=\sum_{j\in{1,\ldots,J}}\pi^j\psi^{j} \}$,
where $\psi^j \in Co(\mathcal{R}^j)$, with $Co(\mathcal{R}^j)$ representing the
convex hull of the set $\mathcal{R}^j$, and $\preceq$
denoting component-wise inequality. Fig.~\ref{fig:capacity-region} shows an example of a simple two link
network with two fading states, with the associated network
stability region under a node-exclusive interference model. Figs.~\ref{fig:capacity-region}(a) and Figs.~\ref{fig:capacity-region}(b) illustrate the achievable rate regions in state $S_1$ and $S_2$ respectively. The network stability region $\Lambda$ is shown in Fig.~\ref{fig:capacity-region}(c).\par In related work,
\cite{Stolyar} considered a queueing model analogous to a cellular
network with $N$ links, where the network channel state followed an
irreducible discrete time Markov chain with a finite state space. It was
shown
that the policy which selects the queue with the highest
weight.
\[\max_{l=1\ldots N} q_{l}^{\beta}(t)c_{l}(t)\]
in each time slot, where $q_l$ is the queue size for link $l$ was throughput optimal for this network. In \cite{Neely}, it was shown that a
Maxweight-type scheduling policy was throughput optimal for power
allocation in wireless networks with time varying channels.
Similarly, \cite{Eryilmaz} also showed throughput optimality of a class of
Maxweight type policies for wireless networks with fading channels. \par
Before we describe our greedy scheduler, we discuss the performance of non-opportunistic schedulers in the following section. In particular, we focus on a scheduler that utilizes only the mean link
rates, instead of instantaneous link rates. For this scheduler, we illustrate that when arrivals are correlated with channel states,
the non-opportunistic policy can potentially keep serving links that
are experiencing poor channel states, leading to a loss in throughput.
\vspace{-0.08in}
\subsection{Performance of Non-opportunistic Schedulers}
We show
that a scheduler that utilizes the mean link
rates, instead of instantaneous link rates could perform arbitrarily
worse in certain cases. To illustrate this, we consider the
two link network graph shown in Fig. \ref{fig:capacity-region}.
In this example, each link $l$ has one queue $Q_l$, into which packets
arrive according to an IID process.
Suppose that the rates of the two links
in each of the channel states are given by $c_{1}^{1}=1,c_{2}^{1}=\epsilon$, and
$c_{1}^{2}=\epsilon,c_{2}^{2}=1$ respectively. Also, let $\pi^1$ be the
probability with which the network assumes channel state 1, and
$\pi^2$ be the probability for network channel state 2. In each time slot,
the greedy non-opportunistic scheduler that we consider serves the
queue which maximizes the quantity $Q_l(t)\bar{c_{l}}$.
We will now construct an arrival traffic for this network under which
the queues for both links grow unbounded under the
non-opportunistic scheduling scheme. \par
Let the initial queue lengths be $Q_1(0)=Q_2(0)=0$.
At the beginning of each time slot, packets arrive according to the following statistics:
\begin{itemize}
\item [{(i)}]If the network channel state is 1, then with probability
$1-\delta$, for an arbitrary $\delta>0$, $\epsilon$ packets arrive into the queue $Q_{1}$, and none
into $Q_2$; With probability
$\delta$, $C/\bar{c_{1}}+\epsilon$ packets arrive into the queue of link 1,
and $C/\bar{c_{2}}$ packets arrive into the queue of link 2 respectively.
$C$ is a fixed positive quantity.
\item [{(ii)}]If the network channel state is 2, then with probability
$1-\delta$, $\epsilon$ packets arrive into the queue $Q_2$, and none into
$Q_1$; With probability
$\delta$, $C/\bar{c_{1}}$ packets arrive into the queue of link 1, and
$C/\bar{c_{2}}+\epsilon$ packets arrive into the queue of link 2 respectively.
\end{itemize}
Under this arrival statistic, we show that the end of each time slot, the length of each queue either remains unchanged or increases by a fixed quantity $C/\bar{c_{i}}$.
At the beginning of the first time slot, all queues are assumed to be empty. The non-opportunistic scheduler then serves the queue with the highest weight,
\emph{i.e.,} the queue into which $\epsilon$ or $C/\bar{c_{i}}+\epsilon$ packets have arrived. At the end of each time slot, the queue lengths remain unchanged with probability $1-\delta$, or increase by a fixed quantity $C/\bar{c_{i}}$ with probability $\delta$. Moreover, the queue lengths are also equal at the end of each time slot and of the form $kC$, where $k$ is a nonnegative
integer. Since the queue length process
is non-decreasing, and the event that the queue length increases by a
fixed positive quantity occurs infinitely often, the network is unstable
under the greedy non-opportunistic scheduler. The arrival rate vector of our
proposed arrival traffic is determined as
$\vec{\lambda}=\pi^1(1-\delta)[\epsilon\quad 0]
+ \pi^1\delta[{\scriptstyle C/\bar{c_1}}\qquad {\scriptstyle C/\bar{c_{2}}}] +
\pi^2(1-\delta)[0\qquad\epsilon]
+\pi^2\delta[{\scriptstyle C/\bar{c_1}} \qquad {\scriptstyle C/\bar{c_{2}}}]$,
which simplifies to $\vec{\lambda}=\epsilon\left[\pi^1 \qquad \pi2] +
\delta[\frac{{\scriptstyle C}(\pi^1+\pi^2)}{\bar{c_1}}-\epsilon\pi^1 \qquad
\frac{{\scriptstyle C}(\pi^1+\pi^2)}{\bar{c_2}}-\epsilon\pi^2\right]$. Thus,
when $\epsilon$ is small, the greedy non-opportunistic scheduler is unable to support
arrival rates that are within a fraction $\epsilon$
of the stability region. Note that in the above example, the arrival process
is correlated with the network channel state process.
\vspace{-0.1in}
\section{ A Greedy scheduler for Networks
with Fading Channels (GFS)} \label{sec:GFS}
The greedy scheduler that we propose is similar to GMS except that it requires each link to have
a virtual queue corresponding to every channel state of the
network, \emph{i.e.,} each link has a set of $J$ virtual queues. In each time slot, packets arriving into a link $l$ are placed into one of the $J$ queues. In practice, each link could maintain only one real first-in first-out queue, into which packets arrive and depart, and counters for the virtual queues which keep track of the number of packets in the virtual queue. The GFS scheduler would use the values of the counters to make the scheduling decision. Using such counters, also known as shadow queues have been effective in reducing queueing complexity and delay \cite{Bui}. Let $\mathrm{Q}_{l}^{j}$ be the
virtual queue of link $l$ corresponding to fading state $j$ and $\mathit{q}_{l}^{j}(t)$
denote its size at time $t$. Let $\mathrm{Q}_{l}$ denote the real FIFO queue of link $l$.
We now
describe our greedy scheduler:
\begin{itemize}
\item[{(1)}] At the beginning of time slot
$t$, packet arrivals $A_l(t-1)$ are placed in queue $\mathrm{Q}_{l}^{j}$ with
probability $\frac{\pi^jc_{l}^{j}}{\overline{c}_l}$.
\item[{(2)}] In time slot $t$, let the network be in fading state $j$.
GFS observes only the queues
corresponding to fading state $j$, in order to select the rate
allocation vector. The scheduler first selects the link with highest
weight $m=\operatornamewithlimits{argmax}_{l\in \mathcal{E}}\mathit{q}_{l}^{j}c_{l}^{j},$ removes all links in $\mathcal{I}_m$ from the set
of potential links to be scheduled at time $t$, and repeats the
process until there are no more non-interfering links that remain to
be selected.
\end{itemize}
At the end of this procedure GFS selects a rate allocation vector
that belongs to $\mathcal{R}^j$, when the network channel state is $j$. Note that the GFS policy becomes identical to GMS in the case of networks with static link rates. Also, the application of the GFS
policy on the queues corresponding to fading state $j$, requires the
knowledge of the network fading state at every node in the network. The departure process for the virtual queues can now be described as follows: For any link $l$, $\min(\vec{r}(l),Q_{l}^{j}(t))$ packets depart from virtual queue $q_{l}^{j}$, while $\min(\vec{r}(i),Q_{l}(t))$ packets depart from the real FIFO queue $q_l$.
\vspace{-0.1in}
\subsection{Performance Analysis of GFS}
We now give the
main result of this paper, which uses the LPF of a
network graph to evaluate the stability region achievable using GFS. Before we state our result, we define the following static
wireless network: given any wireless network graph
$\mathcal{G}=(\mathcal{V},\mathcal{E})$ with time varying link
rates, we associate with $\mathcal{G}$ a static wireless network
$\hat{G}=(\mathcal{V},\mathcal{E})$, whose link rates are fixed at
$\bar{c}_{l},\, \forall l$. Let $\hat{\mathcal{R}}$ denote the set of
all feasible rate allocation vectors for the network graph $\hat{G}$. We also define
$\mathcal{G}^j=(\mathcal{V},\mathcal{E})$ to be a static network
whose link rates are fixed at $c^{j}_{l},\, \forall l,\, j=1,\ldots,J$.
Finally, we let $\Lambda$ and $\widehat{\Lambda}$ denote the network stability regions of the
networks $\mathcal{G}$ and $\hat{\mathcal{G}}$ respectively. Note
that $\widehat{\Lambda}\subseteq\Lambda$, since
The LPF for
the network graph $\hat{\mathcal{G}}$ can then be defined as follows \cite{JooShroff}:
\begin{defn}
\label{def:def1}Let $\mathcal{L}$ be any subgraph of $\hat{\mathcal{G}}$. Then
$\mathcal{L}$ satisfies $\sigma$-local pooling if, for any given pair $\vec{\mu},\vec{\nu}$, where $\vec{\mu}$ and $\vec{\nu}$ are convex combinations
of the rate vectors in $\hat{\mathcal{R}}_{\mathcal{L}}$,
we have $\sigma\vec{\mu}\nprec\vec{\nu}$. \\
The LPF $\sigma^{*}$, for the network is then defined as:
\begin{align*}
\sigma^{*}=\sup\left\{\sigma\mid\forall\, \mathcal{L}\subset
\hat{\mathcal{G}},
\mathcal{L} \text{ satisfies } \sigma\text{-local pooling} \right\}.
\end{align*}
\end{defn}\vspace{-0.06in}
The LPF of a network graph depends only on the topology of the network graph and therefore
is identical for $\hat{\mathcal{G}}$ and
$\mathcal{G}^j, \,j\in\{1,\ldots,J\}$.
\begin{theorem}\label{th1}
Let $\sigma^{*}$ be the LPF of a network $\hat{\mathcal{G}}$. Then,
the network $\mathcal{G}$ is stable under the GFS policy for all arrival rate
vectors $\vec{\lambda}$ satisfying $\vec{\lambda}\in\sigma^{*}
\hat{\Lambda}$, where $\hat{\Lambda}$ is the stability region of the
corresponding network graph $\widehat{\mathcal{G}}$.
\end{theorem}
Theorem 1 provides performance guarantees for our scheduling policy for any wireless network in terms of the
stability region of an associated identical static network whose link rates are fixed at their expected rates. Note that an LPF of 1 implies that the associated greedy policy can guarantee stability for any arrival rate in $\hat{\Lambda}$.
Examples of network graphs which have $LPF=1$ include tree network
graphs under the $k$-hop interference model for $k\geq1$. In
\cite{Birand}, all network graphs with $LPF=1$ under the
node-exclusive interference model are identified.\par
We prove Theorem 1 by first establishing the stability of the virtual queues. We then provide Lemma \ref{lemma3} to establish stability of the real FIFO queues as well.
\begin{IEEEproof}
We consider the fluid limit model of the system. Let
$\vec{A_{l}^{j}(t)}$ denote the cumulative arrival process into
queue $\mathit{q}_{l}^{j}$ and $S_{l}^{j}(t)$ denote the cumulative
service process for $\mathrm{Q}_{l}^{j}$ until time slot $t$. For the
arrival and service processes, we use $A_{l}^{j}(t)=A_{l}(\lfloor
t\rfloor)$, and $S_{l}^{j}(t)=S_{l}^{j}(\lfloor
t\rfloor).$ For the queue process $\mathit{q}_{l}^{j}(t)$, we employ
linear interpolation.\par We now consider a sequence of scaled queuing
systems $(\vec{\mathit{q}}^{n}(\cdot),\vec{A^{n}}(\cdot),\vec{S^{n}}(\cdot))$.
where we apply the scaling ${\mathit{q}_{l}^{j}(nt)/n},\;
{A_{l}^{j}(nt)/n},\mbox{ and } S_{l}^{j}(t)(nt)/n,\,\forall
l\in\mathcal{E}$ with the queue process satisfying
$\sum_{l\in\mathcal{E}}{\mathit{q}_{l}^{j}}(0)\leq n$. Then, using the
techniques to establish fluid limit in \cite{Dai}, one can show that
a fluid limit exists almost surely, \emph{i.e,} for almost all
sample paths and for any positive $n\rightarrow\infty$, there exists
a sub-sequence $n_{k}$ with $n_{k}\rightarrow\infty$ such that
following convergence holds uniformly over compact sets: For all
$l\in\mathcal{E},$
$\frac{1}{n_{k}}\{A_{l}^{j}\}^{n_{k}}(n_{k}t)\rightarrow\frac{\pi^jc_{l}^{j}}{\overline{c}_{l}}\lambda_{l}t,\,j\in1\ldots
J\,$,
$\frac{1}{n_{k}}\{S_{l}^{j}(t)^{n_{k}}(n_{k}t)\rightarrow
S_{l}^{j}(t)$, and
$\frac{1}{n_{k}}\{\mathit{q}_{l}^{j}\}^{n_{k}}(n_{k}t)\rightarrow
\tilde{\mathit{q}}_{l}^{j}(t)$, where $\tilde{\mathit{q}}_{l}^{j}(t)(t)$ and
$S_{l}^{j}(t)$ are the fluid limits for the queue length
processes and the service rate processes respectively. The fluid
limit is absolutely continuous and hence the derivative of
$\tilde{\mathit{q}}_{l}^{j}(t)$ exists almost everywhere \cite{Dai}
satisfying:
\begin{equation}
{\frac{d}{dt}\tilde{\mathit{q}}_{l}^{j}(t)\ =\begin{cases}
\left[\frac{\pi^jc_{l}^{j}}{\overline{c}_l}\lambda_{l}
-\gamma_{l}^{j}(t)\right]^{+} & \mathit{q}_{l}^{j}(t)>0\\
0 & \mbox{otherwise}\end{cases}}\label{diffeq}
\end{equation}
where $\gamma^{j}_{l}(t)=\frac{d}{dt}(s_{l}^{j}(t))$.
Consider the times $t$ when the derivative $\frac{d}{dt}\mathit{q}^{j}_{l}(t)$
exists for all $l\in\mathcal{E},j\in{1,\ldots, J}$. Let $L_{0}(t)$ denote the set of
queues with the largest weight, \emph{i.e.,} $L_{0}(t)=\operatornamewithlimits{argmax}_{\mathrm{Q}_{l}^{j}\in\Psi}
\tilde{\mathit{q}}_{l}^{j}(t)c_{l}^{j},$
where $\Psi$ is the set of all queues in the network. Let $L(t)$
denote the set of queues from $L_{0}(t)$, which have the maximum
derivative of the weights, \emph{i.e.,} $L(t)=\operatornamewithlimits{argmax}_{\mathrm{Q}_{l}^{j}\in L_{0}(t)}\frac{d}{dt}
\tilde{\mathit{q}}_{l}^{j}(t)c_{l}^{j}.$
The set $L(t)$ can then be expressed as
$\underset{j}{\bigcup}L^j(t)$, where \vspace{-0.1in}
\[L^j(t)=\{\mathrm{Q}_{l}^{j},l \in \mathcal{E}\mid
\mathrm{Q}_{l}^{j}\in L(t)\},\,j=1\ldots J.\]
Since $\tilde{\mathit{q}}_{l}^{j}(t)$ is absolutely continuous, there
exists a small $\delta>0$ such that in the interval $(t,t+\delta)$, the weight of queues in $L^j(t)$ dominates
the weight of other queues, whenever the network channel state is $j$. Hence, GFS gives
priority to queues belonging to $L^j(t)$ in
$(t,t+\delta)$.
We now provide the following two lemmas to characterize the arrival
rates and service rates for the queues in $L^j(t)$. Let
$\mathcal{E}_{L^j(t)}\subset\mathcal{G}^j$ denote the set of links whose queues are in $L^j(t)$.
Thus $\mathcal{R}^{j}_{\mathcal{E}_{L^j(t)}}$ denotes the set of all feasible
rate allocation vectors for the subgraph $\mathcal{E}_{L^j(t)}$.
Let $\vec{\lambda}^{j}$ be the $|\mathcal{E}|$ dimensional arrival rate vector whose each element $\lambda^{j}(l)$ represents the arrival rate into queue $\mathrm{q}_{l}^{j}$. For any $|\mathcal{E}|$ vector $\vec{\eta}$, the projection of
$\vec{\eta}$ on a subset of edges $L$, denoted by $\vec{\eta}|_{L}$, is defined as
the $|L|$ dimensional vector obtained by restricting $\vec{\eta}$ to the edges in
$L$.
\begin{lem} \label{lemma1}
Consider any fading state $j \in {1\ldots J}$ such that
$L^j(t)\neq \emptyset$. If the arrival rate vector $\vec{\lambda}\in\sigma^{*}\hat{\Lambda}$, then $\vec{\lambda}^{j}$, the arrival rate into the queues
$\mathrm{q}_{l}^{j} \,\forall l\in\mathcal{E}$, when projected on to the set of links
$\mathcal{E}_{L_{j}(t)}$ can be expressed as $\vec{\lambda}^{j}|_{\mathcal{E}_{L^j(t)}}=\sigma^{*}\pi^j\vec{\mu},$
where $\vec{\mu}$ is a convex combination of the rate allocation vectors in
$\mathcal{R}^{j}_{\mathcal{E}_{L^j(t)}}.$
\end{lem}
\begin{proof}
Since $\vec{\lambda}\in\sigma^{*}\hat{\Lambda}$, it satisfies $\vec{\lambda}\preceq\sigma^{*}\vec{\Phi}$, for some $\vec{\Phi}=\sum_{i}\alpha_{i}\vec{\hat{r}}_{i}$, where $\vec{\hat{r}}_{i}\in\hat{\mathcal{R}}$, and $\sum_i\alpha_i=1$. One can then write the arrival rate into a link $l$ as $\lambda_{l}=\sigma^{*}\overline{c}_{l}\sum_{i}\alpha_i\mathbf{1}_{\{\hat{r}_{i}(l)\neq0\}},$
where $\mathbf{1}_{\{\hat{r}_{i}(l)\neq0\}}$ is the indicator function. The arrival rate into queue $\mathrm{Q}_{l}^{j}$ is then given by
$\lambda_{l}^{j}=\sigma^{*}\frac{\pi^jc_{l}^{j}}{\overline{c}_{l}}\overline{c}_{l}\sum_{i}\alpha_i\mathbf{1}_{\{\hat{r}_{i}(l)\neq0\}},$
which yields
$\lambda_{l}^{j}=\sigma^{*}\pi^{j}c_{l}^{j}\sum_{i}\alpha_i\mathbf{1}_{\{\hat{r}_{i}(l)\neq0\}}$, for all $l\in\mathcal{E},\text{and } j\in\{1\ldots J\}.$
We can then write the arrival rate vector $\vec{\lambda}^{j}$ in terms of rate allocation vectors in $\mathcal{R}^{j}$ as $\vec{\lambda}^{j}=\sigma^{*}\pi^{j}\sum_{i}\alpha_i \vec{r}^{j}_{i},$
since if $\mathbf{1}_{\{\hat{r}_{i}(l)\neq0\}}=1, \text{ then }
\mathbf{1}_{\{r^{j}_{i}(l)\neq0\}}=1$, or $c_{l}^{j}=0$. It follows that $\vec{\lambda}^{j}|_{\mathcal{E}_{L^j(t)}}
\preceq\sigma^{*}\pi^j\vec{\mu}$, where $\vec{\mu}$ is a convex combination of the rate allocation vectors in $\mathcal{R}^{j}_{\mathcal{E}_{L^j(t)}}.$
\end{proof}
\begin{figure}
\subfloat[\label{fig:four-link graph} ]{\noindent \begin{centering}
\includegraphics[bb=20bp 0bp 400bp 400bp,scale=0.3]{fourlink}
\par\end{centering}
\centering{}}\subfloat[\label{fig:GFScomparison} ]{\noindent \begin{centering}
\includegraphics[bb=20bp 170bp 300bp 500bp,scale=0.3]{gfs_four_link}
\par\end{centering}}
\centering{}
\noindent \caption{A four-link network graph is shown in
Fig. ~\ref{fig:four-link graph} and the performance of GFS and GMS is plotted in Fig. ~\ref{fig:GFScomparison}.}
\vspace{-0.25in}
\centering{}
\end{figure}
\begin{lem} \label{lemma2}
Consider any fading state $j \in {1\ldots J}$ such that $L^j(t)\neq \emptyset$. Then the service rate vector $\vec{\gamma}^{j}(t)$, projected onto the links in
$\mathcal{E}_{L_{j}(t)}$, can be expressed as $\vec{\gamma}^{j}|_{\mathcal{E}_{L^j(t)}}=\pi^j\vec{\nu},$
where $\vec{\nu}$ is a convex combination of the rate allocation vectors in
$\mathcal{R}^{j}_{\mathcal{E}_{L^j(t)}}.$
\end{lem}
\begin{proof}
The full proof is similar to that in
\cite{Stolyar} and \cite{JooShroff} and is omitted here.
Consider all queues belonging to $L^j(t)$. Since the queues in $L^j(t)$ have
the highest weight in $(t,t+\delta)$ when the network is in state $j$, the GFS scheduler gives priority
queues in $L^j(t)$ whenever the network channel state enters state $j$ in the time interval $(t,t+\delta)$. Consequently, the rate allocation vectors selected by GFS
in network channel state $j$, when projected on the set of links $\mathcal{E}_{L^j(t)}$
yields an element from the set
$\mathcal{R}^{j}_{L^j(t)}$. Therefore, under the GFS policy,
the service rate vector $\vec{\gamma}^{j}(t)$ for the set of queues $\vec{\mathrm{Q}}^j$,
projected onto $\mathcal{E}_{L^{j}(t)}$ is a convex combination of the elements
of $\mathcal{R}^{j}_{\mathcal{E}_{L^j(t)}}$. From the ergodicity of the network channel
state process, GFS serves elements in $\mathcal{R}^{j}_{L^j(t)}$ a fraction
$\pi^j$ of the time. It follows that $\vec{\gamma}^{j}|_{L_{j}(t)}=\pi^j\vec{\nu}$. \end{proof}
From Lemma \ref{lemma1} and Lemma \ref{lemma2},
the arrival rate $\vec{\lambda}^{j}|_{\mathcal{E}_{L^j(t)}}$ as well as
the service rate $\vec{\gamma}^{j}|_{\mathcal{E}_{L^j(t)}}$ can be
expressed in terms of the convex combinations of elements in
$\mathcal{R}^{j}_{L^j(t)}$. Since $\hat{\mathcal{G}}$ satifies $\sigma^{*}$ local pooling,
$\mathcal{E}_{L^j(t)}$ being a subgraph of $\mathcal{G}^j$ satisfies
$\sigma^{*}$ local pooling. It follows from the definition of
$\sigma$-local pooling that there exists a link
$l\in\mathcal{E}_{L^j(t)}$ such that its queue
$\mathrm{q}_{l}^{j}$ satisfies
$\lambda_{l}^{j}-\gamma_{l}^{j}\leq-\epsilon$,
for some $\epsilon >0$. Since
$\frac{d}{dt}\tilde{\mathit{q}}_{l}^{j}(t)=\frac{d}{dt}
\tilde{\mathit{q}}_{m}^{n}(t)$ for any pair
$\mathrm{Q}_{l}^{j},\mathit{Q}_{m}^{n} \in L(t)$, we obtain
$\frac{d}{dt}\tilde{\mathit{q}}_{l}^{j}(t)<\epsilon$,
for all $\mathrm{Q}_{l}^{j}\in L(t)$. \par
We now consider the Lyapunov function $V(t)=\max_{l\in \mathcal{E},j\in \{1\ldots J\}}
\tilde{\mathit{q}}_{l}^{j}.$
The derivative of $V(t)$ is given by :
\begin{align*}
\frac{d}{dt}V(t)&=\frac{d}{dt}\max_{l\in\mathcal{E},j\in\{1\ldots J\}}\tilde{\mathit{q}}_{l}^{j}c_{l}^{j}
\leq\max_{\tilde{\mathit{q}}_{l}^{j}\in L(t)}\frac{d}{dt}
\tilde{\mathit{q}}_{l}^{j}c_{l}^{j}\leq-\epsilon.
\end{align*}
The negative drift of the Lyapunov function implies that the
fluid limit model of the system is stable and hence by Theorem 4.2
in \cite{Dai}, the original
system is also stable.
\begin{lem} \label{lemma3}
Consider any sequence of arrivals $A_l(t), t=1,2,3\cdots,$ for all $l\in\mathcal{E}$. Then under the GFS policy, we have $q_l(t)\leq \sum_{j=1}^{J}q^{j}_{l}(t) + B,\, \forall t=1,2,3,\cdots,\text{ and } \forall \,l\in\mathcal{E}$, where $B$ is a bounded positive number.
\end{lem}
\begin{proof}
Without loss of generality, we assume $B=0$. Suppose at the beginning of time slot $t, t\geq 0$, we have $q_l(t)\leq \sum_{j=1}^{J}q^{j}_{l}(t),\, \forall \,l\in\mathcal{E}$. Let $j$ denote the network state in time slot $t$. Then, if $D_l(t)$ and $D^{j}_l(t)$ denote the packets departing in time slot $t$ from the real FIFO queue $q_l$ and the virtual queue $q^{j}_l$ respectively , the following must be true:
If $D_l(t)=D^{j}_l(t)$, then in time slot $t+1$, we have $q_l(t+1)\leq \sum_{j=1}^{J}q^{j}_{l}(t+1)$, since both $q_l(t)$ and $\sum_{j=1}^{J}q^{j}_{l}(t)$ are incremented by the same number of arrivals $A_l(t)$. Similarly, if $D_l(t)>D^{j}_l(t)$, then it again implies that $q_l(t+1)< \sum_{j=1}^{J}q^{j}_{l}(t+1)$. Finally, if $D_l(t)<D^{j}_l(t)$, it implies that $q_l(t)< \vec{r}^j_l(t)$. Consequently, $q_l(t)$ empties and $q_l(t+1)=A_l(t)\leq\sum_{j=1}^{J}q^{j}_{l}(t+1)$.
Since $q_l(t)\leq \sum_{j=1}^{J}q^{j}_{l}(t)$ is satisfied at $t=0$, we obtain the desired condition at any time $t$.
\end{proof}
Lemma 3 shows that if the virtual queues are stable then the corresponding real FIFO queue is also stable.
\end{IEEEproof}
\section{Simulation}
In this section we simulate the performance of GFS for the four link network graph shown in Fig.~\ref{fig:four-link graph}. Each link independently assumes one of four different states in each time slot, where the link states correspond to rates 1, 2 , 3 and 4 units per time slot. The probability distribution of the link states are independent and non-identical across links, with the average link rates being $\overline{c}_1=2.7,\, \overline{c}_2=2.1, \overline{c}_3=2.8,\, \text{ and } \overline{c}_4=3.1$ respectively.
In Fig.~\ref{fig:GFScomparison}, we plot the total queue sizes as we uniformly increase the arrival rate into all links. The plots show that GFS is able to sustain a load of atleast 1 unit per link. Since the network in Fig.~\ref{fig:four-link graph} has LPF value of 1, GFS can stabilize the region $\hat{\Lambda}$. GFS therefore guarantees a per-link symmetric rate of at least 1, since the arrival rate $[1\; 1\; 1\; 1]$ lies inside $\hat{\Lambda}$. While the performance of GMS is better than GFS in the plot of Fig.~\ref{fig:GFScomparison}, the current known performance guarantee of GMS is only half the network stability region $\Lambda$ under the one-hop interference model, which corresponds to a symmetric load of 0.5 per link. Based on simulations, we conjecture that the performance of GFS is a lower bound on the performance of Greedy Maximal Scheduling in time varying wireless networks. The performance guarantees for GFS thus motivates the analysis of GMS for time varying networks as our future work.
\section{conclusion}
We develop a greedy scheduler, GFS, for wireless networks with
time varying channel states and provide provable performance guarantees for
this scheduler. Our greedy scheduler, though suboptimal, has low
computational complexity and performs better than
non-opportunistic schedulers that do not exploit instantaneous channel
state information. The performance guarantees, along with simulations, also paint an optimistic picture of the performance of GMS in wireless networks with fading channels, and we conjecture the stability region guaranteed under GFS for any wireless network to be a lower bound on the stability region of GMS.
|
{
"timestamp": "2012-03-12T01:01:03",
"yymm": "1203",
"arxiv_id": "1203.2024",
"language": "en",
"url": "https://arxiv.org/abs/1203.2024"
}
|
\section{Introduction and Preliminaries}\label{S:Introduction}
In dimensions $7$ and $13$ there are two very special families of closed Riemannian manifolds, namely the Eschenburg and Bazaikin spaces. These are defined as quotients of $\operatorname{SU}(3)$ and $\operatorname{SU}(5)/\operatorname{Sp}(2)$ by free, isometric circle actions (see Section \ref{S:EschBaz} and \cite{AW}, \cite{Baz}, \cite{DE}, \cite{Es}, \cite{Ke}, \cite{Zi}), where $\operatorname{SU}(k)$ has been equipped with a left-invariant, right $\operatorname{U}(k-1)$-invariant metric. In each case there are infinitely many (distinct homotopy types of) family members admitting positive sectional curvature. Given that there are so few known examples of closed manifolds with positive sectional curvature, these families has been studied extensively (see, for example, \cite{Baz}, \cite{CEZ}, \cite{DE}, \cite{Es}, \cite{Es2}, \cite{FZ}, \cite{GSZ}, \cite{Ke}, \cite{Kr}, \cite{Zi}).
One particularly intriguing observation, made in \cite{Ta}, is that to each Bazaikin space there can be associated a totally geodesic, embedded Eschenburg space. In fact, as demonstrated in \cite{DE}, a Bazaikin space generically contains ten mutually distinct, totally geodesic, embedded Eschenburg spaces. This led to the question: given an Eschenburg space, does there exist a (non-singular) Bazaikin space containing it as a totally geodesic, embedded submanifold?
\begin{main}
\label{Thm A}
For any given Eschenburg space $E^7_{a,b}$, there exist infinitely many mutually non-homotopy equivalent Bazaikin spaces into which $E^7_{a,b}$ can be embedded as a totally geodesic submanifold.
\end{main}
In \cite{DE} it has also been proven that a Bazaikin space is positively curved if and only if each of the ten embedded, totally geodesic Eschenburg spaces it contains are also positively curved. On the other hand, it is well-known (see \cite{DE}, \cite{Zi}) that all positively curved Aloff-Wallach spaces (see \cite{AW}), that is, the subfamily of homogeneous Eschenburg spaces, and all positively curved cohomogeneity-one Eschenburg spaces (see \cite{GSZ}) can be embedded as totally geodesic submanifolds of positively curved Bazaikin spaces. This raises the question of whether this is true in general. However, at least for the known construction of a totally geodesic embedding, there exist counter-examples already in cohomogeneity-two, see Table \ref{tab:fail} in Section \ref{S:curv}.
The article is organised as follows. In Section \ref{S:Biqs} the basic notation and definitions for biquotients are reviewed. Section \ref{S:EschBaz} provides a brief summary of Eschenburg and Bazaikin spaces, while Section \ref{S:TGSubmBaz} recalls how to each Bazaikin space there correspond ten totally geodesic, embedded Eschenburg spaces. In Section \ref{S:Converse} the proof of Theorem \ref{Thm A} is given and, finally, in Section \ref{S:curv} the ability to embed a positively curved Eschenburg space as a totally geodesic submanifold of a positively curved Bazaikin space is discussed.
\section{Biquotients and induced metrics}
\label{S:Biqs}
The following section provides a review of some material from \cite{Es} which establishes the basic ideas and notation which will be used throughout the remainder of the article.
Let $G$ be a compact Lie group, $U \subset G \times G$ a closed subgroup, and let $U$ act (effectively) on $G$ via
\begin{equation}
\label{eq:action}
(u_1, u_2) \star g = u_1 g u_2^{-1}, \ \ g \in G, (u_1, u_2) \in U.
\end{equation}
The resulting quotient $G / \hspace{-.12cm} / U$ is called a {\it biquotient}. The action (\ref{eq:action}) is free if and only if, for all non-trivial $(u_1, u_2) \in U$, $u_1$ is never conjugate to $u_2$ in $G$. In the event that the action of $U$ is ineffective, the quotient $G / \hspace{-.12cm} / U$ will be a manifold so long as any element $(u_1, u_2) \in U$ which fixes some $g \in G$ fixes all of $G$, that is, $(u_1, u_2)$ lies in the ineffective kernel. A biquotient is \emph{non-singular} if the action (\ref{eq:action}) is free (modulo any ineffective kernel), and \emph{singular} otherwise.
Let $K \subset G$ be a closed subgroup. Suppose we have a biquotient $G / \hspace{-.12cm} / U$, where $U \subset G \times K \subset G \times G$ and $G$ is equipped with a left-invariant, right $K$-invariant metric $\<\, ,\, \>$. Then $U$ acts by isometries on $G$ and therefore the submersion $G \to G / \hspace{-.12cm} / U$ induces a metric on $G / \hspace{-.12cm} / U$ from the metric on $G$. We encode this in the notation $(G, \<\, ,\, \>) / \hspace{-.12cm} / U$.
Now, for $g \in G$ define
\begin{align*}
U^g_L &:= \{(g u_1 g^{-1}, u_2) \ | \ (u_1, u_2) \in U \},\\
U^g_R &:= \{(u_1, g u_2 g^{-1}) \ | \ (u_1, u_2) \in U \},\ \ {\rm and}\\
\widehat U &:= \{(u_2, u_1) \ | \ (u_1, u_2) \in U \}.
\end{align*}
Then $U^g_L, U^g_R$ and $\widehat U$ act freely on $G$, and $G / \hspace{-.12cm} / U$ is isometric to $G / \hspace{-.12cm} / U^g_L$, diffeomorphic to $G / \hspace{-.12cm} / U^g_R$ (isometric if $g \in K$), and diffeomorphic to $G / \hspace{-.12cm} / \widehat U$ (isometric if $U \subset K \times K$).
In the case of $U^g_L$ this follows from the fact that left-translation $L_g : G \to G$ is an isometry which satisfies $g u_1 g^{-1}( L_g g') u_2^{-1} = L_g (u_1 g' u_2^{-1})$. Therefore $L_g$ induces an isometry of the orbit spaces $G/ \hspace{-.12cm} / U$ and $G/ \hspace{-.12cm} / U^g_L$. Similarly we find that $R_{g^{-1}}$ induces a diffeomorphism between $G/ \hspace{-.12cm} / U$ and $G / \hspace{-.12cm} / U^g_R$, which is an isometry if $g \in K$.
Consider now $\widehat U$. The actions of $U$ and $\widehat U$ are equivariant under the diffeomorphism $\tau : G \to G$, $\tau(g) := g^{-1}$. That is, $u_1 \tau(g) u_2^{-1} = \tau(u_2 g u_1^{-1})$. Notice that this is an isometry only if $U \subset K \times K$. In general $G/ \hspace{-.12cm} / U$ and $G/ \hspace{-.12cm} / \widehat U$ are therefore diffeomorphic but not isometric.
For completeness we remark that it is, in fact, simple to equip $G$ with a left-invariant, right $K$-invariant metric. Given $K \subset G$, let $\mathfrak{k} \subset \mathfrak{g} $ be the corresponding Lie algebras and let $\<\, ,\, \>_0$ be a bi-invariant metric on $G$. One can write $\mathfrak{g} = \mathfrak{k} \oplus \mathfrak{p}$ with respect to $\<\, ,\, \>_0$. Recall that $G \cong (G \times K)/ \Delta K$ via $(g,k) \mapsto g k^{-1}$, where $\Delta K = \{(k,k) \mid k \in K\}$ acts diagonally on the right of $G \times K$. Thus we may define a left-invariant, right $K$-invariant metric $\<\, ,\, \>$ on $G$ via the Riemannian submersion
\begin{align*}
(G \times K, \<\, ,\, \>_0 \oplus t \<\, ,\, \>_0 |_\mathfrak{k}) &\to (G, \<\, ,\, \>)\\
(g,k) &\mapsto g k^{-1},
\end{align*}
where $t>0$ and
\begin{equation}
\label{met1}
\<\, ,\, \> = \<\, ,\, \>_0 |_\mathfrak{p} + \lambda \<\, ,\, \>_0 |_{\mathfrak{k}}, \ \ \lambda = \frac{t}{t+1} \in (0,1).
\end{equation}
\section{Eschenburg and Bazaikin spaces}
\label{S:EschBaz}
Given $a = (a_1, a_2, a_3)$, $b = (b_1, b_2, b_3) \in {\mathbb{Z}}^3$, with $\sum a_i = \sum b_i$, recall that the Eschenburg biquotients (see \cite{AW}, \cite{Es}) are defined as $E^7_{a,b} := (\operatorname{SU}(3), \<\, ,\, \>) / \hspace{-.12cm} / S^1_{a,b}$, where $S^1_{a,b}$ acts isometrically on $(\operatorname{SU}(3), \<\, ,\, \>)$ via
$$
z \star A =
\operatorname{diag}(z^{a_1}, z^{a_2}, z^{a_3}) \cdot A \cdot \operatorname{diag}(\bar z^{b_1}, \bar z^{b_2}, \bar z^{b_3}), \ \ A \in \operatorname{SU}(3), z \in S^1,
$$
and the left-invariant, right $\operatorname{U}(2)$-invariant metric $\<\, ,\, \>$ (with $\sec \geq 0$) on $\operatorname{SU}(3)$ is defined as in (\ref{met1}), where $\operatorname{U}(2) \hookrightarrow \operatorname{SU}(3)$ via
$$
A \in \operatorname{U}(2) \mapsto \operatorname{diag}(\overline{\det(A)}, A) \in \operatorname{SU}(3).
$$
The action is free if and only if
\begin{equation}
\label{freeness}
\gcd(a_1 - b_{\sigma(1)}, a_2 - b_{\sigma(2)}) = 1 \ \ \textrm{for all permutations} \ \ \sigma \in S_3,
\end{equation}
in which case $E^7_{a,b}$ is called an \emph{Eschenburg space}.
It is important to remark that the above defined circle subgroup $S^1_{a,b}$ is not, in general, a subgroup of $\operatorname{SU}(3) \times \operatorname{SU}(3)$. Indeed, $S^1_{a,b} \subset \SUU{3}{3} := \{(A,B) \in \operatorname{U}(3) \times \operatorname{U}(3) \mid \det A = \det B\}$. This is not a problem, however, since the bi-invariant metric on $\operatorname{SU}(3)$ can be thought of as the restriction of the the analogously defined bi-invariant metric on $\operatorname{U}(3)$. Hence, an element of $(A,B) \in \SUU{3}{3}$ maps $(\operatorname{SU}(3), \<\, ,\, \>_0)$ isometrically to itself via $X \in \operatorname{SU}(3) \mapsto A X B^{-1}$. In particular, conjugation by an element of the centre of $\operatorname{U}(3)$ is an isometry (namely, the identity map) of $(\operatorname{SU}(3), \<\, ,\, \>_0)$, and remains an isometry with respect to the new metric $\<\, ,\, \>$. Therefore the Eschenburg biquotient $E^7_{a',b'}$ defined by the action of the circle $S^1_{a',b'}$, where $a' = (a_1 + c, a_2 + c, a_3 + c)$ and $b' = (b_1 + c, b_2 + c, b_3 + c)$, with $c \in {\mathbb{Z}}$, is isometric to $E^7_{a,b}$. Furthermore, introducing an ineffective kernel to the circle action will not alter the isometry class of the biquotient. Thus $E^7_{\tilde a, \tilde b}$ defined by $\tilde a = (k a_1, k a_2, k a_3)$ and $\tilde b = (k b_1, k b_2, k b_3)$ is isometric to $E^7_{a,b}$. In particular, it follows that a circle action by $S^1_{a,b} \subset \SUU{3}{3}$ can then be rewritten as the action of a circle subgroup of $\operatorname{SU}(3) \times \operatorname{SU}(3)$ via the change of parameters $(a_1, a_2, a_3, b_1, b_2, b_3) \mapsto (3 a_1 - \kappa, 3 a_2 - \kappa, 3 a_3 - \kappa, 3 b_1 - \kappa, 3 b_2 - \kappa, 3 b_3 - \kappa)$, where $\kappa := \sum a_i = \sum b_i$, without changing the isometry class.
From Section \ref{S:Biqs} it is clear that, for the $S^1_{a,b}$-action, permuting the $a_i$ (via the action of the Weyl group of $\operatorname{SU}(3)$) and permuting $b_2, b_3$ are isometries, while permuting all of the $b_i$ and swapping $a$, $b$ are diffeomorphisms. Indeed, given our fixed choice of embedding $\operatorname{U}(2) \hookrightarrow \operatorname{SU}(3)$, cyclic permutations of the $b_i$ (and, similarly, swapping $a$ and $b$ and considering cyclic permutations of the $a_i$) induce, in general, non-isometric metrics on the quotient $E^7_{a,b}$.
It was shown in \cite{Es} that an Eschenburg biquotient $E^7_{a,b} = (\operatorname{SU}(3), \<\, ,\, \>) / \hspace{-.12cm} / S^1_{a,b}$ has positive curvature if and only if $b_i \not\in [\underline{a}, \overline{a}]$ for all $i=1,2,3$, where $\underline{a} := \min \{a_1, a_2, a_3 \}$, $\overline{a} := \max \{a_1, a_2, a_3 \}$.
\medskip
In order to define the Bazaikin spaces (see \cite{Baz}, \cite{Zi}, \cite{DE}), first let
$$
\operatorname{Sp}(2) \cdot S^1_{q_1, \dots, q_5} = (\operatorname{Sp}(2) \times S^1_{q_1, \dots, q_5})/{\mathbb{Z}}_2,\ \ \ {\mathbb{Z}}_2 = \{\pm(1, I)\},
$$
where $q_1, \dots, q_5 \in {\mathbb{Z}}$ and $\operatorname{Sp}(2)$ is considered as a subgroup of $\operatorname{SU}(4)$ via the inclusion $\operatorname{Sp}(2) \hookrightarrow \operatorname{SU}(4)$ given by
\begin{equation}
\label{eq:embed}
A = S + T j \in \operatorname{Sp}(2) \mapsto \hat A = \begin{pmatrix} S & T \\ - \bar T & \bar S \end{pmatrix} \in \operatorname{SU}(4), \ \ \ S, T \in M_2({\mathbb{C}}).
\end{equation}
Equip $\operatorname{SU}(5)$ with a left-invariant and right $\operatorname{U}(4)$-invariant metric $\<\, ,\, \>$ as defined in (\ref{met1}), where $\operatorname{U}(4) \hookrightarrow \operatorname{SU}(5)$ via $A \in \operatorname{U}(4) \mapsto \operatorname{diag}(\overline{\det A}, A) \in \operatorname{SU}(5)$.
Then $ \operatorname{Sp}(2) \cdot S^1_{q_1, \dots, q_5}$ acts effectively and isometrically on $(\operatorname{SU}(5), \<\, ,\, \>)$ via
$$[A,z] \star B = \operatorname{diag}(z^{q_1}, \dots, z^{q_5}) \cdot B \cdot \operatorname{diag}(\bar z^q, \hat A),$$
with $q := \sum q_i$, $z \in S^1$, $B \in \operatorname{SU}(5)$, and $A \in \operatorname{Sp}(2) \subset \operatorname{SU}(4)$. The quotient $B^{13}_{q_1, \dots, q_5} := (\operatorname{SU}(5), \<\, ,\, \>) / \hspace{-.12cm} / \operatorname{Sp}(2) \cdot S^1_{q_1, \dots, q_5}$ is called a Bazaikin biquotient.
It is not difficult to show that the action of $\operatorname{Sp}(2) \cdot S^1_{q_1, \dots, q_5}$ is free (hence $B^{13}_{q_1, \dots, q_5}$ is a \emph{Bazaikin space}) if and only all $q_1, \dots, q_5$ are odd and
\begin{equation}
\label{freeBaz}
\gcd(q_{\sigma(1)} + q_{\sigma(2)}, q_{\sigma(3)} + q_{\sigma(4)}) = 2 \ \ \textrm{for all permutations} \ \ \sigma \in S_5.
\end{equation}
From the discussion in Section \ref{S:Biqs} it follows that permuting the $q_i$ is an isometry of $B^{13}_{q_1, \dots, q_5}$. Furthermore, it is well-known (see \cite{Zi}, \cite{DE}) that a general Bazaikin biquotient $B^{13}_{q_1, \dots, q_5}$ admits positive curvature if and only if $q_i + q_j > 0$ (or $<0$) for all $1 \leq i < j \leq 5$.
\section{Totally geodesic submanifolds of Bazaikin spaces}
\label{S:TGSubmBaz}
Consider the Lie group $\operatorname{SU}(5)$ equipped with the left-invariant, right $\operatorname{U}(4)$-invariant metric $\<\, ,\, \>$ as described in Section \ref{S:EschBaz}. Let $\sigma : (\operatorname{SU}(5), \<\, ,\, \>) \to (\operatorname{SU}(5), \<\, ,\, \>)$ be the isometric involution defined by
\begin{equation}
\label{eq:invol}
\sigma(A) = \operatorname{diag}(1,1,1,-1,-1)\cdot A \cdot \operatorname{diag}(1,1,1,-1,-1), \ \ \ A \in \operatorname{SU}(5).
\end{equation}
The fixed point set of this involution is totally geodesic in $(\operatorname{SU}(5), \<\, ,\, \>)$ and given by
$$
\SUU{3}{2} = \{(\operatorname{diag}(B,C) \in \operatorname{U}(3) \times \operatorname{U}(2) \mid \det B = \overline{\det C} \}.
$$
Given the embedding (\ref{eq:embed}) of $\operatorname{Sp}(2)$ into $\operatorname{SU}(5)$, it follows that the intersection $\SUU{3}{2} \cap \operatorname{Sp}(2)$ is $\Delta\operatorname{U}(2) := \{(\operatorname{diag}(1,C,\bar C) \mid C \in \operatorname{U}(2) \}$.
Hence the image of $\SUU{3}{2}$ in the quotient $\operatorname{SU}(5)/\operatorname{Sp}(2)$ is the totally geodesic submanifold $\SUU{3}{2}/ \Delta\operatorname{U}(2)$. Every element of $\SUU{3}{2}/ \Delta\operatorname{U}(2)$ has a unique representative of the form
$\operatorname{diag}(B, I)$, where
\begin{equation}
\label{eq:SU(3)}
B = X \left(\begin{smallmatrix} 1 & \\ & Y^t \end{smallmatrix} \right) \in \operatorname{SU}(3), \ \ \
\textrm{with } \ X \in \operatorname{U}(3), Y \in \operatorname{U}(2) \ \
\textrm{and } \det Y = \overline{\det X}.
\end{equation}
Therefore $\SUU{3}{2}/ \Delta\operatorname{U}(2)$ is a totally geodesic copy of $\operatorname{SU}(3)$ inside $\operatorname{SU}(5)/\operatorname{Sp}(2)$.
The induced metric on $\operatorname{SU}(3) \cong \SUU{3}{2}/ \Delta\operatorname{U}(2) \subset \operatorname{SU}(5)/\operatorname{Sp}(2)$ is invariant under $\operatorname{SU}(3) \times \operatorname{U}(2)$ and, after the identification (\ref{eq:SU(3)}), one may consider this metric as a left-invariant, right $\operatorname{U}(2)$-invariant metric on $\operatorname{SU}(3)$ of the form described in Section \ref{S:EschBaz}. Indeed, the identity component of the total isometry group of $\operatorname{SU}(3) \cong \SUU{3}{2}/ \Delta\operatorname{U}(2)$ is given by
\begin{align*}
\operatorname{Isom} (\operatorname{SU}(3)) &= \left\{ \left( \begin{pmatrix} Z & \\ & W \end{pmatrix},
\begin{pmatrix} w & \\ & I \end{pmatrix}
\right)
\mid (Z, W) \in U(3) \times U(2), w = (\det Z)(\det W)
\right\}\\
&\cong \operatorname{U}(3) \times \operatorname{U}(2)
\end{align*}
and acts on $\operatorname{SU}(3) \cong \SUU{3}{2}/ \Delta\operatorname{U}(2)$ via
\begin{align}
\nonumber
\left(\begin{pmatrix} Z & \\ & W \end{pmatrix},
\begin{pmatrix} w & \\ & I \end{pmatrix} \right)
\star \left[\begin{pmatrix} B & \\ & I \end{pmatrix} \right] &=
\left[\begin{pmatrix} Z B \left(\begin{smallmatrix} \bar w & \\ & I \end{smallmatrix} \right) & \\ & W \end{pmatrix} \right]\\
\label{eq:Isomgp}
&= \left[\begin{pmatrix} ZB \left(\begin{smallmatrix} \bar w & \\ & W^t \end{smallmatrix} \right) & \\ & I \end{pmatrix} \right]
\end{align}
where $[\operatorname{diag}(B,I)] \in \operatorname{SU}(3)$, $(Z,W) \in \operatorname{U}(3) \times \operatorname{U}(2)$ and $w = (\det Z)(\det W)$.
Now, if the isometric left-action of $S^1_{q_1, \dots, q_5}$ on $\operatorname{SU}(5)/\operatorname{Sp}(2)$ is free with ineffective kernel $\{\pm 1\}$, the same must be true of the induced action on the totally geodesic submanifold $\operatorname{SU}(3) \cong \SUU{3}{2}/ \Delta\operatorname{U}(2)$. From (\ref{eq:Isomgp}) it is clear that the action on $\operatorname{SU}(3)$ is given by
$$
z \star B =
\operatorname{diag}(z^{q_1}, z^{q_2}, z^{q_3}) B \operatorname{diag}(\bar z^{q}, z^{q_4}, z^{q_5}), \ \ B \in \operatorname{SU}(3), z \in S^1,
$$
where $q = \sum q_i$. The quotient is an Eschenburg space $\operatorname{SU}(3) / \hspace{-.12cm} / S^1_{q_1, \dots, q_5}$ totally geodesically embedded in the Bazaikin space $B^{13}_{q_1, \dots, q_5}$. The circle action defining the Eschenburg space can be made effective. Indeed, since the $q_i$ are odd and from the discussion of reparameterisations in Section \ref{S:EschBaz}, it follows that the Eschenburg space $\operatorname{SU}(3) / \hspace{-.12cm} / S^1_{q_1, \dots, q_5}$ is isometric to the Eschenburg space $E^7_{a,b}$, where $a = (a_1, a_2, a_3) = (\tfrac{1}{2}(q_1 - 1), \tfrac{1}{2}(q_2 - 1), \tfrac{1}{2}(q_3 - 1))$ and $b = (b_1, b_2, b_3) = (\tfrac{1}{2}(q - 1), - \tfrac{1}{2}(q_4 + 1),- \tfrac{1}{2}(q_5 + 1))$.
It is now simple to recover the remaining nine totally geodesic, embedded Eschenburg spaces in $B^{13}_{q_1, \dots, q_5}$. As every permutation $\sigma \in S_5$ of the $q_i$ is an isometry, it follows that $B^{13}_{q_1, \dots, q_5} = B^{13}_{q_{\sigma(1)}, \dots, q_{\sigma(5)}}$ and hence, by the same reasoning as before, the Eschenburg space $E^7_{a_\sigma,b_\sigma}$, where $a_\sigma = (\tfrac{1}{2}(q_{\sigma(1)} - 1), \tfrac{1}{2}(q_{\sigma(2)} - 1), \tfrac{1}{2}(q_{\sigma(3)} - 1))$ and $b_\sigma = (\tfrac{1}{2}(q - 1), - \tfrac{1}{2}(q_{\sigma(4)} + 1),- \tfrac{1}{2}(q_{\sigma(5)} + 1))$, is a totally geodesic, embedded submanifold. That generically there are ten such submanifolds follows since permutations of the entries of $a_\sigma$, and of the last two entries of $b_\sigma$, are isometries. This, of course, is equivalent to fixing the order of the $q_i$ and permuting the signs of the entries along the diagonal in the involution (\ref{eq:invol}) acting on $\operatorname{SU}(5)$, thus achieving ten (isometric) copies of $\operatorname{SU}(3)$ which quotient to the desired Eschenburg spaces (see \cite{DE}).
\section{Totally geodesic embeddings of Eschenburg spaces}
\label{S:Converse}
In Section \ref{S:TGSubmBaz} it was shown that every Bazaikin space contains a totally geodesically embedded Eschenburg space. The converse statement, namely that every Eschenburg space can be totally geodesically embedded into a Bazaikin space, is also true. Indeed, it will now be shown that every Eschenburg space can be totally geodesically embedded into infinitely many Bazaikin spaces.
Let $E^7_{a,b}$ be the Eschenburg space given by $a = (a_1, a_2, a_3)$, $b = (b_1, b_2, b_3) \in {\mathbb{Z}}^3$ with $\sum a_i = \sum b_i$ and satisfying the freeness condition (\ref{freeness}):
$$
\gcd(a_1 - b_{\sigma(1)}, a_2 - b_{\sigma(2)}) = 1 \ \ \textrm{for all permutations} \ \ \sigma \in S_3.
$$
By the discussion in Section \ref{S:TGSubmBaz}, a candidate for a Bazaikin space into which to embed is given by the $5$-tuple $(q_1, \dots, q_5) := (2 a_1 + 1, 2 a_2 + 1, 2 a_3 + 1, -(2 b_2 + 1), -(2 b_3 + 1))$, with $q = \sum q_i = 2 b_1 + 1$.
As each of the $q_i$ is odd, one need only check the condition (\ref{freeBaz}):
$$
\gcd(q_{\sigma(1)} + q_{\sigma(2)}, q_{\sigma(3)} + q_{\sigma(4)}) = 2 \ \ \textrm{for all permutations} \ \ \sigma \in S_5.
$$
It follows from the Eschenburg freeness condition (\ref{freeness}) (and from $\sum a_i = \sum b_i$) that $\gcd(q_4 + q_i, q_5 + q_j) = 2$ for all $i \neq j \in \{1,2,3\}$. Furthermore, if $\{i,j,k\} = \{1,2,3\}$ and $\{\ell,m\} = \{4,5\}$, then since $q = \sum q_i$ it follows that
\begin{align*}
\gcd(q_i + q_j, q_k + q_\ell) &= \gcd(q_i + q_j, q - q_m)\\
&= \gcd(q - q_m, q_k + q_\ell)\\
\textrm{and that }\ \ \gcd(q_i + q_j, q_4 + q_5) &= \gcd(q_i + q_j, q - q_k).
\end{align*}
Hence the $5$-tuple $(q_1, \dots, q_5) = (2 a_1 + 1, 2 a_2 + 1, 2 a_3 + 1, -(2 b_2 + 1), -(2 b_3 + 1))$ defines a Bazaikin space if and only if
\begin{equation}
\label{eq:NonSing}
d_{k \ell} := \gcd(a_i + a_j + 1, a_k - b_\ell) = 1 \ \ \textrm{for all} \ \ \ell \in \{i,j,k\} = \{1,2,3\}.
\end{equation}
\begin{lem}
\label{lem:oddprimes}
Let $a = (a_1, a_2, a_3), b = (b_1, b_2, b_3) \in {\mathbb{Z}}^3$ satisfy (\ref{freeness}) and $\sum a_i = \sum b_i$. Suppose that $p$ is a prime divisor of $\gcd(a_i + a_j + 1, a_k - b_\ell)$, where $\ell \in \{i,j,k\} = \{1,2,3\}$. Then $p$ is odd.
\end{lem}
\begin{proof}
By (\ref{freeness}) together with $\sum a_i = \sum b_i$ it follows that either all $a_i$ or all $b_i$ have the same parity. Hence either all $a_i + a_j + 1$ or all $b_m + b_n + 1$ are odd, where $1 \leq i < j \leq 3$, $1 \leq m < n \leq 3$. Since $\gcd(a_i + a_j + 1, a_k - b_\ell) = \gcd(b_m + b_n + 1, a_k - b_\ell)$, where $\{i,j,k\} = \{\ell,m,n\} = \{1,2,3\}$, the conclusion follows.
\end{proof}
Recall now that one can (isometrically) rewrite the Eschenburg space $E^7_{a,b}$ as $E^7_{a_c,b_c}$, where $a_c = (a_1 + c, a_2 + c, a_3 + c)$ and $b_c = (b_1 + c, b_2 + c, b_3 + c)$, for any $c \in {\mathbb{Z}}$. In this case the candidate Bazaikin biquotient is given by the $5$-tuple $(q^c_1, \dots, q^c_5) := (2 (a_1 + c) + 1, 2 (a_2 + c) + 1, 2 (a_3 + c) + 1, -(2 (b_2 + c) + 1), -(2 (b_3 + c) + 1))$, with $q^c = \sum q^c_i = 2 (b_1 + c) + 1$. Analogously to above, this Bazaikin biquotient is non-singular if and only if
\begin{equation}
\label{eq:modNonSing}
d^c_{k \ell} := \gcd(a_i + a_j + 1 + 2c, a_k - b_\ell) = 1 \ \ \textrm{for all} \ \ \ell \in \{i,j,k\} = \{1,2,3\}.
\end{equation}
\begin{lem}
\label{lem:free}
Let $a = (a_1, a_2, a_3), b = (b_1, b_2, b_3) \in {\mathbb{Z}}^3$ satisfy (\ref{freeness}) and $\sum a_i = \sum b_i$. For all $\ell \in \{i,j,k\} = \{1,2,3\}$, let $p_{k\ell 1}, \dots, p_{k\ell r_{k \ell}}$ be the prime divisors, if any, of $a_k - b_\ell$ which satisfy $\gcd(p_{k \ell \tau}, a_i + a_j + 1) = 1$, for all $1 \leq \tau \leq r_{k \ell}$. Set
\begin{equation}
\label{eq:cvalue}
c = c_\mu := \pm 2^{\mu - 1} \left(\prod_{k, \ell = 1}^3 \prod_{\tau = 1}^{r_{k \ell}} p_{k \ell \tau} \right)^\mu,
\end{equation}
where $\mu$ is an arbitrary positive integer. Then $d^c_{k \ell} = \gcd(a_i + a_j + 1 + 2c, a_k - b_\ell) = 1$.
\end{lem}
\begin{proof}
Suppose that $p$ is a prime divisor of $d^c_{k \ell}$. As $p$ divides $a_k - b_\ell$, then $p$ divides either $a_i + a_j + 1$ or $c$. But $p$ divides $a_i + a_j + 1 + 2c$, hence must divide both $a_i + a_j + 1$ and $c$, and furthermore must be odd by Lemma \ref{lem:oddprimes}. Now, by definition of $c$, $p$ must divide $a_t - b_u$, for some $(t,u) \neq (k, \ell)$, and must satisfy $\gcd(p, a_r + a_s + 1) = 1$, where $\{r,s,t\} = \{1,2,3\}$. However, because of the freeness condition (\ref{freeness}) for Eschenburg spaces, either $t = k$ or $u = \ell$. If $t = k$, then $1 = \gcd(p, a_r + a_s + 1) = \gcd(p, a_i + a_j + 1) = p$. On the other hand, if $u = \ell$ then $k \neq t \in \{1,2,3\}$, that is, it may assumed without loss of generality that $(t,u) = (i,\ell)$. As $p$ divides $a_i + a_j + 1$, $a_k - b_\ell$ and $a_i - b_\ell$, it follows that $1 = \gcd(p, a_r + a_s + 1) = \gcd(p, a_j + a_k + 1) = \gcd(p, (a_i + a_j + 1) + (a_k - b_\ell) - (a_i - b_\ell)) = p$.
\end{proof}
\begin{cor}
\label{cor:embed}
With notation as above, for all $c = c_\mu$, $\mu \in {\mathbb{Z}}$, $\mu > 0$, the Bazaikin biquotient $B^{13}_{q^c_1, \dots, q^c_5}$ is non-singular and contains $E^7_{a,b} = E^7_{a_c,b_c}$ as a totally geodesic submanifold.
\end{cor}
\begin{rem}
The conditions (\ref{eq:NonSing}) and (\ref{freeness}) ensuring that the Bazaikin biquotient is non-singular are precisely the conditions which ensure that all ten of the totally geodesic, embedded Eschenburg biquotients are themselves non-singular. The equivalence of the non-singularity of the Bazaikin biquotient and these ten submanifolds was already observed in \cite{DE}.
\end{rem}
\begin{rem}
Given a Bazaikin space $B^{13}_{q_1, \dots, q_5}$, with $(q_1, \dots, q_5) = (2 a_1 + 1, 2 a_2 + 1, 2 a_3 + 1, -(2 b_2 + 1), -(2 b_3 + 1))$ and $q = \sum q_i = 2 b_1 + 1$, into which the Eschenburg space $E^7_{a, b}$ has a totally geodesic embedding, that is, condition (\ref{eq:NonSing}) holds, then the diffeomorphic, but in general non-isometric, Eschenburg space $E^7_{b,a}$ totally geodesically embeds into the Bazaikin space $B^{13}_{q, -q_4, -q_5, -q_2, -q_3}$. This follows easily from condition (\ref{eq:NonSing}) together with $\sum a_i = \sum b_i$. The Bazaikin spaces $B^{13}_{q_1, \dots, q_5}$ and $B^{13}_{q, -q_4, -q_5, -q_2, -q_3}$ are diffeomorphic but not isometric in general (see \cite{EKS}, \cite{FZ}).
\end{rem}
The integral cohomology rings of Eschenburg spaces and Bazaikin spaces are well-known (see \cite{Es2}, \cite{CEZ}, \cite{Baz}, \cite{FZ}). In particular, if $\sigma_i (x) := \sigma_i (x_1, \dots, x_m)$ denotes the $i^{\rm th}$ symmetric polynomial in $x = (x_1, \dots, x_m)$, then $H^4(E^7_{a,b}) = {\mathbb{Z}}_r$, where $r = |\sigma_2(a) - \sigma_2(b)|$ is always odd \cite[Remark 1.3]{Kr}, and $H^6(B^{13}_{q_1, \dots, q_5}) = {\mathbb{Z}}_s$, where $s = \tfrac{1}{8}|\sigma_3(q_1, q_2, q_3, q_4, q_5, -\sum q_i)|$.
\begin{lem}
\label{lem:order}
Let $a:=(a_1, a_2, a_3), b:=(b_1, b_2, b_3) \in {\mathbb{Z}}^3$, with $\sum a_i = \sum b_i$, and for $c \in {\mathbb{Z}}$ define the $6$-tuple
$$
q_c := (2(a_1 + c) + 1, 2(a_2 + c) + 1, 2(a_3 + c) + 1, -2(b_1 + c) - 1, -2(b_2 + c) - 1, -2(b_3 + c) - 1).
$$
If $c \neq d \in {\mathbb{Z}}$, then $|\sigma_3(q_c)| = |\sigma_3(q_d)|$ if and only if either $\sigma_2(a) = \sigma_2(b)$ or
$$
c+d = \frac{\sigma_3(a)-\sigma_3(b)}{\sigma_2(a) - \sigma_2(b)} - \sigma_1(a) - 1.
$$
\end{lem}
\begin{proof}
Recall that $\sigma_i(x)$, the $i^{\rm th}$ symmetric polynomial in $x = (x_1, \dots, x_m)$, is defined as the coefficient of $y^{m-i}$ in the product $\prod_{j=1}^m (y + x_j)$. Then, with $y_+ = y + 1 + 2c$ and $y_- = y - 1 - 2c$, $\sigma_3 (q_c)$ is given by the coefficient of $y^3$ in the product
\begin{align*}
\prod_{j=1}^3 (y + 2a_j + 1 + 2c) \prod_{j=1}^3 (y - 2b_j - 1 - 2c)
&= \prod_{j=1}^3 (y_+ + 2a_j) \prod_{j=1}^3 (y_- -2 b _j) \\
&= \left(\sum_{j = 0}^3 \sigma_j (a) y_+^{3-j} \right) \left(\sum_{j = 0}^3 \sigma_j (b) y_-^{3-j} \right).
\end{align*}
Hence, since $\sigma_1(a) = \sum a_j = \sum b_j = \sigma_1(b)$,
$$
\sigma_3 (q_c) = 8(\sigma_3(a)-\sigma_3(b)) - 8(\sigma_1(a) + 2c+1)(\sigma_2(a)-\sigma_2(b)).
$$
Now $|\sigma_3(q_c)| = |\sigma_3(q_d)|$ if and only if $\sigma_3(q_c) = \pm \sigma_3(q_d)$, that is, if and only if either
\begin{align*}
(c - d)(\sigma_2(a)-\sigma_2(b)) &= 0 \ \ \textrm{ or} \\
(c + d)(\sigma_2(a)-\sigma_2(b)) &= (\sigma_3(a)-\sigma_3(b)) - (\sigma_1(a) + 1)(\sigma_2(a)-\sigma_2(b)),
\end{align*}
from which the claim follows.
\end{proof}
\begin{cor}
\label{cor:topology}
The collection of Bazaikin spaces into which a particular Eschenburg space can be totally geodesically embedded consists of infinitely many distinct homotopy types.
\end{cor}
\begin{proof}
Given an Eschenburg space $E^7_{a,b}$ and $c$ of the form (\ref{eq:cvalue}), it has been shown that there is a totally geodesic embedding of $E^7_{a,b}$ into each Bazaikin space $B^{13}_{q^c_1, \dots, q^c_5}$, where $(q^c_1, \dots, q^c_5) = (2 a_1 + 2c + 1, 2 a_2 + 2c + 1, 2 a_3 + 2c + 1, -2 b_2 - 2c - 1, -2 b_3 -2 c - 1)$. Fix one such value of $c$, that is, fix some $\mu > 0$. Since $2^{\mu-1} > 1$ for $\mu > 1$, we may assume without loss of generality that $|c| > 1$. Let $c_1 = c^{\alpha_1}$ and $c_2 = c^{\alpha_2}$ for some $0 < \alpha_1 < \alpha_2$. With the same notation as in Lemma \ref{lem:order}, the order of $H^6$ for the Bazaikin space corresponding to $c_i$ is given by $s_i = \tfrac{1}{8} |\sigma_3(q_{c_i})|$, for $i = 1,2$ respectively. By Lemma \ref{lem:order}, $s_1 = s_2$ if and only if either $\sigma_2(a) - \sigma_2(b) = 0$ or
\begin{equation}
\label{eq:diffcohom}
c_1 + c_2 = \frac{\sigma_3(a)-\sigma_3(b)}{\sigma_2(a) - \sigma_2(b)} - \sigma_1(a) - 1.
\end{equation}
But, by \cite[Remark 1.3]{Kr}, $\sigma_2(a) - \sigma_2(b)$ is odd, hence non-zero. Therefore (\ref{eq:diffcohom}) must hold. However, as the right-hand side is constant, it is clear that (\ref{eq:diffcohom}) can hold for at most one pair $0 < \alpha_1 < \alpha_2$.
\end{proof}
The following example shows that the $c$ values given in Lemma \ref{lem:free} do not achieve all Bazaikin spaces into which an Eschenburg space can be totally geodesically embedded.
\begin{eg}
\label{eg:cohom2}
Consider the positively curved Eschenburg space $E^7_{a,b}$ given by $a = (2,0,0)$ and $b = (15, -2, -11)$. The corresponding candidate for a Bazaikin space is given by $q = (5, 1, 1, 3, 21)$, which is singular because $\gcd(q_1 + q_2, q_4 + q_5) = 6$.
By the expression in (\ref{eq:cvalue}), $E^7_{a,b}$ can be totally geodesically embedded into each of the Bazaikin spaces given by $(5 + 2c, 1 + 2c, 1 + 2c, 3 - 2c, 21 - 2c)$, with $c = \pm 2^{\mu - 1}(2^3 \cdot 5^2 \cdot 11^2 \cdot 13^2)^\mu$, for $\mu > 0$. None of these spaces is positively curved, since it is clear that any value of $\mu > 0$ leads to $q_i + q_j$ of mixed signs.
On the other hand, note that when $c = -1$ the Eschenburg space $E^7_{a,b}$ can be rewritten as $E^7_{\tilde a, \tilde b}$, with $\tilde a = (1, -1, -1)$ and $\tilde b = (14, -3, -12)$, and the corresponding Bazaikin biquotient $B^{13}_{\tilde q}$ with $\tilde q = (3, -1, -1, 5, 23)$ is non-singular. Note that $B^{13}_{\tilde q}$ does not have positive curvature.
However, when $c = 2$ and when $c = 5$, the resulting biquotients are both non-singular and positively curved, namely the Bazaikin spaces $B^{13}_{q'}$, with $q' = (9, 5, 5, -1, 17)$ (corresponding to $a' = (4, 2, 2)$, $b' = (17, 0, -9)$), and $B^{13}_{q''}$, with $q'' = (15, 11, 11, -7, 11)$ (corresponding to $a' = (7, 5, 5)$, $b' = (20, 3, -6)$).
Finally, notice that the order of $H^6$ is $503$, $1541$ and $2579$ for $B^{13}_{\tilde q}$, $B^{13}_{q'}$ and $B^{13}_{q''}$ respectively, hence these spaces are not even homotopy equivalent.
\end{eg}
\section{Embeddings inducing positive sectional curvature}
\label{S:curv}
Given that every Eschenburg space can be totally geodesically embedded into infinitely many Bazaikin spaces, it is natural to ask whether a positively curved Eschenburg space admits a totally geodesic embedding into a positively curved Bazaikin space.
Recall that an Eschenburg space $E^7_{a,b}$ has positive curvature if and only if $b_i \not\in [\underline{a}, \overline{a}]$ for all $i=1,2,3$, where $\underline{a} := \min \{a_1, a_2, a_3 \}$, $\overline{a} := \max \{a_1, a_2, a_3 \}$. Because $\sum a_j = \sum b_j$, this is equivalent to the requirement that two of the $b_i$ lie on one side of $[\underline{a}, \overline{a}]$, and one on the other. Indeed, given the metric on $E^7_{a,b}$ defined in Section \ref{S:EschBaz}, it turns out that $b_2$ and $b_3$ must lie on the same side of $[\underline{a}, \overline{a}]$, see \cite{Es}, \cite{Ke}. Since permuting the $a_i$ and permuting $b_2$ and $b_3$ are isometries, the condition for positive curvature is, after relabelling if necessary, equivalent to
\begin{equation}
\label{poscond}
b_3 \leq b_2 < a_3 \leq a_2 \leq a_1 < b_1 \ \
\textrm{ or } \ \
b_1 < a_1 \leq a_2 \leq a_3 < b_2 \leq b_3.
\end{equation}
In fact, the second chain of inqualities is equivalent to the first via the reparametrization of $S^1_{a,b}$ via $z \mapsto \bar z$, and therefore one may restrict attention to the first chain.
Now $E^7_{a,b}$ can be totally geodesically embedded into the (possibly singular) Bazaikin biquotient $B^{13}_{q_1, \dots, q_5}$, with $(q_1, \dots, q_5) = (2 a_1 + 1, 2 a_2 + 1, 2 a_3 + 1, -(2 b_2 + 1), -(2 b_3 + 1))$. As discussed in Section \ref{S:EschBaz}, $B^{13}_{q_1, \dots, q_5}$ has positive curvature if and only if $q_i + q_j > 0$ (or $<0$) for all $1 \leq i < j \leq 5$. Since the first chain of inequalities in (\ref{poscond}) implies that $q_i + q_j = 2(a_i - b_{j-2}) > 0$, for all $i = 1,2,3$, $j = 4,5$, it follows that $B^{13}_{q_1, \dots, q_5}$ has positive curvature if $a_i + a_j + 1 > 0$, for all $1 \leq i < j \leq 3$, and $b_2 + b_3 + 1 < 0$. But $a_3 \leq a_2 \leq a_1$, hence $B^{13}_{q_1, \dots, q_5}$ has positive curvature if and only if
\begin{equation}
\label{eq:Bazcurv}
b_2 + b_3 + 1 < 0 < a_2 + a_3 + 1.
\end{equation}
In particular, it is necessary that $0 \leq a_2 \leq a_1 < b_1$ and $b_3 \leq -1$. Clearly the special case $b_2 < 0 \leq a_3$ ensures that the inequalities in (\ref{eq:Bazcurv}) are satisfied.
If $B^{13}_{q_1, \dots, q_5}$ does not have positive curvature, then one can find a $c \in {\mathbb{Z}}$ such that the Bazaikin biquotient $B^{13}_{q^c_1, \dots, q^c_5}$, with
$$
(q^c_1, \dots, q^c_5) :=
(2 (a_1 + c) + 1, 2 (a_2 + c) + 1, 2 (a_3 + c) + 1, -(2 (b_2 + c) + 1), -(2 (b_3 + c) + 1)),
$$
is positively curved. Indeed, $B^{13}_{q^c_1, \dots, q^c_5}$ has positive curvature if and only if $c \in Z$ satisfies
\begin{equation}
\label{eq:Bazcurvmod}
-\frac{1}{2}(a_2 + a_3 + 1) < c < -\frac{1}{2}(b_2 + b_3 + 1).
\end{equation}
It then remains only to examine the values of $c$ given by (\ref{eq:Bazcurvmod}) to determine which, if any, of the Bazaikin biquotients $B^{13}_{q^c_1, \dots, q^c_5}$ are non-singular.
One particular example, where an embedding into a positively curved Bazaikin space exists, is that of a positively curved Eschenburg space of cohomogeneity-one, given by $a = (p, 1, 1)$, $b = (p+2, 0, 0)$, with $p \geq 1$. Here (\ref{eq:Bazcurvmod}) yields $-1 \leq c \leq 0$. Indeed, $c = -1$ ensures that $B^{13}_{q^c_1, \dots, q^c_5}$ is non-singular and positively curved. This was first observed by W.\ Ziller and appeared in \cite{DE}.
Note that in this example $c = - a_3 = -1$. Indeed, for an arbitrary Eschenburg space $E^7_{a,b}$, the choice $c = -a_3$ ensures $B^{13}_{q^c_1, \dots, q^c_5}$ has positive curvature, although in general one cannot hope that this space will be non-singular. Example \ref{eg:cohom2} illustrates this phenomenon and, furthermore, that the values of $c$ suggested by the expression (\ref{eq:cvalue}) are not of much use when it comes to finding a Bazaikin space of positive curvature.
It turns out, in fact, that there are examples of positively curved Eschenburg spaces for which none of the values of $c$ coming from (\ref{eq:Bazcurvmod}) yield a non-singular Bazaikin biquotient, that is, a totally geodesic embedding constructed as in Section \ref{S:Converse} cannot be into a positively curved Bazaikin space. A list of such examples is given in Table \ref{tab:fail}. Notice, in particular, that the first and last examples in Table \ref{tab:fail} are Eschenburg spaces of cohomogeneity-two (see \cite{GSZ}). In fact, there are infinitely many such cohomogeneity-two examples, for example, the infinite families given by $a = (15015k + 39, 0, 0)$, $b = (15015k + 55, -3, -13)$, $k \geq 0$, and $a = (15015k + 12909, 0, 0)$, $b = (15015k + 12925, -3, -13)$, $k \geq 0$, respectively.
\begin{rem}
\label{rem:reduce}
One might hope that introducing an ineffective kernel into the $S^1_{a,b}$ action defining a positively curved Eschenburg space $E^7_{a,b}$ would yield a Bazaikin space with positive curvature into which it can be totally geodesically embedded. However, such a modification reduces to the case discussed above. Indeed, if $\tilde a = (\lambda a_1 + d,\lambda a_2 + d, \lambda a_3 + d)$ and $\tilde b = (\lambda b_1 + d, \lambda b_2 + d, \lambda b_3 + d)$, where without loss of generality $\lambda > 0$, then, by (\ref{freeness}), $\gcd(\tilde a_i - \tilde b_\ell, \tilde a_j - \tilde b_m) = \lambda$. Hence the candidate for a positively curved Bazaikin space $B^{13}_{\tilde q}$ is given by $\tilde q = (2 \tilde a_1 + 1, 2 \tilde a_2 + 1, 2 \tilde a_3 + 1, -(2 \tilde b_2 + 1), -(2 \tilde b_3 + 1))$, where $\gcd(\tilde a_i + \tilde a_j + 1, \tilde a_k - \tilde b_\ell) = \lambda$, for all $\ell \in \{i,j,k\} = \{1,2,3\}$, and $\tilde a_2 + \tilde a_3 + 1 > 0 > \tilde b_2 + \tilde b_3 + 1$. Consequently $\lambda$ must be odd and there is $c \in {\mathbb{Z}}$ such that $2 d + 1 = \lambda (2 c + 1)$, in which case $B^{13}_{\tilde q}$ is non-singular if and only if (\ref{freeness}) and (\ref{eq:modNonSing}) hold, and has positive curvature if and only if (\ref{eq:Bazcurvmod}) is satisfied.
\end{rem}
\begin{table}[!ht]
\begin{tabular}{|c|c|c|c|} \hline
\multicolumn{2}{|c|}{$\mathbf{ E^7_{a,b}}$} & \multicolumn{2}{|c|}{$\mathbf{ B^{13}_{q^c_1, \dots, q^c_5}}$}\\ \hline
$\mathbf{ a}$ & $\mathbf{ b}$ & $\mathbf{ (q^c_1, \dots, q^c_5)}$ & $\mathbf{ \sec > 0}$ \\ \hline \hline
$(39, 0, 0)$ & $(55, -3, -13)$ & $(79 + 2c, 1 + 2c, 1 + 2c, 5 - 2c, 25 - 2c)$ & $0 \leq c \leq 7$ \\ \hline
$(77, 2, 0)$ & $(93, -3, -11)$ & $(155 + 2c, 5 + 2c, 1 + 2c, 5 - 2c, 21 - 2c)$ & $-1 \leq c \leq 6$ \\ \hline
$(171, 2, 0)$ & $(187, -3, -11)$ & $(343 + 2c, 5 + 2c, 1 + 2c, 5 - 2c, 21 - 2c)$ & $-1 \leq c \leq 6$ \\ \hline
$(225, 4, 0)$ & $(247, -5, -13)$ & $(451 + 2c, 9 + 2c, 1 + 2c, 9 - 2c, 25 - 2c)$ & $-2 \leq c \leq 8$ \\ \hline
$(281, 3, 0)$ & $(294, -2, -8)$ & $(563 + 2c, 7 + 2c, 1 + 2c, 3 - 2c, 15 - 2c)$ & $-1 \leq c \leq 4$ \\ \hline
$(309, 6, 0)$ & $(323, -3, -5)$ & $(619 + 2c, 13 + 2c, 1 + 2c, 5 - 2c, 9 - 2c)$ & $-3 \leq c \leq 3$ \\ \hline
$(664, 2, 0)$ & $(678, -3, -9)$ & $(1329 + 2c, 5 + 2c, 1 + 2c, 5 - 2c, 17 - 2c)$ & $-1 \leq c \leq 5$ \\ \hline
$(827, 4, 0)$ & $(843, -3, -9)$ & $(1655 + 2c, 9 + 2c, 1 + 2c, 5 - 2c, 17 - 2c)$ & $-2 \leq c \leq 5$ \\ \hline
$(12909, 0, 0)$ & $(12925, -3, -13)$ & $(25819 + 2c, 1 + 2c, 1 + 2c, 5 - 2c, 25 - 2c)$ & $0 \leq c \leq 7$ \\ \hline
\end{tabular}
\vspace{.1cm}
\caption{Eschenburg spaces which do not totally geodesically embed into a positively curved, non-singular Bazaikin space via the construction of Section \ref{S:Converse}.}
\label{tab:fail}
\end{table}
|
{
"timestamp": "2012-03-12T01:02:02",
"yymm": "1203",
"arxiv_id": "1203.2124",
"language": "en",
"url": "https://arxiv.org/abs/1203.2124"
}
|
\section{Introduction}\label{s:intro}
The stellar spectra even at modest resolution contain wealth of information
on stellar parameters. In fact, most of the classification work has been
done using medium/low resolution spectra. Hydrogen lines are good indicators
of temperature and luminosity for a good range in spectral types; although for hotter
end stars lines of neutral and ionized helium, carbon and nitrogen
are used while strengths
of molecular features are employed for the cool stars. A recent summary of the
advances in classification can be found in Giridhar (2010).
Additional features such as near IR triplet at 7771-74\AA~ and Ca II lines
in 8490-8670 \AA~ region are also used for luminosity calibration.
Many large telescopes are now equipped with
multi-object spectrometers enabling coverage of a large
number of objects per frame for stellar systems like
clusters. Instruments such as 6df on the UK Schmidt
telescope and AAOMEGA at the AAT can provide very
large number of spectra per night. On-going and future
surveys, and space missions would collect a large number
of spectra for stars belonging to different components of
our Galaxy. Such large volume of data can be handled
only with automatic procedures which would also have the
advantage of being objective and providing homogeneous
data set most suited for Galactic structure and
evolutionary studies. Another outcome would be detection
of stellar variability and finding of peculiar objects.
\section{ Automated methods for parametrization}\label{s: Automated methods}
Several methods have been developed to estimate atmospheric parameters from
medium-resolution stellar spectra in a fast, automatic, objective fashion.
The most commonly adopted approaches are based upon
the minimum distance method (MDM) and
those using Artificial Neural Network or ANN.
Both the approaches use reference libraries to make comparison
with object spectra. Other methods use
correlations between broadband colors or the strength
of prominent metallic lines and the atmospheric parameters
e.g. Stock and Stock (1999).
\subsection{Comparison between empirical and synthetic libraries}
The observed stellar spectra are assigned a given spectral type and Luminosity Class (LC)
based upon the appearance of spectral features and hence these classifications
are not model dependent.
Synthetic spectra depend on model atmospheres mostly assuming
local thermodynamic equilibrium (LTE), are affected by inadequacy of atomic
and molecular database and non-LTE effects are severe for certain temperature/metallicity
domain. Empirical spectra however may not have the required uniform range in
the parameter space.
\subsection { MDM based approaches }
The basic concept is to minimize the distance metric between the reference
spectrum and spectrum to be classified/parametrized. The accuracy depends
upon the density of reference spectra in parameter space.
We need to construct a stellar spectral template library for
stars of known parameters. The software
TGMET developed by Katz et al. (1998) is based upon direct comparison
with a reference library of stellar spectra. Soubiran et al (2003) used
this approach to estimate the T$_{\rm eff}$ , log~$g$ and [Fe/H] with
very good accuracy 86 K, 0.28 dex and 0.16 dex respectively
for good S/N ratio spectra of F, G and K stars. Instead of reference spectra
synthetic spectra using the model atmospheres were used by
Zwitter et al.(2008) and others. In SPADES (Posbic et al. 2011) the comparison
is made of specific lines allowing abundance determination of various elements.
\subsection { Artificial Neural Network }
A very good account of this approach can be found in numerous papers
e.g. Bailer-Jones (2002), von Hippel (1994) and others. It is a computational
method which can provide non-linear mapping between the input vector
(a spectrum for example) and one or more outputs like T$_{eff}$, log~$g$ and [M/H].
A network need to be trained with the help of spectra of stars of known parameters.
The trained network is used to parametrize the unclassified spectra.
We have used the back-propagation ANN code by Ripley (1993).
The chosen configuration of ANN is described in Giridhar, Muneer and Goswami (2006).
\section{ Analysis of VBT spectra}
We had initiated a modest survey program for exploration of metal-poor
candidate stars from HK Survey (Beers, Shectmann and Preston 1992), EC survey
(Stobie et al. 1997) and high proper motion list of Lee (1984).
The semi-empirical approach based upon the strengths of prominent lines
and line ratio adopted in Giridhar and Goswami (2002) resulted in detection
of several new metal-poor stars. We therefore chose to explore the use
of ANN on a larger sample of candidate metal-poor stars.
The medium resolution spectra (R$\sim$2000) were obtained
using OMR spectrometer with 2.3m telescope at VBO, Kavalur.
The spectra cover 3800-6000\AA~ region. Our spectral analysis, alignment
procedure etc. are described in Giridhar, Muneer and Goswami (2006).
A few representative spectra arranged in increasing temperatures
are presented in (Fig. \ref{sample_spectra}).
\begin{figure}
\centerline{\includegraphics[height=9.8cm,width=12.5cm]{sample_spectra.eps}}
\caption[]{Sample spectra arranged in temperature sequence are presented.
The stars with normal metallicity are plotted as dotted lines while metal-poor stars
are shown as continuous lines.}
\label{sample_spectra}
\end{figure}
\subsection { Calibration accuracies of stellar parameters}
Our training set containing 143 stars of known atmospheric parameters
were chosen from Allende Prieto \& Lambert (1999), Gray, Grahm \& Hoyt(2001),
Snider et al. (2001) and ELODIE data base (Soubiran, Karz \& Cayrel 1998).
\begin{figure}
\centerline{\includegraphics[height=6.8cm]{Temp_metal.eps} \quad
\includegraphics[height=6.8cm]{Grav_Mv.eps}}
\caption[]{The parameters estimated from ANN are compared with those
from literature.}
\label{Temp_metal}
\end{figure}
Figure 2 shows the ANN results compared with calibrating values.
We have shown in Figure 2a [Fe/H] ANN results for 76 calibrating stars plotted against
those from literature. For the metallicity range of $-$3.0 to $+$0.3 dex. the RMS scatter
about the line of unity is 0.3 dex which is similar to the intrinsic uncertainties
metallicities for calibrating stars.
To avoid using the same spectra for training and testing
purposes, we divided the training set into two parts and trained ANN for each part.
Then the weights for part~1 were used to estimate [Fe/H] for stars in part~2 while
those of part~2 were used to estimate [Fe/H] for stars in part ~1. The errors
shown in the Figure 2 are therefore realistic estimate of errors.
This approach of dividing calibrating sample into two separate training
and testing sets has been adopted for T$_{\rm eff}$ and log~$g$ calibration also.
We had good T$_{\rm eff}$ and log~$g$ estimates for 143 stars for calibrations
and among them 110 stars had nearly solar metallicities while 33 were hard core metal-poor
stars. While training the networks for temperature we found that usage of the same ANN
for normal metallicity stars as well as metal-poor stars was giving large calibration errors
(250 to 300K for T$_{\rm eff}$).
It is understandable as the spectra of metal-poor stars and also those of hot stars have
weak metallic lines. To overcome this degeneracy
we used separate ANNs for each metallicity subgroup for the temperature (as well as gravity)
calibration. The temperatures estimated by AAN are
compared with the literature values in Figure 2b. The RMS error is now reduced to 150K.
The Figure 2c shows the result for gravity calibration adopting the
procedure mentioned above.
The RMS error is about 0.35 for log ~$g$ range of 1 to $+$4.5 dex.
A large fraction of stars observed by us have good parallax estimates (errors less than 20\%).
Combining the V magnitudes with parallaxes the distances and hence M$_{V}$ could be
estimated. Most of these objects were nearby objects so the effect of
interstellar extinction could be assumed as negligible.
Our spectral region contains many luminosity sensitive features like hydrogen lines,
Mg I lines at 5172-83\AA~, G bands etc.
However, the same feature cannot serve the whole range of spectral types. We have divided
the sample stars into two temperature groups and yet another group for metal-poor objects.
The usage of three separate networks helped in attaining calibration accuracy
$\sim$ $\pm$0.3 mag for M$_{V}$. The M$_{V}$ estimated by ANN are compared with those
estimated from parallaxes as shown in Figure 2d.
\section{ Stellar parameters for metal-poor candidate stars}
A different set of
ANNs for each atmospheric parameter were trained for metal-poor candidate stars.
A preliminary estimation of metallcity was made using ANN trained on the full range of
metallicity. Then, we refined the measurements by using two different ANN sets; one for
estimating the atmospheric parameters for stars of near solar metallicties and the other for
the significantly metal-poor stars ([Fe/H] $<$ $-$0.7 dex). The (B-V) colours were available for
many of them which were used to verify the T$_{\rm eff}$ estimated by ANN. In most cases
the temperature estimated using ANN were in close agreement with colour temperatures.
A sizeable fraction of the candidate stars belonged to [Fe/H] $-$0.5 to $-$2.5 dex range.
\section{ Conclusions}
We have demonstrated that using ANN we can measure atmospheric parameters
with an accuracy of $\pm$ 0.3dex in [Fe/H], $\pm$200K in temperature and $\pm$0.35 in log~$g$
with the help of training set of stars of known parameters. We find that independent
calibrations for near solar metallicity stars and metal-poor stars decrease the
errors in T$_{\rm eff}$ and log~$g$ by a factor of two. We have extended the application of this
method to estimation of absolute magnitude using nearby stars with well determined parallaxes.
Better M$_{V}$ calibration accuracy can be obtained by using two separate ANNs for cool
and warm stars. The present accuracy of M$_{V}$ calibration is $\sim$$\pm$0.3mag.
\section {Acknowledgment}
This work was partially funded by the National Science
Foundation's Office of International Science and
Education, Grant Number 0554111: International Research
Experience for Students, and managed by the National Solar
Observatory's Global Oscillation Network Group.
|
{
"timestamp": "2012-03-12T01:00:58",
"yymm": "1203",
"arxiv_id": "1203.2014",
"language": "en",
"url": "https://arxiv.org/abs/1203.2014"
}
|
\section{INTRODUCTION}
Ever since the monolayer graphene was first successfully produced experimentally,\cite{first}
intriguing properties from its strictly two-dimensional structure and
massless Dirac-like behavior of low-energy
excitation have been intensively investigated.\cite{review1,review5}
Of particular interest are the
graphene nanoribbons (GNRs) that are strips of graphene obtained by different
methods, e.g., high-resolution lithography,\cite{Kim1,Kim2} chemical
means,\cite{Dai1,Cai1} or most recently the
unzipping of carbon nanotubes.\cite{Dai2,Kosynkin} Their semiconducting character with a tunable band
gap sensitive to the structural size and geometry makes them good candidates for
future electric and spintronic devices.\cite{Frank} GNRs are classified into two basic
groups, namely, armchair and zigzag ones, according to edge termination
types.\cite{Fujita1,Fujita2,Fujita3}
In the framework of the nearest neighbor tight binding model, the zigzag
GNRs are always metallic and exhibit special spin-polarized edge
states.\cite{Fujita1,Fujita2,Fujita3} For the armchair GNRs with
width $M$ (as defined in Fig.~\ref{figtw1}), they are metallic when $M=3n+2$,
with $n$ being an integer, and semiconducting otherwise.\cite{Fujita2,Fujita3}
Graphene field-effect transistors have been experimentally realized
by making use of the band gap introduced in GNRs.\cite{Dai1,Dai3} However,
large switching voltage up to several volts is
needed due to the thick back-gate oxides used in these devices. Moreover, the
excellent feature of a tunable band gap in GNRs has not been used in the
overall back-gate configuration.\cite{Dai1,Dai3}
The electronic transport in GNR-based nanodevices has also be investigated
theoretically.\cite{Areshkin,Nguyen,Orellana,Jauho}
Particular energy dependences of
conductance, resulting from the interference effects, are reported in metallic
GNRs\cite{Areshkin,Nguyen} or in semiconducting
ones out of the gap regime.\cite{Nguyen,Jauho} However, the robustness of these
transport properties against disorder has been shown to be
questionable.\cite{Areshkin,Jauho,Roche,Schubert,Mucciolo}
In this work, we introduce two classes of structures based on gapped armchair
GNRs by using either sidearm or on-site gate voltage, which allow ``on'' and
``off'' operations in the gap regime. A schematic view of the armchair GNR with
one sidearm is shown in Fig.~\ref{figtw1}. Other configurations,
i.e., with two sidearms, one or two on-site gate voltages, are shown together
with the numerical results of transport behaviors in the following
figures. Such structures are within the reach of nowaday technology,
i.e., the patterned GNR can be obtained through high-resolution lithography\cite{Kim1,Kim2} and
the contact\cite{Kim1,Kim2} and top-gate\cite{Liao} technologies are also well developed. It is
noted that throughout this work, the width of the GNR is
taken as $M=21$, so there is a band gap in the pristine
GNR.\cite{Fujita2,Fujita3} Here the corresponding band gap is about
$400$~meV. The two terminals of the GNR are connected to semi-infinite metal
leads,\cite{Nguyen,Martin,Robinson} which simulates the real experimental
condition.\cite{Huard} Such a
configuration is crucial to access possible states
in the gap regime of the GNR, due to the propagation modes in the metal leads
which are otherwise absent in graphitic ones.\cite{Robinson,Martin}
Meanwhile, the effective length of the sidearm can be electrically
adjusted by a gate voltage (not shown in the figure).\cite{Shen}
We show that by increasing the length of the sidearm $N_s$, or by increasing the
strength of positive or negative on-site gate voltage in the configuration
shown in Fig.~\ref{figtw4}(a), conduction peaks are
introduced into the originally switched-off gap regime.
The positions of these conduction peaks are determined by the
gate voltage, which, in addition to the common property of switching on and off,
allows us to selectively choose electrons of a particular energy while filter out
the others. We further propose two
schemes of structures with markedly improved robustness against disorder by
employing two sidearms or on-site gates [see
Fig.~\ref{figtw5}(a) and (d)]. Due to the
resonance between the two conduction peaks induced by the dual structure, a
conduction plateau, i.e., a broad energy window in which the transmission is close
to one, is formed. This conduction window is very robust against disorder, which
makes our proposal highly feasible for real applications.
\begin{figure}[t]
\begin{center}
\includegraphics[width=8cm]{figtw1.eps}
\end{center}
\caption{Schematic view of the armchair GNR with zigzag edged sidearm and
metal leads. The insets show how the width and length of the specific
structures are defined. Throughout this work, the width of armchair GNR is
taken to be $M=21$. }
\label{figtw1}
\end{figure}
\section{Model and Hamiltonian}
We describe the structures consisting of an armchair GNR coupled with metal leads
by using the tight-binding Hamiltonian with the nearest-neighbor approximation,
\begin{equation}
H=H_L+H_C+H_R+H_T,
\end{equation}
where $H_{L,R}$ are the Hamiltonians of the left and right leads, respectively,
$H_C$ is the Hamiltonian of the GNR and $H_T$ stands for the coupling between
the GNR and the leads. These terms are written as
\begin{eqnarray}
H_\alpha&=&-t_\alpha\sum_{\langle i_\alpha,j_\alpha
\rangle}c^\dagger_{i_\alpha}c_{j_\alpha}, \hspace{1cm} \alpha=L,R\\
H_C&=&\sum_{i_c}\varepsilon_{i_c} c^\dagger_{i_c}c_{i_c}-t\sum_{\langle i_c,j_c
\rangle}c^\dagger_{i_c}c_{j_c}, \label{Eq:GNR}\\
H_T&=&-t_T\sum_{\alpha=L,R}\sum_{\langle i_\alpha,j_c
\rangle}(c^\dagger_{i_\alpha}c_{j_c}+H.c.) .
\end{eqnarray}
Here, the index $i_c$ ($i_\alpha$) is the site coordinate in the GNR (metal
leads) and $\langle i, j\rangle$ denotes pair of nearest neighbors. $t_\alpha$
and $t_T$ are hopping parameters in the metal leads and between the leads and
the GNR, respectively, which are taken to be equal to the hopping element $t$ in the
GNR.\cite{Nguyen,note1} The on-site energy in the GNR $\varepsilon_{i_c}$ is
modulated by the on-site gate voltage, which equals $U_g$ in the gated region
and zero elsewhere.
Within the Landauer-B\"uttiker framework,\cite{Buttiker} the transmission amplitude is given by
\begin{equation}
T(E)={\rm tr}[\Gamma_L(E)G_{C}^r(E)\Gamma_R(E)G_{C}^a(E)]
\end{equation}
in which $\Gamma_{L/R}$ denotes the self-energy of the isolated ideal leads and
$G_{C}^{r/a}(E)$ represents the retarded/advanced Green's
function for the GNR.\cite{Datta} Here, $E$ is the Fermi energy in the leads.
\section{Results and Discussion}
\subsection{Electronic transport in GNRs with one sidearm or on-site gate voltage}
\begin{figure}[b]
\includegraphics[width=7cm]{figtw2.eps}
\caption{(Color online) Transmission $T$ as function of the Fermi energy of
in the GNR with one sidearm shown in
Fig.~\ref{figtw1}: (a) dependence on the length of the sidearm $N_s$. The two gray
arrows here and hereafter are plotted to indicate the evolution of
quantities with the varying of parameters, accordingly; (b) dependence on the total length
of the GNR $N_a$; (c) dependence on the width of the
sidearm $N_{\rm ws}$. All necessary parameters
are indicated in the corresponding figures.}
\label{figtw2}
\end{figure}
\begin{figure}[t]
\includegraphics[width=7cm]{figtw3a.eps}
\includegraphics[width=8.5cm]{figtw3b.eps}
\caption{(Color online) (a) Eigenenergies in the isolated GNR with
configurations being the same as those in Fig.~\ref{figtw2}(a). The gap regime is indicated
between the two dashed lines. Local density of states in the GNR at Fermi energy
$E_F=51.511$~meV: (b) corresponding to the
right peak in the gap shown in Fig.~\ref{figtw2}(b) for $N_a=15$, $N_{\rm
ws}=10$ and $N_s=4$; and (c) in the same condition but without a
sidearm. $a_{cc}$ is the carbon-carbon bond distance. Note that
the scale in the $y$-axis is elongated to make the figures clearer and
the local density of states in the leads are not included.}
\label{figtw3}
\end{figure}
We first investigate the transport properties in the GNR with one sidearm as shown
in Fig.~\ref{figtw1}. In Fig.~\ref{figtw2}(a)-(c), the transmissions are plotted
as function of the Fermi energy. The transmission in the pristine GNR is
indicated by the red solid curve in
Fig.~\ref{figtw2}(a). It is seen that the band gap manifests
itself in the electronic transport
behavior that the transmission is well below $10^{-3}$ in the gap regime, i.e.,
$E_F \in (-200, 200)$~meV. This may serve as the ``off'' state of the device
with excellent switch-off character.
By increasing the length of the sidearm $N_s$ with
fixed width $N_{\rm ws}=10$, one notices that two conduction peaks from the
positive and negative energy sides are introduced into the gap regime and moving
towards the Dirac point symmetrically. They correspond to n-
and p-type channels, respectively.
In order to elucidate this behavior, we calculate the
eigenstates and eigenenergies of the isolated GNRs with configurations employed
in Fig.~\ref{figtw2}(a) and the results are plotted in Fig.~\ref{figtw3}(a). The
eigenenergies are indicated by points with the same
color as in Fig.~\ref{figtw2}(a) for the corresponding length of the sidearm $N_s$. It
is noted that the states with eigenenergies at the
Dirac point are the localized edge states in the zigzag
terminals,\cite{Fujita1,Fujita2,Fujita3} which do not really exist when the
terminals are connected to the metal leads. Apart from these fake states, as
indicated by the gray arrows, two states (even more for $N_s \ge 8$) come into the gap and
move towards the Dirac point symmetrically. By comparing Fig.~\ref{figtw2}(a) and
Fig.~\ref{figtw3}(a), close correspondence between the positions of the conduction peaks and
those of the eigenenergies is seen. We hence conclude that the transport behavior in this
structure of a small size can be understood as resonant tunneling through the GNR
via the confined states therein. We then examine how the conduction peaks induced by
the sidearm are influenced by the total length of the GNR.
From Fig.~\ref{figtw2}(b), one observes that the positions of
conduction peaks are insensitive to the total length of the GNR.
This suggests that the states which contribute
to the conduction peaks distribute mainly in the sidearm region.
Moreover, with the increase of the total length of the GNR, the confined states
are less coupled to the leads. As a result, the
conduction peaks are narrowed and the
transmissions in the gap regime other than the conduction peaks are suppressed.
In addition, as shown in Fig.~\ref{figtw2}(c), a wider
sidearm is more effective in bringing conduction peaks into the gap regime.
It is illustrative to perform a spatial analysis of the conductance. In
Fig.~\ref{figtw3}(b), we plot the local density of states in the GNR
corresponding to the right conduction peak shown in Fig.~\ref{figtw2}(b) for
$E_F=51.511$~meV, $N_a=15$, $N_{\rm ws}=10$ and $N_s=4$; and in
Fig.~\ref{figtw3}(c) for the same condition but without the sidearm. One
observes that distinct from the case without the sidearm where the
electronic state is restricted in the vicinities of the two terminals, the
state contributing to the conduction peak indeed mostly distributes in
the sidearm region.
This kind of bound states have been discussed by
Sevin\c{c}li {\em et al}.\cite{Sevincli} and Prezzi {\em et al}.\cite{Prezzi}
in the superlattice structures of GNR.
The underlying physics is that, since the energy
band gap of armchair GNR shows strong dependence on the ribbon width, one can
fabricate structures similar to the conventional semiconductor
heterojunctions
by joining GNRs with different widths. In the condition shown in
Fig.~\ref{figtw3}(b), segments of armchair GNRs with widths $M=21$, 25 and
21 are joined together. The band gaps of the corresponding infinite GNRs
with the same widths are $(-200, 200)$, $(-140, 140)$ and
$(-200, 200)$~meV, respectively. Therefore,
the GNR with one sidearm resembles the quantum
well structure in semiconductors with conduction- and valence-band
offset $\Delta=60$~meV, and hence the bound states are formed therein.
It is further noted that due to the finite lengths of the
GNR segments and the detailed joining
condition, the actual band offset is different from the above simple
estimation.
\begin{figure}[t]
\includegraphics[width=3.2cm]{figtw4a.eps}
\includegraphics[width=5.3cm]{figtw4b.eps}
\includegraphics[width=8.5cm]{figtw4c.eps}
\caption{(Color online) (a) Schematic
view of the armchair GNR with on-site gate voltage deposited in the region
labelled with dashed box. (b) Transmission $T$ as function of the Fermi
energy of in the GNR with a gate voltage shown in
(a), for different values of on-site energy $U_g$. (c) Local density of
states in the GNR corresponding to the
conduction peak induced by an on-site gate voltage $U_g=-0.12t$ in (b), at
Fermi energy $E_F=33.405$~meV. All necessary parameters
are indicated in the corresponding figures.}
\label{figtw4}
\end{figure}
Following the idea of introducing bound states into the gap regime, we then
propose another way of accessing states inside the band gap by using an on-site gate
voltage.\cite{Silvestrov,Recher} As illustrated in Fig.~\ref{figtw4}(a), we apply a
positive (negative) voltage in the framed region by a top gate\cite{Liao}
which acts as a well potential to electrons (holes). The transmissions as function
of the Fermi energy are plotted in Fig.~\ref{figtw4}(b) for different values of
the gate voltage. One observes that by increasing the strength of positive (negative) gate
voltage, a conduction peak enters the gap regime from the right (left) and moves
towards the left (right).
These conduction peaks are from the tunneling via the
bound states in the gapped region.\cite{Silvestrov} We demonstrate this by
plotting the local density of states in
Fig.~\ref{figtw4}(c), which corresponds to the conduction peak induced by a gate
voltage $U_g=-0.12t$.
In the sense that here one can introduce only one
conduction peak into gap regime with its position fully determined by the gate voltage, this configuration
has the advantage to serve as an energy filter. The influences of the width of
the gate region and the total length of the GNR on the induced conduction peaks
resemble the case with one sidearm, and are not explicitly plotted.
\begin{widetext}
\begin{figure}[b]
\begin{minipage}[]{20cm}
\hspace{-2.5 cm}\parbox[t]{6cm}{
\includegraphics[width=4.8cm]{figtw5a.eps}}
\hspace{-0.5 cm}\parbox[t]{6cm}{
\includegraphics[width=5.8cm]{figtw5b.eps}}
\hspace{-0.5 cm}\parbox[t]{6cm}{
\includegraphics[width=5.8cm]{figtw5c.eps}}
\end{minipage}
\begin{minipage}[]{20cm}
\hspace{-2.5 cm}\parbox[t]{6cm}{
\includegraphics[width=4.8cm]{figtw5d.eps}}
\hspace{-0.5 cm}\parbox[t]{6cm}{
\includegraphics[width=5.8cm]{figtw5e.eps}}
\hspace{-0.5 cm}\parbox[t]{6cm}{
\includegraphics[width=5.8cm]{figtw5f.eps}}
\end{minipage}
\begin{minipage}[]{17.5cm}
\begin{center}
\caption{(Color online) (a) and (d) Schematic
view of the armchair GNR with two sidearms and two on-site gates,
respectively. Transmission $T$ as function of the Fermi energy
in the GNR shown in (a): (b) dependence on the spacing
between the two sidearms $N_b$; (c) dependence on the length
of the sidearms $N_s$. Transmission $T$ as function of the Fermi energy
in the GNR shown in (d): (e) dependence on the spacing
between the two gates $N_b$; (f) dependence on the values of on-site energy $U_g$ induced
by the gate voltage. All necessary parameters
are indicated in the corresponding figures.}
\label{figtw5}
\end{center}
\end{minipage}
\end{figure}
\end{widetext}
\subsection{Electronic transport in GNRs with two sidearms or on-site gate voltages}
Due to the fact that the conduction peaks introduced into the gap regime are
extremely sharp in the above two configurations and hence are easily destroyed by disorders
as we will show below, we further propose two schemes of structures to improve
the robustness, i.e., by employing two sidearms or gate voltages shown in
Fig.~\ref{figtw5}(a) and (d).\cite{Shen,Wang,Bliokh,Michael} It is noted that here we fix the total length of
the GNRs $N_{\rm tot}=40$ and the total width of the two sidearms or the gates
is set to be the same width of the sidearm or gate region in the previous configurations.
In searching for the best performance of the devices, i.e., a
wide conduction window, we vary the
spacing length $N_b$ between the two sidearms or gate
regions to modify the interference between tunnelings via the two (quasi-)bound
states from the dual structure. The results are plotted in Fig.~\ref{figtw5}(b)
and (e), for the situations with two sidearms and two gates, respectively. It is
found that for both cases with $N_b=14$, a conduction plateau is formed with a
wide energy window up to 50~meV
centered around $E_F=125$~meV (red solid curves in the figure). Such a wide conduction
window in the original gate
region allows a large current in the ``on'' state of the proposed device
which is of potential use for high performance field-effect
transistors. Moreover, as shown in Fig.~\ref{figtw5}(c) and (f), the positions
of the conduction plateaus can be controlled by the length of the sidearms and by
on-site gate voltage, respectively. So the excellent feature of controlled
modification of the conduction window is preserved in the dual structures.
\subsection{Disorder analysis}
\begin{figure}[t]
\includegraphics[width=7cm]{figtw6a.eps}\\
\includegraphics[width=7cm]{figtw6b.eps}
\caption{(Color online) Transmission $T$ as function of the Fermi
energy with different Anderson disorder strength $W$ in the GNR: (a) and (b) with
one and two sidearms [corresponding to GNR structures shown in
Fig.~\ref{figtw1} and Fig.~\ref{figtw5}(a)], respectively; (c) and (d) with
one and two one-site gate voltages [corresponding to GNR structures shown in
Fig.~\ref{figtw4}(a) and Fig.~\ref{figtw5}(d)], respectively. All necessary parameters
are indicated in the corresponding figures.}
\label{figtw6}
\end{figure}
We now show the feasibility of the above proposed devices for real application
by analyzing the robustness of the switch-off character, and more importantly the
conduction peaks and plateaus against the Anderson disorder.\cite{Schubert,Shen} In our
simulation, the Anderson disorder is created out by introducing random on-site
energy at the carbon atoms:
$\varepsilon^\prime_{i_c}=\varepsilon_{i_c}+\lambda W$ [see Eq.~(\ref{Eq:GNR})]. Here $W$ is the
disorder strength and $\lambda$ is a random number with a uniform probability
distribution in the range $(-1, 1)$. The converged
transmissions are obtained by averaging over 100 random
configurations.\cite{Shen} The
results of pristine GNR without a sidearm or on-site gate voltage are plotted in the inset
of Fig.~\ref{figtw6}(a). One notices that under different strength of disorder,
all five curves almost coincide with each other. This indicates that the gap behavior is very
robust against disorder, which ensures extremely small leakage current in the
``off'' state of the device. We then turn to check the robustness of the ``on''
state of the proposed structures.
By comparing the corresponding curves in
Fig.~\ref{figtw6}(a) and (b), one finds that the conduction peaks from one
sidearm rapidly decrease with the strength of the disorder whereas the conduction
plateaus from two sidearms are more sustained. The latters are only reduced by about
$50\%$ for the largest disorder strength $W=0.06t$.
Therefore, the robustness is
immensely improved by using two sidearms to introduce a wide conduction window into the
gap regime. The situation for
configurations with one or two gate voltages is similar. So we only
plot our results in
Fig.~\ref{figtw6}(c) and (d) without more discussions. In this way, we
demonstrate the robustness of the proposed devices for both ``on'' and ``off''
states.
\section{Summary}
In summary, we have proposed two schemes for field-effect transistor, which may
also work as energy filter, by studying transport properties in the GNR-based
structures with sidearms or on-site gate voltages. Gapped armchair GNRs
are employed with the band gap used as a natural ``off''
state of the transistor. Metal leads are employed so that by further
introducing a sidearm or on-site gate voltage to the GNR, one is able to
access the gap regime with conduction peaks. Moreover, by employing two sidearms
or on-site gate voltages, we
obtain much wider conduction windows with the transmission close to one, which
allows a large ``on'' current. We show that the positions of the conduction
peaks or plateaus can be controlled by the length of the sidearm (which can be modulated by a
gate voltage), or by the voltage of the on-site gates on the GNR. This property
enables the proposed devices not only serve as a common transistor with large on/off ratio, but
also as an energy filter.
We further demonstrate the robustness of both the ``off'' and ``on'' states of the
devices against disorder. The excellent switch-off ability, the
wide conduction window of ``on'' state which allows controlled modifications
and the high robustness against disorder suggest that the proposed structures have great
potential to work as high performance
field-effect transistor in reality.
\begin{acknowledgments}
This work was supported by the National Basic Research
Program of China under Grant No. 2012CB922002 and
the National Natural Science Foundation of China under Grant No. 10725417.
\end{acknowledgments}
|
{
"timestamp": "2012-05-08T02:02:24",
"yymm": "1203",
"arxiv_id": "1203.1989",
"language": "en",
"url": "https://arxiv.org/abs/1203.1989"
}
|
\section{\label{sec:intro}Introduction}
Properties of the interacting, harmonically trapped, ultracold gas are much more interesting in one dimension than in two and three dimensions.
In 2D and 3D a phase transition occurs whereas in 1D does not (surprisingly in quasi-1D systems two-step condensation is possible \cite{ketterle1997}).
Moreover, in symmetric 3D systems it was shown experimentally \cite{stenger1999,philliphs1999} that nearly all the way up to critical temperature the phase of the cloud is spatially uniform, equivalently: a coherence length of the system is equal to its size.
On the contrary, Petrov et al. have shown that the observation of the quasicondensate (condensate with fluctuating phase) is possible in 1D \cite{petrov2000}, 2D \cite{petrov2D2000} and very elongated 3D \cite{petrov2001} repulsive Bose gas.
Those predictions were confirmed experimentally \cite{dettmer2001, gerbier2003, hellweg2003}.
Recently density and phase properties of elongated systems were investigated in a number of experiments: \cite{esteve2006,YangYang, armijo2011, manz2010}.
In spite of many theoretical attempts \cite{andersen2002, khawaja2003, decoherentToQuasi, cockburn2011} the theory of quasicondesates is still not as mature as the theory of condensates.
In this paper we shed a new light on the quasicondensation phenomenon.
In \cite{bienias2011} we stressed that shortening of the coherence length is not only the property of a repulsive gas but also of an attractive one.
There exists a direct connection between the quasicondensation and the spectrum of a one-body density matrix
\footnote{In the description of partially coherent light spatial coherence modes are used also as eigenfunctions of the first order correlation function of the light field.
See for instance B. Saleh \textit{Photoelectron Statistics: With Applications to Spectroscopy and Optical Communication} (Springer-Verlag, New York, 1978)
}.
More precisely, the quasicondensation occurs when occupation of more than one eigenmode is comparable with the total population
\cite{penrose1956}.
In some sense it is similar to the "fragmented" spinor condensate \cite{leggett2001}.
In Section \ref{sec:inter} we show that properties of one dimensional Bose gas are similar regardless the sign of the interaction.
In Section \ref{sec:ideal} we explore coherence properties of the ideal gas in 1D, 2D, 3D and with arbitrary ratio of the trap frequencies.
\section{\label{sec:inter}Interacting gas}
Firstly, we study a one dimensional, weakly interacting Bose gas confined in a harmonic trap.
Thus, our Hamiltonian of the one dimensional Bose gas has a form:
\begin{eqnarray}
\nonumber H&=& \int \hat{\Psi}^{\dagger }(x)\left( \frac{p^2}{2m}+\frac{1}{2}m\,\omega_0^2 x^2 \right) \hat{\Psi}(x) dx +\\
& & +\,\frac{g}{2} \int \hat{\Psi}^{\dagger }(x) \hat{\Psi}^{\dagger }(x) \hat{\Psi}(x) \hat{\Psi}(x).
\label{eqn:hamiltonian}
\end{eqnarray}
The Hamiltonian is a sum of the single particle oscillator energy with mass $m$ and angular frequency $\omega$ and a conventional contact interaction with the coupling constant $g$.
Throughout this paper we use the oscillator units of position, energy and temperature, $\sqrt{\frac{\hbar}{m \omega_0}} $, $\hbar \omega_0$ and $\frac{\hbar \omega_0}{k_B}$ respectively.
Hence a dimensionless coupling $g$ is in units of $\sqrt{\frac{\hbar^3\omega_0}{m}}$.
Almost all results are calculated for 1000 atoms.
At the beginning we study a one dimensional, repulsive, weakly interacting Bose gas confined in a harmonic trap.
Our results are for the canonical statistical ensemble, thus the temperature is a control parameter.
In previous works we showed that in a wide range of temperatures such a system can be efficiently described using the so called classical field approximation \cite{bienias2011, bienias2011attr}.
To analyze the coherence properties of the system we use the first-order correlation function defined as:
\begin{eqnarray}
g_1(-x, x) & = \frac{\langle \Psi^*(-x) \Psi(x) \rangle}{\langle |\Psi(x)|^2 \rangle} \label{eqn:g1} \\
\nonumber & = \frac{\langle \sqrt{n(-x)}e^{-i\phi(-x)}\sqrt{n(x)}e^{i\phi(x)} \rangle}{\langle n(x)\rangle}.
\end{eqnarray}
\begin{figure}
\includegraphics{comparisonOdTempPub_1a}
\includegraphics{comparisonOdTempPubM_1b}
\caption{(Color online) Correlation length, width of $\tilde{g}_1$ and size of the whole cloud.
All lengths are defined as full width at half maximum.
Results for (a) repulsive and (b) attractive gas.
$1000$ atoms considered.
}
\label{fig:g1_vs_T}
\end{figure}
Where $\langle \ldots\rangle$ is an ensemble average, $\phi(x)$ and $\sqrt{n(x)}$ are respectively: the phase and the absolute value of the wave function $\Psi(x)$ describing the whole system.
As a coherence length $l_{\phi}$ of the system we take the full-width at half maximum of the $g_1(-x,x)$.
Analogously the width of the atomic cloud is a full-width at half maximum of $|\Psi(x)|^2$.
In recent papers we showed that in both repulsive \cite{bienias2011} and attractive \cite{bienias2011attr} gas two regimes exist.
One between the zero temperature and $T_{ph}$ in which the size of the condensate is smaller than the coherence length.
The second one, above $T_{ph}$, in which the opposite condition occurs - a quasicondensate regime.
Such a situation is shown in Figure
\ref{fig:g1_vs_T} for weakly interacting repulsive gas.
To check how important are fluctuations of the phase in the behavior of $l_{\phi}$ we compared $g_1$ with:
\begin{equation}
\tilde{g}_1 (-x, x) = \langle e^{-i\phi(-x)}e^{i\phi(x)} \rangle
\label{eqn:g1mod}
\end{equation}
Function $\tilde{g}_1$ unlike $g_1$ accounts for phase fluctuations only.
In Figure
\ref{fig:g1_vs_T} we see that in our regime of parameters $g_1(-x,x)$ is nearly the same as $\tilde{g}_1(-x,x)$, so density fluctuations do not contribute to the first-order correlation function regardless the sign of the interaction.
\begin{figure}
\includegraphics{delta2VsDens_2a}
\includegraphics{{delta2VsDens2_2b}}
\caption{(Color online) Behavior of the density fluctuations at the center of the trap versus interaction for 1000 atoms.
In (a) fluctuations divided by the density and in (b) divided by the square of density.}
\label{fig:densityFluct2}
\end{figure}
Moreover, density fluctuations are very similar for both attractive and repulsive cases.
In Figure
\ref{fig:densityFluct2} we show $\delta^2n/n$ (square of the density fluctuations divided by the density of atoms) at the center of the trap for changing $g$.
We see that for the repulsive gas the fluctuations are smaller than for the attractive one.
The reason is that for the repulsive case the mean-field interparticle interaction energy is positive and proportional to the density squared.
Because of this, the system prefers even distribution of the density rather than concentration of atoms.
In the attractive gas we have the opposite situation.
The interaction energy is lower for local bunching of atoms.
Nevertheless, we don't see any qualitative difference between positive and negative $g$, as the dependence is nearly linear.
What is more: the value of the fluctuations divided by the square of the density is the same for every $g$ what is presented in Figure
\ref{fig:densityFluct2}.
Finally, we see that coherence properties as well as density fluctuations are very similar for attractive and repulsive gas.
Therefore next, we take a closer look at the properties of the ideal gas.
\section{\label{sec:ideal}Ideal gas}
Detailed exact calculations are possible for the ideal gas. In the grand canonical ensemble results can be found in \cite{glauber1999,barnett2000,zyl2003}.
We know that $\phi_i(x)$ in \eref{eqn:rho} are the eigenstates of harmonic oscillator.
In the canonical ensemble we used the recursion derived in \cite{wilkens} to calculate the partition function of $N$ atoms:
\begin{eqnarray}
\nonumber Z_0 & = 1,\\
Z_1(\beta) &= \sum_{\nu}exp( - \beta \epsilon_{\nu}) \label{partFunc},\\
\nonumber Z_N(\beta)&= \frac{1}{N} \sum_{n=1}^N Z_1(n\beta) Z_{N-n}(\beta),
\end{eqnarray}
where $\epsilon_{\nu}$ is a single particle energy of the state $\nu$, $N$ is a number of atoms.
Knowing $Z_N$ we can calculate probability of finding $n$ atoms in a state of energy $\epsilon_{\nu}$.
Probability of finding at least $n$ atoms there is:
\begin{equation}
P_{\nu}^{\ge}(n|N)= e^{-n\beta \epsilon_{\nu}}\frac{Z_{N-n}}{Z_N}.
\end{equation}
Therefore:
\begin{equation}
P_{\nu}(n|N)= e^{-n\beta \epsilon_{\nu}}\frac{Z_{N-n}}{Z_N}-e^{-(n+1)\beta \epsilon_{\nu}}\frac{Z_{N-n-1}}{Z_N}.
\label{prob}
\end{equation}
Then we can find the average number of atoms in the $\nu$ mode:
\begin{equation}
N_{\nu}=\sum_n P_{\nu}(n|N)n.
\end{equation}
For deeper understanding of coherence properties lets go back to the definition \eref{eqn:g1} of $g_1(-x,x)$.
We know that the one-body density matrix has the form:
\begin{equation}
\rho(x, y) = \langle \Psi^* (x)\Psi(y) \rangle = \sum_{i=0}\frac{N_i}{N}\phi_i^*(x)\phi_i(y),
\label{eqn:rho}
\end{equation}
where $N_i/N$ and $\phi_i(x)$ are eigenvalues and eigenvectors of $\rho(x,y)$.
Hence,
\begin{equation}
g_1(-x, x)= \frac{\sum_{i=0}\frac{N_i}{N}\phi_i^*(-x) \phi_i(x)}{\sum_{i=0}\frac{N_i}{N}\phi_i^*(x) \phi_i(x) }.
\label{eqn:g1_rho}
\end{equation}
We see a strong connection between the coherence length of the system and the spectrum of one-body density matrix.
Existence of a few comparable eigenvalues of the one-body density matrix signifies fragmented condensate \cite{gunn1998}.
It follows from the Penrose-Onsager criterion that the quasicondensate transition is caused by "sticking" of one-body density matrix eigenvalues
- the situation when $N_0$ is comparable to $N_1$.
There are two important remarks which enable us to gain better intuition before further analysis.
First, when all atoms are in the condensate, this means that $N_0=N$, then $g_1(-x,x)$ is everywhere equal to one so we have a fully coherent system.
On the other hand if all
eigenvalues are equal then $g_1(-x, x) \simeq \delta (x-x^{\prime})$ and coherence length is equal to zero.
\begin{figure}
\includegraphics{{NiVsT_3}}
\caption{(Color online) Comparison of 1D and 3D ideal Bose gas in a harmonic trap.
Populations of the ground state and the first excited state in both geometries for 1000 atoms are shown.
}
\label{fig:1Dvs3D}
\end{figure}
To show a notable difference in the form of density matrix spectrum in 1D and 3D for the ideal gas we present in Figure
\ref{fig:1Dvs3D} a population of two lowest modes ($N_0$, $N_1$ respectively) as a function of temperature.
We see a qualitative difference between 1D and 3D.
In 3D $N_0$ is much bigger than $N_1$ all the way up to $T_c$.
Additionally the "sticking" of $N_0$ and $N_1$ takes place for almost vanishing $N_0$, $N_0/N \ll 1$, i.e.
almost at $T_c$.
On the other hand, in 1D we see the "sticking" for lower relative temperatures and, what is important, for much higher values of $N_0/N$.
Using the intuition gained, we analyze more precisely one, two and three dimensions.
The most intuitive way of comparing temperature effects for different regular geometries is to present temperature in $T_c$ units:
\begin{eqnarray}
\nonumber 1D: & T_c& = \frac{\hbar \omega}{k_B}\frac{N}{\log(2 N)},\\
2D: & T_c& = \frac{\hbar \omega}{k_B} \left(\frac{N}{\zeta(2)}\right)^{1/2} \label{Tc},\\
\nonumber 3D: & T_c& = \frac{\hbar \omega}{k_B} \left(\frac{N}{\zeta(3)}\right)^{1/3}.
\end{eqnarray}
For 2D and 3D $T_c$ is equal to the critical temperature in thermodynamical limit.
In 1D the phase transition doesn't occur.
This is why only characteristic temperature is used \cite{ketterle1996}.
However to make a comparison for different aspect ratios with finite number of particles it is more convenient to keep a relative population of the condensate $N_0/N$ constant rather than the corresponding relative temperature.
\begin{figure}
\includegraphics{{canVsGrand_4}}
\caption{(Color online) Ratio of populations of the ground state and the first excited state for 1D, 2D and 3D in canonical ensemble (points) for $N_0/N=0.2$.
Comparison with grand canonical results is presented (lines).
Note a good agreement in 2D and 3D.
Grand canonical results for 1D are calculated numerically using \eref{aeqn:averN}.
In the inset is shown, that the discrepancy between both ensembles in 1D exists even for huge number of atoms.
}
\label{fig:N1VsN}
\end{figure}
In Figure
\ref{fig:N1VsN} we show $N_1/N_0$ for different numbers of atoms in 1, 2 and 3 dimensions.
All results are for $N_0/N=0.2$.
In 1D `sticking' is definitely much stronger than in 2D and 3D.
Results in canonical ensemble are presented up to 1600 atoms - the number used in the experiment \cite{armijo2011}.
Because of numerical limitations the ratio of $N_1/N_0$ for large $N$ can be obtained in the grand canonical ensemble only.
We see a good agreement between canonical and grand canonical ensembles in crossover range of $N$ for 2D and 3D.
The agreement is worse in 1D.
This is because the ensembles are equivalent in the thermodynamic limit only.
Results of both ensembles (inset in Figure \ref{fig:N1VsN}) approach each other on the logarithmic scale.
Even for extremely large number of atoms the ratio $N_1/N_0$ in 1D is above $0.1$, thus one would expect quasicondensation even for very large one dimensional system.
Evidently, the finite size corrections in 1D systems are quite important.
The asymptotic behavior is not reached even for $N \simeq 10^{15}$ particles.
Moreover, the grand canonical approach allows also for finding the asymptotic value of the ratio of two dominant eigenvalues of the one-body density matrix.
The quantum partition function for a system of noninteracting bosons in the grand canonical ensemble may be written as follows:
\begin{equation}
\mathcal{Z}(z, T) = \sum_{n_0=0}^{\infty}\sum_{n_1=0}^{\infty}\ldots \prod_{\lambda} z^{n_{\lambda}}\exp(-E_{\lambda} n_{\lambda} / k_B T),
\label{aeqn:part}
\end{equation}
where $E_{\lambda}$ is the energy of the single particle state, and $z = \exp\left({\mu/k_B T}\right)$ is a fugacity.
For atoms trapped in a harmonic potential with frequency $\omega$ we have
in 1D: $E_\lambda = \hbar \omega \lambda$, 2D: $E_\lambda = \hbar \omega (\lambda_x + \lambda_y)$ and in 3D: $E_\lambda = \hbar \omega (\lambda_x + \lambda_y + \lambda_z)$.
Knowing that $\langle N \rangle~=~z\frac{\partial} {\partial z} \ln \mathcal{Z}(z, T)$ we get an expression for the average number of atoms in the system:
\begin{equation}
N=\frac{z}{1-z} + \sum_{\lambda\neq 0}\frac{z e^{-\beta E_{\lambda}}}{1-z e^{-\beta E_{\lambda}}},
\label{aeqn:averN}
\end{equation}
where we have extracted the term corresponding to the lowest energy $E_\lambda=0$.
In 2D and 3D we can safely set $z=1$ in the sum over all excited states \eref{aeqn:averN}.
Next, we can expand the formula \eref{aeqn:averN} considering only leading terms in a small parameter $\beta \hbar \omega$ \cite{castain},
it leads to the expression for the atom number:
\begin{equation}
N=\frac{z}{1-z}+\zeta(D)\left(\frac{k_B T}{\hbar \omega}\right)^{D},
\label{aeqn:3D}
\end{equation}
where $D=2~(3)$ in 2D~(3D).
Remembering that we are interested in results for the constant value $N_0=C N=\frac{z}{1-z}$
we get $z=\frac{C N}{1+C N}$ and $\frac{k_B T}{\hbar \omega} = \left( \frac{N(1-C)}{\zeta(D)} \right)^{1/D}$.
In the 1D in the thermodynamic limit we have
\begin{equation}
N=\frac{z}{1-z}-\ln(1-z) \frac{k_B T}{\hbar \omega}.
\label{aeqn:1D}
\end{equation}
Finally, we get that $\frac{k_B T}{\hbar \omega} $ is equal to $ \frac{N(1-C)}{\ln(C N+1)}$.
The asymptotic behavior of $N_1/N_0$ can be obtained from the relation:
\begin{equation}
\lim_{N\to\infty} \frac{N_1}{N_0} = \frac{z e^{-\hbar \omega \beta}}{1- z e^{-\hbar \omega \beta}} \frac{1-z}{z}.
\end{equation}
Finally, in 1D we get the following scaling of the `sticking' ratio:
\begin{equation}
\lim_{N\to\infty} \frac{N_1}{N_0} \sim \frac{1}{\ln(N)},
\end{equation}
while in higher dimensions we have:
\begin{equation}
\lim_{N\to\infty} \frac{N_1}{N_0} \sim \frac{1}{N^{(D-1)/D}};
\label{aeqn:lim2D}
\end{equation}
that is
\begin{displaymath}
\frac{1}{N^{1/2}} \quad\mathrm{for}\quad D=2
\end{displaymath}
and
\begin{displaymath}
\frac{1}{N^{2/3}} \quad\mathrm{for}\quad D=3\,.
\end{displaymath}
The sticking ratio has totally different behavior in the asymptotic limit depending on the dimensionality of the system.
In higher dimensions it goes to zero with dimension-depend power while in 1D the decay of the sticking ratio is logarithmically slow.
This is the reason while in 1D system there is always a finite range of temperatures below the quantum degeneracy temperature where occupation of the higher orbitals of the one-body density matrix is relatively high.
\begin{figure}
\includegraphics{{quasiT_VsN_5a}}
\includegraphics{{quasiN1_VsN_5b}}
\caption{(Color online) Parameters of the system at the point of the transition from true condensate to quasicondensate versus number of atoms.
In (a) temperature of the transition is presented, in (b) fraction of atoms in the ground state at the transition temperature.}
\label{fig:N0ph}
\end{figure}
The spectrum of one-body density matrix enables us to look at the dependence of $T_{ph}$ (the temperature when the width of the system is equal to the coherence length) as a function of $N$.
In Figure
\ref{fig:N0ph}(a) for better comparison we divided $T_{ph}$ by $T_c$ \eref{Tc}.
Once more a significant difference between 1D and higher dimensions is seen.
For 2D and 3D the temperature of quasicondensation is really near $T_c$ whereas for 1D the temperature $T_{ph}$ is much less than $T_c$.
Because of the ambiguity of the $T_c$ definition in 1D we looked at $N_{ph}^0$ (a number of atoms in the ground state at the temperature $T_{ph}$).
In Figure
\ref{fig:N0ph} (b) we show $N_{ph}^0/N$ as a function of $N$.
The ratio $N_{ph}^0/N$ is a fast decreasing function tending to zero in 2D and 3D, while for 1D it decreases much slower exceeding the value $0.5$ in the whole presented region.
\begin{figure}
\includegraphics{{N_1000_0_0_400_Tc_6}}
\caption{(Color online) Relative number of atoms in the ground state and in the first excited state versus the aspect ratio $\omega_z/\omega_{\perp}$ for $N=1000$ and $N_0/N=0.4$.
The left vertical line denotes point when $k_B T_{ph} = \hbar \omega_{\perp}$ and the right one when $k_B T_{ph} = \hbar \omega_z$.
}
\label{fig:cross}
\end{figure}
Having analyzed 1D, 2D and 3D symmetric cases we study the ratio of $N_1/N_0$ and $N_2/N_0$ versus trap aspect ratio $\omega_Z/\omega_{\perp}$.
In Figure \ref{fig:cross} we present results for fixed: $N=1000$ and $N_0/N=0.4$.
This way we can analyze transition between different geometries of the system.
When $\omega_Z\ll\omega_{\perp}$ the system is nearly 1D, when $\omega_Z\approx\omega_{\perp}$ is nearly 3D, and when $\omega_z\gg\omega_{\perp}$ is nearly 2D.
Once more we see a clear difference between dimensions.
However, we do not observe any sharp transition between them.
The left vertical line corresponds to $k_B T_{ph} = \hbar \omega_{\perp}$ and the right to $k_B T_{ph} = \hbar \omega_z$, so to the situations when system can be considered respectively as one and two dimensional.
We see that these points agree with the beginning of the flattening of $N_1/N_0$ and $N_2/N_0$.
Moreover 2D case is much more similar to 3D than to 1D.
In experiments we never have exactly symmetric systems, however relatively broad range of aspect ratios around $\omega_z=\omega_{\perp}$ corresponds to systems which can be considered as 3D because of the flattening of $N_1/N_0$ in this region.
We checked that this region gets broader with increasing $N$.
Summarizing, we have shown that quasicondensation phenomenon is strongly related to the dimensionality of the system rather then to interactions.
Using the sticking ratio $N_1/N_0$ as the quasicondensation criterion we found the asymptotic behavior of this quantity in the ideal gas case.
The 1D asymptotic of the sticking ratio differs drastically from asymptotic in higher dimensions.
Vanishing of the second largest eigenvalue of the one-body density matrix is logarithmically slow in 1D.
Some decreasing of the correlation length in 2D and 3D can be also observed but this effect results from the finite number of atoms in the system and occurs just below the critical temperature.
\begin{ack}
\label{sec:acknowl}
This work was supported by Polish Government Funds for the years 2010-2012.
Two of us (K.P.and K.Rz.) acknowledge financial support of the project "Decoherence in long range interacting
quantum systems and devices" sponsored by the Baden-W\"urtenberg Stiftung".
\end{ack}
\section*{References}
|
{
"timestamp": "2012-03-12T01:02:08",
"yymm": "1203",
"arxiv_id": "1203.1811",
"language": "en",
"url": "https://arxiv.org/abs/1203.1811"
}
|
\section{Introduction} \label{sec:intro}
The exact role the environment plays in the evolution of galaxies has been a long standing question in astronomy. It has been widely observed that the properties of galaxies depend on environment. High density environments in the local Universe, such as galaxy groups and clusters, are dominated by red early-type galaxies \citep{dressler1980,butcher1984}. This is in contrast with the lower density environments where blue star-forming late-type galaxies are more frequent. Furthermore, elliptical galaxies in cluster environments are generally older than their field counterparts \citep{clemens2006,sanchez2006,vandokkum2007,gobat2008} and the red sequence extends down to fainter magnitudes in denser environments indicating it is formed earlier there \citep{tanaka2005,tanaka2007,tanaka2008}. In addition, cD galaxies, the most massive galaxies known, are located exclusively in galaxy clusters.
In order to adequately explain the differences between low- and high-density environments it is essential to study galaxy clusters at all epochs. By doing so it may be possible to identify what exact physical processes constitute the more general term of 'environmental influence'. Unfortunately, the search for galaxy clusters at $>1.5$ is difficult and only a handful of spectroscopically confirmed galaxy clusters at $z>1.5$ with X-ray emission are currently known \citep{wilson2008,papovich2010,tanaka2010,henry2010,gobat2011}.
One of the most successful methods to push the search for galaxy clusters beyond $z=2$ is targeting high-z radio galaxies \citep[hereafter HzRGs,][]{miley2008}. With large observed $K$ band luminosities, these galaxies are thought to have large stellar masses of the order of $10^{11}-10^{12}$~M$_{\odot}$~\citep{roccavolmerange2004,seymour2007}. In the model of hierarchical galaxy formation these massive galaxies should be located in dense environments and are thus possible members of galaxy cluster progenitors. These structures are often referred to as 'protoclusters', because at these redshifts galaxy clusters are likely still in the process of formation and therefore have not yet virialised \citep[e.g.][]{kuiper2011a}.
By targeting a HzRG field with a narrowband filter chosen such that it contains a strong emission line at the redshift of the radio galaxy, it is possible to select galaxies in a narrow redshift interval around the radio galaxy. This has resulted in evidence that HzRGs indeed probe overdense regions in the early Universe \citep[e.g.][]{pascarelle1996,knopp1997,pentericci2000,kurk2004a,kurk2004b,venemans2007,matsuda2011,kuiper2011b}.
Although this method is efficient in locating overdensities, it is limited in the number and type of galaxies that can be selected. The emission line most commonly used for these searches is Ly$\alpha$, thus only galaxies with strong Ly$\alpha$ emission are selected. As a consequence, a large number of galaxies that reside in the overdensity are missed altogether. This method is therefore not suitable for in-depth studies that attempt to obtain a complete picture of the protocluster.
There is a variety of methods that are aimed at selecting high-$z$ galaxies. The most well-known uses the Lyman break to select UV-bright star-forming galaxies (Lyman Break Galaxies or LBGs) in a relatively broad redshift range compared to the narrowband technique. This method was pioneered by \citet{steidel1995} and has been succesfully used in many studies since then. The Lyman break selection technique selects a much larger sample of star forming galaxies than methods relying on Ly$\alpha$ narrowband data, as only 20 per cent of LBGs at a given luminosity also qualify as Ly$\alpha$ emitters taking into account current selection criteria \citep{steidel2000,steidel2011}.
The Lyman break method has been used only sparingly on HzRG fields \citep{intema2006,overzier2008}. This is mainly because the redshift range probed by the LBG criterion is $\sim0.3-0.7$, significantly larger than the redshift range spanned by a typical protocluster ($\Delta z\sim0.03$ or $\Delta v\sim2000$~km~s$^{-1}$ at $z\sim3$). One of the HzRG fields for which the LBG selection criterion has been used is MRC~0316-257 at $z=3.13$. This is one of the best studied HzRGs at $z>2.5$ and it has been shown to host an overdensity of Ly$\alpha$ emitters \citep[][hereafter V05]{venemans2005}. A study by \citet[][hereafter M08]{maschietto2008} has found tentative evidence for a similar overdensity of [O{\sc iii}] emitters. In an attempt to obtain a complete galaxy census of the overdensity, \citet[][hereafter K10]{kuiper2010} assembled photometry of the field in 18 bands ranging from $U$ band to {\it Spitzer} 8~$\mu$m. K10 selected LBGs in the 40~\arcmin$^2$ field around the radio galaxy and detected a small surface overdensity. The most massive and actively star-forming LBGs were found to be located near to the radio galaxy indicating the presence of environmental influence. However, the inability to distinguish protocluster galaxies from field galaxies is likely to diminish any real trend in the data.
In this work we present spectroscopic follow-up of the sample of LBGs composed in K10. By spectroscopically confirming the redshifts of the individual LBG candidates we can determine which galaxies are truly in the protocluster and which are in the field. This will therefore give us a better estimate of the volume overdensity. Also, it allows us to compare the properties of field and protocluster LBGs in a fully self-consistent manner. This is particularly important in determining whether the protocluster environment influences the evolution of its constituent galaxies.
The paper is structured as follows: a brief summary of the sample selection of K10, a description of the data and its reduction are given in Sect.~\ref{sec:data}. Spectroscopic redshifts and the resulting velocity distribution are discussed in Sect.~\ref{sec:results} and further discussion concerning the presence of a possible superstructure is presented in Sect.~\ref{sec:disc}. Finally, conclusions and a future outlook are presented in Sect.~\ref{sec:conc}. Throughout this paper a standard $\Lambda$ cold dark matter ($\Lambda$CDM) cosmology is used, with $H_{\rm 0}=71$~km~s$^{-1}$, $\Omega_{\rm M}=0.27$ and $\Omega_{\Lambda}=0.73$. All magnitudes are given in the AB magnitude system.
\section{Sample selection \& data} \label{sec:data}
In K10 a $UVR$ colour criterion was introduced that was designed to select star-forming galaxies in the redshift range $3.0<z<3.3$.
\begin{eqnarray} \label{eq:crit}
& U-V \ge 1.9, & \nonumber \\
& V-R \le 0.51, & \\
& U-V \ge 5.07\times (V -R)+2.43, & \nonumber \\
& R \le 26. & \nonumber
\end{eqnarray}
A total of 52 galaxies were found to satisfy the criterion. Photometric redshifts were derived for this sample using the {\sc eazy} code \citep{brammer2008} and broadband photometry in 18 bands ranging from $U$ band to {\it Spitzer} 8~$\mu$m. The initial sample was then reduced to 48 by applying a photometric redshift cut of $2.8<z_{\rm phot}<3.5$.
K10 also constructed an additional sample of 55 potential LBGs (pLBGs). These objects satisfy all selection criteria, except that they are too blue to make the $U-V \ge 1.9$ cut. However, all these objects are undetected in the $U$ band used and deeper $U$ band data may yield redder $U-V$ colours. These objects are thus not strictly LBGs when considering the selection criterion of K10, but deeper data may show that they do in fact satisfy all criteria.
Only objects with $R<25.5$ were considered for follow-up spectroscopy, because the continuum and absorption lines of fainter objects are unlikely to be detected. This reduced the samples to 29 LBGs and 27 pLBGs, respectively. The samples were subsequently divided in three brightness categories: objects with $R<24.5$ are classified as `bright', objects with $24.5<R<25.0$ as `intermediate' and objects with $R>25.0$ as `faint'. To ensure the most detections, the objects in the mask were prioritised according to their brightness. Further restrictions were imposed by the locations of the individual objects as slits in the mask are not allowed to overlap. The final mask contained 13 LBGs and 11 pLBGs of which 10 are classified as `bright', 9 as 'intermediate' and 5 as `faint'. Therefore a total of 24 protocluster candidates were observed spectroscopically. One of these objects is the Ly$\alpha$ emitter \#1867 from V05. This galaxy has been spectroscopically confirmed to be at the redshift of the protocluster.
The spectroscopy was performed with the FOcal Reducer and low dispersion Spectrograph (FORS2) in the mask multi-object spectroscopy mode (MXU) at the Very Large Telescope during the nights of 10 and 11 December 2010. The seeing varied during the two nights between 0.7\arcsec and 1.2\arcsec. The width of the slits in the mask was 1.0\arcsec. The objects were observed through the ``300V'' grism and GG435 blocking filter, with a resolution of 440. The spectral range covered is approximately $4500<\lambda<8500$~\AA. The pixels were binned $2\times2$, which resulted in a spatial scale of 0.25\arcsec~pixel$^{-1}$ and a dispersion of 3.36~\AA~pixel$^{-1}$. A total of 25 exposures of 1560 seconds each were obtained. Between the individual exposures, the pointing of the telescope was shifted in steps of 0.25\arcsec~along the slit to enable more accurate sky subtraction and cosmic ray removal. The total integration time per object was 39,000 seconds (10.83 hr).
Data reduction was performed with various {\sc iraf}\footnote[1]{{\sc iraf} is distributed by the National Optical Astronomy Observatory, which is operated by the Association of Universities for Research in Astronomy, Inc., under cooperative agreement with the National Science Foundation.} routines. The reduction included the following steps: individual frames were bias subtracted and flat fielded using lamp flats. Cosmic rays were identified and removed before the background was subtracted. The background subtracted two-dimensional frames were combined and one-dimensional spectra were extracted. Wavelength calibration was performed using arc lamp spectra and night sky lines in the science frames. The uncertainty in the wavelength calibration is $\sim0.3$~\AA, which corresponds to a systematic redshift uncertainty of $\sigma_{z}\sim0.0002$.
\section{Results} \label{sec:results}
\subsection{Redshift determination}
Spectroscopic redshifts are obtained for 20 objects out of the 24 observed. For the other 4 objects no continuum or emission lines are detected. Examining the data we find artefacts in two of the 2D spectra indicating slit defects as the possible cause of these two non-detections. The remaining two undetected objects either have a low surface brightness or are very faint ($R\sim25.5$) making it impossible to obtain a spectroscopic redshift. Properties of the objects for which a spectrum was extracted are given in Table~\ref{table1}.
The spectra of the objects that do allow for a redshift determination are shown in Fig.~\ref{fig:spec}. The spectra have been plotted in the restframe to facilitate comparison between the different objects. Also, the locations of the most important spectral features have been marked. The 2D spectra have also been included because the Ly$\alpha$ break is more obviously apparent in the 2D spectra.
The results obtained from the spectra are summarised in Table~\ref{table1b}. Spectroscopic redshifts based on both emission and absorption lines are listed as outflows may affect the spectroscopic redshift determined from Ly$\alpha$ \citep[e.g.][]{shapley2003}. For the spectroscopic redshift we take the mean value of the redshifts obtained for the individual discernible absorption features. Uncertainties listed in Table~\ref{table1} are calculated by varying the spectra according to a normal distribution characterised by the rms noise level. The individual lines are subsequently fitted again. This process is repeated 1000 times for each of the spectral features.
Approximately half of the objects show an emission line, which is assumed to be Ly$\alpha$. This is consistent with other spectroscopic studies of LBGs at $z\sim3$ \citep[e.g.][]{shapley2003,steidel2011}. The emission line is used for determining a preliminary redshift. Based on this redshift the spectrum is searched for consistent absorption lines. For objects that do not show an emission line the redshift is determined by identifying multiple interstellar absorption lines such as Si{\sc ii}$\lambda1260$, C{\sc ii}$\lambda1335$ or C{\sc iv}$\lambda1549$ in combination with a possible spectral break. Almost all objects show either a combination of an emission and an absorption line or multiple absorption lines, resulting in robust redshifts.
There is one object where the redshift is possibly ambiguous. Object \#12 has only one identifiable absorption feature and evidence for a break. Since the Lyman break is a poor redshift indicator, this makes it difficult to set an accurate redshift. The single absorption feature is very strong and based on the approximate location of the break, the line can be identified as either the O{\sc i}/Si{\sc ii} doublet at $\sim1303$~\AA~or C{\sc ii} at 1335~\AA. This indicates its redshift is either $z\sim3.11$ or $z\sim3.01$. To ascertain which is the more likely, the 2D spectrum of \#12 is compared to the 2D spectra of other objects with strong breaks and clear O{\sc i}/Si{\sc ii} and C{\sc ii} features. The former option better resembles the other 2D spectra and therefore we conclude that the redshift of \#12 is $z=3.1127$.
\begin{figure*}
\resizebox{\hsize}{!}{\includegraphics{figure1a.eps}}
\caption{\label{fig:spec} 1D and 2D restframe spectra of individual objects for which a redshift can be determined. The 1D spectra have been scaled to a common arbitrary flux scale. Vertical dotted lines denote the most important spectral features in this wavelength range. From left to right these features are: Ly$\alpha$, N{\sc v}$\lambda1240$, Si{\sc ii}$\lambda1260$, O{\sc i}/Si{\sc ii}$\lambda1303$, C{\sc ii}$\lambda1335$, Si{\sc iv}$\lambda1392,1402$, Si{\sc ii}$\lambda1527$ and C{\sc iv}$\lambda1549$. The symbols next to the ID number indicate whether the object is located in the 0316 structure ($\ast$), the foreground structure ($\dagger$) or the field ($\ddagger$).}
\end{figure*}
\begin{figure*}
\resizebox{\hsize}{!}{\includegraphics{figure1b.eps}}
\contcaption{\label{fig:spec2}}
\end{figure*}
\begin{table}
\caption{\label{table1} Properties of all LBGs and pLBGs for which spectra can be extracted.}
\small
\begin{tabular}{c|c|c|c|c}
\hline
Object ID & RA & Dec. & Type & $R$ \\
\hline
\#1 & 03:18:01.10 & -25:35:56.1 & pLBG & $25.09$ \\
\#2 & 03:18:12.30 & -25:35:42.2 & pLBG & $24.80$ \\
\#3 & 03:18:08.72 & -25:35:22.4 & LBG & $24.11$ \\
\#4 & 03:18:11.53 & -25:35:08.2 & LBG & $24.52$ \\
\#5 & 03:18:08.94 & -25:34:59.6 & LAE/LBG & $23.77$ \\
\#6 & 03:18:05.35 & -25:34:40.8 & LBG & $24.95$ \\
\#7 & 03:18:07.75 & -25:34:26.1 & pLBG & $24.37$ \\
\#8 & 03:18:18.40 & -25:34:17.1 & LBG & $24.54$\\
\#9 & 03:18:20.28 & -25:34:02.0 & LBG & $25.06$ \\
\#10 & 03:17:58.79 & -25:33:49.9 & pLBG & $24.12$ \\
\#11 & 03:18:08.39 & -25:33:39.6 & LBG & $24.69$ \\
\#12 & 03:17:58.93 & -25:33:27.7 & pLBG & $25.19$ \\
\#13 & 03:18:04.10 & -25:33:09.1 & pLBG & $25.42$ \\
\#14 & 03:18:07.43 & -25:32:51.2 & pLBG & $24.47$ \\
\#15 & 03:18:19.41 & -25:38:16.9 & pLBG & $24.42$ \\
\#16 & 03:18:13.19 & -25:38:06.1 & pLBG & $24.83$ \\
\#17 & 03:17:58.33 & -25:37:59.0 & pLBG & $24.94$ \\
\#18 & 03:17:59.20 & -25:37:37.9 & LBG & $23.80$ \\
\#19 & 03:18:19.73 & -25:37:26.8 & LBG & $23.76$ \\
\#20 & 03:18:03.91 & -25:37:14.1 & LBG & $25.44$ \\
\#21 & 03:18:02.78 & -25:37:05.5 & pLBG & $24.99$ \\
\#22 & 03:18:13.25 & -25:36:39.7 & LBG & $23.94$ \\
\hline
\end{tabular}
\end{table}
\begin{table*}
\caption{\label{table1b} Redshifts, equivalent width of the Ly$\alpha$ line and discernible spectral features of all LBGs and pLBGs for which spectra can be extracted. $^{a}$ Only given for galaxies that show Ly$\alpha$ in emission.}
\small
\begin{tabular}{c||c|c|c|c}
\hline
Object ID & $z_{\rm spec,Ly\alpha}$ & $z_{\rm spec,abs}$ & $EW_{\rm 0,Ly\alpha}$ (\AA)$^{a}$ & Spectral features \\
\hline
\#1 & $3.4004^{+0.0003}_{-0.0003}$ & $3.3965^{+0.0011}_{-0.0009}$ & $17.9\pm1.1$ & Break, Ly$\alpha$, O{\sc i}/Si{\sc ii}, Si{\sc iv} \\
\#2 & $3.1258^{+0.0004}_{-0.0004}$ & $3.1282^{+0.0023}_{-0.0026}$ & $14.9\pm0.6$ & Break, Ly$\alpha$, O{\sc i}/Si{\sc ii} \\
\#3 & $3.2306^{+0.0004}_{-0.0005}$ & $3.2303^{+0.0016}_{-0.0015}$ & $43.9\pm2.8$ & Break, Ly$\alpha$, Si{\sc ii}, O{\sc i}/Si{\sc ii}, Si{\sc iv}\\
\#4 & $3.0521^{+0.0005}_{-0.0005}$ & $3.0442^{+0.0011}_{-0.0010}$ & $5.9\pm0.7$ & Break, Ly$\alpha$, O{\sc i}/Si{\sc ii}\\
\#5 & $3.1343^{+0.0001}_{-0.0001}$ & $3.1266^{+0.0011}_{-0.0007}$ & $39.3\pm1.3$ & Break, Ly$\alpha$, O{\sc i}/Si{\sc ii}, C{\sc ii}, C{\sc iv} \\
\#6 & $3.1251^{+0.0010}_{-0.0010}$ & $3.1219^{+0.0016}_{-0.0015}$ & $17.8\pm1.8$ & Break, Ly$\alpha$, Si{\sc ii}, C{\sc ii} \\
\#7 & - & $3.0324^{+0.0007}_{-0.0007}$ & - & Break, Si{\sc ii}, O{\sc i}/Si{\sc ii}, C{\sc ii}, C{\sc iv}, Si{\sc ii}, C{\sc iv}, Fe{\sc ii}, Al{\sc ii} \\
\#8 & - & $2.9352^{+0.0017}_{-0.0006}$ & - & Break, Si{\sc ii}, C{\sc iv}\\
\#9 & $3.2257^{+0.0042}_{-0.0040}$ & $3.2181^{+0.0034}_{-0.0019}$ & $12.8\pm3.3$ & Break, Ly$\alpha$, C{\sc ii}\\
\#10 & - & $3.1121^{+0.0010}_{-0.0011}$ & - & Break, O{\sc i}/Si{\sc ii}, C{\sc ii}, C{\sc iv}\\
\#11 & - & $2.7795^{+0.0017}_{-0.0019}$ & - & Break, O{\sc i}/Si{\sc ii}, C{\sc ii} \\
\#12 & - & $3.1127^{+0.0017}_{-0.0019}$ & - & Break, O{\sc i}/Si{\sc ii} \\
\#13 & $3.1032^{+0.0017}_{-0.0012}$ & $3.0988^{+0.0015}_{-0.0016}$ & $9.2\pm2.0$ & Break, Ly$\alpha$, O{\sc i}/Si{\sc ii} \\
\#14 & - & $3.1041^{+0.0009}_{-0.0012}$ & - & Break, Si{\sc ii}, O{\sc i}/Si{\sc ii}, C{\sc ii}\\
\#15 & $2.9215^{+0.0006}_{-0.0005}$ & $2.9109^{+0.0015}_{-0.0015}$ & $7.8\pm0.8$ & Break, Ly$\alpha$, C{\sc ii} \\
\#16 & - & - & - & - \\
\#17 & $3.2252^{+0.0006}_{-0.0006}$ & $3.2295^{+0.0046}_{-0.0057}$ & $12.6\pm1.4$ & Break, Ly$\alpha$, Si{\sc ii}\\
\#18 & - & $2.9865^{+0.0005}_{-0.0007}$ & - & Break, Si{\sc ii}, O{\sc i}/Si{\sc ii}, C{\sc ii}, C{\sc iv}, Fe{\sc ii}, Al{\sc ii} \\
\#19 & $3.1115^{+0.0007}_{-0.0007}$ & $3.1003^{+0.0016}_{-0.0009}$ & $2.5\pm0.5$ & Break, Ly$\alpha$, Si{\sc ii}, O{\sc i}/Si{\sc ii}, C{\sc ii}, C{\sc iv}, Fe{\sc ii}, Al{\sc ii} \\
\#20 & - & - & - & - \\
\#21 & - & $3.0233^{+0.0023}_{-0.0013}$ & - & Break, O{\sc i}/Si{\sc ii}, Si{\sc ii} \\
\#22 & - & $2.9996^{+0.0006}_{-0.0007}$ & - & Break, Si{\sc ii}, O{\sc i}/Si{\sc ii}, C{\sc ii}, C{\sc iv}, Fe{\sc ii}, Al{\sc ii} \\
\hline
\end{tabular}
\end{table*}
\subsection{Redshift distribution} \label{sec:distz}
The full distribution of spectroscopic redshifts based on the absorption lines is shown in the left panel of Fig.~\ref{fig:histz}. The redshift of the radio galaxy is marked by an arrow. All spectroscopically confirmed LBGs have redshifts consistent with $2.7 < z < 3.5$. The low-$z$ interloper rate is therefore small. Assuming that the four non-detected LBG candidates are low-$z$ galaxies the worst-case success rate is $\sim83$ per cent.
The left panel of Fig.~\ref{fig:histz} also shows a clear concentration of galaxies near the redshift of the radio galaxy, consistent with the presence of a structure. To verify this we compare the observed $z_{\rm spec}$ distribution to the selection efficiency curve also shown in Fig.~\ref{fig:histz}. The curve indicates that the LBG selection criterion of K10 is most efficient in selecting objects between $2.8 < z < 3.5$. A Kolmogorov-Smirnov (KS) test shows that there is a probability of 0.012 that the observed spectroscopic distribution is drawn from the distribution defined by the selection efficiency curve. The two distributions therefore differ at the $2.5\sigma$ level.
\begin{figure*}
\resizebox{\hsize}{!}{\includegraphics{figure2.eps}}
\caption{\label{fig:histz} Left panel: Distribution of all spectroscopic redshifts obtained in this work. The dashed curve shows the selection efficiency of the LBG selection criterion. Right panel: Redshift distribution of [O{\sc iii}] emitters and LBGs that are located between $3.08 < z < 3.18$. Also included is LAE \#1518 of V05 which has been identified as an LBG. The objects in the red dashed histogram are those LBGs that are associated with the 0316 radio galaxy, whereas the blue dashed histogram includes [O{\sc iii}] emitters and LBGs that are likely in a foreground structure. The solid and dotted curves indicate the transmission curves of the narrowband filters used to select the [O{\sc iii}] emitters and LAEs, respectively. The arrow marks the redshift of the radio galaxy in both of the panels. All Ly$\alpha$ based redshifts are corrected for the commonly observed shift between Ly$\alpha$ emission and absorption lines.}
\end{figure*}
As M08 found evidence for a possible foreground structure at $z\sim3.1$, we take a closer look at the redshift interval $3.05<z<3.20$. This is shown in the right panel of Fig.~\ref{fig:histz}. The distribution includes the [O{\sc iii}] emitters of M08 and the relevant LBGs presented in this work. We correct the redshifts of the M08 [O{\sc iii}] emitters that have Ly$\alpha$ based redshifts, because the Ly$\alpha$ line is commonly redshifted with respect to the absorption lines. This redshift is due to outflows and the resonant nature of the Ly$\alpha$ line. The applied correction is taken to be the mean difference in redshift of all galaxies in our sample that show both Ly$\alpha$ emission and absorption lines. This correction is $\Delta z\sim 0.005$ or $\Delta v\sim350$~km~s$^{-1}$, which is roughly consistent with the shift found by \citet{shapley2003}.
The LAEs of V05 are not included in Fig.~\ref{fig:histz} because the narrowband filter used to select this sample does not allow detection of objects at $z\sim3.1$. This is illustrated by the dotted line in Fig.~\ref{fig:histz}. The distribution of LAEs is thus strongly concentrated near $z\sim3.13$ by design and including it would unfairly skew the overall distribution. The exception to this is LAE \#1518. K10 identified this object as an LBG and as such it is included in the right panel of Fig.~\ref{fig:histz}. It is also included in the subsequent analysis where possible.
The work of M08 aimed to identify [O{\sc iii}] emitters in the 0316 protocluster. Spectroscopic follow-up of three of the candidate [O{\sc iii}] emitters showed that these objects are not located at $z\sim3.13$ as expected but at $z\sim3.1$. Taking into account the correction for the Ly$\alpha$ redshifts, this amounts to a $\sim1700$~km~s$^{-1}$ blueshift with respect to the radio galaxy. The confirmation of an additional 5 objects at $z\sim3.1$ presented in this work further strengthen the notion that a structure exists in front of the 0316 protocluster.
The question is whether the structures are separate or whether they belong to one larger protocluster. When we look in detail at the LAE distribution of V05 we find that the latter is a possibility. The velocity dispersion of this sample is 535~km~s$^{-1}$ and the median redshift is similar to the median redshift of the red distribution shown in Fig.~\ref{fig:histz}. If we use a KS test to compare this to the expected distribution based on the transmission of the Ly$\alpha$ narrowband filter (dotted line), then we find a probability of $1.2\times10^{-6}$ that the LAE distribution is drawn from the expected distribution. This indicates that the blueshift of the LAE distribution with respect to the central redshift targeted by the narrowband filter is real. This in turn implies that the distribution of LAEs may extend to lower redshifts, but that these objects have been missed due to the location of the narrowband filter.
On the other hand, a normalized tail index \citep{bird1993} of 0.66 implies that the composite distribution more closely resembles a uniform distribution rather than a single Gaussian. For this work we will follow the approach of M08 and assume that the foreground structure is a separate structure. More evidence for this will be provided in Sect.~\ref{sec:spatial}.
Assuming that there are two subgroups we find mean redshifts of $z=3.1039$ and $z=3.1262$ which implies a velocity difference of $\sim1620$~km~s$^{-1}$. Using a Gapper scale estimator \citep{beers1990} we find a velocity dispersion of $965\pm112$~km~s$^{-1}$ for the composite distribution and individual velocity dispersions of $492\pm120$~km~s$^{-1}$ and $364\pm120$~km~s$^{-1}$ for the blue- and redshifted subgroups, respectively.
Although we cannot determine, based on the redshift distribution, whether the two groups are separate structures or one larger protocluster, for the remainder of this work we will refer to the objects associated with the possible foreground structure as foreground objects, whereas those objects at $z\sim3.13$ will be classified as 0316 galaxies. Objects not associated with either of the $z\sim3.1-3.13$ structures will be referred to as field galaxies.
\section{Discussion} \label{sec:disc}
\subsection{A possible superstructure and implications for the overdensity} \label{sec:odens}
In K10, the LBG surface density of the 0316 field was found to be a factor $1.6\pm0.3$ larger than the control field used. Using a volume argument and assuming that the surface overdensity is caused by a single structure connected to the $z=3.13$ radio galaxy, K10 translated this surface density to a volume density which is $8\pm4$ larger than the field density.
Based on the surface overdensity, one expects spectroscopic follow-up to reveal approximately 1 out of 3 objects to be in the 0316 protocluster. Instead, 3 out of 20 objects are found to be associated with the radio galaxy and an additional 5 objects are found to be located at $\sim3.1$. We therefore confirm the surface overdensity, but it is not solely caused by the 0316 protocluster. The volume overdensity around the 0316 radio galaxy found by K10 is thus not as large as previously estimated and must be adjusted.
To correct the volume overdensity we must assess what fraction of the overdensity is due to the foreground structure. Due to the small samples that are considered, this cannot be more than a rough estimate. In the sample of 13 [O{\sc iii}] emitters found by M08 a total of eight have spectroscopic redshifts; five due to overlap with the Ly$\alpha$ emitter sample of V05 and three through spectroscopic confirmation of the [O{\sc iii}] line. The latter three were all found to be at $z\sim3.1$. Taking into account that the five unconfirmed [O{\sc iii}] emitters may be in either of the two structures, the fraction of foreground objects is thus 25--60~per~cent.
In this work, five out of eight objects (or $\sim60$~per~cent) are found to be in the foreground structure. Based on these numbers we assume that half of the surface overdensity can be attributed to the foreground structure. We therefore estimate the volume overdensity of the 0316 protocluster to be $\sim4$, rather than 8. This is very similar to the overdensity of Ly$\alpha$ emitters of $3.3^{+0.5}_{-0.4}$ found in V05. Based on these numbers we also expect that the foreground structure is similar in richness and mass as the 0316 protocluster.
\subsection{Spatial distribution} \label{sec:spatial}
Figure~\ref{fig:spatdist} shows the spatial distribution of all objects that are spectroscopically confirmed to be either in the 0316 protocluster or in the $z=3.1$ foreground structure. This includes the 32 Ly$\alpha$ emitters of V05 and the three [O{\sc iii}] emitters of M08. The blue objects are those identified to be in the foreground structure.
Interestingly, four out of five of the foreground LBGs are located in a small area in the North-West region of the field. This specific region is also mostly devoid of $z=3.13$ Ly$\alpha$ emitters. The North-West region is thus dominated by foreground objects. Eight of the non-confirmed LBGs are also located in that general region, four of which are strongly clustered.
The distribution of the foreground [O{\sc iii}] emitters does not reflect this apparent concentration of objects. This is, however, due to the small size of the narrowband image used for the detection of these objects, as illustrated by the outline shown in Fig.~\ref{fig:spatdist}. This limits the location of the foreground [O{\sc iii}] emitters to the central region of the field.
\begin{figure}
\resizebox{\hsize}{!}{\includegraphics{figure3.eps}}
\caption{\label{fig:spatdist} Spatial distribution of all spectroscopically confirmed objects that reside in the 0316 protocluster (red symbols) or the foreground structure (blue symbols). Ly$\alpha$ emitters are denoted by plus signs, [O{\sc iii}] emitters by triangles and spectroscopically confirmed LBGs/pLBGs by diamonds. Also shown are the locations of the unconfirmed LBGs and pLBGs as small open circles. The location of the radio galaxy is marked by the star. The dotted lines denote the size and location of the narrowband image used for the detection of the [O{\sc iii}] emitters. The contours indicate the surface density of unconfirmed LBGs and the LBGs that are shown to be either at $z\sim3.10$ or $z\sim3.13$.}
\end{figure}
To further illustrate the subclustering, Fig.~\ref{fig:spatdist} also shows LBG surface density contours. The LBGs that have been shown to be field galaxies are not considered, but LBGs that have not been spectroscopically confirmed or have been confirmed to be in either of the two $z\sim3$ structures are included. The contours have been obtained using a grid with a gridsize of 3\arcsec. For each cell of the grid the surface density was calculated by determining the distance to the $N$th nearest neighbour and subsequently using $\sigma=N/\pi r_N^2$. The contours shown have been obtained with $N=8$. The resulting surface density map is smoothed with a smoothing length of 0.5\arcmin. A clear peak in the surface density map is located in the North-West region, near the concentration of foreground objects and $\sim3$\arcmin~or $\sim1.4$~Mpc from the radio galaxy. A second, less pronounced peak in the surface density is located in the centre of the field, near the radio galaxy. This provides further evidence that the foreground structure is offset from the 0316 protocluster.
The significance of the spatial subclustering can be quantified using a 2D KS test. First we determine whether the distribution of LBGs in the foreground structure is consistent with being drawn from a random distribution. We find a probability of 0.033, which implies that the distribution is different from random at the $\sim2\sigma$ level.
When the spatial distribution of foreground objects is compared to that of the 0316 objects, we find a probability of 0.0034 that both originate from the same parent distribution. The distributions of the two structures therefore differ at the $\sim3\sigma$ significance level. The foreground [O{\sc iii}] emitters have not been taken into account in this comparison. The foreground structure thus seems to be offset from the 0316 structure. We consider this evidence that the two structures are two separate groups of galaxies and not one single protocluster.
\subsection{Influence of environment on galaxy properties} \label{sec:galprop}
The influence of environment on galaxy evolution is an important topic in present day astronomy. Local galaxies in dense environments are generally older, redder and have lower star formation rates (SFRs) than those in less dense environments. There has, however, been mounting evidence that the decrease in star formation observed locally in dense regions turns around at earlier cosmic times \citep{elbaz2007,cooper2008,tran2010,hilton2010,popesso2011}.
Protocluster fields make excellent targets for studying environmental effects at $z>2$ and several studies have presented ample evidence that environment influences galaxy properties at $z\sim2$. \citet{tanaka2010b} showed that galaxies in the $z\sim2.15$ protocluster around PKS~1138-262 assembled their mass earlier than field galaxies. \citet{hatch2011b} found that H$\alpha$ emitters in the protocluster around the radio galaxy 4C+10.48 at $z=2.35$ are twice as massive as their field counterparts. Similarly, \citet{steidel2005} showed that galaxies in a serendipitously discovered protocluster at $z=2.3$ are approximately twice as old and twice as massive as their field counterparts.
For the 0316 protocluster, no significant differences have been found between the field and the protocluster galaxies in terms of mass or SFR (K10). There are, however, trends of decreasing mass and SFR with increasing distance from the radio galaxy. One of the main problems is that the field interlopers in the LBG sample of K10 possibly dilute any differences that may be apparent in a pure protocluster sample. With the spectroscopy presented in this work a first division between protocluster LBGs and field LBGs can be made.
\subsubsection{SED fitting}
We first examine the properties of the LBGs in different environments by fitting their spectral energy distributions using photometry presented in K10. For this we use the {\sc fast} SED fitting code \citep{kriek2009} in combination with the \citet{bruzual2003} evolutionary population synthesis models. Originally, all objects were assumed to be located at $z=3.13$. The addition of spectroscopic redshifts allows for a fully self-consistent comparison between field and protocluster galaxies. The free parameters in the fitting routine are the age, mass, SFH and the extinction by dust, but as in K10 we only focus on the stellar mass, because this is the only property that can be determined with reasonable accuracy.
Briefly summarising the details of the SED fitting process employed in K10: we consider exponentially declining SFHs with decay times, $\tau$, ranging from 10~Myr to 10~Gyr with steps of 0.1~dex. The ages we consider range from log(age/yr)=7 to the age of the Universe at $z\sim3.13$ which is log(age/yr)=9.3. The \citet{calzetti2000} extinction law is used for the internal dust extinction, with $A_{\rm V}$ ranging from 0 to 3 with steps of 0.1. For all cases a Salpeter mass function and solar metallicity are assumed.
The SED fitting results show a marginally larger mean stellar mass of $4.8\times10^{10}$~M$_{\odot}$~for the 0316 galaxies compared to $2.9\times10^{10}$~M$_{\odot}$~for the foreground structure and $1.7\times10^{10}$~M$_{\odot}$~for the field galaxies. When we combine the 0316 and foreground samples we obtain a mean mass of $3.7\times10^{10}$~M$_{\odot}$~for LBGs in dense environments. However, the small samples considered in this work imply that this difference is not significant. This is also apparent from the stellar mass distributions shown in Fig.~\ref{fig:masshisto}. The small samples make it impossible to distinguish the distributions. KS tests also reflect this, yielding probablities of $0.75-0.9$ that the various distributions are drawn from the same parent distribution. There is therefore no discernible difference in stellar mass between the field and protocluster populations.
\begin{figure}
\resizebox{\hsize}{!}{\includegraphics{figure4.eps}}
\caption{\label{fig:masshisto} Stellar mass distributions of galaxies residing in the field (black), the 0316 protocluster (red dashed) and the foreground structure (blue dashed).}
\end{figure}
\subsubsection{Stacked spectra}
The presence or absence of environmental dependence is studied further using the stacked spectra. In Fig.~\ref{fig:stacks} we show a series of stacked spectra of the galaxies in each of the environmental categories. These stacked spectra have been obtained by shifting the individual spectra to a common restframe wavelength scale using the absorption line redshifts. The individual spectra are then scaled to the same mean flux level in the restframe wavelength range $1300<\lambda_{\rm rest}<1500$~\AA~and subsequently added together. Here the observed wavelength range $5565<\lambda<5590$~\AA~is excluded due to the presence of strong night-skyline residuals. Since we are dealing with small samples no other outliers are excluded in the stacking process.
\begin{figure}
\centering
\includegraphics[width=83mm]{figure5.eps}
\caption{\label{fig:stacks} Stacked LBG spectra including, from top to bottom: 0316 objects with $3.12<z<3.13$, foreground structure objects with $3.1<z<3.12$, combined sample of 0316 and foreground objects, field objects and all spectroscopically confirmed objects. Dotted vertical lines indicate the location of the most important spectral features as in Fig.~\ref{fig:spec}. Spectral features above 1600~\AA~are Fe{\sc ii}$\lambda1608$, He{\sc ii}$\lambda1640$ and Al{\sc ii}$\lambda1670$.}
\end{figure}
The properties of all detectable spectral lines in the stacked spectra are listed in Table~\ref{table2}. Uncertainties on the derived properties were obtained by repeating the stacking process, but with a number of the spectra replaced by randomly drawn spectra from the same sample. This is to obtain a measure of the intrinsic scatter between the different spectra. For the field galaxies we replace three of the spectra, for the combined 0316+foreground sample we replace two spectra, whereas for the 0316 and foreground structures only one spectrum is replaced. In total 10, 15, 20 and 50 fake spectra are constructed for the 0316, foreground, 0316+foreground and field galaxies, respectively. The fake stacked spectra are subsequently varied according to their rms noise after which all properties are recalculated. For each fake spectrum this process is repeated 100 times. The standard deviations of the subsequent distributions are taken as $1\sigma$ uncertainties. The Ly$\alpha$ FWHM are measured from stacked spectra using the Ly$\alpha$ redshifts rather than the absorption line redshifts. Using the latter increases the FWHM by a factor of 1.5-2.
\begin{table*}
\centering
\caption{\label{table2} Properties of the emission and absorption lines found in the stacked spectra of the 0316 protocluster, the foreground structure and the field. Restframe equivalent widths are taken to be positive for emission lines and negative for absorption lines. Velocity offsets are given with respect to the O{\sc i}/Si{\sc ii} doublet. FWHM values are corrected for the instrumental resolution. $^a$ Based on a stack of the three objects that do show Ly$\alpha$ emission. $^b$ These values are obtained from stacked spectra created using the Ly$\alpha$ redshift where available. $^c$ This value cannot be constrained and is therefore not listed. $^{d}$ The UV slope $\beta$ is calculated using the $R$ and $I$ band data of K10.}
\begin{tabular}{l|c|c|c|c}
\hline
& 0316 & Foreground & 0316+Foreground & Field \\
\hline
\hline
$\Delta v_{\rm Ly\alpha}$ (km s$^{-1}$) & $+442\pm142$ & $+734\pm225^{a}$ & $+451\pm156$ & $+396\pm162$\\
$EW_{\rm 0,Ly\alpha}$ (\AA) & $26.4\pm3.8$ & $-13.3\pm5.3$ & $11.3\pm3.7$ & $7.4\pm2.1$\\
FWHM$_{\rm Ly\alpha}$ (km s$^{-1}$)$^b$ & $561\pm118$ & -$^c$ & $493\pm122$ & $803\pm241$\\
\hline
$\Delta v_{\rm SiII}$ (km s$^{-1}$) & - & $+308\pm178$ & $+340\pm167$ & $+281\pm152$ \\
$EW_{\rm 0,SiII}$ (\AA) & - & $-1.9\pm0.8$ & $-1.5\pm0.8$ & $-2.2\pm0.8$ \\
FWHM$_{\rm SiII}$ (km s$^{-1}$) & - & $400\pm352$ & -$^c$ & $558\pm212$ \\
\hline
$\Delta v_{\rm OI/SiII}$ (km s$^{-1}$) & $0$ & $0$ & $0$ & $0$ \\
$EW_{\rm 0,OI/SiII}$ (\AA) & $-2.1\pm0.7$ & $-4.5\pm1.1$ & $-3.5\pm0.9$ & $-2.4\pm0.5$ \\
FWHM$_{\rm OI/SiII}$ (km s$^{-1}$) & $365\pm233$ & $799\pm233$ & $708\pm223$ & $623\pm201$\\
\hline
$\Delta v_{\rm CII}$ (km s$^{-1}$) & $-263\pm169$ & $+103\pm360$ & $-117\pm202$ & $-61\pm188$ \\
$EW_{\rm 0,CII}$ (\AA) & $-3.0\pm1.1$ & $-3.1\pm1.1$ & $-2.7\pm0.8$ & $-1.8\pm0.6$ \\
FWHM$_{\rm CII}$ (km s$^{-1}$) & $726\pm309$ & $1802\pm545$ & $1127\pm559$ & $509\pm205$\\
\hline
$\Delta v_{\rm SiIV}$ (km s$^{-1}$) & - & - & $+151\pm196$ & $-61\pm204$ \\
$EW_{\rm 0,SiIV}$ (\AA) & - & - & $-1.4\pm0.8$ & $-0.9\pm0.5$\\
FWHM$_{\rm SiIV}$ (km s$^{-1}$) & - & - & -$^c$ & $102\pm226$ \\
\hline
$\Delta v_{\rm SiII}$ (km s$^{-1}$) & - & $-75\pm263$ & - & $+88\pm216$ \\
$EW_{\rm 0,SiII}$ (\AA) & - & $-2.1\pm0.9$ & - & $-1.7\pm0.5$ \\
FWHM$_{\rm SiII}$ (km s$^{-1}$) & - & $711\pm266$ & - & $315\pm231$ \\
\hline
$\Delta v_{\rm CIV}$ (km s$^{-1}$) & $-369\pm267$ & $-194\pm220$ & $-213\pm226$ & $-120\pm324$ \\
$EW_{\rm 0,CIV}$ (\AA) & $-5.3\pm1.5$ & $-3.1\pm0.9$ & $-3.4\pm1.1$ & $-2.7\pm0.8$ \\
FWHM$_{\rm CIV}$ (km s$^{-1}$) & $1881\pm414$ & $800\pm334$ & $1408\pm607$ & $887\pm286$ \\
\hline
$\Delta v_{\rm FeII}$ (km s$^{-1}$) & - & - & - & $+24\pm252$ \\
$EW_{\rm 0,FeII}$ (\AA) & - & - & - & $-1.2\pm0.5$\\
FWHM$_{\rm FeII}$ (km s$^{-1}$) & - & - & - & $414\pm278$ \\
\hline
$\beta^{d}$ & $-1.7\pm0.2$& $-0.8\pm0.4$ & $-1.2\pm0.3$ & $-1.3\pm0.1$ \\
\hline
\end{tabular}
\end{table*}
The top panel of Fig.~\ref{fig:stacks} shows the stacked spectrum of the three objects identified to be in the 0316 protocluster. The main feature of the spectrum is the strong Ly$\alpha$ emission with $EW_{\rm 0}=26.4\pm3.8$~\AA, but this is mostly driven by the LAE included in the sample. Removing this galaxy lowers the $EW_{\rm 0}$ to 17.9~\AA. The field galaxies, in the fourth panel, also show Ly$\alpha$ emission, but with an $EW_{\rm 0}=7.4\pm2.1$~\AA~it is on average not as strong as in the 0316 protocluster.
The average spectrum of the foreground galaxies shows no Ly$\alpha$ emission, but considering the small sample size we are unable to determine whether this is a significant difference. In fact, \citet{shapley2003} show that approximately half of all LBGs show Ly$\alpha$ in emission and the other half shows Ly$\alpha$ in absorption. With sample sizes of 3 for the 0316 protocluster and 5 for the foreground structure, the chance that the observed difference between the 0316 and foreground structure is a mere statistical fluctuation should be considered significant. We can therefore draw no conclusions based on this sample, but It would be interesting to see whether this difference persists in a larger sample.
We can slightly alleviate the problem with the sample size by combining the 0316 and foreground structure samples and comparing this to the field population. Any difference would be a strong indication of environmental influence at $z\sim3$. The stacked spectrum is shown in the third panel of Fig.~\ref{fig:stacks} and the relevant properties are listed in Table~\ref{table2}. The spectrum shows Ly$\alpha$ emission with $EW_{\rm 0}=11.3\pm3.7$~\AA. This is consistent with the field population within the $1\sigma$ uncertainties. The strength of the absorption lines of the combined sample are also consistent with that found for the field population. The composite spectrum of galaxies in the $z\sim3$ overdense structures therefore does not differ from field galaxies, so we find no evidence for environmental influence at $z\sim3$.
The properties listed in Table~\ref{table2} can be compared to the results of \citet[][hereafter S03]{shapley2003}. In S03 the spectra of $\sim1000$ LBGs were stacked to perform a detailed study of the average properties of these galaxies. In general, the LBG properties in Table~\ref{table2} are similar to those in \citet{shapley2003}. There is a velocity difference of $\sim400-900$~km~s$^{-1}$ between the Ly$\alpha$ line and the absorption lines, where the Ly$\alpha$ line is redshifted with respect to the absorption. This is also seen in the individual LBG spectra of S03 and is indicative of outflows. The LBGs in the 0316 field therefore also show evidence of outflows.
\citet{shapley2003} divided their sample of LBGs into four bins based on Ly$\alpha$ equivalent width, ranging from Ly$\alpha$ in absorption to strong Ly$\alpha$ emission. LBGs with strong Ly$\alpha$ emission were found to have weaker low-ionisation lines, bluer UV slopes and smaller kinematic offsets between Ly$\alpha$ and interstellar absorption lines. Based on the stacked spectra presented here we can make a similar division between the various samples. For this purpose we will only consider the field population and the combined 0316+foreground population as the individual 0316 and foreground samples are too small to make a meaningful comparison. Both populations fall into the moderate emission category or group 3 of S03.
Little difference is seen between the composite field spectrum and the results of S03. All $EW_0$ values are consistent within 1$\sigma$ with the properties of group 3 in S03. Since we make no distinction in $EW_{\rm Ly\alpha}$ when stacking the spectra we also expect this spectrum to match closely to the full LBG stack of S03. Indeed, all absorption line equivalent widths are fully consistent with the average LBG of S03.
When comparing the 0316+foreground objects with group 3 of S03 we see that the O{\sc i}/Si{\sc ii} doublet and the C{\sc ii} line of the combined sample is stronger. This is partially due to the inclusion of \#12 in the stack, which has an exceptionally strong O{\sc i}/Si{\sc ii} doublet. Removing this object from the summed spectrum reduces the equivalent width to $-2.7\pm0.6$~\AA~which is formally consistent with the results of S03. This, however, does not explain the strong C{\sc ii} feature. The other spectral features are consistent with S03.
Following the physical picture presented by S03, stronger absorption lines may be explained by a larger covering fraction of the outflowing gas. This would, however, also diminish the Ly$\alpha$ flux. Since the 0316+foreground sample does show significant Ly$\alpha$ emission there must be something compensating for the larger covering fraction. This could be related to a lower than expected dust content. The UV slopes of galaxies are sensitive to dust content, but the values listed in~\ref{table2} show no significant difference between this work and S03. Furthermore, we must consider that the samples used in this study are much smaller than the samples presented in S03. This could indicate that the strong C{\sc ii} absorption line is due to statistical fluctuation. It is therefore necessary to increase the number of spectroscopically confirmed galaxies in order to put proper constraints on any possible differences between the various populations in this work and the results of S03.
The stacked spectra can also be used to determine whether IGM absorption blueward of the Ly$\alpha$ line is more prevalent in either of the structures or in the field. To do this we assess the mean flux level for $\lambda_{\rm rest}<1185$~\AA~and compare it to the mean flux level for $\lambda_{\rm rest}>1280$~\AA. The ratio of these flux levels is highest for 0316 at $0.67\pm0.02$, whereas the foreground and field galaxies show ratios of $0.53\pm0.02$ and $0.50\pm0.01$, respectively. The combined sample of the two structures yields $0.59\pm0.01$. The Ly$\alpha$ break is thus less pronounced in the protocluster galaxies indicating that there is less IGM absorption in the overdense structures.
\subsection{Interacting or unrelated structures?}
The presence of a foreground structure is not the first indication that HzRG-selected protoclusters are part of superstructures. \citet{kuiper2011a} has shown that the well-studied protocluster around PKS~1138-262 (1138) at $z=2.15$ exhibits a broad bimodal velocity structure which has been independently found for both the megaparsec scale structure and the central kiloparsec scale structure. This is best explained by a line-of-sight merger scenario of two massive halos. Could this also be the case for the 0316 protocluster and its foreground companion?
In Sect.~\ref{sec:spatial} we show that the spatial distributions of the two structures in the 0316 field are not drawn from the same distribution and Fig.~\ref{fig:spatdist} indicates a projected separation of $\sim1.4$~Mpc. If there is a merger it is not along the line of sight.
The velocity difference between the 0316 protocluster and the foreground structure is $\sim1600$~km~s$^{-1}$, but this is only the line-of-sight velocity component. If this is indeed a merging or interacting system, then additional transverse velocity components may be present. This implies that the true relative velocity may be larger than 1600~km~s$^{-1}$. The relative velocity is thus similar to the 1600~km~s$^{-1}$ found in the 1138 system. \citet{kuiper2011a} showed that the Millennium simulation \citep{springel2005,delucia2007} could reproduce such a velocity difference, but only for the largest halo masses. Doing the same analysis at $z\sim3$ as in \citet{kuiper2011a} reveals no such mergers in the Millennium simulation.
In order to determine whether such a merger is possible at $z\sim3$, we calculate how the relative velocity evolves with decreasing distance $d$ in the case of two merging massive halos. For this we use the equations described in \citet{sarazin2002}. Conservation of energy dictates
\begin{equation}
\frac{1}{2}m v^{2}-\frac{GM_{\rm 1}M_{\rm 2}}{d}=-\frac{GM_{\rm 1}M_{\rm 2}}{d_{\rm 0}}
\end{equation}
with $m=M_{1}M_{2}/(M_{1}+M_{2})$ and $d_{\rm 0}$ the separation between the structures when they drop out of the Hubble flow. For simplicity it is assumed that the transverse velocity is zero. This yields
\begin{equation}
v=\sqrt{2G(M_{1}+M_{2})\left(\frac{1}{d}-\frac{1}{d_{0}}\right)}.
\end{equation}
In Fig.~\ref{fig:mergvelos} we show how the velocity increases with decreasing distance in a merger scenario. The curves shown are for a variety of values for $d_{\rm 0}$. The largest value for $d_0$ was chosen such that the time it takes to reach a relative velocity of 1600~km~s$^{-1}$ is equal to the age of the Universe at $z=3.13$. We also consider two specific halo masses, but in all cases it is assumed that the 0316 and foreground structure are of equal mass. We see that for masses of $10^{14}$~M$_{\odot}$~a velocity of $\sim1600$~km~s$^{-1}$ is only reached at small separations of the order of $<0.6$~Mpc. For masses closer to the estimated mass of the 0316 protocluster, the distance at which $1600$~km~s$^{-1}$ is reached ranges between $1.0 < d < 1.6$~Mpc.
\begin{figure}
\resizebox{\hsize}{!}{\includegraphics{figure6.eps}}
\caption{\label{fig:mergvelos} Evolution of the relative velocity of two massive structures with distance for a variety of starting distances $d_0$ and halo masses. The starting separations are 1.5, 2.0, 2.5 and 2.95~Mpc. These are denoted by the solid, dotted, dashed and dash-dotted curves respectively. For each starting distance two halo masses are considered. The halo mass ratio is fixed at 1:1 and the massed considered are 1 and $5\times10^{14}$~M$_{\odot}$~with the steeper curves corresponding to the larger halo masses. The horizontal and vertical lines indicate the relative velocity of 1600~km~s$^{-1}$ and the projected separation between the two structures as observed in the 0316 field.}
\end{figure}
As we have shown in Sect.~\ref{sec:spatial}, the projected distance between the structures is of the order of $\sim1.4$~Mpc. If the two structures are interacting then the true distance is likely larger. The relative velocity of 1600~km~s$^{-1}$ is also likely a lower limit due to projection effects. If the merger scenario is possible we therefore expect the curves to cross through the upper right quadrant of Fig.~\ref{fig:mergvelos}. There are two curves that meet this requirement, with starting separations of 2.5 and 2.95~Mpc and masses of $5\times10^{14}$~M$_{\odot}$. It is therefore possible that the system is undergoing a merger, but only if the two structures are massive and the starting separation is roughly 2.5 to 3~Mpc.
The other option is that the 0316 protocluster and the foreground structure are `unrelated' structures in the Hubble flow at the time of observing. Assuming that the Hubble flow dominates the relative motions of the two structures we use the Hubble law to find that a velocity difference along the line of sight of 1620~km~s$^{-1}$ implies a distance of at least 23~comoving~Mpc. Following \citet{bahcall2004} we assume that the mean distance between two clusters in the Hubble flow at $z\sim3$ is $\sim50$~Mpc. What is the chance of encountering such a line-of-sight alignment as possibly witnessed here? In a field of $100\times100$~Mpc$^2$ we expect a total of $\sim15$ structures. The 0316 field covers approximately 9~Mpc${^2}$. So the probability of having one additional structure directly in front of the main structure is $\sim1$~per~cent. If we lower the mean distance to $\sim30$~Mpc, then the chance increases to $\sim4$~per~cent. These probabilities are small, but they are not negligible. It is thus not unreasonable to find an unrelated foreground structure and the possibility that this is a change alignment of two unrelated structures in the Hubble flow cannot be rejected.
If the two structures are a chance alignment then they are separated by $\sim23$~Mpc. This does not preclude that at some later epoch the two structures will interact. We try to determine whether two massive haloes separated by $\sim23$~Mpc at $z\sim3$ will merge before $z=0$ by using the Millennium simulation. Only the most massive haloes at $z\sim3$ are considered, because the estimated mass of the 0316 protocluster is large ($\sim5\times10^{14}$~M$_{\odot}$, V05) and the foreground structure seems to have a similar mass. Using a lower limit of $2\times10^{13}$~M$_{\odot}$~we find a total of 20 halo pairs with relative distances between 20 and 25~Mpc. For the majority of the pairs the separation decreases, but none reach separations smaller than 8~Mpc. None of the pairs therefore interact or merge before $z=0$. Thus, if the two structures in the 0316 field are not interacting at $z\sim3$, then it is likely that there will be no interaction between the structures on the timescale of $\sim10$~Gyr.
\section{Conclusions} \label{sec:conc}
We have presented spectroscopic follow-up of LBGs in the field surrounding the 0136 protocluster at $z\sim3.13$. We observed a total of 24 LBG candidates. Using spectroscopic redshifts we distinguish between field and protocluster galaxies. This in turn allows us to make a self-consistent comparison between the field and protocluster galaxy samples.
\begin{enumerate}
\item{
We determine redshifts for 20 out of 24 objects, finding that all objects are located between $2.7 < z < 3.5$. This implies an interloper fraction of at most $\sim17$~per~cent. Out of the 20 confirmed objects, 5 are located at $z\sim3.10$ and 3 at $z\sim3.13$. The number of 0316 protocluster objects is too small to account for the surface overdensity presented in K10, but is consistent with the presence of two structures: the 0316 protocluster at $z\sim3.13$ and a foreground structure at $z\sim3.1$. The presence of such a foreground structure was already hypothesised by \citet{maschietto2008}. Based on the redshift distribution, however, it is not clear whether the two structures should be considered as separate or as part of one larger structure.
}
\item{
The spatial distribution of the foreground and 0316 LBGs shows two distinct density peaks: one centred on the radio galaxy and a stronger peak located in the North-West corner of the field. This latter stronger peak coincides with a concentration of foreground objects indicating that the foreground structure is not directly in front of the 0316 protocluster. A 2D Kolmogorov-Smirnov test confirms this, indicating that the spatial distributions of the 0316 and foreground LBGs differ at the $3\sigma$ level. This implies that the two structures are likely indeed separated and not part of a larger protocluster.
}
\item{
The presence of the foreground structure implies that the volume overdensity of LBGs of K10 is overestimated. Instead of the previously determined overdensity of 8, the volume density is only a factor $\sim4$ larger than the field. We also estimate that the foreground structure is of similar mass and richness to the 0316 protocluster.
}
\item{
There is no systematic difference in mass between the protocluster galaxies and field galaxies. Stacking the spectra shows that the galaxies associated with the 0316 protocluster have stronger Ly$\alpha$ emission than the field galaxies, whereas the galaxies in the foreground structure show very little Ly$\alpha$ emission. However, considering the limited sample size this may be due to statistical fluctuation. Combining the galaxies in the two structures in one composite sample shows that galaxies in dense environments do not differ from field galaxies. This implies that there is no discernible evidence for environmental effects on galaxy evolution at $z\sim3$. However, the Lyman break is less pronounced in the combined 0316+foreground sample indicating that there is less absorption by the IGM in these structures with respect to the field galaxies.
}
\item{
The relative velocity of the 0316 and foreground structures can be reproduced if the structures are merging and have large masses of $5\times10^{14}$~M$_{\odot}$. Alternatively, the structures may not be merging in which case the relative velocity translates into a radial separation 23~Mpc. Such a large separation at $z\sim3$ means it is unlikely that the two structures will interact by the present day.
}
\end{enumerate}
The results presented here for the different samples of galaxies should be considered preliminary. Further spectroscopic observations are necessary to get a better census of which galaxies are in which of the structures. This will also give better constraints on the potential differences between the protocluster and field galaxies at $z\sim3$. Furthermore, extra data will provide stronger constraints on the merger scenario considered in this work. If additional spectroscopic redshifts result in smaller separations on the sky and in redshift space, then this may increase the likelihood of the merger scenario.
\section*{Acknowledgements}
We would like to thank the anonymous referee for the very useful comments that have helped improve this paper. This research is based on observations carried out at the European Southern Observatory, Paranal, Chile, with program number 086.A-0930(A). The authors wish to thank the staff at the VLT for their excellent support during the observations. EK acknowledges funding from Netherlands Organization for Scientific Research (NWO). NAH acknowledges support from STFC and the University of Nottingham Anne McLaren Fellowship.
|
{
"timestamp": "2012-03-13T01:00:20",
"yymm": "1203",
"arxiv_id": "1203.2196",
"language": "en",
"url": "https://arxiv.org/abs/1203.2196"
}
|
\section{Introduction}\label{sec:intro}
Wireless mesh networks (WMNs) have lately been recognized as having great
potential to provide the necessary networking infrastructure for communities and
companies, as well as to help address the problem of providing last-mile
connections to the Internet \cite{Nandiraju2007,Siekkinen2007}. However, mutual
radio interference among the network's nodes can easily reduce the throughput as
network density grows above a certain threshold \cite{Balachandran2005} and
therefore compromise the entire endeavor. Such interference is caused by the
attempted concomitant communication among nodes of the same network and
constitutes the most common cause of the network's throughput's falling short of
being satisfactory (hardly reaching a fraction of that of a wired network
\cite{Gupta2000}). A promising approach to tackle the reduction of mutual
interference seems to be to combine routing algorithms with some interference
avoidance approach, such as power control, link scheduling, or the use of
multi-channel radios \cite{Akyildiz2005}. In fact, this type of network
interference problem has been addressed by a considerable number of different
strategies to be found in the literature
\cite{Ying2000,Cruz2003,Abolhasan2004,Sheriff2006,Campista2008,Wang2008,Srikanth2010,Augusto2011}.
An alternative approach that presents itself naturally is the use of multi-path
routing to distribute traffic among multiple paths sharing the same origin and
the same destination, since in principle it can help with both path recovery and
load balancing better than the use of single-path strategies. It may, in
addition, lead to better throughput values over the entire network
\cite{Salma2006,Augusto2010}. But while these benefits accrue only insofar as
they relate to how the multiple paths interfere with one another
\cite{Tsai2006,Tarique2009}, unfortunately this aspect of the problem is not
commonly addressed by multi-path strategies. What happens as a consequence is
that, though promising by virtue of adopting multiple paths to accommodate the
same end-to-end traffic, in general such strategies fail to perform as desired
because they do not tackle the interference problem during path discovery. The
single noteworthy exception here seems to be the algorithm reported in
\cite{Waharte2006}, but it uses geographic information (like localization aided
by GPS) to find paths with sufficient spatial separation so as not to interfere
with one another. In our view this weakens the approach somewhat, since such
type of information may not always be available \cite{Demirkol2006}. Moreover,
the corresponding algorithm relies on the solution of an NP-hard problem on an
input that has the size of the network \cite{Waharte2008}, so the solution may
be unattainable in practice.
Here we propose a different approach to alleviate the effects of interference in
multi-path routing. Our approach is based on two general principles. First, that
it is to work as a refinement phase over existing routing algorithms, thereby
inherently preserving, to the fullest possible extent, the advantages of any
given routing method. Second, that it is to rely only on information that is
locally available to the common origin of any given set of multiple paths
leading to the same destination. That is, only information that the origin can
obtain by communicating with its direct neighbors in the WMN should be used. One
intended consequence of the latter, in particular, is that refining the set of
paths departing from any common origin should be easily implementable by
straightforward message passing, and moreover, that any required calculation by
that node should be amenable to being carried out efficiently even if it
involves the solution of a computationally difficult problem. In order to comply
with these two principles, our approach operates on a previously established set
of paths leading from a common origin, say $i$, to a common destination, say
$j$. It operates exclusively on the neighborhood information stored at node $i$
itself or at any of its neighbors, say $k$, such that $k$ participates in some
of the $i$-to-$j$ paths, as well as on the information stored at these same
nodes regarding the routing of packets to node $j$. Once node $i$ has acquired
all this information, an undirected graph $G_{ij}$ is constructed that
represents every possible interference that can occur as packets get forwarded
toward $j$ by those of $i$'s neighbors that are on $i$-to-$j$ paths. Solving a
well-known NP-hard problem (that of finding a maximum weighted independent set)
on this typically small graph serves as a heuristic to decide which of the
$i$-to-$j$ paths to keep and which to discard.
It is important to note that, being determined with reference to graph $G_{ij}$,
the resulting set contains no two paths that interfere with each other as far as
node $i$'s neighbors are concerned, except of course for the inevitable
interference that may occur as packets leave $i$ or reach $j$. In our view, this
provides a sharp contrast between our approach and others that aim at weaker
forms of independence between the paths, for example by seeking paths that are
merely edge- or node-disjoint
\cite{Lee2001,Sung2001,Tsirigos2001,Cruz2003,Alicherry2006,Sheriff2006,Wang2006,Xiaojun2007,Wang2008,Wang2009}.
This is so because, as remarked elsewhere (e.g., \cite{Pearlman2000}),
independence by edge-disjointness encompasses independence by node-disjointness,
which in turn encompasses independence by noninterference. Of course, the
highest an independence relation's level in this hierarchy the easiest it is to
implement it as the multiple paths are discovered (not coincidentally, the
simple exchange of tokens between nodes suffices to produce edge- or
node-disjoint path sets \cite{Xuefei2004}).
We proceed in the following manner. First we state the problem in
graph-theoretic terms and give our solution in Section~\ref{sec:mra}. Then we
move, in Section~\ref{sec:methods}, to a presentation of the methodology we
followed in conducting our computational experiments. Our results are given in
Section~\ref{sec:results} and involve comparisons with some prominent routing
algorithms, viz.\ AODV \cite{Perkins1999}, AOMDV \cite{Marina2002}, OLSR
\cite{Jacquet2001}, and MP-OLSR \cite{Yi2011}. We used these algorithms both as
stand-alone methods and as bases to our own heuristic. Our results include
throughput and fairness \cite{Jain1998} comparisons based both on NS2.34
\cite{ns2} simulations and on the SERA link scheduling algorithm
\cite{Fabio2012}. We continue in Section~\ref{sec:discussion} with further
discussions and conclude in Section~\ref{sec:conclusion}.
\section{Problem formulation and heuristic}\label{sec:mra}
For $i$ and $j$ any two distinct nodes of the WMN, we begin by assuming that
some routing protocol already established a set $\mathcal{P}_{ij}$ of paths
directed from $i$ to $j$. Another key element is that, since we seek to
establish independence by noninterference, an interference model, along with its
assumptions, must be selected. Our choice is the protocol-based interference
model, together with the assumption that a node's communication and interference
radii are the same. Should a different model be selected or the two radii be
significantly different, the only effect would be for the graph construction
process outlined below, once adapted accordingly, to produce a different graph
(in particular, the interference radius could be chosen appropriately in order
for the protocol-based interference model to mimic the physical interference
model \cite{Shi2009}).
Under the assumptions of the protocol-based interference model, the
communication/interference radius is fixed at some value $R$, which we take to
be the same for all nodes. It follows that two nodes are neighbors of each other
in the WMN if and only if the Euclidean distance between them is no greater than
$R$. Moreover, since every link may transmit in both directions for error
control, it also follows that two links can interfere with each other if either
one's transmitter or receiver is a neighbor of the other's transmitter or
receiver \cite{Balakrishnan2004}. As will become apparent shortly, this has
important implications when modeling interference, since links that share no
nodes can still interfere with one another.
In keeping with the locality principle outlined in Section~\ref{sec:intro}, we
work on the premise that a node $k$'s knowledge is limited to the set $N_k$ of
its neighbors and the set $\mathrm{Next}_k(i,j)\subseteq N_k$ of neighbors to
which it may forward packets sent by node $i$ to node $j$ (assuming $k\neq j$).
Set $\mathrm{Next}_k(i,j)$, obviously, depends on the $j$-bound paths that leave
$i$ and go through node $k$. The problem we study is that of eliminating from
$\mathcal{P}_{ij}$ the fewest possible paths (in a weighted sense, to be
discussed later) so that the remaining path set, henceforth denoted by
$\mathcal{P}^\mathrm{R}_{ij}$, contains no mutually interfering paths except at
$i$ or $j$. However, owing once again to the issue of locality, we forgo both
optimality and feasibility a priori and settle for a heuristic instead. That is,
for the sake of locality we admit the possibility that, in the end, neither will
the selected paths be collectively optimal nor will the absence of interference
among them be guaranteed (except at those paths' second hops, which will be
non-interfering relative to one another necessarily).
Before proceeding, we tackle two special cases. The first one is that in which
$\mathcal{P}_{ij}$ contains the single-link path that connects $i$ directly to
$j$. In this case, we let $\mathcal{P}^\mathrm{R}_{ij}$ be the singleton that
contains that path, since no other arrangement can possibly do better. The
second special case is that in which $\vert\mathcal{P}_{ij}\vert=1$, provided
the single path contained in $\mathcal{P}_{ij}$ has at least two links, and is
motivated by the situations in which $\mathcal{P}_{ij}$ originates from
single-path routing. In this case, we enlarge $\mathcal{P}_{ij}$ before feeding
it to our path-selection heuristic. Letting node $k$ be such that
$\mathrm{Next}_i(i,j)=\{k\}$, we do this enlargement of $\mathcal{P}_{ij}$ by
including in it as many paths from $\mathcal{P}_{kj}$ as possible, each suitably
prefixed by the new origin $i$, provided none of the new paths is weightier than
the one initially in $\mathcal{P}_{ij}$. If enlargement turns out to be
impossible, then the problem becomes moot for $\mathcal{P}_{ij}$ and we let
$\mathcal{P}^\mathrm{R}_{ij}=\mathcal{P}_{ij}$.
We are then in position to introduce our refinement algorithm for multi-path
routing, henceforth referred to by the acronym MRA (for multi-path refinement
algorithm). The goal of MRA is to create an undirected graph $G_{ij}$
corresponding to the path set $\mathcal{P}_{ij}$ and to extract from it the
information necessary to determine $\mathcal{P}^\mathrm{R}_{ij}$. This graph's
node set, henceforth denoted by $V$, has one node for each of the paths in
$\mathcal{P}_{ij}$. Its edge set, denoted by $E$, is constructed in such a way
as to represent every interference possibility that can be inferred solely from
the sets $N_k$ and $\mathrm{Next}_k(i,j)$ for every $k\in N_i$ that participates
in at least one of the paths in $\mathcal{P}_{ij}$. Once graph $G_{ij}$ is
built, finding a maximum weighted independent set in it (i.e., a subset of $V$
containing no two nodes joined by an edge in $E$ and being as weighty as
possible) provides the best possible approximate decision on which of the paths
in $\mathcal{P}_{ij}$ should constitute $\mathcal{P}^\mathrm{R}_{ij}$, namely
those corresponding to the nodes in the maximum weighted independent set that
was found.
MRA proceeds as given next, following the introduction of some auxiliary
nomenclature. Given two neighboring WMN nodes, say $k$ and $k'$, we are
interested in the following two possibilities for the pair $k,k'$. A type-A pair
is part of at least one path in $\mathcal{P}_{ij}$ in such a way that
$k'\in\mathrm{Next}_k(i,j)$ with $k\in N_i$ or $k\in\mathrm{Next}_{k'}(i,j)$
with $k'\in N_i$. In a type-B pair, both $k$ and $k'$ belong to at least one
path in $\mathcal{P}_{ij}$ as well, but not the same path. Moreover, at least
one of them is a member of $N_i$ while the other, if not in $N_i$ as well, is in
$\mathrm{Next}_l(i,j)$ for some other $l\in N_i$. Note that type-A and -B pairs
constitute all the structural information that node $i$ can gather by strictly
local communication from its neighborhood $N_i$ in the WMN. In the case of a
type-A pair, we let $\mathrm{Paths}(k,k')$ be the set of $i$-to-$j$ paths to
which the pair belongs.
\begin{enumerate}
\item[(1)] Let $V$ have one node for each path in $\mathcal{P}_{ij}$. Add one
further node to $V$ for each type-B pair of neighboring WMN nodes. We refer to
these additional nodes as temporary nodes.
\item[(2)] Construct $E$ as follows:
\begin{enumerate}
\item[i.] Let $k,k'$ and $l,l'$ be two pairs of neighboring WMN nodes, each
pair being of type A or B. If it holds that $k=l$, $k=l'$, $k'=l$, or $k'=l'$,
then add an edge to $E$ between each node corresponding to a path in
$\mathrm{Paths}(k,k')$ (if the pair is of type A) or the corresponding temporary
node (if the pair is of type B) to each node corresponding to a path in
$\mathrm{Paths}(l,l')$ (if the pair is of type A) or the corresponding temporary
node (if the pair is of type B).
\item[ii.] Connect any two nodes in $V$ by an edge if, after the previous step,
the distance between them is $2$.
\item[iii.] Remove all temporary nodes from $V$ and all edges that touch them
from $E$.
\end{enumerate}
\item[(3)] Find a maximum weighted independent set of $G_{ij}$ and output
$\mathcal{P}^\mathrm{R}_{ij}$ accordingly.
\end{enumerate}
In these steps, graph $G_{ij}$ starts out as a graph with
$\vert\mathcal{P}_{ij}\vert$ nodes and gets enlarged by the addition of
temporary nodes that represent some of the possibilities of off-path
interference as nodes in $N_i$ engage in transmitting packets (Step~(1)). Then
it receives edges to account for the assumptions of the protocol-based
interference model (Steps~(2).i and~(2).ii) and is after that stripped of all
temporary nodes to end up with $\vert\mathcal{P}_{ij}\vert$ nodes once again
(Step~(2).iii). Step~(2).ii, in particular, accounts for interference in the
WMN when a link's transmitter or receiver does not coincide with (but is a
neighbor of) another link's transmitter or receiver. The last MRA step,
Step~(3), is the determination of a maximum weighted independent set of
$G_{ij}$. We use node weights such that, for the node corresponding to path
$p\in\mathcal{P}_{ij}$, the weight is $1/C_p$, where $C_p$ is the path's hop
count. In other words, shorter paths tend to be favored over longer ones as
$\mathcal{P}^\mathrm{R}_{ij}$ is extracted from $\mathcal{P}_{ij}$. Clearly,
though, any other desired criterion can be used as well. An illustration of how
MRA works is given in Fig.~\ref{figure1}.
\begin{figure}[p]
\centering
\scalebox{0.75}{\includegraphics{figure1.eps}}
\caption{Construction of graph $G_{ij}$ for the path set
$\mathcal{P}_{ij}=\{a,b,c,d,e\}$. The paths in $\mathcal{P}_{ij}$ are shown in
panel (a), where each directed edge leads to a node in some $\mathrm{Next}(i,j)$
set (for example, $\mathrm{Next}_{k_1}(i,j)=\{j,k_5\}$) and each dashed edge
joins neighboring nodes that share none of the $i$-to-$j$ paths. We see in this
panel that $\mathrm{Paths}(k_2,k_6)=\{b,c\}$, and also that there are five
type-A node pairs ($k_1,j$; $k_2,k_6$; $k_2,k_3$; $k_1,k_5$; and $k_4,k_6$) as
well as one single type-B node pair ($k_3,k_5$, labeled $x$). Panel (b) shows
graph $G_{ij}$ as it stands after Steps~(1) and~(2).i, respectively for creating
its node set $V$ and initializing its edge set $E$ as a function of node
coincidence among all type-A or -B pairs. The distance-$2$ closure determined in
Step~(2).ii is shown in panel (c), with the additional edges represented as
dashed lines. Panel (d), finally, shows $G_{ij}$ as it stands at the end, after
Step~(2).iii. Notice, in particular, the fundamental role played by node $x$ in
identifying the interference between paths $c$ and $d$ under the protocol-based
interference model (it is through $x$, as the distance-$2$ closure is
determined, that $c$ and $d$ become connected). If equal weights were used for
all nodes, then clearly any of $\{a,b\}$, $\{a,c\}$, $\{a,e\}$, $\{b,d\}$, or
$\{d,e\}$ would qualify as a maximum weighted independent set of $G_{ij}$ in
Step~(3). However, using $1/C_p$ as the weight of the node corresponding to path
$p$ leads to $\{a,b\}$ and $\{a,e\}$ as the only possibilities, each of total
weight $1/2+1/3=5/6$, since $C_a=2$ and $C_b=C_e=3$. In the end, then, we have
either
$\mathcal{P}^\mathrm{R}_{ij}=\{a,b\}$ or $\mathcal{P}^\mathrm{R}_{ij}=\{a,e\}$,
each set containing paths whose second hops do not interfere with one another,
each set as large as possible in the weighted sense provided by the use of
$1/C_p$.}
\label{figure1}
\end{figure}
\section{Methods}\label{sec:methods}
We evaluated the performance of MRA through extensive experimentation with the
following routing algorithms: AODV \cite{Perkins1999}, AOMDV \cite{Marina2002},
OLSR \cite{Jacquet2001}, and MP-OLSR \cite{Yi2011}. For the purpose of
conciseness, we henceforth refer to the combination of each of these algorithms
with MRA as R-AODV, R-AOMDV, R-OLSR, and R-MP-OLSR, respectively. We remark
that, since both AODV and OLSR are single-path routing algorithms, handling the
paths they generate falls into one of the special cases discussed in
Section~\ref{sec:mra}. Our experiments were run in the network simulator NS2.34
(NS2 henceforth) \cite{ns2} and in a simulator that employs the SERA link
scheduling algorithm \cite{Fabio2012}, briefly described later in this section.
We used two different configurations of routing-algorithm parameters, and
likewise two different configurations of NS2 parameters (one for the
path-discovery process and another for performance evaluation). These
configurations were selected during initial tuning experiments and will be
presented shortly.
\subsection{Topology generation}
We generated four types of network according to the maximum number of neighbors,
$\Delta$, a node may have. Each network was generated by placing $n$ nodes
inside a square of side $1500$. The first of these nodes was positioned at the
center of the square and the remaining nodes were placed randomly as a function
of the communication/interference radius $R$ introduced at the beginning of
Section~\ref{sec:mra}. Their placement was subject to the constraints that each
node would have at least one neighbor, that no node would be closer to any other
than $25$ units of Euclidean distance, and that no node would have more than
$\Delta$ neighbors. No more than $1000$ attempts at positioning nodes were
allowed; if this limit was reached then the growing network was discarded and
the generation of a new one was started. The value of $R$ was determined so
that the expected density of nodes inside a radius-$R$ circle would be
proportional to $\Delta/R^2$ (assuming some uniformly random form of placement),
and be moreover about the same density as that of the whole network. It follows
that $\Delta/R^2 \propto n$. We chose the proportionality constant to yield
$R=200$ for $n=80$ and $\Delta=4$, whence $R=200\sqrt{20\Delta/n}$. Of all the
networks generated, there are $100$ networks for each combination of
$n\in\{60,80,100,120\}$ and $\Delta\in\{4,8,16,32\}$, thus totaling $1600$
networks.
\subsection{Path discovery}
For each of the $1600$ networks we randomly generated $100$ sets of node pairs
to function as origin-destination pairs (instances of the $i,j$ pair we have
been using throughout). Each set comprises $n$ pairs and no node was allowed to
appear more than once in any set as an origin. For each of the node-pair sets
and each of the four routing algorithms (AODV, AOMDV, OLSR, and MP-OLSR) we
obtained $n$ path sets (instances of $\mathcal{P}_{ij}$). Likewise, for each of
the node-pair sets and each of the four refined routing algorithms (R-AODV,
R-AOMDV, R-OLSR, and R-MP-OLSR) we obtained another $n$ path sets (instances of
$\mathcal{P}^\mathrm{R}_{ij}$).
For each routing algorithm, the discovery of each $\mathcal{P}_{ij}$ instance
(i.e., for a single origin and a single destination) proceeded as follows. After
loading the network topology onto NS2 we conducted a $15$-second simulation with
one flow agent for the single origin $i$ and the single destination $j$, using a
CBR of one $1000$-byte packet per second. The remaining pairs were handled
likewise after resetting the simulator. We remark that this one-pair-at-a-time
strategy, as opposed to generating paths for all $n$ pairs in the same set
concomitantly, was meant to minimize packet loss due to path overload and also
to avoid the possible interference of a previously discovered path with the
discovery of a new one. Of course, this argument is only valid for on-demand
routing algorithms (AODV and its variants depend on network load, while the OLSR
variants always find identical paths for the same network topology), but we
proceeded in this way in all cases. As a consequence, our experiments are
entirely reproducible.
We set NS2 to its default configuration, but employed the DRAND MAC protocol
\cite{Rhee2006} to avoid collisions in the path-discovery process. To adjust the
radius $R$ we also set the parameter \emph{RXThresh\_} (RXT) to the appropriate
value given by the program \emph{threshold.cc} (cf.\ the NS2 manual). We used
the implementations of the AODV, OLSR, and AOMDV routing agents available in
version 2.34 of NS2 and the MP-OLSR routing agent available at \cite{mpolsr}.
For AOMDV, we made the small modifications proposed by \cite{YuHua2005} to
discover only node-disjoint paths with at most $K$ paths for each
origin-destination pair of nodes. We adopted these modifications because
node-disjoint paths are clearly more interference-free than otherwise. We chose
$K=5$ because it achieved the best throughput values for $2\le K\le 7$. The same
modifications were effected on MP-OLSR (as proposed by \cite{Xun2005}). Out of
the same range for $K$, and for the same reason as above, we used $K=3$ for
$\Delta\in\{4,8\}$ and $K=5$ for $\Delta \in \{16,32\}$.
\subsection{Performance evaluation}
Once we fix a value for $n$ and a value for $\Delta$, there are $10^4$ path sets
on which to evaluate the performance of MRA. Each of these sets is relative to
$n$ origin-destination pairs. In our experiments, we randomly grouped these
pairs into $n$ sets, each containing a different number of origin-destination
pairs (i.e., one set containing a single pair, another containing two pairs, and
so on), and simulated the network's behavior in transporting predominantly heavy
traffic from the origins to the destinations. We use $OD$ to denote the set of
pairs in question, hence $1\le\vert OD\vert\le n$, and let
$\mathcal{P}=\bigcup_{ij\in OD}\mathcal{P}_{ij}$ and
$\mathcal{P}^\mathrm{R}=\bigcup_{ij\in OD}\mathcal{P}^\mathrm{R}_{ij}$. For the
sake of normalization, all our performance results are presented against the
pair density $\theta=\vert OD\vert/n\in(0,1]$.
Each experiment began by loading the corresponding network onto NS2, with MAC
set to 802.11 and the routing agent to NOAH \cite{noah}. NOAH works only with
fixed paths that have to be configured manually and therefore does not send
routing-related packets, thus providing the ideal setting for a performance
evaluation free of any interference from control packets. Next we started a CBR
traffic flow from each origin to each destination and measured the number of
successfully delivered packets during the last $120$ of the $135$ seconds of
simulation (following, therefore, a warm-up period of $15$ seconds).
During the initial, tuning experiments we varied three of the NS2 parameters
widely. These were the carrier sense threshold (CST), aiming to increase the
spatial reuse and consequently the throughput \cite{Kim2006}, and the CBR
parameter as well as the network transmission rate, aiming to obtain as much
throughput and fairness as possible (cf.\ Section \ref{subsec:theresults}).
Table~\ref{table1} presents the parameter ranges of our tuning experiments,
during which the highest CBR rate that afforded some gain in throughput was
identified for each value of $\Delta$, and also lowest rate that afforded some
gain in fairness.\footnote{We observed in NS2 simulations that, within limits,
increasing the CBR \emph{rate\_} parameter tends to lead to an increase in
throughput while decreasing it tends to lead to an increase in fairness.} Our
choices for use thereafter while conducting performance evaluation are shown in
Table~\ref{table2}, where CBR1 and CBR2 refer to such highest and lowest rate,
respectively.
\begin{table}[t]
\centering
\caption{Parameters used for tuning.}
\input{table1}
\label{table1}
\end{table}
\begin{table}[t]
\centering
\caption{Parameters used for performance evaluation.}
\input{table2}
\label{table2}
\end{table}
All our NS2 experiments were carried out under 802.11, which is a CSMA protocol.
As an alternative setting that might provide some insight into the performance
of MRA under some TDMA scheme (an approach fundamentally distinct from CSMA
\cite{Gummalla2000}), we selected the SERA link scheduling algorithm
\cite{Fabio2012}. SERA seeks to schedule the links of a set of paths while
striving to maximize throughput on those paths. It is therefore quite well
suited to the task at hand. The throughput that our SERA simulator provides is
given in terms of time slots, so in order to achieve a meaningful basis for
comparing CSMA- and TDMA-based results a translation is needed of such
throughput figures into those provided by NS2 in the CSMA experiments. We did
this by resorting to a very simple NS2 simulation to determine the duration of
a time slot. In this experiment, a node sends packets to another and the time
for successful deliveries is recorded. Since this time is conceptually the same
that in SERA is taken to be a time slot, the translation from one setting to the
other can be accomplished easily. Using the parameter values shown in
Table~\ref{table2}, we found a time-slot duration of $0.002$ seconds. This is
the duration we use, together with the ND-BF numbering scheme for SERA and its
$B$ parameter set to $2$ (cf.\ \cite{Fabio2012}).
\section{Computational results}\label{sec:results}
We divide our results into two categories. First we present a statistical
analysis of the networks generated and the path sets obtained by the original
routing algorithms and by their refinements through the use of MRA. Then we
present the ratios of the refined algorithms' throughputs to those of their
corresponding originals (absolute values are given in
Figs.~\ref{figureS1}--\ref{figureS3}) and also fairness figures. For the purpose
of conciseness we report only on the $n=120$ results, since they are
qualitatively similar to those related to the other three values of $n$ we used.
In Section~\ref{sec:discussion}, though, we do discuss some of the quantitative
differences that were observed.
\subsection{Properties of the networks generated}
The $1600$ networks we generated are that same that were used in
\cite{Fabio2012}. We refer the reader to Section~8.1 of that publication for a
variety of the networks' statistical properties, such as the occurrence of
topologies structured in some particular way and some of their
structure-related distributions. Here we concentrate on presenting those
properties that pertain to routing both before and after refinement, since they
are the ones we have found useful in helping explain the throughput results we
present later.
The average path multiplicity (number of paths) per origin-destination pair in
the path sets $\mathcal{P}$ and $\mathcal{P}^\mathrm{R}$ is given in
Table~\ref{table3} for every combination of $\Delta$ and $\vert OD\vert$.
Note that MP-OLSR is absent from the table in spite of being a multi-path
algorithm, the reason being that in this case the $K$ parameter does not work
as an upper bound (as it does for AOMDV), but rather as the fixed number of
paths to be found. We observe in the table that the average path multiplicity
increases monotonically with $\Delta$ for fixed $\vert OD\vert$, which is
expected from the well-known fact that the number of possible paths between the
same two nodes grows with $\Delta$ in arbitrary graphs \cite{Hoffman1963}. On
the other hand, increasing $\vert OD\vert$ for fixed $\Delta$ causes very little
variation, probably owing to the method we used to discover the paths in the
first place (i.e., by handling each origin-destination pair independently of the
others). The overall pruning effect of MRA is also clearly manifest in the
table, since in all cases the average for an algorithm's refined version is less
than that of the original algorithm (i.e., on average we have
$\vert\mathcal{P}^\mathrm{R}\vert<\vert\mathcal{P}\vert$ for all routing
algorithms).
\begin{table}[t]
\centering
\caption{Average path multiplicity per origin-destination pair for $n=120$. Data
are averages over $10^4$ instances of $\mathcal{P}$ or $\mathcal{P}^\mathrm{R}$
for each combination of $\Delta$ and $\vert OD\vert$.}
\small
\input{table3}
\label{table3}
\end{table}
The path-size distributions for $\mathcal{P}$ and $\mathcal{P}^\mathrm{R}$ with
$\vert OD\vert=n$ (that is, for the original algorithms and their refinements,
using the full path sets, as generated) are given in Figs.~\ref{figure2}
and~\ref{figure3}, respectively, for every value of $\Delta$. Note first that,
as expected by virtue of the well-known dependency of path sizes on the average
degree of nodes \cite{Dirac1952}, increasing $\Delta$ leads the path-size
distribution for a given algorithm to peak at ever smaller values. Another
expected result is that the OLSR variants all allow for longer paths than those
of AODV. The reason for this is that the multi-point relay (MPR) concept at the
heart of OLSR tends to cause longer paths to be produced than the shortest-path
algorithm (SPA) used by AODV. It is also noteworthy that the application of MRA
to yield the refined algorithms has no impact on the distributions other than
causing some of the greatest path sizes to occur more frequently.
\begin{figure}[p]
\centering
\scalebox{0.55}{\includegraphics{figure2.eps}}
\caption{Distribution of path sizes for the original algorithms with $n=120$,
for $\Delta=4$ (a), $\Delta=8$ (b), $\Delta=16$ (c), and $\Delta=32$ (d). For
each value of $\Delta$ the distribution refers to $100$ networks and $100$ path
sets per network, each corresponding to $n$ origin-destination pairs.}
\label{figure2}
\end{figure}
\begin{figure}[p]
\centering
\scalebox{0.55}{\includegraphics{figure3.eps}}
\caption{Distribution of path sizes for the refined algorithms with $n=120$, for
$\Delta=4$ (a), $\Delta=8$ (b), $\Delta=16$ (c), and $\Delta=32$ (d). For each
value of $\Delta$ the distribution refers to $100$ networks and $100$ path sets
per network, each corresponding to $n$ origin-destination pairs.}
\label{figure3}
\end{figure}
\subsection{Results}\label{subsec:theresults}
Our computational results relating to throughput are summarized in
Fig.~\ref{figure4} for CBR1, Fig.~\ref{figure5} for CBR2, and Fig.~\ref{figure6}
for SERA. Each figure is organized as a set of four panels, each for one of the
four values of $\Delta$. All three figures show the behavior of the ratio, here
denoted by $\sigma$, of each refined algorithm's throughput to that of its
original version. All plots are given against the ratio $\theta$ introduced
earlier, which indicates what fraction of the $n$ origin-destination pairs
corresponding to each path set is being taken into account. In all panels a
highlighted horizontal line is used to mark the $\sigma=1$ threshold, above
which a refined algorithm can be said to have overtaken its original version.
\begin{figure}[p]
\centering
\scalebox{0.55}{\includegraphics{figure4.eps}}
\caption{Throughput ratio for CBR1 with $n=120$, for $\Delta=4$ (a), $\Delta=8$
(b), $\Delta=16$ (c), and $\Delta=32$ (d). Data are averages over the $10^4$
path sets that correspond to each value of $\Delta$ for each value of $\theta$.
Confidence intervals are less than $1\%$ of the mean at the $95\%$ level, so
error bars are omitted.}
\label{figure4}
\end{figure}
\begin{figure}[p]
\centering
\scalebox{0.55}{\includegraphics{figure5.eps}}
\caption{Throughput ratio for CBR2 with $n=120$, for $\Delta=4$ (a), $\Delta=8$
(b), $\Delta=16$ (c), and $\Delta=32$ (d). Data are averages over the $10^4$
path sets that correspond to each value of $\Delta$ for each value of $\theta$.
Confidence intervals are less than $1\%$ of the mean at the $95\%$ level, so
error bars are omitted.}
\label{figure5}
\end{figure}
\begin{figure}[p]
\centering
\scalebox{0.55}{\includegraphics{figure6.eps}}
\caption{Throughput ratio for SERA with $n=120$, for $\Delta=4$ (a), $\Delta=8$
(b), $\Delta=16$ (c), and $\Delta=32$ (d). Data are averages over the $10^4$
path sets that correspond to each value of $\Delta$ for each value of $\theta$.
Confidence intervals are less than $1\%$ of the mean at the $95\%$ level, so
error bars are omitted.}
\label{figure6}
\end{figure}
Except for a few cases of $\theta<0.1$ (in which only a few origin-destination
pairs coexist in the network and therefore refinement to achieve
path-independence by noninterference is probably pointless to begin with), it
follows from Figs.~\ref{figure4}--\ref{figure6} that MRA was effective to some
extent in all cases. Fixing the simulation scenario (CBR1, CBR2, or SERA)
reveals that the behavior of $\sigma$ depends very little on the value of
$\Delta$ or $\theta$ (provided $\theta$ is sufficiently large). Overall, the
values of $\sigma$ seem best for the coupling of OLSR with SERA, followed by
CBR1 coupled with either OLSR or MP-OLSR, and lastly for CBR2 without any marked
preference for any routing method. Recall that the SERA scenario is TDMA-based,
while both CBR1 and CBR2 are CSMA-based, with CBR1 operating at the higher
rates.
Another perspective from which it is worth examining performance is that of
fairness in the distribution of traffic through the paths. In other words, given
an origin-destination pair and the multiple paths leading from the origin to the
destination, we look at how traffic gets distributed through the various paths.
One way of quantifying this is by means of the fairness index \cite{Jain1998}.
Given a set of paths $\mathcal{Q}$,\footnote{We use $\mathcal{Q}$ as a place
holder for either $\mathcal{P}$ or $\mathcal{P}^\mathrm{R}$, depending on
whether the routing algorithm in question is one of the originals or one of the
refinements through MRA.} the corresponding fairness index can be defined as
$(\sum_{p\in\mathcal{Q}}x_p)^2/\vert\mathcal{Q}\vert\sum_{p\in\mathcal{Q}}x_p^2$,
where $x_p$ is the number of packets delivered to $j$ through path $p$ during
the experiment. The fairness index ranges from $1/\vert\mathcal{Q}\vert$ to $1$,
indicating when equal to $1$ that traffic is evenly distributed among the paths.
We give results on the fairness index in Figs.~\ref{figure7}--\ref{figure9},
respectively for CBR1, CBR2, and SERA. Note initially that, somewhat
unexpectedly (owing to the algorithms' markedly different strategies), AODV and
OLSR are statistically indistinguishable from each other as far as fairness is
concerned. The same holds for their refined versions, respectively R-AODV and
R-OLSR. Note also that the fairness index, in all cases, tends to decrease as
$\theta$ is increased. This means that, as might be expected, the presence of
denser end-to-end traffic tends to disrupt the balance between paths more
easily. AODV and OLSR have the best figures overall, better even than their
refined versions. So, unlike throughput, fairness does not seem to improve as we
extend a single-path algorithm's set of paths and then apply MRA for refinement.
All multi-path strategies, on the other hand, can be seen to benefit from the
use of MRA.
\begin{figure}[p]
\centering
\scalebox{0.55}{\includegraphics{figure7.eps}}
\caption{Fairness index for CBR1 with $n=120$, for $\Delta=4$ (a), $\Delta=8$
(b), $\Delta=16$ (c), and $\Delta=32$ (d). Data are averages over the $10^4$
path sets that correspond to each value of $\Delta$ for each value of $\theta$.
Confidence intervals are less than $1\%$ of the mean at the $95\%$ level, so
error bars are omitted.}
\label{figure7}
\end{figure}
\begin{figure}[p]
\centering
\scalebox{0.55}{\includegraphics{figure8.eps}}
\caption{Fairness index for CBR2 with $n=120$, for $\Delta=4$ (a), $\Delta=8$
(b), $\Delta=16$ (c), and $\Delta=32$ (d). Data are averages over the $10^4$
path sets that correspond to each value of $\Delta$ for each value of $\theta$.
Confidence intervals are less than $1\%$ of the mean at the $95\%$ level, so
error bars are omitted.}
\label{figure8}
\end{figure}
\begin{figure}[p]
\centering
\scalebox{0.55}{\includegraphics{figure9.eps}}
\caption{Fairness index for SERA with $n=120$, for $\Delta=4$ (a), $\Delta=8$
(b), $\Delta=16$ (c), and $\Delta=32$ (d). Data are averages over the $10^4$
path sets that correspond to each value of $\Delta$ for each value of $\theta$.
Confidence intervals are less than $1\%$ of the mean at the $95\%$ level, so
error bars are omitted.}
\label{figure9}
\end{figure}
Another interesting trend that can be observed in
Figs.~\ref{figure7}--\ref{figure9} is that, though always decaying with $\theta$
as we noted, in general the fairness index is best for SERA, followed by CBR2,
then by CBR1. While we believe the position of SERA in this rank to be closely
related to its TDMA-based nature, it is curious to observe that the relative
positions of CBR1 and CBR2 are exchanged with respect to what we observed for
throughput. It seems, then, that in selecting between the rates associated with
CBR1 and CBR2 one is automatically forced to favor throughput over fairness or
conversely.
Our definition of the fairness index given above is only one of the
possibilities in the context of multi-path routing. Another alternative is to
coalesce all packets delivered from one origin, say $i$, to one destination, say
$j$, into a single number $x_{ij}$ and then compute the fairness index as
$(\sum_{ij\in OD}x_{ij})^2/\vert OD\vert\sum_{ij\in OD}x_{ij}^2$. Proceeding in
this way would shift the focus of the fairness index from paths to
origin-destination pairs. We give no detailed results on this alternative, but
do provide an example with SERA in Fig.~\ref{figureS4}. There is clearly great
similarity with the results in Fig.~\ref{figure9}, but the numbers tend to be
higher.
\section{Discussion}\label{sec:discussion}
As we remarked earlier, we have provided no results for $n\in\{60,80,100\}$
because, essentially, they are indistinguishable from those of $n=120$ in
qualitative terms. We do remark, however, that a few noteworthy quantitative
differences were observed. For example, in the case of SERA a higher throughput
ratio $\sigma$ was sometimes observed for reduced $n$ but the same value of the
pair density $\theta$. This stems not only from the fact that the average path
size decreases as $n$ decreases, but more generally from the fact that the
number of links for the same $\theta$ is lower in the smaller networks, thus
fewer links interfere with one another and fewer links have to be scheduled.
Such improvement in the value of $\sigma$, therefore, seems to depend on the
network's path-size distribution.
As we noted briefly in Section~\ref{subsec:theresults}, often OLSR and MP-OLSR
turned out to be the routing methods most prone to benefit from the refinement
provided by MRA. One clue as to why this is so may be already present in
Table~\ref{table3}, where the refined versions of these two methods have some
of the lowest path multiplicities overall, thence a tendency to incur less
interference. However, this holds also for the refined version of AODV, so there
has to be some other distinguishing aspect. While our data are not sufficient to
provide a definitive answer, we believe the two methods' superiority to be owed
to a combination of MRA (which attempts to reduce path multiplicity to lower
interference) with the MPR concept that is intrinsic to the OLSR variants (which
in turn attempts to provide an initial set of paths as spatially distributed as
possible). As for AODV, the SPA at its core probably produces paths that are
less spatially separated \cite{Jiazi2008,Biradar2010}.
In a related vein, the results of Section~\ref{subsec:theresults} also point at
SERA as providing superior throughput results vis-\`{a}-vis those of CBR1 and
CBR2, and similarly for CBR1 with respect to CBR2. Because such results refer to
throughput gains after refinement by MRA, more detailed data are needed for a
direct comparison of throughputs. These are shown in
Figs.~\ref{figureS5}--\ref{figureS7}, which clearly confirm the rank. Once again
the reason for this is not totally clear, but it may be a manifestation of
SERA's properties rather than of the superiority of its underlying TDMA scheme
over the CSMA protocols of CBR1 and CBR2. After all, a considerable amount of
research has been directed toward the TDMA versus CSMA question, without however
reaching an agreement
\cite{Ding2002,Gupta2007,Ashutosh2009,Banaouas2009}.
Another curiosity related to this issue of how SERA compares to CBR1 and CBR2
has to do with the total absence of traffic on some paths. While SERA precludes
this from occurring as a matter of design principle, unused paths do occur in
the other two cases. During the corresponding NS2 simulations this persisted
even if simulation times were extended or the synchronization of multiple CBRs
was reordered or entirely removed.\footnote{As these appear to be commonly
occurring problems of NS2 agents.} As it turns out, it seems that certain paths
remained unused so that throughput could be increased on other paths. Data on
the most critical cases, viz.\ those in which no packets at all were delivered
for an origin-destination pair, are shown in Figs.~\ref{figureS8}
and~\ref{figureS9}. Examining these data confirms our expectation that the
single-path algorithms should be less prone to the occurrence of such extreme
cases. It also reveals that both R-AOMDV and R-MP-OLSR had fewer such
occurrences than their corresponding originals (i.e., MRA seems to have
attenuated the problem).
\section{Concluding remarks}\label{sec:conclusion}
MRA is a heuristic for the refinement of routing paths in WMNs. It was developed
with multi-path routing algorithms in mind but works also on single-path
algorithms (through a manipulation of its input to obtain multiple paths from
the overall set of single paths). MRA is fully local, in the sense that it
depends only on information that is readily available to each node and its
immediate neighborhood in the network. Being local means that the extra control
traffic it may entail is negligible, especially if we consider that it need
happen only once for a fixed set of paths. It involves the solution of an
NP-hard problem, that of finding a maximum weighted independent set in a graph,
but the inputs involved are typically very small, leading to negligible running
times if compared to arbitrary graphs \cite{Pardalos1991}.
Our computational experiments with MRA on a TDMA setting (SERA) and two CSMA
settings (CBR1 and CBR2) revealed improvements in throughput of up to $30\%$ for
both the AODV and OLSR routing methods and their multi-path variants. The latter
were also improved by MRA in terms of fairness. Finding out just how close these
improvements come to the very best that can be achieved is an open problem that
involves solving the path-pruning problem globally. This problem is NP-hard as
the one solved by MRA, but its input, relating to the WMN in a global scale, is
much more sizable.
Another interesting aspect for further investigation is the effect of MRA on
multi-radio networks. In such schemes the network's path density can be
drastically reduced for a given frequency, thus potentially benefiting MRA.
Conversely, it may also be possible to use MRA to achieve the desirable goal of
minimizing the number of radios \cite{Bahl2004,Raniwala2005}. Further research
is also needed on the trade-off that clearly exists between obtaining spatially
separated paths or relatively short ones. While the former is good for
non-interference, disregarding the latter may lead to throughput loss on the
relatively longer paths.
\section*{Acknowledgments}
We acknowledge partial support from CNPq, CAPES, a FAPERJ BBP grant, and a
scholarship grant from Universit\'{e} Pierre et Marie Curie. All computational
experiments were carried out on the Grid'5000 experimental testbed, which is
being developed under the INRIA ALADDIN development action with support from
CNRS, RENATER, and several universities as well as other funding bodies (see
\texttt{https://www.grid5000.fr}).
|
{
"timestamp": "2012-03-09T02:04:44",
"yymm": "1203",
"arxiv_id": "1203.1905",
"language": "en",
"url": "https://arxiv.org/abs/1203.1905"
}
|
\subsection{Sorting}
\label{sec:Sorting}
The problem of sorting a collection of comparable elements, generally taken to be numbers, has been extensively researched in the parallel algorithms domain. In BSP, two main algorithms exist. The first one, by Shi and Schaeffer \cite{ShiSchaeffer}, is known as Parallel Sorting by Regular Sampling (PSRS) and is asymptotically optimal for $n \geq p^3$. For $n < p^3$, Goodrich \cite{GoodrichSorting} proposed a different algorithm, the MapReduce version of which is given in \cite{Goodrich}.
\paragraph{}
The algorithm in \cite{ShiSchaeffer} is more suited for today's data sizes, and is very straightforward. It can be split up into two parts, sampling and sorting, and proceeds as follows. Given a collection of $n$ items, this is partitioned across the $p$ processors, each receiving about $\sfrac{n}{p}$ elements. Every processor sorts its data using an optimal sequential algorithm, and selects $p + 1$ regularly spaced primary samples, including the first and last elements, from this sorted data. A single processor then receives all $p \cdot (p + 1)$ primary samples, sorts them and again chooses $p + 1$ regular secondary samples from them. These secondary samples divide the $n$ input elements into $p$ buckets. Each processor is then assigned a bucket and every processor distributes its input data between the processors, based on their assigned bucket. Upon receiving all its data, each processor sequentially sorts it and outputs it. The algorithm has cost $W = O(\sfrac{(n \log n)}{p})$, $H = O(\sfrac{n}{p})$, $S = O(1)$. The extra cost measured in BSPMR is $F = O(\sfrac{n}{p})$, and $H_{n} = O(\sfrac{n}{p})$ so the algorithm can be efficiently implemented in MapReduce.
\paragraph{}
In MapReduce the algorithm is just as straightforward and is again split into two parts. The algorithm can be implemented using the simulation described above, with each reduce task simulating a single processor in the BSP system. However, advantage can be taken of the MapReduce framework to allow for better implementation. Since input to MapReduce is in the form of $\langle key;value \rangle$ pairs, the input values are divided into $q$ partitions each having about $\sfrac{n}{q}$ elements, with each partition having a unique key from $1$ to $q$. The number of map tasks $q$ and reduce tasks $r$ should be set to the largest possible values allowed by the system, so we assume that $q \geq r$ as stated in \cite{MapReduceOrig} and \cite{HadoopTutorial}.
\paragraph{}
Every one of the $q$ available map tasks reads its input data, sorts the values and selects $r + 1$ regularly spaced primary samples. All the primary samples are sent to a single reduce task by setting the key to a common value, and are sorted by the shuffle function. The $r + 1$ regularly spaced secondary samples are then chosen, which determine the buckets to be used in the sorting phase. In the sorting phase, the input is again split amongst the available map tasks which produce for each input element $x_i$ a $\langle key;value \rangle$ pair with $key = value = x_i$. The shuffle function is changed so that every reduce task is assigned a bucket and every value is sent to the reduce task responsible for the bucket it falls into. All the values in each bucket are combined and automatically sorted by key, by the shuffle function, and the reduce function simply outputs the values in this sorted order. The cost of the algorithm in MR is $T = O(n \log n)$, $C = O(n)$, and $D = O(1)$.
\paragraph{}
Use of the MapReduce framework for sorting was first discussed in the original paper \cite{MapReduceOrig}. Hadoop has been used in sorting benchmarking\footnote{\url{http://sortbenchmark.org/}} \cite{TeraSort2008,TeraSort2009,TritonSort} with very good results, when sorting 1 Terabyte and 1 Petabyte of data. Google have also independently achieved very good results sorting large data \cite{GoogleSort}. The algorithms used for such sorting benchmarks are very similar to the one described above, varying mostly in the sampling phase, since deterministic sampling is replaced by random sampling. This reduces the actual computation time, since no sorting is performed in the map phase of the sampling phase.
\subsection{Dense Matrix Multiplication}
Given $n \times n$ matrices $A$, $B$, the aim is to compute the $n \times n$ matrix $C = A \cdot B$. The problem is defined as
\[\begin{array}{ccc}
c_{ik} = \sum_{j=1}^{n} a_{ij} \cdot b_{jk}& & 1 \leq i,k \leq n
\end{array}\]
The standard sequential algorithm requires $\Theta(n^3)$ elementary operations.
\paragraph{}
A BSP algorithm for standard matrix multiplication is presented in \cite{McColl}, due to McColl and Valiant. The algorithm revolves around partitioning the input and output matrices across the $p$ available processors. The $n^3$ elementary products can be represented by a cube $V$ on axes $i, j, k$. Matrix $C$ can then be calculated by adding the elementary products in groups along the $j$-axis. The same operations can be performed in blocks, facilitating the distribution of all the elementary operations over the available processors.
\paragraph{}
Matrices $A,$ $B$ and $C$ are each divided into $p^{\sfrac{2}{3}}$ regular square blocks of size $\sfrac{n}{p\sfrac{1}{3}}$, with the blocks denoted by $A[i,j], B[j,k]$ and $C[i,k]$ for $1 \leq i, j, k \leq p^{\sfrac{1}{3}}$. Every processor is responsible for computing $V[i,j,k] = A[i,j] \cdot B[j,k]$, for a particular $i, j, k$, after receiving blocks $A[i,j]$ and $B[j,k]$ as input. The matrix $C$ is then divided between the $p$ processors and each of the $\sfrac{n^2}{p\sfrac{2}{3}}$ elements of the resulting $V[i,j,k]$ blocks is sent to the processor responsible for that summing up these elements to obtain $C[i,k]$. The cost of the algorithm is asymptotically optimal, with $W = O(\sfrac{n^3}{p})$, $H = O(\sfrac{n^2}{p^{\sfrac{2}{3}}})$ and $S = O(1)$. In BSPMR, $F = O(\sfrac{n^2}{p})$, with $H_{n} = O(\sfrac{n^2}{p^{\sfrac{2}{3}}})$ meaning the algorithm can be implemented efficiently on MapReduce.
\paragraph{}
Using the simulation presented above, the algorithm can be implemented in two rounds using $r = p$ reduce tasks to simulate the $p$ BSP processors. However, since the algorithm only runs in two supersteps, it can be implemented in MapReduce in a single round, using the whole framework, with $q = r = p$. Matrices $A, B, C$ are split into $q^{\sfrac{2}{3}}$ regular square blocks and with $A, B$ distributed amongst the $q$ available map tasks. Each map task is assigned the respective submatrices of $A$ and $B$, along with the indices of these submatrices. These indices are used as \textit{keys} to determine which of the $r$ reduce tasks will be sent the computed matrix operations. Each reduce task, upon receiving the elements, sums them up and outputs the respective submatrix of $C$. The number of map and reduce tasks are equal, with the algorithm's cost in MR being $W = O(n^3)$, $C = O(n^2 \cdot q^{\sfrac{1}{3}})$, and $D = O(1)$.
\paragraph{}
Matrix multiplication has been studied in the MapReduce context. As part of the Apache Software Foundation's efforts on Hadoop, HAMA was developed as a framework for massive matrix and graph computations using MapReduce and BSP, as detailed in \cite{Seo}.
\subsection{Breadth First Search}
The Breadth First Search (BFS) algorithm provides a simple way of searching for vertices in a graph $G = (V, E)$. The result is a tree, rooted at a vertex $s$, to all the vertices reachable from $s$. Each vertex's distance from $s$ is also noted. The typical approach taken for the parallelisation of the BFS algorithm \cite{Xia,Scarpazza,Yoo} is a level synchronous one, in line with the BSP model. The vertices of the graph are partitioned across the processors and then acted upon, with each processor also assigned the edges incident to each assigned vertex. A processor is only allowed to process the vertices it owns. Each vertex is marked as either processed or not processed. Initially, all the vertices are marked as not processed.
\paragraph{}
Given $G = (V, E)$, where $|V| \gg p$, the parallel BFS algorithm proceeds as follows. In the first superstep, the processor responsible for the root vertex identifies its neighbouring vertices and marks the root vertex as processed. Typically, most of the identified neighbouring vertices are not owned by the processor. Therefore, the processors responsible for each neighbouring vertex are identified and notified that these vertices are currently at the frontier of the search and need to be processed. In the following superstep, the next level in the BFS tree is processed. Each vertex reads the list of vertices that need to be processed in the current superstep. If any of these vertices is marked as not processed, their status is set to processed and their neighbours are identified. The processor responsible for each vertex is identified and notified and a new superstep can start. This process is repeated until all vertices have been processed. The cost of the algorithm is $W = O(\sfrac{|V|^2}{p})$, $H = O(\sfrac{|V|^2}{p})$, $S = O(d)$, where $d$ is the diameter of the graph $G$. The cost of not having storage, as measured in BSPMR, is $F = O(\sfrac{|V|^2}{p})$, while $H_{n} = O(|V|)$.
\paragraph{}
Since $F > H_{n}$, the algorithm cannot be efficiently implemented on MapReduce. When the primary local memory is cleared, the whole graph structure has to be stored and read from global memory in each superstep. Real life graphs exhibit the power law property \cite{Chakrabarti}. Therefore, the number of edges stored per processor varies depending on the assigned vertices. This makes load balancing very hard to achieve. For a graph with a fixed maximum degree $k$, $F = O(\sfrac{k|V|}{p})$, which could lead to an efficient MapReduce implementation for small values of $k$.
\paragraph{}
The inefficiency of implementing some graph algorithms on MapReduce is recognised in \cite{Seo}, which presents a framework that works in conjunction with Hadoop to perform matrix and graph computations. The authors claim that MapReduce is not appropriate for graph traversal algorithms, stating that algorithms based on their BSP engine would be better suited for such tasks.
\section{The BSP Model}
The traditional von Neumann model has served, for a long time, as the main model for designing and reasoning about sequential algorithms. It has also served as a reference model for hardware design. In the context of parallel algorithm design, no such ubiquitous model exists. One of the earliest attempts at defining such a model was the PRAM \cite{JaJa}. While it allowed for better theoretical reasoning about parallel algorithms, PRAM made a number of assumptions that cannot be fulfilled in hardware; mainly because the cost of communication is greater than that of computation and the number of processors is limited.
\paragraph{}
Valiant \cite{Valiant} introduced the BSP model to better reflect the hardware design features of mainstream parallel computers, through the direct mode of BSP (assumed in this paper). The BSP model allows for efficient parallel algorithm design without any overspecification requiring the use of a large number of parameters. The underlying parallel computer implementation is similarly not overspecified. A BSP computer can be defined by $p$ processors, each with its local memory, connected via some means of point-to-point communication. BSP algorithms proceed in supersteps in each of which processors receive input at the beginning, perform some computation asynchronously, and communicate any output at the end. Barrier synchronisation is used at the end of every superstep to synchronise all the $p$ processors in the system. Each processor can communicate directly with every other processor, providing complete control over how the data is distributed between the processors in every superstep.
\paragraph{}
A BSP system is also defined by its bandwidth inefficiency $g$. Every $l$ time steps an attempt is made to synchronise the processors. If they have finished their supersteps then the processors are synchronised, otherwise they keep on working for another $l$ time units.
\paragraph{}
An algorithm designed in BSP can be measured by three main features: the computation time and communication cost for each superstep, and the number of supersteps. Let $w_s$ be the maximum number of arithmetic operations performed by each of the $p$ processors in a superstep $s$. Let $h'_s$ be the maximum input, and $h''_s$ the maximum output data units over all $p$ processors in superstep $s$. The total communication cost of superstep $s$ is $h_s = h'_s + h''_s$. An algorithm running in $S$ supersteps, therefore, has costs $W = \sum_{s=1}^{S}{w_s}$ and $H = \sum_{s=1}^{S}{h_s}$. An estimate running time on any physical system is defined as $W + H \cdot g + S \cdot l$.
\paragraph{}
The synchronisation periodicity $l$ , is typically much higher than the bandwidth inefficiency $g$. Therefore, the number of supersteps required by an algorithm should be minimised as much as possible, while aiming to achieve optimal computation and communication costs. Balanced algorithms are achieved by dividing computation and communication equally amongst the available processors. Since $g$ and $l$ vary between system implementations, algorithm design revolves around improving $W$, $H$ and $S$.
\section{BSP on MapReduce}
Having determined that the BSP model can simulate any MapReduce algorithm having tasks with known execution times, and that it allows for better parallel algorithm design, it is important to determine the role of MapReduce. MapReduce offers simplicity in the development process by abstracting intricate programming details. Thus, it is important to determine whether any algorithm design in BSP can be implemented in MapReduce, and if the framework's simplicity comes at a price.
\paragraph{}
Goodrich et al. \cite{Goodrich} give a simulation for any BSP algorithm on their MapReduce model. The memory size $M$ of each processor is limited to $\lceil N/P \rceil$, where $N$ is total memory size of the BSP system and $P$ is the number of BSP processors. The simulation is also correct for the MR model presented in this paper, and works as follows:
\begin{itemize}
\item The map function is set to the identity function.
\item Each reduce task simulates a BSP processor, with each reduce phase simulating a single superstep.
\item Every reduce task receives the input of the BSP processor it is simulating, performs the required computation and outputs the results to the relevant reduce tasks.
\item Any data that in BSP is stored in local memory between supersteps is stored in global memory and read in the following round.
\end{itemize}
Simulating a BSP algorithm running in $R$ supersteps with $N$ total memory has a cost in MapReduce of $O(R)$ rounds and $O(RN)$ communication. While the result of \cite{Goodrich} is very important, no distinction is made between algorithms whose asymptotic costs are preserved when implemented in MapReduce and those for which the costs do increase.
\paragraph{}
The previously discussed differences between MapReduce and BSP lead to certain inefficiencies when implementing parallel algorithms designed in BSP on the MapReduce framework. The most evident, as also shown in the above simulation, is that the map phase is mostly ignored, with map tasks assigned only the identity function. This is due to the fact that it is not possible to pass data to specific map tasks. The only exception is the first round of computation, since data can be specifically partitioned in such a way as to have all relevant data passed to the same map task. However, in a general setting, if in BSP two processors each send data to a third processor, this cannot be simulated using map tasks. Such direct communication can be achieved in the reduce phase since the shuffle step merges all the data with the same key and sends it to a single reduce task. The reduce task's id can be used as a key to achieve direct communication. Therefore, unless the algorithm only has two supersteps, the map phase is not used for computation.
\paragraph{}
Using only the reduce phase does not increase the theoretical costs of an algorithm, and given the simplicity provided by the MapReduce framework, incurring some extra cost is allowed in practice. However, when certain BSP algorithms are implemented on the MapReduce framework, the asymptotic costs of the algorithm do increase. When a map or reduce task is finished, the local primary memory is cleared and the new map or reduce task's input data is loaded. This allows the framework to achieve automatic load balancing and fault tolerance. However, any data that needs to be used in later rounds has to be stored in global memory, inherently increasing the communication cost of the algorithm. If this extra cost is less than the algorithm's actual communication cost, then the algorithm can be efficiently implemented in MapReduce, otherwise it should not. Also, for certain BSP algorithms the communication cost is dominated by the cost of reading the input, with the cost in the following supersteps being significantly less. BSP algorithms do not cater for this feature, and the size of data stored on local memory for use in the following supersteps is not measured in the cost model.
\paragraph{}
A simple extension to the BSP model is proposed to allow classification of BSP algorithms into those that can and cannot be efficiently implemented on the MapReduce framework. $BSPMR(p,g,l)$ is a cost model exactly the same as BSP, with the same definitions for measuring an algorithm's efficiency, except for an extra cost $f_s$. This cost $f_s$ is the maximum size of data, over all the processors, that has to be stored in local memory, in superstep $s$, for use in later supersteps. The total cost of storing local data over all supersteps is $F = \sum_{s=1}^{S} f_s$. The cost of the algorithm is still $W + H \cdot g + S \cdot l$ when implemented on a conventional parallel system, but changes to $W + (H + F) \cdot g + S \cdot l$ when implemented on MapReduce.
\paragraph{}
Let the communication cost of a BSPMR algorithm not including the cost of reading the input and writing the output be $H_{n}$.
\begin{theorem}
\label{lemmaEfficiency}
Any $\mathrm{BSPMR}$ algorithm implementable on a $\mathrm{BSP}$ computer having $p$ processors with costs $W$, $H$, $F$, $S$ can be efficiently implemented in $\mathrm{MR}(v,g,l)$ with $v = p$ using the simulation provided in \cite{Goodrich} only if $(i)$ $O(T) = W \cdot p$, $(ii)$ $O(C) = H \cdot p$, $(iii)$ $O(D) = S$, and $(iv)$ $O(F) = H_{n}$.
\end{theorem}
\begin{proof}
$(i)$, $(ii)$, $(iii)$ Trivial.
$(iv)$ If the cost of reading the input on a BSP computer is $n$, the cost in MR is $O(n)$, since at most each value is given a key of constant size. The same applies to the output of the algorithm. This cost is included in $H$, but not in $H_{n}$, which only includes the cost of communicating local data between supersteps. Therefore, if $O(F) > H_{n}$, then the cost of implementing the algorithm on MapReduce is higher than implementing the algorithm on a conventional parallel system. \Square
\end{proof}
\section{Conclusion}
The MapReduce framework has generated great interest in the area of parallel algorithm design due to the simplicity with which parallel algorithms can be developed. This wide use of the framework demands that it is put on sound theoretical foundations, which is essential if the potential and limitations of the framework are to be fully understood.
\paragraph{}
A number of theoretical models for the framework have already been proposed, but each poses limitations on the framework to enforce parallel algorithm design. These are not necessary for efficient design, as shown by previously existing models such as BSP. This work shows that the BSP model can in fact be used to model MapReduce algorithms, with the same asymptotic cost for conventional systems with identical number of processors. It is, therefore, determined that MapReduce should be used purely as an implementation framework with BSP used to design the algorithms.
\paragraph{}
This work also discusses the differences between MapReduce and BSP, highlighting the fact that BSP algorithms which require data to be stored and accessed in multiple supersteps cannot be efficiently implemented using MapReduce if the number of supersteps is not constant. This is due to the nature of the framework which does not allow for storage between rounds but requires that this data is communicated between such rounds. However, since the MapReduce framework provides a simple abstraction on top of the intricate parallel development challenges, it may still be deemed more feasible in practice to use the framework despite its inefficiencies.
\section{Introduction}
Efficient algorithms are of fundamental importance in a world driven by computation. The amount of data in today's world is constantly increasing at a dramatic rate, and this poses a great challenge to algorithm design. The web graph today is much larger than could have probably been imagined 20 or even 10 years ago. Other examples of such large datasets abound, from search logs to financial data.
\paragraph{}
One framework of processing massive datasets, called MapReduce, has been attracting great interest. It was developed and widely used by Google \cite{MapReduceOrig}, while its open source implementation Hadoop \cite{Hadoop} is currently used by more than 100 companies worldwide including eBay, IBM, Yahoo!, Facebook, and Twitter, along with a number of universities \cite{HadoopUsers}. Some companies, such as Amazon and Microsoft, allow users to run MapReduce programs on their cloud \cite{AmazonElastic,Daytona}. MapReduce algorithms have been used to solve a number of non-trivial problems in diverse areas including data processing, data mining and graph analysis (see \cite{MapReduceAlg1,MapReduceAlg2} for an introductory list of works).
\paragraph{}
The popularity of MapReduce means that it now plays a prominent role in the field of parallel computation. This creates the demand to put MapReduce on sound theoretical foundations, and to establish its relationship to other major parallel computation models such as BSP and PRAM. In this regard, major work has already been done by Feldman et al. \cite{Feldman}, Karloff et al. \cite{Karloff} and Goodrich et al. \cite{Goodrich}. Between them, these authors have shown that MapReduce can be used to simulate the PRAM and BSP model, as well as link MapReduce to a subclass of streaming algorithms.
\paragraph{}
This work aims to further analyse the connection between MapReduce and BSP. Section 2 introduces the MapReduce framework, followed by an introduction to the BSP model in section 3. In section 4, a model for MapReduce is developed and the links between this model and BSP are discussed. A simulation of MapReduce on BSP is given in section 5, while section 6 defines a subclass of algorithms that can be efficiently implemented in MapReduce along with a few examples. The final section provides some concluding views.
\section{MapReduce framework}
Parallel programming is, unfortunately, not straightforward. The developer needs to cater for a number of non-trivial problems such as fault tolerance, load balancing and synchronisation. In their seminal 2004 paper, Dean and Ghemawat \cite{MapReduceOrig} introduced the parallel computation framework MapReduce. This framework allows for simplified programming on large clusters of low-end systems. The simplicity of MapReduce is due to the fact that it acts as an abstraction on top of the complex details that need to be catered for when writing parallel code. This allows the programmer to focus on the functions that actually manipulate the data at hand.
\paragraph{}
The main idea for MapReduce comes from functional programming languages such as Lisp \cite{Lisp}. The Lisp function Map takes as input a function and a sequence, and applies this function to all the elements in the sequence. The function Reduce, given a sequence of values and a binary function, uses this function to combine the elements of the sequence into a single output value. These two functions, performed in rounds, form the basis of MapReduce.
\paragraph{}
MapReduce was originally designed to run on large clusters of low-end commodity machines. Every machine has a processor, a fast primary memory and a slower secondary memory, and is connected via an underlying network to the rest of the cluster. The secondary memory is used as part of a global shared memory, with each machine allowed to access other machines' secondary memory remotely, although only during synchronisation. The framework works on data in the form of $\langle key;value \rangle$ pairs, with $n$ initial such pairs stored in global memory as the algorithm's input. A MapReduce algorithm proceeds in rounds, with two phases in every round, a map and a reduce phase. Each phase is composed of an input, a computation and an output step, with the output of each phase used as input to the next phase. When a phase is finished, i.e. when every machine has written its output data to shared memory, the data is synchronised, i.e. each machine is allowed to read the data written in the previous phase. No other communication is allowed between machines, except with the master processor as described later. The local primary memory is cleared before each synchronisation. Parallelism is achieved by having different machines perform the same functions on different data.
\paragraph{}
System failures are common in clusters of hundreds or thousands of low-end systems, and therefore automatic fault tolerance and load balancing play a crucial role in the design of the MapReduce framework. These are achieved by having each machine work on multiple tasks, making it easier to reprocess and reassign these tasks in case of machine failure. One system processor is assigned the role of master processor and controls how these tasks are assigned across the other worker processors. Each task reads its input and processes it using either a map or reduce function designed by the developer. Each function acts on a single $\langle key;value \rangle$ pair, with tasks computing a function for every input pair assigned to them. MapReduce was initially designed to cater for algorithms where the output was much smaller than the input, so the number of map tasks $q$ was much larger than the number of reduce tasks $r$. Typically, in a system with $p$ processors, $q$ was between 10 and 100 times $p$ and $r$ was around 2 to 5 times $p$, as described in the original paper \cite{MapReduceOrig} and the official Hadoop tutorial \cite{HadoopTutorial}. However, subsequently algorithms have been designed that use the same number of map and reduce tasks.
\paragraph{}
The first round of MapReduce proceeds as follows. The algorithm's input is placed in global memory, split into $q$ parts. Each part will be processed by a separate map task, one pair at a time. The master then assigns map tasks to workers. The number of map tasks is typically greater than the number of processors, so one task is initially assigned to each worker processor. When a processor finishes computing its task a new map task is assigned to it. Upon being assigned a new task, a processor reads the input data pertaining to the task from global memory to primary memory and processes it. The output of each map task, in the form of $\langle key;value \rangle$ pairs, is stored in the worker's secondary memory.
\paragraph{}
While in the map phase, every map task processes various $\langle key;value \rangle$ pairs, in the reduce phase all the values for a given key are processed by a single reduce task. This is achieved by logically partitioning the secondary memory of each worker processing a map task into $r$ partitions, and then determining in which particular partition an output pair should be stored, a process accomplished by the shuffle step. This step can be viewed as a data routing step, determining which reduce task will process a data pair based on the pair's key. This function is performed by the workers while processing the map tasks. Typically, a function such as $(hash(key)\bmod r)$ is used, where the $hash$ function is a simple function, computable in a small constant time, used to map the keys to a more manageable domain. Other partitioning functions can be defined by the user, especially if the keys are in numeric form, such as partitioning the keys into $r$ logical partitions representing various ranges of values.
\paragraph{}
When all the map tasks have finished, the $r$ reduce tasks are assigned to the available workers using the same process as for the map tasks. Each reduce task accesses the data assigned to it, stored across the workers responsible for computing the $q$ map tasks. All the pairs with the same key are stored in the same partition, and each partition can have pairs with different keys. All this data is sorted by key and combined such that all values associated with a key are grouped together in a single $\langle key;value \rangle$ pair. This is sometimes considered as being a second part of the shuffle step.
\paragraph{}
Each reduce task then reads its assigned data and processes it one $\langle key;value \rangle$ pair at a time using the reduce function. The task's output is then written to global memory, and can either be the final output of the algorithm or used as input to a new round of MapReduce. The input for a new round of MapReduce is partitioned into $q$ parts by the master processor and the process just described is repeated.
\paragraph{}
The relationship between the system processors and the map and reduce tasks leads to some interesting aspects of the MapReduce framework. A number of map and reduce tasks can be performed, in sequence, by a single processor in each round. In each of the map and reduce phases, tasks are assigned to workers as these finish their previously assigned task. Therefore, if computation is equally divided between tasks, then every processor will perform about $\sfrac{q}{p}$ map tasks and $\sfrac{r}{p}$ reduce tasks. If on the other hand the computation time differs for each task, then load balancing is automatically achieved. It also allows for the efficient handling of fault tolerance. However, the task assignment strategy also places some limitations on the framework. Between rounds the data is split up into $q$ parts and each is assigned to a map function. After any task finishes, the worker's primary memory is cleared, so data cannot be associated with a single processor and accessed at will in different rounds. Therefore, any data that is required in multiple rounds should be specifically stored in global memory.
\paragraph{}
Given a multiset of $n$ $\langle key ; value \rangle$ pairs as an input, the above process describing a single MapReduce round is defined by two functions: map and reduce, and the shuffle step. These are defined as follows:
\begin{itemize}
\item Given a single input pair from the multiset of the round's input pairs $\{\langle k_1 ; v_1 \rangle, \langle k_2 ; v_2 \rangle, \ldots, \langle k_n ; v_n \rangle\}$, the \textit{map} function performs some computation to produce a new intermediate multiset of $\langle key; value \rangle$ pairs $\{\langle l_1 ; w_1 \rangle, \langle l_2 ; w_2 \rangle, \ldots, \langle l_m ; w_m \rangle\}$.
\item The union of all the intermediate multisets produced by the \textit{map} functions is acted upon in the \textit{shuffle} step. All the pairs with the same key $l_i$ are combined to produce a new set of lists of the form $\langle l_i;w_1,w_2, \ldots \rangle$.
\item Each of the lists produced by the shuffle step is passed to a separate \textit{reduce} function that performs some computation to produce a new list $\langle j_i; x_1, x_2, \ldots \rangle$.
\end{itemize}
\section{A Model for MapReduce}
The popularity of the MapReduce framework requires that the theoretical limitations of the system be analysed, and theoretical models developed for it. The connection between MapReduce and existent models of parallel algorithm design also needs to be investigated. Pioneering work in the field has already been done, linking MapReduce to other parallel models, as well as to other fields of computer science.
\paragraph{}
The first work \cite{Feldman}, by Feldman et al., proposed a model for the framework aimed at linking it to the data stream model. A subclass of MapReduce algorithms, called mud (massive, unordered and distributed) algorithms, is defined and shown to be closely related to a subclass of streaming algorithms. In fact, mud algorithms can simulate any symmetric (order invariant) streaming algorithms with comparable communication and storage costs. The computation cost is exponential due to the use of Savitch's theorem \cite{Sipser}. Symmetric mud algorithms are limited in space and communication to polylogarithmic cost in terms of the algorithm's input size. The model is very restrictive in the type of algorithms that can be designed, since only one round of MapReduce is allowed and the input is assumed to be independent separate streams.
\paragraph{}
Karloff et al. \cite{Karloff}, propose a model that better captures the specifics of the MapReduce framework. The map and reduce phases are clearly defined, and algorithms can have multiple rounds. A limit is placed on the number of processors in the system, such that the input size $n$ is greater than the number of workers. The primary and secondary memory of each worker is limited to $n^{1-\epsilon}$, for some $\epsilon > 0$, as is the size of the input and output data for each worker in a specific round. The size of global shared memory is in turn bound by the size of the secondary memory of each worker. The authors show that a subclass of EREW PRAM algorithms can be simulated using the model.
\paragraph{}
The most recent model (to the best of our knowledge) is due to Goodrich et al. \cite{Goodrich} which is based on BSP. The size of data that can be sent or received by each reduce task is limited to a value $M$ determined by the algorithm designer. Apart from this restriction, the model differs from BSP in the way the communication cost is calculated. Simulations of CRCW PRAM and BSP algorithms are presented.
\paragraph{}
The idea behind placing limits on I/O size and storage size is to enforce parallelism in the algorithm design, as discussed in \cite{Karloff} and \cite{Goodrich}. If this is not enforced, algorithms could be designed that trivially place all the data on a single processor and run the sequential algorithm. This should not be of any concern if the algorithm designer is aiming to move from sequential to parallel design, and such limitations are not posed in existent models like PRAM and BSP.
\paragraph{}
From the work presented in \cite{Goodrich} and the discussions in the previous two sections it is evident that the relationship between BSP and MapReduce is very strong. Both handle parallel algorithm design in a coarse-grained fashion, interleaving phases of computation and communication. Both can be used to design algorithms running on clusters of low-end systems connected with point-to-point communication, and both make use of synchronisation between rounds/supersteps. The aim of the rest of this work is to further investigate this relationship.
\paragraph{}
Goodrich et al. \cite{Goodrich} showed that all BSP algorithms are implementable in MapReduce by only using the reduce phase, setting the map function to the identity function. However, they did not discuss the efficiency of implementing BSP algorithms on the framework, given its differences to conventional parallel systems. The main difference is that after every map or reduce task finishes, the worker on which they were computed clears the primary local memory. Any data which might need to be used in the following rounds has to be stored in global memory, increasing the communication costs in the process.
\paragraph{}
Another aspect of MapReduce that differs from BSP is the use of map and reduce tasks running on physical processors. These tasks can be viewed as virtual processors, with multiple virtual processors running on top of physical processors. In the original BSP paper \cite{Valiant}, multiple virtual processors were introduced to design BSP with automatic memory management. Work is divided between the $v$ virtual processors, with each of the $p$ physical processors performing the work of $v/p$ virtual processors. If the work and communication are equally divided between the virtual processors, then work and communication will also be balanced amongst the physical processors. Similarly, if the work is balanced between the map and reduce functions in each MapReduce round, then every processor will process $\sfrac{q}{p}$ map tasks and $\sfrac{r}{p}$ reduce tasks.
\paragraph{}
BSP does not allow any communication between processors in between synchornisation. Similarly, in MapReduce, map and reduce tasks are not allowed to communicate between each other in the same phase. However, asynchronous communication is used between the master processor and the workers to assign these tasks. This allows for dynamic load balancing to be achieved when the individual computation time of the tasks are not balanced. Also, for tasks with unknown exact computation time an offline balanced distribution between processors cannot be found, so dynamic load balancing is used.
\paragraph{}
Finally, while one BSP superstep involves a single computation step, and corresponding input and output communication phases, a round of MapReduce is made up of the map and reduce computation phases with their respective input and output communication. Also, the BSP model assumes that data can be sent directly to any processor in the system, but while data can be directed to specific reduce tasks using specific keys, there is no way of determining which map task will process which part of the input data.
\paragraph{}
Given the similarities between BSP and MapReduce, a model for MapReduce, $MR(p,g,l)$, is proposed, that is based on BSP. The model has three parameters: the number of physical processors $p$ in the system, the inverse bandwidth $g$ and the system latency $l$ which is the time taken by the shuffle network to set up communication between the map and reduce phases. The number of map tasks $q$ and reduce tasks $r$ to be used are chosen by the algorithm designer, and can vary depending on the algorithm being designed. In specific rounds, a subset of the $r$ reduce tasks can be used instead of all the available tasks. This is not allowed in the case of the map tasks. For the rest of this work, it is assumed that for input size $n$, $n \gg q$ and $n \gg r$, and $q > p$ and $r > p$.
\paragraph{}
Let $q_1, q_2, \ldots$ and $r_1, r_2, \ldots$ denote the specific map and reduce tasks respectively in a single round, and let $R_d$ denote the number of reduce tasks used in a round $d$. The size of data written to or read from shared memory between rounds is denoted by $c_{q_1}, c_{q_2}, \ldots$ for map tasks and $c_{r_1}, c_{r_2}, \ldots$ for reduce tasks. Finally, $t_{q_1}, t_{q_2}, \ldots$ and $t_{r_1}, t_{r_2}, \ldots$ denote the running time of the specific map and reduce tasks respectively.
\paragraph{}
Given the definitions above, for a single round $d$, the computation cost is defined as $T_d = (\sum_{i=1}^{q} t_{q_i}) + (\sum_{i=1}^{R_d} t_{r_i})$. The communication cost for round $d$ is $C_d = (\sum_{i=1}^{q} c_{q_i}) + (\sum_{i=1}^{R_d} c_{r_i})$. A MapReduce algorithm therefore has cost $T + C \cdot g + D \cdot l$, where $T = \sum_{d=1}^{D} T_d$, $C = \sum_{d=1}^{D} C_d$, and $D$ is the number of rounds.
\paragraph{}
It is assumed that since the work done by each task involves at least reading and writing its input and output data from shared memory, then $T \geq C$.
\paragraph{}
The model allows for cost measurements that better reflect the load balancing properties of the MapReduce framework. The MapReduce dynamic load balancing framework leads to a maximum processing time on each processor of $(2 - \sfrac{1}{p}) OPT$, as discussed in \cite{Graham}, where $OPT$ is the optimal maximum processing time on each processor. For balanced algorithms in which the computation and communication costs are equally divided amongst the map and reduce tasks, each machine will handle around $\lceil \sfrac{q}{p} \rceil$ map and $\lceil \sfrac{R_d}{p} \rceil$ reduce tasks in a single round $d$, given that no machines in the cluster fail.
\section{MapReduce on BSP}
The rise in popularity of the MapReduce framework has renewed interest in parallel computing. Still, irrespective of this popularity, it has not been determined whether MapReduce allows for better algorithm design than previously existent models such as BSP, which has been one of the most prominent models for parallel algorithm design for over two decades. It is, therefore, important to ask whether any MapReduce algorithm can be simulated in BSP while preserving its asymptotic costs?
\begin{theorem}
Any round $d$ of an $\mathrm{MR}(p,g,l)$ algorithm with known individual task times and a maximum execution time $t_{max}$ on any processor, can be simulated by a $\mathrm{BSP}(p,g,l)$ machine in $O(1)$ supersteps, such that the maximum execution time on any processor is $O(t_{max})$.
\end{theorem}
\begin{proof}
An MR round can be split into two separate phases, the map phase and the reduce phase. Each phase involves reading input data, performing some computation and outputting any results, in order. These two phases can be simulated by two BSP supersteps $s'$ and $s''$.
\paragraph{}
The $q$ map tasks and $R_d$ reduce tasks in MapReduce are distributed amongst the $p$ processors as these become idle. Since the computation time for each task is known, then a optimal distribution of tasks amongst the processors exists which reduces the maximum processing time on each processor, also known as the makespan. Let this optimal makespan be $t_{OPT}$, then Graham \cite{Graham} states that any distribution of the tasks amongst the processors, as these become idle, will lead to a makespan of $(2 - \sfrac{1}{p}) t_{OPT}$, i.e. $t_{max} = O(t_{OPT})$. Such a distribution can be found offline, by using the same method employed by MapReduce or using the PTAS presented \cite{Hochbaum}, and the tasks distibuted amongst the BSP processors makespan of $O(t_{max})$.
\paragraph{}
The shuffle phase in MapReduce determines how data is distributed amongst the reduce tasks in the map phase. This can be performed while simulating the $q$ map tasks. The data assigned to each reduce task is then sorted and combined by key in the reduce phase, which can be performed during the simulation of the $R_d$ reduce tasks.
\paragraph{}
If the MR round $d$ is the last round of the algorithm, then the results are stored in global memory and computation stops. When the output of a reduce phase serves as input to a new MR round, the input is stored in global memory and divided into $q$ equal parts by the master processor. Each map task is then assigned one such part. This distribution can be achieved in BSP by adding an extra superstep. Each processor sends the size of its data to a designated master processor, which determines how the data has to be evenly distributed amongst the processors and communicates this to the rest of the processors. The processors then communicate the actual data between them according to the determined distribution. The cost of this procedure is $w = O(p)$, $h = O(p)$ and $s = O(1)$. Since the computation and communication costs of the algorithm are functions of $n$, and $n \gg p$, then the total cost of the algorithm does not increase due to this procedure. \Square
\end{proof}
\paragraph{}
Looking at the simulations and the way the MapReduce framework works it is evident that, for algorithms with known computation time for the individual tasks, designing such algorithms in MapReduce does not provide any asymptotic speed-up over design in BSP. MapReduce forces the designer to cater for features such as the shuffle step, the lack of physical memory storage between rounds, the lack of specifiable association between data and map tasks in the map phase, and the intrinsic distinction between map and reduce tasks and how these process and output data. BSP does not enforce such limitations, but these can be introduced by the designer if necessary, allowing a much more flexible algorithm design.
\paragraph{}
This flexibility provided by BSP over MapReduce makes it a natural choice for algorithm design. Performing algorithm design in BSP also means that the algorithms can be implemented on several different systems, not just MapReduce. It would also be an important step towards consolidating the field of parallel algorithm design to use a single theoretical model. For tasks with unknown execution time, however, MapReduce should still be used since BSP does not have dynamic load balancing capabilities.
|
{
"timestamp": "2012-06-19T02:00:58",
"yymm": "1203",
"arxiv_id": "1203.2081",
"language": "en",
"url": "https://arxiv.org/abs/1203.2081"
}
|
\section{Introduction}
\label{seq:intro}
The relaxation of an abundant spin population is affected by a rare spin population owing to inter- and intramolecular magnetization transfer processes mediated by scalar or dipolar couplings or chemical exchange \cite{wolff_nmr_1990}. As a consequence, by selective radio frequency (\textit{rf}\xspace) irradiation of a coupled rare population not only the relaxation dynamics, but also the steady-state magnetization of the abundant population can be manipulated. Due to this preparation, the NMR signal of the abundant population contains additional information on the rare population and its interactions.
In this context, we analyze two experiments : chemical exchange saturation transfer (CEST\xspace) \cite{zhou_chemical_2006} and off-resonant spin-lock (SL\xspace).
CEST\xspace and SL\xspace experiments are commonly applied to enhance the NMR sensitivity of protons in diluted metabolites \invivo \cite{zhou_using_2003,cai_magnetic_2012,jin_magnetic_2012,ling_assessment_2008} yielding an imaging contrast for different pathologies \cite{jia_amide_2011,schmitt_new_2011,zhou_amide_2011,gerigk_7_2012,schmitt_cartilage_2011}.
The normalized z-magnetization after irradiation at different frequencies, the so-called Z-spectrum, is affected by relaxation and irradiation parameters. In the following, the large pool of water protons is called pool \textit{a}\xspace and the pool of \add{dilute} protons pool \textit{b}\xspace.
To obtain a pure contrast that depends only on the exchanging pool \textit{b}\xspace, concomitant effects like direct water saturation or partial labeling of the exchanging proton pool must be taken into account in modeling of Z-spectra.
Similarities between CEST and SL have been noticed before \cite{jin_spin-locking_2011,vinogradov_pcest:_2012}. Here we consider the projection factors which are required for application of static and dynamic solutions derived for SL\xspace to CEST\xspace experiments and vice versa. We demonstrate how the experimental data have to be normalized that the dynamics of CEST\xspace and SL\xspace can be described by one single eigenvalue, namely \ensuremath{R_{1\rho}}\xspace, the longitudinal relaxation rate in the rotating frame. A first approximation for \ensuremath{R_{1\rho}}\xspace including chemical exchange was published by Trott and Palmer \cite{trott_r1rho_2002}. \add{In the present article, this approach is extended by inclusion of \R{2b}, the transverse relaxation rate of pool \textit{b}\xspace.}
An interesting CEST effect is amide proton transfer (APT) of $^1H$ in the backbone of proteins, because quantitative determination of the exchange rate may allow noninvasive pH mapping \cite{sun_imaging_2008}.
The exchange rate \ensuremath{k_{b}}\xspace for APT is relatively small (\ensuremath{k_{b}}\xspace=$28.6\pm 7.4$~Hz \cite{zhou_chemical_2006}) compared to the transversal relaxation rate of the amide proton pool $\R{2b}=1/T_{2b}$. Sun et al. measured $T_{2b}$ of 8.5~ms ( $\R{2b}=90.9~\text{Hz}$) for amine protons of aqueous creatine at \ensuremath{B_{0}}\xspace=9.4~T. For amino protons in ammonium chloride dissolved in agar gel, $T_{2b}=40 \text{ ms } (\R{2b}=25 \text{ Hz}$) was found at \ensuremath{B_{0}}\xspace=3~T \cite{desmond_understanding_2012}. Thus, \R{2b} in tissue may be in the range of or even surpass \ensuremath{k_{b}}\xspace and must be taken into account for quantification of \ensuremath{k_{b}}\xspace.
For systems with strong hierarchy in the eigenvalues - as it is the case for diluted spin populations - we present an approximation for \ensuremath{R_{1\rho}}\xspace that includes \R{2b} and provide an analytical solution for CEST\xspace and SL\xspace experiments valid for exchange rates in the range of \R{2b}.
\section{Theory}
\label{seq:theo}
CEST\xspace and SL\xspace experiments for coupled spin systems can be described by classical magnetization vectors \ensuremath{\vec{M}}\xspace in Euclidean space governed by the Bloch-McConnell (BM\xspace) equations \cite{mcconnell_reaction_1958}. We consider a system of two spin populations: pool \textit{a}\xspace (abundant pool) and pool \textit{b}\xspace (rare pool) in a static magnetic field $\vec{\ensuremath{B_{0}}\xspace} = (0, 0, \ensuremath{B_{0}}\xspace)$, with \add{forward rate} \ensuremath{k_{b}}\xspace and thermal equilibrium magnetizations \ensuremath{M_{\mathit{0,a}}}\xspace and \ensuremath{M_{\mathit{0,b}}}\xspace, respectively. The relative population fraction $\frac{\ensuremath{M_{\mathit{0,b}}}\xspace}{\ensuremath{M_{\mathit{0,a}}}\xspace}=\ensuremath{f_{b}}\xspace$ is conserved by the back exchange rate $\ensuremath{k_{a}}\xspace = \ensuremath{f_{b}}\xspace\ensuremath{k_{b}}\xspace$.
The 2-pool BM\xspace equations are six coupled first-order linear differential equations
\begin{align}
\label{eqn:BM_diff}
\dot{\vec{M}}=\mathbf{A}\cdot \vec{M} + \vec{C}, \quad
\ensuremath{\mathbf{A}}\xspace=
\begin{bmatrix}
\mathbf{L}_a-\ensuremath{f_{b}}\xspace \mathbf{K} & +\mathbf{K} \\
+\ensuremath{f_{b}}\xspace \mathbf{K} & \mathbf{L}_b-\mathbf{K}
\end{bmatrix},
\end{align}
where (i = a,b)
\begin{align}
\mathbf{L}_i=
\begin{pmatrix}
-\R{2{i}} & -\ensuremath{\Delta\omega}\xspace_{i} &0 \\
+\ensuremath{\Delta\omega}\xspace_{i} &-\R{2{i}} &-\ensuremath{\omega_{1}}\xspace \\
0 &+\ensuremath{\omega_{1}}\xspace &-\R{1{i}}
\end{pmatrix}, \quad
\label{eqn:BM_KL}
\mathbf{K}=
\begin{pmatrix}
\ensuremath{k_{b}}\xspace &0 &0 \\
0 &\ensuremath{k_{b}}\xspace &0 \\
0 &0 &\ensuremath{k_{b}}\xspace
\end{pmatrix}, \\
\label{eqn:BM_C}
\vec{C}=\begin{pmatrix}
&0, &0, &\R{1a}\ensuremath{M_{\mathit{0,a}}}\xspace, &0, &0, &\R{1b}\ensuremath{M_{\mathit{0,b}}}\xspace
\end{pmatrix}^\text{T},
\end{align}
given in the rotating frame $(x,y,z)$ defined by \textit{rf}\xspace irradiation with frequency \ensuremath{\omega_{rf}}\xspace. $\ensuremath{\Delta\omega}\xspace=\ensuremath{\Delta\omega}\xspace_a=\ensuremath{\omega_{rf}}\xspace-\ensuremath{\omega_{a}}\xspace$ is the frequency offset relative to the Larmor frequency \ensuremath{\omega_{a}}\xspace of pool \textit{a}\xspace (for $^1H \; \ensuremath{\omega_{a}}\xspace/\ensuremath{B_{0}}\xspace = \gamma = \mathrm{267.5~\frac{rad}{\mu T s}}$). The offset of pool \textit{b}\xspace $\ensuremath{\Delta\omega_{b}}\xspace=\ensuremath{\omega_{rf}}\xspace-\ensuremath{\omega_{b}}\xspace=\ensuremath{\Delta\omega}\xspace-\ensuremath{\delta_{b}}\xspace\ensuremath{\omega_{a}}\xspace$ is shifted by \ensuremath{\delta_{b}}\xspace (chemical shift) relative to the abundant-spin resonance. \add{In contrast to Ref. \cite{trott_r1rho_2002}, we allow different relaxation rates \R{1} and \R{2} for the pools. The assumption of their equality is only valid if $|\R{1a}-\R{1b}|\ll\ensuremath{k_{b}}\xspace$ or $|\R{2a}-\R{2b}|\ll\ensuremath{k_{b}}\xspace$ \cite{miloushev_r1_2005}.} Longitudinal relaxation rates $\R{1,a/b} = 1/T_{1,a/b}$ are in the order of Hz, while transverse relaxation rates $\R{2,a/b} =1/T_{2,a/b}$ are 10-100~Hz. For semisolids \R{2b} can take values up to $10^6$~Hz. The \textit{rf}\xspace irradiation field $\vec{\ensuremath{B_{1}}\xspace} = (\ensuremath{B_{1}}\xspace, 0, 0)$ in the rotating frame, with $\ensuremath{B_{1}}\xspace \approx \mu T$, induces a precession of the magnetization with frequency $\ensuremath{\omega_{1}}\xspace=\gamma\cdot\ensuremath{B_{1}}\xspace$ around the x-axis in the order of several 100 Hz. The population fraction \ensuremath{f_{b}}\xspace is assumed to be $<1~\%$, hence \ensuremath{k_{a}}\xspace is 0.01 to 10~Hz.
\subsection{\add{Solution of the Bloch-McConnell equations for asymmetric populations}}
The BM equations \eqref{eqn:BM_diff} are solved in the eigenspace of the matrix \ensuremath{\mathbf{A}}\xspace leading to the general solution for the magnetization
\begin{align}
\label{eqn:BM_generalsol}
\vec{M}(t)=\sum_{n=1}^{6}e^{\lambda_n t}\vec{v}_n+\ensuremath{\vec{M}^{\mathit{ss}}}\xspace ,
\end{align}
where $\lambda_n $ is the nth eigenvalue with the corresponding eigenvector $\vec{v}_n$ of matrix \ensuremath{\mathbf{A}}\xspace and \ensuremath{\vec{M}^{\mathit{ss}}}\xspace is the stationary solution. Two eigenvalues are real and four are complex \cite{trott_r1rho_2002}. They describe precession and, since all real parts of the eigenvalues are negative, the decay of the magnetization towards the stationary state in each pool.
As shown before \cite{trott_theoretical_2004}, if \ensuremath{\Delta\omega}\xspace or \ensuremath{\omega_{1}}\xspace are large compared to the relaxation rates \R{1} and \R{2} and exchange rate \ensuremath{k_{b}}\xspace, the eigensystem of pool \textit{a}\xspace is mainly unaffected. One \add{eigenvector} $\vec{v}_1$ is closely aligned with the effective field $\ensuremath{\vec{\omega}_{\mathit{eff}}}\xspace=(\ensuremath{\omega_{1}}\xspace,0,\ensuremath{\Delta\omega}\xspace)$ which defines the longitudinal direction (\ensuremath{z_{\mathit{eff}}}\xspace) in the effective frame (\ensuremath{x_{\mathit{eff}}}\xspace,\ensuremath{y_{\mathit{eff}}}\xspace,\ensuremath{z_{\mathit{eff}}}\xspace) and is tilted around the y-axis by the angle $\theta = \tan^{-1}(\frac{\ensuremath{\omega_{1}}\xspace}{\ensuremath{\Delta\omega}\xspace})$ off the z-axis (Fig. \ref{fig:EVweff}a). \add{Mathematical derivation (\ref{seq:App_EV}) as well as} numerical evaluations (Fig.\ref{fig:EVweff}b-d) demonstrate that $\vec{v}_1$ and \ensuremath{\vec{\omega}_{\mathit{eff}}}\xspace are collinear in good approximation if $(\R{2a}-\R{1a})$ is much smaller than \ensuremath{\omega_{\mathit{eff}}}\xspace.
\begin{figure}[H]
\includegraphics{1_EVweff.eps}
\caption{(a) Geometry of the vectors in the rotating frame. (b-d) \add{Cosine of the angle $\beta$ between} the eigenvector of the smallest eigenvalue and \ensuremath{\vec{\omega}_{\mathit{eff}}}\xspace. (b) In the far off-resonant case both vectors are parallel. Near resonance \ensuremath{\omega_{1}}\xspace has to be strong to keep them parallel. (c) The assumption of collinearity is still valid if pool \textit{b}\xspace with relative concentration $\ensuremath{f_{b}}\xspace < 10\%$ is coupled to the water pool. (d) Large differences in \R{2a} and \R{1a} lead to an increasing angle between the vectors, but even for $\R{2a}\approx 50Hz$ and $\R{1a}\approx 1$Hz both vectors are still collinear in good approximation.
The eigenvector and the effective field vector are collinear if \ensuremath{\omega_{\mathit{eff}}}\xspace is large compared to $(\R{2a}-\R{1a})$ \add{(\ref{seq:App_EV})} and $\ensuremath{f_{b}}\xspace < 10\%$ -- both is fulfilled for CEST experiments since metabolite concentrations are small and frequency offsets of interest are mostly larger than several 100 rad/s. }
\label{fig:EVweff}
\end{figure}
The collinearity of the corresponding eigenvector and the effective field is the principal reason why off-resonant SL\xspace and CEST\xspace exhibit the same dynamics. For an appropriate analysis of a saturation experiment it is mandatory to identify the initial projections on the eigenvectors and the measured components.
\ensuremath{\vec{B}_{0}}\xspace and \ensuremath{\vec{M}_0}\xspace are parallel to the z-axis, the preparation is a projection of the longitudinal magnetization along~z onto the effective frame
\begin{align}
\label{eqn:hintrafoz}
M_{\ensuremath{z_{\mathit{eff}}}\xspace}(t=0)&=\cos{\theta} \cdot M_z(t=0)=\ensuremath{P_{\zeff}}\xspace \cdot \ensuremath{M_0}\xspace , \\
\label{eqn:hintrafoxy}
M_{\ensuremath{x_{\mathit{eff}}}\xspace}(t=0)&=\sin{\theta} \cdot \ensuremath{M_0}\xspace ;\quad M_{\ensuremath{y_{\mathit{eff}}}\xspace}(t=0)=0 .
\end{align}
The transversal components induce an oscillation decaying with $T_{2\rho}$ \add{\cite{moran_near-resonance_1995}} which can be neglected in the case of small $\theta$, by averaging over a complete cycle of \ensuremath{\omega_{\mathit{eff}}}\xspace, or by measuring after a delay of $~5\cdot T_{2\rho}$. This simplification leads to the relation for the back projection, via \ensuremath{P_{z}}\xspace, from \ensuremath{z_{\mathit{eff}}}\xspace to~z
\begin{align}
\label{eqn:backtrafoz}
M_{z}(t)=\cos{\theta} \cdot M_{\ensuremath{z_{\mathit{eff}}}\xspace}(t)=\ensuremath{P_{z}}\xspace \cdot M_{\ensuremath{z_{\mathit{eff}}}\xspace}(t).
\end{align}
Since we identified the effective frame as the eigenspace of the magnetization, Eq.~\eqref{eqn:BM_generalsol} can be written as an exponential decay law with the eigenvalue \ensuremath{\lambda_{1}}\xspace associated with the \ensuremath{z_{\mathit{eff}}}\xspace~direction. Let the normalized magnetization be $\ensuremath{Z}\xspace=\frac{M_{\mathit{z,a}}}{\ensuremath{M_{\mathit{0,a}}}\xspace}$ and, for the stationary solution, $\ensuremath{{Z^{\mathit{ss}}}}\xspace=\frac{\ensuremath{{M}^{\mathit{ss}}}\xspace_{\mathit{z,a}}}{\ensuremath{M_{\mathit{0,a}}}\xspace}$. Then Eq.~\eqref{eqn:BM_generalsol}, taken for the \ensuremath{z_{\mathit{eff}}}\xspace direction, yields the dynamic solution for the z-magnetization
\begin{align}
\label{eqn:Z_full_solution_apex}
\ensuremath{Z}\xspace(\ensuremath{\Delta\omega}\xspace,\ensuremath{\omega_{1}}\xspace,t)=(\ensuremath{P_{z}}\xspace\ensuremath{P_{\zeff}}\xspace-\ensuremath{{Z^{\mathit{ss}}}}\xspace)\cdot e^{\ensuremath{\lambda_{1}}\xspace\cdot t}+\ensuremath{{Z^{\mathit{ss}}}}\xspace
\end{align}
Without preparation pulses $\ensuremath{P_{z}}\xspace = \ensuremath{P_{\zeff}}\xspace \approx \cos{\theta}$ (CEST\xspace experiment).
If a preparation pulse with flip angle $\theta$ is applied before and after cw irradiation the projection factors are $\ensuremath{P_{z}}\xspace = \ensuremath{P_{\zeff}}\xspace \approx 1$ (SL\xspace experiment), hence oscillations are suppressed (Fig.~\ref{fig:1}), but still persist since \ensuremath{z_{\mathit{eff}}}\xspace is not perfectly collinear with the eigenvector.
Transformation of Eq.~\eqref{eqn:BM_diff} into the effective frame and setting $\frac{\text{d}}{\text{d}t}\ensuremath{\vec{M}}\xspace = 0$ yields the steady-state solution \add{(\ref{seq:App_EV})}
\begin{align}
\label{eqn:Zss}
\ensuremath{{Z^{\mathit{ss}}}}\xspace(\ensuremath{\Delta\omega}\xspace,\ensuremath{\omega_{1}}\xspace)=-\frac{\ensuremath{P_{z}}\xspace\cdot R_{1a}\cdot\cos{\theta}}{\lambda_1}.
\end{align}
It is important to note that in the case where the steady-state is non-zero, it is locked along the corresponding eigenvector. Equations \eqref{eqn:Z_full_solution_apex} and \eqref{eqn:Zss} agree with the full solution previously found for SL\xspace by Jin et al. \cite{jin_magnetic_2012} but extend it for CEST\xspace.
\add{To obtain a pure dynamic quantity independent of the steady-state we rearrange Eq.\eqref{eqn:Z_full_solution_apex} \add{and define}}
\begin{align}
\label{eqn:Zs}
\ensuremath{\tilde{Z}}\xspace(\ensuremath{\Delta\omega}\xspace,\ensuremath{\omega_{1}}\xspace,t)\equiv\frac{Z-\ensuremath{{Z^{\mathit{ss}}}}\xspace}{\ensuremath{P_{z}}\xspace\ensuremath{P_{\zeff}}\xspace-\ensuremath{{Z^{\mathit{ss}}}}\xspace}=e^{\ensuremath{\lambda_{1}}\xspace\cdot t}.
\end{align}
\add{Eqs. \eqref{eqn:Zss} and \eqref{eqn:Zs} are the central formulas in this article.}
In fact, the description of SL\xspace and CEST\xspace experiments differs in the projection factors \ensuremath{P_{z}}\xspace and \ensuremath{P_{\zeff}}\xspace. The intuitive solution $Z_{CEST}=\cos{\theta}\cdot Z_{SL}$ is valid for the steady-state, but not for the transient-state. If the initial magnetization $M_i$ is not fully relaxed and flipped before the saturation pulse by an angle $\beta$, \ensuremath{P_{\zeff}}\xspace changes to $\cos(\theta-\beta)\cdot{M_i/\ensuremath{M_0}\xspace}$.
After understanding of the transition between the two experiments we will now solve the dynamics of CEST and SL experiments by finding the corresponding eigenvalue and verify it numerically.
\begin{figure}[H]
\begin{center}
\includegraphics{2_SL_CEST.eps}
\end{center}
\caption{The full numerical Bloch-McConnell solution (dots) with the proposed normalization (Eq.~\eqref{eqn:Zs}) demonstrates the equivalence of two experiments: chemical exchange saturation transfer (CEST\xspace) without preparation pulses (a); spin-lock (SL\xspace) with preparation and measurement in the effective frame (b). \ensuremath{\tilde{Z}}\xspace of CEST\xspace undergoes oscillations because of residual transversal magnetization in the effective frame. SL\xspace shows no oscillations since the transversal magnetization in the effective frame is zero [\ensuremath{\Delta\tilde{Z}}\xspace= 0, see text] (Eq.~\eqref{eqn:dZs}). Both, SL\xspace and CEST\xspace, show the same monoexponential decay of the z-magnetization with \ensuremath{\lambda_{1}}\xspace (Eq.~\eqref{eqn:SOL_l1_leff_lstrich}) (solid red). For full BM simulations \cite{woessner_numerical_2005} parameters were taken from the amide proton system \cite{zhou_using_2003} in brain white matter \cite{stanisz_t_2005} at \ensuremath{B_{0}}\xspace = 3 T: If not varied, $\R{2a} = 14.5$~Hz, $\R{1a}= \R{1b} = 0.954$~Hz, $\R{2b} = 66.6$~Hz, $\ensuremath{f_{b}}\xspace = 1$~\%, $\ensuremath{k_{b}}\xspace = 25$~Hz, $\ensuremath{\delta_{b}}\xspace = 3.5$~ppm, $\ensuremath{B_{1}}\xspace= 1~\mathrm{\mu T}$, $\ensuremath{t_{\mathit{sat}}}\xspace=1$~s.}
\label{fig:1}
\end{figure}
As already demonstrated for the SL\xspace experiment \cite{trott_r1rho_2002}, the eigenvalue, which corresponds to the eigenvector along the \ensuremath{z_{\mathit{eff}}}\xspace-axis, is the smallest eigenvalue in modulus of the system.
Assuming that all eigenvalues of an arbitrary full-rank matrix \ensuremath{\mathbf{A}}\xspace are much larger in modulus than the smallest eigenvalue, i.e. $|\lambda_1|\ll|\lambda_{2...n}|$, we obtain (see \ref{seq:App})
\begin{align}
\label{eqn:c0c1}
\lambda_1\approx-\frac{c_0}{c_1} ,
\end{align}
where $c_0$ and $c_1$ are the coefficients of the constant and the linear term of the normalized characteristic polynomial, respectively.
We derive the full solution for the smallest eigenvalue by employing the solution of the unperturbed system $(\ensuremath{f_{b}}\xspace=0)$. The solution is $\ensuremath{\lambda_{\mathit{eff}}}\xspace=-\ensuremath{R_{\mathit{eff}}}\xspace$ with the decay rate in the effective frame \ensuremath{R_{\mathit{eff}}}\xspace which was shown to be approximately \cite{trott_theoretical_2004}
\begin{align}
\label{eqn:reff}
-\ensuremath{R_{\mathit{eff}}}\xspace = \R{1a}\cos^2{\theta}+\R{2a}\sin^2{\theta}.
\end{align}
With this eigenvalue of the unperturbed system we can rescale the system by
\begin{align}
\label{eqn:Astrich_rescale}
\ensuremath{\mathbf{A'}}\xspace=\ensuremath{\mathbf{A}}\xspace-\mathbf{I}\cdot\ensuremath{\lambda_{\mathit{eff}}}\xspace
\end{align}
\add{thus shifting the smallest eigenvalue by \ensuremath{R_{\mathit{eff}}}\xspace. The smallest eigenvalue of \ensuremath{\mathbf{A'}}\xspace, still contains terms of \R{1a} and \R{2a}, but represents the exchange-induced perturbation of \ensuremath{R_{1\rho}}\xspace.}
\begin{figure}[H]
\begin{center}
\includegraphics{3_EW1.eps}
\end{center}
\caption{Hierarchy of numerically calculated BM eigenvalues (lines) of the standard system (see caption of Fig. \ref{fig:1}). (b) The rescaled matrix \ensuremath{\mathbf{A'}}\xspace (eq. \eqref{eqn:Astrich_rescale}) has a much stronger hierarchy in the eigenvalues than matrix \ensuremath{\mathbf{A}}\xspace (a). This improves the approximation of the smallest eigenvalue (x, Eq. \eqref{eqn:c0c1}). }
\label{fig:EW}
\end{figure}
The result is a strong hierarchy (Fig. \ref{fig:EW}) in the eigenvalues of \ensuremath{\mathbf{A'}}\xspace if the coupling is small ($\ensuremath{f_{b}}\xspace\ll 1$). Now Eq.~\eqref{eqn:c0c1} can be employed to calculate the eigenvalue $\lambda_1'$ of the matrix \ensuremath{\mathbf{A'}}\xspace to obtain the full solution:
\begin{align}
\label{eqn:SOL_l1_leff_lstrich}
\lambda_{1}=\ensuremath{\lambda_{\mathit{eff}}}\xspace+\lambda_{1}' .
\end{align}
Here $\lambda_{1}' =-c_0'/c_1'$ is the ratio of the coefficients of the characteristic polynomial of the matrix \ensuremath{\mathbf{A'}}\xspace. This analytical procedure gives us a very good approximation of the dynamics of the BM\xspace system.
For further simplification we assume that relaxation of pool \textit{a}\xspace is well described by \ensuremath{R_{\mathit{eff}}}\xspace and the perturbation is dominated by the exchange and relaxation of pool \textit{b}\xspace. We call the exchange-dependent relaxation rate $\ensuremath{R_{\mathit{ex}}}\xspace=-\lambda_1'$. The eigenvalue \ensuremath{\lambda_{1}}\xspace is associated with \ensuremath{z_{\mathit{eff}}}\xspace \add{and} is therefore an approximation of the relaxation rate in the rotating frame $\ensuremath{R_{1\rho}}\xspace\approx-\ensuremath{\lambda_{1}}\xspace$ given by Eqs. \eqref{eqn:reff} and \eqref{eqn:SOL_l1_leff_lstrich}
\begin{align}
\label{eqn:SOL_Rho_reff_rex}
\ensuremath{R_{1\rho}}\xspace(\ensuremath{\Delta\omega}\xspace) = \ensuremath{R_{\mathit{eff}}}\xspace(\ensuremath{\Delta\omega}\xspace) +\ensuremath{R_{\mathit{ex}}}\xspace(\ensuremath{\Delta\omega}\xspace) .
\end{align}
To derive a useful approximation of \ensuremath{R_{\mathit{ex}}}\xspace, we neglect all relaxation terms of pool \textit{a}\xspace in matrix \ensuremath{\mathbf{A'}}\xspace. Furthermore, we assume that \R{1b} is much smaller than \R{2b} and \ensuremath{k_{b}}\xspace and therefore \R{1b} can be neglected in \ensuremath{\mathbf{A'}}\xspace. In contrast to Trott and Palmer \cite{trott_r1rho_2002}, we do not neglect \R{2b}, but \ensuremath{k_{a}}\xspace. By this means, the obtained eigenvalue approximation by using Eq.~\eqref{eqn:c0c1} is linearized in the small parameter \ensuremath{f_{b}}\xspace giving
\begin{align}
\label{eqn:SOL_Lorentz}
\ensuremath{R_{\mathit{ex}}}\xspace(\ensuremath{\Delta\omega_{b}}\xspace)= \frac{\ensuremath{R_{\mathit{ex}}^{\mathit{max}}}\xspace\frac{\Gamma^2}{4}}{\frac{\Gamma^2}{4}+\ensuremath{\Delta\omega_{b}}\xspace^2}
\end{align}
with maximum value
\begin{align}
\label{eqn:SOL_Rexmaxfull}
\ensuremath{R_{\mathit{ex}}^{\mathit{max}}}\xspace= \ensuremath{f_{b}}\xspace \ensuremath{k_{b}}\xspace\sin^2{\theta}
\frac{ (\ensuremath{\omega_{b}}\xspace-\ensuremath{\omega_{a}}\xspace)^2+ \frac{\R{2b}}{\ensuremath{k_{b}}\xspace}(\ensuremath{\omega_{1}}\xspace^2+\ensuremath{\Delta\omega}\xspace^2)+ \R{2b}(\ensuremath{k_{b}}\xspace+\R{2b}) }{\frac{\Gamma^2}{4}}
\end{align}
and full width at half maximum (FWHM)
\begin{align}
\label{eqn:SOL_RexGamma}
\Gamma=2\sqrt{\frac{\ensuremath{k_{b}}\xspace+\R{2b}}{\ensuremath{k_{b}}\xspace}\ensuremath{\omega_{1}}\xspace^2+(\ensuremath{k_{b}}\xspace+\R{2b})^2 } .
\end{align}
For large $|\ensuremath{\omega_{b}}\xspace-\ensuremath{\omega_{a}}\xspace|$
\begin{align}
\label{eqn:SOL_RexmaxLS}
\ensuremath{R_{\mathit{ex}}^{\mathit{max}}}\xspace\approx\ensuremath{f_{b}}\xspace \ensuremath{k_{b}}\xspace\cdot \frac{\ensuremath{\omega_{1}}\xspace^2}{\ensuremath{\omega_{1}}\xspace^2+\ensuremath{k_{b}}\xspace(\ensuremath{k_{b}}\xspace+\R{2b})} .
\end{align}
The \ensuremath{\omega_{1}}\xspace-dependent factor yields the amount of labeling of pool \textit{b}\xspace. Hence, we call this factor labeling efficiency, refering to \cite{sun_correction_2007}:
\begin{align}
\label{eqn:alpha}
\alpha=\frac{\ensuremath{\omega_{1}}\xspace^2}{\ensuremath{\omega_{1}}\xspace^2+\ensuremath{k_{b}}\xspace(\ensuremath{k_{b}}\xspace+\R{2b})} .
\end{align}
For strong \ensuremath{B_{1}}\xspace and small \R{2b} and \ensuremath{k_{b}}\xspace, $\alpha$ is approximately one and we obtain the \textit{full-saturation} limit
\begin{align}
\label{eqn:SOL_Rexmax_ka}
\ensuremath{R_{\mathit{ex}}^{\mathit{max}}}\xspace\approx\ensuremath{f_{b}}\xspace \ensuremath{k_{b}}\xspace=\ensuremath{k_{a}}\xspace .
\end{align}
\section{Results}
\label{seq:res}
We obtained numerical values for the eigenvalues computed by means of the full numerical BM matrix solution \cite{woessner_numerical_2005} and compared them to the proposed approximations via
\begin{align}
\label{eqn:numanaexp_rex}
\ensuremath{R_{\mathit{ex}}}\xspace =-|\lambda_{1,\text{numerical}}|-\ensuremath{R_{\mathit{eff}}}\xspace .
\end{align}
To verify equations (\ref{eqn:Zs},\ref{eqn:Zss},\ref{eqn:SOL_l1_leff_lstrich},\ref{eqn:SOL_Rexmaxfull}) the dynamics of the magnetization vectors of the exchanging spin pools were simulated. The decay rate \ensuremath{R_{\mathit{ex}}}\xspace is obtained from \ensuremath{\tilde{Z}}\xspace (Eq. \eqref{eqn:Zs} ) and \ensuremath{R_{\mathit{eff}}}\xspace via
\begin{align}
\label{eqn:exp_rex}
\ensuremath{R_{\mathit{ex}}}\xspace =-\frac{\log(\ensuremath{\tilde{Z}}\xspace)}{\ensuremath{t_{\mathit{sat}}}\xspace}-\ensuremath{R_{\mathit{eff}}}\xspace .
\end{align}
The simulation parameters for the abundant pool were chosen according to published data for brain white matter \cite{stanisz_t_2005} including a rare pool attributed to amide protons \cite{zhou_chemical_2006}.
\begin{figure}[H]
\begin{center}
\includegraphics{4_EW_trott_zaiss.eps}
\end{center}
\caption{(a,c,e) \ensuremath{R_{\mathit{ex}}}\xspace on-resonant on pool \textit{b}\xspace from smallest eigenvalue in modulus (Eq. \eqref{eqn:numanaexp_rex}) , calculated numerically (line) and analytically by the approximations of Eq. \eqref{eqn:SOL_Lorentz} (squares) and the asymmetric population limit of Trott and Palmer \cite{trott_r1rho_2002} (Eq. \eqref{eqn:SOL_Rho_Trott}, diamonds). x and + mark the relative error $(1 - (\ensuremath{R_{\mathit{ex}}}\xspace^{ana}/\ensuremath{R_{\mathit{ex}}}\xspace^{num}))$ when it is larger than 0.1 \%. (a) For small \R{2b}, both solutions for \ensuremath{R_{\mathit{ex}}}\xspace agree with the numerical value; if \R{2b} is larger than \ensuremath{k_{b}}\xspace the proposed solution still matches the numerical value. The extension by \R{2b} is important if the CEST pool is not fully saturated, which is the case for small \ensuremath{B_{1}}\xspace (c) or large \ensuremath{k_{b}}\xspace (e). But also for large \ensuremath{B_{1}}\xspace the solution, that includes \R{2b} fits the numerical value with higher accuracy. (b,d,f) Imaginary parts of the numerical eigenvalues. \ensuremath{B_{1}}\xspace and \R{2b} ranges where Eq. \eqref{eqn:SOL_Lorentz} (squares) shows deviations from the numerical solution correlate with ranges where the imaginary part becomes small or even zero. In this case, the assumption of a strong hierarchy in the eigenvalues is not valid anymore.}
\label{fig:EW_trott_zaiss}
\end{figure}
The proposed approximation of \ensuremath{R_{\mathit{ex}}}\xspace by Eq.\eqref{eqn:SOL_Lorentz} was compared to the asymmetric population solution of Ref.\cite{trott_r1rho_2002} (Fig. \ref{fig:EW_trott_zaiss}). If \R{2b} is non-zero, \ensuremath{R_{\mathit{ex}}}\xspace proposed by Eq.\eqref{eqn:SOL_Lorentz} matches the numerical value better than the \ensuremath{R_{\mathit{ex}}}\xspace \add{given in Ref.\cite{trott_r1rho_2002}} \add{(see Eq. \eqref{eqn:SOL_Rho_Trott} below)}.
Especially the dependence of \ensuremath{R_{\mathit{ex}}}\xspace on \ensuremath{B_{1}}\xspace (Fig. \ref{fig:EW_trott_zaiss}c) changes by taking \R{2b} into account.
\begin{figure}[H]
\begin{center}
\includegraphics{5_Z_Zs_spectra.eps}
\end{center}
\caption{Numerical simulations (dots) of an CEST\xspace experiment evaluated for dynamic \ensuremath{\tilde{Z}}\xspace-spectra (a,\,b) and steady-state Z-spectra (c,\,d) are in agreement with Eqs.~\eqref{eqn:Zs} and \eqref{eqn:Zss} (solid red), respectively. Plots demonstrate high correlation for different $\ensuremath{B_{1}}\xspace (0.1 - 20~\mathrm{\mu T}; a, c)$ and $\ensuremath{k_{b}}\xspace(1 - 2000~\mathrm{Hz}; b, d)$. Deviations near resonance of pool \textit{a}\xspace for large \ensuremath{B_{1}}\xspace (a) are caused by oscillations of the magnetization in the transverse plane of the effective frame. In the SL\xspace experiment these oscillations are suppressed (Fig.~\ref{fig:1}).}
\label{fig:ZZs_spectra}
\end{figure}
For an CEST\xspace experiment, the normalized numerical solution agrees with the theory of dynamic \ensuremath{\tilde{Z}}\xspace-spectra (Fig.~\ref{fig:ZZs_spectra}a,\,b) and steady-state Z-spectra (Fig.~\ref{fig:ZZs_spectra}c,\,d) for different values of \ensuremath{B_{1}}\xspace and \ensuremath{k_{b}}\xspace. The competing direct and exchange-dependent saturation -- a central problem in proton CEST \cite{sun_imaging_2008,sun_correction_2007,zaiss_quantitative_2011} -- is modeled correctly. Deviations in Fig.~\ref{fig:ZZs_spectra}a for strong \ensuremath{B_{1}}\xspace and $\ensuremath{\Delta\omega}\xspace \rightarrow 0$ result from transversal magnetization in the effective frame which was neglected before. By projection on the transverse plane of the effective frame using Eqs.~\eqref{eqn:hintrafoxy} we obtained the resulting deviation of \ensuremath{\tilde{Z}}\xspace
\begin{align}
\label{eqn:dZs}
\ensuremath{\Delta\tilde{Z}}\xspace(\ensuremath{\Delta\omega}\xspace,\ensuremath{\omega_{1}}\xspace,t)=\frac{\ensuremath{P_{x}}\xspace\ensuremath{P_{\xeff}}\xspace}{\ensuremath{P_{z}}\xspace\ensuremath{P_{\zeff}}\xspace-Z_{ss}}\cdot Re(e^{\lambda_2\cdot t}) ,
\end{align}
with projections \ensuremath{P_{\xeff}}\xspace and \ensuremath{P_{x}}\xspace into the transverse plane of the effective frame and back. For MT $\ensuremath{P_{x}}\xspace=\ensuremath{P_{\xeff}}\xspace=\sin{\theta}$. Real and imaginary parts of the complex eigenvalue $\lambda_2$ are given by $-\R{2\rho}\approx-\frac{1}{2}(\R{2a}+\R{1a}\sin^2{\theta}+\R{2a}\cos^2{\theta})$ \cite{moran_near-resonance_1995} and \ensuremath{\omega_{\mathit{eff}}}\xspace, respectively. The implicit neglect of \ensuremath{\Delta\tilde{Z}}\xspace in Eq.~\eqref{eqn:Zs} is justified if $\ensuremath{t_{\mathit{sat}}}\xspace\gg T_{2\rho}$ or \ensuremath{P_{x}}\xspace and \ensuremath{P_{\xeff}}\xspace are small. This can be realized either by SL\xspace preparation or by $\ensuremath{\omega_{1}}\xspace\ll\ensuremath{\Delta\omega}\xspace$.
The on-resonant case of CEST\xspace $(\theta = 90^\circ) $ is not defined, because \ensuremath{{Z^{\mathit{ss}}}}\xspace in Eq.~\eqref{eqn:Zss} and thus the denominators in Eqs.~\eqref{eqn:Zs} and \eqref{eqn:dZs} vanish. Then the z-axis lies in the transverse plane of the effective frame and Z is described by $\ensuremath{M_{\mathit{0,a}}}\xspace\cdot Re(e^{(-(\R{2\rho}+i\ensuremath{\omega_{1}}\xspace) t)}) $.
Therefore, near resonance SL\xspace is preferable to CEST\xspace; it also yields in general a higher SNR (given by the projection factors \ensuremath{P_{z}}\xspace, \ensuremath{P_{\zeff}}\xspace). Regarding the experimental realization, CEST\xspace is simpler than SL\xspace, because \ensuremath{\Delta\omega}\xspace and \ensuremath{\omega_{1}}\xspace and thus $\theta$ can be corrected effectively after the measurement by \ensuremath{B_{0}}\xspace and \ensuremath{B_{1}}\xspace field mapping \cite{sun_correction_2007,kim_water_2009}. In contrast, SL\xspace requires knowledge of \ensuremath{B_{1}}\xspace and \ensuremath{B_{0}}\xspace during the scan for proper preparation or techniques that are insensitive to field inhomogeneities such as adiabatic pulses \cite{mangia_rotating_2009,michaeli_transverse_2004}.
\begin{figure}[H]
\includegraphics{6_Rex_l1.eps}
\caption{Numerical \ensuremath{R_{\mathit{ex}}^{\mathit{max}}}\xspace (dots), employing Eq.~\eqref{eqn:exp_rex}, fit to $-\lambda_1'$ of Eq.~\eqref{eqn:SOL_l1_leff_lstrich} (solid blue) and to \ensuremath{R_{\mathit{ex}}^{\mathit{max}}}\xspace of Eq.~\eqref{eqn:SOL_Rexmaxfull} (dashed green) as a function of \ensuremath{f_{b}}\xspace, \ensuremath{k_{b}}\xspace, \ensuremath{B_{1}}\xspace, and \R{2b}. x and + mark the relative error $(1 - (\ensuremath{R_{\mathit{ex}}}\xspace^{ana}/\ensuremath{R_{\mathit{ex}}}\xspace^{num}))$ when it is larger than 1 \%. (a) As expected, the eigenvalue approximation is insufficient for $\ensuremath{f_{b}}\xspace > 5\% $. (b) Contrary to the approximation of Eq.\eqref{eqn:SOL_Rexmax_ka} (solid red) the full solution follows the decrease of \ensuremath{R_{\mathit{ex}}^{\mathit{max}}}\xspace with large \ensuremath{k_{b}}\xspace. The decrease of deviations for small \ensuremath{B_{1}}\xspace (c) and high \R{2b} (d) may be caused by overdamping in pool b, i.e., eigenvalues become real which reduces the required hierarchy in the set of the eigenvalues. However, the deviation in \ensuremath{R_{\mathit{ex}}}\xspace is too small (Fig. \ref{fig:EW_trott_zaiss}) to explain this deviation leading to the conclusion that other eigenvectors are contributing to the relaxation. (d) Inclusion of \R{2b} is relevant for $\R{2b} > 100$ Hz. }
\label{fig:Rex}
\end{figure}
The values of the rate \ensuremath{R_{\mathit{ex}}^{\mathit{max}}}\xspace obtained by simulations fit well to the full (Eq.~\eqref{eqn:SOL_l1_leff_lstrich}) and approximate (Eq.~\eqref{eqn:SOL_Rexmaxfull}) solution for the observed parameters (Fig.~\ref{fig:Rex}). Deviations of simulation and analytical solution were smaller than 1\% for rates varied in the ranges: $\R{1b}=0.1-10$ Hz, $\R{1a}=0.1-10$ Hz and $\R{2a}=2-100$ Hz (data not shown).
\section{Discussion}
\subsection{General solution}
We showed that our formalism, established by Eqs.~\eqref{eqn:Zs},\eqref{eqn:Zss} and \eqref{eqn:Z_full_solution_apex} together with the eigenvalue approximation of Eq.~\eqref{eqn:SOL_l1_leff_lstrich}, is a general solution for CEST experiments. This now allows us to discuss from a general point of view the techniques and theories proposed in the field of chemical exchange saturation transfer. For the SL\xspace solution this was already accomplished by Jin et al. \cite{jin_spin-locking_2011,jin_magnetic_2012}.
The proposed eigenvalue approximation assumes the case of asymmetric populations. This restricts its application to systems where the water proton pool is much larger than the exchanging pools -- which is the case for CEST experiments.
There are many analytical approaches for the smallest eigenvalue (\ensuremath{R_{1\rho}}\xspace) of the BM matrix besides our approach. They use pertubation theory \cite{trott_theoretical_2004}, the stochastic Liouville equation \cite{abergel_markov_2005}, an average magnetization approach \cite{trott_average-magnetization_2003}, and the polynomial root finding algorithm of Laguerre \cite{miloushev_r1_2005}. The latter is even valid in the case of symmetric populations.
However, all these treatments neglect the transverse relaxation of the exchanging pool. Since in CEST experiments the exchange rates are often quite small (e.g., $\ensuremath{k_{b}}\xspace\approx 28 Hz$ for APT), \R{2b} cannot be neglected against \ensuremath{k_{b}}\xspace.
We chose therefore a simple approach which is suitable for the condition of asymmetric populations and took \R{2b} into account.
Our approach to find the eigenvalue including \R{2b} is similar to that of Trott and Palmer \cite{trott_r1rho_2002}.
However, different \R{1} and \R{2} were allowed for the involved pools. In addition, an alternative justification of the relation $\lambda_1=-\frac{c_0}{c_1}$ was obtained, which uses the intrinsic hierarchy of the eigenvalues (\ref{seq:App}) instead of linearization of the characteristic polynomial.
By this means, it turned out that a strong hierarchy of the eigenvalues is necessary for the approximation. The hierarchy was increased by rescaling the system by the unperturbed eigenvalue \ensuremath{R_{\mathit{eff}}}\xspace (Fig.\ref{fig:EW}). Thus the accuracy of the approximation was improved. As the parameter \R{2b} was included and equations were linearized directly in the small parameter \ensuremath{k_{a}}\xspace, a formula was obtained (Eq.~\eqref{eqn:SOL_Rho_reff_rex}) that differs from the asymmetric population limit of Ref. \cite{trott_r1rho_2002} reading
\begin{align}
\label{eqn:SOL_Rho_Trott}
\ensuremath{R_{1\rho}}\xspace=\ensuremath{R_{\mathit{eff}}}\xspace+ \underbrace{\sin^2{\theta} \frac{(\ensuremath{\omega_{b}}\xspace-\ensuremath{\omega_{a}}\xspace)^2\frac{\ensuremath{k_{a}}\xspace\ensuremath{k_{b}}\xspace}{\ensuremath{k_{a}}\xspace+\ensuremath{k_{b}}\xspace}}{\ensuremath{\Delta\omega_{b}}\xspace^2+\ensuremath{\omega_{1}}\xspace^2+(\ensuremath{k_{a}}\xspace+\ensuremath{k_{b}}\xspace)^2}}_{\ensuremath{R_{\mathit{ex}}}\xspace} .
\end{align}
Equality is reached if \R{2b} is neglected in our approximation and if Eq. \eqref{eqn:SOL_Rho_Trott} is linearized in \ensuremath{k_{a}}\xspace.
With our extension simulated CEST Z-spectra could be predicted well in a broad range of parameters. Moreover, it turned out that \R{2b} is important if it is in the range of \ensuremath{k_{b}}\xspace (Fig. \ref{fig:Rex}d). Inclusion of \R{2b} also allows to model macromolecular magnetization transfer effects with large \R{2b} values (Fig. \ref{fig:IOPA}c).
Our solution agrees for SL\xspace with the existing treatment \cite{jin_spin-locking_2011}, but only with the correct projection factors SL\xspace and CEST\xspace can be described by the same theory. This is contrary to the conclusion of Jin et al. \cite{jin_spin-locking_2011} that SL\xspace theory can be used directly to describe CEST experiments. The deviation is not large for small $\theta$, but for $\ensuremath{\omega_{1}}\xspace\approx\ensuremath{\Delta\omega}\xspace$ the projection factors are crucial as shown in Fig. \ref{fig:wrongP}.
With the correct projections the transition to CEST\xspace is straightforward and provides a much broader range of validity than previous models developed for CEST\xspace which are either appropriate only for small \ensuremath{B_{1}}\xspace \cite{zaiss_quantitative_2011} or large \ensuremath{B_{1}}\xspace \cite{baguet_off-resonance_1997} or only \add{for the case of on-resonant irradiation of pool \textit{b}\xspace} \cite{sun_correction_2007,sun_imaging_2008}. The proposed theory (Eq. \eqref{eqn:Z_full_solution_apex}) gives a model for full Z-spectra for transient and steady-state CEST\xspace experiments which enables analytical rather than numerical fitting of experimental data.
\begin{figure}[H]
\includegraphics{7_Pz1.eps}
\caption{The same plot of \ensuremath{R_{\mathit{ex}}^{\mathit{max}}}\xspace (Eq. \eqref{eqn:exp_rex}) as in Fig.~\ref{fig:Rex}c, but now for CEST (dots) and SL (diamonds) employing the corresponding projection factors in Eq. \eqref{eqn:Zs} ($\ensuremath{P_{z}}\xspace=\ensuremath{P_{\zeff}}\xspace=1$ for SL and $\ensuremath{P_{z}}\xspace=\ensuremath{P_{\zeff}}\xspace=\cos\theta$ for CEST). Additionally, the result of an evaluation is shown employing the projection factors of SL for a CEST experiment (circles) and employing the projection factors of CEST for a SL experiment (squares). Only with the correct projection factors both experiments are described by the same theory and yield \ensuremath{R_{\mathit{ex}}}\xspace (solid blue). }
\label{fig:wrongP}
\end{figure}
\subsection{Extension to other systems}
As verified for SL\xspace \cite{trott_theoretical_2004}, the theory can be extended to $n$-site exchanging systems. By simply superimposing the exchange-dependent relaxation rates of several pools one obtains the Z-spectra for a multi-pool system. We applied this to the contrast agent iopamidol in water, which has two exchanging amide proton groups \cite{longo_iopamidol_2011}, considering a three pool system: water, amide proton B at 4.2 ppm and amide proton C at 5.5 ppm. Assuming for the exchange rates $k_c=6\cdot\ensuremath{k_{b}}\xspace$, the superposition of \ensuremath{R_{\mathit{eff}}}\xspace and the two corresponding \ensuremath{R_{\mathit{ex}}}\xspace yields the Z-spectrum of the iopamidol system (Fig. \ref{fig:IOPA}a). A three-pool system relevant for \textit{in vivo} CEST studies includes water protons, amide protons and a macromolecular proton pool. Modeling the macromolecular pool by $\ensuremath{R_{\mathit{ex}}}\xspace^{m}$(\R{2m} = 5000 Hz, \ensuremath{k_{m}}\xspace = 40 Hz) with an offset of -2.6 ppm and again superimposing it with $\ensuremath{R_{\mathit{ex}}}\xspace^{amide}$ we are able to model analytically Z-spectra of APT with an underlying symmetric and asymmetric MT effect up to 5\% relative concentration \ensuremath{f_{m}}\xspace (Fig. \ref{fig:IOPA}b). Hence, the model is able to describe the \textit{in vivo} situation of several CEST pools and underlying MT competing with direct water saturation. Using the superimposed \ensuremath{R_{\mathit{ex}}}\xspace including $\ensuremath{R_{\mathit{ex}}}\xspace^{m}$ and $\ensuremath{R_{\mathit{ex}}}\xspace^{amide}$ and fitting the obtained Z-spectra $\ensuremath{R_{\mathit{ex}}}\xspace^{amide}$ can be isolated. For macromolecular MT the extension of \ensuremath{R_{\mathit{ex}}}\xspace by \R{2b} is crucial, since \R{2b} can be as large as $\approx 10^5$ Hz. The implicitly assumed Lorentzian lineshape of the macromolecular pool is only valid around the water proton resonance, for large offsets a super-Lorentzian lineshape must be included in $\ensuremath{R_{\mathit{ex}}}\xspace^{m}$ \cite{stanisz_t_2005}.
\begin{figure}[H]
\includegraphics{8_IOPA_MT.eps}
\caption{Three applications of the proposed theory:(a) The system of iopamidol with corresponding \ensuremath{\mathrm{MTR}_{\mathrm{asym}}}\xspace evaluation. (b) The system of APT with an asymmetric macromolecular MT pool \add{(for concentration fractions \ensuremath{f_{m}}\xspace=1\%, 3\%, 5\%)}. (c) The system of exchanging hyperpolarized xenon soluted or encapsulated in cryptophane cages, a biosensor method called HyperCEST.}
\label{fig:IOPA}
\end{figure}
Hyperpolarized xenon spin ensembles exchanging between the dissolved phase and cryptophane cages (HyperCEST experiment, \cite{schroeder_molecular_2006}) can also be described by Eq.~\eqref{eqn:Zs}. Since the initial hyperpolarized magnetization $M_i$ is in the order up to $10^5...10^6\ensuremath{M_0}\xspace$, the steady-state can be neglected for depolarization. This yields $\ensuremath{Z}\xspace\approx\ensuremath{\tilde{Z}}\xspace=M_i \cdot e^{-\ensuremath{R_{1\rho}}\xspace\ensuremath{t_{\mathit{sat}}}\xspace}$ in agreement with the result in Ref. \cite{zaiss_analytical_2012}. Figure ~\ref{fig:IOPA}c shows the simulated Z-spectrum around the cage peak in the HyperCEST experiment of a Xe-cryptophane system for different \ensuremath{k_{b}}\xspace.
Pulsed irradiation, employed for saturation in SAR limited clinical scanners \cite{schmitt_optimization_2011}, was shown to have similar effects on \ensuremath{\mathrm{MTR}_{\mathrm{asym}}}\xspace as cw irradiation with effective \ensuremath{B_{1}}\xspace \cite{zu_optimizing_2011,sun_simulation_2011}. The presented solution for CEST Z-spectra can therefore be used for optimization of pulsed saturation transfer experiments.
\subsection{Proton transfer ratio}
For a CEST\xspace experiment the parameters of particular interest are the exchange rate \ensuremath{k_{b}}\xspace of the metabolite proton pool and the relative concentration \ensuremath{f_{b}}\xspace. The former is often pH catalyzed and permits pH-weighted imaging; the latter allows molecular imaging with enhanced sensitivity. The ultimate method must allow -- with high spectral selectivity -- the generation of \ensuremath{k_{b}}\xspace and \ensuremath{f_{b}}\xspace maps separately and for different exchanging groups. Unfortunately, both parameters occur in the water pool BM equations as product, i.e. the back-exchange rate $\ensuremath{k_{a}}\xspace=\ensuremath{k_{b}}\xspace\cdot\ensuremath{f_{b}}\xspace$. There are some approaches which are able to separate \ensuremath{k_{b}}\xspace and \ensuremath{f_{b}}\xspace for specific cases like rotation transfer of amid protons \cite{zu_multi-angle_2012} or the method of Dixon et al. \cite{dixon_concentration-independent_2010} applicable to PARACEST agents. \add{CEST experiments are commonly evaluated to yield the proton transfer ratio PTR. PTR is an ideal parameter in the sense that it reflects the decrease of the water pool signal owing to exchange from a labeled exchanging pool only, thus neglecting any direct saturation.}
In the following, we assume one CEST pool resonance on the positive \ensuremath{\Delta\omega}\xspace axis.
Employing Eq. \eqref{eqn:Zss} with the limit $\theta \rightarrow 0$ we obtain for PTR in steady-state:
\begin{align}
\label{eqn:QUREXtheo_PTR}
\text{PTR}=1-\ensuremath{{Z^{\mathit{ss}}}}\xspace(\ensuremath{\Delta\omega}\xspace) \approx \frac{\ensuremath{R_{\mathit{ex}}}(+\dw)\xspace}{\R{1a}+\ensuremath{R_{\mathit{ex}}}(+\dw)\xspace}
\end{align}
which yields the maximal value $\frac{\ensuremath{k_{a}}\xspace}{\R{1a}+\ensuremath{k_{a}}\xspace}$ \cite{zhou_chemical_2006} in the full-saturation limit ($\ensuremath{R_{\mathit{ex}}}\xspace\approx\ensuremath{k_{a}}\xspace$ ). Eq. \eqref{eqn:QUREXtheo_PTR} is consistent with PTR including the labeling efficiency $\alpha$ introduced in Ref. \cite{sun_correction_2007}.
\subsection{Z-spectra evaluation - \ensuremath{\mathrm{MTR}}\xspace and \ensuremath{\mathrm{MTR}_{\mathrm{asym}}}\xspace}
Methods using asymmetry implicitly assume that the full width at half maximum of \ensuremath{R_{\mathit{ex}}}\xspace(\ensuremath{\Delta\omega_{b}}\xspace) is narrow compared to the chemical shift of the corresponding pool. This means that \ensuremath{R_{\mathit{ex}}}\xspace(\ensuremath{\Delta\omega_{b}}\xspace) can be neglected for the reference scan $\ensuremath{Z}\xspace(-\ensuremath{\Delta\omega}\xspace)$ what is only true in the slow-exchange limit \cite{zhou_chemical_2006}. This limit can be defined more generally by the width of \ensuremath{R_{\mathit{ex}}}\xspace(\ensuremath{\Delta\omega_{b}}\xspace) (Eq. \eqref{eqn:SOL_RexGamma}):
\begin{align}
\label{eqn:slow-ex-limit}
\Gamma=2\sqrt{\frac{\ensuremath{k_{b}}\xspace+\R{2b}}{\ensuremath{k_{b}}\xspace}\cdot\ensuremath{\omega_{1}}\xspace^2+(\ensuremath{k_{b}}\xspace+\R{2b})^2} \ll |\ensuremath{\Delta\omega_{b}}\xspace-\ensuremath{\Delta\omega}\xspace_a|
\end{align}
This new limit depends on \ensuremath{B_{1}}\xspace which affects the ability to distinguish different peaks in the Z-spectrum (Fig. \ref{fig:ZZs_spectra}c). The limit is therefore a useful parameter for exchange-regime characterization in saturation spectroscopy.
For CEST\xspace the common evaluation parameters are the magnetization transfer rate $\ensuremath{\mathrm{MTR}}\xspace(\ensuremath{\Delta\omega}\xspace) = 1-Z(\ensuremath{\Delta\omega}\xspace)$ and the asymmetry of the Z-spectrum $\ensuremath{\mathrm{MTR}_{\mathrm{asym}}}\xspace(\ensuremath{\Delta\omega}\xspace) = Z(-\ensuremath{\Delta\omega}\xspace) - Z(+\ensuremath{\Delta\omega}\xspace)$. \ensuremath{\mathrm{MTR}_{\mathrm{asym}}}\xspace is generally employed to estimate PTR.
Using Eq.~\eqref{eqn:Zss} together with Eq. \eqref{eqn:SOL_Rho_reff_rex} we obtain for steady-state Z-spectrum asymmetry
\begin{align}
\label{eqn:asymSS}
\begin{split}
\ensuremath{\mathrm{MTR}_{\mathrm{asym}}^{\mathit{ss}}}\xspace(\ensuremath{\Delta\omega}\xspace) &= \ensuremath{{Z^{\mathit{ss}}}}\xspace(-\ensuremath{\Delta\omega}\xspace)-\ensuremath{{Z^{\mathit{ss}}}}\xspace(+\ensuremath{\Delta\omega}\xspace) \\
&= \frac{(\ensuremath{R_{\mathit{ex}}}(+\dw)\xspace-\ensuremath{R_{\mathit{ex}}(-\dw)}\xspace)\cdot\R{1a}\ensuremath{P_{z}}\xspace\cos{\theta}}{(\ensuremath{R_{\mathit{eff}}}\xspace+\ensuremath{R_{\mathit{ex}}(-\dw)}\xspace)(\ensuremath{R_{\mathit{eff}}}\xspace+\ensuremath{R_{\mathit{ex}}}(+\dw)\xspace) }.
\end{split}
\end{align}
The comparison shows that \ensuremath{\mathrm{MTR}_{\mathrm{asym}}^{\mathit{ss}}}\xspace yields PTR of Eq. \eqref{eqn:QUREXtheo_PTR} only if $\theta=0$.
Sun et al. \cite{sun_imaging_2008} found $\ensuremath{\mathrm{MTR}_{\mathrm{asym}}^{\mathit{ss}}}\xspace=PTR\cdot\alpha\cdot(1-\sigma)$ which combines the labeling efficiency $\alpha$ found by the weak-saturation-pulse approximation and a spillover coefficient $\sigma$ from the strong-saturation-pulse approximation. This formula is only valid on resonance of pool \textit{b}\xspace in contrast to Eq. \eqref{eqn:asymSS}.
Another approach, applicable for small \ensuremath{B_{1}}\xspace, eliminates the spillover effect by a probabilistic approach \cite{zaiss_quantitative_2011}.
This Z-spectrum model taken from \cite{zaiss_quantitative_2011} yields
\begin{align}
\begin{split}
\text{PTR}(\ensuremath{\Delta\omega}\xspace)&\approx \frac{\ensuremath{{Z^{\mathit{ss}}(-\dw)}}\xspace -\ensuremath{{Z^{\mathit{ss}}(+\dw)}}\xspace}{\ensuremath{{Z^{\mathit{ss}}(-\dw)}}\xspace -\ensuremath{{Z^{\mathit{ss}}(+\dw)}}\xspace+\ensuremath{{Z^{\mathit{ss}}(-\dw)}}\xspace\cdot\ensuremath{{Z^{\mathit{ss}}(+\dw)}}\xspace}\\
&=\frac{\ensuremath{R_{\mathit{ex}}}(+\dw)\xspace}{\cos^2\theta\cdot\R{1a}+\ensuremath{R_{\mathit{ex}}}(+\dw)\xspace}
\end{split}
\end{align}
which turns out, after substitution of \ensuremath{{Z^{\mathit{ss}}}}\xspace by Eq.\eqref{eqn:Zss}, to be an approximation of PTR if $\theta$ is small.
The asymmetry normalized by the reference scan was proposed for spillover correction \cite{liu_high-throughput_2010}.
Applying eq. \eqref{eqn:Zss} yields
\begin{align}
\label{eqn:reddyrefnorm}
\frac{\ensuremath{{Z^{\mathit{ss}}}}\xspace(-\ensuremath{\Delta\omega}\xspace)-\ensuremath{{Z^{\mathit{ss}}(+\dw)}}\xspace}{\ensuremath{{Z^{\mathit{ss}}(-\dw)}}\xspace} = \frac{\ensuremath{R_{\mathit{ex}}}(+\dw)\xspace}{\ensuremath{R_{\mathit{eff}}}\xspace(+\ensuremath{\Delta\omega}\xspace)+\ensuremath{R_{\mathit{ex}}}(+\dw)\xspace}
\end{align}
which again approximates PTR if $\theta$ is small.
By use of Eqs. \eqref{eqn:Zss} and \eqref{eqn:Z_full_solution_apex} we obtain for the asymmetry in transient-state a bi-exponential function
\begin{equation}
\label{eqn:SOL_MTRasym}
\begin{split}
\ensuremath{\mathrm{MTR}_{\mathrm{asym}}}\xspace(\ensuremath{\Delta\omega}\xspace,t)=& \ensuremath{\mathrm{MTR}_{\mathrm{asym}}^{\mathit{ss}}}\xspace(\ensuremath{\Delta\omega}\xspace)\\
+e^{-\ensuremath{R_{1\rho}}\xspace(-\ensuremath{\Delta\omega}\xspace) t}\cdot & (\ensuremath{P_{z}}\xspace\ensuremath{P_{\zeff}}\xspace-Z^{ss}(-\ensuremath{\Delta\omega}\xspace)) \\
-e^{-\ensuremath{R_{1\rho}}\xspace(+\ensuremath{\Delta\omega}\xspace) t}\cdot & (\ensuremath{P_{z}}\xspace\ensuremath{P_{\zeff}}\xspace-Z^{ss}(+\ensuremath{\Delta\omega}\xspace)) .
\end{split}
\end{equation}
Neglecting direct saturation of pool \textit{a}\xspace and assuming \ensuremath{P_{z}}\xspace~=~\ensuremath{P_{\zeff}}\xspace~=~1 yields the mono-exponential approximation at the CEST resonance \cite{zhou_chemical_2006,mcmahon_quantifying_2006}
\begin{align}
\ensuremath{\mathrm{MTR}_{\mathrm{asym}}}\xspace(\ensuremath{\Delta\omega_{b}}\xspace=0,t)=\ensuremath{\mathrm{MTR}_{\mathrm{asym}}^{\mathit{ss}}}\xspace(\ensuremath{\Delta\omega_{b}}\xspace=0)\cdot(1-e^{-(\R{1a}+\ensuremath{k_{a}}\xspace)t}) ,
\end{align}
with the rate constant $\ensuremath{R_{1\rho}}\xspace = \R{1a} + \ensuremath{k_{a}}\xspace$.
This is valid if $\theta$ is small, leading to $\ensuremath{R_{\mathit{eff}}}\xspace\approx\R{1a}$ and, with the limit of Eq.~\eqref{eqn:SOL_Rexmax_ka}, $\ensuremath{R_{\mathit{ex}}}\xspace \approx \ensuremath{k_{a}}\xspace$ (solid red, Fig.~\ref{fig:Rex}b).
The ratiometric analysis approach QUESTRA \cite{sun_simplified_2011} includes direct saturation and is independent of steady-state. It can be expressed by means of Eq.~\eqref{eqn:Zs} under the same assumptions \ensuremath{P_{z}}\xspace=\ensuremath{P_{\zeff}}\xspace=1 and $\ensuremath{R_{\mathit{eff}}}\xspace\approx\R{1a}$ and $\ensuremath{R_{\mathit{ex}}}\xspace\approx\ensuremath{k_{a}}\xspace$
\begin{align}
\text{QUESTRA}(t)=\frac{\ensuremath{\tilde{Z}}\xspace(+\ensuremath{\Delta\omega}\xspace,t)}{\ensuremath{\tilde{Z}}\xspace(-\ensuremath{\Delta\omega}\xspace,t)}\approx e^{-\ensuremath{k_{a}}\xspace t}.
\end{align}
\add{Another method, pCEST \cite{vinogradov_pcest:_2012}, employs \ensuremath{R_{1\rho}}\xspace in an inversion recovery experiment.
The pCEST signal obeys the negative of Eq.~\eqref{eqn:SOL_MTRasym} if the initial inversion is introduced by
$\ensuremath{P_{\zeff}}\xspace = -\cos\theta$. Hence, the full dynamics of the \ensuremath{R_{1\rho}}\xspace inversion recovery signal is }
\begin{align}
\add{\text{pCEST}(\ensuremath{\Delta\omega}\xspace,t)=-\ensuremath{\mathrm{MTR}_{\mathrm{asym}}}\xspace(\ensuremath{\Delta\omega}\xspace,t,\ensuremath{P_{\zeff}}\xspace = -\cos\theta)}
\end{align}
\add{The pCEST signal can be positive in transient-state, but is negative in steady-state.}
\add{ This inversion recovery approach was suggested first to increase SNR for MT effect by Mangia et al. \cite{mangia_magnetization_2011} and for SL already by Santyr et al. \cite{santyr_off-resonance_1994} and again by Jin and Kim \cite{jin_quantitative_2012}. Their \textit{iSL} signal is in our notation equal to \ensuremath{Z}\xspace(\ensuremath{\Delta\omega}\xspace,\ensuremath{\omega_{1}}\xspace,t) (Eq.\eqref{eqn:Z_full_solution_apex}) with \ensuremath{P_{\zeff}}\xspace=-1 and their projection factors for CEST and SL are identical with \ensuremath{P_{z}}\xspace and \ensuremath{P_{\zeff}}\xspace. For \ensuremath{R_{\mathit{ex}}}\xspace the approximation of Ref. \cite{trott_r1rho_2002} is used, assuming $\R{2b}=\R{2a}$. Especially for the quantification employing different \ensuremath{B_{1}}\xspace their approach will benefit from our approximation of \ensuremath{R_{\mathit{ex}}}\xspace. By irradiation with Toggling Inversion Preparation (iTIP) Jin and Kim were able to remove \ensuremath{{Z^{\mathit{ss}}}}\xspace which allows for direct exponential fit of the difference signal of SL and iSL and thus promises reduced scanning time \cite{jin_quantitative_2012}.}
\subsection{Separation for \ensuremath{R_{\mathit{ex}}}\xspace}
The dependence of CEST and SL on exchange is mediated by \ensuremath{R_{\mathit{ex}}}\xspace, the exchange-dependent relaxation rate in the rotating frame. Since the discussed evaluation algorithms for PTR depend on direct water saturation, we propose methods which use the underlying structure of the Z-spectrum and solve the solutions for \ensuremath{R_{\mathit{ex}}}\xspace.
For the transient state QUESTRA can be extended by inclusion of \ensuremath{R_{\mathit{ex}}}\xspace and the projection factors \add{(in \ensuremath{\tilde{Z}}\xspace, Eq. \eqref{eqn:Zs})} :
\begin{multline}
\label{eqn:ZZ}
\text{QUESTRA}_{R_{ex}}(t)=\frac{\ensuremath{\tilde{Z}}\xspace(+\ensuremath{\Delta\omega}\xspace,t)}{\ensuremath{\tilde{Z}}\xspace(-\ensuremath{\Delta\omega}\xspace,t)}= e^{-\ensuremath{R_{\mathit{ex}}}\xspace(\ensuremath{\Delta\omega}\xspace) t}.
\end{multline}
which provides direct access to \ensuremath{R_{\mathit{ex}}}\xspace. Even without creating \ensuremath{\tilde{Z}}\xspace one can measure the experimental \ensuremath{R_{1\rho}}\xspace(\ensuremath{\Delta\omega}\xspace) decay rate and obtains $\ensuremath{R_{\mathit{ex}}}\xspace(\ensuremath{\Delta\omega}\xspace) \approx \ensuremath{R_{1\rho}}\xspace(+\ensuremath{\Delta\omega}\xspace)-\ensuremath{R_{1\rho}}\xspace(-\ensuremath{\Delta\omega}\xspace)$ by asymmetry analysis of the rate \ensuremath{R_{1\rho}}\xspace(\ensuremath{\Delta\omega}\xspace).
For the evaluation of steady-state measurements we suggest an extension of Eq. \eqref{eqn:reddyrefnorm}
\begin{multline}
\label{eqn:MTnorm}
\text{MTR}_{R_{ex}}(+\ensuremath{\Delta\omega}\xspace)=\frac{\ensuremath{{Z^{\mathit{ss}}(-\dw)}}\xspace-\ensuremath{{Z^{\mathit{ss}}(+\dw)}}\xspace}{\ensuremath{{Z^{\mathit{ss}}(-\dw)}}\xspace\cdot\ensuremath{{Z^{\mathit{ss}}(+\dw)}}\xspace} = \\
= \frac{1}{\ensuremath{{Z^{\mathit{ss}}(+\dw)}}\xspace} - \frac{1}{\ensuremath{{Z^{\mathit{ss}}(-\dw)}}\xspace} = \frac{\ensuremath{R_{\mathit{ex}}}(+\dw)\xspace}{\cos\theta\cdot \ensuremath{P_{z}}\xspace \cdot \R{1a}}
\end{multline}
which yields \ensuremath{R_{\mathit{ex}}}\xspace in units of \R{1a} and is independent of spillover. \ensuremath{R_{\mathit{ex}}}\xspace can be calculated by determination of \R{1a} and the projection factors . $\theta$ can be determined by \ensuremath{B_{1}}\xspace mapping and \R{1a} can be measured, however \R{1a} is not the same as the observed relaxation rate \ensuremath{R_{\mathit{obs}}}\xspace in a inversion or saturation recovery experiment, especially if a macromolecular pool is present \cite{desmond_understanding_2012}.
Since MTR$_{R_{ex}}$ and QUESTRA$_{R_{ex}}$ evaluations employ directly Z-spectra data, they are useful saturation transfer evaluation methods for determination of \ensuremath{R_{\mathit{ex}}}\xspace with correction of direct saturation.
However, they are still asymmetry-based and are not applicable to systems with pools with opposed resonance frequencies. In this case, the most reliable evaluation is fitting whole Z-spectra by using Eq.\eqref{eqn:Z_full_solution_apex} including a superimposed \ensuremath{R_{\mathit{ex}}}\xspace of the contributing pools.
\subsection{Determination of \R{2b}, \ensuremath{k_{b}}\xspace and \ensuremath{f_{b}}\xspace}
As proposed by Jin et al. \cite{jin_spin-locking_2011} the width $\Gamma$ (Eq. \eqref{eqn:SOL_RexGamma}) of \ensuremath{R_{\mathit{ex}}}\xspace(\ensuremath{\Delta\omega_{b}}\xspace) can be used to obtain \ensuremath{k_{b}}\xspace directly. But especially for small \ensuremath{k_{b}}\xspace the extension by \R{2b} is necessary. Fitting \ensuremath{R_{\mathit{ex}}^{\mathit{max}}}\xspace for different \ensuremath{B_{1}}\xspace yields \ensuremath{f_{b}}\xspace and \ensuremath{k_{b}}\xspace separately similar to the QUESP method \cite{mcmahon_quantifying_2006} and Dixons Omega Plots \cite{dixon_concentration-independent_2010} , but again the neglect of \R{2b} in Eq. \eqref{eqn:SOL_RexmaxLS} will distort the values for \ensuremath{k_{b}}\xspace and \ensuremath{f_{b}}\xspace.
The width of \ensuremath{R_{\mathit{ex}}}\xspace is a linear function of $\ensuremath{\omega_{1}}\xspace^2$:
\begin{align}
\frac{\Gamma^2}{4}(\ensuremath{\omega_{1}}\xspace^2)= \frac{\ensuremath{k_{b}}\xspace+\R{2b}}{\ensuremath{k_{b}}\xspace}\cdot (\ensuremath{\omega_{1}}\xspace^2) + (\ensuremath{k_{b}}\xspace+\R{2b})^2
\end{align}
and 1/\ensuremath{R_{\mathit{ex}}^{\mathit{max}}}\xspace is a linear function of $\ensuremath{\omega_{1}}\xspace^{-2}$
\begin{align}
\frac{1}{\ensuremath{R_{\mathit{ex}}^{\mathit{max}}}\xspace (\ensuremath{\omega_{1}}\xspace^{-2})} = \frac{\ensuremath{k_{b}}\xspace+\R{2b}}{\ensuremath{f_{b}}\xspace}\cdot (\ensuremath{\omega_{1}}\xspace^{-2}) + \frac{1}{\ensuremath{f_{b}}\xspace\ensuremath{k_{b}}\xspace}.
\end{align}
Hence, also the fit of Z-spectra for different \ensuremath{B_{1}}\xspace yields \ensuremath{f_{b}}\xspace, \ensuremath{k_{b}}\xspace and \R{2b}, separately.
\section{Conclusion}
\label{seq:conc}
We extended the analytical solution of the BM\xspace equations for SL by the relaxation rate \R{2b} and identified the projection factors necessary for application of the theory to CEST experiments. Temporal evolution as well as steady-state magnetization of CEST\xspace and SL\xspace experiments can be described by one single model governed by the smallest eigenvalue in modulus of the BM\xspace equation system which is $-\ensuremath{R_{1\rho}}\xspace$. \ensuremath{R_{1\rho}}\xspace contains the exchange-dependent relaxation rate \ensuremath{R_{\mathit{ex}}}\xspace. We extended \ensuremath{R_{\mathit{ex}}}\xspace by the transversal relaxation \R{2b} which allows application of the theory to slow exchange, where \R{2b} is in the order of \ensuremath{k_{b}}\xspace and not negligible. \ensuremath{R_{\mathit{ex}}}\xspace of different pools can be superimposed to a multi-pool model even for a macromolecular MT pool. Compared to methods designed to estimate PTR, estimators of \ensuremath{R_{\mathit{ex}}}\xspace are less dependent on water proton relaxation. Finally, we showed that determination of \ensuremath{R_{\mathit{ex}}}\xspace as a function of \ensuremath{\omega_{1}}\xspace and \ensuremath{\Delta\omega}\xspace allows to determine concentration, exchange rate, and transverse relaxation of the exchanging pool.
|
{
"timestamp": "2013-01-01T02:04:05",
"yymm": "1203",
"arxiv_id": "1203.2067",
"language": "en",
"url": "https://arxiv.org/abs/1203.2067"
}
|
\section{Asymptotic behavior and types of representations}\label{asymptotic}
In this section we shall give function-theoretic conditions for a function in $ \mathcal{L}_n$ to have a representation of a given type. These conditions will be in terms of the asymptotic behavior of the function at $\infty$.
Every function in $\mathcal{L}_n$ has a type $4$ representation, by Theorem \ref{thm2.4}. Let us characterize the functions that possess a type $3$ representation. We denote by $\chi$ the vector $(1,\dots,1)$ of ones in $\mathbb{C}^n$.
The following statement contains Theorem \ref{type3intro}.
\begin{theorem}\label{type3asymp}
The following three conditions are equivalent for a function $h \in \mathcal{L}_n$.
\begin{enumerate}
\item The function $h$ has a Nevanlinna representation of type $3$;
\item \beq\label{type3liminf}
\liminf_{s\to\infty} \frac{1}{s} \im h(is\chi)=0;
\eeq
\item \beq\label{type3lim}
\lim_{s\to\infty} \frac{1}{s} \im h(is\chi)=0.
\eeq
\end{enumerate}
\end{theorem}
\begin{proof}
(1)$\Rightarrow$(3) Suppose that $h$ has a Nevanlinna representation of type $3$:
\beq\label{type3bis}
h(z)= a + \ip{(1-iA)(A-z_Y)^{-1}(1+z_YA)(1-iA)^{-1} v}{v}
\eeq
for suitable $a\in\mathbb{R}, \h, A, Y$ and $v\in\h$.
Since
\[
(is\chi)_Y =\sum_j isY_j = is
\]
we have
\[
h(is\chi) =a+\ip{(1-iA)(A-is)^{-1}(1+isA)(1-iA)^{-1} v}{v}.
\]
Let $\nu$ be the scalar spectral measure for $A$ corresponding to the vector $v\in\h$. By the Spectral Theorem
\begin{align*}
h(is\chi)&=a+\int (1-it)(t-is)^{-1}(1+ist)(1-it)^{-1} \ \dd\nu(t)\\
&= a+\int \frac {1+ist}{t-is} \ \dd\nu(t).
\end{align*}
Since
\[
\im\frac{1+ist}{t-is} = \frac{s(1+t^2)}{s^2+t^2},
\]
we have
\[
\frac{1}{s} \im h(is\chi) = \int \frac{1+t^2}{s^2+t^2} \ \dd\nu(t).
\]
The integrand decreases monotonically to $0$ as $s\to\infty$ and so, by the Monotone Convergence Theorem, equation \eqref{type3lim} holds.
(3)$\Rightarrow$(2) is trivial.
(2)$\Rightarrow$(1)
Now suppose that $h \in \mathcal{L}_n$ and
\[
\liminf_{s\to\infty}\frac{1}{s}\IM h(is\chi)=0.
\]
By Theorem \ref{thm2.4}, $h$ has a Nevanlinna representation of type $4$: that is, there exist $a, \h, \mathcal{N}\subset\h$, operators $A, \, Y$ on $\mathcal{N}^\perp$ and a vector $v\in\h$ with the properties described in Definition \ref{def3.1} such that
\[
h(z) = a+ \ip{M(z)v}{v}
\]
for all $z\in\Pi^n$, where
\beq\label{defM4bis}
M(z)=\bbm -i&0\\0&1-iA \ebm
\left( \bbm 1&0 \\ 0 & A \ebm -
z_P\bbm 0&0\\0& 1 \ebm \right)^{-1}
\left(z_P\bbm 1&0\\0& A\ebm +
\bbm 0&0\\ 0&1 \ebm \right)
\bbm -i&0\\0&1-iA \ebm^{-1}.
\eeq
Thus, for $s>0$, since once again $(is\chi)_P=is$,
\begin{align*}
M(is\chi) &= \bbm -i&0\\0&1-iA \ebm
\bbm 1&0 \\ 0 & (A-is)^{-1} \ebm
\bbm is&0\\0& 1+isA \ebm
\bbm i&0\\0&(1-iA)^{-1} \ebm\\
&=\bbm is&0\\0& (1-iA)(A-is)^{-1}(1+isA)(1-iA)^{-1}\ebm.
\end{align*}
Let the projections of $v$ onto $\mathcal{N},\ \mathcal{N}^\perp$ be $v_1, v_2$ respectively. Then
\begin{align*}
h(is\chi) &= a+\ip{M(is\chi)v}{v} \\
&= a +is\norm{v_1}^2 + \ip{(1-iA)(A-is)^{-1}(1+isA)(1-iA)^{-1} v_2}{v_2}
\end{align*}
and therefore
\begin{align*}
\frac{1}{s} \im h(is\chi) &= \norm{v_1}^2 + \frac{1}{s} \im\ip{(1-iA)(A-is)^{-1}(1+isA)(1-iA)^{-1} v_2}{v_2}\\
&\geq \norm{v_1}^2
\end{align*}
by Corollary \ref{Mpick}. Hence
\begin{align*}
0 &= \liminf_{s\to\infty}\frac{1}{s}\im h(is\chi) \\
& \geq \norm{v_1}^2.
\end{align*}
It follows that $v_1 = 0$.
Let the compression of the projection $P_j$ to $\mathcal{N}^\perp$ be $Y_j$: then $Y=(Y_1,\dots,Y_n)$ is a positive decomposition of $\mathcal{N}^\perp$, and the compression of $z_P$ to $\mathcal{N}^\perp$ is $z_Y$.
By Remark \ref{3.2} the (2,2) block $M_{22}(z)$ in $M(z)$ is
\begin{align*}
M_{22}(z)&= (1-iA)(A-z_Y)^{-1} (1+z_YA)(1-iA)^{-1}.
\end{align*}
Since $v_1=0$ it follows that
\begin{align*}
h(z)&= a+ \ip{M(z)v}{v} \\
&= a+ \ip{M_{22}(z)v_2}{v_2} \\
&= a+ \ip{(1-iA)(A-z_Y)^{-1} (1+z_YA)(1-iA)^{-1} v_2}{v_2},
\end{align*}
which is the desired type $3$ representation of $h$. Hence (2)$\Rightarrow$(1).
\end{proof}
In \cite{BKV2} it is shown that condition (3) in the above theorem is also a necessary and sufficient condition that $-ih$ have a {\em $\Pi^n$-impedance-conservative realization.}
Type $2$ representations were characterized by the following theorem in \cite{ATY} in the case of two variables. The following result, which contains Theorem \ref{type2intro}, shows that the result holds generally.
\begin{theorem}\label{type2asymp}
The following three conditions are equivalent for a function $h\in\mathcal{L}_n$.
\begin{enumerate}
\item The function $h$ has a Nevanlinna representation of type $2$;
\item \beq \label{type2liminf}\liminf_{s\to\infty} s\IM h(is\chi)<\infty;\eeq
\item \beq \label{type2lim} \lim_{s\to\infty} s\IM h(is\chi)<\infty.\eeq
\end{enumerate}
\end{theorem}
\begin{proof}
\nin (1)$\Rightarrow$(3) Suppose that $h$ has the type $2$ representation $h(z)=a+\ip{(A-z_Y)^{-1} v}{v}$ for a suitable real $a$, self-adjoint $A$, positive decomposition $Y$ and vector $v$. Let $\nu$ be the scalar spectral measure for $A$ corresponding to the vector $v$. Then, for $s>0$, $A-(is\chi)_Y = A-is$ and so
\begin{align*}
s \im h(is\chi) &= s \im \int \frac{\dd \nu(t)}{t-is} \\
&= \int \frac{ s^2 \ \dd\nu(t)}{t^2+s^2}.
\end{align*}
The integrand is positive and increases monotonically to $1$ as $s\to\infty$. Hence, by the Dominated Convergence Theorem
\[
\lim_{s\to\infty} s \im h(is\chi) = \nu(\mathbb{R}) = \|v\|^2 < \infty.
\]
Hence (1)$\Rightarrow$(3).
\nin (3)$\Rightarrow$(2) is trivial.
\nin (2)$\Rightarrow$(1) Suppose (2) holds. {\em A fortiori},
\[
\liminf_{s\to\infty} \frac{1}{s} \im h(is\chi) = 0.
\]
By Theorem \ref{type3asymp} $h$ has a type $3$ representation \eqref{type3bis}
for suitable $a\in\mathbb{R}, \h, A, Y$ and $v\in\h$. Let $\nu$ be the scalar spectral measure for $A$ corresponding to the vector $v$. Then for $s>0$
\begin{align*}
s\im h(is\chi) &= s\im \int\frac{1+ist}{t-is} \ \dd\nu(t) \\
&= \int \frac{s^2(1+t^2)}{t^2+s^2}\ \dd\nu(t).
\end{align*}
As $s\to\infty$ the integrand increases monotonically to $1+t^2$. Condition (2) now implies that
\[
\int 1+t^2 \ \dd\nu(t) < \infty.
\]
It follows that $v\in\mathcal{D}(A)$. Hence, by Proposition \ref{2gives3}, $h$ has a representation of type $2$.
\end{proof}
\begin{comment}
\begin{theorem}
Let $h \in \mathcal{L}_n$, and for $s \in \mathbb{R}$, let $m_s$ be defined by
\[
m_s(z) = \frac{1 +sz}{s - z}.
\]
$m_s \circ h$ has a Nevanlinna representation of type $2$ for all but a countable number of $s$.
\end{theorem}
\end{comment}
In \cite{ATY} we proved Theorem \ref{type2asymp} for $n=2$ using a different approach from the present one.
From this theorem the characterization of type $1$ representations follows just as in the one-variable case. We obtain a strengthening of Theorem \ref{thm1.2}.
\begin{theorem}\label{type1asymp}
The following three conditions are equivalent for a function $h\in\mathcal{L}_n$.
\begin{enumerate}
\item The function $h$ has a Nevanlinna representation of type $1$;
\item
\[
\liminf_{s\to\infty} s\abs{h(is\chi)}<\infty;
\]
\item
\beq
\label{type1lim} \lim_{s\to\infty} s\abs{h(is\chi)}<\infty.
\eeq
\end{enumerate}
\end{theorem}
\begin{proof}
We follow Lax's treatment \cite{lax02} of the one-variable Nevanlinna theorem.
(1)$\Rightarrow$(3) Suppose that $h$ has a type $1$ representation as in equation \eqref{type1formula} for some $\h,\ A,\ Y$ and $v$. Then
\begin{align}
h(is\chi) &= \ip{(A - is)^{-1}\alpha}{\alpha} \notag \\
&= \ip{(A + is)(A^2 + s^2)^{-1} \alpha}{\alpha} \notag,
\end{align}
and so
\[
\re sh(is\chi) = \ip{sA (A^2 + s^2)^{-1} \alpha}{\alpha}, \quad \im sh(is\chi) = \ip{s^2 (A^2 + s^2)^{-1} \alpha}{\alpha}.
\]
Let $\nu$ be the scalar spectral measure for $A$ corresponding to the vector $\al\in\h$. Then
\[
\re sh(is\chi) = \int \frac{st}{t^2+s^2}\ \dd\nu(t), \quad \im sh(is\chi) = \int \frac{s^2}{t^2+s^2} \ \dd\nu(t).
\]
The integrand in the first integral tends pointwise in $t$ to $0$ as $s\to\infty$, and by the inequality of the means it is no greater than $\half$; thus the integral tends to $0$ as $s\to\infty$ by the Dominated Convergence Theorem. The integrand in the second integral increases monotonically to $1$ as $s\to \infty$. Thus
\[
\re sh(is\chi) \to 0, \qquad \im sh(is\chi) \to \|\al\|^2 \quad \mbox{ as } s\to \infty.
\]
Hence the inequality \eqref{type1lim} holds. Thus (1)$\Rightarrow$(3).
(3)$\Rightarrow$(2) is trivial.
(2)$\Rightarrow$(1) Suppose that
\beq\label{toomany}
\liminf_{s \to \infty} s\abs{h(is\chi)} < \infty.
\eeq
As
\beq \notag
\liminf_{s \to \infty} s \IM h(is\chi) \leq \liminf_{s \to \infty} s\abs{h(is\chi)} < \infty,
\eeq
$h$ satisfies condition \eqref{type2liminf} of Theorem \ref{type2asymp}. Therefore $h$ has a representation of type $2$, say
\[
h(z) = a + \ip{(A - z_Y)^{-1}\alpha}{\alpha}.
\]
It remains to show that $a = 0$. The inequality \eqref{toomany} implies that there exists a sequence $s_n$ tending to $\infty$ such that $h(is_n\chi) \to 0$. But
\beq \notag
\RE h(is_n\chi) = a + \ip{A(A^2 + s_n^2)^{-1}\alpha}{\alpha} \to a.
\eeq
Hence $a = 0$ and $h$ has a type $1$ representation. This establishes (2)$\Rightarrow$(1).
\end{proof}
\section{Carapoints at infinity}\label{caraInfty}
How can we recognise from function-theoretic properties whether a given function in the $n$-variable Loewner class admits a Nevanlinna representation of a given type? In the preceding section it was shown that it depends on growth along a single ray through the origin. In this section we describe the notion of carapoints at infinity for a function in the Pick class, and in the next section we shall give succinct criteria for the four types in the language of carapoints.
Carapoints (though not with this nomenclature) were first introduced by Carath\'eodory in 1929 \cite{car29} for a function $\varphi$ on the unit disc, as a hypothesis in the ``Julia-Carath\'eodory Lemma". For any $\tau\in\mathbb{T}$, a function $\varphi$ in the Schur class {\em satisfies the Carath\'eodory condition at $\tau$} if
\beq\label{julia}
\liminf_{\lambda\to\tau} \frac{1 - \abs{\varphi(\lambda)}}{1 - \abs{\lambda}} < \infty.
\eeq
The notion has been generalized to other domains by many authors. Consider domains $U\subset\mathbb{C}^n$ and $V\subset \mathbb{C}^m$ and an analytic function $\varphi$ from $U$ to the closure of $V$. The function $\varphi$ is said to satisfy Carath\'eodory's condition at $\tau\in\partial U$ if
\[
\liminf _{\la\to\tau}\frac{ \dist(\varphi(\la), \partial V)}{\dist(\la, \partial U)} < \infty.
\]
Thus, for example, when $U=\Pi^n, V=\Pi$, a function $h\in \Pick_n$ satisfies Carath\'eodory's condition at the point $x\in\mathbb{R}^n$ if
\beq \label{caragen}
\liminf_{z\to x} \frac {\im h(z)}{\min_j \im z_j } < \infty.
\eeq
This definition works well for finite points in $\partial U$, but for our present purpose we need to consider points at infinity in the boundaries of $\Pi^n$ and $\Pi$. We shall introduce a variant of Carath\'eodory's condition for the class $\Pick_n$ with the aid of the Cayley transform
\beq\label{cayleypar}
z = i\frac{1+\lambda}{1-\lambda}, \qquad \la = \frac{z-i}{z+i},
\eeq
which furnishes a conformal map between $\mathbb{D}$ and $\Pi$, and hence a biholomorphic map between $\mathbb{D}^n$ and $\Pi^n$ by co-ordinatewise action. We obtain a one-to-one correspondence between $\mathcal{S}_n\setminus \{\bf {1}\}$ and $\Pick_n$ via the formulae
\beq \label{cayleyfunc}
h(z) = i\frac{1+\varphi(\lambda)}{1-\varphi(\lambda)}, \quad \varphi(\la)= \frac{h(z)-i}{h(z)+i}
\eeq
where $\bf{1}$ is the constant function equal to $1$ and $\la, z$ are related by equations \eqref{cayleypar}. For $\varphi\in\mathcal{S}_n$ we define $\tau\in\mathbb{T}^n$ to be a {\em carapoint} of $\varphi$ if
\beq\label{juliaBis}
\liminf_{\lambda\to\tau} \frac{1 - |\varphi(\lambda)|}{1 - \norm{\lambda}_\infty} < \infty.
\eeq
We can now extend the notion of carapoints to points at infinity. The point $(\infty,\dots,\infty)$ in the boundary of $\Pi^n$ corresponds to the point $\chi$ in the closed unit disc;
as in the last section, $\chi$ denotes the point $(1,\dots,1)\in\mathbb{C}^n$.
\begin{definition}
Let $h$ be a function in the Pick class $\Pick_n$ with associated function $\varphi$ in the Schur class $\mathcal{S}_n$ given by equation \eqref{cayleyfunc}. Let $\tau \in \mathbb{T}^n, \ x \in (\mathbb{R}\cup\infty)^n$ be related by
\beq\label{xandtau}
x_j=i\frac{1+\tau_j}{1-\tau_j} \quad \mbox{ for } j=1,\dots,n.
\eeq
We say that $x$ is a {\em carapoint} for $h$ if $\tau$ is a carapoint for $\varphi$. We say that $h$ has a {\em carapoint at} $\infty$ if $h$ has a carapoint at $(\infty,\dots,\infty)$, that is, if $\varphi$ has a carapoint at $\chi $.
\end{definition}
Note that, for a point $x\in\mathbb{R}^n$, to say that $x$ is a carapoint of $h$ is {\em not} the same as saying that $h$ satisfies the Carath\'eodory condition \eqref{caragen} at $x$. Consider the function $h(z)=-1/z_1$ in $\Pick_n$. Clearly $h$ does not satisfy Carath\'eodory's condition at $0\in\mathbb{R}^n$. However, the function $\varphi$ in $\mathcal{S}_n$ corresponding to $h$ is $\varphi(\la) = -\la_1$, which does have a carapoint at $-\chi$, the point in $\mathbb{T}^n$ corresponding to $0 \in\mathbb{R}^n$. Hence $h$ has a carapoint at $0$.
We shall be mainly concerned with carapoints at $0$ and $\infty$. The following observation will help us identify them. For any $h\in\Pick_n$ we define $h^\flat\in\Pick_n$ by
\[
h^\flat(z)= h\left(-\frac{1}{z_1},\dots, -\frac{1}{z_n}\right)\quad\mbox{ for }z\in\Pi^n.
\]
For $\varphi\in\mathcal{S}_n$ we define
\[
\varphi^\flat(\la)= \varphi(-\la).
\]
If $h$ and $\varphi$ are corresponding functions, as in equations \eqref{cayleyfunc}, then so are $h^\flat$ and $\varphi^\flat$.
\begin{proposition}\label{caracriter}
The following conditions are equivalent for a function $h\in\Pick_n$.
\begin{enumerate}
\item $\infty$ is a carapoint for $h$;
\item $0$ is a carapoint for $h^\flat$;
\item
\[
\liminf_{y\to 0+} \frac{\im h^\flat(iy\chi)}{y|h^\flat(iy\chi)+i|^2} < \infty;
\]
\item
\[
\liminf_{y\to\infty} \frac{y\im h(iy\chi)}{|h(iy\chi)+i|^2} < \infty.
\]
\end{enumerate}
\end{proposition}
\begin{proof}
(1)$\Leftrightarrow$(2)
Since $-\chi\in\mathbb{T}^n$ corresponds under the Cayley transform to $0\in \mathbb{R}^n$, we have
\begin{align*}
\infty \mbox{ is a carapoint of } h\quad &\Leftrightarrow \quad \chi \mbox{ is a carapoint of } \varphi \\
&\Leftrightarrow \quad -\chi \mbox{ is a carapoint of }\varphi^\flat \\
&\Leftrightarrow \quad 0 \mbox{ is a carapoint of } h^\flat.
\end{align*}
\nin (2)$\Leftrightarrow$(3)
A consequence of the $n$-variable Julia-Carath\'eodory Theorem \cite{jafari,abate},
is that $\tau\in\mathbb{T}^n$ is a carapoint of $\varphi\in\mathcal{S}_n$ if and only if
\[
\liminf_{r\to 1-} \frac{1-|\varphi(r\tau)|}{1-r} < \infty.
\]
It follows that
\begin{align*}
0 \mbox{ is a carapoint for } h^\flat \quad &\Leftrightarrow \quad -\chi \mbox{ is a carapoint for } \varphi^\flat \\
&\Leftrightarrow \quad \liminf_{r\to 1-} \frac{1-|\varphi^\flat(-r\chi)|}{1-r} < \infty \\
&\Leftrightarrow \quad \liminf_{r\to 1-} \frac{1-|\varphi^\flat(-r,-r)|^2}{1-r^2} < \infty. \\
\end{align*}
Let $iy\in\Pi$ be the Cayley transform of $-r \in (-1,0)$, so that $y\to 0+$ as $r\to 1-$.
In view of the identity
\beq\label{caracond}
\frac{1-|\varphi(\la)|^2}{1-\|\la\|^2_\infty}= \left(\max_j \frac{|z_j+i|^2}{\im z_j}\right) \frac{\im h(z)}{|h(z)+i|^2}
\eeq
we have
\begin{align*}
0 \mbox{ is a carapoint for } h^\flat \quad &\Leftrightarrow \quad \liminf_{y\to 0+} \frac{|iy+i|^2}{y} \frac{\im h^\flat(iy\chi)}{|h^\flat(iy\chi)+i|^2} < \infty \\
&\Leftrightarrow \quad \liminf_{y\to 0+} \frac{\im h^\flat(iy\chi)}{y|h^\flat(iy\chi)+i|^2} < \infty.
\end{align*}
\nin (3)$\Leftrightarrow$(4) Replace $y$ by $1/y$.
\end{proof}
\begin{corollary}\label{suffcond}
If $f\in\Pick_n$ satisfies Carath\'eodory's condition
\beq\label{ccond}
\liminf_{z\to x} \frac{\im f(z)}{\im z} < \infty
\eeq
at $x\in\mathbb{R}^n$ then $x$ is a carapoint for $f$. If
\[
\liminf_{y\to\infty} y\im f(iy\chi) < \infty
\]
then $\infty$ is a carapoint for $f$.
\end{corollary}
\begin{proof}
Let $h=f^\flat \in\Pick_n$. Clearly $|h^\flat(z)+i| \geq 1$ for all $z\in\Pi^n$. If the condition \eqref{ccond} holds for $x=0$ then
\[
\liminf_{z\to 0} \frac{\im h^\flat(z)}{|h^\flat(z)+i|^2 \min_j \im z_j}\leq \liminf_{z\to 0}\frac {\im h^\flat(z)}{\min_j\im z_j}< \infty
\]
and hence, by (2)$\Leftrightarrow$(3) of Proposition \ref{caracriter}, $0$ is a carapoint for $h^\flat=f$. The case of a general $x\in\mathbb{R}^n$ follows by translation.
\end{proof}
If $h \in \Pick_n$ has a carapoint at $x\in(\mathbb{R}\cup\infty)^n$ then it has a value at $x$ in a natural sense. If $\varphi\in\mathcal{S}_n$ has a carapoint at $\tau\in\mathbb{T}^n$, then by \cite{jafari} there exists a unimodular constant $\varphi(\tau)$ such that
\beq\label{ntlimit}
\lim_{\lambda \stackrel{\mathrm {nt}}{\to} \tau} \varphi(\lambda) = \varphi(\tau).
\eeq
Here $\la\stackrel{\mathrm {nt}}{\to}\tau$ means that $\la$ tends nontangentially to $\tau$ in $\mathbb{D}^n$.
\begin{definition}
If $h\in\Pick_n$ has a carapoint at $x\in (\mathbb{R}\cup\infty)^n$ then we define
\[
h(x) = \threepartdef{\infty}{\varphi(\tau)=1}{}{}{\ds i\frac{1+\varphi(\tau)}{1-\varphi(\tau)}}{\varphi(\tau)\neq 1}
\]
where $\tau\in\mathbb{T}^n$ corresponds to $x$ as in equation \eqref{xandtau}.
\end{definition}
Thus $h(\infty)\in \mathbb{R}\cup\{\infty\}$ when $\infty$ is a carapoint of $h$.
In the example $h(z)=-1/z_1$, since the value of $\varphi(-\la)$ at $-\chi$ is $1$, we have $h(0)=\infty$.
Although the value of $h(\infty)$ is defined in terms of the Schur class function $\varphi$, it can be expressed more directly in terms of $h$.
\begin{proposition} \label{hinfty}
If $\infty$ is a carapoint of $h$ then
\beq\label{ntlimatinfty}
h(\infty)= h^\flat(0) = \lim_{z \stackrel{\mathrm {nt}}{\to} \infty} h(z).
\eeq
\end{proposition}
Here we say that $z \stackrel{\mathrm {nt}}{\to} \infty$ if $z\to (\infty,...,\infty)$ in the set $\{z\in\Pi^n: (-1/z_1,\dots, -1/z_n) \in S\}$ for some set $S\subset \Pi^n$ that approaches $0$ nontangentially, or equivalently, if $z\to (\infty,\dots,\infty)$ in a set on which $\|z\|_\infty /\min_j \im z_j$ is bounded.
\begin{proof}
Clearly
\[
h(\infty) =\infty \quad \Leftrightarrow \quad \varphi(\chi)=1 \quad \Leftrightarrow \quad \varphi^\flat(-\chi)=1 \quad \Leftrightarrow \quad h^\flat(0) =\infty.
\]
Similarly, for $\xi\in\mathbb{R}$,
\[
h(\infty)=\xi \quad \Leftrightarrow \quad\varphi(\chi) = \frac{\xi -i}{\xi+i} \quad\Leftrightarrow \quad \varphi^\flat(-\chi) = \frac{\xi -i}{\xi+i} \quad \Leftrightarrow \quad h^\flat(0) =\xi.
\]
Thus, whether $h(\infty)$ is finite or infinite, $h(\infty)=h^\flat(0)$.
Equation \eqref{ntlimatinfty} follows from the relation \eqref{ntlimit}.
\end{proof}
\section{Types of functions in the Loewner class}\label{caraTypes}
In this section we shall show that the type of a function $h\in\mathcal{L}_n$ is entirely determined by whether or not $\infty$ is a carapoint of $h$ and by the value of $h(\infty)$.
Let us make precise the notion of the {\em type} of a function in $\mathcal{L}_n$.
\begin{definition}
A function $h \in\mathcal{L}_n$ is of {\em type $1$} if it has a Nevanlinna representation of type $1$. For $n=2, 3$ or $4$ we say that $h$ is of {\em type $n$} if $h$ has a Nevanlinna representation of type $n$ but has no representation of type $n-1$.
\end{definition}
Clearly every function in $\mathcal{L}_n$ is of exactly one of the types $1$ to $4$. We shall now prove Theorem \ref{typeoffcn}. Recall that it states the following, for any function $h \in \mathcal{L}_n$.
\begin{enumerate}
\item $h$ is of type $1$ if and only if $\infty$ is a carapoint of $h$ and $h(\infty) = 0$;
\item $h$ is of type $2$ if and only if $\infty$ is a carapoint of $h$ and $h(\infty)\in \mathbb{R}\setminus\{0\}$;
\item $h$ is of type $3$ if and only if $\infty$ is not a carapoint of $h$;
\item $h$ is of type $4$ if and only if $\infty$ is a carapoint of $h$ and $h(\infty) = \infty$.
\end{enumerate}
\begin{proof}
(2) Let $h\in\mathcal{L}_n$ have a type $2$ representation $h(z)=a+\ip{(A-z_Y)^{-1} v}{v}$ with $a\neq 0$. By Theorem \ref{type2asymp},
\[
\liminf_{y\to\infty} y \im h(iy\chi) < \infty.
\]
By Corollary \ref{suffcond}, $\infty$ is a carapoint for $h$. Furthermore, by Proposition \ref{hinfty}
\[
h(\infty) =\lim_{y\to\infty} h(iy\chi) = a \in\mathbb{R}\setminus\{0\}.
\]
Conversely, suppose that $\infty$ is a carapoint for $h$ and $h(\infty)\in \mathbb{R}\setminus\{0\}$. By Proposition \ref{caracriter}
\[
\liminf_{y\to\infty}\frac{y\im h(iy\chi)}{|h(iy\chi)+i|^2} < \infty
\]
while by Proposition \ref{hinfty}
\[
\lim_{y\to\infty} |h(iy\chi)+i|^2 = h(\infty)^2+1 \in (1,\infty).
\]
On combining these two limits we find that
\[
\liminf_{y\to\infty} y\im h(iy\chi) < \infty,
\]
and so, by Theorem \ref{type2asymp}, $h$ has a representation of type $2$. Since $h(\infty)\neq 0$ it is clear that $h$ does not have a representation of type $1$. Thus (2) holds.
A trivial modification of the above argument proves that (1) is also true.\\
\nin (4) Let $h$ be of type 4. Then $h$ has no type $3$ representation, and so, by Theorem \ref{type3asymp}, there exists $\delta > 0$ and a sequence $(s_n)$ of positive numbers tending to $ \infty$ such that
\[
\frac{1}{s_n} \IM h(is_n\chi) \geq \delta > 0.
\]
Let $y_n=1/s_n$; then $-1/(is_n) = iy_n$, and we have
\beq\label{hflatyn}
y_n \im h^\flat(iy_n\chi) \geq \delta \quad \mbox{ for all } n\geq 1.
\eeq
Since $|h^\flat(z)+i| > \im h^\flat (z)$ for all $z$, we have
\begin{align*}
\liminf_{z\to 0} \frac{\im h^\flat(z)}{|h^\flat(z)+i|^2 \min_j \im z_j} & \leq \liminf_{z\to 0} \frac{1}{ \im h^\flat(z) \min_j \im z_j} \\
&\leq \liminf_{n\to\infty} \frac{1}{y_n \im h^\flat(iy_n\chi)} \\
& \leq 1/\delta.
\end{align*}
Hence $(0,0)$ is a carapoint of $h^\flat$, and so $\infty$ is a carapoint of $h$.
Since $y_n \to 0$ it follows from the inequality \eqref{hflatyn} that $\im h^\flat(iy_n\chi) \to \infty$, hence that $h^\flat(0) =\infty$, and therefore that $h(\infty)=\infty$.
Conversely, suppose that $\infty$ is a carapoint of $h$ and that $h(\infty) = \infty$. We shall show that
\beq \label{nolimit}
\lim_{s\to\infty}\frac{1}{s}\IM h(is\chi) \neq 0,
\eeq
and it will follow from Theorem \ref{type3asymp} that $h$ does not have a representation of type $3$, that is, $h$ is of type $4$.
Let $\varphi\in\mathcal{S}_n$ correspond to $h$ and let $r\in (0,1)$ correspond to $is\in\Pi$. Then
\begin{align}\label{need1}
\frac{1}{s}\im h(is\chi) &= \frac{1-r}{1+r} \quad \frac{1-|\varphi(r\chi)|^2}{|1-\varphi(r\chi)|^2} \nn \\
&= \frac{1-|\varphi(r\chi)|^2}{1-r^2} \, \, \frac{(1-r)^2}{|1-\varphi(r\chi)|^2}.
\end{align}
By hypothesis, $\chi$ is a carapoint for $\varphi$ and $\varphi(\chi)=1$. By definition of carapoint,
\[
\liminf_{z\to\chi} \frac{1-|\varphi(z)|^2}{1-\|z\|_\infty^2} = \al < \infty \quad \mbox{ for all } s > 0.
\]
The $n$-variable Julia-Carath\'eodory Lemma (see \cite{jafari,abate})
now tells us that $\al > 0$ and
\beq\label{need2}
\frac{|1-\varphi(r\chi)|^2}{|1-r|^2} \leq \al \frac{1-|\varphi(r\chi)|^2}{1-r^2} \quad \mbox{ for all } r\in (0,1).
\eeq
On combining equations \eqref{need1} and \eqref{need2} we obtain
\[
\frac{1}{s}\im h(is\chi) \geq \frac{1}{\al} > 0 \quad \mbox{ for all } s>0.
\]
Thus the relation \eqref{nolimit} is true, and so, by Theorem \ref{type3asymp}, $h$ is of type $4$.
Statement (3) now follows easily. The function $h\in\mathcal{L}_n$ is of type $3$ if and only if it is not of types $1,2$ or $4$, hence if and only if it is not the case that $\infty$ is a carapoint for $h$ and $h(\infty) \in \mathbb{R}\cup\{\infty\}$, hence if and only if $\infty$ is not a carapoint of $h$.
\end{proof}
We now show that there are functions in the Pick class $\Pick_2$ of all four types. We return to Example \ref{4types} and show that the functions in $\Pick_2$ which we presented there are indeed of the stated types.
\begin{example}\label{fourtypes} \rm
(1) The function
\[
h(z)= -\frac{1}{z_1+z_2} = \ip{(0-z_Y)^{-1} v}{v}_{\mathbb{C}},
\]
where $Y=\half$ and $v=1/\sqrt{2}$,
is obviously of type $1$. Let us nevertheless check that $\infty$ is a carapoint of $h$ and $h(\infty) = 0$, in accordance with Theorem \ref{typeoffcn}. We have $h(iy,iy) = \half i/y$
and hence
\[
\liminf_{y\to 0+} y\im h(iy,iy)= \half.
\]
Thus $\infty$ is a carapoint for $h$ by Proposition \ref{caracriter}. Moreover $h(iy,iy) \to 0$ as $y\to \infty$, and therefore $h(\infty)=0$.\\
\nin (2) It is immediate that the function $1+h$, with $h$ as in (1), is of type 2, and that $\infty$ is a carapoint of $1+h$ with value $1$.\\
\nin (3) We have seen that the function
\beq\label{truehbis}
h(z) =\threepartdef{ \ds \frac{1}{1+z_1z_2}\left(z_1-z_2 + \frac{iz_2(1+z_1^2)}{\sqrt{z_1z_2}}\right)}{z_1z_2 \neq -1}{}{}{\half (z_1+z_2)}{z_1z_2 = -1}
\eeq
has a representation of type $3$. To show that $h$ is indeed of type $3$ we must prove that $\infty$ is not a carapoint of $h$.
For all $y>0$ we have $h(iy,iy)=i$. Hence
\[
\liminf_{y\to \infty} \frac{y\im h(iy,iy)}{|h(iy,iy)+i|^2} = \liminf_{y\to\infty} \frac{y}{4} = \infty.
\]
By Proposition \ref{caracriter}, $\infty$ is not a carapoint for $h$. Thus $h$ is of type $3$.\\
\nin (4) The function
\[
h(z)= \frac{z_1z_2}{z_1+z_2} = -1 \left/ \left(-\frac{1}{z_1} - \frac{1}{z_2}\right) \right.
\]
is clearly in $\Pick_2$. We gave a type $4$ representation of $h$ in Example \ref{4types}.
We claim that $\infty$ is a carapoint of $h$. We have $h(iy,iy)= \half iy$, and thus
\begin{align*}
\liminf_{y\to \infty} \frac{y \im h(iy,iy)}{|h(iy,iy)+i|^2} &= \liminf_{y\to \infty} \frac{\half y^2}{|\half iy+i|^2}=2.
\end{align*}
Hence $\infty$ is a carapoint for $h$. Furthermore $h(iy,iy)= \half iy \to \infty$ as $y\to \infty$, and so $h(\infty)=\infty$. Thus $h$ is of type $4$.
\end{example}
Another example of a function of type $4$ is $h(z)=\sqrt{z_1z_2}$.
\section{ Rates of growth in the Loewner class } \label{growth}
The Nevanlinna representation formulae give rise to growth estimates for functions in the $n$-variable Loewner class. It turns out that growth is mild, both at infinity and close to the real axis. Even though the type of a function is determined by its growth on the single ray $\{iy\chi:y>0\}$, in turn the growth of the function on the entire polyhalfplane is constrained by its type.
Consider first the one-variable case. If $h$ is the Cauchy transform of a finite positive measure $\mu$ then
\[
|h(z)| \leq \int \frac{\dd \mu(t)}{|t-z|} \leq \int \frac{\dd\mu(t)}{\im z} = \frac{C}{\im z}
\]
for some $C>0$ and for all $z\in\Pi$. For a general function $h$ in the Pick class, by Nevanlinna's representation (Theorem \ref {thm1.1}) there exist $a\in\mathbb{R}, b\geq 0$ and a finite positive measure $\mu$ on $\mathbb{R}$ such that, for all $z\in\Pi$,
\begin{align*}
h(z) &= a + bz + \int \frac{1+tz}{t-z} \ \dd\mu(t)\\
&= a+bz +\int \frac{1+z^2}{t-z} + z \ \dd\mu(t)
\end{align*}
and therefore
\begin{align*}
|h(z)| &\leq |a| + b|z| + \left(\frac{1+|z|^2}{\im z} + |z|\right) \mu(\mathbb{R})\\
& \leq C\left(1+ |z| + \frac{1+|z|^2}{\im z}\right)
\end{align*}
for some $C>0$.
Similar estimates hold for the Loewner class.
\begin{proposition}
For any function $h\in\mathcal{L}_n$ there exists a non-negative number $C$ such that, for all $z\in\Pi^n$,
\beq\label{type4gr}
|h(z)| \leq C\left( 1+\|z\|_1 + \frac{1+\|z\|_1^2}{\min_j \im z_j}\right).
\eeq
For any function $h\in\mathcal{L}_n$ of type $2$ there exists a non-negative number $C$ such that, for all $z\in\Pi^n$,
\beq\label{type2gr}
|h(z)| \leq C\left(1+ \frac{1}{\min_j\im z_j}\right).
\eeq
For any function $h\in\mathcal{L}_n$ of type $1$ there exists a non-negative number $C$ such that, for all $z\in\Pi^n$,
\beq\label{type1gr}
|h(z)| \leq \frac{C}{\min_j\im z_j}.
\eeq
\end{proposition}
\begin{proof}
Let $h\in\mathcal{L}_n$. Let $\mathcal{N},\mathcal{M}, A, P, a$ and $v$ be as in Theorem \ref{thm2.4}, so that
\[
h(z)= a+\ip{M(z)v}{v}
\]
for all $z\in\Pi^n$, where $M(z)$ is the matricial resolvent given by equation \eqref{defM}. By Proposition \ref{2x2resolv} we have, for all $z\in\Pi^n$,
\begin{align*}
\|M(z)\| & \leq (1+\sqrt{10} \|z\|_1)\left(1 + \frac{1+\sqrt{2}\|z\|_1}{\min_j \im z_j}\right) \\
&\leq 1+\sqrt{10} \|z\|_1 + B \frac{1+\|z\|_1+\|z\|_1^2}{\min_j \im z_j}
\end{align*}
for a suitable choice of $B \geq 0$. Hence
\begin{align*}
|h(z)| &\leq |a| + \|M(z)\| \|v\|^2 \\
&\leq |a| + \left( 1+\sqrt{10}\|z\|_1 + B \frac{1+\|z\|_1+\|z\|_1^2}{\min_j \im z_j}\right) \|v\|^2.
\end{align*}
Since
\[
1+\|z\|_1+\|z\|_1^2 \leq \tfrac 32 (1+\|z_1\|^2),
\]
we have
\[
|h(z)| \leq C\left( 1+\|z\|_1 + \frac{1+\|z\|_1^2}{\min_j \im z_j}\right)
\]
for some choice of $C>0$ and for all $z\in\Pi^n$. Thus the estimate \eqref{type4gr} holds.
Similarly, the estimates \eqref{type2gr} and \eqref{type1gr} follow easily from the simple resolvent estimate \eqref{boundAzYinv}.
\end{proof}
\section{Introduction}\label{intro}
In a classic paper \cite{nev22} of 1922 R. Nevanlinna solved the problem of the determinacy of solutions of the Stieltjes moment problem. {\em En route} he proved several other theorems that have since been influential; in particular, the following theorem, which characterizes the Cauchy transforms of positive finite measures $\mu$ on $\mathbb{R}$, has had a profound impact on the development of modern analysis. Let $\Pick$ denote the Pick class, that is, the set of analytic functions on the upper halfplane,
\[
\Pi \df \{z \in \mathbb{C}: \im z > 0\},
\]
that have non-negative imaginary part on $\Pi$.
\begin{theorem}[Nevanlinna's Representation] \label{thm1.0}
Let $h$ be a function defined on $\Pi$. There exists a finite positive measure $\mu$ on $\mathbb{R}$ such that
\beq \label{cauchyt}
h(z) = \int \frac{d\mu}{t-z}
\eeq
if and only if $h \in \Pick$ and
\beq \label{cauchycond}
\liminf_{y\to\infty} y\abs{h(iy)}< \infty.
\eeq
\end{theorem}
A closely related theorem, also referred to in the literature as Nevanlinna's Representation, provides an integral representation for a general element of $\Pick$.
\begin{theorem}\label{thm1.1}
A function $h:\Pi\to \mathbb{C}$ belongs to the Pick class $ \mathcal P$ if and only if there exist $a\in\mathbb{R}, \ b\geq 0$ and a finite positive Borel measure $\mu$ on $\mathbb{R}$ such that
\beq\label{classical}
h(z) = a+bz+\int\frac{1+tz}{t-z} \ \dd\mu(t)
\eeq
for all $z\in\Pi$. Moreover, for any $h\in\Pick$, the numbers $a\in\mathbb{R}, \ b\geq 0$ and the measure $\mu \geq 0$ in the representation \eqref{classical} are uniquely determined.
\end{theorem}
What are the several-variable analogs of Nevanlinna's theorems? In this paper we shall propose four types of Nevanlinna representation for various subclasses of the $n$-variable Pick class $\Pick_n$, where $\Pick_n$ is defined to be the set of analytic functions $h$ on the polyhalfplane $\Pi^n$ such that $\im h \geq 0$. In addition, we shall present necessary and sufficient conditions for a function defined on $\Pi^n$ to possess a representation of a given type in terms of asymptotic growth conditions at $\infty$.
The integral representation \eqref{cauchyt} of those functions in the Pick class that satisfy condition \eqref{cauchycond} can be written in the form
\[
h(z) = \ip{(A-z)^{-1}\mathbf{1}}{\mathbf{1}}_{L^2(\mu)},
\]
where $A$ is the operation of multiplication by the independent variable on $L^2(\mu)$ and $\mathbf{1}$ is the constant function $1$. We propose that an appropriate $n$-variable analog of the Cauchy transform is the formula
\beq\label{ctn}
h(z_1,\dots,z_n) = \ip{(A-z_1Y_1-\dots -z_nY_n)^{-1} v}{v}_\h \qquad \mbox{ for } z_1,\dots,z_n \in\Pi,
\eeq
where $\h$ is a Hilbert space, $A$ is a densely defined self-adjoint operator on $\h$, $Y_1,\dots,Y_n$ are positive contractions on $\h$ summing to $1$ and $v$ is a vector in $\h$.
Theorem \ref{thm1.2} below characterizes those functions on $\Pi^n$ that have a representation of the form \eqref{ctn}. To state this theorem we require a notion based on the following classical result of Pick \cite{Pick}.
\begin{theorem}\label{pickcond1}
A function $h$ defined on $\Pi$ belongs to $\Pick$ if and only if the function $A$ defined on $\Pi \times \Pi$ by
\[
A(z, w) = \frac{h(z) - \cc{h(w)}}{z - \cc{w}}
\]
is positive semidefinite, that is, for all $n \geq 1, z_1, \dots, z_n \in \Pi, c_1, \dots, c_n \in \mathbb{C}$,
\[
\sum A(z_j, z_i)\cc{c_i} c_j \geq 0.
\]
\end{theorem}
The following theorem, proved in \cite{ag90}, leads to a generalization of Theorem \ref{pickcond1} to two variables. The {\em Schur class of the polydisc}, denoted by $\mathcal{S}_n$, is the set of analytic functions on the polydisc $\mathbb{D}^n$ that are bounded by $1$ in modulus.
\begin{theorem}\label{schurmodel}
A function $\varphi$ defined on $\mathbb{D}^2$ belongs to $\mathcal S_2$ if and only if there exist positive semidefinite functions $A_1$ and $A_2$ on $\mathbb{D}^2$ such that
\beq
1 - \cc{\varphi(\mu)}\varphi(\la) = (1 - \cc\mu_1\la_1)A_1(\la, \mu)+ (1 - \cc\mu_2\la_2)A_2(\la, \mu).
\eeq
\end{theorem}
By way of the transformations
\beq\label{cayleyin}
z = i\frac{1+\lambda}{1-\lambda}, \quad \la = \frac{z-i}{z+i},
\eeq and
\beq\label{cayleyout}
h(z) = i\frac{1+\varphi(\lambda)}{1-\varphi(\lambda)}, \quad \varphi(\la)= \frac{h(z)-i}{h(z)+i},
\eeq
there is a one-to-one correspondence between functions in the Schur and Pick classes. Under these transformations, Theorem \ref{schurmodel} becomes the following generalization of Pick's theorem to two variables.
\begin{theorem}\label{pickcond2}
A function $h$ defined on $\Pi^2$ belongs to $ \Pick_2$ if and only if there exist positive semidefinite functions $A_1$ and $A_2$ on $\Pi^2$ such that
\[
h(z) - \cc{h(w)} = (z_1 - \cc{w_1})A_1(z,w) + (z_2 - \cc{w_2})A_2(z,w).
\]
\end{theorem}
In the light of Theorems \ref{pickcond1} and \ref{pickcond2} we define the {\em Loewner class} $\mathcal{L}_n$ to be the set of analytic functions $h$ on $\Pi^n$ with the property that there exist $n$ positive semidefinite functions $A_1,\dots,A_n$ on $\Pi^n$ such that
\beq \label{loewnercond}
h(z) - \overline{h(w)} = \sum_{j=1}^n (z_j-\overline{w_j}) A_j(z,w)
\eeq
for all $z,w\in\Pi^n$.
The Loewner class $\mathcal{L}_n$ played a key role in \cite{AMY}, which gave a generalization to several variables of Loewner's characterization of the one-variable operator-monotone functions \cite{lo34}. As the following theorem makes clear, $\mathcal{L}_n$ also has a fundamental role to play in the understanding of Nevanlinna representations in several variables.
\begin{theorem} \label{thm1.2}
A function $h$ defined on $\Pi^n$ has a representation of the form \eqref{ctn} if and only if $h \in \mathcal{L}_n$ and
\beq\label{strong}
\liminf_{y\to\infty} y|h(iy,\dots,iy)| < \infty.
\eeq
\end{theorem}
In the cases when $n=1$ and $n=2$, Theorems \ref{pickcond1} and \ref{pickcond2} assert that that $\mathcal{L}_n = \Pick_n$, and so for $n=1$, Theorem \ref{thm1.2} is Nevanlinna's classical Theorem \ref{thm1.0}, and when $n=2$, Theorem \ref{thm1.2} is a straightforward generalization of that result to two variables. When there are more than two variables, it is known that the Loewner class is a proper subset of the Pick class, $\mathcal{L}_n \neq \Pick_n$ \cite{par70,var74}. Nevertheless, Nevanlinna's result survives as a theorem about the representation of elements of $\mathcal{L}_n$. Other than the work in \cite{gkvw} very little is known about the representation of functions in $\Pick_n$ for three or more variables.
For a function $h$ on $\Pi^n$, we call the formula \eqref{ctn} a {\em Nevanlinna representation of type $1$}. Thus, Theorem \ref{thm1.2} can be rephrased as the assertion that $h$ has a Nevanlinna representation of type $1$ if and only if $h \in \mathcal{L}_n$ and $h$ satisfies condition \eqref{strong}. Somewhat more complicated representation formulae are needed to generalize Theorem \ref{thm1.1}. We identify three further representation formulae, of increasing generality, and show that every function in $\mathcal{L}_n$ has a representation of one or more of the four types.
For a function $h$ defined on $\Pi^n$, we refer to a formula
\beq
h(z_1,\dots,z_n) = a + \ip{(A-z_1Y_1-\dots -z_nY_n)^{-1} v}{v}_\h \qquad \mbox{ for } z_1,\dots,z_n \in\Pi,
\eeq
where $a$ is a constant, $\h$ is a Hilbert space, $A$ is a densely defined self-adjoint operator on $\h$, $Y_1,\dots,Y_n$ are positive contractions on $\h$ summing to $1$ and $v$ is a vector in $\h$, as a \emph{ Nevanlinna representation of type $2$}.
\begin{theorem}\label{type2intro}
A function $h$ defined on $\Pi^n$ has a Nevanlinna representation of type $2$ if and only if $h \in \mathcal{L}_n$ and
\beq
\liminf_{y\to\infty} y\im h(iy,\dots,iy) < \infty.
\eeq
\end{theorem}
A {\em Nevanlinna representation of type $3$} of a function $h$ defined on $\Pi^n$ is of the form
\[
h(z) = a+ \ip{(1-iA)(A - z_Y)^{-1} (1+z_YA)(1-iA)^{-1} v}{v} \quad\mbox{ for all } z\in\Pi^n
\]
for some real $a$, some self-adjoint operator $A$ and some vector $v$, where $Y_1, \dots,Y_n$ are operators as in equation \eqref{ctn} above and $z_Y=z_1Y_1+\dots+z_nY_n$.
\begin{theorem}\label{type3intro}
A function $h$ defined on $\Pi^n$ has a Nevanlinna representation of type $3$ if and only if $h \in \mathcal{L}_n$ and
\[
\liminf_{y\to\infty} \frac{1}{y} \im h(iy, \dots, iy)=0.
\]
\end{theorem}
Finally, {\em Nevanlinna representations of type $4$} are given by the formula
\beq \label{type4intro}
h(z) = \ip{M(z)v}{v},
\eeq where $M(z)$ is an operator of the form
\beq \label{Mz}
\bbm -i&0\\0&1-iA \ebm
\left( \bbm 1&0 \\ 0 & A \ebm -
z_P\bbm 0&0\\0& 1 \ebm \right)^{-1}
\left(z_P\bbm 1&0\\0& A\ebm +
\bbm 0&0\\ 0&1 \ebm \right)
\bbm -i&0\\0&1-iA \ebm^{-1},
\eeq acting on an orthogonal direct sum of Hilbert spaces $\mathcal N \oplus \mathcal{M}$. In \eqref{type4intro}, $v$ is a vector in $\mathcal N \oplus \mathcal{M}$. In \eqref{Mz}, $A$ is a densely-defined self-adjoint operator acting on $\mathcal M$ and $z_P$ is the operator acting on $\mathcal N \oplus \mathcal{M}$ via the formula
\[
z_P = \sum z_i P_i
\]
where $P_1, \dots, P_n$ are pairwise orthogonal projections acting on $\mathcal{N} \oplus \mathcal{M}$ that sum to $1$.
\begin{theorem}\label{thm2.4}
Let $h$ be a function defined on $\Pi^n$. Then $h$ has a Nevanlinna representation of type $4$ if and only if $h \in \mathcal{L}_n$.
\end{theorem}
A weaker, ``generic" version of Theorem \ref{thm2.4} appeared in \cite[Theorem 6.9]{AMY}, where it was used to show that elements in $\mathcal{L}_n$ are locally operator-monotone.
It turns out that for $1 \leq k \leq 4$, if $h$ is a function on $\Pi^n$ and $h$ has a Nevanlinna representation of type $k$, then for $k \leq j \leq 4$, $h$ also has a Nevanlinna representation of type $j$. Thus, it is natural to define the \emph{type} of a function in $\mathcal{L}_n$ to be the smallest $k$ such that $h$ has a Nevalinna representation of type $k$.
For $h \in \mathcal{L}_n$ the type of $h$ can be characterized in function-theoretic terms through the use of a geometric idea due to Carath\'eodory. A \emph{carapoint} for a function $\varphi$ in the Schur class $\mathcal{S}_n$ is a point $\tau \in \mathbb{T}$ such that
\[
\liminf_{\lambda \to \tau} \frac{1 - \abs{\varphi(\lambda)}}{1 - \norm{\lambda}_\infty} < \infty,
\]
where
\[
\norm{\lambda}_\infty = \max_{1\leq i \leq n} \abs{\lambda_i}.
\]
Carath\'eodory introduced this notion in one variable in \cite{car29}, along the way to refining earlier results of Julia \cite{ju20}. The following was Carath\'eodory's main result; the notation $\la\stackrel{\mathrm {nt}}{\to}\tau$ means that $\la$ tends nontangentially to $\tau$.
\begin{theorem}
Let $\varphi \in \mathcal S_1, \tau \in \mathbb{T}$. If $\tau$ is a carapoint for $\varphi$, then $\varphi$ is nontangentially differentiable at $\tau$, that is, there exist values $\varphi(\tau)$ and $\varphi'(\tau)$ such that
\[
\lim_{\lambda \stackrel{\mathrm {nt}}{\to} \tau} \frac{\varphi(\lambda) - \varphi(\tau) - \varphi'(\tau)(\lambda - \tau)}{\lambda - \tau} = 0.
\]
In particular, if $\tau$ is a carapoint for $\varphi$ then there exists a unique point $\varphi(\tau) \in \mathbb{T}$ such that $\varphi(\lambda) \to \varphi(\tau)$ as $\lambda \stackrel{\mathrm {nt}}{\to} \tau$.
\end{theorem}
In several variables, carapoints have been studied in \cite{abate,jafari,amy10a}. The strong conclusion of nontangential differentiability is lost in several variables; however, at a carapoint $\tau$, there still exists a unimodular nontangential limit $\varphi(\tau)$.
As the point $\chi = (1, \dots, 1)$ is transformed to the point $\infty = (\infty, \dots, \infty)$ by \eqref{cayleyin}, it is natural to say that a function $h \in \mathcal{L}_n$ has a carapoint at $\infty$ if the associated Schur function $\varphi$, given by the transformation in \eqref{cayleyout}, has a carapoint at $\chi$, and in that case to define $h(\infty)$ by
\beq
h(\infty) = i\frac{1 + \varphi(\chi)}{1 - \varphi(\chi)}.
\eeq
\black
The connection between carapoints and function types is given in the following theorem.
\begin{theorem}\label{typeoffcn}
For a function $h \in \mathcal{L}_n$,
\begin{enumerate}
\item $h$ is of type $1$ if and only if $\infty$ is a carapoint of $h$ and $h(\infty) = 0$;
\item $h$ is of type $2$ if and only if $\infty$ is a carapoint of $h$ and $h(\infty)\in \mathbb{R}\setminus\{0\}$;
\item $h$ is of type $3$ if and only if $\infty$ is not a carapoint of $h$;
\item $h$ is of type $4$ if and only if $\infty$ is a carapoint of $h$ and $h(\infty) = \infty$.
\end{enumerate}
\end{theorem}
The paper is structured as follows. As is clear from the formulae used to define the various Nevanlinna representations, Nevanlinna representations are generalizations of the resolvent of a self-adjoint operator. These \emph{structured resolvents}, studied in Sections \ref{structured} and \ref{matricial}, are analytic operator-valued functions on the polyhalfplane $\Pi^n$ with non-negative imaginary part, fully analogous to the familiar resolvent operator. There are also {\em structured resolvent identities} for them, studied in Section \ref{resolvident} of the paper.
In modern texts Nevanlinna's representation is derived from the Herglotz Representation with the aid of the Cayley transform \cite{lax02,don74}. In Section \ref{type4} we introduce the $n$-variable \emph{strong Herglotz class} and then prove Theorem \ref{type4intro} by applying the Cayley transform to Theorem 1.8 of \cite{ag90}.
In Section \ref{type321} we derive the Nevanlinna representations of type $3, 2$, and $1$, we show how they arise naturally from the underlying Hilbert space geometry and we prove slight strengthenings of Theorems \ref{thm1.2}, \ref{type2intro} and \ref{type3intro}. In Section \ref{asymptotic} we give function-theoretic conditions for a function $h \in \mathcal{L}_n$ to possess a representation of a given type.
In Section \ref{caraInfty} we introduce the notion of carapoints for functions in the Pick class and in Section \ref{caraTypes} we establish the criteria in Theorem \ref{typeoffcn} for the type of a function using the language of carapoints.
In Section \ref{growth} we give the growth estimates for functions in $\mathcal{L}_n$ that flow from our analysis of structured resolvents, and in Section \ref{resolvident} we present resolvent identities for structured resolvents.
Results related to ours from a system-theoretic perspective have been obtained in ongoing work of J. A. Ball and D. Kalyuzhnyi-Verbovetzkyi \cite{BKV1,BKV2}. See also \cite{BS}, where Krein space methods are applied to similar problems.
\section{The matricial resolvent} \label{matricial}
The third and last form of structured resolvent that we consider has a $2\times 2$ matricial form. As will become clear, this extra complication is needed for the description of the most general type of function in the several-variable Loewner class.
By an {\em orthogonal decomposition} of a Hilbert space $\h$ we shall mean an $n$-tuple $P=(P_1,\dots,P_n)$ of orthogonal projection operators with pairwise orthogonal ranges such that $\sum_{j=1}^n P_j$ is the identity operator.
\begin{proposition}\label{2x2resolv}
Let $\mathcal H$ be the orthogonal direct sum of Hilbert spaces $\mathcal{ N, M}$, let $A$ be a densely defined self-adjoint operator on $\mathcal M$ with domain $\mathcal{D}(A) $ and let $P$ be an orthogonal decomposition of $\mathcal H$. For every $z\in\Pi^n$ the operator on $\mathcal H$ given with respect to the decomposition $\mathcal {N \oplus M}$ by the matricial formula
\begin{align}\label{defM}
M(z)&=\bbm -i&0\\0&1-iA \ebm
\left( \bbm 1&0 \\ 0 & A \ebm -
z_P\bbm 0&0\\0& 1 \ebm \right)^{-1}
\left(z_P\bbm 1&0\\0& A\ebm +
\bbm 0&0\\ 0&1 \ebm \right)
\bbm -i&0\\0&1-iA \ebm^{-1}
\end{align}
is a bounded operator defined on all of $\h$, and
\beq\label{estNormM}
\|M(z)\| \leq (1+\sqrt{10} \|z\|_1)\left(1 + \frac{1+\sqrt{2}\|z\|_1}{\min_j \im z_j}\right)
\eeq
\end{proposition}
\begin{proof}
Let $z\in\Pi^n$.
Let the projection $P_j$ have operator matrix
\beq\label{expP}
P_j = \begin{bmatrix} X_j & B_j \\ B_j^* & Y_j \end{bmatrix}
\eeq
with respect to the decomposition $\mathcal{H = N \oplus M}$. Then
\[
X=(X_1,\dots,X_n), \quad Y=(Y_1, \dots, Y_n)
\]
are positive decompositions of $\mathcal{N}, \ \mathcal{M}$ respectively, and
\[
B=(B_1, \dots, B_n), \quad B^*=(B_1^*, \dots, B_n^*)
\]
are $n$-tuples of contractions summing to $0$, from $\mathcal{M}$ to $\mathcal{N}$ and from $\mathcal{N}$ to $\mathcal{M}$ respectively. Since the $B_j$ are contractions we have
\[
\|z_B\| \leq \|z\|_1.
\]
For any $z\in\mathbb{C}^n$,
\beq\label{formzP}
z_P = \begin{bmatrix} z_X & z_B \\ z_{B^*} & z_Y \end{bmatrix}.
\eeq
Consider the third and fourth factors in the product on the right hand side of equation \eqref{defM}; the product of these two factors is well defined as an operator on $\h$ since $(1-iA)^{-1}$ maps $\mathcal{M}$ to $\mathcal D(A)$. It is even a bounded operator, since, by virtue of equation \eqref{formzP},
\begin{align}\label{last2}
\left(z_P\bbm 1&0\\0& A\ebm +
\bbm 0&0\\ 0&1 \ebm \right)
\bbm -i&0\\0&1-iA \ebm^{-1} &=
\bbm iz_X & z_BA(1-iA)^{-1} \\ iz_{B^*} & (1+z_YA)(1-iA)^{-1} \ebm.
\end{align}
Since
\[
\|A(1-iA)^{-1}\|= \|i\left(1-(1-iA)^{-1}\right)\| \leq 2
\]
we can immediately see that the operator \eqref{last2} is bounded. We can get an estimate of the norm of the operator matrix \eqref{last2} if we replace each of the four operator entries by an upper bound for its norm. We find that
\begin{align}\label{normlast2}
\left\|\left(z_P\bbm 1&0\\0& A\ebm +
\bbm 0&0\\ 0&1 \ebm \right)
\bbm -i&0\\0&1-iA \ebm^{-1}\right\| &\leq \left\| \bbm \|z\|_1 & 2\|z\|_1 \\ \|z\|_1 & 1+2\|z\|_1 \ebm \right\| \nn \\
&\leq 1+\|z\|_1\left\|\bbm 1 & 2\\1& 2\ebm \right\|\nn \\
&= 1+\sqrt{10} \|z\|_1.
\end{align}
Now consider the second factor in the definition \eqref{defM} of $M(z)$. We find that
\begin{align}\label{needThis}
\left( \bbm 1&0 \\ 0 & A \ebm -
z_P\bbm 0&0\\0& 1 \ebm\right)^{-1} &= \bbm 1 & -z_B \\ 0 & A-z_Y \ebm^{-1} \nn \\
&= \bbm 1& z_B(A-z_Y)^{-1} \\ 0& (A-z_Y)^{-1} \ebm,
\end{align}
which maps $\h$ into $\mathcal{N}\oplus \mathcal D(A)$. Hence the product of the first two factors in the product on the right hand side of equation \eqref{defM} is
\beq\label{first2}
\bbm -i&0\\0&1-iA \ebm
\left( \bbm 1&0 \\ 0 & A \ebm -
z_P\bbm 0&0\\0& 1 \ebm \right)^{-1} = \bbm -i & -iz_B(A-z_Y)^{-1} \\ 0& (1-iA)(A-z_Y)^{-1} \ebm.
\eeq
Since
\begin{align*}
\|(1-iA)(A-z_Y)^{-1}\| &= \|(1-iz_Y)(A-z_Y)^{-1} -i\| \\
&\leq 1+ \|1-iz_Y\| \, \|(A-z_Y)^{-1}\| \\
&\leq 1+\frac{1+\|z\|_1}{\min_j \im z_j}
\end{align*}
we deduce from equation \eqref{first2} that
\begin{align}\label{normfirst2}
\left\| \bbm -i&0\\0&1-iA \ebm
\left( \bbm 1&0 \\ 0 & A \ebm -
z_P\bbm 0&0\\0& 1 \ebm \right)^{-1} \right\| &\leq \left\|\bbm 1& \|z\|_1 \ \|(A-z_Y)^{-1} \| \\ 0 & 1+(1+\|z\|_1) \|(A-z_Y)\|^{-1}\ebm \right\| \nn \\
& \leq 1 +\left\|\bbm 0&\|z\|_1 \\ 0& 1+\|z\|_1 \ebm \bbm 0&0\\0& \|(A-z_Y)^{-1}\| \ebm \right\| \nn \\
&\leq 1 + \frac{1+\sqrt{2}\|z\|_1}{\min_j \im z_j}.
\end{align}
On combining the estimates \eqref{normfirst2} and \eqref{normlast2} we obtain the bound \eqref{estNormM} for $\|M(z)\|$.
\end{proof}
\begin{remark}\label{3.2} \rm
On multiplying together the expressions \eqref{first2} and \eqref{last2} we obtain the formula
\[
M(z)= \bbm z_X+z_B(A-z_Y)^{-1} z_{B^*} & -iz_B(A-z_Y)^{-1}(1+iA) \\
i(1-iA)(A-z_Y)^{-1} z_{B^*} & (1-iA)(A-z_Y)^{-1}(1+z_YA)(1-iA)^{-1} \ebm.
\]
Notice in particular that the $(2,2)$ entry (that is, the compression of $M(z)$ to $\mathcal{M}$) is the structured resolvent of $A$ of type $3$ corresponding to $Y$, the compression of $P$ to $\mathcal{M}$, as in equation \eqref{defM3}.
\end{remark}
\begin{definition}\label{defMatResolv}
Let $\mathcal H$ be the orthogonal direct sum of Hilbert spaces $\mathcal{ N, M}$, let $A$ be a densely defined self-adjoint operator on $\mathcal M$ with domain $\mathcal{D}(A) $ and let $P$ be an orthogonal decomposition of $\mathcal H$. The {\em structured resolvent of $A$ of type $4$} corresponding to $P$ is the operator-valued function $M: \Pi^n \to \L(\h)$ given by equation \eqref{defM}.
\end{definition}
We shall also refer to $M(z)$ as the {\em matricial resolvent of $A$ with respect to $P$}. The important property that $\im M(z)\geq 0$ is not at once apparent from the formula \eqref{defM}; as with structured resolvents of type $3$, there are alternative formulae from which this property is more easily shown. Once again the alternatives suffer the minor drawback that they give $M(z)$ only on a dense subspace of $\h$.
\begin{proposition}\label{alternM}
With the notation of Definition {\rm \ref{defMatResolv}}, as operators on $ \mathcal{N}\oplus\mathcal D(A)$,
\begin{align}\label{2ndM}
M(z)&=\bbm -i&0\\0&1-iA \ebm \left(\bbm 1&0\\0& A(1+A^2)^{-1}\ebm z_P + \bbm 0&0\\0&(1+A^2)^{-1} \ebm\right)
\times \nn \\ &\hspace*{3cm}
\left(\bbm 1&0\\0&A \ebm - \bbm 0&0\\0&1\ebm z_P \right)^{-1} \bbm i&0\\0&1+iA\ebm\\
&= \bbm -i&0\\0&1-iA \ebm \left(\bbm 1&0\\0&0\ebm z_P + \bbm 0&0\\0&1 \ebm\right)
\left( \bbm 1&0\\0&A \ebm - \bbm 0&0\\0&1 \ebm z_P\right)^{-1} \bbm i&0\\0&1+iA\ebm \nn \\
& \hspace*{3cm} - \bbm0&0\\0&A \ebm \label{simpler} \\
&= \bbm -i&0\\0&1-iA \ebm \left( \bbm 1&0\\0&A \ebm - z_P \bbm 0&0\\0&1 \ebm \right)^{-1} \left(z_P\bbm 1&0\\0&0\ebm + \bbm 0&0\\0&1 \ebm\right)
\bbm i&0\\0&1+iA\ebm \nn \\
& \hspace*{3cm} - \bbm0&0\\0&A \ebm \label{simpler2}
\end{align}
for all $z\in\Pi^n$.
Moreover, for all $z,\ w\in\Pi^n$,
\begin{align}\label{imM}
M(z) - M(w)^*&= \bbm -i & 0\\ 0&1-iA \ebm \left(\bbm 1&0\\0&A \ebm - w_P^* \bbm 0&0\\0&1 \ebm\right)^{-1} \times \nn\\
&\hspace*{2cm} ( z_P- w_P^*) \left(\bbm 1&0\\0&A \ebm - \bbm 0&0\\0&1 \ebm z_P\right)^{-1} \bbm i&0\\0&1+iA \ebm
\end{align}
on $\mathcal{N}\oplus \mathcal D(A)$.
\end{proposition}
\begin{proof}
By Lemma \ref{basicUnbdd} the operators $(1+A^2)^{-1}$ and
\[
C\df \im (1-iA)^{-1} = A(1+A^2)^{-1}
\]
are self-adjoint contractions defined on all of $\mathcal{M}$. Furthermore,
\[
\ran (1+A^2)^{-1} = \mathcal D(A^2), \qquad \ran C \subset \mathcal D(A).
\]
We claim that, as operators on $\mathcal{N}\oplus\mathcal D(A)$,
\begin{align}\label{swap}
\left( \bbm 1&0\\ 0&A \ebm \right . & \left . -z_P \bbm 0&0\\0&1 \ebm \right)^{-1} \left(z_P\bbm 1&0\\0&A\ebm +\bbm 0&0\\0&1 \ebm \right) = \nn \\
&\left(\bbm 1&0\\0& C\ebm z_P + \bbm 0&0\\0&(1+A^2)^{-1} \ebm\right)\left( \bbm 1&0\\0&C\ebm - \bbm 0&0\\0&(1+A^2)^{-1} \ebm z_P\right)^{-1}.
\end{align}
We have
\begin{align*}
\left(z_P\bbm 1&0\\0&A\ebm +\bbm 0&0\\0&1 \ebm \right)&\left( \bbm 1&0\\0&C\ebm - \bbm 0&0\\0&(1+A^2)^{-1} \ebm z_P\right) \\
& \hspace{-2cm} =\bbm 0&0\\0&C\ebm + z_P \bbm 1&0\\ 0&AC \ebm - \bbm 0&0\\0&(1+A^2)^{-1} \ebm z_P - z_P\bbm 0&0\\0&C \ebm z_P \\
& \hspace{-2cm} =\bbm 0&0\\0&C\ebm + z_P \left(\bbm 1&0\\ 0&AC \ebm-1\right)+\left(1 - \bbm 0&0\\0&(1+A^2)^{-1} \ebm\right) z_P - z_P\bbm 0&0\\0&C \ebm z_P \\
& \hspace{-2cm} =\bbm 0&0\\0&C\ebm - z_P \bbm 0&0\\0&(1+A^2)^{-1} \ebm + \bbm 1&0\\0& AC \ebm z_P -z_P\bbm 0&0\\0&C\ebm z_P \\
& \hspace{-2cm} =\left(\bbm 1&0\\0&A\ebm - z_P \bbm 0&0\\0&1\ebm\right)
\left(\bbm 1&0\\0&C \ebm z_P + \bbm 0&0 \\0& (1+A^2)^{-1} \ebm \right).
\end{align*}
This is an identity between operators on $\h$, in both cases a composition $\h\to\mathcal{N}\oplus\mathcal D(A)\to \h$, and moreover the first factor on the left hand side and the second factor on the right hand side are invertible, from $\mathcal{N}\oplus\mathcal D(A)$ to $\h$ and from $\h$ to $\mathcal{N}\oplus\mathcal D(A)$ respectively.
We may pre- and post-multiply appropriately to obtain equation \eqref{swap}, but note that the equation is then only valid as an identity between operators on $\mathcal{N}\oplus\mathcal D(A)$.
On combining equations \eqref{defM} and \eqref{swap} we deduce that
\begin{align*}
M(z)&=\bbm -i&0\\0&1-iA \ebm \left(\bbm 1&0\\0& C\ebm z_P + \bbm 0&0\\0&(1+A^2)^{-1} \ebm\right) \times \\
&\hspace*{3cm}\left( \bbm 1&0\\0&C\ebm - \bbm 0&0\\0&(1+A^2)^{-1} \ebm z_P\right)^{-1} \bbm -i&0\\0&1-iA\ebm^{-1}.
\end{align*}
Since
\[
\bbm -i&0\\0&1-iA\ebm^{-1} = \bbm 1&0\\0&1+A^2 \ebm^{-1} \bbm i&0\\0&1+iA \ebm
\]
and
\[
\bbm 1&0\\0&1+A^2 \ebm \left( \bbm 1&0\\0&C\ebm - \bbm 0&0\\0&(1+A^2)^{-1} \ebm z_P\right) =
\bbm 1&0\\0&A \ebm - \bbm 0&0\\0&1\ebm z_P,
\]
we deduce further that
\begin{align} \label{halfway}
M(z)&=\bbm -i&0\\0&1-iA \ebm \left(\bbm 1&0\\0& C\ebm z_P + \bbm 0&0\\0&(1+A^2)^{-1} \ebm\right)
\times \nn \\ &\hspace*{3cm}
\left(\bbm 1&0\\0&A \ebm - \bbm 0&0\\0&1\ebm z_P \right)^{-1} \bbm i&0\\0&1+iA\ebm,
\end{align}
which proves equation \eqref{2ndM}.
It is straightforward to verify that
\begin{align}\label{stfwd}
\left(\bbm 1&0\\0& C\ebm z_P + \bbm 0&0\\0&(1+A^2)^{-1} \ebm\right)
&\left(\bbm 1&0\\0&A \ebm - \bbm 0&0\\0&1\ebm z_P \right)^{-1} \\
&\hspace*{-4cm}=
\left(\bbm 1&0\\0&0\ebm z_P + \bbm 0&0\\0&1 \ebm\right)
\left( \bbm 1&0\\0&A \ebm - \bbm 0&0\\0&1 \ebm z_P\right)^{-1} -
\bbm0&0\\0&A(1+A^2)^{-1}\ebm.
\end{align}
Clearly
\[
\bbm -i&0\\0&1-iA\ebm \bbm0&0\\0&A(1+A^2)^{-1}\ebm \bbm i&0\\0&1+iA\ebm = \bbm 0&0\\0&A \ebm,
\]
and so on suitably pre- and post-multiplying equation \eqref{stfwd}, we obtain equation \eqref{simpler}.
To prove equation \eqref{simpler2}, check first that
\begin{align*}
\left( \bbm 1&0\\0&A \ebm - z_P \bbm 0&0\\0&1\ebm \right)\left(\bbm 1&0\\0&0 \ebm z_P + \bbm 0&0\\0&1\ebm \right)&= \\
\left(z_P\bbm 1&0\\0&0\ebm+\bbm 0&0\\0&1\ebm \right)&\left(\bbm 1&0\\0&A\ebm - \bbm 0&0\\0&1 \ebm z_P\right)
\end{align*}
as operators on $\mathcal{N}\oplus\mathcal{D}(A)$.
It follows that
\begin{align*}
\left(\bbm 1&0\\0&0 \ebm z_P + \bbm 0&0\\0&1\ebm \right)\left(\bbm 1&0\\0&A\ebm - \bbm 0&0\\0&1 \ebm z_P\right)^{-1} &= \\
\left( \bbm 1&0\\0&A \ebm - z_P \bbm 0&0\\0&1\ebm \right)^{-1} &\left(z_P\bbm 1&0\\0&0\ebm+\bbm 0&0\\0&1\ebm \right)
\end{align*}
as operators from $\h$ to $\mathcal{N}\oplus \mathcal{D}(A)$. On combining this relation with equation \eqref{simpler} we derive the expression \eqref{simpler2} for $M(z)|\mathcal{N}\oplus\mathcal{D}(A)$.
We now derive the identity \eqref{imM}. Let
\[
D=\bbm i&0\\0&1+iA \ebm
\]
and consider $z,\ w\in\Pi^n$. By equation \eqref{2ndM}
\beq\label{MandW}
M(z)=D^* W(z) D
\eeq
on $\mathcal{N}\oplus \mathcal D(A)$, where
\begin{align}\label{defW}
W(z) &= R(z)S(z)^{-1} -
\begin{bmatrix}
0 & 0 \\ 0 & A(1 + A^2)^{-1}
\end{bmatrix}
\end{align}
and
\[
R(z)= \begin{bmatrix}
1 & 0 \\ 0 & 0
\end{bmatrix} z_P+
\begin{bmatrix}
0 & 0 \\ 0 & 1
\end{bmatrix}, \quad S(z) = \begin{bmatrix}
1 & 0 \\ 0 & A
\end{bmatrix} -
\begin{bmatrix}
0 & 0 \\ 0 & 1
\end{bmatrix} z_P.
\]
We have seen that $S(z)$ is invertible for any $z\in \Pi^n$, so that $W(z)$ is a bounded operator on $\h$.
Clearly
\begin{align*}
M(z)-M(w)^*&=D^*\left(R(z)S(z)^{-1} - S(w)^{*-1}R(w)^*\right)D\\
&= D^*S(w)^{*{-1}}\left( S(w)^*R(z) - R(w)^*S(z)\right) S(z)^{-1}D.
\end{align*}
Here
\begin{align*}
S(w)^*R(z) - R(w)^*S(z) &= \begin{bmatrix} 1&0\\0&0\end{bmatrix} z_P +\begin{bmatrix}0&0\\0&A\end{bmatrix}- w_P^*\begin{bmatrix} 0&0\\ 0&1\end{bmatrix} -\\
&\hspace{2cm} \left( w_P^*\begin{bmatrix}1&0\\0&0\end{bmatrix} + \begin{bmatrix} 0&0\\0&A\end{bmatrix} - \begin{bmatrix} 0&0\\0&1\end{bmatrix}z_P \right)\\
&=z_P-w_P^*.
\end{align*}
Hence
\[
M(z)-M(w)^*= D^*S(w)^{*{-1}}(z_P-w_P^*) S(z)^{-1}D,
\]
which is equation \eqref{imM}.
\end{proof}
The next result shows that the matricial resolvent belongs not just to the operator Pick class, but to the smaller {\em operator Loewner class}.
\begin{proposition}\label{Mloewner}
With the notation of Definition {\rm \ref{defMatResolv}}, there exists an analytic operator-valued function
$F:\Pi^n \to \mathcal{L}(\h)$ such that
for all $z,\ w\in\Pi^n$,
\beq\label{loewDecomp}
M(z)-M(w)^* = F(w)^*(z-\bar w)_PF(z)
\eeq
on $\h$.
\end{proposition}
\begin{proof}
The identity \eqref{imM} shows that such a relation holds on $\mathcal{N}\oplus \mathcal{D}(A)$; we must extend it to all of $\h$.
Write $P_j$ as an operator matrix with respect to the decomposition $\h=\mathcal{N}\oplus\mathcal{M}$, as in equation \eqref{expP}. Then $z_P$ has the matricial expression \eqref{formzP}. For $z\in\Pi^n$ let
\[
F^\sharp(z) = \left(\bbm 1&0\\0&A\ebm - \bbm 0&0\\0&1\ebm z_P\right)^{-1} \bbm i&0\\0&1+iA\ebm.
\]
Then $F^\sharp(z)$ is an operator from $\mathcal{N}\oplus\mathcal{D}(A)$ to $\h$, and we find that
\begin{align*}
F^\sharp(z) &= \bbm 1& 0\\-z_{B^*} & A-z_Y\ebm^{-1} \bbm i&0\\0&1+iA\ebm \\
&=\bbm i&0 \\ i(A-z_Y)^{-1} z_{B^*} & (A-z_Y)^{-1} (1+iA)\ebm : \mathcal{N}\oplus\mathcal{D}(A) \to \h.
\end{align*}
Let
\beq\label{defFz}
F(z) = \bbm i&0 \\ i(A-z_Y)^{-1} z_{B^*} & i+(A-z_Y)^{-1} (1+iz_Y)\ebm : \mathcal{N}\oplus\mathcal{M} \to \h.
\eeq
Since
\[
(A-z_Y)^{-1} (1+iA)=i+ (A-z_Y)^{-1} (1+iz_Y)
\]
on $\mathcal{N}\oplus\mathcal{D}(A)$ and the right hand side of the last equation is a bounded operator on all of $\h$, it is clear that, for every $z\in\Pi^n$, $F(z)$ is a continuous extension to $\h$ of $F^\sharp(z)$ and is a bounded operator. Furthermore $F$ is analytic on $\Pi^n$.
By Proposition \ref{alternM}, equation \eqref{imM}, the relation \eqref{loewDecomp} holds on the dense subspace $\mathcal{N}\oplus\mathcal{D}(A)$ of $\h$ for every $z,\ w\in\Pi^n$. Since the operators on both sides of equation \eqref{loewDecomp} are continuous on $\h$, the equation holds throughout $\h$.
\end{proof}
\begin{corollary}\label{MatPick}
A matricial resolvent has a non-negative imaginary part at every point of $\Pi^n$.
\end{corollary}
\begin{proof}
In the notation of Proposition \ref{Mloewner}, on choosing $w=z$ in equation \eqref{loewDecomp} and dividing by $2i$ we obtain the relation
\[
\im M(z)= F(z)^*(\im z_P )F(z)
\]
on $\h$. We have
\[
\im z_P =\sum_j (\im z_j)P_j \geq 0,
\]
and so $\im M(z) \geq 0$ on $\h$ for all $z\in\Pi^n$.
\end{proof}
Here is a concrete example of a matricial resolvent.
\begin{example}\label{type4MatR}
\rm
The function
\beq\label{formtype4}
M(z)= \frac{1}{z_1+z_2}\bbm 2z_1z_2 & i(z_1-z_2) \\ -i(z_1-z_2) & -2 \ebm
\eeq
is the matricial resolvent corresponding to
\[
\h=\mathbb{C}^2, \quad \mathcal{N}=\mathcal{M}=\mathbb{C},\quad A=0 \mbox{ on }\mathbb{C},\quad P_1= \half \bbm 1&1\\1&1 \ebm, \quad P_2=1-P_1.
\]
\end{example}
\section{ Structured resolvent identities }\label{resolvident}
To conclude the paper we point out that there are structured analogs of the classical resolvent identity
\[
(A-z)^{-1} -(A-w)^{-1} = (z-w)(A-z)^{-1} (A-w)^{-1}
\]
for any $z,w$ in the resolvent set of an operator $A$.
\begin{proposition} \label{resIdent}
Let $A$ be a densely defined self-adjoint operator on a Hilbert space $\h$ and let $Y$ be a positive decomposition of $\h$. For all $z,w \in \Pi^n$
\beq\label{resId2}
(A-z_Y)^{-1} - (A-w_Y)^{-1}= (A-z_Y)^{-1}(z-w)_Y(A-w_Y)^{-1}.
\eeq
If $M(z)$ is the structured resolvent of type $3$ corresponding to $A$ and $Y$ then
\beq\label{resId3}
M(z)-M(w)\left| \mathcal D(A) \right. =(1-iA)(A-z_Y)^{-1}(z-w)_Y(A-w_Y)^{-1}(1+iA).
\eeq
\end{proposition}
\begin{proof}
The first of these identities is immediate. For the second, by equation \eqref{3rdtype3},
\begin{align*}
M(z)-M(w)\left| \mathcal D(A) \right. =(1-iA)\left((A-z_Y)^{-1}-(A-w_Y)^{-1}\right)(1+iA),
\end{align*}
and the identity \eqref{resId3} follows from \eqref{resId2}.
\end{proof}
\begin{proposition}\label{10.2}
Let $\mathcal H$ be the orthogonal direct sum of Hilbert spaces $\mathcal{ N, M}$, let $A$ be a densely defined self-adjoint operator on $\mathcal M$ with domain $\mathcal{D}(A) $ and let $P$ be an orthogonal decomposition of $\mathcal H$. For every $z, \ w\in\Pi^n$, as operators on $\mathcal{N}\oplus\mathcal{D}(A)$,
\begin{align}\label{resId4}
M(z)-M(w) &= \bbm -i&0\\0&1-iA \ebm\left(\bbm1&0\\0&A\ebm -z_P\bbm 0&1\\0&0\ebm\right)^{-1} (z-w)_P\nn \\
&\hspace{2cm} \times \left(\bbm 1&0\\0&A\ebm - \bbm 0&0\\0&1\ebm w_P\right)^{-1}\bbm i&0\\0&1+iA \ebm.
\end{align}
\end{proposition}
\begin{proof}
Let
\[
D= \bbm i&0\\0&1+iA \ebm : \mathcal{N}\oplus \mathcal{D}(A) \to \h.
\]
By equations \eqref{simpler} and \eqref{simpler2} we have
\begin{align*}
M(z)&-M(w)\left| \mathcal{N}\oplus\mathcal{D}(A)\right.\\
& = D^*\left\{\left(\bbm1&0\\0&A\ebm -z_P\bbm 0&1\\0&0\ebm\right)^{-1}\left(z_P\bbm 0&0\\0&1\ebm+ \bbm 0&0 \\ 0&1\ebm \right)\right. \\
&\hspace*{1cm}\left. -\left(\bbm 1&0\\0&0\ebm w_P + \bbm 0&0\\0&1\ebm\right)\left(\bbm 1&0\\0&A\ebm - \bbm 0&0\\0&1\ebm w_P\right)^{-1} \right\} D \\
& = D^*\left(\bbm1&0\\0&A\ebm -z_P\bbm 0&1\\0&0\ebm\right)^{-1} \left\{\left(z_P\bbm 0&0\\0&1\ebm+ \bbm 0&0 \\ 0&1\ebm \right)\left(\bbm 1&0\\0&A\ebm - \bbm 0&0\\0&1\ebm w_P\right)\right. \\
&\left. \hspace*{0.3cm} - \left(\bbm1&0\\0&A\ebm -z_P\bbm 0&1\\0&0\ebm\right)\left(\bbm 1&0\\0&0\ebm w_P + \bbm 0&0\\0&1\ebm\right) \right\} \left(\bbm 1&0\\0&A\ebm - \bbm 0&0\\0&1\ebm w_P\right)^{-1} D.
\end{align*}
The term in braces in the last expression reduces to $(z-w)_P$, and the identity \eqref{resId4} follows.
\end{proof}
\begin{corollary}
With the assumptions of Proposition {\rm \ref{10.2}}, there exists an analytic function $F:\Pi^n\to \mathcal{L}(\h)$ such that, for all $z,\ w\in\Pi^n$,
\beq\label{MFP}
M(z)- M(w) = F(\bar z)^*(z-w)_P F(w).
\eeq
\end{corollary}
The statement follows from Proposition \ref{10.2} just as Proposition \ref{Mloewner} follows from Proposition \ref{alternM}. If $F$ is defined by equation \eqref{defFz} then $F(z)$ is a bounded operator on $\h$, $F$ is analytic on $\Pi^n$ and Proposition \ref{10.2} states that equation \eqref{MFP} holds on $\mathcal{N}\oplus \mathcal{D}(A)$. It follows by continuity that equation \eqref{MFP} holds on $\h$.
\section{Structured resolvents of operators}\label{structured}
The resolvent operator $(A-z)^{-1}$ of a densely defined self-adjoint operator $A$ on a Hilbert space plays a prominent role in spectral theory. It has the following properties.
\begin{enumerate}
\item It is an analytic bounded operator-valued function of $z$ in the upper halfplane $\Pi$;
\item it satisfies the growth estimate $\|(A-z)^{-1}\| \leq 1/ \im z$ for $z\in\Pi$;
\item $(A-z)^{-1}$ has non-negative imaginary part for all $z\in\Pi$;
\item it satisfies the ``resolvent identity".
\end{enumerate}
Here we are interested in several-variable analogs of the resolvent. These will again be operator-valued analytic functions with non-negative imaginary part, but now on the polyhalfplane $\Pi^n$. Because of the additional complexities in several variables we encounter three different types of resolvent; all of them have the four listed properties, with very slight modifications, and therefore deserve the name {\em structured resolvent}.
For any Hilbert space $\h$, a {\em positive decomposition} of $\h$ will mean an $n$-tuple $Y=(Y_1,\dots,Y_n)$ of positive contractions on $\h$ that sum to the identity operator. For any $z=(z_1,\dots,z_n)\in\mathbb{C}^n$ and any $n$-tuple $T=(T_1,\dots,T_n)$ of bounded operators we denote by $z_T$ the operator $\sum_j z_j T_j $. Here each $T_j$ is a bounded operator from $\h_1$ to $\h_2$, for some Hilbert spaces $\h_1, \ \h_2$, so that $z_T$ is also a bounded operator from $\h_1$ to $\h_2$.
\begin{definition} \label{resolv2}
Let $A$ be a closed densely defined self-adjoint operator on a Hilbert space $\h$ and let $Y$ be a positive decomposition of $\h$. The {\em structured resolvent of $A$ of type 2} corresponding to $Y$ is the operator-valued function
\[
z\mapsto (A-z_Y)^{-1} : \Pi^n \to \L(\h).
\]
\end{definition}
The following observation is essentially \cite[Lemma 6.25]{AMY}.
\begin{proposition}\label{AzY}
For $A$ and $Y$ as in Definition {\rm \ref{resolv2}} the structured resolvent $(A-z_Y)^{-1}$ is well defined on $\Pi^n$ and satisfies, for all $z\in\Pi^n$,
\beq\label{boundAzYinv}
\|(A-z_Y)^{-1}\| \leq \frac{1}{\min_j \im z_j}.
\eeq
Moreover
\begin{align} \label{imPos}
\im \left((A-z_Y)^{-1}\right) &= (A-z_Y^*)^{-1} \left(\im z_Y\right) (A-z_Y)^{-1} \\
&= (A-z_Y)^{-1} \left(\im z_Y\right) (A-z_Y^*)^{-1} \nn\\
&\geq 0. \nn
\end{align}
\end{proposition}
The range of the bounded operator $(A-z_Y)^{-1}$ is of course $\mathcal D(A)$, the domain of $A$.
\begin{proof}
For any vector $\xi$ in the domain of $A$,
\begin{align*}
\|(A-z_Y)\xi\| \ \|\xi\| &\geq |\ip{(A-z_Y)\xi}{\xi}| \\
&\geq |\im \ip{(A-z_Y)\xi}{\xi}| \\
&= \ip{(\im z_Y) \xi}{\xi} \\
&= \sum_j (\im z_j)\ip{Y_j\xi}{\xi} \\
&\geq (\min_j \im z_j) \ip{\sum_j Y_j\xi}{\xi}\\
&= (\min_j \im z_j) \|\xi\|^2.
\end{align*}
Thus $A-z_Y$ has lower bound $\min_j\im z_j > 0$, and so has a bounded left inverse. A similar argument with $z$ replaced by $\bar z$ shows that $(A-z_Y)^*$ also has a bounded left inverse, and so $A-z_Y$ has a bounded inverse and the inequality \eqref{boundAzYinv} holds.
The identities \eqref{imPos} are easy.
\end{proof}
Resolvents of type 2 are the simplest several-variable analogues of the familiar one-variable resolvent but they are not sufficient for the analysis of the several-variable Pick class. To this end we introduce two further generalizations. Let us first recall some basic facts about closed unbounded operators.
\begin{lemma}\label{basicUnbdd}
Let $T$ be a closed densely defined operator on a Hilbert space $\h$, with domain $\mathcal D(T)$. The operator $1+T^*T$ is a bijection from $\mathcal D(T^*T)$ to $\h$, and the operators
\[
B\df (1+T^*T)^{-1}, \qquad C\df T(1+T^*T)^{-1}
\]
are everywhere defined and contractive on $\h$. Moreover $B$ is self-adjoint and positive, and
$ \ran C \subset \mathcal D(T^*)$.
\end{lemma}
\begin{proof}
All these statements are proved in \cite[Sections 118, 119]{RN}, although the final statement about $\ran C$ is not explicitly stated. We must show that for all $v\in\h$ there exists $y\in\h$ such that, for all $h\in\h$,
\[
\ip{Th}{Cv}=\ip{h}{y}.
\]
It is straightforward to check that this relation holds for $y=v-Bv$, and so $\ran C\subset \mathcal D(T^*)$.
\end{proof}
\begin{definition} \label{resolv3}
Let $A$ be a closed densely defined self-adjoint operator on a Hilbert space $\h$ and let $Y$ be a positive decomposition of $\h$. The {\em structured resolvent of $A$ of type 3} corresponding to $Y$ is the operator-valued function
$M:\Pi^n\to \L(\h)$ given by
\beq\label{defM3}
M(z) = (1-iA)(A - z_Y)^{-1} (1+z_Y A)(1-iA)^{-1}.
\eeq
\end{definition}
We denote the $\ell_1$ norm on $\mathbb{C}^n$ by $\|\cdot\|_1$. Note that $\|z_Y\| \leq \|z\|_1$ for all $z\in\mathbb{C}^n$ and all positive decompositions $Y$.
\begin{proposition}\label{existResolv3}
For $A$ and $Y$ as in Definition {\rm \ref{resolv3}} the structured resolvent $M(z)$ of type $3$ given by equation \eqref{defM3} is well defined as a bounded operator on $\h$ for all $z\in\Pi^n$ and satisfies
\beq\label{boundM3}
\|M(z)\| \leq (1+2\|z\|_1) \left(1+ \frac{1+\|z\|_1}{\min_j \im z_j}\right).
\eeq
\end{proposition}
\begin{proof}
Since
\[
1+z_YA = 1-iz_Y + iz_Y(1-iA) : \mathcal D(A) \to \h
\]
and $(1-iA)^{-1}$ is a contraction on all of $\h$, with range $\mathcal D(A)$, the operator $(1+z_YA)(1-iA)^{-1}$ is well defined as an operator on $\h$ and
\begin{align}\label{3&4}
\|(1+z_YA)(1-iA)^{-1}\| &= \| (1-iz_Y)(1-iA)^{-1} + iz_Y\| \nn \\
&\leq \|1-iz_Y\| + \|z_Y\| \nn\\
&\leq 1+2\|z_Y\| \nn\\
&\leq 1+2\|z\|_1.
\end{align}
Similarly $(1-iA)(A-z_Y)^{-1}$ is well defined on $\h$, and since
\[
i(A-z_Y)= -(1-iA)+(1-iz_Y): \mathcal D(A) \to \h
\]
we have
\[
i=-(1-iA)(A-z_Y)^{-1} + (1-iz_Y)(A-z_Y)^{-1}:\h\to\h.
\]
Thus, by virtue of the bound \eqref{boundAzYinv},
\begin{align}\label{1&2}
\|(1-iA)(A-z_Y)^{-1}\| &= \|i - (1-iz_Y)(A-z_Y)^{-1}\| \nn \\
&\leq 1+ \|1-iz_Y\| \ \|(A-z_Y)^{-1}\| \nn \\
&\leq 1+ \frac{1+\|z\|_1}{\min_j \im z_j}.
\end{align}
On combining the estimates \eqref{1&2} and \eqref{3&4} we obtain the bound \eqref{boundM3}.
\end{proof}
The following alternative formula for the structured resolvent of type 3, valid on the dense subspace $\mathcal D(A)$ of $\h$, allows us to show that $\im M(z) \geq 0$.
\begin{proposition}\label{2ndResolv3}
For $A$ and $Y$ as in Definition {\rm \ref{resolv3}} and $z\in\Pi^n$
\begin{align}
M(z)|\mathcal D(A) &= (1-iA)\left\{(A-z_Y)^{-1}-A(1+A^2)^{-1}\right\}(1+iA) \label{Resolv3.2}\\
&= (1-iA)(A-z_Y)^{-1}(1+iA) - A: \mathcal D(A) \to \h. \label{3rdtype3}
\end{align}
Moreover, for every $v\in\mathcal D(A)$,
\beq\label{ImMpos}
\im \ip{M(z)v}{v} = \ip {(1-iA)(A-z_Y^*)^{-1} (\im z_Y) (A-z_Y)^{-1}(1+iA)v}{v} \geq 0.
\eeq
\end{proposition}
\begin{proof}
By Lemma \ref{basicUnbdd} the operator $A(1+A^2)^{-1}$ is contractive on $\h$ and has range contained in $\mathcal D(A)$. On $\mathcal D(A^2)$ we have the identity
\[
1+z_YA= 1+A^2- (A-z_Y)A.
\]
Since $(1+A^2)^{-1}$ maps $\h$ into $\mathcal D(A^2)$ we have
\[
(1+z_Y A)(1+A^2)^{-1} = 1-(A-z_Y)A(1+A^2)^{-1} : \h \to \h,
\]
and therefore
\beq\label{3f}
(A-z_Y)^{-1}(1+z_Y A)(1+A^2)^{-1} = (A-z_Y)^{-1}-A(1+A^2)^{-1} : \h \to \mathcal D(A).
\eeq
Clearly
\[
(1+A^2)^{-1} (1+iA)=(1-iA)^{-1} \quad \mbox{ on }\mathcal D(A)
\]
and so, on multiplying equation \eqref{3f} fore and aft by $1\pm iA$, we deduce that, as operators from $\mathcal D(A)$ to $\h$,
\begin{align*}
M(z) | \mathcal D(A)&= (1-iA)(A-z_Y)^{-1} (1+z_YA)(1-iA)^{-1} \nn\\
&= (1-iA)(A-z_Y)^{-1} (1+z_YA)(1+A^2)^{-1}(1+iA) \nn\\
&=(1-iA)\left\{ (A-z_Y)^{-1}-A(1+A^2)^{-1} \right\}(1+iA).
\end{align*}
This establishes equation \eqref{Resolv3.2}.
The expression \eqref{3rdtype3} follows from equation \eqref{Resolv3.2} since
\[
(1-iA)A(1+A^2)^{-1} (1+iA) =A \quad \mbox{ on } \mathcal D(A).
\]
By equation \eqref{3rdtype3} we have, for any $z\in\Pi^n$ and $v\in \mathcal D(A)$,
\begin{align*}
\im \ip{M(z)v}{v} &= \im\ip{(1-iA)(A-z_Y)^{-1}(1+iA)v}{v} -\im \ip{Av}{v}\\
&= \im\ip{(A-z_Y)^{-1}(1+iA)v}{(1+iA)v}
\end{align*}
and hence, by equation \eqref{imPos},
\[
\im \ip{M(z)v}{v} = \ip{(A-z_Y^*)^{-1} (\im z_Y) (A-z_Y)^{-1}(1+iA)v}{(1+iA)v},
\]
and so equation \eqref{ImMpos} holds.
\end{proof}
\begin{corollary} \label{Mpick}
For $A$ and $Y$ as in Definition {\rm \ref{resolv3}} the structured resolvent $M(z)$ given by equation \eqref{defM3} satisfies $\im M(z) \geq 0$ for all $z\in\Pi^n$.
\end{corollary}
For, by Propositions \ref{existResolv3} and \ref{2ndResolv3}, $M(z)$ is a bounded operator on $\h$ and $\im \ip{M(z)v}{v} \geq 0$ for $v\in\mathcal D(A)$. The conclusion follows by density of $\mathcal D(A)$ and continuity.
In the case of bounded $A$ there is yet another expression for the structured resolvent of type $3$.
\begin{proposition}
If $A $ is a {\em bounded} self-adjoint operator on $\h$ and $Y$ is a positive decomposition of $\h$ then, for $z\in\Pi^n$,
\beq\label{Resolv3.3}
M(z)=(1+iA)^{-1}(1+Az_Y)(A-z_Y)^{-1}(1+iA)
\eeq
\end{proposition}
\begin{proof}
Since $A$ is bounded it is defined on all of $\h$. We have
\[
1+Az_Y = 1+ A^2 - A(A-z_Y)
\]
and hence
\[
(1+Az_Y)(A-z_Y)^{-1} = (1+ A^2)(A-z_Y)^{-1} -A.
\]
Thus
\begin{align*}
(1+iA)^{-1}(1+Az_Y)(A-z_Y)^{-1}(1+iA) &= (1-iA)(A-z_Y)^{-1}(1+iA) -A \\
&=M(z)
\end{align*}
by equation \eqref{3rdtype3}.
\end{proof}
\begin{remark} \rm
In the case of unbounded $A$ the expression \eqref{Resolv3.3} for $M(z)$ is valid wherever it is defined, but it is
not to be expected that this will be a dense subspace of $\h$ in general.
\end{remark}
Here are two examples of structured resolvents of type 3, one on $\mathbb{C}^2$ and one on an infinite-dimensional space.
\begin{example} \label{type3ex1} \rm
Let
\[
\h=\mathbb{C}^2,\quad A=\bbm 1&0\\0&-1 \ebm, \quad Y_1=\half \bbm 1&1\\1&1 \ebm, \quad Y_2 =1-Y_1, \quad Y=(Y_1,Y_2).
\]
Then
\begin{align*}
M(z) &= (1-iA)(A-z_Y)^{-1}(1+z_Y A)(1-iA)^{-1} \\
&= \frac{1}{1-z_1z_2}\bbm (1+z_1)(1+z_2) & -i(z_1-z_2) \\ i(z_1-z_2) & -(1-z_1)(1-z_2) \ebm.
\end{align*}
\end{example}
\begin{example}\label{type3ex2} \rm
Let $\h=L^2(\mathbb{R})$, let $A$ be the operation of multiplication by the independent variable $t$ and let $Y=(P,Q)$ where $P, Q$ are the orthogonal projection operators onto the subspaces of even and odd functions respectively in $L^2$. Thus
\[
Pf(t)=\half\left\{f(t)+f(-t)\right\}, \qquad Qf(t)=\half\left\{f(t)-f(-t)\right\}.
\]
Let $Y'=(Q,P)$. Note that
\[
PA=AQ, \qquad QA=AP
\]
and hence
\[
z_YA=Az_{Y'}, \qquad z_{Y'}A=Az_Y, \qquad z_Yz_{Y'}=z_1z_2=z_{Y'}z_Y.
\]
It follows that $z_Y$ and $z_{Y'}$ commute with $A^2$, and it may be checked that
\[
(A-z_Y)^{-1}=(A^2-z_1z_2)^{-1}(z_{Y'}+A)=(z_{Y'}+A)(A^2-z_1z_2)^{-1}
\]
and hence
\[
(A-z_Y)^{-1}(1+z_YA)= (A^2-z_1z_2)^{-1}\left( (1+A^2)z_{Y'} + (1+z_1z_2)A\right).
\]
A straightforward calculation now shows that the structured resolvent $M(z)$ of $A$ corresponding to $Y$ is given by
\[
(M(z)f)(t)= \frac{ \left( \half (z_1+z_2)(1+t^2)+(1+z_1z_2)t\right)f(t) + \half(z_2-z_1)(1-it)^2f(-t)}{t^2-z_1z_2}
\]
for all $z\in\Pi^2,\ f\in L^2(\mathbb{R})$ and $t\in\mathbb{R}$.
In particular, we note for future use that if $f$ is an even function,
\beq\label{fcnoftype3}
(M(z)f)(t)= \frac{t(1+z_1z_2)+(1-it)(itz_1+z_2)}{t^2-z_1z_2}f(t).
\eeq
\end{example}
\section{Nevanlinna representations of types 3, 2 and 1}\label{type321}
Nevanlinna representations of type $4$ have the virtue of being general for functions in $\mathcal{L}_n$, but they are undeniably cumbersome.
In this section we shall show that there are three simpler representation formulae, corresponding to increasingly stringent growth conditions on $h \in \mathcal{L}_n$.
In Nevanlinna's one-variable representation formula of Theorem \ref{thm1.1},
\beq\label{OneVar}
h(z) = a+bz+\int \frac{1+tz}{t-z} \ \dd\mu(t),
\eeq
it may be the case for a particular $h\in\Pick$ that the $bz$ term is absent. The analogous situation in two variables is that the space $\mathcal{N}$ in a type 4 representation may be zero. Equivalently, in the corresponding Herglotz representation, the unitary operator $L$ does not have $1$ as an eigenvalue. This suggests the following notion.
\begin{definition}\label{def3.1}
A {\em Nevanlinna representation of type} $3$ of a function $h$ on $\Pi^n$ consists of a Hilbert space $\mathcal H$, a self-adjoint densely defined operator $A$ on $\h$, a positive decomposition $Y$ of $\h$, a real number $a$ and a vector $v \in \mathcal H$ such that, for all $z\in\Pi^n$,
\beq\label{type3rep}
h(z) = a+ \ip{(1-iA)(A - z_Y)^{-1} (1+z_YA)(1-iA)^{-1} v}{v}.
\eeq
\end{definition}
Thus $h$ has a type 3 representation if $h(z) = a+ \ip{ M(z)v}{v}$ where $ M(z)$ is the structured resolvent of $A$ of type $3$ corresponding to $Y$, as given by equation \eqref{defM3}.
In \cite{ATY} the authors derived a somewhat simpler representation which can also be regarded as an analog of the case $b=0$ of Nevanlinna's one-variable formula \eqref{OneVar}.
\begin{definition} \label{defType2ref}
A {\em Nevanlinna representation of type} $2$ of a function $h$ on $\Pi^n$ consists of a Hilbert space $\mathcal H$, a self-adjoint densely defined operator $A$ on $\h$, a positive decomposition $Y$ of $\h$, a real number $a$ and a vector $\alpha \in \mathcal H$ such that, for all $z\in\Pi^n$
\beq \label{type2rep}
h(z)=a+\ip{(A-z_Y)^{-1} \al}{\al}.
\eeq
\end{definition}
This means of course that, for all $z\in\Pi^n$,
\[
h(z) = a + \ip{M(z)\alpha}{\alpha}
\]
where $M(z)$ is the structured resolvent of $A$ of type $2$ corresponding to $Y$ (compare equation \eqref{resolv2}).
We wish to understand the relationship between type 3 and type 2 representations.
\begin{proposition}\label{2gives3}
If $h \in \mathcal P_n$ has a type $2$ representation then $h$ has a type $3$ representation. Conversely,
if $h \in \mathcal P_n$ has a type $3$ representation as in equation \eqref{type3rep} with the additional property that $v \in \mathcal D(A)$ then $h$ has a type $2$ representation.
\end{proposition}
\begin{proof}
Suppose that $h \in \Pick_n$ has the type 2 representation
\[
h(z) = a_0 + \ip{(A - z_Y)^{-1}\alpha}{\alpha}
\]
for some $a_0\in\mathbb{R}$, positive decomposition $Y$ and $\al\in \h$.
We must show that $h$ has a representation of the form \eqref{type3rep}
for some $a\in\mathbb{R}$ and $v\in \h$. By Proposition \ref{2ndResolv3}, it suffices to find $a\in\mathbb{R}$ and $v\in\mathcal D(A)$ such that
\[
h(z) = a+ \ip{(1-iA)\left\{(A - z_Y)^{-1}-A(1+A^2)^{-1}\right\}(1+iA)v}{v}
\]
for all $z\in\Pi^n$.
To this end, let $C=A(1+A^2)^{-1}$ and let
\beq\label{aanda_0}
a = a_0 + \ip{C\alpha}{\alpha}.
\eeq
Since $1+iA$ is invertible on $\h$ and $\ran (1+iA)^{-1}\subset \mathcal D(A)$ we may define
\beq\label{vandalpha}
v = (1+iA)^{-1} \alpha \in \mathcal D(A).
\eeq
Then
\begin{align*}
h(z) &= a_0 + \ip{(A - z_Y)^{-1}\alpha}{\alpha} \notag \\
&= a-\ip{C\al}{\al} + \ip{(A - z_Y)^{-1}\alpha}{\alpha} \notag \\
&= a+ \ip{\left\{(A - z_Y)^{-1}-C\right\}(1+iA)v}{(1+iA)v} \notag \\
&= a+ \ip{(1-iA)\left\{(A - z_Y)^{-1}-C\right\}(1+iA)v}{v}
\end{align*}
as required. Thus $h$ has a type $3$ representation.
Conversely, let $h$ have a type 3 representation \eqref{type3rep} such that $v \in \mathcal D(A)$, that is
\[
h(z)=a+\ip{M(z)v}{v}
\]
where $a\in\mathbb{R}$ and $M$ is the structured resolvent of $A$ of type 3 corresponding to $Y$, as in equation \eqref{defM3}.
Since $v \in \mathcal D(A)$ we may define the vector $\alpha \df (1+iA)v \in \h$, and furthermore, by Proposition \ref{2ndResolv3},
\begin{align*}
h(z) &= a + \ip{(1-iA)\left\{(A - z_Y)^{-1} - C\right\}(1+iA)v}{v} \\
&= a + \ip{\left\{(A - z_Y)^{-1} - C\right\}\alpha}{\alpha} \\
&= a - \ip{C\alpha}{\alpha} + \ip{(A - z_Y)^{-1}\alpha}{\alpha} \\
&= a_0 + \ip{(A - z_Y)^{-1}\alpha}{\alpha} ,
\end{align*}
where $a_0\in\mathbb{R}$ is given by equation \eqref{aanda_0}. Thus $h$ has a representation of type 2.
\end{proof}
A special case of a type 2 representation occurs when the constant term $a$ in equation \eqref{type2rep} is $0$. In one variable, this corresponds to Nevanlinna's characterization of the Cauchy transforms of positive finite measures on $\mathbb{R}$.
Accordingly we define a {\em type $1$ representation} of $h\in \mathcal{L}_n$ to be the special case of a type 2 representation of $h$ in which $a=0$ in \eqref{type2rep}.
\begin{definition}\label{def4.1}
An analytic function $h$ on $\Pi^n$ has a {\em Nevanlinna representation of type $1$} if there exist a Hilbert space $\h$, a densely defined self-adjoint operator $A$ on $\h$, a positive decomposition $Y$ of $\h$ and a vector $\alpha\in\h$ such that, for all $z\in\Pi^n$,
\beq\label{type1formula}
h(z) = \ip{(A - z_Y)^{-1}\alpha}{\alpha}.
\eeq
\end{definition}
A representation of type 1 is obviously a representation of type 2. The following proposition is an immediate corollary of Proposition \ref{2gives3}.
\begin{proposition}\label{1equiv3}
A function $h \in \mathcal{L}_n$ has a type $1$ representation if and only if $h$ has a type $3$ representation as in equation \eqref{type3rep} with the additional properties that $v \in \mathcal D(A)$ and
\[
a - \ip{A(1+A^2)^{-1}\alpha}{\alpha} =0.
\]
\end{proposition}
For consistency with our earlier terminology for structured resolvents and representations we should have to define a structured resolvent of type 1 to be the same as a structured resolvent of type 2. We refrain from making such a confusing definition.
We conclude this section by giving examples of the four types of Nevanlinna representation in two variables.
\begin{example}\label{4types}
\rm
(1) The formula
\[
h(z)= -\frac{1}{z_1+z_2} = \ip{(0-z_Y)^{-1}v}{v}_{\mathbb{C}},
\]
where $Y=(\half,\half)$ and $v=1/\sqrt{2}$, exhibits a representation of type $1$, with $A=0$.\\
\nin (2) Likewise
\[
h(z)= 1-\frac{1}{z_1+z_2} =1+ \ip{(0-z_Y)^{-1}v}{v}_{\mathbb{C}}
\]
is a representation of type $2$.\\
\nin (3) Let
\beq\label{trueh}
h(z) =\threepartdef{ \ds \frac{1}{1+z_1z_2}\left(z_1-z_2 + \frac{iz_2(1+z_1^2)}{\sqrt{z_1z_2}}\right)}{z_1z_2 \neq -1}{}{}{\half (z_1+z_2)}{z_1z_2 = -1}
\eeq
where we take the branch of the square root that is analytic in $\mathbb{C}\setminus [0,\infty)$ with range $\Pi$.
We claim that $h\in\Pick_2$ and that $h$ has the type $3$ representation
\beq\label{temp}
h(z)= \ip{M(z)v}{v}_{L^2(\mathbb{R})},
\eeq
where $M(z)$ is the structured resolvent of type $3$ given in Example \ref{type3ex2} and $v(t)=1/\sqrt{\pi(1+t^2)}$. To see this, let $h$ be temporarily defined by equation \eqref{temp}. Since $v$ is an even function in $L^2(\mathbb{R})$, equation \eqref{fcnoftype3} tells us that
\[
h(z) = \int_{-\infty}^\infty \frac {t(1+z_1z_2)+(1-it)(itz_1+z_2)}{\pi(t^2-z_1z_2)(1+t^2)} \ \dd t.
\]
Since the denominator is an even function of $t$, the integrals of all the odd powers of $t$ in the numerator vanish, and we have, provided $z_1z_2\neq -1$,
\begin{align*}
h(z) &= \frac{2}{\pi}\int_{0}^\infty \frac {z_2 +t^2z_1}{(t^2-z_1z_2)(1+t^2)} \ \dd t\\
&= \frac{2}{\pi}\int_{0}^\infty \frac {z_2(1+z_1^2)}{1+z_1z_2}\, \, \frac{1}{t^2 -z_1z_2} + \frac{z_1-z_2}{1+z_1z_2} \, \,\frac{1}{1+t^2} \ \dd t.
\end{align*}
Now, for $w\in\Pi$,
\[
\int_0^\infty \frac{\dd t}{t^2-w^2} = \frac{i\pi}{2w},
\]
and so we find that $h$ is indeed given by equation \eqref{trueh} in the case that $z_1z_2 \neq -1$. When $z_1z_2=-1$ we have
\begin{align*}
h(z) &= \frac{2}{\pi}\int_0^\infty \frac{z_2+z_1t^2}{(1+t^2)^2} \dd t \\
&= \frac{2}{\pi}\int_0^\infty \frac{z_1}{1+t^2} + \frac{z_2-z_1}{(1+t^2)^2} \dd t\\
&= \half(z_1+z_2).
\end{align*}
Thus equation \eqref{temp} is a type $3$ representation of the function $h$ given by equation \eqref{trueh}. This function is {\em constant} and equal to $i$ on the diagonal $z_1=z_2$.\\
\nin (4) The function
\[
h(z)= \frac{z_1 z_2}{z_1+z_2} = -\left(-\frac{1}{z_1}-\frac{1}{z_2}\right)^{-1}
\]
clearly belongs to $\Pick_2$. It has the representation of type $4$
\[
h(z)= \ip{M(z)v}{v}_{\mathbb{C}^2}
\]
where $M(z)$ is the matricial resolvent given in Example \ref{type4MatR} and
\[
v=\frac{1}{\sqrt{2}}\begin{pmatrix} 1 \\ 0 \end{pmatrix}.
\]
\end{example}
We claim that each of the above representations is of the simplest available type for the function in question; for example, the function $h$ in part (4) does not have a Nevanlinna representation of type 3. To prove this claim (which we shall do in Example \ref{fourtypes} below) we need characterizations of the types of functions -- the subject of the next two sections.
\section{Nevanlinna representations of type $4$} \label{type4}
In this section we derive a multivariable analog of the most general form of Nevanlinna representation for functions in the one-variable Pick class (Theorem \ref{thm1.1}). We start with a multivariable Herglotz theorem \cite[Theorem 1.8]{ag90}. We shall say that an analytic operator-valued function $F$ on $\mathbb{D}^n$ is a {\em Herglotz function} if $\re F(\la) \geq 0$ for all $\la\in\mathbb{D}^n$. For present purposes we need the following modification of the notion.
\begin{definition}\label{strongHerg}
An analytic function $F: \mathbb{D}^n \to \mathcal {L(K)}$, where $\mathcal K$ is a Hilbert space, is a {\em strong Herglotz function} if, for every commuting $n$-tuple $T=(T_1,\dots,T_n)$ of operators on a Hilbert space and for $0\leq r < 1, \, \re F(rT) \geq 0$.
\end{definition}
In \cite{ag90} these functions were called $\mathcal{F}_n$-Herglotz functions. The class of strong Herglotz functions has also been called the {\em Herglotz-Agler class} (for example \cite{kaluzh,BKV2}).
It is clear that every strong Herglotz function is a Herglotz function, and in the cases $n=1$ and $2$ the converse is also true \cite{ag90}.
\begin{theorem}\label{thm2.1}
Let $\mathcal K$ be a Hilbert space and let $F: \mathbb{D}^2 \to \mathcal {L(K)}$ be a strong Herglotz function such that $F(0)=1$. There exist a Hilbert space $\mathcal H$, an orthogonal decomposition $P$ of $\mathcal H$, an isometric linear operator $V: \mathcal K \to \mathcal H$ and a unitary operator $U$ on $\mathcal H$ such that, for all $\la\in\mathbb{D}^n$,
\beq\label{aglerRep}
F(\lambda) = V^\ast \frac{1+U\lambda_P}{1 - U\lambda_P} V.
\eeq
Conversely, every function $F: \mathbb{D}^n \to \mathcal{ L( K)}$ expressible in the form \eqref{aglerRep} for some $\mathcal{H},\ P, \ V$ and $U$ with the stated properties is a strong Herglotz function and satisfies $F(0)=1$.
\end{theorem}
Note that $\la_P=\sum_j \la_j P_j$ has operator norm at most $\|\la\|_\infty < 1$ for $\la\in\mathbb{D}^n$, and hence equation \eqref{aglerRep} does define $F$ as an analytic operator-valued function on $\mathbb{D}^n$.
On specialising to scalar-valued functions in the $n$-variable Herglotz class we obtain the following consequence.
\begin{corollary}\label{thm2.2}
Let $f$ be a scalar-valued strong Herglotz function on $\mathbb{D}^n$. There exists a Hilbert space $\mathcal H$, a unitary operator $L$ on $\mathcal H$, an orthogonal decomposition $P$ of $\mathcal H$, a real number $a$ and a vector $v \in \mathcal H$ such that, for all $\la\in\mathbb{D}^n$,
\beq\label{scalarHerg}
f(\lambda) = -ia + \ip{(L-\lambda_P)^{-1}(L+\lambda_P)v}{v}.
\eeq
Conversely, for any $ \mathcal{H}, L, P, a$ and $v$ with the properties described, equation \eqref{scalarHerg} defines $f$ as an $n$-variable strong Herglotz function.
\end{corollary}
Again, the right hand side of equation \eqref{scalarHerg} is an analytic function of $\la\in\mathbb{D}^n$ since
\[
(L-\lambda_P)^{-1}= L^{-1}(1-\la_P L^{-1})^{-1}
\]
is a bounded operator and is analytic in $\la$.
\begin{definition}\label{def2.1}
A {\em Nevanlinna representation of type $4$} of a function $h:\Pi^n\to\mathbb{C}$ consists of an orthogonally decomposed Hilbert space $\mathcal {H=N\oplus M}$, a self-adjoint densely defined operator $A$ on $\mathcal {M}$, an orthogonal decomposition $P$ of $\h$, a real number $a$ and a vector $v \in \mathcal H$ such that
\beq\label{formh}
h(z)=a+\ip{ M(z)v}{v}
\eeq
for all $z\in\Pi^n$, where $M(z)$ is the structured resolvent of $A$ of type $4$ corresponding to $P$ (given by the formula \eqref{defM}).
\end{definition}
We wish to convert Corollary \ref{thm2.2} to a representation theorem for suitable analytic functions on $\Pi^n$. The fact that the corollary only applies to {\em strong} Herglotz functions results in representation theorems for a subclass of the Pick class $\Pick_n$. Recall from the introduction:
\begin{definition}\label{loewner}
The {\em Loewner class} $\mathcal{L}_n$ is the set of analytic functions $h$ on $\Pi^n$ with the property that there exist $n$ positive semi-definite functions $A_1,\dots,A_n$ on $\Pi^n$, analytic in the first argument, such that
\[
h(z) - \overline{h(w)} = \sum_{j=1}^n (z_j-\overline{w_j}) A_j(z,w)
\]
for all $z,w\in\Pi^n$.
\end{definition}
A function $h$ on $\Pi^n$ belongs to $\mathcal{L}_n$ if and only if it corresponds under conjugation by the Cayley transform to a function in the Schur-Agler class of the polydisc \cite[Lemma 2.13]{AMY}. Another characterization: $h\in\mathcal{L}_n$ if and only if, for every commuting $n$-tuple $T$ of bounded operators with strictly positive imaginary parts, $h(T)$ has positive imaginary part.
We can now prove Theorem \ref{thm2.4} from the introduction: {\em a function $h$ defined on $\Pi^n$ has a Nevanlinna representation of type $4$ if and only if $h \in \mathcal{L}_n$.}
\begin{proof}
Let $h\in\mathcal{L}_n$. Define an $n$-variable Herglotz function $f:\mathbb{D}^n \to\mathbb{C}$ by
\beq\label{deff}
f(\la) =-i h(z)
\eeq
where
\beq
z_j= i\frac{1+\la_j}{1-\la_j} \qquad \mbox{ for } j=1,\dots,n.
\eeq
When $\la\in \mathbb{D}^n$ the point $z$ belongs to $\Pi^n$, and so $f(\la) $ is well defined, and since $\im h(z) \geq 0$ we have $\re f(\la) \geq 0$, so that $f$ is indeed a Herglotz function. In fact $f$ is even a strong Herglotz function: since $h\in\mathcal{L}_n$, the function $\varphi\in\mathcal{S}_n$ corresponding to $h$ lies in the Schur-Agler class of the polydisc, and so $f=(1+\varphi)/(1-\varphi)$ is a strong Herglotz function.
By Corollary \ref{thm2.2} there exist a real number $a$, a Hilbert space $\mathcal H$, a vector $v \in \mathcal H$, a unitary operator $L$ on $\h$ and an orthogonal decomposition $P$ on $\h$ such that, for all $z\in\Pi^n$,
\begin{align}\label{hif}
h(z) &= if(\la)= a+ \ip{i(L-\la)^{-1}(L+\la)v}{v} \nn \\
& = a + \ip{i[L - (z-i)(z+i)^{-1}]^{-1}[L + (z-i)(z+i)^{-1}]v}{v}.
\end{align}
Here and in the rest of this section $z,\ \la$ are identified with the operators $z_P,\ \la_P$ on $\h$, and in consequence the relation
\[
\la= \frac{z-i}{z+i}
\]
is meaningful and valid.
For $z\in\Pi^n$ let
\beq\label{defML}
M(z)= i\left(L - \la\right)^{-1}\left(L + \la\right)= i\left(L - \frac{z-i}{z+i}\right)^{-1}\left(L + \frac{z-i}{z+i}\right).
\eeq
\begin{comment}
\red I reversed the order of the two factors on the right hand side here. This way it comes out that the operator $M(z)$ is defined and bounded on all of $\h$, not just on $\mathcal{N}\oplus \mathcal D(A)$. Note that
\begin{align*}
M_{\mbox{Jim}}(z) &= i(L+\la)(L-\la)^{-1} \\
&= iL(1+L^*\la)(1-L^*\la)^{-1} L^*\\
&=iL (1-L^*\la)^{-1} (1+L^*\la)L^*\\
&=iL (1-L^*\la)^{-1} L^*L(1+L^*\la)L^*\\
&=iL(L-\la)^{-1}(L+\la)L^*\\
&=L M_{\mbox{Nicholas}}(z) L^*.
\end{align*}
\end{comment}
Since $L$ is unitary on $\h$ and $\la\in\mathbb{D}^n$, the operator $M(z)$ is bounded on $\h$ for every $z\in\Pi^n$
and, by equation \eqref{hif}, we have
\beq\label{repofh}
h(z)=a+\ip{ M(z)v}{v}
\eeq
for all $z\in\Pi^2$. Theorem \ref{thm2.4} will follow provided we can show that $ M(z)$ is given by equation \eqref{defM} for a suitable self-adjoint operator $A$.
Observe that
\begin{align}\label{exp2.1}
M(z) &= i((z+i)L - (z-i))^{-1} ((z+i)L + (z-i)) \notag \\
&= i\left(z(L-1) + i(L+1)\right)^{-1}\left(z(L+1) +i (L-1)\right).
\end{align}
We wish to take out a factor $1-L$ from both factors in equation \eqref{exp2.1}, but this may be impossible since $1-L$ can have a nonzero kernel. Accordingly we decompose $\h$ into $\mathcal{N}\oplus\mathcal{M}$ where $\mathcal{N} = \ker (1-L), \ \mathcal{M}= \mathcal{N}^\perp$. With respect to this decomposition we can write
$L$ as an operator matrix
\[
L = \begin{bmatrix}
1 & 0 \\ 0 & L_0
\end{bmatrix},
\]
where $L_0$ is unitary and $\ker (1 - L_0) = \{0\}$. Substituting into equation \eqref{exp2.1} we have
\begin{align} \label{exp2.1.1}
M(z) &= i\left(z\begin{bmatrix}
0 & 0 \\ 0 & L_0-1
\end{bmatrix} +
i\begin{bmatrix}
2 & 0 \\ 0 & L_0 +1
\end{bmatrix}\right)^{-1} \left(
z\begin{bmatrix}
2 & 0 \\ 0 & L_0 +1
\end{bmatrix} +
i\begin{bmatrix}
0 & 0 \\ 0 & L_0 -1
\end{bmatrix} z\right) \nn \\
&=\left(-z \bbm 0&0 \\0& 1-L_0 \ebm +\bbm 2i & 0 \\ 0 & i(1+L_0)\ebm \right)^{-1}
\left( z\bbm 2i &0 \\ 0&i(1+L_0) \ebm + \bbm 0&0\\0 & 1-L_0\ebm \right)
\end{align}
Formally we may now write
\begin{align} \label{newtilde}
M(z) &= \begin{bmatrix}
-\half i & 0 \\ 0 & (1 - L_0)^{-1}
\end{bmatrix}
\left( -z \begin{bmatrix}
0 & 0 \\ 0 &1
\end{bmatrix} +
\begin{bmatrix}
1 & 0 \\ 0 & i\frac{1 + L_0}{1 - L_0}
\end{bmatrix}\right)^{-1} \times \nn \\
& \hspace{2cm} \left(
z\begin{bmatrix}
1 & 0 \\ 0 & i\frac{1 + L_0}{1 - L_0}
\end{bmatrix} +
\begin{bmatrix}
0 & 0 \\ 0 & 1
\end{bmatrix} \right)
\begin{bmatrix}
2 i & 0 \\ 0 & 1 - L_0
\end{bmatrix},
\end{align}
but whereas equation \eqref{exp2.1.1} is a relation between bounded operators defined on all of $\h$, equation \eqref{newtilde} involves unbounded, partially defined operators and we must verify that the product of operators on the right hand side is meaningful.
Let
\[
A = i\frac{1 + L_0}{1 - L_0}.
\]
Since $L_0$ is unitary on $\mathcal{M}$ and $\ker (1-L_0) =\{0\}$, the operator $A$ is self-adjoint and densely defined on $\mathcal{M}$ \cite[Section 121]{RN}. The domain $\mathcal{D}(A)$ of $A$ is the dense subspace $\ran (1-L_0)$ of $\mathcal{M}$. It follows from the definition of $A$ that
\beq\label{useful}
(1-L_0)^{-1} = \half (1-iA),
\eeq
which is an equation between bijective operators from $\mathcal{D}(A)$ to $\mathcal{M}$. Likewise
\beq\label{useful2}
1+L_0 = -2iA(1-iA)^{-1} :\mathcal{M}\to\mathcal{D}(A)
\eeq
are bounded operators.
Let us continue the calculation from the first factor on the right hand side of equation \eqref{exp2.1.1}. Since $\ker (1-L_0) = \{0\}$, the right hand side of the relation
\begin{align*}
-z \bbm 0&0\\0& 1-L_0 \ebm+ \bbm 2i&0\\0& i(1+L_0)\ebm &= \left( -z \bbm 0&0\\0&1 \ebm + \bbm 1 &0\\ 0&A \ebm \right) \bbm 2i&0\\0& 1-L_0 \ebm
\end{align*}
comprises a bijective map from $\h$ to $\mathcal{N} \oplus \mathcal D(A)$ followed by a bijection from $\mathcal{N} \oplus \mathcal D(A)$ to $\h$ (recall the equation \eqref{needThis}).
We may therefore take inverses in the equation to obtain
\begin{align}\label{2nd}
\left( -z \bbm 0&0\\0& 1-L_0 \ebm+ \bbm 2i&0\\0& i(1+L_0)\ebm \right)^{-1}
&= \bbm -\half i&0\\0&(1-L_0)^{-1} \ebm \left( \bbm1&0\\0&A \ebm - z\bbm 0&0\\0&1 \ebm \right)^{-1} \nn \\
&= \bbm -\half i&0\\0& \half (1-iA) \ebm \left( \bbm1&0\\0&A \ebm - z\bbm 0&0\\0&1 \ebm \right)^{-1}
\end{align}
as operators on $\mathcal{N}\oplus \mathcal{D}(A)$.
Similar reasoning applies to the equation
\begin{align}\label{1st}
z\bbm 2i & 0\\0& i(1+L_0) \ebm + \bbm 0&0\\0& 1-L_0 \ebm &= \left(z\bbm 1&0\\0& A \ebm + \bbm0&0\\0&1 \ebm \right) \bbm 2i&0\\ 0&1-L_0 \ebm \nn\\
&=\left(z\bbm 1&0\\0& A \ebm + \bbm0&0\\0&1 \ebm \right) \bbm -\half i &0\\0&\half (1-iA) \ebm^{-1};
\end{align}
it is valid as an equation between operators on $\h$. The right hand side comprises an operator from $\h$ to $\mathcal{N}\oplus \mathcal{D}(A)$ followed by an operator from $\mathcal{N}\oplus \mathcal{D}(A)$ to $\h$,
and so both sides of the equation denote an operator on $\h$.
On combining equations \eqref{exp2.1.1}, \eqref{2nd} and \eqref{1st} we obtain
\begin{align*}
M(z) &= \bbm -\half i&0\\0& \half (1-iA) \ebm \left( \bbm1&0\\0&A \ebm - z\bbm 0&0\\0&1 \ebm \right)^{-1} \left(z\bbm 1&0\\0& A \ebm + \bbm0&0\\0&1 \ebm \right) \bbm -\half i &0\\0&\half (1-iA) \ebm^{-1}.
\end{align*}
Premultiply this equation by $2$ and postmultiply by $\half$ to deduce that $M(z)$ is indeed the structured resolvent of $A$ of type $4$ corresponding to $P$, as defined in equation \eqref{defM}. Thus the formula \eqref{repofh} is a Nevanlinna representation of $h$ of type $4$.
Conversely, let $h\in\mathcal{L}_n$ have a type 4 representation \eqref{formh}.
\begin{comment}
Then $h$ is analytic on $\Pi^n$ and by Corollary \ref{MatPick}, for all $z\in\Pi^n$,
\[
\im h(z) =\ip{\im M(z) v}{v} \geq 0
\]
since $\im M(z) \geq 0$ on $\h$. Thus $h\in\Pick_n$.
\end{comment}
By Proposition \ref{Mloewner} there exists an analytic operator-valued function
$F:\Pi^n \to \mathcal{L}(\h)$ such that,
for all $z,\ w\in\Pi^n$,
\beq
M(z)-M(w)^* = F(w)^*(z-\bar w)_PF(z)
\eeq
on $\h$. Hence
\begin{align*}
h(z)-\overline{h(w)} &= \ip{ (M(z)-M(w)^*)v}{v} \\
&=\ip{F(w)^*(z-\bar w)_P F(z)v}{v} \\
&=\sum_{j=1}^n (z_j- \bar w_j)A_j(z,w)
\end{align*}
for all $z, \ w\in\Pi^n$, where
\[
A_j(z,w) = \ip{P_jF(z)v}{F(w)v}.
\]
The $A_j$ are clearly positive semidefinite on $\Pi^n$, and hence $h$ belongs to the Loewner class $\mathcal{L}_n$.
\end{proof}
|
{
"timestamp": "2012-06-26T02:04:29",
"yymm": "1203",
"arxiv_id": "1203.2261",
"language": "en",
"url": "https://arxiv.org/abs/1203.2261"
}
|
\section{Introduction}
\label{sec:intro}
\input{intro.tex}
\section{Experimental apparatus, data sample, event reconstruction and selection}
\label{sec:detector}
\input{detector.tex}
\section{D meson reconstruction and selection}
\label{sec:signal}
\input{signalextraction.tex}
\section{Corrections}
\label{sec:corrections}
\input{corrections.tex}
\section{Reference pp cross section at $\sqrt{s}=2.76~\mathrm{TeV}$}
\label{sec:reference}
\input{reference.tex}
\section{Systematic uncertainties}
\label{sec:systematics}
\input{systematics.tex}
\section{Results}
\label{sec:results}
\input{results.tex}
\section{Summary}
\label{sec:conclusions}
\input{conclusions.tex}
\vspace{1cm}
\section*{Affiliation notes}
\renewcommand\theenumi{\roman{enumi}}
\begin{Authlist}
\item \Adef{M.V.Lomonosov Moscow State University, D.V.Skobeltsyn Institute of Nuclear Physics, Moscow, Russia}Also at: M.V.Lomonosov Moscow State University, D.V.Skobeltsyn Institute of Nuclear Physics, Moscow, Russia
\item \Adef{Institute of Nuclear Sciences, Belgrade, Serbia}Also at: "Vin\v{c}a" Institute of Nuclear Sciences, Belgrade, Serbia
\end{Authlist}
\section*{Collaboration Institutes}
\renewcommand\theenumi{\arabic{enumi}~}
\begin{Authlist}
\item \Idef{org1279}Benem\'{e}rita Universidad Aut\'{o}noma de Puebla, Puebla, Mexico
\item \Idef{org1220}Bogolyubov Institute for Theoretical Physics, Kiev, Ukraine
\item \Idef{org1262}Budker Institute for Nuclear Physics, Novosibirsk, Russia
\item \Idef{org1292}California Polytechnic State University, San Luis Obispo, California, United States
\item \Idef{org14939}Centre de Calcul de l'IN2P3, Villeurbanne, France
\item \Idef{org1197}Centro de Aplicaciones Tecnol\'{o}gicas y Desarrollo Nuclear (CEADEN), Havana, Cuba
\item \Idef{org1242}Centro de Investigaciones Energ\'{e}ticas Medioambientales y Tecnol\'{o}gicas (CIEMAT), Madrid, Spain
\item \Idef{org1244}Centro de Investigaci\'{o}n y de Estudios Avanzados (CINVESTAV), Mexico City and M\'{e}rida, Mexico
\item \Idef{org1335}Centro Fermi -- Centro Studi e Ricerche e Museo Storico della Fisica ``Enrico Fermi'', Rome, Italy
\item \Idef{org17347}Chicago State University, Chicago, United States
\item \Idef{org1288}Commissariat \`{a} l'Energie Atomique, IRFU, Saclay, France
\item \Idef{org1294}Departamento de F\'{\i}sica de Part\'{\i}culas and IGFAE, Universidad de Santiago de Compostela, Santiago de Compostela, Spain
\item \Idef{org1106}Department of Physics Aligarh Muslim University, Aligarh, India
\item \Idef{org1121}Department of Physics and Technology, University of Bergen, Bergen, Norway
\item \Idef{org1162}Department of Physics, Ohio State University, Columbus, Ohio, United States
\item \Idef{org1300}Department of Physics, Sejong University, Seoul, South Korea
\item \Idef{org1268}Department of Physics, University of Oslo, Oslo, Norway
\item \Idef{org1145}Dipartimento di Fisica dell'Universit\`{a} and Sezione INFN, Cagliari, Italy
\item \Idef{org1270}Dipartimento di Fisica dell'Universit\`{a} and Sezione INFN, Padova, Italy
\item \Idef{org1315}Dipartimento di Fisica dell'Universit\`{a} and Sezione INFN, Trieste, Italy
\item \Idef{org1132}Dipartimento di Fisica dell'Universit\`{a} and Sezione INFN, Bologna, Italy
\item \Idef{org1285}Dipartimento di Fisica dell'Universit\`{a} `La Sapienza' and Sezione INFN, Rome, Italy
\item \Idef{org1154}Dipartimento di Fisica e Astronomia dell'Universit\`{a} and Sezione INFN, Catania, Italy
\item \Idef{org1290}Dipartimento di Fisica `E.R.~Caianiello' dell'Universit\`{a} and Gruppo Collegato INFN, Salerno, Italy
\item \Idef{org1312}Dipartimento di Fisica Sperimentale dell'Universit\`{a} and Sezione INFN, Turin, Italy
\item \Idef{org1103}Dipartimento di Scienze e Tecnologie Avanzate dell'Universit\`{a} del Piemonte Orientale and Gruppo Collegato INFN, Alessandria, Italy
\item \Idef{org1114}Dipartimento Interateneo di Fisica `M.~Merlin' and Sezione INFN, Bari, Italy
\item \Idef{org1237}Division of Experimental High Energy Physics, University of Lund, Lund, Sweden
\item \Idef{org1192}European Organization for Nuclear Research (CERN), Geneva, Switzerland
\item \Idef{org1227}Fachhochschule K\"{o}ln, K\"{o}ln, Germany
\item \Idef{org1122}Faculty of Engineering, Bergen University College, Bergen, Norway
\item \Idef{org1136}Faculty of Mathematics, Physics and Informatics, Comenius University, Bratislava, Slovakia
\item \Idef{org1274}Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University in Prague, Prague, Czech Republic
\item \Idef{org1229}Faculty of Science, P.J.~\v{S}af\'{a}rik University, Ko\v{s}ice, Slovakia
\item \Idef{org1184}Frankfurt Institute for Advanced Studies, Johann Wolfgang Goethe-Universit\"{a}t Frankfurt, Frankfurt, Germany
\item \Idef{org1215}Gangneung-Wonju National University, Gangneung, South Korea
\item \Idef{org1212}Helsinki Institute of Physics (HIP) and University of Jyv\"{a}skyl\"{a}, Jyv\"{a}skyl\"{a}, Finland
\item \Idef{org1203}Hiroshima University, Hiroshima, Japan
\item \Idef{org1329}Hua-Zhong Normal University, Wuhan, China
\item \Idef{org1254}Indian Institute of Technology, Mumbai, India
\item \Idef{org36378}Indian Institute of Technology Indore (IIT), Indore, India
\item \Idef{org1266}Institut de Physique Nucl\'{e}aire d'Orsay (IPNO), Universit\'{e} Paris-Sud, CNRS-IN2P3, Orsay, France
\item \Idef{org1277}Institute for High Energy Physics, Protvino, Russia
\item \Idef{org1249}Institute for Nuclear Research, Academy of Sciences, Moscow, Russia
\item \Idef{org1320}Nikhef, National Institute for Subatomic Physics and Institute for Subatomic Physics of Utrecht University, Utrecht, Netherlands
\item \Idef{org1250}Institute for Theoretical and Experimental Physics, Moscow, Russia
\item \Idef{org1230}Institute of Experimental Physics, Slovak Academy of Sciences, Ko\v{s}ice, Slovakia
\item \Idef{org1127}Institute of Physics, Bhubaneswar, India
\item \Idef{org1275}Institute of Physics, Academy of Sciences of the Czech Republic, Prague, Czech Republic
\item \Idef{org1139}Institute of Space Sciences (ISS), Bucharest, Romania
\item \Idef{org27399}Institut f\"{u}r Informatik, Johann Wolfgang Goethe-Universit\"{a}t Frankfurt, Frankfurt, Germany
\item \Idef{org1185}Institut f\"{u}r Kernphysik, Johann Wolfgang Goethe-Universit\"{a}t Frankfurt, Frankfurt, Germany
\item \Idef{org1177}Institut f\"{u}r Kernphysik, Technische Universit\"{a}t Darmstadt, Darmstadt, Germany
\item \Idef{org1256}Institut f\"{u}r Kernphysik, Westf\"{a}lische Wilhelms-Universit\"{a}t M\"{u}nster, M\"{u}nster, Germany
\item \Idef{org1246}Instituto de Ciencias Nucleares, Universidad Nacional Aut\'{o}noma de M\'{e}xico, Mexico City, Mexico
\item \Idef{org1247}Instituto de F\'{\i}sica, Universidad Nacional Aut\'{o}noma de M\'{e}xico, Mexico City, Mexico
\item \Idef{org23333}Institut of Theoretical Physics, University of Wroclaw
\item \Idef{org1308}Institut Pluridisciplinaire Hubert Curien (IPHC), Universit\'{e} de Strasbourg, CNRS-IN2P3, Strasbourg, France
\item \Idef{org1182}Joint Institute for Nuclear Research (JINR), Dubna, Russia
\item \Idef{org1143}KFKI Research Institute for Particle and Nuclear Physics, Hungarian Academy of Sciences, Budapest, Hungary
\item \Idef{org1199}Kirchhoff-Institut f\"{u}r Physik, Ruprecht-Karls-Universit\"{a}t Heidelberg, Heidelberg, Germany
\item \Idef{org20954}Korea Institute of Science and Technology Information, Daejeon, South Korea
\item \Idef{org1160}Laboratoire de Physique Corpusculaire (LPC), Clermont Universit\'{e}, Universit\'{e} Blaise Pascal, CNRS--IN2P3, Clermont-Ferrand, France
\item \Idef{org1194}Laboratoire de Physique Subatomique et de Cosmologie (LPSC), Universit\'{e} Joseph Fourier, CNRS-IN2P3, Institut Polytechnique de Grenoble, Grenoble, France
\item \Idef{org1187}Laboratori Nazionali di Frascati, INFN, Frascati, Italy
\item \Idef{org1232}Laboratori Nazionali di Legnaro, INFN, Legnaro, Italy
\item \Idef{org1125}Lawrence Berkeley National Laboratory, Berkeley, California, United States
\item \Idef{org1234}Lawrence Livermore National Laboratory, Livermore, California, United States
\item \Idef{org1251}Moscow Engineering Physics Institute, Moscow, Russia
\item \Idef{org1140}National Institute for Physics and Nuclear Engineering, Bucharest, Romania
\item \Idef{org1165}Niels Bohr Institute, University of Copenhagen, Copenhagen, Denmark
\item \Idef{org1109}Nikhef, National Institute for Subatomic Physics, Amsterdam, Netherlands
\item \Idef{org1283}Nuclear Physics Institute, Academy of Sciences of the Czech Republic, \v{R}e\v{z} u Prahy, Czech Republic
\item \Idef{org1264}Oak Ridge National Laboratory, Oak Ridge, Tennessee, United States
\item \Idef{org1189}Petersburg Nuclear Physics Institute, Gatchina, Russia
\item \Idef{org1170}Physics Department, Creighton University, Omaha, Nebraska, United States
\item \Idef{org1157}Physics Department, Panjab University, Chandigarh, India
\item \Idef{org1112}Physics Department, University of Athens, Athens, Greece
\item \Idef{org1152}Physics Department, University of Cape Town, iThemba LABS, Cape Town, South Africa
\item \Idef{org1209}Physics Department, University of Jammu, Jammu, India
\item \Idef{org1207}Physics Department, University of Rajasthan, Jaipur, India
\item \Idef{org1200}Physikalisches Institut, Ruprecht-Karls-Universit\"{a}t Heidelberg, Heidelberg, Germany
\item \Idef{org1325}Purdue University, West Lafayette, Indiana, United States
\item \Idef{org1281}Pusan National University, Pusan, South Korea
\item \Idef{org1176}Research Division and ExtreMe Matter Institute EMMI, GSI Helmholtzzentrum f\"ur Schwerionenforschung, Darmstadt, Germany
\item \Idef{org1334}Rudjer Bo\v{s}kovi\'{c} Institute, Zagreb, Croatia
\item \Idef{org1298}Russian Federal Nuclear Center (VNIIEF), Sarov, Russia
\item \Idef{org1252}Russian Research Centre Kurchatov Institute, Moscow, Russia
\item \Idef{org1224}Saha Institute of Nuclear Physics, Kolkata, India
\item \Idef{org1130}School of Physics and Astronomy, University of Birmingham, Birmingham, United Kingdom
\item \Idef{org1338}Secci\'{o}n F\'{\i}sica, Departamento de Ciencias, Pontificia Universidad Cat\'{o}lica del Per\'{u}, Lima, Peru
\item \Idef{org1316}Sezione INFN, Trieste, Italy
\item \Idef{org1271}Sezione INFN, Padova, Italy
\item \Idef{org1313}Sezione INFN, Turin, Italy
\item \Idef{org1286}Sezione INFN, Rome, Italy
\item \Idef{org1146}Sezione INFN, Cagliari, Italy
\item \Idef{org1133}Sezione INFN, Bologna, Italy
\item \Idef{org1115}Sezione INFN, Bari, Italy
\item \Idef{org1155}Sezione INFN, Catania, Italy
\item \Idef{org1322}Soltan Institute for Nuclear Studies, Warsaw, Poland
\item \Idef{org36377}Nuclear Physics Group, STFC Daresbury Laboratory, Daresbury, United Kingdom
\item \Idef{org1258}SUBATECH, Ecole des Mines de Nantes, Universit\'{e} de Nantes, CNRS-IN2P3, Nantes, France
\item \Idef{org1304}Technical University of Split FESB, Split, Croatia
\item \Idef{org1168}The Henryk Niewodniczanski Institute of Nuclear Physics, Polish Academy of Sciences, Cracow, Poland
\item \Idef{org17361}The University of Texas at Austin, Physics Department, Austin, TX, United States
\item \Idef{org1173}Universidad Aut\'{o}noma de Sinaloa, Culiac\'{a}n, Mexico
\item \Idef{org1296}Universidade de S\~{a}o Paulo (USP), S\~{a}o Paulo, Brazil
\item \Idef{org1149}Universidade Estadual de Campinas (UNICAMP), Campinas, Brazil
\item \Idef{org1239}Universit\'{e} de Lyon, Universit\'{e} Lyon 1, CNRS/IN2P3, IPN-Lyon, Villeurbanne, France
\item \Idef{org1205}University of Houston, Houston, Texas, United States
\item \Idef{org20371}University of Technology and Austrian Academy of Sciences, Vienna, Austria
\item \Idef{org1222}University of Tennessee, Knoxville, Tennessee, United States
\item \Idef{org1310}University of Tokyo, Tokyo, Japan
\item \Idef{org1318}University of Tsukuba, Tsukuba, Japan
\item \Idef{org21360}Eberhard Karls Universit\"{a}t T\"{u}bingen, T\"{u}bingen, Germany
\item \Idef{org1225}Variable Energy Cyclotron Centre, Kolkata, India
\item \Idef{org1306}V.~Fock Institute for Physics, St. Petersburg State University, St. Petersburg, Russia
\item \Idef{org1323}Warsaw University of Technology, Warsaw, Poland
\item \Idef{org1179}Wayne State University, Detroit, Michigan, United States
\item \Idef{org1260}Yale University, New Haven, Connecticut, United States
\item \Idef{org1332}Yerevan Physics Institute, Yerevan, Armenia
\item \Idef{org15649}Yildiz Technical University, Istanbul, Turkey
\item \Idef{org1301}Yonsei University, Seoul, South Korea
\item \Idef{org1327}Zentrum f\"{u}r Technologietransfer und Telekommunikation (ZTT), Fachhochschule Worms, Worms, Germany
\end{Authlist}
\endgroup
\subsection{Feed-down subtraction}
The prompt D meson production yields ${\rm d}N/{\rm d}p_{\rm t}$ in Pb--Pb
collisions were obtained by subtracting the contribution of D mesons
from B decays with the same procedure used for the measurement of
the production cross sections in pp collisions~\cite{Dpp7paper}.
In detail, the feed-down contribution was estimated using
the beauty production cross section from the FONLL calculation~\cite{fonllpriv},
the B$\rightarrow$D decay kinematics from the EvtGen package~\cite{evtgen},
and the Monte Carlo efficiencies for feed-down D mesons.
For Pb--Pb collisions,
the FONLL feed-down cross section in pp at $\sqrt{s}=2.76~\mathrm{TeV}$
was scaled by the average nuclear overlap function $\langle T_{\rm AA} \rangle$ in each centrality class.
Thus, omitting for brevity the symbol of the $p_{\rm t}$-dependence $(p_{\rm t})$,
the fraction of prompt D mesons reads:
\begin{equation}
\label{eq:fcNbMethod}
\begin{split}
f_{\rm prompt} &= 1-(N^{\rm D~feed-down~raw}/N^{\rm D~raw})=\\
&= 1 - \langle T_{\rm AA} \rangle
\cdot \left( \frac{{\rm d}^2 \sigma}{{\rm d}y \, {\rm d}p_{\rm t} } \right)^{{\sf FONLL}} _{{\rm feed-down}} \cdot R_{\rm AA}^{\rm feed-down} \cdot \frac{({\rm Acc}\times\epsilon)_{\rm feed-down}\cdot\Delta y \, \Deltap_{\rm t} \cdot {\rm BR} \cdot N_{\rm evt} }{ N^{\rm D~raw } / 2} \, ,
\end{split}
\end{equation}
where $({\rm Acc}\times\epsilon)_{{\rm feed-down}}$ is the
acceptance-times-efficiency for feed-down D mesons.
The nuclear modification factor of the feed-down D mesons, $R_{\rm AA}^{\rm feed-down}$,
is related to the nuclear modification of beauty production in Pb--Pb
collisions, which is currently unknown.
We therefore assumed for the correction that the nuclear modification factors
for feed-down and prompt D mesons are equal
($R_{\rm AA}^{\rm feed-down}=R_{\rm AA}^{\rm prompt}$) and varied this hypothesis
in the range $1/3<R_{\rm AA}^{\rm feed-down}/R_{\rm AA}^{\rm prompt}<3$
to determine the systematic uncertainty.
This hypothesis is justified by the range of the model predictions for the charm and beauty $R_{\rm AA}$~\cite{whdg,adsw} and, as discussed in Section~\ref{sec:systematics}, by the CMS Collaboration results on $R_{\rm AA}$ for
non-prompt $\rm J/\psi$~\cite{CMSquarkonia}.
The value of $f_{\rm prompt}$ depends on the D meson species, the transverse
momentum interval, the applied cuts, the parameters used in the FONLL B
prediction, and the hypothesis on $R_{\rm AA}^{\rm feed-down}$.
The resulting values, for the case $R_{\rm AA}^{\rm feed-down}=R_{\rm AA}^{\rm prompt}$,
range from $\approx 0.95$ in the lowest transverse momentum interval
($2<p_{\rm t}<3~\mathrm{GeV}/c$) to $\approx 0.85$ at high $p_{\rm t}$.
\subsection{D meson $p_{\rm t}$ spectra and $R_{\rm AA}$}
The transverse momentum distributions $\d N/\dp_{\rm t}$
of prompt $\rm D^0$, $\rm D^+$, and $\rm D^{*+}$ mesons
are presented in Fig.~\ref{fig:spectra},
for the centrality classes 0--20\% and 40--80\%.
The spectra from Pb--Pb
collisions, defined as the feed-down corrected production yields per event (see
Eq.~(\ref{eq:dNdpt})), are compared to the reference spectra from pp
collisions, which are constructed as $\av{T_{\rm AA}}\,\d\sigma/\dp_{\rm t}$,
using the $\sqrt{s}$-scaled pp measurements at 7~TeV~\cite{Dpp7paper}
and the average nuclear overlap function values from Table~\ref{tab:centbins}.
A clear suppression is observed in Pb--Pb collisions, which is stronger in central than in peripheral collisions.
The ratio of the Pb--Pb to the reference spectra provides the
nuclear modification factors $R_{\rm AA}(p_{\rm t})$
of prompt $\rm D^0$, $\rm D^+$, and ${\rm D^{*+}}$ mesons, which are shown
for central
(0--20\%) and semi-peripheral (40--80\%) collisions in Fig.~\ref{fig:RAApt}.
The vertical bars represent the statistical uncertainties, typically about 20--25\%
for ${\rm D^0}$ and about 30--40\% for ${\rm D^+}$ and ${\rm D^{*+}}$ mesons in central collisions. The total
$p_{\rm t}$-dependent systematic uncertainties, shown as empty boxes, include all the contributions
described in the previous section, except for the normalization uncertainty, which
is displayed as a filled box at $R_{\rm AA}=1$.
The results for the three D meson species are in agreement within statistical uncertainties and they
show a
suppression reaching a factor 3--4 ($R_{\rm AA}\approx 0.25$--0.3) in central collisions for $p_{\rm t}>5~\mathrm{GeV}/c$.
For decreasing $p_{\rm t}$, the ${\rm D^0}$
$R_{\rm AA}$ in central collisions shows a tendency to less suppression.
\begin{figure}[!t]
\begin{center}
\includegraphics[width=\textwidth]{figures/ThreeYieldsInOne.eps}
\caption{(colour online) Transverse momentum distributions $\d N/\dp_{\rm t}$ of
prompt $\rm D^0$ (left) and $\rm D^+$ (centre), and $\rm D^{*+}$ (right)
mesons in the 0--20\% and 40--80\% centrality classes in Pb--Pb collisions
at $\sqrt{s_{\rm \scriptscriptstyle NN}}=2.76~\mathrm{TeV}$.
The reference pp distributions $\av{T_{\rm AA}}\,\d \sigma/\dp_{\rm t}$ are shown as well.
Statistical uncertainties (bars) and systematic uncertainties from data
analysis (empty boxes) and from feed-down subtraction (full boxes) are shown.
For Pb--Pb, the latter includes the uncertainties from the FONLL feed-down
correction and from the variation of the hypothesis on
$R_{{\rm AA}}^{\rm prompt} / R_{{\rm AA}}^{\rm feed-down}$.
Horizontal error bars reflect bin widths, symbols were placed at the centre of the bin.}
\label{fig:spectra}
\end{center}
\end{figure}
\begin{figure}[!t]
\begin{center}
\includegraphics[width=\textwidth]{figures/Dmeson_Raa_210612.eps}
\caption{(colour online) $R_{\rm AA}$
for prompt $\rm D^0$, $\rm D^+$, and ${\rm D^{*+}}$ in the 0--20\% (left) and 40--80\% (right) centrality classes. Statistical (bars),
systematic (empty boxes), and normalization (full box) uncertainties are shown.
Horizontal error bars reflect bin widths, symbols were placed at the centre of the bin.}
\label{fig:RAApt}
\end{center}
\end{figure}
The centrality dependence of the nuclear modification factor was studied
in the two wider transverse momentum intervals $2<p_{\rm t}<5~\mathrm{GeV}/c$,
for ${\rm D^0}$, and $6<p_{\rm t}<12~\mathrm{GeV}/c$, for the three D meson species.
This study was performed in five centrality classes from 0--10\% to 60--80\%
(see Table~\ref{tab:centbins}).
The invariant mass analysis and all the corrections were carried out
as described in Sections~\ref{sec:signal} and~\ref{sec:corrections}.
The systematic uncertainties are essentially the same as for the
$p_{\rm t}$-dependence analysis, except for the contribution from the D meson
$p_{\rm t}$-shape in the simulation, which is larger in the wide intervals.
It amounts to 8\% for ${\rm D^0}$, 10\% for ${\rm D^+}$, and
5--15\% (depending on centrality) for ${\rm D^{*+}}$ mesons in $6<p_{\rm t}<12~\mathrm{GeV}/c$.
In the transverse momentum interval 2--5~$\mathrm{GeV}/c$, this uncertainty is larger
(8--17\%, depending on centrality) due to the larger contribution from the
$p_{\rm t}$ dependence of the nuclear modification factor.
The resulting $R_{\rm AA}$ is shown in Fig.~\ref{fig:RAAcentr} as a function
of the average number of participants, $\langleN_{\rm part}\rangle$.
The contribution to the systematic uncertainty that is fully correlated
between centrality classes (normalization and pp reference cross-section) and
the remaining, uncorrelated, systematic uncertainties are displayed
separately, by the filled and empty boxes, respectively.
The contribution from feed-down correction was considered among the
uncorrelated sources because it is dominated by the variation of the
ratio $R_{\rm AA}^{\rm feed-down}/R_{\rm AA}^{\rm prompt}$, which may depend on
centrality.
For the $p_{\rm t}$ interval 6--12~$\mathrm{GeV}/c$, the suppression increases with
increasing centrality.
It is interesting to note that the suppression of prompt D mesons at central
rapidity and high transverse momentum, shown in the right-hand panel of
Fig.~\ref{fig:RAAcentr} is very similar, both in size and centrality
dependence, to that of prompt J/$\psi$ mesons in a similar $p_{\rm t}$ range and
$|y|<2.4$, recently measured by the CMS Collaboration~\cite{CMSquarkonia}.
\begin{figure}[!t]
\begin{center}
\includegraphics[width=\textwidth]{figures/Dmesons_RaavsNpart_260412.eps}
\caption{Centrality dependence of $R_{\rm AA}$ for prompt D mesons.
Left: ${\rm D^0}$ mesons with $2<p_{\rm t}<5~\mathrm{GeV}/c$.
Right: ${\rm D^0}$, ${\rm D^+}$, and ${\rm D^{*+}}$ mesons with $6<p_{\rm t}<12~\mathrm{GeV}/c$.
${\rm D^+}$ and ${\rm D^{*+}}$ points are displaced horizontally for better visibility.}
\label{fig:RAAcentr}
\end{center}
\end{figure}
\subsection{Comparisons to light-flavour hadrons and with models}
\label{sec:comparisons}
In this section, the average nuclear modification factor of the three D meson species is compared to
that of charged particles~\cite{chargedRAA}, mainly light-flavour hadrons, and to model calculations. The contributions of
${\rm D^0}$, ${\rm D^+}$, and ${\rm D^{*+}}$ to the average were weighted by their statistical uncertainties. Therefore,
the resulting $R_{\rm AA}$ is close to that of the ${\rm D^0}$~meson, which has the
smallest uncertainties.
The systematic errors were calculated by propagating the uncertainties through
the weighted average, where the contributions from the tracking efficiency,
from the B feed-down correction, and from the FONLL scaling of 7 TeV data to
2.76 TeV were taken as fully correlated among the three D meson species.
The possible statistical correlation between the ${\rm D^0}$ and ${\rm D^{*+}}$ $R_{\rm AA}$,
induced by the ${\rm D^{*+}\to D^0\pi^+}$ decay, is negligible, because the
statistical uncertainties, used as weights, are mainly determined by the
background uncertainties, which are uncorrelated.
The resulting values are shown in Table~\ref{tab:averD1}
for the two centrality classes where $R_{\rm AA}$ was
measured as a function of $p_{\rm t}$, and in Table~\ref{tab:averD2}
for the $R_{\rm AA}$ as a function of centrality in the transverse momentum
range $6<p_{\rm t}<12~\mathrm{GeV}/c$.
\begin{table}[!t]
\caption{Average $R_{\rm AA}$ as a function of $p_{\rm t}$ for prompt D mesons in the
0--20\% and 40--80\% centrality classes. The systematic error does
not include the normalization uncertainty, which is $\pm$5.3\%~($\pm$7.5\%)
for the 0--20\%~(40--80\%) centrality class.}
\label{tab:averD1}
\centering
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{|c|c|c|}
\hline
$p_{\rm t}$ interval & \multicolumn{2}{c|}{$R_{\rm AA}~\pm$~stat~$\pm$~syst}\\
($\mathrm{GeV}/c$) & 0--20\% centrality & 40--80\% centrality\\
\hline
2--3 & $0.51\pm0.10_{-0.22}^{+0.18}$ & $0.75\pm0.13_{-0.32}^{+0.23}$\\
3--4 & $0.37\pm0.06_{-0.13}^{+0.11}$ & $0.59\pm0.09_{-0.21}^{+0.15}$\\
4--5 & $0.33\pm0.05_{-0.11}^{+0.10}$ & $0.55\pm0.07_{-0.18}^{+0.14}$\\
5--6 & $0.27\pm0.07_{-0.09}^{+0.08}$ & $0.54\pm0.08_{-0.17}^{+0.13}$\\
6--8 & $0.28\pm0.04_{-0.08}^{+0.07}$ & $0.60\pm0.08_{-0.18}^{+0.14}$\\
\phantom{0}8--12 & $0.26\pm0.03_{-0.07}^{+0.06}$ & $0.66\pm0.08_{-0.20}^{+0.16}$\\
12--16 & $0.35\pm0.06_{-0.12}^{+0.10}$ & $0.64\pm0.16_{-0.18}^{+0.16}$\\
\hline
\end{tabular}
\end{table}
\begin{table}[!t]
\caption{Average $R_{\rm AA}$ as a function of centrality for prompt D mesons in the
transverse momentum interval $6<p_{\rm t}<12~\mathrm{GeV}/c$.}
\label{tab:averD2}
\centering
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{|c|c|}
\hline
Centrality & $R_{\rm AA}$$\pm$~stat~$\pm$~syst(uncorr)~$\pm$~syst(corr)\\
\hline
\phantom{0}0--10\% & $0.23\pm0.03~_{-0.06}^{+0.05}~_{-0.03}^{+0.03}$\\
10--20\% & $0.28\pm0.04~_{-0.07}^{+0.06}~_{-0.04}^{+0.03}$\\
20--40\% & $0.42\pm0.04~_{-0.11}^{+0.08}~_{-0.06}^{+0.05}$\\
40--60\% & $0.54\pm0.05~_{-0.13}^{+0.10}~_{-0.08}^{+0.07}$\\
60--80\%& $0.81\pm0.10~_{-0.21}^{+0.16}~_{-0.12}^{+0.11}$\\
\hline
\end{tabular}
\end{table}
In addition to final state effects, where parton energy loss would be predominant,
also initial-state effects are expected to influence the measured $R_{\rm AA}$.
In particular,
the nuclear modification of the parton distribution functions of the nucleons in the two colliding
nuclei modifies the initial hard scattering probability and, thus, the production yields of
hard partons, including heavy quarks. In the kinematic range relevant for charm production
at LHC energies, the main expected effect is nuclear shadowing, which reduces the parton distribution functions for partons with nucleon momentum fraction $x$ below $10^{-2}$.
The effect of shadowing on the D meson $R_{\rm AA}$ was estimated using the
next-to-leading order (NLO) perturbative QCD calculation by Mangano,
Nason, and Ridolfi (MNR)~\cite{mnr} with CTEQ6M parton distribution
functions~\cite{cteq6} and the EPS09NLO parametrization~\cite{eps09}
of their nuclear modification. The uncertainty band determined by the EPS09 uncertainties
is shown in the left-hand panel of Fig.~\ref{fig:RAApt_eps_charged}, together with the average D meson
$R_{\rm AA}$. The shadowing-induced effect on the $R_{\rm AA}$
is limited to $\pm 15\%$ for $p_{\rm t}>6~\mathrm{GeV}/c$, suggesting that
the strong suppression observed in the data is a final-state effect.
The expected colour charge and parton mass dependences of parton energy loss
should be addressed by comparing
the nuclear modification factor of D and $\pi$ mesons. Since final results on the
pion $R_{\rm AA}$ at the LHC are not yet available, we compare here to
charged particles.
Preliminary results~\cite{harryQM} have shown that the charged-pion $R_{\rm AA}$
coincides with that of charged particles above $p_{\rm t}\approx 5~\mathrm{GeV}/c$
and it is lower by 30\% at $3~\mathrm{GeV}/c$.
The comparison between D meson and charged particle $R_{\rm AA}$, reported in the
right-hand panel of Fig.~\ref{fig:RAApt_eps_charged}, shows
that the average D meson nuclear modification factor is close to that of
charged particles~\cite{chargedRAA}.
However, considering that the systematic uncertainties of D mesons are not fully correlated with $p_{\rm t}$, there is an indication for $R_{\rm AA}^{\rm D}>R_{\rm AA}^{\rm charged}$.
In the same figure, the nuclear modification factor measured by the CMS Collaboration for non-prompt
J/$\psi$ mesons (from B decays)
with $p_{\rm t}>6.5~\mathrm{GeV}/c$~\cite{CMSquarkonia} is also shown. Their suppression is clearly weaker
than that of charged particles, while the comparison with D mesons is not conclusive and
would require more differential and precise measurements of the transverse momentum dependence.
\begin{figure}[!t]
\begin{center}
\includegraphics[width=0.49\textwidth]{figures/RAA_shadowing.eps}
\includegraphics[width=0.49\textwidth]{figures/RAA_charged.eps}
\caption{Average $R_{\rm AA}$ of D mesons in the 0--20\% centrality class compared to: left,
the expectation from NLO pQCD~\cite{mnr} with nuclear shadowing~\cite{eps09}; right,
the nuclear modification factors of charged particles~\cite{chargedRAA} and non-prompt $\rm J/\psi$
from B decays~\cite{CMSquarkonia} in the same centrality class.
The charged particle $R_{\rm AA}$ is shown only for \mbox{$2<p_{\rm t}<16~\mathrm{GeV}/c$}. The three normalization uncertainties shown
in the right-hand panel are almost fully correlated.}
\label{fig:RAApt_eps_charged}
\end{center}
\end{figure}
\begin{figure}[!t]
\begin{center}
\includegraphics[width=\textwidth]{figures/RAADandChargedwithModels_2.eps}
\caption{(colour online) Average $R_{\rm AA}$ of D mesons (left) and $R_{\rm AA}$ of charged particles (right)~\cite{chargedRAA} in the \mbox{0--20\%} centrality class compared to model calculations: (I)~\cite{vitev,vitevjet}, (II)~\cite{whdg2011}, (III)~\cite{horowitzAdSCFT}, (IV)~\cite{beraudo}, (V)~\cite{gossiaux}, (VI)~\cite{bamps}, (VII)~\cite{cujet}, (VIII)~\cite{adsw}. The two normalization uncertainties are almost fully correlated.}
\label{fig:RAApt_models}
\end{center}
\end{figure}
Several theoretical models based on parton energy loss compute the charm
nuclear modification factor: (I)~\cite{vitev,vitevjet}, (II)~\cite{whdg2011}, (III)~\cite{horowitzAdSCFT}, (IV)~\cite{beraudo}, (V)~\cite{gossiaux}, (VI)~\cite{bamps}, (VII)~\cite{cujet}, (VIII)~\cite{adsw}. Figure~\ref{fig:RAApt_models} displays the comparison of these
models
to the average D meson $R_{\rm AA}$, for central Pb--Pb collisions (0--20\%), along with the
comparison to the charged-particle $R_{\rm AA}$~\cite{chargedRAA}, for those models that also compute
this observable: (I)~\cite{vitev}, (II)~\cite{whdg2011}, (III)~\cite{horowitzAdSCFT}, (VII)~\cite{cujet}. Among the models that compute both observables,
radiative energy loss supplemented with in-medium D meson dissociation (I)~\cite{vitev}
and radiative plus collisional energy loss in the WHDG (II)~\cite{whdg2011} and CUJET1.0 (VII)~\cite{cujet} implementations describe reasonably well at the same time the charm and
light-flavour suppression.
While in the former calculation the medium density is tuned to describe the inclusive jet
suppression at the LHC~\cite{vitevjet}, for the latter two it is extrapolated to LHC conditions starting from the
value that describes the pion suppression at RHIC energy ($\sqrt{s_{\rm \scriptscriptstyle NN}}=200~\mathrm{GeV}$). This could explain why these two models are somewhat low with respect to the charged-particle $R_{\rm AA}$ data.
A model based on AdS/CFT drag coefficients (III)~\cite{horowitzAdSCFT} underestimates significantly the charm $R_{\rm AA}$ and
has very limited predictive power for the light-flavour $R_{\rm AA}$.
|
{
"timestamp": "2012-10-15T02:01:09",
"yymm": "1203",
"arxiv_id": "1203.2160",
"language": "en",
"url": "https://arxiv.org/abs/1203.2160"
}
|
\section{Definition and First Properties}\label{sec:definitions}
First, we review the definition of the Bruhat-Tits building of the projective linear group over a non-archimedean field. Let $R$ be a discrete valuation ring, $K$ its field of fractions, and $V$ a $K$-vector space of dimension $d$. A \emph{lattice} is a free $R$-submodule of $V$ of rank $d$. Two lattices $L'$ and $L$ are called \emph{homothetic} if $L' = cL$ for some $c \in K^\times$. Homothety is an equivalence relation, and an equivalence class $[L]$ is called a \emph{vertex}. The set of all vertices is denoted by $\mathfrak{B}_d^0$. Two vertices $[L_1]$ and $[L_2]$ are called \emph{adjacent} if there are representatives $L_1' \in [L_1]$ and $L_2' \in [L_2]$ such that $\pi L_1' \subsetneq L_2' \subsetneq L_1'$. This is a symmetric relation and thus defines an undirected graph on the vertex set $\mathfrak{B}_d^0$. The clique complex of this graph, where each set of pairwise adjacent vertices forms a simplex, is a simplicial complex denoted by $\mathfrak{B}_d$. It is called the \emph{Bruhat-Tits building} of $\PGL(V)$. See \cite[Section~6.9]{abramenko-brown} for details of the construction and some properties of $\mathfrak{B}_d$.
Let $\mathcal{F} = (k_1 < k_2 < \cdots < k_r)$ be a tuple of ascending integers satisfying $1 \leq k_i \leq d-1$. We call this a \emph{flag type}.
To a vertex $[L] \in \mathfrak{B}_d^0$, we assign the flag variety $\mathcal{F}(L)$, which is defined as the $R$-scheme representing the functor which sends a scheme $h \colon T \to \Spec R$ to the set
\[\{h^*(\widetilde L) \supset \mathscr{U}_1 \supset \mathscr{U}_2 \supset \dots \supset \mathscr{U}_r : h^*(\widetilde L)/\mathscr{U}_i \text{ is a locally free $\mathscr{O}_T$-module of rank $k_i$}\}\text{.}\]
$\mathcal{F}(L)$ is an integral scheme, projective and smooth over $R$. If $\mathcal{F} = (k)$, $\mathcal{F}(L)$ is the Grassmannian $\calG_k(L)$ parametrizing quotients of dimension $k$, and in particular, we recover the projective space $\mathbb{P}(L) \cong \mathbb{P}_R^{d-1}$ (parametrizing hyperplanes) for $k=1$ and the dual projective space $\mathbb{P}^\vee(L)$ (parametrizing lines) for $k=d-1$. In these cases, we also write $\mathcal{F} = \calG_k$, $\mathcal{F} = \mathbb{P}$ and $\mathcal{F} = \mathbb{P}^\vee$ instead of $\mathcal{F} = (k)$, $\mathcal{F} = (1)$, and $\mathcal{F} = (d-1)$, respectively.
It will become important in \cref{sec:combinatorics} to observe that an inclusion
\[\mathcal{F}' = (k_{i_1} < k_{i_2} < \cdots < k_{i_{r'}}) \subseteq (k_1 < k_2 < \cdots < k_r) = \mathcal{F}\]
of flag types induces a surjective morphism $\mathcal{F}(L) \twoheadrightarrow \mathcal{F}'(L)$, given on $T$-valued points by sending a flag $\{\mathscr{U}_1 \supset \mathscr{U}_2 \supset \dots \supset \mathscr{U}_r\}$ as above to the subflag $\{\mathscr{U}_{i_1} \supset \mathscr{U}_{i_2} \supset \dots \supset \mathscr{U}_{i_{r'}}\}$.
Since, by definition, any two lattices $L$ and $L'$ are isomorphic as $R$-modules, the associated flag varieties are isomorphic as $R$-schemes. Note, however, that the isomorphism is not canonical, since it depends on a choice of basis for each lattice. On the other hand, their generic fibers $\mathcal{F}(L) \times_R K$ and $\mathcal{F}(L') \times_R K$ are canonically isomorphic to $\mathcal{F}(V)$, the $K$-scheme representing the functor which sends $h \colon T \to \Spec K$ to
\[\{h^*(\widetilde V) \supset \mathscr{U}_1 \supset \mathscr{U}_2 \supset \dots \supset \mathscr{U}_r : h^*(\widetilde V)/\mathscr{U}_i \text{ is a locally free $\mathscr{O}_T$-module of rank $k_i$}\}\text{.}\]
This allows us to make the following definition:
\begin{Definition}\label{def:mustafin-degeneration}
Given a flag type $\mathcal{F}$ as above and a finite subset $\Gamma = \{[L_1], \ldots, [L_n]\} \subset \mathfrak{B}_d^0$, we define the \emph{Mustafin degeneration} $\mathcal{M}_\mathcal{F}(\Gamma)$ to be the join of the schemes $\mathcal{F}(L_1), \ldots, \mathcal{F}(L_n)$, which is the scheme-theoretic image (i.e., the closure of the image with the induced reduced subscheme structure) of the map
\begin{equation}\label{eqn:def-mustafin}
\mathcal{F}(V) \longrightarrow \mathcal{F}(L_1) \times_R \cdots \times_R \mathcal{F}(L_n)\text{.}
\end{equation}
\end{Definition}
We can also describe $\mathcal{M}_\mathcal{F}(\Gamma)$ by giving equations. Recall that the flag variety $\mathcal{F}(L)$ with $\mathcal{F} = (k_1 < \cdots < k_r)$ can be embedded into the product of projective spaces $\prod_{t=1}^r \mathbb{P}^{\binom{d}{k_t}-1}_R$ by the Plücker embedding. The equations cutting out the image of this embedding can be found in \cite[Equations~(1) and~(3)]{youngtableaux}. If, for $1 \leq j \leq n$, $e_1^{(j)}, \ldots, e_d^{(j)}$ is a basis for $L_j$, then the respective multihomogeneous coordinates on $\prod_{t=1}^r \mathbb{P}^{\binom{d}{k_t}-1}_R$ are
\[ (p_{i_1, \ldots, i_{k_t}}^{(j)} : 1 \leq i_1 < \cdots < i_{k_t} \leq d)_{t=1,\ldots,r}\text{,} \]
where $p_{i_1, \ldots, i_{k_t}}^{(j)} = e_{i_1}^{(j)} \wedge \dots \wedge e_{i_{k_t}}^{(j)}$. Let $A^{(j)}_t$ be the matrix satisfying $A^{(j)}_tp_{i_1, \ldots, i_{k_t}}^{(1)} = p_{i_1, \ldots, i_{k_t}}^{(j)}$. $\mathbold{A}^{(j)} := (A_1^{(j)}, \ldots, A_r^{(j)})$ is an isomorphism $\mathcal{F}(L_1) \to \mathcal{F}(L_j)$ inducing an automorphism $\mathcal{F}(V) \to \mathcal{F}(V)$. The diagram
\[ \xymatrix@C+5cm{
\mathcal{F}(V) \ar[r]^{((\mathbold{A}^{(1)})^{(-1)}, \ldots, (\mathbold{A}^{(n)})^{(-1)}) \circ \Delta} \ar[d] & \mathcal{F}(V)^n \ar[d] \\
\prod_{[L] \in \Gamma} \mathcal{F}(L) \ar[r]^{((\mathbold{A}^{(1)})^{(-1)}, \ldots, (\mathbold{A}^{(n)})^{(-1)})} & \mathcal{F}(L_1)^n
} \]
commutes, where $\Delta \colon \mathcal{F}(V) \to \mathcal{F}(V)^n$ is the diagonal map and the vertical arrow on the left is the map from \cref{eqn:def-mustafin}; hence, the Mustafin degeneration is cut out in $\prod_{[L] \in \Gamma}\prod_{t=1}^r \mathbb{P}^{\binom{d}{k_t}-1}_R$ by the ideal $\mathfrak{a} \cap R[\ldots, p_{i_1, \ldots, i_{k_t}}^{(j)}, \ldots]$, where $\mathfrak{a}$ is
the ideal generated over $K$ by all $2 \times 2$-minors of the matrices
\[\begin{pmatrix}
\vdots & & \vdots \\
p_{i_1, \ldots, i_{k_t}}^{(1)} & \cdots & (A^{(j)}_t)^{-1} p_{i_1, \ldots, i_{k_t}}^{(j)} & \cdots \\
\vdots & & \vdots
\end{pmatrix} \qquad (t = 1, \ldots, r)\text{,}\]
together with the equations cutting out the product of flag varieties $\prod_{[L] \in \Gamma} \mathcal{F}(L)$.
Mustafin degenerations have the following geometric properties:
\begin{Theorem}\label{thm:basic-properties}
For a finite subset $\Gamma$ of $\mathfrak{B}_d^0$, any Mustafin degeneration $\mathcal{M}_\mathcal{F}(\Gamma)$ is an integral scheme which is flat and projective over $R$. Its generic fiber is equal to $\mathcal{F}(V)$, and its special fiber $\mathcal{M}_\mathcal{F}(\Gamma)_s$ is connected and equidimensional of dimension $\dim \mathcal{F}(V)$.
\end{Theorem}
\begin{Proof}
By construction, the Mustafin degeneration $\mathcal{M}_\mathcal{F}(\Gamma)$ is a reduced, irreducible, projective scheme over $R$ with generic fiber $\mathcal{F}(V)$. Since $R$ is a discrete valuation ring, flatness follows from the fact that $\mathcal{M}_\mathcal{F}(\Gamma)$ is reduced with non-empty generic fiber by \cite[Proposition~4.3.9]{kinglouis}.
In order to show that the special fiber is connected, we apply Zariski's Connectedness Principle \cite[Theorem~5.3.15]{kinglouis}. This amounts to showing that the structure sheaf $\mathscr{O}_{\mathcal{M}_\mathcal{F}(\Gamma)}$ evaluates to $K$ on the generic fiber and to $R$ globally. But $\mathscr{O}_{\mathcal{M}_\mathcal{F}(\Gamma)}(\mathcal{M}_\mathcal{F}(\Gamma)_\eta) = \mathscr{O}_{\mathcal{F}(V)}(\mathcal{F}(V))$, which equals $K$ by \cite[Corollary~3.3.21]{kinglouis}, whence the former claim. For the latter claim, note that $\mathscr{O}_{\mathcal{M}_\mathcal{F}(\Gamma)}(\mathcal{M}_\mathcal{F}(\Gamma))$ is a finitely generated $R$-module by \cite[Theorem~5.3.2\,(a)]{kinglouis}, and it is contained in $\mathscr{O}_{\mathcal{F}(V)}(\mathcal{F}(V)) = K$, since $\mathcal{M}_\mathcal{F}(\Gamma)$ is integral, and therefore equal to $R$, since $R$ is integrally closed.
Equidimensionality of the special fiber follows from \cite[Corollaire~14.2.2]{EGAIV3}.
\end{Proof}
\begin{Remark}\label{rem:reduced}
It has been proved in \cite[Theorem~2.3]{mustafinvarieties} that for $\mathcal{F} = \mathbb{P}$, the special fiber of a Mustafin degeneration is always reduced. The methods used there do not readily apply to arbitrary flag varieties. However, since any Mustafin degeneration is a degeneration of the diagonal in $\mathcal{F}(V)^n$, up to a change of coordinates, the special fiber is generically reduced as long as this diagonal is \emph{multiplicity-free} as defined, e.g., in the introduction of \cite{brion-multfree}. This is a condition on $\mathcal{F}$, $d = \dim V$, and $n = |\Gamma|$. \cref{tab:multiplicity-free} lists some instances where this is the case; one example where it is \emph{not} the case is $\mathcal{F} = \calG_2$, $d = 4$ and $n = 4$.
\begin{table}
\centering
\begin{tabular}{lll}
\toprule
Flag type $\mathcal{F}$ & $d = \dim V$ & $n = |\Gamma|$ \tabularnewline
\midrule
arbitrary & arbitrary & $2$ \tabularnewline
$\mathbb{P}$ & arbitrary & arbitrary \tabularnewline
$(1 < 2)$ & $3$ & arbitrary \tabularnewline
\bottomrule
\end{tabular}
\caption{Some cases where the diagonal in $\mathcal{F}(V)^n$ is multiplicity-free.}
\label{tab:multiplicity-free}
\end{table}
\end{Remark}
\section{Components of the Special Fiber}\label{sec:combinatorics}
If $\Gamma \subseteq \Gamma'$ are finite subsets of $\mathfrak{B}_d^0$, then the following commutative diagram gives a natural surjective morphism $\mathcal{M}_\mathcal{F}(\Gamma') \twoheadrightarrow \mathcal{M}_\mathcal{F}(\Gamma)$:
\[ \xymatrix{& \mathcal{M}_\mathcal{F}(\Gamma') \ar@{^(->}[r] \ar@{-->>}[dd] & \prod_{[L] \in \Gamma'} \mathcal{F}(L) \ar@{->>}[dd] \\
\mathcal{F}(V) \ar@{^(->}[ur] \ar@{^(->}[dr] \\
& \mathcal{M}_\mathcal{F}(\Gamma) \ar@{^(->}[r] & \prod_{[L] \in \Gamma} \mathcal{F}(L) } \]
The existence of the dashed morphism follows from the fact that under the projection $\prod_{[L] \in \Gamma'} \mathcal{F}(L) \twoheadrightarrow \prod_{[L] \in \Gamma} \mathcal{F}(L)$, the image of $\mathcal{F}(V)$ in the former product is mapped to the image of $\mathcal{F}(V)$ in the latter product, and hence, by continuity, its closure in $\prod_{[L] \in \Gamma'} \mathcal{F}(L)$ is mapped to its closure in $\prod_{[L] \in \Gamma} \mathcal{F}(L)$. Surjectivity follows from the fact that the morphism is dominant, since its image contains the generic fiber, and projective, hence closed.
From now on, denote by $X_s$ the special fiber of a scheme $X$ over $\Spec R$.
\begin{Lemma}\label{lem:birational}
Let $\Gamma \subseteq \Gamma'$ be finite subsets of $\mathfrak{B}_d^0$. For each irreducible component $C$ of $\mathcal{M}_\mathcal{F}(\Gamma)_s$, there is a unique irreducible component $C'$ of $\mathcal{M}_\mathcal{F}(\Gamma')_s$ that maps onto $C$ via the natural projection $\mathcal{M}_\mathcal{F}(\Gamma') \twoheadrightarrow \mathcal{M}_\mathcal{F}(\Gamma)$. Furthermore, the map $C' \twoheadrightarrow C$ is birational.
\end{Lemma}
\begin{Proof}
First, let us note that the special fiber of $\mathcal{M}_\mathcal{F}(\Gamma)$ has codimension $1$, because it corresponds to the prime ideal $\langle \pi \rangle$ for a uniformizer $\pi$ of $R$, and the only prime ideal contained in $\langle \pi \rangle$ is $\langle 0 \rangle$. By \cite[Corollary~4.4.3(b)]{kinglouis}, there is a non-empty open subset $V \subseteq \mathcal{M}_\mathcal{F}(\Gamma)$ such that the natural projection $\mathcal{M}_\mathcal{F}(\Gamma') \twoheadrightarrow \mathcal{M}_\mathcal{F}(\Gamma)$ is an isomorphism over $V$, and $\mathcal{M}_\mathcal{F}(\Gamma) \setminus V$ has codimension $\geq 2$. Therefore, $\mathcal{M}_\mathcal{F}(\Gamma) \setminus V$ has codimension $1$ in the special fiber. The claim follows from this.
\end{Proof}
\begin{Definition}
If $\Gamma$ is a finite subset of $\mathfrak{B}_d^0$, then, for any $[L] \in \Gamma$, we call the unique component of $\mathcal{M}_\mathcal{F}(\Gamma)_s$ that maps birationally to $\mathcal{M}_\mathcal{F}(L)_s = \mathcal{F}(L)_s$ the \emph{primary component of $\mathcal{M}_\mathcal{F}(\Gamma)_s$ corresponding to $[L]$ (or $L$)}\index{component!primary}, or simply \emph{$L$-primary}. An irreducible component $C$ of $\mathcal{M}_\mathcal{F}(\Gamma)_s$ not mapping birationally to any $\mathcal{M}_\mathcal{F}(L)_s$ with $[L] \in \Gamma$ is called a \emph{secondary component} if there is a finite subset $\Gamma'$ of $\mathfrak{B}_d^0$ containing $\Gamma$, such that some primary component of $\mathcal{M}_\mathcal{F}(\Gamma')_s$ is mapped birationally to $C$. Any other irreducible component of $\mathcal{M}_\mathcal{F}(\Gamma)_s$ is called \emph{tertiary}.\index{component!tertiary}
\end{Definition}
In \cite[Lemma~5.8]{mustafinvarieties}, we showed that for $\mathcal{F}=\mathbb{P}$, all components are primary or secondary. In what follows, we will show that tertiary components can appear for more general flag types.
\begin{Lemma}[Flag Projection Lemma]\label{lem:flag-projection}\index{flag projection}
Let $\mathcal{F}' \subseteq \mathcal{F}$ be flag types and $\Gamma \subset \mathfrak{B}_d^0$ a finite subset. There is a natural surjective morphism $\mathcal{M}_\mathcal{F}(\Gamma) \twoheadrightarrow \mathcal{M}_{\mathcal{F}'}(\Gamma)$, under which, for any $[L] \in \Gamma$, the primary component of $\mathcal{M}_\mathcal{F}(\Gamma)_s$ corresponding to $L$ is mapped onto the primary component of $\mathcal{M}_{\mathcal{F}'}(\Gamma)_s$ corresponding to $L$.
\end{Lemma}
\begin{Proof}
Since under the natural projection $\prod_{[L] \in \Gamma} \mathcal{F}(L) \twoheadrightarrow \prod_{[L] \in \Gamma} \mathcal{F}'(L)$, the diagonal of the generic fiber on the left is mapped onto the diagonal of the generic fiber on the right, we obtain the desired natural surjective morphism $\mathcal{M}_\mathcal{F}(\Gamma) \twoheadrightarrow \mathcal{M}_{\mathcal{F}'}(\Gamma)$.
Now consider the following cube of morphisms:
\[\xymatrix{
& C \ar@{~}[dd] |\hole \ar@{^(->}[dl] \ar @ {.>>} [rr] & & C' \ar@{~}[dd] \ar@{^(->}[dl] \\
\mathcal{M}_\mathcal{F}(\Gamma)_s \ar@{^(->}[dd] \ar@{-->} [rr] && \mathcal{M}_{\mathcal{F}'}(\Gamma)_s \ar@{^(->}[dd] \\
& \mathcal{F}(L)_s \ar@{->>} [rr] |(.49)\hole && \mathcal{F}'(L)_s \\
\prod\limits_{[L] \in \Gamma} \mathcal{F}(L)_s \ar@{->>}[rr] \ar@{->>}[ur] && \prod\limits_{[L] \in \Gamma} \mathcal{F}'(L)_s \ar@{->>}[ur]
}\]
$C$ and $C'$ are the primary components corresponding to $L$ in $\mathcal{M}_\mathcal{F}(\Gamma)_s$ and $\mathcal{M}_{\mathcal{F}'}(\Gamma)_s$, respectively, the squiggly lines $\xymatrix{\ar@{~}[r]&}$ indicate birationality, and the left and the right face of the cube are commutative by definition; the bottom face consists of natural projections and is trivially commutative; the dashed arrow $\xymatrix{\ar@{-->}[r]&}$ (making the front face commutative) is the special fiber of the natural projection from above; and from this follows the existence of the dotted arrow $\xymatrix{\ar@{.>>}[r]&}$ making everything commutative, which is dominant because the other arrows on the back face are, and therefore is surjective.
\end{Proof}
We refer to this type of projection as \emph{flag projection}, as opposed to the projection defined at the beginning of this section stemming from an inclusion of vertex sets $\Gamma \subseteq \Gamma'$.
\begin{Lemma}\label{lem:unique-primaries}
Let $\Gamma \subset \mathfrak{B}_d^0$ be finite, $[L_1], [L_2] \in \Gamma$ two distinct vertices, and $C_1, C_2$ the corresponding primary components in $\mathcal{M}_\mathcal{F}(\Gamma)_s$. Then $C_1 \neq C_2$.
\end{Lemma}
\begin{Proof}
Both projections $\mathcal{M}_\mathcal{F}(\Gamma) \twoheadrightarrow \mathcal{M}_\mathcal{F}(L_1)$ and $\mathcal{M}_\mathcal{F}(\Gamma) \twoheadrightarrow \mathcal{M}_\mathcal{F}(L_2)$ factor through $\mathcal{M}_\mathcal{F}(\Gamma) \twoheadrightarrow \mathcal{M}_\mathcal{F}(L_1, L_2)$, so it suffices to show that the images of $C_1$ and $C_2$ in $\mathcal{M}_\mathcal{F}(L_1, L_2)$ are different. So assume $\Gamma = \{ [L_1], [L_2] \}$.
Now if $\mathcal{F} = \calG_k$ for some $k$, an explicit calculation using the equations given in \cref{sec:definitions} shows that the (unique) components mapping birationally to $\mathcal{M}_{\calG_k}(L_1)$ and $\mathcal{M}_{\calG_k}(L_2)$, respectively, are distinct.
But if, for general $\mathcal{F}$, we had $C_1 = C_2$,
then by the preceeding Flag Projection Lemma, the same would be true in $\mathcal{M}_{\calG_k}(L_1, L_2)$ for any $\calG_k \subset \mathcal{F}$, which we just ruled out.
\end{Proof}
\begin{Lemma}
Let $\Gamma \subset \mathfrak{B}_d^0$ be finite, $C \subset \mathcal{M}_\mathcal{F}(\Gamma)_s$ a secondary irreducible component, and let $\Gamma'$ and $\Gamma''$ be two finite subsets of $\mathfrak{B}_d^0$ such that $C$ becomes primary in both $\mathcal{M}_\mathcal{F}(\Gamma')_s$ and $\mathcal{M}_\mathcal{F}(\Gamma'')_s$. Explicitly, this means that the unique irreducible components $C'$ and $C''$ of $\mathcal{M}_\mathcal{F}(\Gamma')_s$ and $\mathcal{M}_\mathcal{F}(\Gamma'')_s$, respectively, which map birationally to $C$, are $L'$-primary and $L''$-primary, respectively, for some $[L'] \in \Gamma'$ and $[L''] \in \Gamma''$.
Then we have $[L'] = [L'']$.
\end{Lemma}
\begin{Proof}
Let $\tilde \Gamma = \Gamma' \cup \Gamma''$, and let $\tilde C'$ and $\tilde C''$ be the components of $\mathcal{M}_\mathcal{F}(\tilde \Gamma)_s$ mapping to $C'$ and $C''$ under the respective projections. We thus have the following diagram, squiggly lines $\xymatrix{\ar@{~}[r]&}$ indicating birationality:
\[ \xymatrix{
\tilde C'\; \ar@{^(->}[rr] \ar@{~}[ddd] && \mathcal{M}_\mathcal{F}(\tilde \Gamma) \ar@{->>}[dl] \ar@{->>}[dr] \ar@{->>}[dd] && \; \tilde C'' \ar@{_(->}[ll] \ar@{~}[ddd] \\
& \mathcal{M}_\mathcal{F}(\Gamma') \ar@{->>}[d] \ar@{->>}[dr] & & \mathcal{M}_\mathcal{F}(\Gamma'') \ar@{->>}[d] \ar@{->>}[dl] & \\
& \mathcal{M}_\mathcal{F}(L') & \mathcal{M}_\mathcal{F}(\Gamma) & \mathcal{M}_\mathcal{F}(L'') & \\
\bigl.C'\bigr. \ar@{~}[rr] \ar@/^/@{_(->}[uur] \ar@{ ~}[ur] &*{}& \; \bigl.C\bigr. \; \ar@<-1.5pt>@{^ (->}[u] && \bigl.C''\bigr. \ar@/_/@{^(-<}[luu] \ar@{~}[ul] \ar@{~}[ll] &
}\]
Now $\tilde C'$ and $\tilde C''$ both map birationally to $C$, so they are equal. But since they are primary components of $\mathcal{M}_\mathcal{F}(\tilde \Gamma)_s$ corresponding to $[L']$ and $[L'']$ respectively, $[L']$ and $[L'']$ must be equal.
\end{Proof}
This allows us to make the following definition:
\begin{Definition}
Let $\Gamma \subset \mathfrak{B}_d^0$ be finite, und $C$ a secondary component of $\mathcal{M}_\mathcal{F}(\Gamma)_s$. We call $C$ \emph{the secondary component corresponding to $[L]$ (or $L$)}, or simply \emph{$L$-secondary}, if $[L]$ is the vertex in $\mathfrak{B}_d^0$ uniquely determined by the property that the $L$-primary component of $\mathcal{M}_\mathcal{F}(\Gamma \cup \{[L]\})$ maps birationally to $C$.
\end{Definition}
We have seen in \cref{lem:flag-projection} that $L$-primarity is preserved under flag projections. The same is true for $L$-secondarity, but only under an additional precondition:
\begin{Proposition}\label{prop:sec-comps-under-flag-projection}
Let $\Gamma \subset \mathfrak{B}_d^0$ be finite, $\mathcal{F}' \subseteq \mathcal{F}$ flag types. Let $C$ be a secondary component of $\mathcal{M}_\mathcal{F}(\Gamma)_s$ corresponding to $[L] \in \mathfrak{B}_d^0$. Then $C$ is mapped onto an irreducible component $C'$ of $\mathcal{M}_{\mathcal{F}'}(\Gamma)_s$ by the flag projection if and only if there is a secondary component in $\mathcal{M}_{\mathcal{F}'}(\Gamma)_s$ corresponding to $[L]$. In this case, $C'$ is mapped to this $L$-secondary component.
\end{Proposition}
\begin{Proof}
Let $\tilde C$ be the $L$-primary component of $\mathcal{M}_\mathcal{F}(\Gamma \cup \{L\})_s$, which is mapped birationally to $C$ under the projection $\mathcal{M}_\mathcal{F}(\Gamma \cup \{L\}) \twoheadrightarrow \mathcal{M}_\mathcal{F}(\Gamma)$. By \cref{lem:flag-projection}, $\tilde C$ is mapped onto the $L$-primary component $\tilde C'$ of $\mathcal{M}_{\mathcal{F}'}(\Gamma \cup \{L\})_s$. By commutativity of the diagram
\[ \xymatrix{
\mathcal{M}_\mathcal{F}(\Gamma \cup \{L\}) \ar@{->>}[r] \ar@{->>}[d] & \mathcal{M}_{\mathcal{F}'}(\Gamma \cup \{L\}) \ar@{->>}[d] \\
\mathcal{M}_\mathcal{F}(\Gamma) \ar@{->>}[r] & \mathcal{M}_{\mathcal{F}'}(\Gamma)\text{,}
}\]
$\tilde C'$ is mapped onto a component $C'$ of $\mathcal{M}_{\mathcal{F}'}(\Gamma)_s$ if and only if $C$ is. Now if there is an $L$-secondary component in $\mathcal{M}_{\mathcal{F}'}(\Gamma)_s$, it is unique and $\tilde C'$ must be mapped onto it. Conversely, if $\tilde C'$ is mapped onto a component $C'$, it must be birationally by \cref{lem:birational}, making $C'$ an $L$-secondary component.
\end{Proof}
\begin{Definition}
An irreducible component $C$ in the special fiber of a Mustafin degeneration $\mathcal{M}_\mathcal{F}(\Gamma)$ is called \emph{mixed} if there are flag types $\mathcal{F}_1, \mathcal{F}_2 \subseteq \mathcal{F}$ and vertices $[L_1], [L_2] \in \mathfrak{B}_d^0$, $[L_1] \neq [L_2]$, such that $C$ is mapped onto a (primary or secondary) component corresponding to $L_i$ under the flag projection to $\mathcal{M}_{\mathcal{F}_i}(\Gamma)$, for $i \in \{1,2\}$.
\end{Definition}
\cref{lem:flag-projection,prop:sec-comps-under-flag-projection} together imply that every mixed component is tertiary.
The condition that $C$ is mapped onto an irreducible component in \cref{prop:sec-comps-under-flag-projection} is not automatic. In fact, the following example shows that there might be no secondary components at all in $\mathcal{M}_{\mathcal{F}'}(\Gamma)_s$ for $C$ to be mapped to!
\begin{Example}[Secondary and mixed components]\label{ex:mixed-component}
Take $R = \mathbb{Q}[t]_{(t)}$ with field of fractions $K = \mathbb{Q}(t)$ and residue field $\mathbb{Q}$, $d = 3$, and $\mathcal{F} = ( 1 < 2 )$. For a basis $\{e_1, e_2, e_3\}$ of $V$, let
\begin{align*}
L_1 &= Re_1 + Re_2 + Re_3\text{,} \\
L_2 &= R\pi e_1 + Re_2 + Re_3\text{,} \\
L_3 &= Re_1 + Re_2 + R\pi e_3\text{,} \\
\end{align*}
and $\Gamma = \{[L_1], [L_2], [L_3]\}$. By an explicit calculation, using the equations from \cref{sec:definitions}, we find that $\mathcal{M}_\mathcal{F}(\Gamma)_s$ possesses one secondary component $C$ corresponding to $L_4 = R\pi e_1 + Re_2 + R\pi e_3$, four mixed components and (of course) three primary components.
In this example, the non-primary components are all toric threefolds: the secondary component $C$ is isomorphic to $(\mathbb{P}^1_\mathbb{Q})^3$, two of the mixed components are isomorphic to $\mathbb{P}^1_\mathbb{Q} \times \widetilde \mathbb{P}^2_\mathbb{Q}$, where $\widetilde\mathbb{P}^2_\mathbb{Q}$ is the blow-up of of $\mathbb{P}^2_\mathbb{Q}$ in a point, and the remaining two mixed components are mutually isomorphic non-singular toric threefolds corresponding to a fan whose intersection with the unit sphere results in a pentagonal bipyramid (with $f$-vector $(7,15,10)$).
Note, however, that the whole Mustafin degeneration $\mathcal{M}_\mathcal{F}(\Gamma)$ is not toric since the primary components are not toric.
We can depict the configuration $\Gamma \cup \{[L_4]\}$ as follows in the apartment corresponding to the basis $\{e_1, e_2, e_3\}$, which is the subcomplex of $\mathfrak{B}_d$ spanned by all vertices of the form $[\sum_iR\pi^{a_i}e_i]$:
\begin{center}
\begin{tikzpicture}
\draw[step=1cm,gray,very thin] (-2.4,-.4) grid (2.4,1.9);
\begin{scope}
\clip (-2.4,-.4) rectangle (2.4,1.9);
\foreach \x in {-3,...,5} \draw[gray,very thin,rotate=45] (-10,\x*0.707106781) -- (10,\x*0.707106781);
\end{scope}
\filldraw (0,0) circle (3pt) node[anchor=north west] {$L_1$};
\filldraw (-1,0) circle (3pt) node[anchor=north east] {$L_2$};
\filldraw (1,1) circle (3pt) node[anchor=south west] {$L_3$};
\filldraw[fill=white] (0,1) circle (3pt) node[anchor=south east] {$L_4$};
\end{tikzpicture}
\end{center}
By \cite[Theorem~2.10]{mustafinvarieties}, $\mathcal{M}_{\mathbb{P}^\vee}(\Gamma)_s$ does not contain secondary components, since $\Gamma$ is convex in the sense of \cite[Theorem~2.10]{mustafinvarieties}. Therefore, $C$ must be mapped to a proper irreducible closed subset of an irreducible component of $\mathcal{M}_{\mathbb{P}^\vee}(\Gamma)_s$ under the flag projection $\mathcal{M}_\mathcal{F}(\Gamma) \twoheadrightarrow \mathcal{M}_{\mathbb{P}^\vee}(\Gamma)$. However, for the projective space $\mathbb{P}$, where the dual notion of convexity has to be applied, $\Gamma$ is not convex, but has $\Gamma \cup \{[L_4]\}$ as its convex hull, so $\mathcal{M}_{\mathbb{P}}(\Gamma)_s$ contains a secondary component correspondig to $L_4$, and $C$ is mapped onto this component.
The mixed components in this example are identified by the fact that they map to components (primary or secondary) corresponding to different vertices in $\mathcal{M}_{\mathbb{P}}(\Gamma)_s$ and $\mathcal{M}_{\mathbb{P}^\vee}(\Gamma)_s$, respectively.
\end{Example}
\begin{Example}
The example above shows another interesting phenomenon. If we remove $[L_1]$ from $\Gamma$ and consider $\Gamma' = \{[L_2], [L_3]\}$, rather than observing two secondary components corresponding to $L_1$ and $L_4$, as one might expect, we only get four mixed components (besides the obligatory primary components).
\end{Example}
We conclude this \lcnamecref{sec:combinatorics} by giving upper and lower bounds for the total numbers of components of a Mustafin degeneration.
\begin{Lemma}\label{lem:number-of-components}
Let $\mathcal{F} = (k_1 < \dots < k_r)$ be a flag type and $\Gamma \subset \mathfrak{B}_d^0$ finite with $|\Gamma| = n$. Denote by $c$ the number of irreducible components in the special fiber of $\mathcal{M}_\mathcal{F}(\Gamma)$. Then we have the lower bound $c \geq n$. If $n=2$, we also have the upper bound
\begin{align*} c &\leq \binom{d}{d-k_r, \; k_r-k_{r-1}, \; \ldots, \; k_2-k_1, \; k_1} \\
&= \text { the number of Schubert cells in $\mathcal{F}(V)$. }
\end{align*}
\end{Lemma}
\begin{Proof}
$c \geq n$ holds because there is one primary component for each vertex in $\Gamma$, and primary components corresponding to different vertices cannot coincide by \cref{lem:unique-primaries}.
In order to prove the upper bound for $n=2$, we consider the class of $\mathcal{M}_\mathcal{F}(\Gamma)_s$ in the Chow ring $A^\ast(\mathcal{F}(\kappa^d)^2)$, where $\kappa$ denotes the residue field of $R$.
Up to a change of coordinates, $\mathcal{M}_\mathcal{F}(\Gamma)_K$ is embedded in $\mathcal{F}(V)^2$ as the diagonal, so their classes in the Chow ring $A^\ast(\mathcal{F}(V)^2)$ are the same.
Since $\mathcal{M}_\mathcal{F}(\Gamma)_s$ is a specialization of $\mathcal{M}_\mathcal{F}(\Gamma)_K$, as in \cite[Section~20.3]{fulton}, they have the same class in $A^\ast(\mathcal{F}(\kappa^d)^2)$.
Now each irreducible component of $\mathcal{M}_\mathcal{F}(\Gamma)_s$ adds a term with non-negative coefficients, so there can at most be as many components as the sum of the coefficients in $[\Delta]$.
Since $n = 2$, this sum is exactly the number of Schubert varieties by the duality property of Schubert classes. That this number equals the stated multinomial coefficient follows from the definition of the Schubert varieties.
\end{Proof}
\begin{Question}
The upper bound in the $n=2$ case is sharp for $\mathcal{F} = \mathbb{P}$ by \cite[Proposition~4.6]{mustafinvarieties}. It is attained for vertices in \emph{general position} as defined before the cited proposition. We would be interested to know if this is also true for general flag types.
\end{Question}
\section{Introduction}
In this paper, we discuss certain degenerations of flag varieties, which we call \emph{Mustafin degenerations}.
By definition, a Mustafin degeneration is induced by a vertex configuration in the Bruhat-Tits building of the projective linear group over a discretely valued field $K$.
Since these vertices can be described by matrices with entries in $K$, our degenerations can also be thought of as being induced by sets of such matrices, and thus also arise naturally from the perspective of linear algebra. In addition, this point of view gives rise to explicit equations for Mustafin degenerations.
We give a brief account of the definition. Details can be found at the beginning of \cref{sec:definitions}. Fix a vector space $V$ of finite dimension over a discretely valued field $K$ with valuation ring $R$. A vertex in the Bruhat-Tits building is represented by a lattice $L$ in $V$, to which we can assign the flag variety $\mathcal{F}(L)$ parametrizing flags of a specified type in $L$. Given a finite set $\Gamma$ of vertices, represented by lattices $L_1, \ldots, L_n$, we define the Mustafin degeneration $\mathcal{M}_\mathcal{F}(\Gamma)$ as the join of the flag varieties $\mathcal{F}(L_1), \ldots, \mathcal{F}(L_n)$ along their common generic fiber $\mathcal{F}(V)$.
These schemes $\mathcal{M}_\mathcal{F}(\Gamma)$ are natural generalizations of similarly constructed degenerations of projective space, called \emph{Mustafin varieties} in \cite{mustafinvarieties}, which were introduced by Mumford in his influential work \cite{mumford} on the uniformization of curves, and later generalized by Mustafin \cite{mustafin} to higher dimensions.
In \cref{thm:basic-properties}, we show that Mustafin degenerations are integral schemes, flat and projective over $R$, with connected special fiber. In \cref{rem:reduced}, we give a sufficient condition for the special fiber to be reduced, and in \cref{lem:number-of-components} we prove an upper bound for the number of components of the special fiber in certain cases.
In \cite{mustafinvarieties}, we introduced a distinction of the irreducible components of the special fiber of a Mustafin variety into primary and secondary components, where primary components correspond to vertices in $\Gamma$ and secondary components correspond to vertices in the convex hull of $\Gamma$ that are not contained in $\Gamma$.
However, for arbitrary flag types, the situation is different, since there may be components that are neither primary nor secondary. We show how such so-called tertiary components appear by studying the behavior of the different kinds of components under a natural projection morphism between Mustafin degenerations, which arises from the inclusion of flags.
This work is organized as follows. In \cref{sec:definitions}, we present our framework and give the definition of a Mustafin degeneration, as well as the basic geometric properties.
\cref{sec:combinatorics} is devoted to the study of the components of the special fiber of a Mustafin degeneration.
|
{
"timestamp": "2013-03-08T02:02:04",
"yymm": "1203",
"arxiv_id": "1203.1733",
"language": "en",
"url": "https://arxiv.org/abs/1203.1733"
}
|
\section{Introduction}
\label{intro}
The detection of high energy cosmic neutrinos can help solve the
problem of the origin of high energy cosmic rays and be a new tool to
elucidate the mechanisms of hadronic acceleration in astrophysical
objects. In the low energy domain (few MeV to several GeV) the
observation of extraterrestrial and atmospheric neutrinos gave rise to
the discovery of neutrino oscillations and to one of the most direct
experimental tests of our models of supernova explosions. In the high
energy regime (several GeV to EeV), neutrinos have several advantages
as cosmic messengers and can provide information on the particle
acceleration mechanisms in the Universe. Experimental methods to
detect them exist and have been technologically proven. The major
challenge in the field of Neutrino Astronomy is at present to reach a
sensitivity high enough to detect the first cosmic neutrino sources.
Let us briefly summarize the advantages of neutrinos as cosmic
messengers. They are neutral particles, therefore they are not
deflected by magnetic fields and point back to their sources. They are
weakly interacting and thus can escape from very dense astrophysical
objects and travel long distances without being absorbed by matter or
background radiation. Moreover, in cosmic sites where hadrons are
accelerated, it is likely that neutrinos are generated in the decay of
charged pions produced in the interaction of those hadrons with the
surrounding matter or radiation, being therefore a smoking gun of
hadronic acceleration mechanisms.
The observation of neutrinos in a Cherenkov neutrino telescope is
based on the detection of the muons produced by the neutrino charged
current interactions with the matter surrounding the telescope by
means of the Cherenkov light induced by the muons when crossing the
detector medium, natural ice or water. Cascades produced in charged or
neutral current interactions of neutrinos inside or nearby the
detector can also be detected.
A typical neutrino telescope consists of a three dimensional array of
light sensors, photomultipliers (PMTs), that record the position and
time of the emitted Cherenkov photons, enabling the reconstruction of
the muon track or the cascade. To avoid the huge background of muons
produced by cosmic ray showers in the atmosphere, the telescopes look
at the other side of the Earth, i.e. they use it as a shield against
the muons produced in normal atmospheric showers. The increase in the
range of muons in the rock at high energies (from kilometres to
several kilometres) together with the increase of the neutrino cross
section gives rise to an approximately exponential increase of the
effective areas of these devices in the GeV to PeV energy range. Above
a few TeV the telescopes can determine the direction of the incoming
neutrinos with angular resolutions better than 1$^\circ$, hence the
name ``telescope''. At energies above the PeV, the Earth becomes
opaque to neutrinos, but the atmospheric muon flux decreases
dramatically so that the neutrino telescopes can look for downgoing
neutrinos. Other neutrino flavours can be observed through the
detection of hadronic or electromagnetic showers or, in the case of
tau neutrinos, via the observation of its interaction and the
subsequent decay of the produced tau lepton.
The first attempt to build a neutrino telescope in natural water,
namely the DUMAND project, dates back to the 60's~\cite{dumand}.
DUMAND paved the way for subsequent projects. NT200 in Lake Baikal
and then ANTARES in the Mediterranean Sea benefited from the
experience of DUMAND. We will cover in this article these two last
experiments which are the first underwater neutrino telescopes ever
built.
The first efforts to install a neutrino telescope in Lake Baikal
started in the 80's~\cite{baifirst}. After some site tests, the first
single string arrays were operated between 1984 and 1990. Already
since 1987 the construction of a telescope with 200~PMTs was
envisaged. Between 1993 and 1994, the so-called NT-36 version of this
telescope with 36 PMTs was operated. Since then the detector has been
growing gradually: NT-72 (1995-1996), NT-96 (1996-1997), NT-144
(1997-1998) and NT-200 (since 1998). In 2005 three outer strings were
added to form the so-called NT-200+.
The Baikal neutrino telescope NT-200 is located in Lake Baikal at a
latitude of around 52$^{\circ}$ North~\cite{bainim}. The detector is
around 3.6~km from the shore and at a depth between 1115~m and
1185~m. The NT200 configuration is composed of 8 strings that are held
by an umbrella shaped mechanical structure (see Fig.~\ref{nt200}).
Each string has 24 optical modules (OMs) arranged in pairs adding up
to a total of 192 OMs. Each OM contains a 37-cm diameter QUASAR
photomultiplier~\cite{quasar} specifically designed for this
detector. The two photomultipliers (PMTs) of a pair are operated in
coincidence in order to supress background from bioluminiscence and
PMT noise. The upgraded version NT200+ includes three new lines that
surround the old NT200 detector and are located at 100~m from its
centre, thereby increasing its sensitivity by a factor four for very
high energy cosmic neutrinos.
The ANTARES neutrino telescope is located 40~km offshore from Toulon
at 2475~m depth at a latitude around 43$^{\circ}$
North~\cite{antaresdet}. It consists of 12 mooring lines anchored to
the sea bed and held taut by means of buoys (see Fig.~\ref{antares}).
\begin{figure}[bh]
\begin{center}
\includegraphics[scale=0.38]{nt200.eps}
\end{center}
\caption{\label{nt200} Schematic view of the Baikal Neutrino Telescope}
\end{figure}
\noindent
Each line contains 25 storeys. The lowest storey is 100~m above the
sea bed and the vertical distance between consecutive storeys is
14.5~m. The total line length is 480~m. Each storey has a triplet of
OMs and an electronics module. The OMs contain a 10-inch
photomultiplier looking 45$^\circ$ downwards~\cite{om, pmt}. In
addition, several optical beacons~\cite{obs} are distributed
throughout the lines for calibration purposes~\cite{timecalib}. The
horizontal separation between lines is between 60 and 80~m. Each line
is connected to a junction box by means of interlink cables and the
junction box is connected to the shore by the main electro-optical
cable.
The ANTARES initiative started in 1998 and after a period of site
evaluation, detector design, tests and construction, the first line
was deployed in 2006. The detector was operated with 5-lines during
several months in 2007 and was fully deployed in 2008. Taking
advantage of the possibility of detector maintenance offered by water,
the ANTARES collaboration has recovered and repaired some of the lines
and fixed problems in some of the detector's interlink cables.
\begin{figure}[t]
\begin{center}
\includegraphics[scale=0.42]{antares2.eps}
\end{center}
\caption{\label{antares} Schematic view of the Antares Neutrino Telescope}
\end{figure}
\noindent
\section{Search for a diffuse flux of cosmic neutrinos}
The term diffuse flux refers to the search of cosmic neutrinos without
requiring precise directional information. An excess over the
expected atmospheric neutrino background is looked for and if none is
found limits are customarily set on the normalization of signal fluxes
with energy spectra of the type $E^{-2}$.
The Baikal collaboration has performed several searches for diffuse
fluxes of cosmic neutrinos~\cite{baidiffuse1,baidiffuse2,
baidiffuse3}. They looked for cascades produced both in charged and
neutral current interactions of neutrinos in the medium surrounding
the detector. Initial cuts were applied to the energy of the
reconstructed cascades in order to select neutrino events. The number
of upward going cascades detected agrees with those expected from
background. A cut on energy of 10~TeV and 130~TeV for upgoing and
downgoing cascades, respectively, was then introduced to select the
final neutrino signal. No events were observed for an expected
background of around 2 events. From this lack of signal an upper
limit of E$^{-2}\Phi < 2 \times 10^{-7}$ GeV cm$^{-2}$ s$^{-1}$
sr$^{-1}$ was set for the flux of all flavours of neutrinos of cosmic
origin in the energy interval 20~TeV $<$ E$_{\nu}$ $<$ 20~PeV. As
usual, a flavour ratio $\nu_{e}$:$\nu_{\mu}$:$\nu_{\tau}$ = 1:1:1 was
assumed.
Using the data collected by ANTARES during the period from December
2007 to December 2009, corresponding to a total live time of 334 days
with different detector configurations (9, 10 and 12 lines), a search
for a diffuse flux of astrophysical muon neutrinos was
performed~\cite{antdiffuse}. In addition to the cuts on the quality of
the reconstructed track and on the number of hits, an energy cut was
also applied. A novel technique based on the repetition rate of
photoelectrons on a given PMT averaged over all the PMTs was used.
This variable is a good proxy of the energy of the track and is well
described by the Monte Carlo simulation. After unblinding of the data,
the number of events above the optimized cut in repetition rate was
found to agree with background expectations. From the compatibility
of the observed number of events with the expected background and
assuming an E$^{-2}$ flux spectrum for the signal, a 90\% C.L. upper
limit on the $\nu_{\mu }+\bar{\nu}_{\mu }$ diffuse flux of E$^2 \,
\Phi <$ 5.3 $\times$10$^{-8}$ GeV cm$^{-2}$ s$^{-1}$ in the energy
range 20~TeV to 2.5~PeV was obtained.
The 90\% C.L. upper limits on a diffuse flux of muon neutrinos (plus
antineutrinos) from several experiments~\cite{otherdiffuse} are given
in Fig.~\ref{antdiffuse}. The original limits given by BAIKAL NT-200
and Amanda-II UHE are for all flavours and are divided by 3 in this
plot for the sake of comparison. The recent limit on the diffuse flux
of astrophysical $\nu_{\mu}$ from the IceCube experiment, E$^2 \, \Phi
<$ 8.9 $\times$10$^{-9}$ GeV cm$^{-2}$ s$^{-1}$~\cite{icediffuse}, is
not shown in this figure. The grey band represents the expected
variation of the atmospheric $\nu_{\mu}$ flux: the minimum is the
Bartol flux from the vertical direction and the maximum is the
Bartol+RQPM flux from the horizontal direction. The central line is
averaged over all directions. The phenomenological upper bounds of
W\&B and MPR~\cite{WB} are also given, dividing by 2 the original
values in order to take into account neutrino oscillations.
\begin{figure}[hbt]
\begin{center}
\includegraphics[scale=0.85]{antdiffuse}
\end{center}
\caption{90\% C.L. upper limits for an E$^{-2}$ high energy cosmic
neutrino diffuse flux from several experiments(see text for
explanations). }
\label{antdiffuse}
\end{figure}
\section{Search for point-like sources}
\label{pointsources}
A search for point sources was performed by ANTARES using the data
taken from 2007 to 2010. After the selection of data runs requiring
that most of the detector was operating and that the optical
background from bioluminiscence was low, the final data sample
amounted to a total of 813 live days. Only events with upgoing muons
were kept for further analysis, requiring in addition that the
corresponding track had a good reconstruction quality and an estimated
angular error lower than 1$^{\circ}$. The cut in quality was chosen so
as to optimize the discovery potential. A total of 3058 events were
selected. According to Monte Carlo simulations around 15\% of them
were atmospheric muons wrongly reconstructed as upgoing tracks.
Clusters of events with a large enough significance above that
expected from background fluctuations were looked for with a likehood
ratio method. The likehood used the distribution in declination of
the atmospheric background obtained by scrambling the data in right
ascension and an angular resolution of (0.46$\pm$0.15)$^{\circ}$, as
given by Monte Carlo simulation. The full sky was searched for
possible sources and then a list of 51 pre-selected directions in the
sky corresponding to possible astrophysical neutrino sources were
scrutinized. No significant excess was found in either case. An
alternative search method~\cite{em} was used as a cross-check
obtaining similar results.
In Fig.~\ref{skymap} the direction in Galactic coordinates of all the
selected tracks are shown as (blue) dots. The hue of the yellow
background of the figure indicates the percentage of visibility of the
corresponding region of the sky, white corresponds to no visibility
and dark yellow to 100\%.
\begin{figure}[htb]
\begin{center}
\includegraphics[scale=0.45]{skymap.eps}
\end{center}
\caption{\label{skymap} Skymap in Galactic coordinates. The grade in
the hue of the (yellow) background indicates the visibility of the
corresponding region according to the scale on the right (white:
0\%; darkest yellow: 100\%). Blue dots: position in the sky of the
3058 selected neutrinos candidates. Red stars: position of the 51
pre-selected sources. Red ellipse: the most significant cluster of
events.}
\end{figure}
\begin{figure}[htb]
\begin{center}
\includegraphics[scale=0.45]{limits.eps}
\end{center}
\caption{\label{pslimits} 90\% C.L. upper limits for a neutrino flux
with an E$^{-2}$ spectrum for 51 candidates sources (blue points)
and the corresponding sensitivity (dashed blue curve). Results from
the MACRO, Amanda~II, Super Kamiokande and IceCube
telescopes~\cite{psothers} are also shown. }
\end{figure}
The most significant cluster in the full-sky search was found at
$\alpha = -46.5^{\circ}$ and $\delta= -65.0^{\circ}$ and is indicated
in Fig.~\ref{skymap} by a (red) ellipse. The post-trial $p$-value for
this cluster was 2.5\%, a value not significant enough to claim a
signal.
The (red) stars in the figure correspond to the sky position of the 51
pre-selected sources for which the dedicated candidate search was
carried out. The most significant source of the predefined list (HESS
J1023-575) was fully compatible with a background fluctuation
($p$=41\%). The corresponding limits for neutrino sources emitting
with an $E^{-2}$ energy spectrum are given in Fig.~\ref{pslimits}.
This limit is 2.5 times better than the one previously
published~\cite{psprevious}. Limits from other experiments are also
given~\cite{psothers}. As can be seen, these results are at present
the most stringent for the Southern Sky, except for the case of the
IceCube detector for which in this hemisphere very high energy
neutrinos can be looked for (E$>$ 1~PeV). Note that, even though
neutrino sources in the Galaxy with a UHE component are not
discarded~\cite{pevatrons}, for the more plausible Galactic neutrino
sources (e.g. young SNRs) most of the neutrino signal is expected to
lie below a few hundred TeV~\cite{galneut, galneutb}.
\section{Multimessenger searches}
The search of neutrinos in coincidence with other messengers has
several advantages. Sources already known to have high-energy
emission, e.g. gamma-rays, can be investigated, increasing the chance
to observe sites of hadronic acceleration. In addition, the
restriction of the search to limited time windows and sky directions
highly reduces the atmospheric neutrino background and therefore
increases the sensitivity to possible signals, so that a handful of
events can be enough to claim a signal. In the case of neutrino events
coincident with gravitational waves, the same astrophysical phenomena
are expected to produce both types of signals. We give below a couple
of examples of the multimessenger program, which is too broad to be
fully reported here.
A selection of flares from blazars observed by the LAT detector of the
Fermi satellite during 2008 was carried out and the data taken by
ANTARES in the same period was investigated for neutrino coincidences
with the flaring period of the blazars~\cite{blazars}. The selected
blazars are shown in Table~\ref{tab:blazars} together with the number
of events required to claim a 5$\sigma$ signal. Only one event
--during a flare of 3C279-- was detected. The post-trial $p$-value of
such a coincidence is 10\%, compatible with a background
fluctuation. The 90\% C.L. limits on the neutrino fluence from these
blazars are given in Table~\ref{tab:blazars} for the 100~GeV to 1~PeV
region and assuming an $E^{-2}$ spectrum.
\begin{table}[ht]
\begin{center}
\begin{tabular}{|l|c|c|}
\hline
Source & N($5\sigma$) & Fluence\\ \hline \hline
PKS0208-512 & 4.5 & 2.8 \\ \hline
AO0235+164 & 4.3 & 18.7 \\ \hline
PKS1510-089 & 3.8 & 2.8 \\ \hline
3C273 & 2.5 & 1.1 \\ \hline
3C279 & 5.0 & 8.2 \\ \hline
3C454.3 & 4.4 & 23.5 \\ \hline
OJ287 & 3.9 & 3.4 \\ \hline
PKS0454-234 & 3.3 & 2.9 \\ \hline
WComae & 3.8 & 3.6 \\ \hline
PKS2155-304 & 3.7 & 1.6 \\ \hline
\end{tabular}
\caption{List of blazars for which neutrinos were looked for in
coincidence with their flares. {\it N(5$\sigma$)} is the average
number of events required for a 5$\sigma$ discovery (50\%
probability) and {\it Fluence} is the upper limit (90\% C.L.) on the
neutrino fluence in GeV$\cdot$cm$^{-2}$.}
\label{tab:blazars}
\end{center}
\end{table}
Several models predict the production of high energy neutrinos during
gamma-ray bursts. As in the previous analysis, restricting the search
to a short time window sizeably reduces the atmospheric background so
that only a few events would be enough to claim a discovery. Using the
2007 ANTARES data, a search for neutrinos coming from 40 GRBs events
was performed. No neutrino event was found in the corresponding time
windows and within the defined search cone around each source. The
limits obtained from this lack of signal are shown in Fig.~\ref{grbs},
where the 90\% C.L. limits on the total fluence of the 40 GRBs are
shown as a function of the neutrino energy for three different energy
spectra.
\begin{figure}[t]
\begin{center}
\includegraphics[scale=0.60]{antgrbs2.eps}
\end{center}
\caption{\label{grbs} 90\% C.L. limits on the total muon neutrino
fluence from the selected 40~GRBs as a function of the neutrino
energy for three different energy spectra: E$^{-2}$ (thick solid
line), Waxman and Bahcall GRB model~\cite{grbwb} and Guetta et
al.~\cite{grbguet}. The black dashed line is the expected neutrino
fluence for 40 GRBs with the Waxman and Bahcall energy spectrum and
the grey dashed line is the sum of the 40 expected individual GRB
fluences according to Guetta et al. The total prompt emission
duration of the 40 GRBs is 2114~s. The grey dash--dotted line is the
90\% C.L. limit set by IceCube using 117 GRBs~\cite{grbice}. }
\end{figure}
The Baikal collaboration has also performed a search for neutrinos
associated to 303 GRBs alerts provided by the BATSE detector from 1998
to 2000~\cite{grbbai}. From the absence of neutrino events in
coincidence with the GRBs a 90\% C.L. limit on the neutrino flux of
E$^{-2} \Phi_{\nu} <$ 1.1 $\times$ 10$^{-6}$ GeV cm$^{-2}$ s$^{-1}$
was set for a Waxman and Bahcall--type energy spectrum~\cite{grbwb}.
\section{Indirect search for dark matter}
If dark matter is made up of weakly interacting massive particles
(WIMPs), some of these will slow down by elastic scattering and end up
gravitationally trapped in heavy astrophysical objects such as the
centre of the Galaxy, the Sun or the Earth. They will then
self-annihilate and high energy neutrinos will be emitted in the decay
chain of their products.
ANTARES is analysing the data taken during 2007 and 2008, a total live
time of around 300 days, looking for high energy neutrinos coming from
the Sun. The analysis is based on the optimization of the cuts based
on the reconstruction quality of the muon tracks and the size of the
half-cone angle around the Sun direction to select neutrino
candidates. The sensitivity, i.e. the expected average 90\%
C.L. upper limit, for the muon flux coming from the Sun is shown in
Fig.~\ref{wimps} for the CMSSM framework. As expected the more
stringent limits will come from the hard channels, i.e those that
produce $W^+ W^-$ or $\tau^+ \tau^-$ in the annihilation process.
\begin{figure}[htb]
\begin{center}
\includegraphics[scale=0.43]{wimps.eps}
\end{center}
\caption{\label{wimps} ANTARES 90\% sensitivity on the muon flux as a
function of the WIMP mass using the 2007-2008 data. Also shown are
the 90\% C.L. limits on the same flux set by different experiments:
with Baksan~\cite{dmbak}, Macro~\cite{dmmac},
SuperKamiokande~\cite{dmsk}, and IceCube-22~\cite{dmic} for the $b
\bar{b}$ and $W^+ W^-$ channels). }
\end{figure}
Using the data recorded during the period 1998--2002, a total of 1007
live days, the Baikal collaboration was able to set a 90\% C.L. upper
limit of $\Phi < 3 \times 10^3$ km$^{-2}$ yr$^{-1}$ on an excess in
the muon flux coming from the Sun for neutralino masses larger than
100~GeV~\cite{dmbai}. Similarly using a total of 1038 live days, the
corresponding 90 \% C.L. upper limit for a flux coming from the
Earth's core was found to be $\Phi < 1.2 \times 10^3$ km$^{-2}$
yr$^{-1}$ for neutralino masses greater than 100~GeV.
\section{Search for exotic particles}
The existence of monopoles has been put forward in the context of
several theories. To date there is no clear evidence of their
existence and several limits have been set on the flux of monopoles
crossing the Earth.
Relativistic monopoles with masses above 10$^7$ GeV can cross the
Earth and leave a conspicuous signal in neutrino telescopes. Magnetic
charges crossing water at a speed larger than their Cherenkov
threshold ($\beta > 0.74$ in water) would produce a huge amount of
light. For one unit of magnetic charge this radiation would be 8550
times larger than that of a muon. Moreover, even below the threshold,
for $\beta > 0.52$, the high energetic ionization electrons
($\delta$-rays) produced by the monopole would also radiate a large
amount of light.
\begin{figure}[htb]
\begin{center}
\includegraphics[scale=0.47]{monopoles.eps}
\end{center}
\caption{\label{monopoles} 90\% C.L. upper limit on a flux of magnetic
monopoles set by the ANTARES~\cite{antmonopoles},
Amanda~II~\cite{amanmonopoles}, Baikal~\cite{baimonopoles} and
MACRO~\cite{macromonopoles} are also shown. }
\end{figure}
\noindent
The Baikal collaboration used the data taken between April 1998 and
February 2003, which includes the NT36, NT96 and NT200 configurations,
to perform a search for relativistic monopoles~\cite{baimonopoles}.
Events were selected on the basis of a high number of hits and good
reconstruction quality as determined by a $\chi^2$ test. Only upgoing
tracks (zenith angle greater than 100$^{\circ}$) were kept. A cut on
the radial distance depending on the exact detector configuration was
also applied. No candidate was found when simulations indicated that
around 4 background events were expected. The 90 \% C.L. limits on
the flux obtained from this negative result are shown in
Fig.~\ref{monopoles}.
Using the data taken during 2007 and 2008, ANTARES performed a search
for magnetic monopoles based also on the quality of the track and the
number of hits as well as on the track reconstructed velocity,
$\beta$. A special reconstruction was performed in which the $\beta$
of the particle was a free parameter and the $\chi^2$ values for the
hypotheses of $\beta$ equal or different from one were compared. The
selection criteria were optimized for discovery in eight velocity
intervals in the region 0.625 $\le \beta \le$ 0.995. Only one
candidate was found, compatible with the total expected background. In
Fig.~\ref{monopoles} the 90\% C.L. upper limit on the flux of upgoing
monopoles obtained is shown~\cite{antmonopoles}. As can be seen, this
limit is more stringent than the previous existing
limits~\cite{amanmonopoles, macromonopoles}.
A search for nuclearites, massive aggregates of up, down and strange
quarks, has also been performed by ANTARES. Nuclearites would produce
in water a thermal shock wave emitting a large amount of radiation at
visible wavelengths. No clear indication of nuclearites was observed
using the 2007-2008 data sample and a 90\% C.L. upper limit of
10$^{-16}$ cm$^{-2}$ sr$^{-1}$ s$^{-1}$ for a flux of nuclearites with
masses between 10$^{-14}$ and 10$^{-17}$ GeV was established.
\section{Summary}
The underwater neutrino telescopes ANTARES and NT-200 not only have
shown the technical and scientific feasibility of this sort of devices
in sea and lake waters, but have also produced interesting limits in
the search of cosmic neutrino sources, dark matter and exotic
particles. They are the precursors of much larger telescopes that will
be operating in the coming years.
\section*{Acknowledgements}
We gratefully acknowledge the financial support of the Spanish
Ministerio de Ciencia e Innovaci\'on (MICINN), grants
FPA2009-13983-C02-01, ACI2009-1020 and Consolider MultiDark
CSD2009-00064 and of the Generalitat Valenciana, Prometeo/2009/026.
\section*{References}
|
{
"timestamp": "2012-03-12T01:02:12",
"yymm": "1203",
"arxiv_id": "1203.2143",
"language": "en",
"url": "https://arxiv.org/abs/1203.2143"
}
|
\section{Introduction}
The aim of this note is to present a proof of the (classical)
Oort Conjecture, which is a question about lifting Galois
covers of curves from characteristic~$p>0$ to characteristic
zero. In one form or the other, this kind of question might
well be considered math folklore, and it was also well known
that in general the lifting is not possible. The problem was
systematically addressed and formulated by \nmnm{Oort}
at least as early as~1987, see~\cite{Oo1} or rather~\cite{Oo2}.
The general context of the lifting question/problem is as
follows: Let $k$ be an algebraically closed field of characteristic
$p>0$, and $W(k)$ be the ring of Witt vectors over~$k$. Let
$Y \to X$ a $G$-cover of complete smooth $k$-curves, where
$G$ is a finite group. Then the question is whether there
exists a finite extension of discrete valuation rings $W(k)\hookrightarrow R$
and a $G$-cover ${\mathcal Y}_R\to{\mathcal X}_R$ of complete smooth
$R$-curves whose special fiber is the given $G$-cover
$Y\to X$. The answer to this question in general is negative,
because over $k$ there are curves of genus $g>1$ and huge
automorphism groups, see e.g.~\nmnm{Roquette}~\cite{Ro2},
whereas in characteristic zero one has the Hurwitz bound $84(g-1)$
for the order of the automorphism group. The Oort Conjecture
is about a subtle interaction between the ramification structure
and the nature of the inertia groups of generically Galois
covers of curves, the idea being that if the inertia groups of
a $G$-cover $Y\to X$ look like in characteristic zero, i.e.,
they are all cyclic, then the Galois cover should be smoothly
liftable to characteristic zero.
\vskip7pt
{\bf Oort Conjecture}. {\it Let $k$ and $W(k)$ be as above.
Let $Y\to X$ be a possibly ramified
$G$-cover of complete smooth $k$-curves having only
cyclic groups as inertia groups. Then there exists a finite
extension $R$ of $\,W(k)$ over which the $G$-cover $Y\to X$
lifts smoothly, i.e., there exists a $G$-cover ${\mathcal Y}_R\to{\mathcal X}_R$
of complete smooth $R$-curves with special fiber $Y\to X$.\/}
\vskip7pt
There is also the {\bf local Oort conjecture}, which
asserts that every finite cyclic extension $k[[t]]\hookrightarrow k[[z]]$
is the canonical reduction of a cyclic extension
$R[[T]]\hookrightarrow R[[Z]]$ for some finite extension~$R$ of $W(k)$.
The local/global Oort Conjecrures are related as follows,
see~Fact~\ref{equivOC}, where further equivalent forms
of the Oort Conjecture are given: Let $R$ be a
finite extension of $W(k)$, and ${\mathcal X}_R$ be a complete
smooth curve with special fiber $X$. Then a given $G$-cover
$Y\to X$, $y\mapsto x$, with cyclic inertia groups lifts
smoothly over a given $R$ iff the local cyclic
extensions $k[[t_x]]:={\widehat{\clO}}_{X,x}\hookrightarrow{\widehat{\clO}}_{Y,y}=:k[[t_y]]$
lift smoothly over~$R$ for all $x\in X$ and $y\mapsto x$.
\vskip4pt
{\bf Notation:} Let $\deg_p({\fam\eufam\teneu D})$ be the power of~$p$
in the degree of the different of $k[[t]]\hookrightarrow k[[z]]$, and
$\deg_p({\fam\eufam\teneu D}_x)$ be correspondingly defined for the
local extension $k[[t_x]]\hookrightarrow k[[t_y]]$ at $y\mapsto x$.
\begin{theorem}
\label{OC}
The Oort Conjecture holds. Moreover, for every $\delta$
there exists an algebraic integer $\pi_\delta$ such that
for every algebraically closed field $k$, $\chr(k)=p$, the
following hold:
\vskip2pt
{\rm1)} Let $k[[t]]\hookrightarrow k[[z]]$ be a cyclic extension
with $\deg_p({\fam\eufam\teneu D})\leq\delta$.
\vskip2pt
{\rm2)} Let $Y\to X$ be a $G$-cover with cyclic inertia groups
and $\deg_p({\fam\eufam\teneu D}_x)\leq\delta$ for all $x\in X$.
\vskip2pt
\noindent
Then $k[[t]]\hookrightarrow k[[z]]$ and $Y\to X$ are smoothly liftable
over $W(k)[\pi_\delta]$.
\end{theorem}
{\bf Historical Note}: The first evidence for the Oort
Conjecture is the fact that the conjecture holds for
$G$-covers $Y\to X$ which have tame ramification only,
i.e., $G$-covers whose inertia groups are cyclic of the
form ${\fam\lvfam\tenlv Z}/m$ with $(p,m)=1$. Indeed, the lifting of
such $G$-covers follows from the famous
{\it Grothendieck's specialization theorem\/} for the
tame fundamental group, see e.g.,~SGA~I.
The first result which involved typical wild ramification
is \nmnm{Oort--Sekiguchi--Suwa}~\cite{OSS} which
tackled the case of ${\fam\lvfam\tenlv Z}/p$-covers. It was followed
by a quite intensive research
activity, see the survey article by \nmnm{Obus}~\cite{Ob}
as well as the bibliography list at the end of this
note. \nmnm{Garuti}~\cite{Ga1},~\cite{Ga2} contains a
lot of foundational work and beyond that showed that
every $G$-cover $Y\to X$ has (usually) non-smooth
liftings with a well understood geometry. This aspect
of the problem was revisited recently by \nmnm{Saidi}~\cite{Sa},
where among other things a systematic discussion
of the (equivalent) forms of the Oort Conjecture is given. The
paper~\cite{GM1} by \nmnm{Green--Matignon} contains
further foundational work and gives a positive answer
to the Oort Conjecture in the case of inertia groups of
the form ${\fam\lvfam\tenlv Z}/mp^e$ with $(p,m)=1$ and $e\leq2$.
Their result relies on the Sekiguchi--Suwa theory,
see~\cite{SS1} and~\cite{SS2}. The paper
\nmnm{Bertin--M\'ezard}~\cite{B-M} addresses the
deformation theory for covers, whereas
\nmnm{Chinburg--Guralnick--Harbater} \cite{CGH1},
\cite{CGH2} initiated the study of the so called {\it Oort
groups,\/} and showed that the class of Oort groups
is quite restrictive. Last but not least, the very recent
result by \nmnm{Obus--Wewers}~\cite{O--W}, see
rather \nmnm{Obus}~\cite{Ob},~Theorem~6.28, solves
the Oort Conjecture in the case the inertia groups are
of the form ${\fam\lvfam\tenlv Z}/mp^e$ for $(p,m)=1$~and~$e\leq 3$,
and {\it essential for the method of this note,\/} when
the upper ramification jumps are subject to some
explicit (strong) restrictions, see the explanations
at~Remark~\ref{remarkOW} in section~4 for details.
\vskip5pt
{\bf About the proof}: Concerning technical tools,
we use freely a few of the foundational results from the
papers mentioned above. The main novel tools for the
proof are Key Lemma~\ref{keylemma1} and its global
form~Theorem~\ref{charpOC}, and second,
Key~Lemma~\ref{keylemma2}, which is actually a very
special case of Lemma~6.27 from~\nmnm{Obus}~\cite{Ob},
see section~4 for precise details and citations. All these
results will be used as ``black boxes'' in the proof of
the Oort Conjecture, given in section~4. Concerning the
idea of the proof, there is little to say: The point is to
first deform a given $G$-cover $Y\to X$ to a cover
${\mathcal Y}_{\fam\eufam\teneu o}\to{\mathcal X}_{\fam\eufam\teneu o}$ over ${\fam\eufam\teneu o}=k[[\upi]]$ in such a
way that the ramification of the deformed cover has no
\defi{essential upper jumps} as defined/introduced at
the beginning of section~3, then apply the local-global
principles, etc.
\vskip2pt
Maybe it's interesting to mention that the first variant
of the proof (January, 2011) was shorter, but relied
heavily on model theoretical tools and was not effective
(concerning the finite extensions of $W(k)$ over which
the smooth lifting can be realized).
\vskip5pt
{\bf Acknowledgements(...)} If my recollection is correct, during
an MFO Workshop in 2003~or~so, someone asked what
should be the ``characteristic $p$ Oort Conjecture,'' but
it seems that nobody ever followed up (successfully) on
that idea.
\vskip5pt
\hhb{0}
\section{Reviewing well known facts}
Throughout this section, $k$ is an algebraically closed
field with ${\rm char}(k)=p>0$. All the other fields will be
field extensions of $k$, in particular will be fields of
characteristic~$p$.
\vskip7pt
\noindent
A) {\it Reviewing higher ramification for cyclic extensions\/}
\vskip5pt
Let $K$ be a complete discrete valued field of positive
characteristic $p>0$, with valuation ring $R$ and having
as residue field an algebraically closed field $k$. Let
$L|K$ be a finite Galois extension, say with Galois
group $G={\rm Gal}(L|K)$. Since $K$ is complete, the
valuation $v\!_K^\nmi$ of $K$ as a unique prolongation to $L$.
And since the residue field $k$ of $K$ is algebraically
closed, $L|K$ is totally ramified, i.e., $[L:K]=e(L|K)$.
Finally, let $v\!_L^\nmi$ be the normalized valuation of $L$,
hence $v\!_L^\nmi(L^\times)=Z$ and ${1\over[L|K]}v\!_L^\nmi=v\!_K^\nmi$ on $K$.
\vskip2pt
Recall that the lower ramification
groups of $L|K$ or of $G$ are defined as follows:
Let $z\in L$ be a uniformizing parameter. For every
$\jmath$ we set $G_\jmath:=\{\sigma\in G\midv\!_L^\nmi(\sigma z-z)>\jmath\}$
and call it the $\jmath^{\rm th}$ \defi{lower ramification group}
of $L|K$ or of $G$. Clearly, $G=G_0$ is the inertia
group of $L|K$, and $G_1$ is the ramification group
of $L|K$, thus the Sylow $p$-group of $G$, and $G_\jmath=1$
for $\jmath$ sufficiently large. In particular, $G=G_1$ iff $G$
is a $p$-group iff $L|K$ is totally wildly ramified.
\vskip2pt
The first important fact about the lower ramification groups
$(G_\jmath)_\jmath$ is {\it Hilbert's different formula,\/} which gives an
estimate for the degree $\dgr{{\fam\eufam\teneu D}_{L|K}}$ of the different of
$L|K$ in terms of the orders of the lower ramification groups,
see e.g., \nmnm{Serre}~\cite{Se},~IV,~\S1:
\[
\deg({\fam\eufam\teneu D}_{L|K})=\sum_{\jmath=0}^\infty\big(|G_\jmath|-1\big).
\]
We further denote by $\jmath_\alp$ the \defi{lower jumps}
for $L|K$, or of $G$, as being the numbers satisfying
$G_{\jmath_\alp}\neq G_{\jmath_\alp+1}$. In particular,
setting $\jmath_{-1}=-1$ and $\jmath_0=0$, and denoting the
upper jumps for~$L|K$ by $\jmath_0\leq\jmath_1\leq\dots\leq\jmath_r$,
one has that $\jmath_r=\max\,\{\hhb2\jmath \mid G_\jmath\neq1\,\}$.
\vskip2pt
Now suppose that $L|K$ is a cyclic extension with
$G={\fam\lvfam\tenlv Z}/p^e$. Then $G=G_0=G_1$ by the discussion above,
and every subgroup of $G$ is a lower ramification group
for $L|K$. Thus one has precisely $e$ lower positive jumps
$\jmath_1\leq\dots\leq\jmath_e$, and $G_{\jmath_1}\geq\dots\geq G_{\jmath_e}$
are precisely the $e$ non-trivial subgroups of $G={\fam\lvfam\tenlv Z}/p^e$.
Finally, the Hilbert's Different Formula becomes:
\begin{eqnarray*}
\textstyle
\deg({\fam\eufam\teneu D}_{L|K})=p^e-1+\sum_{\alp=1}^e
(\jmath_\alp-\jmath_{\alp-1})\big(|G_{\jmath_\alp}|-1\big)
&=&p^e-1+\sum_{\alp=1}^e
(\jmath_\alp-\jmath_{\alp-1})(p^{e-(\alp-1)}-1)
\end{eqnarray*}
We recall that the lower ramification
subgroups behave functorially in the base field, i.e.,
if $K'|K$ is some finite sub-extension of $L|K$, and
$G'\subseteq G$ is the Galois group of $L|K'$, then
$G'_\jmath=G_\jmath\cap G'$. Since the lower ramification
groups do not behave functorially with respect to
Galois sub-extensions, one introduced the \defi{upper
ramification groups} $G^{^{\,\scriptstyle\imath}}$ for
$\imath\geq-1$ of $L|K$, which behave functorially under
taking Galois sub-extensions, see \nmnm{Serre}~{Se},~IV,~\S3.
\vskip2pt
At least in the case of cyclic extensions $L|K$ with Gallois
group $G\cong{\fam\lvfam\tenlv Z}/p^e$, the formula which relates the
lower ramification groups $G_{\!\jmath}$ to the upper ramification
groups $G^{^{\,\scriptstyle\imath}}$ is explicit via {\it Herbrand's
formula,\/} see e.g.\ \nmnm{Serre}~\cite{Se},~IV,~\S3.
Namely, if $\imath_0:=0$ and $\imath_1\leq \dots\leq \imath_e$
are the \defi{upper jumps} for $L|K$, then one has:
\[
\jmath_\alp-\jmath_{\alp-1}=p^{\,\alp-1}(\imath_\alp-\imath_{\alp-1}),
\quad \alp=1,\dots,e.
\]
\indent
Thus in the case $L|K$ is cyclic with Galois group
$G={\fam\lvfam\tenlv Z}/p^e$, one can express the degree of the different
of $L|K$ in terms of higher ramification groups as follows:
\[
\deg({\fam\eufam\teneu D}_{L|K})=p^e-1+\sum_{\alp=1}^e
(\imath_\alp-\imath_{\alp-1})p^{\alp-1}(p^{e-(\alp-1)}-1)=
\sum_{\alp=1}^e(\imath_\alp+1)(p^{\hhb1\alp}-p^{\alp-1}).
\]
\noindent
B) {\it Explicit formulas via Artin--Schreier--Witt Theory\/}
\vskip5pt
In the above notations, let $t$ be any uniformizing
parameter of the complete discrete valued field $K$
of characteristic $p > 0$. Then $K$ is canonically
isomorphic to the Laurent power series field $K=k\lps t$
in the variable $t$ over $k$.
\vskip2pt
Recall that the Artin--Schreier--Witt theory gives a
description of the cyclic $p$-power extensions of $K$
via finite length Witt vectors as follows, see e.g.,
\nmnm{Lang}~\cite{La}, or \nmnm{Serre}~\cite{Se},~II. Let
${\mathcal A}$ be an integrally closed domain which is a $k$-algebra,
and $W_{\!\nx}({\mathcal A})=\{(a_1,\dots,a_e)\mid a_i\in{\mathcal A}\}$
be the Witt vectors of length $e$ over ${\mathcal A}$.
Then the Frobenius morphism ${\rm Frob}$ of ${\mathcal A}$
lifts to the \defi{Frobenius morphism} ${\rm Frob}_e$
of $W_{\!\nx}({\mathcal A})$, and one defines the
\defi{Artin--Schreier--Witt operator}
$\wp_e:={\rm Frob}_e-{\rm Id}$ of
${\mathcal A}$. If ${\mathcal A}\hookrightarrow{\mathcal A}^{^{\rm nr}}$ is a ind-\'etale
universal cover of ${\mathcal A}$, one has the Artin--Schreier--Witt
exact sequence
\[
0\to W_{\!\nx}({\fam\lvfam\tenlv F}_p)={\fam\lvfam\tenlv Z}/p^e\hor{}W_{\!\nx}({\mathcal A}^{^{\rm nr}})
\hor{\wp_e}W_{\!\nx}({\mathcal A}^{^{\rm nr}})\to0.
\]
of sheaves on ${\rm Et}({\mathcal A})$. In particular, if
${\rm Pic}({\mathcal A})=0$, one gets a canonical isomorphism
\[
W_{\!\nx}({\mathcal A})/{\rm im}(\wp_e)\to
{\rm Hom}\big(\pi_1({\mathcal A}), {\fam\lvfam\tenlv Z}/p^e\big),
\]
which gives a canonical bijection between the cyclic
subgroups $\langle{\undr a}\rangle\subset
W_{\!\nx}({\mathcal A})/{\rm im}(\wp_e)$
and the integral \'etale cyclic extensions ${\mathcal A}\hookrightarrow{\mathcal A}_{\undr a}$
with Galois group a quotient of ${\fam\lvfam\tenlv Z}/p^e$ by via
\[
\langle{\undr a}\rangle\mapsto {\mathcal A}_{\undr a}:={\mathcal A}[{\undr x}]
\]
where ${\undr x}=(x_1,\dots,x_e)$ is any solution of the
equation $\wp_e({\undr x})={\undr a}$.
\vskip2pt
In the special case ${\mathcal A}=K=k\lps t$, one can make
things more precise as follows. First, by Hensel's Lemma,
the class of every element in $W_{\!\nx}(K)/{\rm im}(\wp_e)$
contains a representative of the form ${\undr p}=(p_1,\dots,p_e)$
with $p_\alp=p_\alp({t^{-1}}) \in k[{t^{-1}}] $ a polynomial
in the variable ${t^{-1}}$ over $k$. Second, using the
properties of the Artin--Schreier operator $\wp_e$, one
can inductively ``reduce'' the terms of each $p_\alp({t^{-1}})$
which contain powers of ${t^{-1}}$ to exponent divisible by
$p$. If all the polynomials $p_\alp({t^{-1}})$ have this property,
one says that ${\undr p}=(p_1,\dots,p_e)$ is in \defi{standard
form}. Following \nmnm{Garuti}~\cite{Ga3}, Thm.~1.1, and
\nmnm{Thomas}~\cite{Tho},~Prop.~4.2, see also
\nmnm{Obus--Priess}~\cite{O--P}, one can describe the
upper jumps $\imath_1\leq\dots\leq \imath_e$ for the extension
$L=K_{\undr p}$ with ${\undr p}=(p_1,\dots,p_e)$ in standard
form and $p_1({t^{-1}})\neq0$ as follows:\footnote{Notice that
we use the conventions $\deg(0)=-\infty$ and $\imath_0=0$.}
\[
\imath_\alp=\max\,\{\hhb2p\hhb1\imath_{\alp-1},\dgr{p_\alp({t^{-1}})}\},
\quad \alp=1,\dots,e.
\]
In particular, $\imath_\alp\geq p^{\hhb1\alp}\dgr{p_1({t^{-1}})} $,
and the highest non-zero upper ramification index,
i.e., the Artin conductor of $K_{\undr p}|K$ is
$\imath_e=\max\,\{\hhb2p^{e-\alp}
\dgr{p_\alp({t^{-1}})}\mid \alp=1,\dots,e\}$.
\vskip3pt
We make the following remark for later use: For
${\undr a}=(a_1,\dots,a_e)$ an arbitrary Witt vector of
length $e$ over $K$, and $K_{\undr a}|K$ as above one
has: $[K_{\undr a}:K]=p^m$, where $m$ is minimal such
that $(a_1,\dots,a_{e-m})\in{\rm im}(\wp_{e-m})$. In particular,
if $m<e$, then setting ${\undr b}=(b_1,\dots,b_m)$ with
$b_\alp=a_{\alp+e-m}$, one has: $K_{\undr a}|K$ is
actually the cyclic extension $K_{\undr b}|K$ of degree
$p^m$ of $K$, and one can compute the upper ramification
indices of $K_{\undr a}=K_{\undr b}$ using the discussion
above.
\vskip7pt
\noindent
C) {\it Kato's smoothness criterion\/}
\vskip5pt
Let $k$ be as usual an algebraically closed field of
characteristic $p>0$, and ${\fam\eufam\teneu o}$ a complete discrete
valuation ring with quotient field $\kk={\rm Quot}({\fam\eufam\teneu o})$
and residue field $k$. Let ${\mathcal A}={\fam\eufam\teneu o}[[T]]$ be the power
series ring over ${\fam\eufam\teneu o}$. Then ${\mathcal R}:={\mathcal R}\otimes_{\fam\eufam\teneu o}\kk
=\kk\langle\hhb{-1}\langle T\rangle\hhb{-1}\rangle$
is the ring of power series in $T$ over $\kk$ having
$v_\kk$-bounded coefficients. Thus ${\mathcal R}$ is a Dedekind
ring having $\Spec({\mathcal R})$ in bijection with the points
of the open rigid disc ${\fam\eufam\teneu X}=\Spf\,{\mathcal R}$ of radius~$1$
over the complete valued field~$\kk$. Further, ${\mathcal A}$ is
a two dimensional complete regular ring with maximal
ideal $(\pi,T)$ with ${\mathcal A}\to{\mathcal A}/(\pi,T)=k$, and
${\fam\eufam\teneu X}=\Spec({\mathcal R})$ is nothing but the complement
of $V(\pi)\subset\Spec({\mathcal A})$. Finally,
$A:={\mathcal A}/(\pi)=k[[t]]$ is the power series ring in the
variable $t:=T\,{\rm\big(mod}\,(\pi)\big)$, thus
a complete discrete valuation ring.
\vskip2pt
Let ${\mathcal K}:={\rm Quot}({\mathcal A})$ and $K:={\rm Quot}(A)=k\lps t$
be the fraction fields of ${\mathcal A}$, respectively~$A$. Let
${\mathcal K}\hookrightarrow{\mathcal L}$ be a finite separable field extension,
and ${\mathcal B}\subset{\mathcal S}$ be the integral closures of
${\mathcal A}\subset{\mathcal R}$ in the finite field extension ${\mathcal K}\hookrightarrow{\mathcal L}$.
Then ${\mathcal B}$ is finite ${\mathcal A}$-module, and ${\mathcal S}$ is a finite
${\mathcal R}$-module, in particular a Dedekind ring.
\vskip2pt
Next let ${\fam\eufam\teneu r}_1,\dots,{\fam\eufam\teneu r}_r$ be the prime ideals
of ${\mathcal B}$ above $(\pi)$. Then each ${\fam\eufam\teneu r}_i$ has
height one, and the localizations ${\mathcal B}_{{\fam\eufam\teneu r}_i}$ are
precisely the valuation rings of ${\mathcal L}$ above the
discrete valuation ring ${\mathcal A}_{(\pi)}$ of ${\mathcal K}$. And
since ${\mathcal K}\hookrightarrow{\mathcal L}$ is a finite separable extension,
by the Finiteness Lemma, the fundamental equality
holds:
\[
[{\mathcal L}:{\mathcal K}]=\sum_{i=1}^r e({\fam\eufam\teneu r}_i|\pi)\cdot f({\fam\eufam\teneu r}_i|\pi),
\]
where $e({\fam\eufam\teneu r}_i|\pi)$ and $f({\fam\eufam\teneu r}_i|\pi)=[\kappa({\fam\eufam\teneu r}_i):K]$
is the ramification index, respectively the residual degree
of ${\fam\eufam\teneu r}_i|\pi$. Therefore, if $v_\pi$ is the discrete valuation
of ${\mathcal K}$ with valuation ring ${\mathcal A}_{(\pi)}$, one has
$K=\kappa(\pi)={\rm Quot}\big({\mathcal A}/(\pi)\big)=\kappa(v_\pi)$,
and the following are equivalent:
\vskip3pt
$\hhb2$i) There exists a prolongation $w$ of $v_\pi$ to
${\mathcal L}$ such that $[{\mathcal L}:{\mathcal K}]=[L:K]$, where $L:=\kappa(w)$.
\vskip3pt
ii) The ideal ${\fam\eufam\teneu r}:=\pi{\mathcal B}$ is a prime ideal of ${\mathcal B}$, or
equivalently, $\pi$ is a prime element of ${\mathcal B}$.
\vskip3pt
\noindent
In particular, if the above equivalent conditions i),~ii), hold,
then ${\mathcal B}_{\fam\eufam\teneu r}$ is the valuation ring of~$w$, and one has
$\kappa({\fam\eufam\teneu r})={\rm Quot}\big({\mathcal B}/(\pi)\big)=\kappa(w)=L$.
Further, $w$ is the unique prolongation of $v_\pi$ to ${\mathcal L}$,
and ${\fam\eufam\teneu r}=\pi{\mathcal B}$ is the unique prime ideal of ${\mathcal B}$ above
the ideal $\pi{\mathcal A}$ of ${\mathcal A}$.
\vskip2pt
We conclude by mentioning the following smoothness
criterion, which is a special case of the theory developed
in \nmnm{Kato}~\cite{Ka},~\S5; see also
\nmnm{Green--Matignon}~\cite{GM1},~\S3, especially~3.4.
\begin{fact}
\label{KCR}
{\it In the above notations, suppose that
$v_\pi$ has a prolongation $w$ to ${\mathcal L}$ such that
$[{\mathcal L}:{\mathcal K}]=[L:K]$, where $L:=\kappa(w)$ and
$K=:\kappa(v_\pi)$. Let $A\hookrightarrow B$ be the integral
closure of $A=k[[t]]={\mathcal A}/(\pi)$ in the field extension $K\hookrightarrow L$.
Let ${\fam\eufam\teneu D}_{{\mathcal S}|{\mathcal R}}$ and ${\fam\eufam\teneu D}_{B|A}$ be the differents
of the extensions of Dedekind rings ${\mathcal R}\hookrightarrow{\mathcal S}$,
respectively $A\hookrightarrow B$. The following are equivalent:
\vskip2pt
$\hhb2${\rm i)} $\Spec{\mathcal B}$ is smooth over
$\Spec\,{\fam\eufam\teneu o}$.
\vskip2pt
{\rm ii)} The degrees of ${\fam\eufam\teneu D}_{{\mathcal S}|{\mathcal R}}$ and
${\fam\eufam\teneu D}_{B|A}$ are equal: $\deg({\fam\eufam\teneu D}_{{\mathcal S}|{\mathcal R}})=
\deg({\fam\eufam\teneu D}_{B|A})$.
\vskip3pt
If the above equivalent conditions are satisfied, then
there exists $Z\in{\mathcal B}$ such that ${\mathcal B}={\fam\eufam\teneu o}[[Z]]$ and
$B={\mathcal S}/(\pi)=k[[z]]$, where $z=Z\,\big({\rm mod}\,(\pi)\big)$.
\/}
\end{fact}
\section{The characteristic $p$ Oort Conjecture}
\begin{remark/definition}
\label{essential}
In the context of section~2), A), let $L|K$ be a cyclic
extension of degree $p^e:=[L:K]$ with upper ramification
jumps $\imath_1\leq \dots \leq \imath_e$. Recall that setting
$\imath_0=0$, one has that $p\hhb1\imath_{\alp-1}\leq\imath_{\alp}$
for all~$\alp$ with $0<\alp\leqe$, and the inequality
is strict if and only if $\imath_{\alp}$ is not divisible by $p$.
The division by $p$ gives:
\[
\imath_\alp-p\hhb1\imath_{\alp-1}=
p\hhb1q_{\alp}+\epsilon_{\alp}
\]
with $0\leq q_{\alp}$ and $0\leq\epsilon_{\alp}< p$,
and notice that by the remark above one has:
$0<\epsilon_{\alp}$ if and only if $(p,\imath_{\alp})=1$
if and only if $p\hhb1\imath_{\alp-1} < \imath_{\alp}$.
\vskip2pt
We call $q_{\alp}$ the \defi{essential part} of the upper
jump at $\alp$, and if $0< q_{\alp}$ we say that
$\imath_\alp$ is an \defi{essential upper jump} for $L|K$,
and that $\alp$ is an \defi{essential upper index} for
$L|K$.
\vskip4pt
We introduce terminology as follows: Let ${\mathcal R}\hookrightarrow{\mathcal S}$
be any generically finite Galois extension of Dedekind
rings with cyclic inertia groups, and
${\mathcal K}:={\rm Quot}({\mathcal R})\hookrightarrow{\rm Quot}({\mathcal S})=:{\mathcal L}$
be the corresponding cyclic extension of their quotient
fields. For a maximal ideal ${\fam\eufam\teneu q}\in\Spec{\mathcal S}$ above
${\fam\eufam\teneu p}\in\Spec{\mathcal R}$, let ${\mathcal K}_{\fam\eufam\teneu p}\hookrightarrow{\mathcal L}_{\fam\eufam\teneu q}$ be the
corresponding extension of complete discrete valued
fields. We we will say that ${\mathcal R}\hookrightarrow{\mathcal S}$ has \defi{(no)
essential ramification jumps} at ${\fam\eufam\teneu p}$, if the $p$-part
of the cyclic extension of discrete complete valued fields
${\mathcal K}_{\fam\eufam\teneu p}\hookrightarrow{\mathcal L}_{\fam\eufam\teneu q}$ has (no) essential upper
ramification jumps. And we say that ${\mathcal R}\hookrightarrow{\mathcal S}$ is has
no essential ramification, if ${\mathcal R}\hookrightarrow{\mathcal S}$ is has no
essential ramification jumps at any ${\fam\eufam\teneu p}\in\Spec{\mathcal R}$.
\end{remark/definition}
In the remaining part of this subsection, we will work in
a special case of the situation presented in section~2,~C),
which is as follows: We consider a fixed algebraically
closed field~$k$ with ${\rm char}(k)=p>0$, let
${\fam\eufam\teneu o}=k[[\upi]]$ be the
power series ring in the variable $\upi$ over $k$, thus
$\kk=k\lps\upi={\rm Quot}({\fam\eufam\teneu o})$ is the Laurent power
series in the variable $\upi$ over $k$. Let ${\mathcal A}=k[[\upi,{t}]]$
and ${\mathcal K}=k\lps{\upi,{t}}={\rm Quot}({\mathcal A})$ be its field
of fractions. Then $A={\mathcal A}/(\upi)=k[[t]]$
and $K=k\lps t={\rm Quot}(A)$ is the fraction
field of $A$. Further, ${\mathcal R}:={\mathcal R}\otimes_{\fam\eufam\teneu o}\kk
=\kk\langle\hhb{-1}\langle t\rangle\hhb{-1}\rangle$
is the ring of power series in $t$ over $\kk$ having
$v_\kk$-bounded coefficients. Thus ${\mathcal R}$ is a Dedekind
ring having $\Spec({\mathcal R})$ in bijection with the points
of the open rigid disc ${\fam\eufam\teneu X}=\Spf\,{\mathcal R}$ of radius~$1$
over the complete valued field~$\kk$. And we notice
that ${\fam\eufam\teneu X}=\Spec({\mathcal R})$ is precisely the complement
of $V(\upi)\subset\Spec({\mathcal A})$. Finally, for a finite
separable field extension ${\mathcal K}\hookrightarrow{\mathcal L}$, we let
${\mathcal B}\subset{\mathcal S}$ be the integral closures of
${\mathcal A}\subset{\mathcal R}$ in the finite field extension ${\mathcal K}\hookrightarrow{\mathcal L}$.
Thus ${\mathcal B}$ is finite ${\mathcal A}$-module, and ${\mathcal S}$ is a finite
${\mathcal R}$-module, in particular a Dedekind ring.
\begin{keylemma}
\label{keylemma1}
{\rm (Characteristic $p$ local Oort conjecture)} \ \
In the above notations, let $N:=1+q_1+\dots+q_e$
and $x_1,\dots,x_N\in{\fam\eufam\teneu m}_{\fam\eufam\teneu o}$ be distinct points.
Let $K\hookrightarrow L$ be a cyclic ${\fam\lvfam\tenlv Z}/p^e$-extension with
upper ramification jumps $\imath_1\leq\dots\leq \imath_e$.
Then there exists a cyclic ${\fam\lvfam\tenlv Z}/p^e$-extension
${\mathcal K}\hookrightarrow{\mathcal L}$ such that the integral closure ${\mathcal A}\hookrightarrow{\mathcal B}$
of ${\mathcal A}$ in ${\mathcal K}\hookrightarrow{\mathcal L}$ and the corresponding
extension of Dedekind rings ${\mathcal R}\hookrightarrow{\mathcal S}$ satisfy:
\vskip3pt
{\rm 1)} The morphism $\varphi:\Spec{\mathcal B}\to\Spec{\fam\eufam\teneu o}$
is smooth. In particular, ${\mathcal B}=k[[\upi,Z]]$ and the
special fiber of $\varphi$ is $\Spec B\to k$, where
$B=k[[z]]={\mathcal B}/(\upi)$ and $z=Z\,\big({\rm mod}\,(\upi)\big)$.
\vskip3pt
{\rm 2)} The canonical morphism ${\mathcal R}\hookrightarrow{\mathcal S}$ has
no essential ramification and is ramified only at points
$y_\mu\in\Spec{\mathcal S}$ above the points $x_\mu\in\Spec{\mathcal R}$,
$1\leq\mu\leq N$.
\vskip3pt
{\rm3)} Let $(\imath_{\mu,\alp})_{1\leq\alp\leqe_\mu}$
be the upper ramification jumps at each $y_\mu\mapsto x_\mu$.
Then $(e_\mu)_{1\leq\mu\leq N}$ is decreasing, and the
upper jumps are given by:
\begin{itemize}
\vskip2pt
\item[{\rm i)}] $\imath_{1,{\hhb{.5}\rho}}=p\hhb1\imath_{1,{\hhb{.5}\rho}-1}+\epsilon_{{\hhb{.5}\rho}}$
for $\,1\leq{\hhb{.5}\rho}\leqe$.
\vskip1pt\noindent \ \
\item[{\rm ii)}] $\imath_{\mu,{\hhb{.5}\rho}}=p\hhb1\imath_{\mu,{\hhb{.5}\rho}-1}+p-1$
for $1 < \mu\leq N$ and $1\leq\alp\leq e_\mu$.
\end{itemize}
\vskip2pt
{\rm4)} In particular, the branch locus $\,\{x_1,\dots,x_N\}$
of $\,{\mathcal R}\hookrightarrow{\mathcal S}$ is independent of $k[[t]]\hookrightarrow k[[z]]$, and
the upper ramification jumps
$\uix_\mu:=(\imath_{\mu,1},\dots,\imath_{\mu,e_\mu})$ at
each $y_\mu\mapsto x_\mu$, $1\leq\mu\leq N\!$, depend
only on the upper jumps
$\uix:=(\imath_1,\dots,\imath_e)$ of $k[[t]]\hookrightarrow k[[z]]$.
\end{keylemma}
The proof of the Key Lemma~\ref{keylemma1} will
take almost the whole section. We begin by recalling that
in the notations from section~2,~B), there exists
${\undr p}=\big(p_1({t^{-1}}) ,\dots,p_e({t^{-1}}) \big)$, say
in standard form, such that $L=K_{\undr p}$. The integral
closure $A\hookrightarrow B$ of $A=k[[t]]$ in the field extension
$K\hookrightarrow L$ is of the form $B=k[[z]]$ for any uniformizing
parameter $z$ of $L={\rm Quot}(B)$. And the degree of the
different ${\fam\eufam\teneu D}_{L|K}:={\fam\eufam\teneu D}_{B|A}$ is $\,\deg({\fam\eufam\teneu D}_{L|K})=
\sum_{\alp=1}^e(\imath_\alp+1)(p^{\hhb1\alp}-p^{\alp-1})$.
\vskip7pt
\noindent
A) {\it Combinatorics of the upper jumps\/}
\vskip7pt
\vskip2pt
In the above context, let $e_0$ be the number
of essential upper jumps, which could be zero. If
there exist essential upper jumps, i.e., $0 < e_0$,
let $r_1\leq\dots\leq r_{e_0}$ be the essential upper
indices for $L|K$, and notice that the sequence
$(r_i)_{1\leq i\leqe_0}$ is strictly increasing with
$r_{e_0}\leq e$. For technical reasons (to simplify
notations) we set $r_{e_0+1}:=e+1$, and if we
need to speak about $r_{e_0+1}$, we call
it the \defi{improper upper index}, which for $e_0=0$
would become $r_1=e+1$.
\vskip2pt
We next construct a finite strictly increasing sequence
$(d_i)_{0\leq i\leqe_0}$ as follows: We set $d_0=1$,
and we are done if $e_0=0$. If $e_0>0$, we define
inductively $d_i:=d_{i-1}+q_{r_i}$ for $1\leq i\leqe_0$.
In particular, we see that $N:=1+q_1+\dots q_e$ is
$N=1$ if $e_0=0$, and $N:=d_{e_0}$ otherwise.
\vskip2pt
We define an $N\timese$ matrix of non-negative integers
$(\theta_{\mu,{\hhb{.5}\rho}})_{1\leq\mu\leq N,\,1\leq{\hhb{.5}\rho}\leqe}$
as follows:
\vskip5pt
$\bullet$ If $e_0=0$, then $N=1$, and the $1\timese$
matrix is given by $\theta_{1,{\hhb{.5}\rho}}:=u_{\hhb{.5}\rho}$, $1\leq{\hhb{.5}\rho}\leqe$.
\vskip2pt
$\bullet$ If $e_0>0$, thus $N>1$, we
define:\footnote{Here and elsewhere, we set
$\theta_{\mu,0}:=0$ as well as $\imath_0=0$ and $\imath_{\mu,0}=0$.}
\begin{itemize}
\item[{\rm a)}] $\theta_{1,{\hhb{.5}\rho}}=p\hhb1\theta_{1,{\hhb{.5}\rho}-1}+\epsilon_{{\hhb{.5}\rho}}$
for $\,1\leq{\hhb{.5}\rho}\leqe$.
\vskip5pt\noindent \ \
\item[{\rm b)}] For $i=1,\dots,e_0$ \ and \ $d_{i-1}<\mu\leq d_i$ define:
\vskip2pt\noindent
$\scriptstyle\bullet$ \ $\theta_{\mu,{\hhb{.5}\rho}}=0$ for $1\leq{\hhb{.5}\rho} < r_i$.
\vskip2pt\noindent
$\scriptstyle\bullet$ \ $\theta_{\mu,{\hhb{.5}\rho}}=p\hhb1\theta_{\mu,{\hhb{.5}\rho}-1}+p-1$
for $r_i\leq{\hhb{.5}\rho}\leqe$.
\end{itemize}
\vskip3pt
Notice that in the case $e_0>0$, one has: Let
${\hhb{.5}\rho}$ with $1\leq{\hhb{.5}\rho}\leqe$ be given. Consider the
unique $1\leq i\leqe_0$ such that $r_i\leq{\hhb{.5}\rho}< r_{i+1}$.
(Recall the if $r_i=e$, then $r_{i+1}:=e+1$ by
the convention above!) Then for all $\mu$ with
$1\leq\mu\leq N$ one has: $\theta_{\mu,{\hhb{.5}\rho}}\neq0$
if and only if $\mu\leq d_i$.
The fundamental combinatorial property of
$(\theta_{\mu,{\hhb{.5}\rho}})_{1\leq\mu\leq N,\,1\leq{\hhb{.5}\rho}\leqe}$
is given by the following:
\begin{lemma}
\label{combinlemma}
For $1\leq i\leq e_0$ and $r_i\leq{\hhb{.5}\rho}< r_{i+1}$ one has:
$\imath_{\hhb{.5}\rho}+1=\sum_{1\leq\mu\leq d_i}(\theta_{\mu,{\hhb{.5}\rho}}+1)$.
\end{lemma}
\begin{proof}
The proof follows by induction on ${\hhb{.5}\rho}=1,\dots,e$.
Indeed, if $e_0=0$, then $N=1$, and there is nothing
to prove. Thus supposing that $e_0>0$, one argues as
follows:
\vskip2pt
$\bullet$ The assertion holds for ${\hhb{.5}\rho}=1$: First, if $r_1>1$,
then $\theta_{\mu,1}=0$ for $1<\mu$, thus there is nothing to
prove. Second, if $r_1=1$, then $q_1>0$ and $d_1=1+q_1$.
Further, by the definitions one has: $\theta_{1,1}=\epsilon_1$
and $\theta_{\mu,1}=p-1$ for $1<\mu\leq d_1$, and conclude
by the fact that $u_1=p\hhb1q_1+\epsilon_1$.
\vskip2pt
$\bullet$ If the assertion of Lemma~\ref{combinlemma}
holds for ${\hhb{.5}\rho}<e$, the assertion also holds for ${\hhb{.5}\rho}+1$:
Indeed, let $i$ be such that $r_i\leq{\hhb{.5}\rho}< r_{i+1}$.
\vskip3pt
\underbar{Case 1}: ${\hhb{.5}\rho}+1< r_{i+1}$. \ Then
$r_i\leq{\hhb{.5}\rho}+1<r_{i+1}$, and in particular, ${\hhb{.5}\rho}+1$ is
not an essential jump index. Hence by definitions one has
that $\imath_{{\hhb{.5}\rho}+1}=p\hhb1\imath_{\hhb{.5}\rho}+\epsilon_{{\hhb{.5}\rho}+1}$
with $0\leq\epsilon_{{\hhb{.5}\rho}+1}< p$. On the other hand, by
the induction hypothesis we have that
$\imath_{\hhb{.5}\rho}=\theta_{1,{\hhb{.5}\rho}}+\sum_{1<\mu\leq d_i}(\theta_{\mu,{\hhb{.5}\rho}}+1)$.
Hence taking into account the definitions of $\theta_{\mu,{\hhb{.5}\rho}}$,
we conclude the proof in Case~1 as follows:
\begin{eqnarray*}
\imath_{{\hhb{.5}\rho}+1}+1
&=&p\hhb1\imath_{\hhb{.5}\rho}+\epsilon_{{\hhb{.5}\rho}+1}+1\\
&=&p\hhb1\theta_{1,{\hhb{.5}\rho}}+
\textstyle\sum_{1<\mu\leq d_i}
(\hhb1p\hhb1\theta_{\mu,{\hhb{.5}\rho}}+p)+\epsilon_{{\hhb{.5}\rho}+1}+1\\
&=&(p\hhb1\theta_{1,{\hhb{.5}\rho}+1}+\epsilon_{{\hhb{.5}\rho}+1}+1)+
\textstyle\sum_{1 < \mu\leq d_i}
\big( (\hhb1p\hhb1\theta_{\mu,{\hhb{.5}\rho}}+p-1)+1\big)\\
&=&(\imath_{1,{\hhb{.5}\rho}+1}+1)+ \textstyle\sum_{1 < \mu\leq d_i}
(\theta_{\mu,{\hhb{.5}\rho}+1}+1)\\
&=&\textstyle\sum_{1\leq\mu\leq d_i}
(\theta_{\mu,{\hhb{.5}\rho}+1}+1).
\end{eqnarray*}
\underbar{Case~2}: ${\hhb{.5}\rho} +1 = r_{i+1}$. Then ${\hhb{.5}\rho}+1$
is an essential jump index, thus by definitions one has:
$\imath_{{\hhb{.5}\rho}+1}=p\hhb1\imath_{\hhb{.5}\rho}+p\hhb1q_{{\hhb{.5}\rho} +1}+\epsilon_{{\hhb{.5}\rho}+1}$
with $0< q_{{\hhb{.5}\rho}+1}$ and $0 < \epsilon_{{\hhb{.5}\rho}+1} < p$,
$d_{i+1}=d_i+q_{{\hhb{.5}\rho}+1}$, $r_{i+1}\leq{\hhb{.5}\rho} +1< r_{i+2}$.
On the other hand, by the induction hypothesis one has
$\imath_{\hhb{.5}\rho}=\theta_{1,{\hhb{.5}\rho}}+
\sum_{1< \mu\leq d_i}(\theta_{\mu,{\hhb{.5}\rho}}+1)$.
Therefore, using the definitions of $\theta_{\mu,{\hhb{.5}\rho}}$ we get:
\begin{eqnarray*}
\imath_{{\hhb{.5}\rho}+1} +1
&=&p\hhb1\imath_{\hhb{.5}\rho}+p\hhb1q_{{\hhb{.5}\rho} +1}+\epsilon_{{\hhb{.5}\rho}+1}+1\\
&=&(p\hhb1\theta_{1,{\hhb{.5}\rho}}+\epsilon_{{\hhb{.5}\rho}+1}+1)+
\textstyle\sum_{1<\mu\leq d_i}
(\hhb1p\hhb1\theta_{\mu,{\hhb{.5}\rho}}+p)
+p\hhb1q_{{\hhb{.5}\rho} +1}\\
&=&(\imath_{1,{\hhb{.5}\rho}+1}+1)+
\textstyle\sum_{1 < \mu\leq d_i}
\big( (\hhb1p\hhb1\theta_{\mu,{\hhb{.5}\rho}}+p-1)+1\big)+
\textstyle\sum_{d_i < \mu\leq d_{i+1}}\big((p-1)+1\big)\\
&=&(\imath_{1,{\hhb{.5}\rho}+1}+1)+ \textstyle\sum_{1 < \mu\leq d_{i+1}}
(\theta_{\mu,{\hhb{.5}\rho}+1}+1)\\
&=&\textstyle\sum_{1\leq\mu\leq d_{i+1}}
(\theta_{\mu,{\hhb{.5}\rho}+1}+1).
\end{eqnarray*}
This completes the proof of Lemma~\ref{combinlemma}.
\end{proof}
\vskip5pt
\noindent
B) {\it Generic liftings\/}
\vskip7pt
Let $k[[t]]\hookrightarrow k[[z]]$ be a ${\fam\lvfam\tenlv Z}/p^e$-cyclic extension
with upper ramification jumps $\imath_1\leq\dots\leq\imath_e$.
Recall that setting $K=k\lps t$ and $L=k\lps z$, in the
notations introduced in section~2,~B), there exists
${\undr p}=\big(p_1({t^{-1}}) ,\dots,p_e({t^{-1}}) \big)$ such that
$L=K_{\undr p}$, where ${\undr p}$ is in standard form, i.e.,
either $p_\alp({t^{-1}})=0$ or it contains no non-zero terms in
which the exponent of ${t^{-1}}$ is divisible by~$p$. And by the
discussion in~section~2,~B), one has that
\[
\imath_\alp=\max\,\{\hhb2p\hhb1\imath_{\alp-1},\dgr{p_\alp({t^{-1}})}\},
\quad\alp=1,\dots,e.
\]
\begin{fact}
\label{fact2}
Let ${\undr c}:=\big(h_1({t^{-1}}),\dots,h_e({t^{-1}})\big)$ be an
arbitrary Witt vector with coordinates in~$k[{t^{-1}}]$.
For a fixed polynomial $h({t^{-1}})\in k[{t^{-1}}]$, let ${\undr c}_{h,i}$
be the Witt vector whose $i^{\rm th}$ coordinate
is $h({t^{-1}})$, and all the other coordinates are equal
to~$0\in k[{t^{-1}}]$. Then $\tilde {\undr c}:={\undr c}+{\undr c}_{h,i}$
has coordinates $\tilde{\undr c}=\big(\tilde h_1({t^{-1}}),\dots,
\tilde h_e({t^{-1}})\big)$ satisfying the following:
\begin{itemize}
\item[a)] $\tilde h_j({t^{-1}})=h_j({t^{-1}})$ for $j < i$.
\vskip2pt
\item[b)] $\tilde h_i({t^{-1}})=h_i({t^{-1}})+h({t^{-1}})$.
\vskip2pt
\item[c)] $\dgr{\tilde h_j({t^{-1}})}\leq
\max\,\{\hhb2\dgr{h_j({t^{-1}})},\,p^{j-i}\dgr{h({t^{-1}})}\}$
for all $i < j$.
\end{itemize}
\end{fact}
\vfill\eject
\begin{definition/remark}
\label{normaliz}
$\hhb1$
\vskip2pt
1) In the above context, let
${\undr q}:=\big(q_1({t^{-1}}),\dots,q_e({t^{-1}})\big)$ with
$q_\alp({t^{-1}})\in k[{t^{-1}}]$ be some generator of $L|K$, i.e.,
$L=K_{\undr q}$. We say that ${\undr q}$ is \defi{normalized}
if $\imath_\alp=\dgr{q_\alp({t^{-1}})}$, $\alp=1,\dots,e$. And
we say that ${\undr q}$ is separable, if each $q_\alp({t^{-1}})$
is a separable polynomial (in ${t^{-1}}$).
\vskip2pt
2) We notice that if
${\undr q}:=\big(q_1({t^{-1}}),\dots,q_e({t^{-1}})\big)$
is some given Witt vector and $L:=K_{\undr q}$, then
${\undr q}$ is normalized if and only if it satisfies:
$\dgr{q_1({t^{-1}})}$ is prime to~$p$, and for all $1\leq\alp < e$
one has that $p\hhb1|\hhb1\dgr{q_{\alp+1}({t^{-1}})}$
implies $\dgr{q_{\alp+1}({t^{-1}})}=p\,\dgr{q_\alp({t^{-1}})}$.
\vskip2pt
3) Given a generator
${\undr p}=\big(p_1({t^{-1}}),\dots,p_e({t^{-1}})\big)$
in standard form for $L|K$, one can construct
a separable normalized generator
${\undr q}=\big(q_1({t^{-1}}),\dots,q_e({t^{-1}})\big)$
as follows: Consider Witt vectors of the form
${\undr c}:=\big(h_1({t^{-1}}),\dots,h_e({t^{-1}})\big)$
with $h_\alp({t^{-1}})\in k[{t^{-1}}]$ and
$\dgr{h_\alp({t^{-1}})}=\imath_{\alp-1}$ for $1< \alp\leq e$,
which are ``inductively generic'' with those properties.
Then setting
\[
{\undr q}=:{\undr p}+\wp_e({\undr c})=:
\big(q_1({t^{-1}}),\dots,q_e({t^{-1}})\big),
\]
it follows that $K_{\undr q}=L=K_{\undr p}$, thus ${\undr q}$
is a representative for ${\undr p}$ modulo $\wp_e(K)$.
And applying inductively Fact~\ref{fact2} above,
one gets: If $p\hhb1 \imath_{\alp-1}<\imath_\alp$, then
$\dgr{h_\alp({t^{-1}})}=\dgr{p_\alp({t^{-1}})}=\imath_\alp$.
Second, if $p\hhb1 \imath_{\alp-1}=\imath_\alp$, then
$\dgr{p_\alp({t^{-1}})} < \imath_\alp=p\hhb1\imath_{\alp-1}$. Hence
applying Fact~\ref{fact2} inductively, since $h_\alp({t^{-1}})$
is generic of degree $\imath_{\alp-1}$, we get:
$\dgr{q_\alp({t^{-1}})}=p\hhb1\dgr{h_{\alp-1}({t^{-1}})}
=p\hhb1\imath_{\alp-1}=\imath_\alp$, and
each $q_\alp({t^{-1}})$ is separable.
\end{definition/remark}
Coming back to our general context, let
${\undr p}=\big(p_1({t^{-1}}),\dots,p_e({t^{-1}})\big)$
be a {\it normalized generator\/} for $L|K$. Let
$(\theta_{\mu,{\hhb{.5}\rho}})_{1\leq\mu\leq N,\,1\leq{\hhb{.5}\rho}\leqe}$
be the matrix of non-negative integers produced
in the previous subsection~A). Since $k$ is algebraically
closed, we can write each polynomial $p_\alp({t^{-1}})$
as a product of polynomials $p_{\mu,\alp}({t^{-1}})$ as follows:
\[
p_\alp({t^{-1}})=\textstyle{\prod}_{1\leq\mu\leq N}
\,p_{\mu,\alp}({t^{-1}}),
\]
with $\dgr{p_{1,\alp}({t^{-1}})}=\theta_{1,\alp}$,
$\dgr{p_{\mu,\alp}({t^{-1}})}=\theta_{\mu,\alp}+1$ for
$\mu >1, \theta_{\mu,\alp}\neq0$,
$p_{\mu,\alp}=1$ if $\theta_{\mu,\alp}=0$.
\vskip5pt
For the given elements $x_\mu\in\upi{\fam\eufam\teneu o}$, $\mu=1,\dots,N$
we set ${t}_\mu:={t}-x_\mu\in{\fam\eufam\teneu o}[{t}]$. And for a
fixed choice $p_{\mu,\alp}({t^{-1}})\in k[{t^{-1}}]$, we let
$P_{\mu,\alp}(\TT_\mu^{-1})\in{\fam\eufam\teneu o}[\TT_\mu^{-1}]$
be ``generic'' preimages with
$\dgr{P_{\mu,\alp}(\TT_\mu^{-1})}=
\dgr{p_{\mu,\alp}({t^{-1}})}$ for $\mu=1,\dots,N$
and $\alp=1,\dots,e$. In particular, each
\[
P_\alp:=\textstyle\prod_\mu\,P_{\mu,\alp}(\TT_\mu^{-1})
\in{\fam\eufam\teneu o}[{t}^{-1}_{x_1},\dots,{t}^{-1}_{x_N}]\subset
{\mathcal A}_{x_1,\dots,x_N}, \quad \alp=1,\dots,e
\]
is a linear combination of monomials in
$({t}-x_1)^{-1},\dots,({t}-x_N)^{-1}$ with ``general''
coefficients from ${\fam\eufam\teneu o}$ such that under the specialization
homomorphism ${\mathcal A}_{x_1,\dots,x_N}\to A[{t^{-1}}]$ they
map to $p_\alp=p_\alp({t^{-1}})$. To indicate this, we will write
for short
\[
P_{\mu,\alp}(\TT_\mu^{-1})\mapsto p_{\mu,\alp}({t^{-1}}),
\quad P_\alp\mapsto p_\alp({t^{-1}}).
\]
\vskip4pt
We set ${\undr P}:=(P_1,\dots,P_e)$ and view it as a Witt vector
of length $e$ over ${\mathcal K}$, and consider the corresponding
cyclic field extension ${\mathcal L}:={\mathcal K}_{\undr P}$. Let ${\mathcal A}\hookrightarrow{\mathcal B}$
be the normalization of ${\mathcal A}$ in~${\mathcal K}\hookrightarrow{\mathcal L}$. Since
${\mathcal A}=k[[\upi,t]]$ is Noetherian and ${\mathcal K}\hookrightarrow{\mathcal L}$ is separable,
it follows that ${\mathcal B}$ is a finite ${\mathcal A}$-algebra, thus Noetherian.
And since ${\mathcal A}$ is local and complete, so is ${\mathcal B}$.
\vskip4pt
We next have a closer look at the branching in the
finite ring extension ${\mathcal A}\hookrightarrow{\mathcal B}$. For that we view
${\mathcal A}\hookrightarrow{\mathcal B}$ as a finite morphism of ${\fam\eufam\teneu o}:=k[[\upi]]$
algebras, and introduce geometric language as
follows: ${\mathcal X}=\Spec{\mathcal A}$ and ${\mathcal Y}=\Spec{\mathcal B}$.
Thus ${\mathcal A}\hookrightarrow{\mathcal B}$ defines a finite ${\fam\eufam\teneu o}$-morphism
${\mathcal Y}\to{\mathcal X}$. Further let ${\fam\eufam\teneu Y}:=\Spec{\mathcal S}\to\Spec{\mathcal R}=:{\fam\eufam\teneu X}$
and $Y:=\Spec{\mathcal B}/(\upi)\to\Spec{\mathcal A}/(\upi)=:X$ be
the generic fiber, respectively the special fiber
of ${\mathcal Y}\to{\mathcal X}$. In particular, $X=\Spec A$ and $Y\to X$
is a finite morphism. We further mention the following
general fact for later use:
\begin{fact}
\label{fact3}
Let ${\mathcal K}\hookrightarrow{\mathcal L}$ be a cyclic extension of degree
$[{\mathcal L}:{\mathcal K}]=p^e$ with Galois group $G={\fam\lvfam\tenlv Z}/p^e$,
say defined by some Witt vector
${\undr a}=(a_1,\dots,a_e)\inW_{\!\nx}({\mathcal K})$.
For every $0\leq m\leq e$ let ${\mathcal K}\hookrightarrow{\mathcal L}_m$
be the unique sub-extension of ${\mathcal K}\hookrightarrow{\mathcal L}$ with
$[{\mathcal L}_m:{\mathcal K}]=p^m$, hence ${\mathcal L}_0={\mathcal K}$ and
${\mathcal L}_e={\mathcal L}$. Let $v$ be a discrete valuation of ${\mathcal K}$,
say with valuation ring ${\mathcal O}_v\subset{\mathcal K}$ and
residue field ${\mathcal O}_v\to\kp v$, $f\mapsto\overline f$, and
let $T_v\subseteq Z_v\subseteq G$ be the inertia,
respectively decomposition, subgroups of $v$ in
$G$. Then for all~$m$ with $1\leq m\leq e$ the
following hold:
\vskip2pt
\begin{itemize}
\item[1)] Let $v(a_1),\dots,v(a_m)\geq0$. Then
$T_v\subseteq p^mG$, and $Z_v\subset p^mG$
iff $(\overline a_1,\dots,\overline a_m)\in{\rm im}(\wp_m)$.
\vskip3pt
\item[2)] If $v(a_m)$ is negative and prime to $p$,
then $p^{m-1}G\subseteq T_v$.
\end{itemize}
\end{fact}
\vskip2pt
We notice that since ${\mathcal A}=k[[\upi,t]]$ is a two dimensional
local regular ring, ${\mathcal X}$ is a two dimensional regular
scheme. Therefore, the branch locus of ${\mathcal Y}\to{\mathcal X}$ is
of pure co-dimension one. Thus in order to describe the
branching behavior of ${\mathcal Y}\to{\mathcal X}$ one has to describe
the branching at the generic point $(\upi)$ of the special
fiber $X\subset{\mathcal X}$ of ${\mathcal X}$, and at the closed points
$x$ of the generic fiber ${\fam\eufam\teneu X}\subset{\mathcal X}$ of ${\mathcal X}$.
\vskip5pt
\noindent
$\bullet$ {\it The branching at $(\upi)$\/}
\vskip4pt
We recall that ${\mathcal K}\hookrightarrow{\mathcal L}$ is defined as a cyclic
extension by ${\undr P}:=(P_1,\dots,P_e)$, where
each $P_\alp$ is of the form $P_\alp=\prod_\mu
P_{\mu,\alp}(\TT_\mu^{-1})$ with $P_{\mu,\alp}
(\TT_\mu^{-1})\in{\fam\eufam\teneu o}[\TT_\mu^{-1}]\subset{\mathcal A}_{x_1,\dots,x_N}$
is some generic preimage
of $p_{\mu,\alp}({t^{-1}})$ with degree satisfying
$\dgr{P_{\mu,\alp}(\TT_\mu^{-1})}=\dgr{p_{\mu,\alp}({t^{-1}})}$.
In particular, the elements $P_{\mu,\alp}(\TT_\mu^{-1})$
are not divisible by $\upi$ in the factorial ring
${\mathcal A}_{x_1,\dots,x_N}$. Therefore, the elements
$P_{\mu,\alp}(\TT_\mu^{-1})$ are units ${\mathcal A}_{(\upi)}$.
By Fact~\ref{fact3} we conclude that $\upi$ is not
branched in ${\mathcal K}\hookrightarrow{\mathcal L}$, and therefore, the
special fiber $Y\to X$ of ${\mathcal Y}\to{\mathcal X}$ is reduced.
Moreover, since $P_1\mapsto p_1({t^{-1}})$, and
the latter satisfies $p_1({t^{-1}})\not\in\wp(K)$, it
follows by~Fact~\ref{fact3},~1), that ${\rm Gal}({\mathcal L}|{\mathcal K})$
is contained in the decomposition group of $v_{\upi}$.
In other words, $\upi$ is totally inert in
${\mathcal K}\hookrightarrow{\mathcal L}$. In particular, ${\mathcal Y}\to{\mathcal X}$ is \'etale
above $\upi$, and moreover, the special fiber $Y\to X$
of ${\mathcal Y}\to{\mathcal X}$ is reduced, irreducible, and generically
cyclic Galois of degree $p^e=[L:K]$.
\vskip5pt
\noindent
$\bullet$ {\it The branching at the points of the generic
fiber $x\in{\fam\eufam\teneu X}$\/}
\vskip4pt
Recall that $\kk=k\lps\upi$ and that ${\fam\eufam\teneu X}=\Spec{\mathcal R}$,
where ${\mathcal R}={\mathcal A}\otimes_{k[[\upi]]}\hhb{-1}\kk$ is the
ring of power series in $t$ with bounded coefficients
from the complete discrete valued field $\kk$.
[Thus ${\fam\eufam\teneu X}$ is actually the rigid open unit disc
over $\kk$.] If $x\in{\fam\eufam\teneu X}$ is a closed point
different from $x_1,\dots,x_N$, and ${\mathcal A}_x$
is the local ring of ${\fam\eufam\teneu X}$ at $x$, it follows
that $P_1,\dots,P_e \in {\mathcal A}_{\fam\eufam\teneu p}$. Hence by
Fact~\ref{fact3},~1), it follows that $x$ is not branched
in ${\mathcal K}\hookrightarrow{\mathcal L}$. Thus it is left to analyze the branching behavior of
${\fam\eufam\teneu Y}\to{\fam\eufam\teneu X}$ at the closed points $x_1,\dots,x_N\in{\fam\eufam\teneu X}$.
In this process we will also compute the total contribution
of the ramification above $x_\mu$ to the total
different~${\fam\eufam\teneu D}_{{\mathcal S}|{\mathcal R}}$ for $\mu=1,\dots,N$.
\vskip3pt
Recall that every $x_\mu$ is a $\kk$ rational point
of ${\fam\eufam\teneu X}$, and ${t}_\mu:={t}-x_\mu$ is the ``canonical''
uniformizing parameter at $x_\mu$. Thus
${\mathcal K}_\mu:=\kk\lps{{t}_\mu}$ is the quotient field
of the completion of the local ring at $x_\mu$, and
we denote by $v_\mu:{\mathcal K}_\mu^\times\to{\fam\lvfam\tenlv Z}$ the
canonical valuation at $x_\mu$. We notice that
${t}_{\nu}=x_\mu-x_\nu +{t}_{\mu}$, hence by
the ``genericity'' of $P_{\mu,\alp}(\TT_\mu^{-1})$,
we can and \underbar{will} suppose that
$P_{\nu,\alp}({t}^{-1}_{\nu})$ is a $v_\mu$-unit
in ${\mathcal K}_\mu$. We conclude that there exist
$v_\mu$-units $\eta_1,\dots\eta_e \in {\mathcal K}_\mu$
such that denoting by ${\mathcal L}_\mu$ the compositum
of ${\mathcal K}_\mu$ and ${\mathcal L}$, the cyclic extension
${\mathcal K}_\mu\hookrightarrow{\mathcal L}_\mu$ is defined by the Witt vector:
\vskip3pt
\centerline{${\undr P}_\mu=\big(\eta_1P_{\mu,1}(\TT_\mu^{-1}),
\dots,\eta_e P_{\mu,e}(\TT_\mu^{-1})\big)$.}
\vskip5pt
\indent
\underbar{Case 1}: $\mu=1$. \
\vskip3pt
\noindent
First, by definitions we have
$\dgr{P_{1,\alp}({t}^{-1}_{1})}=
\dgr{p_{1,\alp}({t^{-1}})}=\theta_{1,\alp}$
for~all~$\alp$. Second, by the definitions of
$(\theta_{1,\alp})_{1\leq\alp\leqe}$ it follows that
$\big(p_{1,1}({t^{-1}}),\dots,p_{1,e}({t^{-1}})\big)$ is
actually a normalized system of polynomials
in $k[{t^{-1}}]$. Hence by Definition/Remark~\ref{normaliz},~3),
it follows that the upper ramification jumps of
${\mathcal K}_1\hookrightarrow{\mathcal L}_1$ are precisely
$\theta_{1,1}\leq\dots\leq\theta_{1,e}$, and in particular,
one has $[{\mathcal L}_1:{\mathcal K}_1]=p^e$.
\vskip5pt
\underbar{Case 2}: $1< \mu$, hence one has $0 < N_0$ as well.
\vskip3pt
\noindent
In the notations from the previous subsection~A), let
$1\leq i \leq N_0$ maximal be such that $d_{i-1}<\mu$. Then
by the definition of $(\theta_{\mu,\alp})_{1\leq\alp\leqe}$
we have: $\theta_{\mu,\alp}=0$ for $\alp< r_i$,
$\theta_{\mu,r_i}=p-1$, and $\theta_{\mu,\alp}=
p\hhb1\theta_{\mu,\alp-1}+p-1$
for $r_i\leq\alp\leqe$. Further, again by definitions,
one has $P_{\mu,\alp}(\TT_\mu^{-1})=1$ for
$\theta_{\mu,\alp}=0$, i.e., for $\alp < r_i$. And
$P_{\mu,\alp}(\TT_\mu^{-1})$ is generic of degree
$\theta_{\mu,\alp}+1$ for $r_i\leq\alp\leq e$. Hence
the Witt vector
${\undr P}_\mu:=\big(\eta_1P_{\mu,1}(\TT_\mu^{-1}),
\dots,\eta_e P_{\mu,e}(\TT_\mu^{-1})\big)$
satisfies the following conditions:
\vskip2pt
- \ $\eta_\alp P_{\mu,{\hhb{.5}\rho}}(\TT_\mu^{-1})=\eta_\alp$
for $1\leq\alp< r_i$.
\vskip2pt
- \ $\eta_\alp P_{\mu,{\hhb{.5}\rho}}(\TT_\mu^{-1})$ is a
generic polynomial of degree $\theta_{\mu,\alp}+1$
for $r_i\leq\alp\leqe$.
\vskip3pt
\noindent
In particular, by Fact~\ref{fact3}, one has that
$v_\mu$ is unramified in the sub-extension
${\mathcal K}\hookrightarrow{\mathcal L}_{r_i-1}$ of degree $p^{r_i-1}$ of
${\mathcal K}\hookrightarrow{\mathcal L}$. We claim that ${\mathcal L}_{r_i-1}\hookrightarrow{\mathcal L}$
is actually totally ramified, or equivalently, that
${\rm Gal}({\mathcal L}|{\mathcal L}_{r_i-1})\subseteq {\rm Gal}({\mathcal L}|{\mathcal K})$
is the inertia group of $v_\mu$, hence the ramification
subgroup of $v_\mu$, because there is no tame
ramification involved. Indeed, it is sufficient to
prove that this is the case after base changing
everything to the maximal unramified extension
${\mathcal K}_\mu\hookrightarrow\clK^{^{\rm nr}}_\mu$ of ${\mathcal K}_\mu$. Recall that for
${\hhb{.5}\rho}$ with $r_i\leq\alp\leq e$ we have by definitions that
$P_{\mu,{\hhb{.5}\rho}}(\TT_\mu^{-1})\in{\fam\eufam\teneu o}[\TT_\mu^{-1}]$
is a generic polynomial in $\TT_\mu^{-1}$ over
${\fam\eufam\teneu o}=k[[\upi]]$ of degree $\theta_{\mu,{\hhb{.5}\rho}}+1$. Hence
for ${\hhb{.5}\rho}=r_i$ we have: $\theta_{\mu,r_1}+1$ is
divisible by $p$ and $P_{\mu,r_i}(\TT_\mu^{-1})$
is generic. But then it follows that the standard
representative $Q_{\mu,r_i}(\TT_\mu^{-1})
\in\scl\kk[\TT_\mu^{-1}]$ of $P_{\mu,{\hhb{.5}\rho}}(\TT_\mu^{-1})$
modulo $\wp(\clK^{^{\rm nr}}_\mu)$ has degree $\theta_{\mu,{\hhb{.5}\rho}}$.
Thus by Fact~\ref{fact3},~3), it follows that $v_\mu$ is totally
ramified in the field extension $\clK^{^{\rm nr}}_\mu\hookrightarrow\clL^{^{\rm nr}}_\mu$
and that $p^{e-r_i+1}=[\clL^{^{\rm nr}}_\mu:\clK^{^{\rm nr}}_\mu]$. Combining
this with the fact that $v_\mu$ is unramified in
${\mathcal K}\hookrightarrow{\mathcal L}_{r_i-1}$ and $p^{r_i-1}=[{\mathcal L}_{r_i-1}:{\mathcal K}]$,
it follows that ${\mathcal K}\hookrightarrow{\mathcal L}_{r_i-1}$ is the ramification
field of $v_\mu$ in ${\mathcal K}\hookrightarrow{\mathcal L}$. Equivalently,
${\rm Gal}({\mathcal L}|{\mathcal L}_{r_i-1})\subseteq{\rm Gal}({\mathcal L}|{\mathcal K})$
is the ramification subgroup of $v_\mu$.
\vskip2pt
We next compute the degree of the local different
of ${\mathcal R}\hookrightarrow{\mathcal S}$ above $x_\mu$. Recall that
${\mathcal K}\hookrightarrow{\mathcal L}$ is defined by the Witt vector
\[
{\undr P}_\mu=\big(\eta_1P_{\mu,1}(\TT_\mu^{-1}),
\dots,\eta_e P_{\mu,e}(\TT_\mu^{-1})\big).
\]
On the other hand, ${\undr P}_\mu$ is equivalent modulo
$\wp_e(\clK^{^{\rm nr}})$ to its standard form
\[
{\undr P}'_\mu=\big(Q_{\mu,1}(\TT_\mu^{-1}),
\dots,Q_{\mu,e}(\TT_\mu^{-1})\big)
\]
with $Q_{\mu,\alp}=0$ for $1\leq\alp< r_i$
and $\dgr{Q_{\mu,e}(\TT_\mu^{-1})}=\theta_{\mu,\alp}$
for $r_i\leq\alp\leq e$. Hence setting
\[
{\undr Q}_\mu=\big(Q_{\mu,r_i}(\TT_\mu^{-1}),
\dots,Q_{\mu,e}(\TT_\mu^{-1})\big),
\]
it follows that ${\undr Q}_\mu$ is a Witt vector of length
$e-r_i+1$, and ${\undr Q}_\mu$ is in standard form, and
the cyclic extension $\clK^{^{\rm nr}}_\mu\hookrightarrow\clL^{^{\rm nr}}_\mu$ is defined
by ${\undr Q}_\mu$. Hence setting $e_\mu:=e-r_i+1$,
it follows that the cyclic field extension
$\clK^{^{\rm nr}}_\mu\hookrightarrow\clL^{^{\rm nr}}_\mu$ has degree $p^{e_\mu}$
and upper ramification jumps given by
\vskip9pt
\centerline{\hhb{20}$(*)$\hhb{60}
$\imath_{\mu,\nu}=\dgr{Q_{\mu,r_i+\nu}(\TT_\mu^{-1})}=
\theta_{\mu,\hhb1r_i+\nu-1},\quad \nu=1,\dots, e_\mu$.\hhb{80}}
\vskip15pt
\noindent
\vfill\eject
\noindent
C) \ {\it Finishing the proof of Key Lemma~\ref{keylemma1}\/}
\vskip10pt
Let ${\fam\eufam\teneu D}_\mu$ be the local part above $x_\mu$ of
the global different ${\fam\eufam\teneu D}_{\!{\mathcal S}|{\mathcal R}}$ of the
extension of Dedekind rings ${\mathcal R}\hookrightarrow{\mathcal S}$. Then
if ${\mathcal K}\hookrightarrow{\mathcal L}^Z\hookrightarrow{\mathcal L}^T\hookrightarrow{\mathcal L}$ are the
decomposition/inertia subfields of $v_\mu$ in the
cyclic field extension ${\mathcal K}\hookrightarrow{\mathcal L}$, by the functorial
behavior of the different, it follows that
\[
\deg({\fam\eufam\teneu D}_\mu)=[\kappa(x_\mu):\kk]\cdot[{\mathcal L}^T:{\mathcal K}]\cdot
\deg({\fam\eufam\teneu D}_{\!\clL^{^{\rm nr}}_\mu|\clK^{^{\rm nr}}_\mu}).
\]
On the other hand, since $x_\mu\in{\fam\eufam\teneu m}_{\fam\eufam\teneu o}$, one
has $\kappa(x_\mu)=\kk$. Further, by the discussion
above one has that ${\mathcal L}^T={\mathcal L}_{r_i-1}$,
thus $p^{r_i-1}=[{\mathcal L}^T:{\mathcal K}]$. And
$\deg({\fam\eufam\teneu D}_{\!\clL^{^{\rm nr}}_\mu|\clK^{^{\rm nr}}_\mu})$ can be
computed in terms of upper ramification jumps as
indicated at the end of~section~1),~A):
\[
\deg({\fam\eufam\teneu D}_{\!\clL^{^{\rm nr}}_\mu|\clK^{^{\rm nr}}_\mu})=
\textstyle\sum_{1\leq\nu\leqe_\mu}
(\imath_{\mu,\nu}+1)(p^\nu-p^{\nu-1}).
\]
Hence taking into account the discussion above, we get:
\begin{eqnarray*}
\deg({\fam\eufam\teneu D}_\mu)
&=&[\kappa(x_\mu):\kk]\cdot[{\mathcal L}^T:{\mathcal K}]\cdot
\deg({\fam\eufam\teneu D}_{\!\clL^{^{\rm nr}}_\mu|\clK^{^{\rm nr}}_\mu})\\
&=&p^{r_i-1}\textstyle\sum_{1\leq\nu\leqe_\mu}
(\imath_{\mu,\nu}+1)(p^\nu-p^{\nu-1})\\
&=&\textstyle\sum_{1\leq\nu\leqe_\mu}
(\theta_{\mu,r_i+\nu-1}+1)(p^{\nu+r_i-1}-p^{(\nu+r_i-1)-1})\\
&=&\textstyle\sum_{r_i\leq{\hhb{.5}\rho}\leqe}(\theta_{\mu,{\hhb{.5}\rho}}+1)
(p^{{\hhb{.5}\rho}}-p^{{\hhb{.5}\rho}-1})
\end{eqnarray*}
\vskip5pt
\noindent
Recall that $\imath_{\hhb{.5}\rho}+1=\sum'_\mu(\theta_{\mu,{\hhb{.5}\rho}}+1)$
for all $1\leq\alp\leq e$, where $\sum'_\mu$ is taken
over all $\mu$ with $\theta_{\mu,{\hhb{.5}\rho}}\neq0$. Further,
$\deg({\fam\eufam\teneu D}_1)=\sum_{1\leq{\hhb{.5}\rho}\leqe}
(\theta_{1,{\hhb{.5}\rho}}+1)(p^{\hhb{.5}\rho}-p^{{\hhb{.5}\rho}-1})$
and $\deg({\fam\eufam\teneu D}_\mu)=\sum_{r_i\leq{\hhb{.5}\rho}\leqe}
(\theta_{\mu,{\hhb{.5}\rho}}+1)(p^{\hhb{.5}\rho}-p^{{\hhb{.5}\rho}-1})$
for all $1<\mu\leq N$. Therefore we get the following:
\begin{eqnarray*}
\deg({\fam\eufam\teneu D}_{{\mathcal S}|{\mathcal R}})
&=&\micsm{1\leq\mu\leq N}{}\deg({\fam\eufam\teneu D}_\mu)\\
&=&\micsm{1\leq{\hhb{.5}\rho}\leqe}{}(\theta_{1,{\hhb{.5}\rho}}+1)(p^{\hhb{.5}\rho}-p^{{\hhb{.5}\rho}-1})
+\micsm{\ 1\leq i\leq N_0}{}\,\micsm{d_{i-1}<\mu\leq d_i \ }{}
\micsm{r_i\leq\alp\leqe}{}(\theta_{\mu,{\hhb{.5}\rho}}+1)(p^{\hhb{.5}\rho}-p^{{\hhb{.5}\rho}-1})\\
&=&\micsm{1\leq{\hhb{.5}\rho}\leqe}{}(\theta_{1,{\hhb{.5}\rho}}+1)(p^{\hhb{.5}\rho}-p^{{\hhb{.5}\rho}-1})
+\micsm{\ 1\leq i\leq N_0}{}\,\micsm{r_i\leq\alp< r_{i+1}}{}\,
\micsm{d_0<\mu\leq d_i \ }{}(\theta_{\mu,{\hhb{.5}\rho}}+1)(p^{\hhb{.5}\rho}-p^{{\hhb{.5}\rho}-1})\\
&=&\micsm{\ 1\leq i\leq N_0}{}\,\micsm{\ r_i\leq\alp< r_{i+1}}{}\,
\micsm{1\leq\mu\leq d_i \ }{}(\theta_{\mu,{\hhb{.5}\rho}}+1)(p^{\hhb{.5}\rho}-p^{{\hhb{.5}\rho}-1})\\
&=&\micsm{\ 1\leq i\leq N_0}{}\,\micsm{\ r_i\leq\alp< r_{i+1}}{}\,
(\imath_{\hhb{.5}\rho}+1)(p^{\hhb{.5}\rho}-p^{{\hhb{.5}\rho}-1})\\
&=&\micsm{1\leq\alp\leqe}{}(\imath_{\hhb{.5}\rho}+1)(p^{\hhb{.5}\rho}-p^{{\hhb{.5}\rho}-1})\\
&=&\deg({\fam\eufam\teneu D}_{L|K}).
\end{eqnarray*}
We thus conclude the proof of Key Lemma~\ref{keylemma1}
by applying Kato's criterion~Fact~\ref{KCR}.
\vskip7pt
\noindent
D) {\it Characteristic $p$ global Oort Conjecture\/}
\begin{theorem}
\label{charpOC}
{\rm (Characteristic $p$ global Oort conjecture)} \ \
In the notations from the Key Lemma~\ref{keylemma1},
let $Y\to X$ be a (ramified) Galois cover of complete
smooth $k$-curves having only cyclic groups as inertia
groups, and set ${\mathcal X}_{\fam\eufam\teneu o}:=X\times_k{\fam\eufam\teneu o}$. Then there
exists a $G$-cover of complete smooth ${\fam\eufam\teneu o}$-curves
${\mathcal Y}_{\fam\eufam\teneu o}\to {\mathcal X}_{\fam\eufam\teneu o}$ with special fiber $\,Y\to X$
such that the generic fiber ${\mathcal Y}_\kk\to {\mathcal X}_\kk$ of
${\mathcal Y}_{\fam\eufam\teneu o}\to {\mathcal X}_{\fam\eufam\teneu o}$ has no essential ramification.
\end{theorem}
\begin{proof}
First, as in the case of the classical Oort Conjecture,
the local-global principle for lifting (ramified) Galois
covers, see \nmnm{Garuti}~\cite{Ga1},~\S3, as well
as \nmnm{Saidi}~\cite{Sa},~\S1.2, where the proofs
of Propositions~1.2.2 and~1.2.4 are very detailed,
reduces the proof of the Theorem~\ref{charpOC} to the
corresponding local problem over ${\fam\eufam\teneu o}$. Further,
exactly as in the case of the classical local Oort
Conjecture, the local problem is equivalent to the case
where the inertia groups are cyclic $p$-groups. One
concludes by applying the~Key~Lemma~\ref{keylemma1}.
\end{proof}
\section{Proof of Theorem~\ref{OC}}
\noindent
A) \ {\it Generalities about covers of $\,{\fam\lvfam\tenlv P}^1$\/}
\vskip2pt
\begin{notations}
\label{notanota}
We begin by introducing notations concerning
families of covers of curves which will be used
throughout this section. Let $S$ be a separated,
integral normal scheme, e.g., $S=\Spec A$ with
$A$ and integrally closed domain, and
$\kbm:=\kappa(S)$ its field of rational functions.
Let $\kbm(t)\hookrightarrow F$ be a finite extension of the
rational function field $\kbm(t)$.
\vskip2pt
1) $\lvPt S=\Proj{\fam\lvfam\tenlv Z}[t_0,t_1]\times S$ is the
$t$-projective line over $S$, where $t=t_1/t_0$
is the canonical parameter on $\lvPt S$. In particular,
$\lvPt S$ is the gluing of its canonical affine lines
over~$S$, namely $\lvAt S:=\Spec{\fam\lvfam\tenlv Z}[t]\times S$
and ${\fam\lvfam\tenlv A}^1_{{t^{-1}}\!,\,S}:=\Spec{\fam\lvfam\tenlv Z}[{t^{-1}}]\times S$.
\vskip2pt
2) Let $\kbm(t)\hookrightarrow F$ be a finite extension, and
${\mathcal Y}_{t,S}\to \lvAt S$ and ${\mathcal Y}_{{t^{-1}},S}\to{\fam\lvfam\tenlv A}^1_{{t^{-1}},S}$
the corresponding normalizations in $\kbm(t)\hookrightarrow F$.
Then the normalization ${\mathcal Y}_S\to\lvPt S$ of $\lvPt S$
in $\kbm(t)\hookrightarrow F$ is nothing but the gluing of
${\mathcal Y}_{t,S}\to \lvAt S$ and ${\mathcal Y}_{{t^{-1}}\!,\,S}\to{\fam\lvfam\tenlv A}^1_{{t^{-1}}\!,\,S}$.
\vskip2pt
3) For every ${\fam\eufam\teneu p}\in S$ we denote by $\overline{\fam\eufam\teneu p}\hookrightarrow S$
the closure of ${\fam\eufam\teneu p}$ in $S$ (endowed with the reduced
scheme structure). We denote by ${\mathcal O}_{\fam\eufam\teneu p}:={\mathcal O}_{S,{\fam\eufam\teneu p}}$
the local ring at ${\fam\eufam\teneu p}\in S$. We set $S_{\fam\eufam\teneu p}:=\Spec{\mathcal O}_{\fam\eufam\teneu p}$
and consider the canonical morphism $S_{\fam\eufam\teneu p}\hookrightarrow S$.
We notice that ${\fam\eufam\teneu p}\hookrightarrow S$ is both the generic fiber of
$\overline{\fam\eufam\teneu p}\hookrightarrow S$ and the special fiber of $S_{\fam\eufam\teneu p}\hookrightarrow S$
at ${\fam\eufam\teneu p}$. We get corresponding base changes:
\[
{\mathcal Y}_{\hhb1\overline{\fam\eufam\teneu p}}\to\lvPt{\overline{\fam\eufam\teneu p}}, \quad
{\mathcal Y}_{S_{\fam\eufam\teneu p}}\to\lvPt{S_{\fam\eufam\teneu p}},\quad
{\mathcal Y}_{\fam\eufam\teneu p}\to\lvPt{\fam\eufam\teneu p}
\]
where ${\mathcal Y}_{\fam\eufam\teneu p}\to\lvPt{\fam\eufam\teneu p}$ is both the generic fiber
of ${\mathcal Y}_{\hhb1\overline{\fam\eufam\teneu p}}\to\lvPt{\overline{\fam\eufam\teneu p}}$ and the special
fiber of ${\mathcal Y}_{S_{\fam\eufam\teneu p}}\to\lvPt{S_{\fam\eufam\teneu p}}$.
\vskip2pt
4) Finally, affine schemes will be sometimes replaced
by the corresponding rings. Concretely, if $S=\Spec A$,
and $\kbm={\rm Quot}(A)$, for a finite extension
$\kbm(t)\hookrightarrow F$ one has/denotes:
\begin{itemize}
\vskip2pt
\item[a)] The $t$-projective line over $A$ is
$\lvPt A=\Spec A[t]\cup\Spec A[{t^{-1}}]$,
and the normalization ${\mathcal Y}_A\to\lvPt A$ of $\lvPt A$
in $\kbm(t)\hookrightarrow F$ is obtained as the gluing of
$\Spec{\mathcal R}_t\to\Spec A[t]$ and
$\Spec{\mathcal R}_{{t^{-1}}}\to\Spec A[{t^{-1}}]$, where ${\mathcal R}_t$,
respectively ${\mathcal R}_{{t^{-1}}}$, are the integral closures of $A[t]$,
respectively of $A[{t^{-1}}]$, in the field extension $\kbm(t)\hookrightarrow F$.
\vskip2pt
\item[b)] For ${\fam\eufam\teneu p}\in\Spec(A)$ one has/denotes:
${\mathcal Y}_{A/{\fam\eufam\teneu p}}\to\lvPt{A/{\fam\eufam\teneu p}}$ and
${\mathcal Y}_{A_{\fam\eufam\teneu p}}\to\lvPt{A_{\fam\eufam\teneu p}}$ are the base
changes of ${\mathcal Y}_A\to\lvPt A$ under $A{\hhb1\to\hhb{-16}\to\hhb0} A/{\fam\eufam\teneu p}$,
respectively $A\hookrightarrow A_{\fam\eufam\teneu p}$; and finally, the fiber
${\mathcal Y}_{\kp{\fam\eufam\teneu p}}\to\lvPt{\kp{\fam\eufam\teneu p}}$ of ${\mathcal Y}_A\to\lvPt A$
is both, the special fiber of ${\mathcal Y}_{A_{\fam\eufam\teneu p}}\to\lvPt{A_{\fam\eufam\teneu p}}$
and the generic fiber of ${\mathcal Y}_{A/{\fam\eufam\teneu p}}\to\lvPt{A/{\fam\eufam\teneu p}}$.
\end{itemize}
\end{notations}
In the above notations, suppose that $A={\mathcal O}$ is a
local ring with maximal ideal ${\fam\eufam\teneu m}$ and residue
field $\kp{\fam\eufam\teneu m}$. Let ${\mathcal O}_v$ be a valuation ring
of $\kbm$ domination ${\mathcal O}$ and having $\kp{\fam\eufam\teneu m}=\kp v$.
We denote by ${\mathcal Y}_{\mathcal O}\to\lvPt{\mathcal O}$ and
${\mathcal Y}_{{\mathcal O}_v}\to\lvPt{{\mathcal O}_v}$ the corresponding
normalizations of the the corresponding projective
lines. The canonical morphism $\Spec{\mathcal O}_v\to\Spec{\mathcal O}$
gives canonically a commutative diagrams dominant
morphisms of the form:
\[
\begin{matrix}
{\mathcal Y}_{{\mathcal O}_v}&\to&\lvPt{{\mathcal O}_v}&\hhb{20}&
{\mathcal Y}_\kbm&\to&{\mathcal Y}_{{\mathcal O}_v}&\leftarrow&{\mathcal Y}_v\cr
\dwn{}&&\dwn{}&&\dwn{\cong}&&\dwn{}&&\dwn{}\cr
{\mathcal Y}_{{\mathcal O}}&\to&\lvPt{{\mathcal O}}&&
{\mathcal Y}_\kbm&\to&{\mathcal Y}_{{\mathcal O}}&\leftarrow&{\mathcal Y}_{\fam\eufam\teneu m}\cr
\end{matrix}
\]
We denote by $\eta_{\fam\eufam\teneu m}\in\lvPt{\fam\eufam\teneu m}$ the generic
point of the special fiber of $\lvPt{\mathcal O}$, and by
$\eta_{{\fam\eufam\teneu m},i}\in{\mathcal Y}_{\fam\eufam\teneu m}$ the generic points of the
special fiber of ${\mathcal Y}_{\mathcal O}$. Correspondingly,
$\eta_v\in\lvPt v$ is the generic point of the special
fiber of $\lvPt{{\mathcal O}_v}$, and $\eta_{v,j}\in{\mathcal Y}_v$ are
the generic points of the special fiber of ${\mathcal Y}_{{\mathcal O}_v}$.
We notice that $\lvPt v \to \lvPt{\fam\eufam\teneu m}$ is an isomorphism
(because $\kp{\fam\eufam\teneu m}=\kp v$),
and by the valuation criterion for completeness, for every
$\eta_{{\fam\eufam\teneu m},i}$ there exists some $\eta_{v,j}$ such that
$\eta_{v,j}\mapsto\eta_{{\fam\eufam\teneu m},i}$ under ${\mathcal Y}_v\to{\mathcal Y}_{\fam\eufam\teneu m}$.
The local ring ${\mathcal O}_{\eta_v}$ of $\eta_v\in\lvPt{{\mathcal O}_v}$
is the valuation ring of the so called \defi{Gauss valuation}
$v_t$ of $\kbm(t)$, thus ${\mathcal O}_{\eta_{v,j}}$ are the valuation
rings of the prolongations $v_j$ of $v_t$ to $F$. Finally,
for every complete $k$-curve $C$ we denote by $g_C$ the
geometric genus of $C$.
\begin{lemma}
\label{preplemma}
In the above notations, let $Y_{{\fam\eufam\teneu m},1}\to{\mathcal Y}_{{\fam\eufam\teneu m},1}$
be the normalization of ${\mathcal Y}_{{\fam\eufam\teneu m},1}$. Suppose that
$[\kappa(\eta_{{\fam\eufam\teneu m},1}):\kp{\fam\eufam\teneu m}(t)]\geq[F:\kbm(t)]$ and
that ${\mathcal Y}_\kbm$ is smooth. Then the following hold:
\vskip2pt
{\rm1)} The special fibers ${\mathcal Y}_{\fam\eufam\teneu m}$ and ${\mathcal Y}_v$ are
reduced and irreducible.
\vskip2pt
{\rm2)} If $g_{Y_{{\fam\eufam\teneu m},1}}\geq g_{{\mathcal Y}_\kbm}$, then
${\mathcal Y}_{{\mathcal O}_v}\to\lvPt{{\mathcal O}_v}$ is a cover of smooth
${\mathcal O}_v$-curves.
\end{lemma}
\begin{proof}
To 1): Let $\eta_{v,1}\mapsto\eta_{{\fam\eufam\teneu m},1}$.
Then $\kp{\eta_{{\fam\eufam\teneu m},1}}\hookrightarrow\kp{\eta_{v,1}}$, thus
$[\kp{\eta_{{\fam\eufam\teneu m},1}}:\kp{\fam\eufam\teneu m}(t)]\leq[\kp{\eta_{v,1}}:\kp v(t)]$.
Hence using the fundamental equality and the hypothesis
one gets that
\[
[\kappa(\eta_{v,1}):\kp v(t)]\geq[\kappa(\eta_{{\fam\eufam\teneu m},1}):\kp{\fam\eufam\teneu m}(t)]\geq
[F:\kbm(t)]={\textstyle\sum}_j[\kp{v_j}:\kp{v(t)}]e(v_j|v_t)\delta(v_j|v_t),
\]
where $e(\cdot|\cdot)$ is the ramification index and
$\delta(\cdot|\cdot)$ is the Ostrowski defect. Hence we
conclude that $w_1$ is the only prolongation of $v_t$ to $F$,
and $e(v_1|v_t)=1=\delta(v_1|v_t)$.
\vskip2pt
To 2): Since ${\mathcal Y}_v$ is reduced and irreducible, by
\nmnm{Roquette}~\cite{Ro1}, Satz~I, it follows that the
Euler characteristics of the special fiber ${\mathcal Y}_v$ and
that of the generic fiber ${\mathcal Y}_\kbm$ of ${\mathcal Y}_{{\mathcal O}_v}$
are equal:
\[
\chi({\mathcal Y}_\kbm|\kbm)=\chi({\mathcal Y}_v|\kp v).
\]
Since ${\mathcal Y}_v$ dominates ${\mathcal Y}_{\fam\eufam\teneu m}$, one has
$\kappa({\mathcal Y}_{\fam\eufam\teneu m})\hookrightarrow\kappa({\mathcal Y}_v)$, hence one
has $g_{{\mathcal Y}_{\fam\eufam\teneu m}}\leq g_{{\mathcal Y}_v}$. Thus~2)
implies:
\[
1-g_{Y_{{\fam\eufam\teneu m},1}}\leq1-g_{{\mathcal Y}_\kbm}=\chi({\mathcal Y}_\kbm|\kbm)=
\chi({\mathcal Y}_v|\kp v)\leq\chi(Y_v|\kp v)\leq\chi(Y_{{\fam\eufam\teneu m},1}|\kp{\fam\eufam\teneu m})
=1-g_{Y_{{\fam\eufam\teneu m},1}}.
\]
Hence all the above are equalities, thus finally one has
that $({\mathcal Y}_v|\kp v)\leq\chi(Y_v|\kp v)$. Therefore, the
normalization $Y_v\to{\mathcal Y}_v$ is an isomorphism, and
${\mathcal Y}_{{\mathcal O}_v}$ is smooth.
\end{proof}
In the context above, let $A={\mathcal O}$ be a {\it valuation ring\/}
and $v$ be its valuation. Let $v:=v_0\circ v_1$ be the
valuation theoretical composition of two valuations, say
with valuation rings ${\mathcal O}_1\subset\kbm$, respectively
${\mathcal O}_0\subset\kbm_0$, where $\kbm_0:=\kbm v_1$ is
the residue field of~$v_1$. Then $\kbm v=:k:=\kbm_0v_0$
is the residue field of both $v$ and $v_0$. Let $t \in F$
is a fixed function, and $t_0:=tv_{0,t}$ be the residue
of $t$ with to the Gauss valuation $v_{0,t}$ on $\kbm(t)$.
Suppose that the following hold:
\vskip2pt
$\hhb2$i) The special fiber ${\mathcal Y}_{1,\hhb1s}$ of the normalization
${\mathcal Y}_1\to\lvPt{{\mathcal O}_1}$ of $\lvPt{{\mathcal O}_1}$ in $\kbm(t)\hookrightarrow F$
is irreducible. Thus $v_{1,t}$ has a unique prolongation
$w_1$ to $F$, and $F_0:=Fw_1=\kappa({\mathcal Y}_{1,\hhb1s})$.
\vskip2pt
ii) The special fiber ${\mathcal Y}_{0,\hhb1s}$ of the normalization
${\mathcal Y}_0\to\lvPt{{\mathcal O}_0}$ of $\lvPt{{\mathcal O}_0}$ in
$\kbm_0(t_0)\hookrightarrow F_0$ is irreducible. Thus $v_{0,t_0}$
has a unique prolongation $w_0$ to $F_0$, and
$F_0w_0=\kappa({\mathcal Y}_{0,\hhb1s})$.
\begin{lemma}
\label{transsmooth}
{\rm(Transitivity of smooth covers)}
In the above notations, suppose that the hypotheses~i),~ii)
are satisfied. Set $w:=w_0\circ w_1$, and let
$\,{\mathcal Y}\to\lvPt{{\mathcal O}}$ be the normalization of $\lvPt{{\mathcal O}}$
in $\kbm(t)\hookrightarrow F$. Then $w$ is the unique prolongation
of $v_t$ to $F$, and the following hold:
\begin{itemize}
\item[{\rm 1)}] The base change of $\,{\mathcal Y}$ under
${\mathcal O}\hookrightarrow{\mathcal O}_{v_1}$
is $\,{\mathcal Y}_1={\mathcal Y}\times_{\mathcal O}\clO_{v_1}$ canonically, thus
${\mathcal Y}_{1,\hhb1s}={\mathcal Y}_{{\fam\eufam\teneu m}_1}$ is the fiber of $\,{\mathcal Y}$ at
the valuation ideal ${\fam\eufam\teneu m}_1\in\Spec{\mathcal O}$ of $v_1$.
\vskip2pt
\item[{\rm 2)}] Let ${\mathcal Y}_{{\mathcal O}_0}\to\lvPt{{\mathcal O}_0}$
be the base change of $\,{\mathcal Y}\to\lvPt{\mathcal O}$ under the
${\mathcal O}{\hhb1\to\hhb{-16}\to\hhb0}{\mathcal O}_0$. Then~$\,{\mathcal Y}_{{\fam\eufam\teneu m}_1}$ is the generic fiber
of ${\mathcal Y}_{{\mathcal O}_0}$ and ${\mathcal Y}_0\to\lvPt{{\mathcal O}_0}$ is the
normalization of ${\mathcal Y}_{{\mathcal O}_0}\to\lvPt{{\mathcal O}_0}$.
\end{itemize}
In particular, ${\mathcal Y}$ is a smooth ${\mathcal O}$-curve if and only
if ${\mathcal Y}_1$ is a smooth ${\mathcal O}_1$ curve and ${\mathcal Y}_0$ is a
smooth ${\mathcal O}_0$-curve.
\end{lemma}
\begin{proof} Klar, by the discussion above, and \nmnm{Roquette}
\cite{Ro1},~Satz~I, combined with the fact that a projective
curve is smooth if and only if its arithmetic genus equal its
geometric genus.
\end{proof}
\vskip5pt
\noindent
B) \ {\it A specialization result\/}
\vskip5pt
We begin by recalling the following two well known facts.
The first one is by \nmnm{Katz} (and \nmnm{Gabber})~\cite{Ka}:
Let $k$ be an algebraically closed field with $\chr(k)=p$. Then
the localization at $t=0$ defines a bijection between the finite
Galois $p$-power degree covers of $\lvPt k$ unramified
outside $t=0$ and the finite Galois $p$-power extensions
$k[[t]]\hookrightarrow k[[z]]$, and this bijection preserves the ramification
data. Thus given a cyclic ${\fam\lvfam\tenlv Z}/p^f\,$-cover $k[[t]]\hookrightarrow k[[z]]$,
there exists a unique cyclic ${\fam\lvfam\tenlv Z}/p^f\,$ cover of complete
smooth curves $Y_k\to\lvPt k$ which is branched only at
$t=0$ (thus totally branched there) such that $k[[t]]\hookrightarrow k[[z]]$
is the extension of local rings of $Y\to\lvPt k$ above $t=0$.
We will say that $Y\to\lvPt k$ is the \defi{KG\hhb1-\hhb1cover} for
$k[[t]]\hookrightarrow k[[z]]$.
The second fact is the local-global principle for the Oort
Conjecture, see e.g.\ \nmnm{Garuti}~\cite{Ga1},~\S3,
\nmnm{Saidi}~\cite{Sa},~\S1.2, especially~Proposition~1.2.4,
which among other things imply:
\begin{LGP}
\label{LGP}
Let $k[[t]]\hookrightarrow k[[z]]$ be a ${\fam\lvfam\tenlv Z}/p^f$-extension and
$Y_k\to\lvPt k$ be its KG-cover. Further let $W(k)\hookrightarrow R$ be
a finite extension of $W(k)$. Then the ${\fam\lvfam\tenlv Z}/p^f$-extension
$k[[t]]\hookrightarrow k[[z]]$ has a smooth lifting over $R$ if and only
if the ${\fam\lvfam\tenlv Z}/p^f$-cover $Y_k\to\lvPt k$ has a smooth
lifting over $R$.
\end{LGP}
Next let $f$ be a fixed positive integer, and
consider a finite sequences of positive numbers
$\uix:=(\imath_1\leq\dots\leq\imath_f)$ satisfying: $1\leq\imath_1$
is prime to $p$, and $\imath_{\nu+1}=p\hhb1\imath_\nu+\epsilon$
with $\epsilon\geq0$ and $\epsilon$ prime to $p$ if $\epsilon>0$.
For such a sequence $\uix$, let $|\uix|=\imath_1+\dots+\imath_f$
and consider ${\undr P}_{\uix}=\big(P_1,\dots,P_f)$ a
sequence of generic polynomials $P_\nu=P_\nu({t^{-1}})$ of
degrees $\deg\big(P_\nu\big)=\imath_\nu$ for $1\leq\nu\leqf$.
In other words, all the coefficients $a_{\nu,{\hhb{.5}\rho}}$,
$1\leq\nu\leqf$, $1\leq{\hhb{.5}\rho}\leq\imath_\nu$ of the polynomials
$P_\nu$ are independent free variables over ${k_0}:=\overline{\fam\lvfam\tenlv F}_p$.
Let $A_{\uix}:={k_0}[(a_{\nu,{\hhb{.5}\rho}})_{\nu,{\hhb{.5}\rho}}]$ be the
corresponding polynomial ring and ${\lvA^{\hhb{-1}{\scriptscriptstyle|}\uix{\scriptscriptstyle|}}}=\Spec A_{\uix}$
the resulting affine space over ${k_0}$.
\vskip2pt
For every $x\in{\lvA^{\hhb{-1}{\scriptscriptstyle|}\uix{\scriptscriptstyle|}}}$ let $k_x$ be any algebraically
closed field extension of ${k_0}$, and $\overline x\in{\lvA^{\hhb{-1}{\scriptscriptstyle|}\uix{\scriptscriptstyle|}}}(k_x)$
be a $k_x$-rational point of ${\lvA^{\hhb{-1}{\scriptscriptstyle|}\uix{\scriptscriptstyle|}}}$ defined by a
${k_0}$-embedding $\phi_x:\kp x\hookrightarrow k_x$. Let
${\undr p}_{\uix,x}=(p_{1,x},\dots,p_{f,x})$ and
${\undr p}_{\uix,\overline x}=(p_{1,{\overline x}},\dots,p_{f,{\overline x}})$ be
the images of ${\undr P}_\uix$ over $\kp x$, respectively
$k_x$. Then one has virtually by definitions that
$p_{\nu,{\overline x}}=\phi_x(p_{\nu,x})$, thus
${\undr p}_{\uix,{\overline x}}=\phi_x({\undr p}_{\uix,x})$.
In particular, if $\deg(P_\nu)=\deg(p_\nu)$ for all $\nu$,
then ${\undr p}_{\uix,x}$ gives rise to a cyclic extension
$k_x[[t]]\hookrightarrow k_x[[z_x]]$ of degree $p^f$ and upper
jumps $\uix=(\imath_1,\dots,\imath_f)$, and canonically to
its KG-cover $Y_{k_x}\to\lvPt{k_x}$.
\begin{definition} For ${k_0}\hookrightarrow k_x$ as above, let
$k_x[[t]]\hookrightarrow k_x[[z_x]]$ be a cyclic ${\fam\lvfam\tenlv Z}/p^f$-extension
and $Y_{k_x}\to\lvPt{k_x}$ be its $KG$-cover. We say that
$k_x[[t]]\to k_x[[z_x]]$ is an $\uix$-\defi{extension} at $x\in{\lvA^{\hhb{-1}{\scriptscriptstyle|}\uix{\scriptscriptstyle|}}}$
and that $Y_{k_x}\to\lvPt{k_x}$ is an $\uix$-KG-\defi{cover}
at $x\in{\lvA^{\hhb{-1}{\scriptscriptstyle|}\uix{\scriptscriptstyle|}}}$, if $k_x[[t]]\hookrightarrow k_x[[z_x]]$ has
$\uix=(\imath_1,\dots,\imath_f)$ as upper ramification jumps.
\end{definition}
\begin{notations}
\label{notalast}
We denote by $\Sigma_\uix\subset{\lvA^{\hhb{-1}{\scriptscriptstyle|}\uix{\scriptscriptstyle|}}}$ the set of all
$x\in{\lvA^{\hhb{-1}{\scriptscriptstyle|}\uix{\scriptscriptstyle|}}}$ which satisfy: There exists some mixed
characteristic valuation ring $R_x$ with residue field
$k_x$ such that some $\uix$-KG-cover $Y_{k_x}\to\lvPt{k_x}$
has a smooth lifting over $R_x$.
\end{notations}
\begin{proposition}
\label{specprop}
In Notations~\ref{notalast}, suppose that $\Sigma_\uix\subseteq{\lvA^{\hhb{-1}{\scriptscriptstyle|}\uix{\scriptscriptstyle|}}}$
is Zariski dense. Then there exists an algebraic integer
$\pi_{\uix}$ such that for every algebraically closed field
$k$ of characteristic $\chr(k)=p$ one has: Every $\uix$-KG-cover
$Y_k\to\lvPt k$ has a smooth lifting over $W(k)[\pi_{\uix}]$.
\end{proposition}
\begin{proof} The proof is quite involved, and has
two main steps as follows:
\vskip7pt
\noindent
{\bf Step 1}. {\it Proving that the generic point
$\eta_\uix\in{\lvA^{\hhb{-1}{\scriptscriptstyle|}\uix{\scriptscriptstyle|}}}$ lies in $\Sigma_\uix$\/}
\vskip5pt
Let $\fam\eufam\teneu U$ be an ultrafilter on $\Sigma$
which contains all the Zariski open subsets of $\Sigma$.
(Since $\Sigma$ is Zariski dense in the irreducible
scheme ${\lvA^{\hhb{-1}{\scriptscriptstyle|}\uix{\scriptscriptstyle|}}}\!$, any Zariski open subset of $\Sigma$
is dense as well, thus ultrafilter $\fam\eufam\teneu U$ exist.)
Let $k_x\to\Theta_x\subset R_x$ be any set of
representatives for for $R_x$. Consider the following
ultraproducts index by $\Sigma$:
\[
\str2k:=\ultrb{k_x}\to\str1\Theta:=\ultrb{\Theta_x}
\subset {\str1W}:=\ultrb{W(k_x)}\hookrightarrow\ultrb{R_x}=:\str2R.\ \
\]
By general model theoretical principles, it follows that
$\str2R$ is a valuation ring having residue field equal to
$\str2k$, and $\str1\Theta\subset\str2R$ is a system of
representatives for the residue field $\str2k$ of $\str2R$.
\vskip5pt
Next, coming to geometry, by general model theoretical
principles, it follows that the family of ${\fam\lvfam\tenlv Z}/p^e$-covers
$Y_x\to\lvPt{k_x}$ with upper ramification jumps
$\uix=(\imath_1,\dots,\imath_f)$ gives rise to a ${\fam\eufam\teneu U}$-\defi{generic}
${\fam\lvfam\tenlv Z}/p^f$-cover $Y_{\str2k}\to\lvPt{\str2k}$ of complete
smooth $\str2k$-curves with upper ramification jumps~$\uix$.
Precisely, setting $\str1{p_\nu}:=(p_{\nu,x})_x/{\fam\eufam\teneu U}$,
the system of polynomials
$\str1{\undr p}_{\uix}=
\big(\str1{p_1}({t^{-1}}),\dots,\str1{p_f}({t^{-1}})\big)$
defines the local extension $\str2k[[t]]\hookrightarrow\str2k[[z]]$ of
$Y_{\str2k}\to\lvPt{\str2k}$ at $t=0$. Moreover, consider
the $\str2k$-rational point $\str3\phi_{\eta_\uix}$ of ${\lvA^{\hhb{-1}{\scriptscriptstyle|}\uix{\scriptscriptstyle|}}}$
defined by
\[
\str3\phi_{\eta_\uix}:\kp{\eta_\uix}\to\str2k, \quad
\str3\phi_{\eta_\uix}(a_{\nu,{\hhb{.5}\rho}}):=
\big(\phi_x(a_{\nu,{\hhb{.5}\rho}})\big)_{\!x\,}/{\fam\eufam\teneu U}.
\leqno{\indent(*)}
\]
Then $\str1{\undr p}_\uix=\str3\phi_{\eta_\uix}({\undr P}_\uix)$,
which means that $\str3\phi_{\eta_\uix}$ is the
$\str2k$-rational point of ${\lvA^{\hhb{-1}{\scriptscriptstyle|}\uix{\scriptscriptstyle|}}}$ defining $\str1{\undr p}_\uix$.
\vskip4pt
Again, by general model theoretical principles for
ultraproducs of (covers of) curves, the family of the
${\fam\lvfam\tenlv Z}/p^f$-covers ${\mathcal Y}_{R_x}\to\lvPt{R_x}$ with special
fiber $Y_x\to\lvPt{k_x}$ gives rise to a ${\fam\lvfam\tenlv Z}/p^f$-cover
${\mathcal Y}_{\str2R}\to\lvPt{\str2R}$ of complete smooth
$\str2R$-curves, with $Y_{\str2k}\to\lvPt{\str1k}$ as special
fiber.
\vskip4pt
Let ${\lvA^{\hhb{-1}{\scriptscriptstyle|}\uix{\scriptscriptstyle|}}}\hookrightarrow{\lvP^{{\scriptscriptstyle|$ be the canonical
embedding of the affine ${k_0}$-space
${\lvA^{\hhb{-1}{\scriptscriptstyle|}\uix{\scriptscriptstyle|}}}:=\Spec {k_0}[(a_{\nu,{\hhb{.5}\rho}})_{\nu,{\hhb{.5}\rho}}]$
into the corresponding projective ${k_0}$-space
${\lvP^{{\scriptscriptstyle|:=\Proj {k_0}[t_0, (t_{\nu,{\hhb{.5}\rho}})_{\nu,{\hhb{.5}\rho}}]$
via the $t_0$-dehom\-ogen\-iza\-tion
$a_{\nu,{\hhb{.5}\rho}}=t_{\nu,{\hhb{.5}\rho}}/t_0$. Letting
${\bm Z}_0:={\fam\lvfam\tenlv Z}^{^{\rm nr}}_p$ be the maximal unramified
extension, and
\[
{\lvA^{\hhb{-1=\Spec{\bm Z}_0[(a_{\nu,{\hhb{.5}\rho}})_{\nu,{\hhb{.5}\rho}}]\ \ {\rm and}
\ \ \lvPixZ=\Proj{\bm Z}_0[t_0, (t_{\nu,{\hhb{.5}\rho}})_{\nu,{\hhb{.5}\rho}}],
\]
the embedding ${\lvA^{\hhb{-1}{\scriptscriptstyle|}\uix{\scriptscriptstyle|}}}\hookrightarrow{\lvP^{{\scriptscriptstyle|$ is the special
fiber of ${\lvA^{\hhb{-1\hookrightarrow\lvPixZ$. Notice that
$\str3\phi_{\eta_\uix}:A_\uix\to\str2k$ gives rise via
$\str2k\to\str1\Theta$ canonically to an embedding
of ${\bm Z}_0$-algebras defined by
\[
\str3\phi_{{\scriptscriptstyle{\bm Z}_0}}:A_{\uix,{\scriptscriptstyle{\bm Z}_0}}:=
{\bm Z}_0[(a_{\nu,{\hhb{.5}\rho}})_{\nu,{\hhb{.5}\rho}}]\hookrightarrow\str2R,\quad
a_{\nu,{\hhb{.5}\rho}}\mapsto\str3{\phi_{\eta_\uix}}(a_{\nu,{\hhb{.5}\rho}}).
\]
Let ${V_{\scriptscriptstyle0}}$ be a projective normal ${\bm Z}_0$-scheme with function
field $\kappa({V_{\scriptscriptstyle0}})$ embeddable in ${\rm Quot}(\str2R)$, say
via $\kappa({V_{\scriptscriptstyle0}})\hookrightarrow{\rm Quot}(\str2R)$, such that the following
are satisfied:
\vskip4pt
1) Let ${\hhb{.5}\eup_{\hhb{-.5}\scriptscriptstyle0}\hhb{-.5}}\in {V_{\scriptscriptstyle0}}$ be the center of ${\str1w}$
of $\str2R$ on ${V_{\scriptscriptstyle0}}$ induced by $\kappa({V_{\scriptscriptstyle0}})\hookrightarrow\str2R$, and
${\clO_{\hhb{-.25}\eup}}$ the local ring of ${\hhb{.5}\eup_{\hhb{-.5}\scriptscriptstyle0}\hhb{-.5}}$. Then the ${\fam\lvfam\tenlv Z}/p^f$-cover
of complete smooth $\str2R$-curves ${\mathcal Y}_{\str2R}\to\lvPt{\str2R}$
is defined over ${\clO_{\hhb{-.25}\eup}}$.
\vskip4pt
2) The image of $\str3\phi_{\scriptscriptstyle{\bm Z}_0}:A_{\uix,{\scriptscriptstyle{\bm Z}_0}}\to\str2R$ is
contained in the image of $\kappa({V_{\scriptscriptstyle0}})\hookrightarrow\str2R$, and the
resulting embedding $A_{\uix,{\scriptscriptstyle{\bm Z}_0}}\hookrightarrow\kappa({V_{\scriptscriptstyle0}})$ is defined
by some proper morphism
\[
{V_{\scriptscriptstyle0}}\to\lvPixZ.
\]
We notice that condition~1) means that there exists a
${\fam\lvfam\tenlv Z}/p^f$-cover of complete smooth ${\clO_{\hhb{-.25}\eup}}$-curves
${\mathcal Y}_{{\clO_{\hhb{-.25}\eup}}}\to\lvPt{{\clO_{\hhb{-.25}\eup}}}$ such that
${\mathcal Y}_{\str2R}\to\lvPt{\str2R}$ is the base change
of ${\mathcal Y}_{{\clO_{\hhb{-.25}\eup}}}\to\lvPt{{\clO_{\hhb{-.25}\eup}}}$ under ${\clO_{\hhb{-.25}\eup}}\hookrightarrow\str2R$.
In particular, if ${\clO_{\hhb{-.25}\eup}}\to\kp{{\hhb{.5}\eup_{\hhb{-.5}\scriptscriptstyle0}\hhb{-.5}}}$ is the
residue field of ${\clO_{\hhb{-.25}\eup}}$, then the special fiber
$Y_{{\hhb{.5}\eup_{\hhb{-.5}\scriptscriptstyle0}\hhb{-.5}}}\to\lvPt{{\hhb{.5}\eup_{\hhb{-.5}\scriptscriptstyle0}\hhb{-.5}}}$ of
${\mathcal Y}_{{\clO_{\hhb{-.25}\eup}}}\to\lvPt{{\clO_{\hhb{-.25}\eup}}}$ is a ${\fam\lvfam\tenlv Z}/p^f$-cover
of complete smooth $\kp{{\hhb{.5}\eup_{\hhb{-.5}\scriptscriptstyle0}\hhb{-.5}}}$-curves whose
base change under $\kp{{\hhb{.5}\eup_{\hhb{-.5}\scriptscriptstyle0}\hhb{-.5}}}\hookrightarrow\str2k$ is canonically
isomorphic to $Y_{\str2k}\to\lvPt{\str2k}$. In other words,
the embedding $\str3\phi_{\eta_\uix}: A_\uix\hookrightarrow\str2k$
defined by~$(*)$ above factors
through $A_\uix\hookrightarrow\kp{\eta_\uix}\hookrightarrow\kp{{\hhb{.5}\eup_{\hhb{-.5}\scriptscriptstyle0}\hhb{-.5}}}$,
and ${\hhb{.5}\eup_{\hhb{-.5}\scriptscriptstyle0}\hhb{-.5}}\in {V_{\scriptscriptstyle0}}$ is mapped to the generic point
${\fam\eufam\teneu p}\mapsto\eta_\uix$ of the special fiber
$\eta_\uix\in{\lvP^{{\scriptscriptstyle|\hookrightarrow\lvPixZ$ under ${V_{\scriptscriptstyle0}}\to\lvPixZ$.
\vskip2pt
Recall that $\overline{\hhb{.5}\eup_{\hhb{-.5}\scriptscriptstyle0}\hhb{-.5}}\subset {V_{\scriptscriptstyle0}}$ the Zariski
closure of ${\hhb{.5}\eup_{\hhb{-.5}\scriptscriptstyle0}\hhb{-.5}}$ in ${V_{\scriptscriptstyle0}}$ viewed as a closed
${\bm Z}_0$-subscheme of~${V_{\scriptscriptstyle0}}$ endowed with the
reduced scheme structure.
Since ${\fam\eufam\teneu p}\mapsto\eta_\uix$, one has that
$\kp{\eta_\uix}\hookrightarrow\kp{{\hhb{.5}\eup_{\hhb{-.5}\scriptscriptstyle0}\hhb{-.5}}}$, hence $\kp{{\hhb{.5}\eup_{\hhb{-.5}\scriptscriptstyle0}\hhb{-.5}}}$
has characteristic~$p$, and ${\hhb{.5}\eup_{\hhb{-.5}\scriptscriptstyle0}\hhb{-.5}}$ lies in the special
fiber ${V_{\scriptscriptstyle0}}_{{k_0}}$ of ${V_{\scriptscriptstyle0}}\!$. We conclude that
$\overline{\hhb{.5}\eup_{\hhb{-.5}\scriptscriptstyle0}\hhb{-.5}}\subset {V_{\scriptscriptstyle0}}_{{k_0}}$.
\vskip2pt
Next, if ${\hhb{.5}\eup_{\hhb{-.5}\scriptscriptstyle0}\hhb{-.5}}$ has codimension $>1$, let
${\tilde V} \to {V_{\scriptscriptstyle0}}$ be the normalization of the blowup
of ${V_{\scriptscriptstyle0}}$ along the closed ${\bm Z}_0$-subscheme $\overline{\hhb{.5}\eup_{\hhb{-.5}\scriptscriptstyle0}\hhb{-.5}}$.
Let $E_1,\dots, E_r\subset{\tilde V}$ be the finitely
many irreducible components of the preimage of the
exceptional divisor of the blowup. Then the generic
points ${\fam\eufam\teneu r}_i$ of the $E_i$, $i=1,\dots,r$ are
precisely the points of codimension one of ${\tilde V}$
which map to ${\hhb{.5}\eup_{\hhb{-.5}\scriptscriptstyle0}\hhb{-.5}}$ under ${\tilde V}\to {V_{\scriptscriptstyle0}}$, and
$\cup_i E_i$ is the preimage of $\overline{\hhb{.5}\eup_{\hhb{-.5}\scriptscriptstyle0}\hhb{-.5}}$ in ${V_{\scriptscriptstyle0}}$.
Further, if ${\fam\eufam\teneu r}={\fam\eufam\teneu r}_i$ is fixed, and ${\tilde\clO}$
is the local ring of ${\fam\eufam\teneu r}\in {\tilde V}$ and $\kp{{\fam\eufam\teneu r}}$
is its residue field, it follows that ${\clO_{\hhb{-.25}\eup}}\hookrightarrow{\tilde\clO}$
and $\kp{{\hhb{.5}\eup_{\hhb{-.5}\scriptscriptstyle0}\hhb{-.5}}}\hookrightarrow\kp{{\fam\eufam\teneu r}}$ canonically. Recall that by
the property~1) above, ${\mathcal Y}_{{\clO_{\hhb{-.25}\eup}}}\to\lvPt{{\clO_{\hhb{-.25}\eup}}}$ is a
${\fam\lvfam\tenlv Z}/p^f$-cover of smooth ${\clO_{\hhb{-.25}\eup}}$-curves with special
fiber $Y_{{\hhb{.5}\eup_{\hhb{-.5}\scriptscriptstyle0}\hhb{-.5}}}\to\lvPt{{\hhb{.5}\eup_{\hhb{-.5}\scriptscriptstyle0}\hhb{-.5}}}$, whose base change
under $\kp{{\hhb{.5}\eup_{\hhb{-.5}\scriptscriptstyle0}\hhb{-.5}}}\hookrightarrow\str2k$ is $Y_{\str2k}\to\lvPt{\str2k}$.
Therefore, the base change ${\mathcal Y}_{{\tilde\clO}}\to\lvPt{{\tilde\clO}}$
of ${\mathcal Y}_{{\clO_{\hhb{-.25}\eup}}}\to\lvPt{{\clO_{\hhb{-.25}\eup}}}$ defined by the inclusion
${\clO_{\hhb{-.25}\eup}}\hookrightarrow{\tilde\clO}$ is a ${\fam\lvfam\tenlv Z}/p^f$ cover of proper
smooth ${\tilde\clO}$-curves whose special fiber
$Y_{{\fam\eufam\teneu r}}\to\lvPt{{\fam\eufam\teneu r}}$ is the base change
of $Y_{{\hhb{.5}\eup_{\hhb{-.5}\scriptscriptstyle0}\hhb{-.5}}}\to\lvPt{{\hhb{.5}\eup_{\hhb{-.5}\scriptscriptstyle0}\hhb{-.5}}}$ under
$\kp{{\hhb{.5}\eup_{\hhb{-.5}\scriptscriptstyle0}\hhb{-.5}}}\hookrightarrow\kp{{\fam\eufam\teneu r}}$. Hence choosing any
$\kp{{\hhb{.5}\eup_{\hhb{-.5}\scriptscriptstyle0}\hhb{-.5}}}$-embedding $\kp{{\fam\eufam\teneu r}}\hookrightarrow\str2k$, we get
that the special fiber $Y_{{\fam\eufam\teneu r}}\to\lvPt{{\fam\eufam\teneu r}}$
becomes $Y_{\str2k}\to\lvPt{\str2k}$ under
$\kp{{\fam\eufam\teneu r}}\hookrightarrow\str2k$.
\vskip2pt
Hence by replacing ${V_{\scriptscriptstyle0}}$ by ${\tilde V}$ if necessary, we
can suppose that ${\hhb{.5}\eup_{\hhb{-.5}\scriptscriptstyle0}\hhb{-.5}}\in {V_{\scriptscriptstyle0}}$ has codimension one,
or equivalently, that $\overline{\hhb{.5}\eup_{\hhb{-.5}\scriptscriptstyle0}\hhb{-.5}}\subset {V_{\scriptscriptstyle0}}_{{k_0}}$ is an
irreducible component of ${V_{\scriptscriptstyle0}}_{{k_0}}$.
\vskip4pt
By de Jong's theory of alterations \nmnm{de~Jong}
\cite{dJ},~Theorem~6.5, there exists a finite extension of
discrete valuation rings ${\bm Z}_0\hookrightarrow{\bm Z}:={\bm Z}_0[\pi_0]$ with
$\pi_0$ any uniformizing parameter of ${\bm Z}$ and a dominant
generically finite proper morphism $W\to {V_{\scriptscriptstyle0}}$ of projective
${\bm Z}_0$-schemes with $W$ {\it strictly semi-stable\/} over
${\bm Z}$, i.e., the generic fiber of $W$ is a smooth projective
variety over ${\rm Quot}({\bm Z})$, the special fiber $W_{\!{k_0}}$
is reduced and satisfies: If $W_{\!{k_0},j}$, $j\in J$ is any
set of $|J|$ distinct irreducible components of $W_{\!{k_0}}$,
then $\cap_j W_{\!{k_0},j}$ is a smooth subscheme of $W$
of codimension $|J|$. Hence the sequence of dominant
proper morphisms of projective ${\bm Z}_0$-schemes
\[
W\to {V_{\scriptscriptstyle0}}\to\lvPixZ
\]
satisfies: Let ${\euq}\in W$ denote
a fixed preimage of ${\hhb{.5}\eup_{\hhb{-.5}\scriptscriptstyle0}\hhb{-.5}}$. Then ${\euq}$ has codimension
one, because ${\hhb{.5}\eup_{\hhb{-.5}\scriptscriptstyle0}\hhb{-.5}}$ does so. Further, the local ring
${\clO_{\hhb{-.25}\euq}}$ of ${\euq}\in W$ as a point of $W$ dominates
the local ring ${\clO_{\hhb{-.25}\eup}}$ of ${\hhb{.5}\eup_{\hhb{-.5}\scriptscriptstyle0}\hhb{-.5}}\in {V_{\scriptscriptstyle0}}$, thus one has a
canonical inclusion ${\clO_{\hhb{-.25}\eup}}\hookrightarrow{\clO_{\hhb{-.25}\euq}}$ which gives rise
to a canonical inclusion of the residue fields
$
\kp{\eta_\uix}\hookrightarrow\kp{{\hhb{.5}\eup_{\hhb{-.5}\scriptscriptstyle0}\hhb{-.5}}}\hookrightarrow\kp{{\euq}} \ \
\hbox{corresponding to}\ \ {\fam\eufam\teneu q}\to{\fam\eufam\teneu p}\to\eta_\uix.
$
Recall that ${\mathcal Y}_{{\clO_{\hhb{-.25}\eup}}}\to\lvPt{{\clO_{\hhb{-.25}\eup}}}$ is a ${\fam\lvfam\tenlv Z}/p^f$-cover
of smooth ${\clO_{\hhb{-.25}\eup}}$-curves with special fiber
$Y_{{\hhb{.5}\eup_{\hhb{-.5}\scriptscriptstyle0}\hhb{-.5}}}\to\lvPt{{\hhb{.5}\eup_{\hhb{-.5}\scriptscriptstyle0}\hhb{-.5}}}$ whose base change under
$\kp{{\euq}}\hookrightarrow\str2k$ is $Y_{\str2k}\to\lvPt{\str2k}$.
Let ${\mathcal Y}_{{\clO_{\hhb{-.25}\euq}}}\to\lvPt{{\clO_{\hhb{-.25}\euq}}}$ be the
base change of ${\mathcal Y}_{{\clO_{\hhb{-.25}\eup}}}\to\lvPt{{\clO_{\hhb{-.25}\eup}}}$ under
${\clO_{\hhb{-.25}\eup}}\hookrightarrow{\clO_{\hhb{-.25}\euq}}$. Then ${\mathcal Y}_{{\clO_{\hhb{-.25}\euq}}}\to\lvPt{{\clO_{\hhb{-.25}\euq}}}$ is a
${\fam\lvfam\tenlv Z}/p^f$-cover of proper smooth ${\clO_{\hhb{-.25}\euq}}$-curves
whose special fiber $Y_{{\euq}}\to\lvPt{{\euq}}$
is the base change of $Y_{{\hhb{.5}\eup_{\hhb{-.5}\scriptscriptstyle0}\hhb{-.5}}}\to\lvPt{{\hhb{.5}\eup_{\hhb{-.5}\scriptscriptstyle0}\hhb{-.5}}}$
under $\kp{{\hhb{.5}\eup_{\hhb{-.5}\scriptscriptstyle0}\hhb{-.5}}}\hookrightarrow\kp{{\euq}}$. Again, choosing any
$\kp{{\hhb{.5}\eup_{\hhb{-.5}\scriptscriptstyle0}\hhb{-.5}}}$-embedding of $\kp{{\euq}}\hookrightarrow\str2k$, one
gets that the base change of the special fiber
$Y_{{\euq}}\to\lvPt{{\euq}}$ under $\kp{{\euq}}\hookrightarrow\str2k$
becomes $Y_{\str2k}\to\lvPt{\str2k}$. This means that the
embedding $\str3\phi_{\eta_\uix}: A_\uix\hookrightarrow\str2k$
defined at~$(*)$ above factors through
$A_\uix\hookrightarrow\kp{\eta_\uix}\hookrightarrow\kp{{\hhb{.5}\eup_{\hhb{-.5}\scriptscriptstyle0}\hhb{-.5}}}\hookrightarrow\kp{{\euq}}$,
reflecting the fact that ${\euq}\mapsto{\hhb{.5}\eup_{\hhb{-.5}\scriptscriptstyle0}\hhb{-.5}}\mapsto\eta_\uix$.
In other words, there exists a $\str2k$-rational point
$\str3\phi_{\fam\eufam\teneu q}:\kp{\fam\eufam\teneu q}\to\str2k$ such that the given
$\str2k$-rational point $\str3\phi_{\eta_\uix}:\kp{\eta_\uix}\to\str2k$
defined by $\str3\phi_{\eta_\uix}: A_\uix\to\Spec\str2k$ is of the form
\[
\str3\phi_{\eta_\uix}=
\str3\phi_{\fam\eufam\teneu q}\circ(\kp{\eta_\uix}\hookrightarrow\kp{\fam\eufam\teneu q}).
\leqno{\indent(**)}
\]
\vskip4pt
\noindent
{\bf Step 2}. {\it Finishing the proof of Proposition~\ref{specprop}\/}
\vskip5pt
Let ${\bm\lambda}:=\kappa(W)$ denote the function field of $W$, and
$F:=\kappa({\mathcal Y}_{{\clO_{\hhb{-.25}\euq}}})$ be the function field of ${\mathcal Y}_{{\clO_{\hhb{-.25}\euq}}}$.
Then ${\mathcal Y}_{{\clO_{\hhb{-.25}\euq}}}\to\lvPt{{\clO_{\hhb{-.25}\euq}}}$ has as generic fiber a
${\fam\lvfam\tenlv Z}/p^f$-cover of complete smooth ${\bm\lambda}$-curves
${\mathcal Y}_{\bm\lambda}\to\lvPt{{\bm\lambda}}$, and gives rise to a ${\fam\lvfam\tenlv Z}/p^f$
extension of function field in one variable ${\bm\lambda}(t)\hookrightarrow F$.
Since ${\clO_{\hhb{-.25}\euq}}$ is a (discrete) valuation ring, and
${\mathcal Y}_{{\clO_{\hhb{-.25}\euq}}}\to\lvPt{{\clO_{\hhb{-.25}\euq}}}$ is a cover of smooth
${\clO_{\hhb{-.25}\euq}}$-curves, it follows by the discussion in subsection~A),
that ${\mathcal Y}_{{\clO_{\hhb{-.25}\euq}}}\to\lvPt{{\clO_{\hhb{-.25}\euq}}}$ is precisely the normalization
of $\lvPt{{\clO_{\hhb{-.25}\euq}}}$ in the function field extension ${\bm\lambda}(t)\hookrightarrow F$.
Notice that $\lvPt{{\bm\lambda}}$ is the generic fiber of $\lvPt W$, and
consider
\[
{\mathcal Y}_W\to\lvPt W
\]
the normalization of $\lvPt W$ in the field extension
${\bm\lambda}(t)\hookrightarrow F$. We notice that the base change of
${\mathcal Y}_W\to\lvPt W$ under $\Spec{\clO_{\hhb{-.25}\euq}}\hookrightarrow W$ is
precisely ${\mathcal Y}_{{\clO_{\hhb{-.25}\euq}}}\to\lvPt{{\clO_{\hhb{-.25}\euq}}}$.
\begin{lemma}
\label{speclemma}
Let $x\in{\lvA^{\hhb{-1}{\scriptscriptstyle|}\uix{\scriptscriptstyle|}}}$ be such that the image
${\undr p}_{\uix,x}=(p_{1,x},\dots,p_{f,x})$ of
${\undr P}_\uix=(P_1,\dots,P_f)$ under
$A_\uix\to\kp x$ satisfies $\deg(p_{\nu,x})=\deg(P_\nu)$
for all $\nu=1,\dots,f$. Let $y\in\overline{\fam\eufam\teneu q}$ be a
preimage of $x\in{\lvA^{\hhb{-1}{\scriptscriptstyle|}\uix{\scriptscriptstyle|}}}\subset{\lvP^{{\scriptscriptstyle|$ under
$\overline{\fam\eufam\teneu q}\to\overline{\fam\eufam\teneu p}\to{\lvP^{{\scriptscriptstyle|$ and ${\mathcal O}_v$ be a
valuation ring dominating ${\clO_{\hhb{-.25}y}}$ with $\kp v=\kpy$.
Then ${\mathcal Y}_{{\mathcal O}_v}\to\lvPt{{\mathcal O}_v}$ is a cover of
smooth curves.
\end{lemma}
\begin{proof} Recall that ${\mathcal Y}_{{\clO_{\hhb{-.25}y}}}\to\lvPt{{\clO_{\hhb{-.25}y}}}$
is the base change of ${\mathcal Y}_W\to\lvPt W$ under the
canonical embedding $\Spec{\clO_{\hhb{-.25}y}}\hookrightarrow W$, and in
particular, ${\mathcal Y}_{{\clO_{\hhb{-.25}y}}}\to\lvPt{{\clO_{\hhb{-.25}y}}}$ is the normalization
of $\lvPt{{\clO_{\hhb{-.25}y}}}$ in the field extension ${\bm\lambda}(t)\hookrightarrow F$.
Since $y\in{\fam\eufam\teneu q}$, and the geometric fiber
${\mathcal Y}_{\fam\eufam\teneu q}\to\lvPt{\fam\eufam\teneu q}$ of ${\mathcal Y}_{\overline{\fam\eufam\teneu q}}\to\lvPt{\overline{\fam\eufam\teneu q}}$
is a ${\fam\lvfam\tenlv Z}/p^f$-cover of smooth complete curves,
the same holds correspondingly, if one replaces
${\clO_{\hhb{-.25}y}}={\mathcal O}_{W,y}$ by ${\euo_y}:={\mathcal O}_{\overline{\fam\eufam\teneu q},y}={\clO_{\hhb{-.25}y}}/{\fam\eufam\teneu q}$,
and ${\bm\lambda}(t)\hookrightarrow F$ by $\kp{\fam\eufam\teneu q}(t)\hookrightarrow F_{\fam\eufam\teneu q}$, where
$F_{\fam\eufam\teneu q}:=\kappa({\mathcal Y}_{\fam\eufam\teneu q})$ is viewed as function field
over $\kp{\fam\eufam\teneu q}$. Recall
that the local extension $\kp{\fam\eufam\teneu q}[[t]]\hookrightarrow\kp{\fam\eufam\teneu q}[[z_{\fam\eufam\teneu q}]]$
of ${\mathcal Y}_{\fam\eufam\teneu q}\to\lvPt{\fam\eufam\teneu q}$ at $t=0$ is defined by the
image ${\undr p}_{\uix,{\fam\eufam\teneu q}}$ of ${\undr P}_\uix$
under the canonical embedding
$A_\uix\hookrightarrow\kp{\eta_\uix}\hookrightarrow\kp{\fam\eufam\teneu q}$. On the other hand, if
${\euo_x}$ denotes the local ring of $x\in{\lvA^{\hhb{-1}{\scriptscriptstyle|}\uix{\scriptscriptstyle|}}}\subset{\lvP^{{\scriptscriptstyle|$
then $A_\uix\subset{\euo_x}$ and ${\euo_y}$ dominates~${\euo_x}$.
Hence $A_\uix\hookrightarrow{\euo_y}$ and therefore,
${\undr p}_{\uix,{\fam\eufam\teneu q}}$ is defined over ${\euo_y}$.
Further, by the commutativity of the diagrams
\[
\begin{matrix}
A_\uix&\hookrightarrow&{\euo_y}&&{\undr P}_\uix&\mapsto&{\undr p}_{\uix,{\fam\eufam\teneu q}}\cr
\dwn{}& &\dwn{}&\hhb{20} &\dwn{} & &\dwn{}\cr
\kp x &\hookrightarrow&\kpy & &{\undr p}_{\uix,x}&\mapsto&{\undr p}_{\uix,y}\cr
\end{matrix}
\]
it follows that the image of ${\undr p}_{\uix,{\fam\eufam\teneu q}}$ under
the residue homomorphism ${\euo_y}\to\kpy$ equals the
image of ${\undr p}_{\uix,x}$ under $\kp x\hookrightarrow\kpy$.
Thus by the functoriality of the Artin--Schreier--Witt theory,
it follows that every irreducible component of the special
fiber of $\lvPt{{\euo_y}}$ dominates the KG-cover of
$\lvPt{\kpy}$ defined by ${\undr p}_{\uix,y}$. Since
$\deg(p_{\nu,y})=\deg(p_{\nu,x})=\deg P_\nu$ for all $\nu$,
the latter cover must have degree $p^f$ and upper ramification
jumps $\uix=(\imath_1,\dots,\imath_f)$. In particular, we can
apply~Lemma~\ref{preplemma}, and conclude that
the special fibers ${\mathcal Y}_y$ and ${\mathcal Y}_v$ are reduced
and irreducible.
\vskip2pt
In order to conclude, we notice that by the discussion
above, the normalization $Y_y\to{\mathcal Y}_y$ dominates the
$\uix$-KG-cover of $\lvPty$ defined by ${\undr p}_{\uix,y}$.
Since every $\uix$-KG-cover has as genus a constant
depending on $\uix$ only, thus including the generic fiber
it follows that $g_{Y_y}\geq g_{{\mathcal Y}_{\fam\eufam\teneu q}}$. We thus
conclude the proof of Lemma~\ref{speclemma}
by applying~Lemma~\ref{preplemma}.
\end{proof}
Coming back to the proof of Proposition~\ref{specprop}
we proceed as follows. Let $k$ be any algebraically
closed field with $\chr(k)=p$, and $Y_k\to\lvPt{k}$ be
an $\uix$-KG-cover, say with local ring extension
$k[[t]]\hookrightarrow k[[z]]$ at $t=0$ defined by
${\undr p}_\uix=\big(p_1,\dots,p_f)$.
\vskip2pt
In notations as introduced right before
Lemma~\ref{speclemma}, let $x\in{\lvA^{\hhb{-1}{\scriptscriptstyle|}\uix{\scriptscriptstyle|}}}$ and $\phi_x:\kp x\to k$
be such that $\phi_x({\undr p}_{\uix,x})={\undr p}_\uix$. Since
$\overline{\fam\eufam\teneu q}\to\overline{\fam\eufam\teneu p}\to{\lvP^{{\scriptscriptstyle|$ is dominant and proper,
there exists a preimage $y\in\overline{\euq}$ of $x$ such
that $\kp x\hookrightarrow\kpy$ is finite. Since $k$ is algebraically
closed, there is a $\kp x$-embedding $\phi_y:\kpy\hookrightarrow k$
such that $\phi_x=\phi_y\circ(\kp x\hookrightarrow\kp y)$. In particular,
if ${\undr p}_{\uix,y}$ is the image of ${\undr p}_{\uix,x}$
under $\kp x\hookrightarrow\kpy$, then ${\undr p}_\uix=
\phi_y({\undr p}_{\uix,y})$.
\vskip4pt
Let $W_{{k_0},j}$, $j\in J$, be the irreducible components
of $W_{{k_0}}$ which contain~$y$, and
$W_J:=\cap_j W_{{k_0},j}$. Then $W_J$ is a smooth
${k_0}$-subvariety $W_J\subset W_{\!{k_0}}$, and the
following hold, see e.g., \nmnm{de~Jong}~\cite{dJ},
section~2.16 and explanations thereafter: Let ${\clO_{\hhb{-.25}y}}$
be the local ring of $y\in W$. There exists a system of
regular parameters $(u_1,\dots,u_N)$ of ${\clO_{\hhb{-.25}y}}$ which
satisfy:
\vskip2pt
$\hhb2$i) $u_j$ defines locally at $y$ the equation of
$W_{{k_0},j}$ and $\pi_0=u_1\dots u_{n_y}$.
\vskip2pt
ii) $(u_j)_{n< j \leq N}$ give rise to a regular system of
parameters at $y\in W_J$ in ${\clO_{\hhb{-.25}y}}/(u_1,\dots,u_{n_y})$.
\vskip4pt
Let ${\fam\eufam\teneu r}:=(u_1-u_i)_i\subset{\clO_{\hhb{-.25}y}}$ be the ideal generated
by all the $u_1-u_i$, $1\leq i\leq N$. Then ${\fam\eufam\teneu r}$ is a
regular point with $(u_1-u_i)_i$ a regular system of
parameters, and
\[
{\bm Z}_y:={\clO_{\hhb{-.25}y}}/{\fam\eufam\teneu r}
\]
is a discrete valuation ring having
$\pi_y:=u_1\,({\rm mod}\,{\fam\eufam\teneu r})$ as uniformizing
parameter, and $\pi_y^{n_y}=\pi_0$. In particular, if
$\pi_0$ was an algebraic integer, then so is $\pi_y$.
\vskip2pt
Let $v_1$ be a valuation of $F$ with center ${\fam\eufam\teneu r}$,
and residue field equal to $\kp{\fam\eufam\teneu r}={\rm Quot}({\bm Z}_y)$.
And further, let $v_0$ be the canonical valuation of
${\mathcal O}_0:={\bm Z}_y$. Then the valuation ring ${\mathcal O}_v$ of the
valuation $v:=v_0\circ v_1$ dominates ${\clO_{\hhb{-.25}y}}$ and has
$\kp v=\kpy$. Hence by Lemma~\ref{speclemma}
above, it follows that ${\mathcal Y}_{{\mathcal O}_v}\to\lvPt{{\mathcal O}_v}$
is a ${\fam\lvfam\tenlv Z}/p^f$-cover of smooth ${\mathcal O}_v$-curves.
Hence by Lemma~\ref{transsmooth}, it follows that
${\mathcal Y}_{{\bm Z}_y}\to\lvPt{{\bm Z}_y}$ is a ${\fam\lvfam\tenlv Z}/p^f$-cover
of smooth ${\bm Z}_y$-curves.
\vskip2pt
Let $n_\uix={\rm l.c.m.}(n_y)_y$, and notice that
$n_\uix$ is bounded by $n\hhb1!$, where
$n=\dim(W)-1$. Choose a fixed algebraic integer
$\pi_0$ such that ${\bm Z}={\bm Z}_0[\pi_0]$, and let $\pi_\uix$
be defined by $\pi_\uix^{n_\uix}=\pi_0$. Then there are
canonical embeddings
${\bm Z}_y\hookrightarrow W(\overline\kappa_y)[\pi_y]\hookrightarrow W(k)[\pi_\uix]$,
and the base change of ${\mathcal Y}_{{\bm Z}_y}\to\lvPt{{\bm Z}_y}$
under ${\bm Z}_y\hookrightarrow W(k)[\pi_\uix]$ is a
${\fam\lvfam\tenlv Z}/p^f$-cover of smooth $W(k)[\pi_\uix]$-curves
\[
{\mathcal Y}_{W(k)[\pi_\uix]}\to\lvPt{W(k)[\pi_\uix]}
\]
with special fiber the $\uix$-KG-cover $Y_k\to\lvPt k$
of the given cyclic ${\fam\lvfam\tenlv Z}/p^f$-extension $k[[t]]\hookrightarrow k[[z]]$.
\vskip2pt
This concludes the proof of Proposition~\ref{specprop}.
\end{proof}
\vskip5pt
\noindent
C) \ {\it The strategy of proof for Theorem~\ref{OC}\/}
\vskip2pt
We begin by recalling that there are several forms of the
Oort Conjecture (OC) which are all equivalent,
see e.g.\ \nmnm{Saidi}~\cite{Sa},~\S3.1, for detailed proofs.
\vskip2pt
Let $k$ be an algebraically closed field with
${\rm char}(k)=p>0$. Let $W(k)$ be the ring of Witt
vectors over $k$, and $W(k)\hookrightarrow R$ denote finite
extension of discrete valuation rings. We consider
the following two situations, which are related to two
variants of OC:
\vskip5pt
a) $Y\to X$ is a finite (ramified) $G$-cover of complete smooth
$k$-curves such that the inertia groups at all closed points
$y\in Y$ are cyclic.
\vskip5pt
b) ${\mathcal X}_R$ is a complete smooth $R$-curve with special fiber
$X$, and $Y\to X$ is as a (ramified) $G$-cover of complete
smooth curves as in~case~a) above.
\vskip5pt
We say that {\it OC holds over~$R$\/} in case a) or b), if
there exists a $G$-cover of complete smooth $R$-curves
${\mathcal Y}_R\to{\mathcal X}_R$, with ${\mathcal X}_R$ the given one in~case~b),
having the $G$-cover $Y\to X$ as special fiber. And given
a cyclic extension $k[[t]]\hookrightarrow k[[z]]$, we say that the {\it local
OC holds over $R$\/} for $k[[t]]\hookrightarrow k[[z]]$, if there exists a
smoth smooth lifting $R[[T]]\hookrightarrow R[[Z]]$ of $k[[t]]\hookrightarrow k[[z]]$.
\begin{fact}
\label{equivOC}
(\nmnm{Saidi}~\cite{Sa}, \S3.1) The following hold:
\vskip2pt
{\rm 1)} {\it Local global principle for OC.\/} Let $Y\to X$ be a finite
$G$-cover with cyclic inertia groups, and for $y\mapsto x$,
let $k[[t_x]]\hookrightarrow k[[t_y]]$ be the corresponding extension of
local rings. Let ${\mathcal X}_R$ be some complete smooth $R$-curve
with special fiber $X$. Then the following are equivalent:
\vskip2pt \ \ \
$\hhb2${\rm i)} There is a $G$-cover of complete smooth
$R$-curves ${\mathcal Y}_R\to{\mathcal X}_R$ with special fiber $Y\to X$.
\vskip2pt \ \ \
{\rm ii)} For all $y\mapsto x$, the local cyclic extension
$k[[t_x]]\hookrightarrow k[[t_y]]$ has a smooth lifting over $R$.
\vskip4pt
{\rm 2)} {\it Equivalent forms of OC.\/} The following assertions
are equivalent:
\begin{itemize}
\item[{\rm a)}] OC holds for all $G$-covers $Y\to X$ as in case a), or b).
\vskip2pt
\item[{\rm b)}] OC holds for $G$ cyclic and $X=\lvPt k$.
\vskip2pt
\item[{\rm c)}] OC holds for $G$ cyclic and $X=\lvPt k$
and $Y\to\lvPt k$ branched at $t=0$ only.
\vskip2pt
\item[{\rm d)}] The local OC holds.
\vskip2pt
\item[{\rm e)}] Any of the assertions above, but restricted to cyclic
$p$-groups as inertia groups.
\end{itemize}
\end{fact}
\vskip5pt
Thus in order to prove Theorem~\ref{OC} from Introduction,
we can proceed as follows: Let $Y\to X$ be a given
$G$-cover of projective smooth $k$-curves, with branch
locus $\Sigma\subset X$. Then for a given algebraic
integer $\pi$ and $R:=W(k)[\pi]$, and a smooth model
${\mathcal X}_R$ of $X$ over $R$ one has: The OC holds for
$Y\to X$ over $R$ iff the local OC holds for the local
cyclic extension $k[[t_x]]\hookrightarrow k[[t_y]]$ over $R$ for all
$x\in\Sigma$. Further, the local OC holds for a fixed local
cyclic extension $k[[t_x]]\hookrightarrow k[[t_y]]$ over $R$ if and only
if the local OC holds over $R$ for the $p$-power sub-extension
$k[[t_x]]\hookrightarrow k[[z_x]]$ of $k[[t_x]]\hookrightarrow k[[t_y]]$. Thus the global
assertion of Theorem~\ref{OC} is equivalent to the local
assertion for cyclic $p$-power extensions $k[[x]]\hookrightarrow k[[z]]$.
\vskip5pt
We tackle the case of $p$-power cyclic extensions
$k[[t]]\hookrightarrow k[[z]]$ as follows.
\vskip5pt
\underbar{Step 1}. Let $\uix=(\imath_1,\dots\imath_e)$ be a
fixed upper ramification jumps sequence. By
Key Lemma~\ref{keylemma1} and~Theorem~\ref{charpOC}
there exists some $N$ and sequences
$\uix_\mu=(\imath_1,\dots,\imath_{e_\mu})$, $1\leq\mu\leq N$,
depending on $\uix$ only, such that the following hold:
Let ${\fam\eufam\teneu o}$ be a complete discrete valuation ring over
with residue field $k$, and $x_1,\dots,x_N\in{\fam\eufam\teneu m}_{\fam\eufam\teneu o}$
be distinct points. Then for every $\uix$-KG-cover
$Y\to\lvPt k$ there exists a ${\fam\lvfam\tenlv Z}/p^e$-cover of projective
smooth ${\fam\eufam\teneu o}$-curves ${\mathcal Y}_{\fam\eufam\teneu o}\to\lvPt{\fam\eufam\teneu o}$ satisfying:
\vskip2pt
a) The special fiber of ${\mathcal Y}_{\fam\eufam\teneu o}\to\lvPt{\fam\eufam\teneu o}$ is the given
$\uix$-KG-cover $Y\to\lvPt k$.
\vskip2pt
b) The generic fiber ${\mathcal Y}_\kk\to\lvPt\kk$ of
${\mathcal Y}_{\fam\eufam\teneu o}\to\lvPt{\fam\eufam\teneu o}$ is branched above $x_1,\dots,x_N$
only.\footnote{\hhb2N.B., ${\mathcal Y}_\kk\to\lvPt\kk$ is not always
an $\uix$-KG-cover!}
\vskip3pt
c) The upper ramification jumps above each $x_\mu$ are
$\uix_\mu:=(\imath_1,\dots,\imath_{e_\mu})$, $\mu=1,\dots,N$.
\vskip5pt
\underbar{Step 2}. Let $\kk\hookrightarrow{l}$ be an algebraic
closure, and ${\mathcal Y}_{l} \to \lvPt{l}$ be the base change
of ${\mathcal Y}_\kk \to \lvPt\kk$. Then ${\mathcal Y}_{l} \to {\fam\lvfam\tenlv P}^1_{l}$
is a ${\fam\lvfam\tenlv Z}/p^e$-cover of projective smooth curves with
no essential ramification.
\begin{hypo}
\label{hypothesis}
In the Notations~\ref{notalast}, suppose that for every
$\uix_\mu=(\imath_1,\dots,\imath_{e_\mu})$ with $\mu=1,\dots,N$,
the subset $\Sigma_{\uix_\mu}\subset{\fam\lvfam\tenlv A}^{{\scriptscriptstyle|}
{\uix_\mu}{\scriptscriptstyle|}}$ is Zariski dense.
\end{hypo}
For every fixed $\mu=1,\dots,N$, consider the
$\uix_\mu$-KG-cover $Y_\mu\to\lvPt l$ of the local
${\fam\lvfam\tenlv Z}/p^{e_\mu}$-extension ${l}[[t_\mu]]\hookrightarrow{l}[[z_\mu]]$,
where $z_\mu$ and $t_\mu$ are local parameters at
${y}_\mu\mapsto{x}_\mu$. Then by Proposition~\ref{specprop}
applied for each $\uix_\mu=(\imath_1,\dots,\imath_\mu)$,
there exists some algebraic integer $\pi_\mu$ such that
$\uix_\mu$-KG-cover $Y_\mu\to\lvPt l$ has a smooth lifting
over $W(l)[\pi_\mu]$. Thus by the local-global
principle~Fact~\ref{equivOC}, it follows that if $\pi_\uix$
is an algebraic integer such that $\pi_\mu\in W(l)[\pi_\uix]$
for all $\mu=1,\dots,N$ then $Y_l\to\lvPt l$ has a smooth
lifting ${\mathcal Y}_{{\mathcal O}_1}\to\lvPt{{\mathcal O}_1}$ over ${\mathcal O}_1:=W(l)[\pi_\uix]$.
\vskip2pt
Let $v_1$ be the canonical valuation of ${\mathcal O}_1$,
and $v_0$ be the (unique) prolongation of the valuation
of ${\fam\eufam\teneu o}$ to $l$, say having valuation ring ${\mathcal O}_0$.
Then the base change ${\mathcal Y}_{{\mathcal O}_0}\to\lvPt{{\mathcal O}_0}$ of the
${\fam\lvfam\tenlv Z}/p^e$-cover of complete curves ${\mathcal Y}_{\fam\eufam\teneu o}\to\lvPt{\fam\eufam\teneu o}$
under ${\fam\eufam\teneu o}\hookrightarrow{\mathcal O}_0$ is a ${\fam\lvfam\tenlv Z}/p^e$-cover of complete
smooth ${\mathcal O}_0$-curves with generic fiber ${\mathcal Y}_l\to\lvPt l$.
And the ${\fam\lvfam\tenlv Z}/p^e$-cover of complete smooth $l$-curves
${\mathcal Y}_l\to\lvPt l$ is the special fiber of the ${\fam\lvfam\tenlv Z}/p^e$-cover
of smooth ${\mathcal O}_1$-curves ${\mathcal Y}_{{\mathcal O}_1}\to\lvPt{{\mathcal O}_1}$.
The setting $v:=v_0\circ v_1$ and letting ${\mathcal O}$ be the
valuation ring of $v$, it follows by Lemma~\ref{transsmooth}
that there exists a smooth lifting of $Y\to\lvPt k$ to a
${\fam\lvfam\tenlv Z}/p^e$-cover of smooth ${\mathcal O}$-curves ${\mathcal Y}_{\mathcal O}\to\lvPt{\mathcal O}$.
\vskip2pt
Since the $\uix$-KG-cover $Y\to\lvPt k$ we started with was
arbitrary, it follows that the Hypothesis~\ref{hypothesis} implies
that $\Sigma_\uix={\lvA^{\hhb{-1}{\scriptscriptstyle|}\uix{\scriptscriptstyle|}}}$. Hence by Proposition~\ref{specprop}
we conclude that:
\vskip7pt
\centerline{\it Hypothesis~\ref{hypothesis} implies the
existence of an algebraic integer $\pi_\uix$ such\/}
\centerline{\it that every $\uix$-KG-cover
$Y\to\lvPt k$ has a smooth lifting over $W(k)[\pi_\uix]$.\/}
\vskip7pt
\noindent
D) {\it Concluding the proof of the Oort Conjecture\/}
\vskip5pt
By the observation above, the proof of the Oort Conjecture
is reduced to showing that the Hypothesis~\ref{hypothesis}
holds for every system of upper ramification indices
$\uix=(\imath_1,\dots,\imath_e)$ which has no essential jump
indices, i.e., $p\hhb1\imath_{{\hhb{.5}\rho}-1}\leq\imath_{\hhb{.5}\rho}< p\hhb1\imath_{{\hhb{.5}\rho}-1}+p$ for
${\hhb{.5}\rho}=1,\dots,e$. Via the local-global principle~Fact~\ref{equivOC},
this fact is equivalent to a (very) special case of the local Oort
Conjecture, which follows from a more general (but still partial)
result recently announced by \nmnm{Obus--Wewers}~\cite{O--W},
see~\nmnm{Obus}~\cite{Ob},~Theorem~6.28. Here
is the special case needed here:
\begin{keylemma}
\label{keylemma2}
{\rm (\nmnm{Special case of Obus--Wewers})}
In notations and context as above, let $k[[t]]\hookrightarrow k[[z]]$ be
cyclic extension of degree~$p^e$ which has no essential
ramification. Then the local Oort Conjecture holds for
$k[[t]]\hookrightarrow k[[z]]$, i.e., $k[[t]]\hookrightarrow k[[z]]$ has a smooth
lifting over some finite extension~$R$ of $\,W(k)$ to a
smooth cyclic cover $R[[T]]\hookrightarrow R[[Z]]$.
\end{keylemma}
\begin{proof} Recall that Lemma~6.27 from~\nmnm{Obus}~\cite{Ob}
asserts that the local Oort conjecture holds for cyclic extensions
$k[[t]]\hookrightarrow k[[z]]$ of degree $p^e\!$, provided the upper ramification
jumps $\imath_1\leq\dots\leq \imath_e$ satisfy: For every $1\leq\nu< e$,
there is no integer $m$ such that:
\[
\hhb{10}(*)\hhb{80}\imath_{\nu+1}-p\hhb1\imath_\nu < p\hhb1m \leq
(\imath_{\nu+1}-p\hhb1\imath_\nu)\,
{{\imath_{\nu+1}}_{\phantom|}\over
{\ \imath_{\nu+1}-\imath_\nu}\ }.\hhb{100}
\]
Notice that if $k[[t]]\hookrightarrow k[[z]]$ has no proper essential
ramification jumps, thus by definition
$\imath_{\nu+1} \leq p\hhb1\imath_\nu +p-1$
for all $1\leq\nu < e$, then the hypothesis~$(*)$ above is
satisfied. Indeed, we first notice that $p\hhb1\imath_\nu\leq \imath_{\nu+1}$
implies that $m$ must be positive. Hence setting
$\delta:=\imath_{\nu+1}-p\hhb1\imath_\nu$ and $u:=\imath_\nu$,
the second inequality becomes
$p\hhb1m\big(\delta+(p-1)u\big)\leq\delta(\delta+p\hhb1u)$,
which is equivalent to $p(p-1)mu\leq\delta(\delta+pu-pm)$.
Hence taking into account that $\delta\leq p-1$, we get:
$p(p-1)mu\leq(p-1)(p-1+pu-mp)$, thus dividing by $(p-1)$,
we get: $p\hhb1mu\leq p-1+p\hhb1u-mp$, or equivalently,
$p(m-1)(u+1)+1\leq0$, which does not hold for any
positive integer~$m$.
\end{proof}
This concludes the proof of Theorem~\ref{OC}.
\begin{remark}
\label{remarkOW}
The main result of \nmnm{Obus--Wewers},
see~\cite{Ob},~Theorem~6.28, asserts that if the
hypothesis~$(*)$ above is satisfied for all $3\leq\nu< e$,
then the local Oort conjecture holds for the cyclic extension
$k[[t]]\hookrightarrow[[z]]$. Therefore, the Oort Conjecture {\it holds
unconditionally for $n=3$,\/} and the above hypothesis~$(*)$
kicks in only for exponents $3<e$. Unfortunately, for $3 < e$,
the hypothesis~$(*)$ becomes very restrictive indeed.
Setting namely $\delta:=\imath_{\nu+1}-p\hhb1\imath_\nu$ and
$u:=\imath_\nu$, we have that $\imath_{\nu+1}=\delta+p\hhb1u$
for some $0\leq\delta$. If $0<\delta$, then $\delta$ is
prime to $p$ hence writing $\delta=r\hhb1p-\eta$ with
$1\leq\eta\leq p-1$, the hypothesis~$(*)$ for $m:=r$ implies that:
\vskip5pt
\centerline{\it At least one of the inequalities \
$\delta < r\hhb1p\leq\delta(\delta+p\hhb1u)/\big(\delta+(p-1)u\big)\,$
does not hold.}
\vskip5pt
\noindent
Since the first inequality holds, the second inequality
must \underbar{not} hold. Therefore we must have
$r\hhb1p > \delta(\delta+p\hhb1u)/\big(\delta+(p-1)u\big)$.
Since the LHS equals $\delta+\delta u/\big(\delta+(p-1)u\big)$,
and $p\hhb1r-\delta=\eta$, the above inequality
is equivalent to $\eta > \delta u/\big(\delta+(p-1)u\big)$,
hence to $\eta\delta+\eta(p-1)u>\delta u$, and finally to
$\eta^2(p-1)>\big(\delta-\eta(p-1)\big)(u-\eta)$. Thus
since $\delta-\eta(p-1)=(r-\eta)p$, the last inequality
becomes $\eta^2(p-1)>p(r-\eta)(u-\eta)$. Since $p>p-1$,
the last inequality implies $\eta^2>(r-\eta)(u-\eta)$.
On the other hand, one has
$1\leq\eta <p$ and $p^2\leq u$ for $1<\nu$, thus
$\eta^2<u-\eta$. Therefore, in order to satisfy the inequality,
one must have $r\leq\eta$. Hence the hypothesis~$(*)$
for $1<\nu< e$ is equivalent to:
\vskip5pt
\centerline{\ $(*)'$ \ {\it
If $\,\imath_{\nu+1}=p\hhb1\imath_\nu+p\hhb1r_\nu -\eta_\nu$
with $\,0\leq\eta_\nu< p$, then $\,0\leq r_\nu\leq\eta_\nu$
for all $\,1 < \nu < e$.\/}}
\vskip5pt
\noindent
In particular, $\eta_\nu=1$ implies $\delta_\nu= p-1$,
and $\delta_\nu\leq(p-1)^2$ in general. It seems to me
that the reformulation~$(*)'$ is better/easier than the original
formulation of hypothesis~$(*)$.
\end{remark}
\bibliographystyle{plain}
\begin{bibdiv}
\begin{biblist}
\bibliographystyle{plain}
\bibselect{big}
\end{biblist}
\end{bibdiv}
\end{document}
First, let $\deg_p({\fam\eufam\teneu D})$
be the maximal power of $p$ which divides the degree of the
different of $k[[t]]\hookrightarrow k[[z]]$. And for a $G$-cover $Y\to X$ as
above, let ${\fam\eufam\teneu D}_x$ be the local different above $x\in X$,
and $\deg_p({\fam\eufam\teneu D}_x)$ be correspondingly defined.
Let $e$ be a fixed positive integer. Then there exist
only finitely many finite sequences of positive numbers
$\uix:=(\imath_1\leq\dots\leq\imath_f)$
with $f\leqe$ and satisfying $1\leq\imath_1< p$ and
$p\hhb1\imath_\nu\leq \imath_{\nu+1}<p\hhb1\imath_\nu+p$
for $1\leq\nu<f$. For every such sequence $\uix$,
let $|\uix|=\imath_1+\dots+\imath_f$ and consider
${\undr P}_{\uix}=\big(P_1,\dots,P_f)$ a sequence
of generic polynomials $P_\nu=P({t^{-1}})$ of degrees
$\deg\big(P_\nu\big)=\imath_\nu$ for $1\leq\nu\leqf$.
In other words, all the coefficients $a_{\nu,{\hhb{.5}\rho}}$,
$1\leq i\leqf$, $1\leq{\hhb{.5}\rho}\leq\imath_i$ of the polynomials
$P_\nu$ are independent free variables over ${\fam\lvfam\tenlv F}_p$.
Let $A_{\uix}:={k_0}[(x_{\nu,{\hhb{.5}\rho}})_{\nu,{\hhb{.5}\rho}}]$ be the
corresponding polynomial ring and ${\lvA^{\hhb{-1}{\scriptscriptstyle|}\uix{\scriptscriptstyle|}}}=\Spec A_{\uix}$
the resulting affine space over ${k_0}:=\overline{\fam\lvfam\tenlv F}_p$.
\vskip2pt
For every $x\in{\lvA^{\hhb{-1}{\scriptscriptstyle|}\uix{\scriptscriptstyle|}}}$ let $k_x$ be any
algebraically closed field extension of ${k_0}$, and
$\overline x\in{\lvA^{\hhb{-1}{\scriptscriptstyle|}\uix{\scriptscriptstyle|}}}(k_x)$ be a $k_x$-rational
point of ${\lvP^{{\scriptscriptstyle|$ factoring through $x$. Then
$\overline x$ gives rise to the corresponding ``specialization''
${\undr p}_{\uix,\hhb1x}=\big(p_1,\dots,p_f)$
of ${\undr P}_{\uix}=\big(P_1,\dots,P_f)$ with
$p_\nu\in k_x[{t^{-1}}]$ for all $1\leq\nu\leqf$. Thus if
$\deg(P_\nu)=\deg(p_\nu)$ for all $\nu$, then the
system ${\undr p}_{\uix,\hhb1x}$ gives rise to a cyclic
extension $k_x[[t]]\hookrightarrow k_x[[z_x]]$ of degree $p^f$ and
upper jumps $\imath_1\leq\dots\leq\imath_f$.
|
{
"timestamp": "2012-03-09T02:04:11",
"yymm": "1203",
"arxiv_id": "1203.1867",
"language": "en",
"url": "https://arxiv.org/abs/1203.1867"
}
|
\section{Contemporary Works : Inspiration for our work}
Lagarias \cite{lagarias} describes the history of the Collatz problem and surveys all the major literature.
Lagarias \cite{lagarias} is taken as the primary reference for this paper.
Below mentioned observations are inspiration for this paper, as presented in the subsections.
\subsection {Collatz Sequence In Base 2}
It is a well known fact that computing Collatz sequence
can be seen as an abstract computer computing the sequence in base `2', whose only operations would be left shift, right shift and addition, as explained below.
\newcounter{qcounter}
The machine will perform the following steps on any odd number `n' until the number reduces to ``1'' :-
\begin{list}{\arabic{qcounter}:~}{\usecounter{qcounter}}
\item Left shift `n' by one bit : thus giving 2n;
\item Add (1) to the original number, `n' by binary addition \\ (giving 2n+n = 3n);
\item Add 1 to the right of the new number in (2) by binary addition.
\item Remove all trailing ``0''s (i.e. repeatedly divide by two until the result is odd). This is the right shift operation.
\end{list}
It is also known from Lagarias \cite{lagarias} that the odd numbers generated in the sequence are generated in `almost' random order and that the convergence can be treated probabilistically.
\subsection {General Collatz Class}
Several authors (M\"{o}ller \cite{moller}) have investigated the range of validity of the result that has a finite stopping time for almost all integers ``n'' by considering more general classes of periodicity linear functions. One such class consists of all functions which are given by
\begin{equation}\label{gen_Collatz_2}
U(n) =
\left\{
\begin{array}{lr}
\frac{n}{b} \; , & n \equiv 0 \; mod \; b \\
\frac{mn - r}{b} \;, \; r = mn \; mod \; b \; , & otherwise
\end{array}
\right.
\end{equation}
It has been shown in M\"{o}ller \cite{moller} that iff
$$
m < b^{\frac{b}{b-1}}
$$
then sequence generated by \eqref{gen_Collatz_2} would converge.
\subsection{Stopping Time}
Assuming that the Conjecture is true, one can consider the problem of determining the expected stopping time function. Crandall \cite{crandall} and Shanks \cite{shanks} were guided by probabilistic heuristic arguments to conjecture that for large set of large random integers
the ratio of stopping time to natural logarithm of the integer should approach a constant limit:-
\begin{equation}\label{gen_Collatz_stopping_time}
\frac{\sigma(n)}{ln(n)} = 2 \left( ln \left (\frac{4}{3} \right )\right)^{-1}
\end{equation}
where $\frac{\sigma(n)}{ln(n)}$ is sampled over large set of large integers.
\section{The Work}
\subsection{Overview}
In present work an intuitive general formula for Collatz like sequences is described, basing the formula
on an abstract computing machine operating on different bases.
Based on the digits on the tape of the abstract machine it has been conjectured
that the Collatz like problems are examples of presence of digit-wise chaos in arithmetic operations.
To prove the digit-wise chaos, a very simple pseudorandom bit generator is presented which is based upon
Collatz sequence and which passes all of the diehard battery of randomness tests.
The convergence criterion for Collatz like sequence then become more probable right shift operation than
left shift operations of the abstract machine, and which becomes statistically decreasing number of digits,
which reduces the problem into classical ruin problem on number of digits.
Finally, based upon simple probability, an approximation of the stopping time ratio is established like in
\eqref{gen_Collatz_stopping_time}.
\subsection{The Implicit Base Assumption}
Let us rewrite,
$$ 3n+1 = 2n + (n+1) $$
And generalize it as Left-shift, followed by addition of an even number ($n+1$ is an even number).
This deletes the last bit of the binary representation of the original number $n$.
It evidently means that adding $n+1$ is the trick to round off the last digit of the modified number $n$ to $0$ in base $2$.
This insight is duly confirmed by the observation that,
$$ 3n+3 = 2n + (n+2) $$
also converges to \{1,3\} depending upon whether or not $n$ is a power of $2$.
If it were not, it would converge to $3$, else it would converge to $1$.
It has been known for sometime that these general $3n+k$ forms are nothing but transforms (scaling) of the
original $3n+1$ sequence, where $k$ is an odd number.
Based upon these observations, it can be suggested that the Collatz Sequence has base $2$
implicit within it. The breaking up of $3n + 1 = 2n + (n+1)$ is a clue to this insight.
If this \emph {``theory of implicit base''} holds, then
\emph { on any base, there would exist a general formula of a sequence which would be based upon the left-shift and right-shift paradigm only}.
\subsection {The Generalized Collatz Formula}
We found that if the general formula has to exist, it has to be of the form :-
\begin{equation}\label{gen_Collatz_1}
f(n) =
\left\{
\begin{array}{lr}
\frac{n}{b} \; , & n \equiv 0 \; mod \; b \\
(b+1)n+ (b - n \; mod \; b ) \; , & otherwise
\end{array}
\right.
\end{equation}
the $(b+1)n+ (b - n \; mod \; b )$ part of course is $bn + ( n + (b - n \; mod \; b ) )$.
That is the basic algorithm that evolved from the concept of left and right shift operations in an abstract machine computing Generalized Collatz sequence in base $b$.
This sequence is in fact similar to the sequence mentioned in \cite{moller} as in \eqref{gen_Collatz_2}.
We note that:-
$$
b + 1 < b^{\frac{b}{b-1}}, \forall b \ge 2
$$
so, the convergence criterion is satisfied.
The findings on the $f(n)$ are summarized in the table [\ref{table:results}].
\begin{table}[ht]
\caption{Results of the Tests on Generalized Collatz Sequence}
\centering
\begin{tabular}{ | l | l | l | }
\hline\hline
Base (b) & Convergence Points & Tested Up-to \\ [0.5ex]
\hline
2 & 1 & Maximum Integer \\ \hline
3 & 1,7,? & 100000 \\ \hline
5 & 1 & 1354093827 \\ \hline
7 & 1 & 386581748 \\ \hline
11 & 1,642,? & 100000 \\ \hline
13 & 1 & 24000000000 \\ \hline
17 & 1,79,? & 100000 \\ \hline
19 & 1 & 340000000 \\ \hline
23 & 1,82,? & 100000 \\ \hline
29 & 1,111,? & 100000 \\ \hline
31 & 1,389,? & 100000 \\ \hline
37 & 1 & 100000000 \\ \hline
41 & 1 & 99999999 \\ \hline
43 & 1 & 99999999 \\ \hline
\hline
\end{tabular}
\label{table:results}
\end{table}
\subsection {A Simpler Collatz Sequence }
A simpler and computationally faster formulation of Collatz Type sequence would be:-
\begin{equation}\label{generalized_simplified_Collatz}
f(n) =
\left\{
\begin{array}{lr}
\frac{n}{b} \; , & n \equiv 0 \; mod \; b \\
\lfloor\frac{(b+1)n}{b} \rfloor \; , & otherwise
\end{array}
\right.
\end{equation}
This can be computed in the fastest way in a digital computer,
because no modulus operation is required.
Our original generic sequence formula (\ref{gen_Collatz_1}) was nothing but:-
$$
f(n) =
\left\{
\begin{array}{lr}
\frac{n}{b} \; , & n \equiv 0 \; mod \; b \\
\lfloor\frac{(b+1)n}{b} \rfloor +1 \; , & otherwise
\end{array}
\right.
$$
The simplified actual Collatz Sequence then can be written as
\begin{equation}\label{original_simplified_Collatz}
f(n) =
\left\{
\begin{array}{lr}
\frac{n}{2} \; , & n \equiv 0 \; mod \; 2 \\
\lfloor\frac{3n}{2} \rfloor \; , & otherwise
\end{array}
\right.
\end{equation}
The table [\ref{table:results_special}] shows some of the results of the experiments on small bases.
\begin{table}[!ht]
\caption{Some Results on Simplified Generalized Collatz Sequence}
\centering
\begin{tabular}{ | l | l | l | }
\hline\hline
Base & Convergence Points & Tested Up-to \\ [0.5ex]
\hline
2 & 1,5,17,? & 100000 \\ \hline
3 & 1,2,22,? & 100000 \\ \hline
5 & 1,2,3,4,57,? & 100000 \\ \hline
7 & 1,2,3,4,5,6,? & 100000 \\ \hline
\hline
\end{tabular}
\label{table:results_special}
\end{table}
\subsection {Pseudorandom Sequence Generator and The Generalized Collatz Sequence}
A Linear Congruential Generator (LCG) represents one of the oldest and best-known algorithm for the generation of pseudorandom numbers.
The generator is defined by the recurrence relation:-
$$
X_{n+1} = ( aX_n + c ) mod \; m
$$
where:- \\
$m > 0$ is the modulus \\
$a, 0 < a < m$, is the multiplier \\
$c, 0 \le c < m$, is the increment \\
$X_0, 0 < X_0 < m$, is the start value \\
$n \ge 0 $ is the iteration number \\
Assume we are in base $2$, and we have a tape holding $k$ bits of information. Treat the tape as a register having $k$ initialized bits.\\
Assume also $d_i = 0 \;, \forall i \ge k $.
Then we can define the recurrence relation on time, $t$ (present), and $t+1$, the next time interval as follows:-
$$
d_1(t+1) = d_1(t)
$$
$$
d_2(t+1) = ( d_1(t+1) + d_2(t) + 1 )mod \; 2
$$
$$
d_3(t+1) = ( d_2(t+1) + d_3(t) )mod \; 2
$$
$$
...
$$
$$
d_{k-1}(t+1) = ( d_{k-2}(t+1) + d_{k-1}(t) ) mod \; 2
$$
$$
d_k = Carry \; digit
$$
Specifically
$$
d_{k-1}(t+1) = ( \sum_{i=1}^{k-1}{d_i}(t) ) mod \; 2
$$
We can see it is nothing but a LCG, applied on $d_i$, $(i-1)$ times, to generate the $d_i$ bit.
Specifically:-
$$
d_{k-1}(t+1) = ( d_{k-1}(t) + \sum_{i=1}^{k-2}{d_i}(t) ) mod \; 2
$$
This is an LCG where :- \\
$a=1$ and $c = (\sum_{i=1}^{k-2}{d_i}(t) ) $, where `$c$' itself is generated using the same LCG, on lower bits.
But this recurrence relation is the prescription of the operation $3n + 1$ in binary digits on an abstract machine.
Replacing $ mod \; 2$ with $ mod \; b$ would be
the prescription of calculating $(b+1)n+ (b - n \; mod \; b )$ in base $b$ on an abstract machine, with set of LCGs:-
\begin{equation}\label{Collatz_lcg}
d_{k-1}(t+1) = ( d_{k-1}(t) + \sum_{i=1}^{k-2}{d_i}(t) ) mod \; b
\end{equation}
This observation is in line with Feinstein \cite{feinstein} that the standard mathematical way of proving this conjecture may not possible.
Also note that this observation moves the Collatz Problem - from number theory to Chaos Theory, and shows all generations
of numbers generated by Collatz function would behave randomly. Similarities with cellular automata is apparent,
and is discussed with a cellular automata in base 6 in Wolfram \cite{anks} which simulates the original Collatz Function.
It is also stated in Wolfram \cite{anks} that arithmetic operations can have digit-wise chaos,
and we hypothesize that the convergence on generalized Collatz type sequences might well be statistical, and the system behaves rather chaotically in higher number of digits.
To test the hypothesis we did pass numbers with highly ordered binary representation, for example :-
\begin{itemize}
\item { $n = 2^k - 1$, this immediately reduces to $3^k -1$, and elementary proof exists for this reduction.
The system $f$ from section 1 can be written as $\frac{3}{2}(n+1) -1$ when n is odd number, and hence $f^k = {(\frac{3}{2})}^k (n+1) - 1 $ if all the $k$ iteration produced odd number.
The reduction to $3^k -1$ immediately follows from there.}
\item { $n = 2^k + 1$ where it reduces to $2^k + 2^{k-1} + 2 + 1$. In binary form, it looks like $110000...00011$. If the ``11'' in the left and right are separated enough,
so that they are isolated in the next $3n+1$ operation, (i.e. carry does not reach the leftmost ``11'') then it would reduce to $100100...001$
reducing the $n_0$ by 2, and increasing the $n_1$ by 1. For example: $ 1000001 \rightarrow 1100011 \rightarrow 100101000 \rightarrow 100101 $}
\item $n = 1111111000000000001$ , where the 1's and 0's are in highly ordered state.
\end{itemize}
\subsection{A Random Experiment}
Empowered by this newfound idea that, $n+ (b - n \; mod \; b )$
is nothing but a random seed when added to $bn$
to randomize the abstract machine's tape, we tried to experimentally verify the same.
A program was written where we replaced the generalized Collatz function by
$$
f(n) = bn+ R[n]
$$
where $R[n]$ is a random integer divisible by $b$ , and both $n$ and $R[n]$
(when represented in the base $b$) share the same number of digits.
The assumption of having same number of digits was made to be consistent with the original Collatz problem where $R[n]= n+1$.
We tested this hypothesis with base-2, and the results were positive.
After a long but finite run, numbers do converge to 1, with generalized randomized Collatz sequence.
We also note that adding a fixed even numbers in range $R[n] \in \{2,4,8,..., 2^{k-1}\}$ at every step,
destroys the chaotic pattern, and makes it uniformly convergent.
This was tested with the largest even number in the set, $2(2^{k-1} -1)$.
\subsection{A Binary Pseudorandom Digit Generator}
To show that the binary digits generated by the Collatz sequence are bitwise random, we wrote a Pseudorandom
digit generator based upon Collatz sequence.
\lstset{
basicstyle=\footnotesize\ttfamily,
numberstyle=\tiny,
numbersep=5pt,
tabsize=2,
extendedchars=true, %
breaklines=true,
keywordstyle=\color{red},
frame=b,
stringstyle=\color{white}\ttfamily,
showspaces=false,
showtabs=false,
xleftmargin=17pt,
framexleftmargin=17pt,
framexrightmargin=5pt,
framexbottommargin=4pt,
showstringspaces=false
}
\lstloadlanguages{
Java
}
\begin{lstlisting}
/******************************************************************
* Just run Collatz Sequence and store the generation of numbers in a
* byte array, noting that we only store the result of 3/2 * ( n+1) -1
* Also - we do not go to the end of Convergence
* Because then it would be too predictable.
* What is apparent Collatz Sequence has 3 domains
* 1. Randomization - where the input number would be shuffled and randomized
* 2. Sustained Run - where the randomized tape would endure chaotic behaviour
* 3. Convergence - where it would actually converge to 1.
* In this generator - we want to operate only on [1] and [2]
*******************************************************************/
static byte[] DoBinaryCollatzUntilSmall(BigInteger nB)
{
byte[] arr = new byte[0];
double perc = nB.toByteArray().length;
byte[] tmp =null;
/* The 0.8 is the amount of reduction, bytewise.
* If we let it reduce to 1 byte, then the end bytes would be too predictable as all numbers
* converge to 1.
* The real random parts are on the higher number of bytes range.
* So, we choose a parameter and let the number of bytes reduced to 0.8 of the original.
*/
int reduction = (int)Math.ceil(perc*0.8);
BigInteger radixB = new BigInteger("2");
do
{
while(nB.mod(radixB).compareTo(BigInteger.ZERO) == 0)
{
nB = nB.divide(radixB);
/*
* Why this is done?
* Assume we have generated X000, then the system would
* generate X00X0X, clearly pattern of X.
* with the new code we will not ever generate X again.
* if 3(X+1) is Y0, then we would generate
* XY
*/
}
BigInteger mod = nB.mod(radixB);
mod = radixB.subtract(mod);
mod = mod.add(nB);
nB = nB.multiply(radixB).add(mod);
nB = nB.divide(radixB);
tmp = nB.toByteArray();
arr = add_to_array(tmp,arr);
//System.out.println(tmp.length +":"+ reduction);
}while( tmp.length > reduction );
return arr;
}
\end{lstlisting}
The findings are summarized in the table [\ref{table:pr-results}].
\begin{table}[!ht]
\caption{Results of the Diehard Tests }
\centering
\begin{tabular}{ | l | l | l | }
\hline\hline
Test Name & Resulting p-value[s] & End Result \\ [0.5ex]
\hline
Birthday Spacing & 0.913665 & PASSED \\ \hline
Overlapping 5-Permutation & 0.479927,0.631744 & PASSED \\ \hline
Binary Rank Test (31X31) Matrix & 0.337549 & PASSED \\ \hline
Binary Rank Test (32X32) Matrix & 0.568434 & PASSED \\ \hline
Binary Rank Test (6X8) Matrix & 0.963464 & PASSED \\ \hline
The Bit Stream Test & min 0.05492, max 0.9487 & PASSED \\ \hline
OPSO & min 0.0492, max 0.9579 & PASSED \\ \hline
OQSO & min 0.0496, max 0.9801 & PASSED \\ \hline
DNA & min 0.0631, max 0.9877 & PASSED \\ \hline
Count The 1's On Byte Stream & 0.980913, 0.589621 & PASSED \\ \hline
Count The 1's On Specific Bytes & min 0.015312,max 0.99139 & PASSED \\ \hline
Parking Lot & 0.177727 & PASSED \\ \hline
Minimum Distance Test & 0.833870 & PASSED \\ \hline
3D Sphere Test & 0.954914 & PASSED \\ \hline
Squeeze Test & 0.483180 & PASSED \\ \hline
Overlapping Sums Test & 0.491630 & PASSED \\ \hline
Runs Test Up & 0.256061, 0.528445 & PASSED \\ \hline
Runs Test Down & 0.485818, 0.751651 & PASSED \\ \hline
Craps Test & wins:0.682508,throws/game:0.268146 & PASSED \\ \hline
\hline
\end{tabular}
\label{table:pr-results}
\end{table}
\subsection{General Collatz Sequence As A Random Walk Problem}
The General Collatz Sequence becomes a random walk problem when we see the results of the random bit generator.
We define the problem as follows:- \\
Define an one dimensional discrete space, where a point: $P(x) \in \{ 1,2,3,4,...\}$.
Assume `$k$' is the number of digits in the binary representation of a number, $n$.
Clearly $k \in \{ 1,2,3,4,...\}$ and can be represented as a point in that discrete space.
All the numbers $ \{ x : |x|=k \} $
(where $|x|$ denotes number of digits of ``x'' when expressed in binary)
maps to the same point ($k$) in that space.
If we define the origin of the space as ``1'', then the distance ``$D(n)$'' of the number with digits``$k$'' from origin is: $D(n)=k-1, \; |n| = k $.
\begin{itemize}
\item If by any transform the number of digits increases for a number `n' , the distance $D$ increases.
\item If the number of digits remains same for the number `n' , the distance $D$ remains same.
\item If by any transform the number of digits decreases for a number `n' , the distance $D$ decreases.
\end{itemize}
Adding any random even number from interval $0$ to $2(2^{k -1} -1)$ would do either of the 3 things as follows :-
\begin{list}{Action \arabic{qcounter}:~}{\usecounter{qcounter}}
\item Effective Left-shift by adding $1$ to the left side, and the number of digits moves to $k+1$
\item No movement i.e. $k$ remains as it is.
\item Right-Shift, as effective $0$s would be padded to the right, and the number of digits moves to less than $k$.
\end{list}
Point to be noted here is that for left-shift, we have step-size as only $1$. But for right-shift, the available step size is in the set $\{1,2,3,..,k-1\}$.
So, the random walk is so defined as $D(t+1) = D(t) + X$ where
$$
X \in \{ 1,0,-1,-2,...,-(k-1)\}
$$
``X'' being a random variable.
The logarithm of a number is nothing but the amount of digits needed to represent it.
In that case stopping time is nothing but the time taken to reach some lower number of digits, and can be taken as
\begin{equation}\label{stopping_time}
\sigma(n) \approx \frac{log_b(n)}{E(X)}
\end{equation}
where $E(X)$ is statistical expectation of random variable ``$X$''.
This is the reason why runtime of Collatz like sequences are known to have stopping time that is proportional to
the logarithm of the number as mentioned in Lagarias \cite{lagarias} and comes from the 1st principle, rather than
any heuristic presented in Crandall \cite{crandall} and Shanks \cite{shanks}, and provides a theoretical
basis for further study.
We can also expect then, that if we do random sampling
on numbers with a very high number of digits
(typically 500+ on smaller bases, and 100+ on larger bases)
then take ratio of the average of the empirical stopping time
with the number of digits, the result would be approximately a fixed ratio, no matter how large we make the number of digits.The table [\ref{table:Collatz_Shift_Table}] shows the expectation values remains fixed,
as number of digit goes large.
In other words, we can empirically find expectation $E(X)$ by this formula
$$
{E(X)} \approx \frac{log_b(n)}{\sigma(n)}
$$
and this would have a limit when digits go large.
This behavior is actually seen when we try to analyze general Collatz behavior.
That experiment is the topic of the next section, as the limiting fixed ratio is what is actually observed.
We see that the convergence can be thought about as ``Classical Ruin'' problem, where you started with some fixed amount of digits, and start either winning 1, or loosing or staying same with probabilities $p_l$ and $p_r$ and $p_n$.
If the expectation is negative then it would ensure that after a random run, the system always would
end up with 1 or more digit less than how it started. Hence \emph{starting from $k$ digits, one would always
end in $k-i, i \in \{1,2,3,...\} $ eventually}.
\section{Generalized Collatz Sequence At Large Number of Random Digits}
We define the Probability $\displaystyle\lim_{x\to\infty}P(b,x,s)$ as the probability of moving to $s$ steps when
number of digits ``x'' is approaching infinity, where $s \in \{1,0,-1,-2,...\}$.
For brevity we can write $\displaystyle\lim_{x\to\infty}P(b,x,s) = P(b,s)$ by dropping the ``x'' altogether.
Collatz sequence does not sample the Most Significant digits uniformly, because whenever it left-shifts,
out of the possibility of
$$
\{1,2,3,...,b-1\} \times \{0,1,2,3,...,b-1\}
$$
it can expand only to $(10)_b$ as the two most significant bits.
We did experiments to find the probability of left-shift and right-shift of the Collatz system.
They were done using random digit stream to be feed to the Generalized Collatz sequence as integer input.
The convergence points are never more than two, even on 1000 or more digit long integers.
For example, base 3 has $\{1,7\}$ convergence points, and even at 1000 digit long integer,
the system still converge at only these two points.
Upon fixing the number of digits, 50 sample numbers were created by choosing each digit randomly.
On these 50 numbers the left, neutral (remain on the same digit) and right shift probabilities were observed and averaged.
The rationale of choosing each digit random is to ensure that the Collatz system does not have to do the randomization.
The table [\ref{table:Collatz_Shift_Table}] shows the system dynamics on different bases, and in different number of digits.
(They might add up to more than 1.00 as we have chosen for all the values - most occurring ones).
\begin{table}[!ht]
\caption{ P(b,s) Empirical Data - Showing Probability Left/Right Movement }
\centering
\begin{tabular}{ | l | l | l | l | l | l | }
\hline\hline
Base & Digits & P(b,1) & P(b,0) & P(b,-Any) & Expectation \\ [0.5ex]
\hline
2 & 1000 - 5000 & 0.39090 & 0.27657 & 0.33252 & -0.27149 \\ \hline
3 & 500 - 1000 & 0.19704 & 0.55520 & 0.24775 & -0.17518 \\ \hline
5 & 500 - 1000 & 0.09451 & 0.73837 & 0.16678 & -0.11436 \\ [1ex]
\hline
\end{tabular}
\label{table:Collatz_Shift_Table}
\end{table}
One key thing to be noted here is that we modified the original sequence.
When a number of form $n = pb^s; p \; mod \; b \ne 0$ comes, instead of treating reduction of this number to
``s'' steps into ``p'', we treated it as a single step reduction, an ``s'' step right shift.
Hence, there exist a P(b,-s), that is probability of occurrence of a number of the form $n = pb^s$.
Another table [\ref{table:Collatz_Right_Prob_Table}] shows the P(b,-s) in different bases.
\begin{table}[!ht]
\caption{ P(b,-s) Empirical Data - Showing Probability Of Various Right Shift Movements }
\centering
\begin{tabular}{ | l | l | l | l | }
\hline\hline
Base & P(b,-1) & P(b,-2) & P(b,-3) \\ [0.5ex]
\hline
2 & 0.16699 & 0.08358 & 0.04117 \\ \hline
3 & 0.16500 & 0.05526 & 0.01851 \\ \hline
5 & 0.13310 & 0.02706 & 0.00527 \\ [1ex]
\hline
\end{tabular}
\label{table:Collatz_Right_Prob_Table}
\end{table}
\clearpage
\section {Collatz Probability Formulae For Base 2 - The Original Collatz Sequence}
In the earlier section, we have presented the probabilities as coming from the data itself. We show how they are to be
expected from the 1st principle.
\begin{theorem}\label{right-shift-prob}
\textbf{Total Right Shift Probability.}
Total Right Shift Probability is :
$$
\frac{1}{b+1} .
$$
\end{theorem}
\begin{proof}[Proof of Total Right Shift Probability Theorem]
The Collatz system is 3 state system. It can either do a left shift (increasing digits), remain neutral
(no change in the number of digits), or do a right shift (decrease in 1 or more number of digits).
This is a discrete step process.
Assume that at the $k$'th step the probability
of right shift is $p_r(b,k)$. So the probability that the system \emph{did not do a right shift at k'th step } is
given by:-
$$
1 - p_r(b,k)
$$
Now, assuming Collatz Transform generates the lower significant digits in Uniform distribution,
the probability of having one or more trailing zero would be digit $p( d_0 = 0 ) = \frac{1}{b}$,
which means, probability of having trailing zeros at $k+1$'th step is:-
$$
p_r(b,k+1) = (1 - p_r(b,k)) \times \frac{1}{b}
$$
Now, at equilibrium, we would expect that
$$
p_r(b,k) \approx p_r(b,k+1) \approx p_r(b)
$$
Solving for $p_r(b)$ immediately gives:-
\begin{equation}\label{right-shift-prob}
p_r(b) = \frac{1}{b+1}
\end{equation}
This establishes the theorem.
\end{proof}
\begin{theorem}\label{right-shift-prob-dist}
\textbf{Right Shift Probabilities.}
Right Shift Probabilities : P(b,-s) are geometrically distributed.
\end{theorem}
\begin{proof}[Proof of Right Shift Probabilities Theorem]
We show here that the P(b,-s) has the formula:-
\begin{equation}\label{right-shift-dist}
P(b,-s) = \frac{(b-1)}{(b+1)}b^{-s}
\end{equation}
We start with noting that P(b,-s) is actually the probability of having ``s'' numbers of ``0'' in
the end of a number, feed into the right shift operation of the abstract machine.
Here we note that
$$
P(b,-k-1) = \frac{1}{b}P(b,-k)
$$
as, to generate a ``k+1'' times ``0'' padded number from ``k'' times ``0'' padded number is to add an additional ``0''
to the right, which only 1 out of $\{0,1,2,3,..,b-2,b-1\}$ or $b$ possibilities.
We now deduce
$$
P(b,-1) = \frac{b-1}{b+1}b^{-1}
$$
Assume a number $X=d_{n-1}d_{n-2}...d_1d_0$. It would undergo a single right shift iff:-
The system is doing a right shift, with $d_1$ is nonzero.
Formally:-
$$
P(b,-1) = P(d_1 \neq 0 \; AND \; `System \; Doing \; Right \; Shift')
$$
as they are independent:-
$$
P(b,-1) = P(d_1 \neq 0 ) \times p_r(b)
$$
However, we clearly know that:-
$$
P(d_1 \neq 0 ) = \frac{b-1}{b}
$$
Hence, P(b,-1), that is only a single right shift takes place becomes:-
$$
P(b,-1) = \frac{b-1}{b} \times \frac{1}{b+1} = \frac{b-1}{b+1}b^{-1}
$$
This establishes the theorem, when we note down that $P(b,-k-1) =\frac{1}{b}P(b,-k)$.
This is also clear from the table [\ref{table:Collatz_Right_Prob_Table}]
\end{proof}
We now note a population of original Collatz sequences, defined to be
$$
\{...,f^i(x),...,f^n(x)\}
$$
We note that in any $f^i(x)$, the most significant 2 digits can be either ``10'' or ``11''.
But as the system always generates every left shift ``10'', hence we can be sure that in the population, ``10''
would be higher than that of ``11''. Assume also that we are concerned with only the digit expansion operation,
as digit contraction operation (right shift) does not change the MSBs.
We now derive the proportion of time the system would be in ``11'' and ``10'' states.
We note that the system prefixing ``11'' would always become ``10'' in the next iteration.
While, ``10'' would become ``11'' if an only if there is a carry from the lower digits.
We now deduce the probability of the carry.
\begin{theorem}\label{carry_theorem}
\textbf{Probability of carry to the MSB is a constant for Collatz Sequence.}
Probability of carry to the MSB is :
$$
\frac{1}{3} .
$$
\end{theorem}
\begin{proof}[Proof of Carry Theorem]
If the number can be written as $d_{k-1}d_{k-2}...d_0$ we can effectively write it as
$$
d_{k-1}d_{k-2}...d_0 = d_{k-1}d_{k-2}X
$$
where X is itself a number which is NOT divisible by 2.
Hence, \emph { X is a max $k-2$ digit odd number, there is no restriction of initial 0's on X}.
Now we note that for a combination of $(d_{k-1},d_{k-2}) = (1,0)$, the resultant number looks like
$$
10X0 + 10X + 1
$$
That is to say, to influence the $d_{k-1} = 1$, the $3X+1$ number had to have $|X| + 2$ digits.
If we consider odd numbers X, starting from 1 to $(2^{k} -1)$,
and check how many times applying generalized Collatz transform increased the number of digits by 2,
we would reach the number of cases a carry is generated.
Assume the number is $d_{k-1}d_{k-2}X_{k-2}$.
If $(d_{k-1},d_{k-2}) = (1,1)$, there surely would be a carry.
If $(d_{k-1},d_{k-2}) = (1,0)$, there would be a carry \emph {iff} $X_{k-2}$ has one carry.
Hence comes the recurrence relation:-
\begin{equation}\label{rr_of_carry}
p_c(k) = \frac{1}{4} + \frac{1}{4}p_c(k-2)
\end{equation}
It is not hard to see that the above recurrence relation, at large k, would yield:-
$$
p_c(k) \approx p_c(k - 2)
$$
And from there:-
\begin{equation}\label{probability_of_carry}
P(Carry) = p_c \approx \frac{1}{3}
\end{equation}
And this completes our proof.
\end{proof}
\begin{theorem}\label{density_theorem}
\textbf{The probability of `10' and `11' states are constants.}
The probability of occurrence of `10' and `11' states are 0.60 and 0.40 respectively.
\end{theorem}
\begin{proof}[Proof of State Density Theorem]
Let's define the symbol $\Delta_{x}N_{y}$ as earlier system was in `x' and now in `y',
Clearly then, we have the following behavior:-
$$
P(10|11) =\frac{\Delta_{11}N_{10} (t+1)}{N_{11}(t)} = 1
$$
And, as ``10X0'' when added with ``10X'' would always generate ``11'' system, \emph {iff} there is no carry:-
$$
P(11|10) = \frac{\Delta_{10}N_{11}(t+1)}{N_{10}(t)} = 1 - p_c
$$
In equilibrium state:-
\begin{equation}\label{msb_delta_population_equality}
\Delta_{11}N_{10}(t) = \Delta_{10}N_{11}(t)
\end{equation}
Immediately the below equations follow:-
\begin{equation}\label{msb_population_density_11}
\frac{N_{11}(t)}{N_{10}(t) + N_{11}(t) } = \frac{ 1 - p_c}{1 + (1 - p_c)} \approx \frac{2}{5}
\end{equation}
\begin{equation}\label{msb_population_density_10}
\frac{N_{10}(t)}{N_{10}(t) + N_{11}(t) } = \frac{1}{1 + ( 1 - p_c)} \approx \frac{3}{5}
\end{equation}
And this completes our proof.
\end{proof}
We end the discussion by stating that the empirical data shows the ratio of population
of $N_{11}$ and $N_{10}$ as 0.40667 and 0.5930 approximately.
Now we derive the left shift probability of the base 2 Collatz system - that is
probability that the number of digits would be increased by 1.
\begin{theorem}\label{left-prob-theorem}
\textbf{Left Shift Probability of Collatz Sequence.}
Left Shift Probability of Collatz Sequence is 0.4 approximately.
\end{theorem}
\begin{proof}[Proof of Left Shift Probability Theorem.]
The system left-shifts only when the system has no '0's in the right.
So, assume the probability of right shift is :-
$$
P(Right\;Shift) = p_r
$$
then, the probability of left-shift of the system is
$$
P(Left\;Shift) = p_l = (1-p_r)(P_L(10)P(10) + P_L(11)P(11))
$$
where $P(10) \approx 0.6$ and $P(11)\approx 0.4$ are probabilities
that system would be in ``10'' and ``11'' MSB state.
But then,
$P_L(11) = 1$, that is system always would increase the number of digits, if it is odd.
Right shift probability from \eqref{right-shift-prob}
\begin{equation}\label{probability_rightshift}
p_r(2) = p_r = \frac{1}{2+1} \approx 0.33
\end{equation}
That would mean that
\begin{equation}\label{probability_leftshift}
p_l \approx (1-p_r)( 0.4 + 0.6p_c ) \approx 0.6(1-p_r) \approx 0.40
\end{equation}
While empirically found value was 0.39.
This establishes the theorem.
\end{proof}
Now we find the expectation value for base 2.
\begin{theorem}\label{expectation-value-theorem}
\textbf{Expectation of Collatz Random Walk is constant.}
Expectation of Collatz Random Walk is $-0.270$ approximately.
\end{theorem}
\begin{proof}[Proof of Expectation Value]
We note down it as
$$
E(X) = 1 \times p_l + 0 \times p_n + {\sum_{s=-1} ^{-\infty}} \frac{(b-1)}{(b+1)}sb^{s}
$$
For b=2, that rightmost term becomes:-
$$
{\sum_{s=-1} ^{-\infty}} \frac{(2-1)}{(2+1)}s2^{s} = -\frac{2}{3}
$$
\begin{equation}\label{expectation_2}
E(X) \approx 0.40 - 0.667 \approx -0.267
\end{equation}
This proves the theorem.
\end{proof}
\begin{theorem}\label{expected-runtime-theorem}
\textbf{Expected Stopping Time of Collatz Sequence would be constant times the number of bits.}
Expected runtime (Stopping Time) of Collatz Sequence would be approximately 3.7 times the number of bits.
\end{theorem}
\begin{proof}[Proof of Expected Stopping Time]
The expected time to converge for random digits would be
\begin{equation}\label{converge_time_2}
\sigma_2(n) \approx \frac{log_2(n)}{E(X)} \approx 3.74log_2(n)
\end{equation}
\end{proof}
This is the linearity with respect to $log(n)$ we see in the large numbers (n).
Empirically observed value is $\sigma_2(n) \approx 3.60log_2(n)$.
\section{Summary And Further Work}
We can surely say that the Collatz like sequences are built on top of chaos generated
by the implicit pseudo random generator, $bn + n + (b - n \; mod \; b )$ according to (\ref{Collatz_lcg}).
We also noted that adding fixed even numbers in range (even at random)
$\{2,4,..., 2(2^{k -1} -1) \}$ at every step, destroys the chaotic pattern.
We have shown empirically that the pure random sequence terminates,
and we proved that the statistical convergence exists by enumerating the possibility of the left and right shift and that
the resulting expectation is always negative.
We also found numerous theorems from the basic probability principles, which shows
remarkable accuracy with experimental data.
Research is much needed on this specific type of pseudorandom sequence generator, and predictability of the bases, where
the system would converge to ``1'', and how many attractor/convergence points the system should have.
We end with a quote from the great Von Neumann:\emph{``Anyone who considers arithmetical methods of producing random digits is, of course, in a state of sin.''}.
We humbly beg to differ, by stating that \emph{The Generalized Collatz Sequence} can be used as a
fantastic random symbol generator, passing all but one diehard tests (fails the minimum distance test) and almost all NIST tests. As Simplified Generalized Collatz Sequence (\ref{generalized_simplified_Collatz}) does not need any arithmetic operation other than addition, it is a good candidate for pseudorandom sequence generation in computers.
\section*{Acknowledgments}
The authors thank the colleagues of Microsoft India.
|
{
"timestamp": "2012-03-13T01:00:52",
"yymm": "1203",
"arxiv_id": "1203.2229",
"language": "en",
"url": "https://arxiv.org/abs/1203.2229"
}
|
\section{Introduction}
Modern approaches to unify theories of quantum mechanics and general
relativity, for instance, string theory and loop quantum gravity
predict that a black hole emits thermal radiations whose thermal
spectrum might deviate from
Planck black body spectrum at Planck scale \cite{barton}.
The black hole's event horizon possesses temperature inversely proportional to black hole mass and with an entropy proportional to its horizon's surface area (in units $c=G=\hbar=1$) i.e.
\begin{equation}\label{f61}
S_h=\frac{A_h}{4},
\end{equation}
where $A_h=4\pi R_h^2$ is the area of the black hole's event
horizon. Therefore the horizon entropy (\ref{f61}) becomes
\begin{equation}\label{f6}
S_h=\pi R_h^2.
\end{equation}
These seminal connections between black holes and thermodynamics were
initially made by Hawking and Bekenstein several decades ago \cite{hawking}.
The Hawking temperature and horizon entropy together with the black
hole mass obey the first law of thermodynamics $TdS=dE+pdV$.
Padmanabhan showed that Einstein field equations for a spherically
symmetric spacetime can be recast in the form of first law of
thermodynamics \cite{paddy}. Cai \& Kim applied the similar
formalism and demonstrated that by applying the first law of
thermodynamics to the apparent horizon of a
Friedmann-Robertson-Walker universe the Friedmann equations of the
universe with any spatial curvature can be derived from the first
law \cite{cai}. Recently similar results have been obtained in
scalar tensor, Gauss-Bonnet, Lovelock and $f(R)$ gravities by
different authors \cite{akbar1}. Jacobson showed that Einstein field
equation is nothing but an equation of state of spacetime i.e. the
Einstein equation can be derived by assuming the universality of
(\ref{f61}) on any local Rindler horizons \cite{ted}. In black hole
physics, the generalized second law is a conjecture about black hole
thermodynamics which states that ``the sum of the black hole entropy
(1/4 of the horizon area) and the common (ordinary) entropy in the
black hole exterior never decreases' originally proposed by
Bekenstein \cite{hod,b}. However as discussed by Jacobson
\cite{ted}, the entire framework of black hole thermodynamics and,
in particular, the notion of black hole entropy extends to any
causal horizon. In cosmological spacetime, the corresponding object
is \textit{apparent horizon} \cite{peng}.
The power-law correction to entropy which appear in
dealing with the entanglement of quantum fields in and
out the horizon is given by
is \cite{power}
\begin{equation}\label{1b}
S_h=\frac{A_h}{4}\Big(1-K_\alpha A_h^{1-\frac{\alpha}{2}} \Big),
\end{equation}
where $\alpha$ is a dimensionless constant and $A_h=4\pi R_h^2$ is the area while $R_h$ is the radius of the horizon.
\begin{equation}\label{1c}
K_\alpha=\frac{\alpha(4\pi)^{\frac{\alpha}{2}-1}}{(4-\alpha)r_c^{2-\alpha}}.
\end{equation}
where $r_c$ is a cross-over scale, $R_h$ is the radius and $A_h$ is
area of the cosmological horizon. For entropy to be a well-defined
quantity, we require $\alpha>0$. The second term in (\ref{1b}) can
be regarded as a power-law correction to the area law, resulting
from entanglement, when the wave-function of the field is chosen to
be a superposition of ground state and exited state \cite{p1}.
Several aspects of power-law corrected entropy (\ref{1b}) have been
studied in literature including GSL \cite{jamil3}, power-law entropy
corrected models of dark energy \cite{jamil4}.
The quantum corrections provided to the entropy-area relationship
leads to the curvature correction in the Einstein-Hilbert action
and vice versa. The logarithmic corrected entropy is \cite{jamil5}
\begin{equation}\label{1a}
S_h=\frac{A_h}{4}+\beta\log\Big( \frac{A_h}{4} \Big)+\gamma.
\end{equation}
These corrections arise in the black hole entropy in loop quantum
gravity due to thermal equilibrium fluctuations and quantum
fluctuations. Jamil \& Sadjadi \cite{log} showed that in a (super)
accelerated universe GSL is valid whenever $\beta(<)>0$ leading to a
(negative) positive contribution from logarithmic correction to the
entropy. In the case of super acceleration the temperature of the
dark energy is obtained to be less or equal to the Hawking
temperature. Using the corrected entropy-area relation motivated by
the loop quantum gravity, Karami et al \cite{karami} investigated
the validity of the GSL in the FRW universe filled with an
interacting viscous dark energy with dark matter and radiation. They
showed that GSL is always satisfied throughout the history of the
universe for any spatial curvature regardless of the dark energy
model.
We consider a scenario of a spatially flat, homogeneous, isotropic
universe filled with phantom energy and contain a Schwarzschild
black hole. Since this simple cosmic system consists of three
components, we associate the entropy with each component. The
entropy of Schwarzschild black hole and FRW universe is proportional
to the size (area) of their horizon (only if the entropic
corrections are ignored) while for phantom energy, the entropy is
calculated via the first law of thermodynamics. We assume that the
black hole accretes phantom energy such that the mass of black hole
decreases very slowly while preserving the spherical symmetry. This
kind of accretion is termed as quasi-static accretion, and the
corresponding black hole in quasi-static state (i.e. Schwarzschild
geometry is still valid) \cite{horvath}. While accretion, the
entropies of phantom energy and the black hole vary, but the total
entropy of the system remains non-decreasing. From the argument of
first law of thermodynamics, we notice that the rate of change of
entropy of phantom energy $\dot S_d=T^{-1}\dot H R_h^2$ depends on
its temperature $T$ and the rate of change of Hubble parameter $\dot
H$. The entropy $\dot S_d>0$ if both $T>0(<0)$ and $\dot H>0(<0)$.
According to some earlier thermodynamic approaches to gravity, the
form of entropy-area relation applies the same to different
horizons, here we apply the same principle in different sections.
First we choose the classical Bekenstein-Hawking relation for
entropy for the horizons of both black hole and cosmological and
investigate the GSL. Later we study the same phenomenon by including
the power-law and logarithmic corrections to the entropy of both
black hole and cosmological horizon. Although the horizon of a
Schwarzschild black hole is uniquely defined, the form of
cosmological horizon is not so. We use the two well-known forms of
cosmic horizons i.e. the future event horizon and the apparent
horizon (both will be defined later).
The idea of the
combined effect of a cosmological system involving a dark energy
component and a black hole has also been explored in `entropic
cosmology' \cite{yifu}. In these works, Cai and collaborators
investigated the dynamical thermal balance of a double-screen model
(corresponding to cosmic horizon and black hole horizon) which can
realize both inflation and late time acceleration of the Universe.
Earlier Izquierdo \& Pavon \cite{pavon} investigated the GSL for a
system comprising of a Schwarzschild black accreting phantom energy
in FRW universe. They showed that GSL is violated. Later Sadjadi
\cite{sadjadi} investigated the same problem and showed that GSL
will be satisfied if the temperature is not taken as de Sitter
temperature. Understanding the evolution of a black hole in a FRW
cosmological background is a very old problem starting from Hawking
\& Carr \cite{carr} whose satisfactory resolution is still not
available, however several approximations (like the present
analysis) are available in the literature, e.g.
\cite{sadjadi,pavon,jamil}.
The plan of our paper is as follows: In section-II, we write down
the basic equations of standard model of cosmology and the
definition of generalized second law of thermodynamics in the
present context. In sections-III, IV and V, we study the GSL with
Bekenstein-Hawking entropy-area relation, power-law entropy
correction and logarithmic entropic correction, respectively with
the use of apparent and event horizons. In section-VI, we discuss
the constraints imposed by GSL on the black hole mass for accretion
of phantom energy. In last section, we write down the conclusion
giving a summary of our results.
\section{Basic Equations}
Assuming homogeneous, isotropic and spatially flat Friedmann-Robertson-Walker metric:
\begin{equation}\label{f1}
ds^2=-dt^2+a(t)^2(dr^2+r^2(d\theta^2+\sin^2\theta d\phi^2)).
\end{equation}
The Friedmann equations are
\begin{equation}\label{f2}
H^2=\frac{8\pi }{3}\rho,
\end{equation}
\begin{equation}\label{f3}
\dot H=-4\pi (\rho+p).
\end{equation}
The continuity equation is
\begin{equation}\label{f4}
\dot\rho+3H(\rho+p)=0,
\end{equation}
where $\rho$ and $p$ are the energy density and pressure of phantom
energy. Assuming the phantom energy as a perfect fluid, we specify
it by a phenomenological equation of state
\begin{equation}\label{f5}
p=w\rho,
\end{equation}
where $w<-1$ is the dimensionless state parameter of phantom energy.
The notion of phantom energy was introduced by Caldwell et al
\cite{cald} as a separate candidate of explaining cosmic
acceleration. The phantom energy possesses some esoteric properties:
``Phantom energy rips apart the Milky Way, solar system, Earth, and
ultimately the molecules, atoms, nuclei, and nucleons of which we
are composed, before the death of the Universe in a Big Rip''
\cite{cald}. Although the model of phantom energy is consistent with
the observational data \cite{cald1}, the cosmic doomsday (or Big
Rip) can be avoided in certain theories of modified gravity
\cite{odintsov}. Thermodynamical studies show that phantom energy
possesses negative temperature and positive entropy \cite{sigu}.
However some discussion contrary to \cite{sigu} on phantom
thermodynamics has been performed in \cite{lima1}. Nojiri et al
discussed the occurrence of different types of future singularities
in phantom cosmology \cite{noji}. One of these singularities is Big
Rip described as: (or Type-I): As $t\rightarrow t_s$,
$a\rightarrow\infty$, $\rho\rightarrow\infty$,
$|p|\rightarrow\infty$. A form of scale factor satisfying these
conditions can be $a(t)=a_0(t_s-t)^{n}$, $t_s>t$,
$n=\frac{2}{3(1+w)}<0$, $a_0>0$ where $t_s$ is the Big Rip time. The
corresponding Hubble parameter goes like
\begin{equation}\label{h}
H(t)=\frac{2}{3(1+w)(t-t_s)}. \ \ (w<-1)
\end{equation}
Thus Hubble parameter diverges as $t\rightarrow t_s$.
Babichev et al \cite{babi} demonstrated that the accretion of
phantom energy as a test perfect fluid on a stationary spherically
symmetric black hole gradually reduces the mass of black hole.
Further the mass of black holes approach to zero near the time
called Big Rip. They deduced that the rate of change in black hole
mass due to phantom energy accretion goes like \cite{babi}
\begin{equation}\label{f13}
\dot M=4\pi A r_h^2(\rho+p)<0.
\end{equation}
Above $A$ is a positive dimensional constant, $r_h$ is the
Schwarzschild radius, while energy density and pressure of phantom
energy violates the null energy condition $\rho+p<0$. Later on their
analysis extended in various ways: using bulk viscosity, generalized
Chaplygin gas, Riessner-Nordstrom and Kerr-Newmann black holes
\cite{jamilbh} to list a few.
Making use of (\ref{f3}) in (\ref{f13}), we can write
\begin{equation}
\dot M=-4AM^2\dot H.
\end{equation}
On integration, we obtain
\begin{equation}\label{f14}
M(H)=\frac{1}{C_1+4AH},
\end{equation}
where $C_1$ is a constant of integration. From (\ref{f14}), we
observe that mass of black hole decreases as the rate of cosmic
expansion increases.
For generalization of these results to many black holes, one
can follow the procedure of Khan and Israel \cite{khan} by first
replacing the spherically symmetric black hole by a point mass
located on the $z$-axis. Then we can use the superposition principle
to write the general expression of gravitational potential of the
system. In this case, we can replace the mass of a BH by the total
mass of many point particles. However we are not interested to such
extensions and beyond the scope of our paper.
To calculate the entropy of phantom fluid, we use the first law of
thermodynamics which relates the pressure, energy density, total
energy and temperature of phantom energy i.e.
\begin{equation}\label{f16a}
dS_d=\frac{1}{T}(dE+pdV)=\frac{1}{T}((\rho+p)dV+Vd\rho).
\end{equation}
The form of GSL containing the time derivatives of entropies of
black hole's horizon, phantom energy and cosmological's horizon is
\begin{equation}\label{f16}
\dot S_{tot}\equiv\dot S_{BH}+\dot S_{d}+\dot S_{A}\geq0.
\end{equation}
Above $\dot S_{tot}$ is the rate of change of total entropy which
must be non-decreasing.
Here we would like to comment that
as a result of accretion, the dark energy goes inside the BH event
horizon. Since the major bulk of dark energy density lies outside
the BH horizon than to its interior, we do not associate entropy to
DE lying inside the BH horizon. In other words, the entropy of DE is
solely determined from its quantity contained between the cosmic and
BH horizons while the entropy of DE inside BH horizon is ignored.
Also note that in a DE filled Universe, we assume that the interior
of BH horizon is always filled with DE as a result of accretion.
The analysis in later sections is based on the assumption of thermal
equilibrium: the temperature of black hole's event horizon,
cosmological horizon and the phantom energy are the same. But this
assumption in cosmological setting is very ideal, since major
components of the universe including dark matter, dark energy and
radiation (CMB and neutrinos inclusive) have entirely different
temperatures \cite{lima}. But Karami and Ghaffari \cite{kk} recently
demonstrated that the contribution of the heat flow between dark
energy and dark matter for GSL in non-equilibrium thermodynamics is
very small, $O(10^{-7})$. Therefore the equilibrium thermodynamics
is still preserved. Further, if there is any thermal difference in
the fluid and the horizons, the transfer of energy across the
horizons might change the geometry of horizons \cite{log}.
\section{GSL with Bekenstein-Hawking entropy}
\subsection{Use of Event Horizon}
The \textit{future event horizon} is the distance that light travels from present time till infinity is defined as
\begin{equation}
R_E(t)=a(t)\int\limits_t^\infty\frac{dt'}{a(t')}<\infty,
\end{equation}
whose time derivative is
\begin{equation}\label{f11}
\dot R_E=HR_E-1.
\end{equation}
The temperature of future event horizon is proportional to de Sitter's universe horizon
\begin{equation}\label{f7}
T_h=\frac{bH}{2\pi},
\end{equation}
where $b$ is a constant. Depending on the argument chosen from
thermodynamics of phantom energy \cite{sigu,lima1}, $b$ can be
positive or negative. Integrating (\ref{f11}) and using (\ref{h}),
we obtain a time evolution of $R_E$:
\begin{equation}\label{f12}
R_E=C_2(t-t_s)^{\frac{2}{3(1+w)}}-\frac{3(t-t_s)(1+w)}{1+3w},
\end{equation}
where $C_2$ is a constant of integration. In the present context,
Sadjadi \cite{sadjadi} studied the cosmological thermodynamics
using the cosmic future event horizon $R_E$. However in his detailed
analysis, the author ignored a very important first term on right
hand side of (\ref{f12}), which will change significantly the
results for the validity of GSL.
\subsection{Use of Apparent Horizon}
The \textit{apparent horizon} is a null surface with vanishing
expansion or the boundary surface of anti-trapped region
\cite{peng}. In a spatially flat FRW universe, the apparent horizon
is $R_A=H^{-1}$ (also called Hubble horizon). This horizon is
consistent if we insist on the validity of holography during
inflation i.e. the apparent horizon is the holographic boundary of
the FRW universe. The temperature of apparent horizon is the same as
temperature of a de Sitter's universe horizon \cite{cai}
\begin{equation}\label{f8}
T_h=\frac{H}{2\pi}
\end{equation}
Using (\ref{f16}) the form of GSL at the apparent horizon becomes
\begin{equation}
\dot {S}_{tot}=-32\pi A\dot H M^3 \geq0.
\end{equation}
In terms of Hubble parameter alone,
\begin{equation}\label{final}
\dot {S}_{tot}= \frac{-32\pi A\dot H}{(C_1+4AH)^3}\geq0.
\end{equation}
In the above case (\ref{final}), mathematically GSL will hold under
two situations: Case-I: (1) $\dot H\leq0$, $C_1+4AH>0$ or Case-II:
(1) $\dot H\geq0$, (2) $C_1+4AH<0$. However physically, under
phantom dominated era, only case-II is relevant. Thus the apparent
horizon expands in the phantom phase while the future event horizon
contracts.
\section{GSL with Power-Law Entropy Correction}
We extend our previous study here by taking into account the
correction to horizon's entropy of the power-law form.
\subsection{Use of Event Horizon}
Using the definition of GSL (\ref{f16}) gives
\begin{eqnarray}\label{eh1}
\dot {S}_{tot}&=&2\pi R_E\dot{R}_E \Big[ 1- \frac{\alpha(4\pi)^{\frac{\alpha}{2}-1}}{2r_c^{2-\alpha}} \Big(\pi R_E^2\Big)^{1-\frac{\alpha}{2}} \Big]
\nonumber\\&&-32\pi A\dot H M^3 \Big[ 1- \frac{\alpha(4\pi)^{\frac{\alpha}{2}-1}}{2r_c^{2-\alpha}}\Big(\frac{4\pi}{M^2}\Big)^{1-\frac{\alpha}{2}} \Big]\nonumber\\&&+\frac{2\pi\dot H}{bH}\dot R_E\geq0.
\end{eqnarray}
Its easy to interpret the positivity of (\ref{eh1}) by writing the
above equation as a total derivative form
\begin{eqnarray}
\dot {S}_{tot}&=&\frac{d}{dt}\Big[\pi R_{E}^2-\frac{\pi\alpha}{4-\alpha}(2r_c)^{\alpha-2}R_E^{4-\alpha}\\\nonumber&&+ 4\pi M^2 -\frac{8\pi}{2r_c^{2-\alpha}}M^\alpha \Big] \\\nonumber&&+\frac{2\pi\dot H}{bH}\dot R_E\geq0.
\end{eqnarray}
For phantom $b<0$, and in both cases $\dot{H}<(>0),\dot{R_E}<(>0)$, the general condition to satisfy GSL is
\begin{eqnarray}
&& \frac{d}{dt}\Big[\pi R_{E}^2-\frac{\pi\alpha}{4-\alpha}(2r_c)^{\alpha-2}R_E^{4-\alpha}\\\nonumber&&+ 4\pi M^2 -\frac{8\pi}{2r_c^{2-\alpha}}M^\alpha \Big] \geq0.
\end{eqnarray}
\subsection{Use of Apparent Horizon}
Using the definition of GSL (\ref{f16}) gives
\begin{eqnarray}
\dot {S}_{tot}&=&2\pi\frac{\dot H}{H^3}\frac{\alpha(4\pi)^{\frac{\alpha}{2}-1}}{2r_c^{2-\alpha}}\Big(\frac{\pi}{H^2}\Big)^{1-\frac{\alpha}{2}}
-32\pi A\dot H M^3\nonumber\\&&\times\Big[ 1-\frac{\alpha(4\pi)^{\frac{\alpha}{2}-1}}{2r_c^{2-\alpha}}\Big(\frac{\pi}{M^2}\Big)^{1-\frac{\alpha}{2}} \Big]\geq0.
\end{eqnarray}
We rewrite it in the following total derivative form
\begin{eqnarray}
\dot {S}_{tot}&=&\frac{d}{dt}\Big[\frac{\pi\alpha}{\alpha-4}\frac{(2r_c)^{\alpha-2}}{H^{4-\alpha}}\nonumber\\&&+4\pi M^2-4\pi(2r_c)^{\alpha-2}M^\alpha\Big]\geq0.
\end{eqnarray}
\section{GSL with Logarithmic Entropy Correction}
We extend our previous study here by taking into account the
correction to horizon's entropy of the logarithmic form.
\subsection{Case of Event Horizon}
Using the definition of GSL (\ref{f16}) gives
\begin{eqnarray}
\dot {S}_{tot}&=&2\dot{R}_E\Big( \pi R_E+\frac{\beta}{R_E} \Big)-8A\dot H M\Big( \beta-4\pi M^2 \Big)\nonumber\\&&+\frac{2\pi\dot H}{bH}\dot R_E\geq0.
\end{eqnarray}
For phantom $b<0$, and in both cases $\dot{H}<(>0),\dot{R_E}<(>0)$, the general condition to satisfy GSL is
\begin{eqnarray}
\dot {S}_{tot}&=&\frac{d}{dt}\Big[\pi R_E^2+2\beta\log (MR_E)-4\pi M^2\Big]\nonumber\\&&+\frac{2\pi\dot H}{bH}\dot R_E\geq0.
\end{eqnarray}
\subsection{Case of Apparent Horizon}
Using the definition of GSL (\ref{f16}) gives
\begin{equation}
\dot {S}_{tot}=-2\pi\frac{\dot H}{H}\beta M-8A\dot H M^2\Big( \beta-4\pi M^2 \Big)\geq0.
\end{equation}
Writing the above equation as a total derivative form, we get
\begin{equation}
\dot {S}_{tot}=\frac{d}{dt}\Big(\beta\log(\frac{M}{H})-2\pi M^2\Big)+\frac{2\pi\dot H}{bH}\dot R_E\geq0.
\end{equation}
\section{Phantom energy accretion by black hole: GSL constraints}
Pacheco \& Horvath \cite{hor} investigated the generalized second
law of thermodynamics for a static spherically symmetric black hole
accreting phantom energy. They showed that for a phantom fluid
violating the null energy condition ($\rho+p<0$), the Euler relation
($\rho+p=TS$) and Gibbs relation ($E+pV=TS$, assuming $E=\rho V$ we
get $(\rho+p)V=TS$) allows two different possibilities for the
entropy and temperature of phantom energy: a situation when the
entropy is negative and the temperature is positive or vice versa.
In the former case, if GSL is valid, the accretion of phantom energy
is not allowed while in the later case, there is a critical black
hole mass above which the accretion process is not allowed. In
another study, Lima et al \cite{mu} discussed the thermodynamics of
phantom energy with chemical potential $\mu_0$ and found the EoS
parameter of the form
\begin{equation}
w\geq-1+\frac{\mu_0 n_0}{\rho_0},
\end{equation}
where $n_0$ is the number density and $\rho_0$ is the energy density
of phantom energy. The authors deduced if $\mu_0<0$ then $w$
describes a phantom fluid. Lima and co-workers \cite{limaco} showed
that the temperature of the phantom fluid without chemical potential
is positive definite but entropy is negative $S_d < 0$. But their
claim is problematic since if considering the usual statistical
definition of entropy, than phantom energy must not exist at all.
Later Pacheco \cite{P} ruled out the previous results of phantom
accretion by black holes with chemical potential \cite{mu,limaco}.
Below, we adapt the procedure of \cite{hor} about constraints
imposed by GSL on the mass of a black hole. We find that there
exists a critical value of black hole mass (or the upper bound)
$M_c$ below which accretion of phantom is allowed. Note that in the
subsequent analysis, we deal phantom energy without chemical
potential on account of \cite{P}. Also the foregoing analysis with
Bekenstein-Hawking entropy has been done in \cite{hor}, therefore,
we will continue our work with logarithmic and power-law entropy
corrections only.
\subsection{Using Logarithmic Entropy}
We consider the new entropy as a sum of black hole entropy and entropy of phantom energy as
\begin{equation}
S_n=f(X)+\kappa \rho^{\frac{1}{1+w}}V,
\end{equation}
where $\kappa$ is a constant and $f(X)$ is a given function of area. We first start with logarithmic entropy
\begin{equation}\label{f}
f(X)=X+\beta\log(X)+\gamma,\ \ X\equiv\frac{A}{4}=4\pi M^2.
\end{equation}
On account of accretion, the change in the entropy of black hole and the phantom energy is
\begin{equation}\label{ds}
\Delta S_n=f'(X)\Delta X+ \frac{\kappa}{1+w}\rho^{\frac{-w}{1+w}}V\Delta\rho.
\end{equation}
The total energy conservation for this system
is \cite{hor}
\begin{equation}
\Delta M=-\frac{1}{2}(1+w)V\Delta\rho,
\end{equation}
or
\begin{equation}\label{dm}
V\Delta\rho=-\frac{2\Delta M}{1+w}.
\end{equation}
Using (\ref{dm}) in (\ref{ds}), we get
\begin{equation}\label{delta-s}
\Delta S_n=\Big[ 8\pi Mf'(X)-\frac{2\kappa}{(1+w)^2} \rho^{\frac{-w}{1+w}} \Big]\Delta M.
\end{equation}
Since mass of black hole decreases due to accretion of phantom energy i.e. $\Delta M<0$, we require
\begin{equation}
\Delta S_n >0 \ \Rightarrow 4\pi M_{c}f'(X_c)-\frac{\kappa}{(1+w)^2} \rho^{\frac{-w}{1+w}} <0.
\end{equation}
From (\ref{f}), we have $f'(X)=1+\frac{\beta}{X}$. Thus
\begin{equation}\label{quad}
M_c^2-\Big( \frac{\kappa}{4\pi (1+w)^2}\rho^{\frac{-w}{1+w}} \Big)M_c+\frac{\beta}{4\pi}\geq0.
\end{equation}
The discriminant of (\ref{quad}) is
\begin{equation}
\delta=\Big( \frac{\kappa}{8\pi (1+w)^2}\rho^{\frac{-w}{1+w}} \Big)^2-\frac{\beta}{4\pi}\geq0.
\end{equation}
Here two cases are possible: (1) $\delta=0$ gives $(M_c-M)^2<0$
which is nonphysical and mathematically not possible, (2) $\delta>0$
implies $(M_c-M_+)(M_c-M_-)>0$, where $M_\pm$ are the roots of
(\ref{quad}) given by
\begin{equation}
M_\pm=\frac{\kappa}{8\pi(1+w)^2}\rho^{\frac{-w}{1+w}}\pm\sqrt{\Big( \frac{\kappa}{8\pi (1+w)^2}\rho^{\frac{-w}{1+w}} \Big)^2-\frac{\beta}{4\pi}},
\end{equation}
while the critical black hole mass lies in the range
\begin{equation}
M_-<M_c<M_+,
\end{equation}
where $M_+$ and $M_-$ are the corresponding upper and lower bounds on critical mass of black hole.
\subsection{Using Power-Law Corrected Entropy}
The form of power-law corrected entropy is
\begin{equation}
S(X)=X[1-K_\alpha(4X)^{1-\frac{\alpha}{2}}].
\end{equation}
Here condition (\ref{delta-s}) implies
\begin{eqnarray}
&&8\pi M_c\Big[ 1-K_\alpha \Big( 2-\frac{\alpha}{2} \Big)(16\pi)^{1-\frac{\alpha}{2}}M_c^{2-\alpha} \Big]\nonumber\\&&-\frac{2\kappa}{(1+w)^2}\rho^{\frac{-w}{1+w}}<0.
\end{eqnarray}
In terms of $r_c$, the general equation for critical mass to satisfy is
\begin{eqnarray}\label{imp}
4\pi M_c\Big[1-\frac{\alpha}{2}\Big(\frac{2M_c}{r_c}\Big)^{2-\alpha}\Big]-\frac{\kappa}{(1+w)^2}\rho^{\frac{-w}{1+w}} <0. \label{gen}
\end{eqnarray}
To solve (\ref{imp}), we consider some special cases of (\ref{imp}) for different values of $\alpha=1,2,3,4,5$.
\subsection{$\alpha=1$}
We can write (\ref{imp}) in terms of cross-over scale parameter $r_c$ for convenience:
\begin{equation}\label{alpha1}
M_c-\frac{\alpha}{r_c}M_c^2-\frac{\kappa}{4\pi(1+w)^2}\rho^{-\frac{w}{1+w}}<0.
\end{equation}
Here discriminant of the above expression is
\begin{equation}
\delta=1-\frac{\kappa}{\pi r_c(1+w)^2}\rho^{-\frac{w}{1+w}}\geq0.
\end{equation}
The corresponding roots are
\begin{equation}
M_{\pm}=\frac{r_c}{2\alpha}(1\pm\sqrt{\delta}).
\end{equation}
The critical black hole mass lies in the range
\begin{equation}
M_-<M_c<M_+,
\end{equation}
where $M_+$ and $M_-$ are the corresponding upper and lower bounds on critical mass of black hole.
\subsection{$\alpha=2$}
In this case, we get the lower bound on the mass of black hole:
\begin{equation}
M>M_c=\frac{-\kappa}{4\pi (1+w)^2}\rho^{\frac{-w}{1+w}},
\end{equation}
while $\kappa<0$ since mass can not be negative physically.
\subsection{$\alpha=3$}
Here we obtain an upper bound on the mass of black hole as
\begin{equation}
M<M_c=\frac{3}{4}r_c+\frac{\kappa}{4\pi (1+w)^2}\rho^{\frac{-w}{1+w}}.
\end{equation}
\subsection{$\alpha=4$}
The equation to be satisfied by critical mass is
\begin{equation}
M_c^2-\frac{\kappa}{4\pi(1+w)^2}\rho^{\frac{-w}{1+w}}M_c-\frac{r_c^2}{2}<0
\end{equation}
The discriminant of the above equation is
\begin{equation}
\delta=\Big( \frac{\kappa \rho^{\frac{-w}{1+w}}}{4\pi (1+w)^2} \Big)^2+2r_c^2>0,
\end{equation}
while the roots are
\begin{equation}
M_\pm=\frac{\kappa}{8\pi (1+w)^2}\rho^{\frac{-w}{1+w}}\pm\frac{\sqrt{\delta}}{2}.
\end{equation}
Further critical mass satisfies $M_-<M_c< M_+$.
\subsection{$\alpha=5$}
Here we have the following cubic equation
\begin{eqnarray}
M_c^3-A_1 M_c^2-B_1<0, \label{cubic}
\end{eqnarray}
where
\begin{eqnarray}
A_1=\frac{\kappa}{4\pi(1+w)^2}\rho^{-\frac{w}{1+w}},\ \
B_1=\frac{5}{16}r_c^3
\end{eqnarray}
We define a new variable $y=M+\frac{A_1}{3}$, than the (\ref{cubic}) converts to
\begin{eqnarray}
y^3+py+q<0, \ \ p=-\frac{A_1^2}{3},\ \ q=B+\frac{2}{27}A_1^3
\end{eqnarray}
Since $p<0$, this cubic equation has three distinct solutions which are
\begin{eqnarray}
y_1&=&\sqrt{-\frac{p}{3}}\cos(\psi), \\ y_2&=&\sqrt{-\frac{p}{3}}\cos(\frac{\psi}{3}+\frac{\pi}{3}),\\
y_3&=&\sqrt{-\frac{p}{3}}\cos(\frac{\pi}{3}-\frac{\psi}{3}),
\end{eqnarray}
where
$$
\cos(\psi)=\frac{108B_1+8A_1^3}{8A_1^3}.
$$
Thus there are two possibilities for the value of critical mass
\begin{eqnarray}
M_c<M_{1},\ \ M_{2}<M_c<M_{3},
\end{eqnarray}
where
\begin{eqnarray}
M_{1}&=&\frac{\kappa}{6\pi(1+w)^2}\rho^{-\frac{w}{1+w}}\Big(-\frac{1}{2}+\cos(\psi)\Big),\\
M_{2}&=&\frac{\kappa}{6\pi(1+w)^2}\rho^{-\frac{w}{1+w}}\Big(-\frac{1}{2}+\cos(\frac{\pi}{3}-\frac{\psi}{3})\Big),\\
M_{2}&=&\frac{\kappa}{6\pi(1+w)^2}\rho^{-\frac{w}{1+w}}\Big(-\frac{1}{2}+\cos(\frac{\pi}{3}+\frac{\psi}{3})\Big).
\end{eqnarray}
\section{Conclusion}
In this paper, we studied the generalized second law of
thermodynamics for a system comprising of a Schwarzschild black
accreting a test non-self-gravitating fluid namely phantom energy in
FRW universe. We are interested if the entropy of this whole system
is positive or not. Since second law of thermodynamics is
fundamental law of physics, its validity needs to be checked for a
thermal system. As is known the form of entropy-area relation for
any causal horizon (either black hole or cosmological) is the same,
we discussed entropies of these horizons with classical relation of
Hawking and with quantum corrections, for the sake of consistency.
However the entropy of phantom energy was calculated using first law
of thermodynamics. Using two different forms of cosmological
horizon, we found general conditions for the validity of GSL in the
present context. Next we used the GSL to impose some restrictions
(upper or lower bounds) on the mass of black hole under which a
black hole can accrete phantom energy.
In this article, we
are unable to predict whether the observable Universe is dominated
by phantom energy or not. Rather we assumed that it contains phantom
energy as claimed by Caldwell \cite{cald,cald1}. Moreover so far
there are no theoretical constraints on the model parameters (like
$\beta$ and $\gamma$ etc) except the BH mass coming from our model.
|
{
"timestamp": "2012-07-03T02:05:06",
"yymm": "1203",
"arxiv_id": "1203.2103",
"language": "en",
"url": "https://arxiv.org/abs/1203.2103"
}
|
\section{Introduction}
\paragraph{Model and results}
We consider a system of $N$ point particles $i=1,2,...,N$ on the
interval $[0,L)\in R$, with periodic boundary conditions, that is
on the circle $S_{L}$ of length $L$. Initially, they are situated
at the points
\[
0=x_{1}(0)<...<x_{N}(0)<L
\]
The trajectories $x_{i}(t)$ are defined by the following system of
$N$ equations
\begin{equation}
\frac{d^{2}x_{i}}{dt^{2}}=-\frac{\partial U}{\partial x_{i}}+F(x_{i})\label{main_eq}
\end{equation}
where the interaction between the particles is
\[
U(\{x_{i}\})=\sum_{<i,i-1>}V(x_{i}-x_{i-1})
\]
where summations is over all pairs of nearest neighbors on the circle.
The Coulomb potential $V(x)=V(-x)=\frac{1}{r},r=|x|$ is assumed,
and we denote $f(r)=-\frac{dV(r)}{dr}=r^{-2}$ the interaction force.
Note that the potential is repulsive, infinite at zero and thus, the
particles cannot change their order . $F(x)$ is the external force.
In the papers \cite{fixed_int,fixed_per} we considered fixed points
of such systems. Dynamics is more complicated to understand. It is
standard that the solution of the system (\ref{main_eq}) exists on
all time interval and is unique (for any initial conditions), however
to get more detailed information about particle trajectories (for
sufficiently large $N$) is sufficiently difficult and demands elaboration
of special methods. If moreover $F$ is analytic, then it is well-known
\cite{Golubev}, that the solution can be presented as the convergent
power series in $t$ in some neighborhood of $t=0$.
Here we consider natural initial conditions, that is for all $i$
\begin{equation}
\Delta=\Delta_{i}(0)=x_{i+1}(0)-x_{i}(0)=\frac{L}{N},v_{i}(0)=0\label{initial}
\end{equation}
and moreover, it is convenient to put $x_{1}(0)=0$. Note that this
configuration is a fixed point for zero external force. We are looking
for the solution of the form
\begin{equation}
v_{i}(t)=\sum_{j=1}^{\infty}c_{i,j}t^{j},c_{i,j}=c_{i,j}^{(N)}\label{series_u}
\end{equation}
and get bounds for the convergence radius, dependent on $N$, of these
series under the initial conditions (\ref{initial}).
\begin{theorem}
Let $F$ be analytic on the circle $S_{L}$. Then
\begin{enumerate}
\item for $j=1,2,...$, there exist numbers $b_{j}<\infty$, not depending
on $N$ and such, that for all $i,j$ and any $N$
\[
|c_{ij}|<b_{j}N^{\frac{j-1}{2}}
\]
\item if moreover, for some constant $C_{F}>0$ and all $x$ and $k$
\[
|F^{(k)}(x)|\leq C_{F}^{k+1},
\]
then there exists constant $0<\chi<\infty$, not depending on $N$
and such that for all $i,j$
\[
|c_{ij}|<\chi^{j}N^{\frac{5}{6}j-\frac{3}{2}}
\]
\end{enumerate}
\end{theorem}
It follows that the convergence radius $R=R(N)$ of the series (\ref{series_u})
has a bound from below $R>\chi^{-1}N^{-\frac{5}{6}}$. From the proof
of the first assertion of the theorem (see section 2.2) one can see
that the bound from above could be of the order $\frac{1}{\sqrt{N}}$,
but this not yet rigorously proved. During the proof of the second
assertion of the theorem we give the explicit bound for $\chi$. Also,
explicit formulas for $c_{ij},j=1,2,3,4$ are presented.
\paragraph{Why this model}
Mathematical problems of statistical physics are elaborated sufficiently
deeply for equilibrium systems on the lattice. But on the continuous
(euclidean) space only for particle systems with small inverse temperature
or small density. There are many other cases where the problems are
not even formulated on the mathematical level. One of such cases is
the direct electric current. It is described by Ohm's law on the macro
level. On the micro level, in any textbook on condensed matter physics,
it is described as a flow of free (or almost free) electrons, any
of which is accelerated by the external force and impeded by the external
media (crystal lattice). Physics and mathematics introduced a lot
of such one-particle models with constant accelerating force and various
models (about 20, historically the first such model is the famous
Drude's model of 1900) of external media, where the particles loose
their kinetic energy.
The central question is where the accelerating force comes from. The
problem is that the power lines can have hundreds kilometers of length
but the external force acts only on the length of several meters of
the wire. In fact, there are even more problems with the direct electric
current. For example, one should explain why the speed of the flow
is permanent and sufficiently slow, but this regime is being established
almost immediately. We discussed the first problem in (\cite{why_1}),
but there was no Coulomb interaction there. We shall come back to
this problem in the next paper. Here we discuss the second problem.
Consider first a trivial case with constant $F\geq0$ and with initial
conditions (\ref{initial}). Then it is clear that for any $i$
\[
v_{i}(t)=Ft,x_{i}(t)=x_{i}(0)+\frac{Ft^{2}}{2}
\]
that is $x_{i}(t)$ are analytic in $t$ for any $t$. Moreover, to
reach the speed of order $1$ (for example $v_{i}(t)=1$) the time
of order $1$ is necessary. We shall see however that the situation
is completely different for non-constant $F$. Anticipating the events,
it is important to note that technically the difference occurs because
for constant $F$ all discrete derivatives in the recurrent equations
below are identically zero.
If we could prove that the coefficients $c_{ij}$ in the expansion
$v_{i}=\sum_{j=1}^{\infty}c_{i,j}t^{j}$ grow as $N^{aj}$ for some
$a>0$, then the speed of order $1$ will be achieved much earlier
- for time of the order $t=N^{-a}$, that is almost immediately. We
cannot prove such result as it is but the estimates of the coefficients,
obtained here, make such result quite plausible.
The techniques of the paper is apparently new. There are results for
small time dynamics of multi-particle systems (see, \cite{Lanford}-\cite{mal_dynamicalClusters})
but they are not applicable in our case because of very strong interaction.
\section{Proof}
\subsection{Equations for the coefficients}
Fix initial data $x_{i}(0),v_{i}(0)$ as in (\ref{initial}) and consider
the trajectories $x_{i}(t)\in S_{L}$ on the interval $0\leq t<t_{0}$
for some $t_{0}=t_{0}(N)>0$. Putting
\[
\Delta_{i}(t)=x_{i+1}(t)-x_{i}(t),\Delta=\Delta_{i}(0)=\frac{L}{N}
\]
we have the equations
\[
\frac{dv_{i}}{dt}=f(x_{i}(t)-x_{i-1}(t))-f(x_{i+1}(t)-x_{i}(t))+F(x_{i}(t))
\]
or
\[
\frac{dv_{i}}{dt}=f(\Delta+\int_{0}^{t}[v_{i}(t_{1})-v_{i-1}(t_{1})]dt_{1})-f(\Delta+\int_{0}^{t}[v_{i+1}(t_{1})-v_{i}(t_{1})])dt_{1})+
\]
\[
+F(x_{i}(0)+\int_{0}^{t}v_{i}(t_{1})dt_{1})
\]
\paragraph{Integral equations}
Equivalent system of integral equations is
\[
v_{i}(t)=\int_{0}^{t}[f(\Delta+\int_{0}^{t}[v_{i}(t_{1})-v_{i-1}(t_{1})]dt_{1})-
\]
\begin{equation}
-f(\Delta+\int_{0}^{t}[v_{i+1}(t_{1})-v_{i}(t_{1})]dt_{1})+F(x_{i}(0)+\int_{0}^{t}v_{i}(t_{1})dt_{1})]dt\label{intEqua}
\end{equation}
and can be rewritten as follows
\begin{equation}
v_{i}(t)=\int_{0}^{t}((\Delta+R_{i-1}(t))^{-2}-(\Delta+R_{i}(t))^{-2}+F(x_{i}(0)+\int_{0}^{t}v_{i}(t_{1})dt_{1}))dt\label{intEq_f}
\end{equation}
where
\[
R_{i-1}(t)=\int_{0}^{t}(v_{i}(t_{1})-v_{i-1}(t_{1}))dt_{1}
\]
For the sequel we need some notation for discrete derivatives. Let
on the interval $[0,N]\subset Z$ with periodic boundary conditions
a function $g(i)$ be given (that is a periodic function on $Z$ with
period $N$). Let us call
\begin{equation}
(\nabla g)(i)=(\nabla^{+}g)(i)=g(i+1)-g(i),(\nabla^{-}g)(i)=g(i)-g(i-1)\label{product_diff}
\end{equation}
its right and left derivative correspondingly. Note that they commute
and the Leibnitz formula holds
\begin{equation}
\nabla^{+}(gf)(i)=f(i+1)(\nabla^{+}g)(i)+g(i)(\nabla^{+}f)(i)=(Sf)(\nabla^{+}g)+g(\nabla^{+}f)\label{productDiff}
\end{equation}
where $S$ is the shift operator
\[
(Sf)(i)=f(i+1)
\]
Below discrete derivatives will act on the indices $i$. If the function
$f(i)$ does not depend on $i$, then its (discrete) differentiation
gives zero.
We come back to the main equations and rewrite them as follows
\[
v_{i}(t)=\int_{0}^{t}dt[(-\nabla^{-}((\Delta+R_{i}(t))^{-2})+F(x_{i}(0)+\int_{0}^{t}v_{i}(t_{1})dt_{1})]
\]
The following representation of the integrand will be useful
\[
(\Delta+R_{i-1}(t))^{-2}-(\Delta+R_{i}(t))^{-2}+F(x_{i}(0)+\int_{0}^{t}v_{i}(t_{1})dt_{1})=
\]
\[
=\Delta^{-2}(1+\frac{R_{i-1}}{\Delta})^{-2}-\Delta^{-2}(1+\frac{R_{i}}{\Delta})^{-2}+F(x_{i}(0)+\int_{0}^{t}v_{i}(t_{1})dt_{1})=
\]
\[
=F(x_{i}(0))+\sum_{m=1}^{\infty}d_{m}\Delta^{-2-m}(R_{i-1}^{m}-R_{i}^{m})+[F(x_{i}(0)+\int_{0}^{t}v_{i}(t_{1})dt_{1})-F(x_{i}(0))]
\]
where
\[
d_{m}=(-1)^{m}(m+1)
\]
If $F$ is analytic on $S_{L}$, then there exists $\epsilon>0$ sufficiently
small and such that for any $x_{0}\in S_{L}$ the following series
\[
F(x)=F(x_{0})+\sum_{k=1}^{\infty}\frac{F^{(k)}(x_{0})}{k!}(x-x_{0})^{k}
\]
converges for all $x\in[x_{0}-\epsilon,x_{0}+\epsilon)$. Then finally
\begin{equation}
v_{i}(t)=F(x_{i}(0))t+\int_{0}^{t}\sum_{m=1}^{\infty}d_{m}\Delta^{-2-m}[-\nabla^{-}R_{i}^{m})]dt+\sum_{k=1}^{\infty}\int_{0}^{t}\frac{F^{(k)}(x_{i}(0))(\int_{0}^{t}v_{i}(t_{1})dt_{1})^{k}}{k!}dt\label{Equat_v_i}
\end{equation}
\paragraph{Recurrent equations}
Using (\ref{series_u}) and
\begin{equation}
R_{i-1}(t)=\sum_{j=1}^{\infty}(c_{i,j}-c_{i-1,j})\frac{t^{j+1}}{j+1}\label{Eq_R_1}
\end{equation}
\begin{equation}
R_{i}-R_{i-1}=\sum_{j=1}^{\infty}(c_{i+1,j}-2c_{i,j}+c_{i-1,j})\frac{t^{j+1}}{j+1}\label{Eq_R_2}
\end{equation}
we see, substituting (\ref{series_u}) to (\ref{Equat_v_i}), that
the right-hand side of (\ref{Equat_v_i}) can also be presented as
a convergent series with well-defined coefficients.
We shall find $c_{i,j}$ by equating the coefficients of $t^{j}$.
For $j=1,2$ the equations give immediately
\begin{equation}
c_{i1}=F(x_{i}(0)),c_{i,2}=0\label{eq_j_1}
\end{equation}
as other summands in the right side of (\ref{Equat_v_i}) have larger
order in $t$. For $j\geq3$ the equations for the coefficients of
$t^{j}$ are
\begin{equation}
c_{ij}=\frac{1}{j}[\sum_{m=1}^{\infty}d_{m}\Delta^{-2-m}(-\nabla^{-}R_{i}^{m})+\sum_{k=1}^{\infty}\frac{F^{(k)}(x_{i}(0))(\int_{0}^{t}v_{i}(t_{1})dt_{1})^{k}}{k!}]_{j-1}\label{coeff_1}
\end{equation}
where, for the power series $\phi(t)=\sum_{k=0}^{\infty}a_{k}t^{k}$,
we denote $[\phi(t)]_{j}=a_{j}$. For $j>2$ the coefficients $c_{i,j}$
can be found recursively, moreover $c_{ij}$ depend only on $c_{i,k}$
with $k\leq j-2$. In fact, the right-hand part of the equation for
$c_{i,j}$ does not contain $c_{i,k}$ with $k\geq j-1$, as due to
(\ref{Eq_R_1}), each of $c_{ik}$ appears together with $t^{k+1}$.
Then the main equations will be
\begin{equation}
c_{ij}=\frac{1}{j}\sum_{m=1}^{\infty}d_{m}\Delta^{-2-m}(-\nabla^{-}[(\sum_{j=1}^{\infty}(c_{i+1,j}-c_{i,j})\frac{t^{j+1}}{j+1})^{m}]_{j-1})+\label{coeff_2}
\end{equation}
\[
+\sum_{k=1}^{\infty}\frac{F^{(k)}(x_{i}(0))}{k!}[\sum_{j=1}^{\infty}c_{i,j}\frac{t^{j+1}}{j+1})^{k}]_{j-1}
\]
We have
\begin{equation}
[(\sum_{j=1}^{\infty}c_{i,i}\frac{t^{j+1}}{j+1})^{k}]_{j-1}=\sum_{j_{1}+...+j_{m}=j-m-1}\frac{c_{i,j_{1}}}{j_{1}+1}...\frac{c_{i,j_{k}}}{j_{k}+1}\label{squareBrack_1}
\end{equation}
where $\sum_{j_{1}+...+j_{m}=j-m-1}$ is the sum over all ordered
arrays $j_{1},...,j_{k}$, such that
\begin{equation}
(j_{1}+1)+...+(j_{k}+1)=k+j_{1}+...+j_{k}=j-1\label{sum_j_i_1}
\end{equation}
It follows
\begin{equation}
k\leq j_{1}+...+j_{k}=j-1-k\leq j-2,k\leq[\frac{j-1}{2}]\label{sum_j_i_2}
\end{equation}
Similarly
\[
[\sum_{j=1}^{\infty}(c_{i+1,j}-c_{i,j})\frac{t^{j+1}}{j+1})^{m}]_{j-1}=\sum_{j_{1},...,j_{m}}^{(j-1,m)}\frac{\nabla^{+}c_{i,j_{1}}}{j_{1}+1}...\frac{\nabla^{+}c_{i,j_{m}}}{j_{m}+1}
\]
and both (\ref{sum_j_i_1}) and (\ref{sum_j_i_2}) hold with $k$
instead of $m$.
That is why the equations can be written as
\begin{equation}
c=Gc+c^{(0)}\label{operator_Equat}
\end{equation}
where $c$ is the vector $c=\{c_{ij}\}$, the free term $c^{(0)}=\{c_{ij}^{(0)}\}$
is simple
\begin{equation}
c_{i1}^{(0)}=F(x_{i}(0)),c_{ij}^{(0)}=0,j\geq2\label{c_0_equat}
\end{equation}
and non-linear operator $G$ is defined by
\[
c_{i1}=c_{i1}^{(0)}=F(x_{i}(0)),c_{i2}=0
\]
\begin{equation}
c_{ij}=(Gc)_{ij}=-\sum_{m=1}^{[\frac{j-1}{2}]}\sum_{j_{1}+...+j_{m}=j-m-1}A_{ij}(m;j_{1},...,j_{m})+\sum_{k=1}^{[\frac{j-1}{2}]}\sum_{j_{1}+...+j_{k}=j-k-1}B_{ij}(k;j_{1},...,j_{k})\label{recurrent_equ}
\end{equation}
for $j\geq3$, where
\begin{equation}
A_{ij}(m;j_{1},...,j_{m})=\frac{1}{j}d_{m}\Delta^{-2-m}\nabla^{-}(\frac{\nabla^{+}c_{i,j_{1}}}{j_{1}+1}...\frac{\nabla^{+}c_{i,j_{m}}}{j_{m}+1})\label{A_equat_0}
\end{equation}
\begin{equation}
B_{ij}(k;j_{1},...,j_{k})=\frac{1}{j}\frac{1}{k!}F^{(k)}(x_{i}(0))\frac{c_{i,j_{1}}}{j_{1}+1}...\frac{c_{i,j_{k}}}{j_{k}+1}\label{B_equat_0}
\end{equation}
Further on, $F_{i,k,q}$ will denote any discrete derivative $(\prod_{p=1}^{q}\nabla^{s(p)})F^{(k)}(x_{i}(0))$,
where $s(p)=\pm$. For the estimates the choice of $s(p)$ does not
matter. Put $F_{i,k}=F_{i,k,0}$.
Explicit expression for $c_{i3},c_{i4}$ is immediately obtained if
in the equations (\ref{recurrent_equ}) we take into account only
the terms with $k=1$ and $m=1$, as $k.m\leq[\frac{j-1}{2}]\leq1$)
\[
c_{i3}=-\frac{1}{3}d_{1}\Delta^{-3}\nabla^{-}\nabla^{+}\frac{c_{i1}}{2}+\frac{1}{3}F^{(1)}(x_{i})\frac{c_{i1}}{2}=\frac{1}{6}(d_{1}\Delta^{-3}F_{i,0,2}+F_{i,0,0}F_{i,1,0})
\]
\[
c_{i4}=-\frac{1}{4}d_{1}\Delta^{-3}\nabla^{-}(\nabla^{+}\frac{c_{i1}}{2})^{2}+\frac{1}{4}F^{(1)}(x_{i})\frac{c_{i1}^{2}}{4}=
\]
\[
=\frac{1}{8}(-d_{1}\Delta^{-3}F_{i,0,2}F_{i,0;1}+\frac{1}{2}F_{i,1,0}F_{i,0,0}^{2})
\]
From the formulas
\begin{equation}
F_{i,0,1}=F_{i+1,0,0}-F_{i,0,0}=\int_{x_{i}}^{x_{i+1}}F^{(1)}(x)dx,|F_{i,0,1}|\leq C_{F}^{2}\Delta\label{deriv_bound}
\end{equation}
\[
F_{i,0,2}=(F_{i+2,0,0}-F_{i+1,0,0})-(F_{i+1,0,0}-F_{i,0,0})=\int_{x_{i}}^{x_{i+1}}(\int_{x}^{x+\Delta}F^{(2)}(y)dy)dx,|F_{i,0,2}|\leq C_{F}^{3}\Delta^{2}
\]
it follows that
\[
|c_{i3}|\leq\frac{1}{3}C_{F}^{3}(\Delta^{-1}+\frac{1}{2}),|c_{i4}|\leq\frac{1}{4}C_{F}^{5}+\frac{1}{16}C_{F}^{4}
\]
It is easy to see that for most $i$ the coefficients $c_{i,3}$ are
really of the order $\Delta^{-1}$.
\subsection{Principal exponent}
From the recurrent formulas (\ref{A_equat_0}) and (\ref{B_equat_0})
it follows that $c_{ij}$ are finite and depend on $i,j,N$. First
of all, we shall study them as functions of $N$ for fixed $i,j$.
Otherwise speaking, we shall prove the first part of the theorem.
Define the principal exponent
\[
I(\xi)=\limsup_{N\to\infty}\frac{\ln|\xi|}{\ln N}
\]
for a variable $\xi$ depending on $N$. Roughly speaking, it shows
that the main order of the asymptotics of $\xi$ is $N^{I(\xi)}$.
We shall consider the algebra $\mathbf{A}$ of polynomials of countable
number of (commuting) variables $F_{i,k,q},i=1,...,N;k,q=0,1,2,...$
with real coefficients, not depending on $F$. For any monomial $M$
of this algebra denote
\[
Q(M)=-\sum q
\]
over all $q$ in this monomial. The natural mapping of $\mathbf{A}$
onto its subalgebra $\mathbf{A}_{0}$, generated by all $F_{i,k}=F_{i,k,0}$,
is defined by the subsequent substitutions
\[
F_{i,k,q}=F_{i+1,k,q-1}-F_{i,k,q-1}
\]
or with the following formula
\[
(\nabla^{+})^{n}=(S-1)^{n}=\sum_{k=0}^{n}C_{n}^{k}(-1)^{k}S^{n-k}
\]
\begin{lemma}\label{lemma_I_Q} For any monomial $M\in\mathbf{A}$
\[
I(M)\leq Q(M)
\]
\end{lemma}
It is sufficient to prove that
\[
I(F_{i,q})\leq Q(F_{i,q})=-q
\]
This can be done as in (\ref{deriv_bound}), using induction in $q$.
Put\inputencoding{koi8-r}\foreignlanguage{russian}{
\[
g_{i,n}=\nabla^{n}g_{i},\Delta=\frac{1}{N}
\]
Then
\[
g_{i,n+1}=\nabla^{n+1}g_{i}=g_{i+1,n}-g_{i,n}=\int_{x_{i}}^{x_{i}+\Delta}dy_{1}(\int_{y_{1}}^{y_{1}+\Delta}dy_{2}....(\int_{y_{n-1}}^{y_{n-1}+\Delta}dy_{n}g^{(n)}(y_{n})))
\]
and if $|g^{(n)}(x)|\leq C_{g}^{n+1}$, we have
\[
|g_{i,n}|\leq C_{g}^{n+1}\Delta^{n}
\]
}
\inputencoding{latin9}For any polynomial $P=\sum a_{r}M_{r}$ with
(different) monomials $M_{r}$ and coefficients $a_{r}$, not depending
on $F$, but possibly depending on $N$, define
\[
Q(P)=\max_{r}(I(a_{r})+Q(M_{r}))
\]
in agreement with the previous definition. Then for any polynomial
$P$
\[
I(P)\leq\max_{r}(I(a_{r})+I(M_{r}))\leq\max_{r}(I(a_{r})+Q(M_{r}))
\]
Note that for any two polyonomials
\[
Q(P_{1}P_{2})\leq Q(P_{1})+Q(P_{2})
\]
Also
\begin{equation}
Q(\nabla^{+}P)\leq Q(P)-1,Q(\nabla^{-}\nabla^{+}P)\leq Q(P)-2\label{Q_1_2}
\end{equation}
Denote $\deg P$ the degree of the polynomial $P=\sum a_{r}M_{r}$,
that is the maximal degree of its monomials.
\begin{lemma}\label{lemma_degree}
For $j>2$ $c_{ij}$ is a polynomial in the algebra $\mathbf{A}_{0}$
and has degree not greater than $j-1$.
\end{lemma}
We already saw this for $j=3,4$. Note that
\[
\deg(\nabla^{\pm}P)=\deg P
\]
Now one can use induction: in the formula (\ref{A_equat_0}) the
degree is $j-2$, and in (\ref{B_equat_0}) the degree will be $j-1$.
Note that the recurrent formulas define $c_{ij}$ for all functions
$F(x)$, not necessary analytic. That is why the following assertion
makes sense.
\begin{lemma}\label{lemma_bound} Let $F$ be infinitely differentiable.
Then for all $i,j$
\[
I(c_{ij})\leq Q(c_{ij})\leq\frac{j-1}{2}
\]
\end{lemma}
As
\[
Q(c_{ij})=0,j=1,2,Q(c_{i,3})=1,Q(c_{i,4})=0
\]
then the assertion holds for $j=1,2,3,4$. We shall prove the lemma
by induction in $j$. Assume that
\[
Q(c_{ij})\leq\frac{j-1}{2}
\]
for all $j=1,2,...,J-2$.
Then for given $m,j_{1},...,j_{m}$, accordingly to (\ref{sum_j_i_1})
and (\ref{sum_j_i_2}), we have
\[
Q(A_{iJ}(m;j_{1},...,j_{m}))\leq2+m-1+Q(c_{ij_{1}})+...+Q(c_{ij_{m}})-m\leq1+\frac{1}{2}(j_{1}+...+j_{m})-\frac{m}{2}=
\]
\[
=1+\frac{1}{2}(J-m-1)-\frac{m}{2}
\]
as, according to (\ref{Q_1_2}), $(-1)$ and $(-m)$ are appended
because of the discrete differentiation of the corresponding monomials.
The last expression attains its maximum when $m=1$. It follows
\[
Q(A_{iJ}(m;j_{1},...,j_{m}))\leq\frac{J-1}{2}
\]
Similarly, for $B_{iJ}(k:j_{1},...,j_{k})$ we have the following
inequalities
\[
Q(B_{iJ}(k;j_{1},...,j_{k}))\leq\frac{1}{2}(J-1-k)-\frac{k}{2}<\frac{J-1}{2}
\]
We get thus $Q(c_{iJ})\leq\frac{J-1}{2}$, and $I(c_{iJ})\leq Q(c_{iJ})\leq\frac{J-1}{2}$.
\subsection{Convergence radius}
Here we shall prove the second assertion of the theorem 1. In the
proof it is convenient to write $N$ instead of $\frac{N}{L}$ and
assume that $C_{F}\geq1$.
We shall use the following majorization principle for infinite system
of recurrent equations and inequalities: for example, if two systems
of equations are given
\[
c_{ij}^{(q)}=P^{(q)}(c_{i1}^{(q)},...,c_{i,j-2}^{(q)}),q=1,2
\]
where $P^{(q)}$ are the polynomials with coefficients $p_{\alpha}^{(q)}$,
where $p_{\alpha}^{(2)}\geq0,|p_{\alpha}^{(1)}|\leq p_{\alpha}^{(2)}$
for all $\alpha$, and also $|c_{ij}^{(1)}|\leq c_{ij}^{(2)}$ for
$j=1,2,3,4$, then $|c_{ij}^{(1)}|\leq c_{ij}^{(2)}$ for all $j$.
One of such system (we use it in the proof) corresponds to one-particle
problem (that is with $N=1$) with specially chosen external force,
will be now introduced. Other auxiliary system $\beta(c_{ij})$ with
positive coefficients will be introduced later.
\paragraph{One-particle problem}
For $j=1,2,...$ and fixed $a$ put
\[
g_{j}=g_{j}(\frac{a}{2})=\{\frac{a}{2}\}^{j}\frac{1.3...(2j-1)}{j!}=\{\frac{a}{2}\}^{j}\frac{(2j)!}{2^{j}j!j!}\sim\{\frac{a}{2}\}^{j}\frac{1}{\sqrt{4\pi j}}
\]
Then we have
\begin{lemma}\label{lemma_one_particle}
For $j=5,6,...$ the following inequalities hold
\[
g_{j}\geq\frac{1}{j}\sum_{k=1}^{[\frac{j-1}{2}]}\sum_{j_{1}+...+j_{m}=j-m-1}(\frac{a}{2})^{k+1}\frac{(k+1)(k+2)}{2}\frac{g_{j_{1}}}{j_{1}+1}...\frac{g_{j_{k}}}{j_{k}+1}
\]
\end{lemma}
Proof. Let the particle, situated initially at the point $x(0)=0$,
move with the speed ($a>0$ being arbitrary)
\[
v(t)=\frac{1}{\sqrt{1-at}}=\sum_{j=0}^{\infty}g_{j}t^{j}
\]
in the field of the external force $F(x)$, which we should find.
Then
\[
x(t)=\int_{0}^{t}v(s)ds=(-\frac{2}{a})\sqrt{1-at}+\frac{2}{a}\Longrightarrow1-at=(1-\frac{ax}{2})^{2}
\]
\[
F=\frac{dv}{dt}=\frac{a}{2}\frac{1}{(1-at)^{\frac{3}{2}}}=\frac{a}{2}\frac{1}{(1-\frac{ax}{2})^{3}}
\]
\[
\frac{F^{(k)}}{k!}=(\frac{a}{2})^{k+1}\frac{3.4...(k+2)}{k!}=(\frac{a}{2})^{k+1}\frac{(k+1)(k+2)}{2}
\]
Similarly to the above derivation of the recurrent equations (the
only difference is that here $v(0)=1$ and there are no $A$-terms),
we get
\[
g_{j}=\sum_{k=1}^{[\frac{j-1}{2}]}\sum_{p=0}^{k-1}\sum_{j_{1}+...+j_{m}=j-m-1}\frac{1}{j}\frac{1}{k!}F^{(k)}(x_{i}(0))C_{k}^{p}v^{p}(0)\frac{g_{j_{1}}}{j_{1}+1}...\frac{g_{j_{k-p}}}{j_{k-p}+1}
\]
for $g_{j}$ defined above. Here, as above, $\sum_{j_{1}+...+j_{m}=j-m-1}$
is the summation over all $j_{1},...,j_{k-p}$ such that
\[
j_{1}+...+j_{k-p}=j-k-1
\]
As all coefficients are positive, then, neglecting the terms with
$p>0$, we have for any $a>0$
\[
\frac{1}{j}\sum_{k=1}^{[\frac{j-1}{2}]}\sum_{j_{1}+...+j_{m}=j-m-1}(\frac{a}{2})^{k+1}\frac{g_{j_{1}}}{j_{1}+1}...\frac{g_{j_{k}}}{j_{k}+1}\leq
\]
\[
\leq\frac{1}{j}\sum_{k=1}^{[\frac{j-1}{2}]}\sum_{j_{1}+...+j_{m}=j-m-1}(\frac{a}{2})^{k+1}\frac{(k+1)(k+2)}{2}\frac{g_{j_{1}}}{j_{1}+1}...\frac{g_{j_{k}}}{j_{k}+1}\leq g_{j}
\]
\paragraph{Majorization }
From the recurrent formula for $c_{ij}$ one can see that they can
be written as
\[
c_{ij}=\sum_{r=1}^{d_{ij}}b_{i,j,r}N^{I_{i,j,r}}M_{i,j,r}
\]
where $b_{i,j,r}$ amd $d_{ij}$ are some numbers not depending neither
on $N$ nor on $F$, and $M_{i,j,r}\in\mathbf{A}$. By lemma \ref{lemma_degree}
\[
\deg M_{i,j,r}\leq j-1
\]
We need other preliminary definitions. For any polynomial
\[
P=\sum b_{r}N^{I_{r}}M_{r}
\]
where $b_{r}$ are the numbers, not depending neither on $N$ nor
on $F$, $M_{r}\in\mathbf{A}$, we put
\[
\beta(P)=\sum_{r}|b_{r}|N^{I_{r}+Q(M_{r})}C_{F}^{Q_{0}(M_{r})-Q(M_{r})+\deg M_{r}}
\]
where, for any monomial $M_{r}$, the natural number $Q_{0}(M_{r})$
equals the sum $\sum k$ over all factors (variables) $F_{i,k,q}$.
In particular,
\[
\beta(c_{ij})=\sum_{r}|b_{i,j,r}|N^{I_{i,j,r}+Q(M_{i,j,r})}C_{F}^{Q_{0}(M_{i,j,r})-Q(M_{i,j,r})+\deg M_{i,j,r}}
\]
By definition for any $i,k$
\[
\beta(F_{i,k,0})=C_{F}^{k+1},\beta(\nabla^{\pm}F_{i,k,0})=\beta(F_{i,k,1})=C_{F}^{k+2}N^{-1}=C_{F}N^{-1}\beta(F_{i,k})
\]
Moreover, for any two polynomials $P_{1},P_{2}$
\begin{equation}
\beta(P_{1}+P_{2})\leq\beta(P_{1})+\beta(P_{2}),\beta(P_{1}P_{2})\leq\beta(P_{1})\beta(P_{2})\label{subadditivity}
\end{equation}
and for any monomial $M$
\begin{equation}
\beta(\nabla^{\pm}M)\leq(\deg M)N^{Q(M)-1}C_{F}^{Q_{0}(M)-Q(M)+\deg M+1}=(\deg M)C_{F}N^{-1}\beta(M)\label{beta_deriv_monome}
\end{equation}
It follows that for any polynomial $P$
\begin{equation}
\beta(\nabla^{\pm}P)\leq(\deg P)C_{F}N^{-1}\beta(P)\label{beta_deriv_polinom}
\end{equation}
We call $\beta(P)$ the majorant of $P$ as by
\[
|F_{i,k,1}|=|\nabla^{+}F_{i,k}|\leq\int_{x_{i}}^{x_{i+1}}|F_{i,k+1}(x)|dx\leq C_{F}^{k+2}N^{-1}=\beta(F_{i,k,1})
\]
we have
\[
|P|\leq\beta(P)
\]
From (\ref{subadditivity}) and (\ref{operator_Equat}) it follows
that
\[
\beta(c_{ij})\leq\sum_{m=1}^{[\frac{j-1}{2}]}\sum_{j_{1}+...+j_{m}=j-m-1}\beta(A_{ij}(m;j_{1},...,j_{m}))+\sum_{k=1}^{[\frac{j-1}{2}]}\sum_{j_{1}+...+j_{k}=j-k-1}\beta(B_{ij}(k;j_{1},...,j_{k}))
\]
Our inductive hypothesis will be (\textcyr{\char241} $g_{j}=g_{j}(1)$)
\begin{equation}
\beta(c_{ij})\leq\chi^{j}N^{\frac{5}{6}j-\frac{3}{2}}g_{j},j=1,2,...,J-2\label{inductive}
\end{equation}
\paragraph{Initial data}
One can choose $\chi_{0}>0$ so that for $j-1,2,3,4$
\[
\chi_{0}^{j}N^{\frac{5}{6}J-\frac{3}{2}}g_{j}\geq\beta(c_{ij})
\]
In fact, only for $j=3$ there is dependence on $N$, but $\frac{5}{6}3-\frac{3}{2}$
is exactly $N$.
\paragraph{Inductive step for $A$-terms with $m>1$}
To estimate $A$-terms we distinguish two cases: $m=1$ and $m>1$.
For $m>1$ we use obvious bounds
\[
\beta(\nabla^{\pm}c_{ij})\leq2\beta(c_{ij}),\beta(\nabla^{-}(\nabla^{+}c_{i,j_{1}}...\nabla^{+}c_{i,j_{m}}))\leq2^{m+1}\prod_{p}\beta(c_{i,j_{p}})
\]
Then from (\ref{subadditivity}) and (\ref{A_equat_0}) we get
\[
\beta(A_{iJ}(m;j_{1},...,j_{m}))\leq\frac{m+1}{J}N^{2+m}2^{m+1}\frac{\beta(c_{i,j_{1}})}{j_{1}+1}...\frac{\beta(c_{i,j_{m}})}{j_{m}+1}\leq
\]
\[
\leq\frac{m+1}{J}N^{2+m}2^{m+1}\chi^{J-m-1}N^{\frac{5}{6}(J-m-1)-m\frac{3}{2}}\prod_{p=1}^{m}\frac{g_{j_{p}}}{j_{p}+1}\leq
\]
\[
\leq2^{m+1}\chi^{J-m-1}N^{\frac{5}{6}J-\frac{3}{2}}\frac{m+1}{J}(\prod_{p=1}^{m}\frac{g_{j_{p}}}{j_{p}+1})
\]
as the exponent over $N$ for $m\geq2$ has the bound
\[
2+m+\frac{5}{6}(J-m-1)-m\frac{3}{2}=\frac{5}{6}J-m\frac{8}{6}+\frac{7}{6}\leq\frac{5}{6}J-\frac{3}{2}
\]
Then by lemma \ref{lemma_one_particle} with $a=2$ (if $\chi\geq2$)
\[
\sum_{m=2}^{[\frac{j-1}{2}]}\sum_{j_{1}+...+j_{m}=j-m-1}\beta(A_{iJ}(m;j_{1},...,j_{m}))\leq
\]
\[
\leq N^{\frac{5}{6}J-\frac{3}{2}}\chi^{J}\sum_{m=2}^{\frac{j-1}{2}}2^{m+1}\chi^{-m-1}\sum_{j_{1}+...+j_{m}=j-m-1}\frac{m+1}{J}(\prod_{p=1}^{m}\frac{g_{j_{p}}}{j_{p}+1})\leq
\]
\begin{equation}
\leq(\frac{2}{\chi})^{3}N^{\frac{5}{6}J-\frac{3}{2}}\chi^{J}g_{j}\label{final_A}
\end{equation}
\paragraph{Inductive step for $A$-terms with $m=1$}
In case $m=1$ we shall use the following bounds for $j=J-2\geq3$.
By (\ref{beta_deriv_polinom}), (\ref{inductive}) and lemma \ref{lemma_degree}
\[
\beta(\nabla^{+}c_{ij})\leq(j-1))N^{-1}C_{F}\beta(c_{ij})\leq(j-1))N^{-1}C_{F}\chi^{j}N^{\frac{5}{6}j-\frac{3}{2}}g_{j}
\]
Similarly
\[
\beta(\nabla^{-}\nabla^{+}c_{i,j})|\leq((j-1)C_{F})^{2}N^{-2}\chi^{j}N^{\frac{5}{6}j-\frac{3}{2}}g_{j}
\]
This gives additional summand $(-2)$ in the exponent over $N$,
equal to
\[
3-2+\frac{5}{6}(J-2)-\frac{3}{2}\leq\frac{5}{6}J-\frac{3}{2}
\]
that is
\[
A_{ij}(1;j_{1})=A_{ij}(1;J-2)=\frac{1}{j}|d_{1}|N^{3}\frac{\nabla^{-}\nabla^{+}c_{i,J-2}}{J-1}\leq
\]
\begin{equation}
\leq2C_{F}^{2}\chi^{j-2}N^{\frac{5}{6}J-\frac{3}{2}}g_{j-2}\leq\frac{2C_{F}^{2}g_{j-2}}{\chi^{2}g_{j}}\chi^{j}N^{\frac{5}{6}J-\frac{3}{2}}g_{j}\label{final_A_1}
\end{equation}
\paragraph{Inductive step for $B$-terms}
For $B$-terms the inductive bound is easier, but here the degree
of monomials increases
\[
\beta(B_{ij}(k;j_{1},...,j_{k}))=\frac{1}{j}\frac{1}{k!}\beta(F^{(k)}(x_{i}(0))\frac{c_{i,j_{1}}}{j_{1}+1}...\frac{c_{i,j_{k}}}{j_{k}+1})\leq
\]
\[
\leq\frac{1}{j}\frac{C_{F}^{k+1}}{k!}\frac{1}{j_{1}+1}...\frac{1}{j_{k}+1}\beta(c_{i,j_{1}})...\beta(c_{i,j_{k}})\leq\frac{1}{j}\frac{C_{F}^{k+1}}{k!}\frac{g_{j_{1}}}{j_{1}+1}...\frac{g_{j_{k}}}{j_{k}+1}\chi^{j_{1}+---+j_{k}}N^{j_{1}+---+j_{k}}
\]
By lemma \ref{lemma_one_particle}
\[
\sum_{k=1}^{[\frac{j-1}{2}]}\sum_{j_{1}+...+j_{k}=j-k-1}\beta(B_{ij}(k;j_{1},...,j_{k}))\leq\frac{1}{j}\sum_{k=1}^{[\frac{j-1}{2}]}\frac{C_{F}^{k+1}}{k!}\sum_{j_{1}+...+j_{k}=j-k-1}\frac{g_{j_{1}}}{j_{1}+1}...\frac{g_{j_{k}}}{j_{k}+1}\chi^{J-k-1}N^{\frac{5}{7}j}\leq
\]
\begin{equation}
\leq C_{F}e^{C_{F}}\chi^{J-2}N^{\frac{J}{2}}\frac{1}{j}\sum_{k=1}^{[\frac{j-1}{2}]}\sum_{j_{1}+...+j_{k}=j-k-1}\frac{g_{j_{1}}}{j_{1}+1}...\frac{g_{j_{k}}}{j_{k}+1}\leq C_{F}e^{C_{F}}\chi^{J-2}N^{\frac{J}{2}}g_{j}\label{final_B}
\end{equation}
Finally, to end the proof of the theorem, we sum up three obtained
expression (\ref{final_A}),(\ref{final_A_1}),(\ref{final_B}), and
choose $\chi=\chi_{1}>0$ so that
\[
(8\chi^{-1}+\frac{2C_{F}^{2}g_{j-2}}{g_{j}}+C_{F}e^{C_{F}})\chi^{-2}\leq1
\]
Then for any $\chi\geq\max(\chi_{0},\chi_{1})$ we will have
\[
|c_{ij}|\leq\chi^{J}N^{\frac{J}{2}}g_{j}\leq\chi^{J}N^{\frac{J}{2}}
\]
The theorem is proved.
\pagebreak
|
{
"timestamp": "2012-03-09T02:03:02",
"yymm": "1203",
"arxiv_id": "1203.1805",
"language": "en",
"url": "https://arxiv.org/abs/1203.1805"
}
|
\section{INTRODUCTION}
Close to the transition to superconductivity fluctuations manifest in several physical quantities like the specific heat, the magnetization, and the resistivity. The Ginzburg number is a measure of the importance of thermal effects for the respective superconductor. The larger the Ginzburg number the more important thermal fluctuations become for the calculation of the physical properties of the superconductor. In layered superconductors extreme physical properties such as large anisotropy, short coherence length, and high transition temperatures combine to enhance thermal fluctuations. A quantity for the estimation of the width of the critical region, where fluctuations are of importance, is the Ginzburg temperature $\tau_G$, which is related to the Ginzburg number $G_i$ by $\tau_G = T_{c0} G_i$. According to Thouless \emph{et al.}~\cite {Thouless1960}, the fluctuation specific heat of clean bulk superconductors is not expected to be observable until $(T-T_{c0})/T_{c0} \approx 10^{-11}$. The situation improves with dirty samples and thin films, since they have increased Ginzburg numbers. In the specific heat these fluctuations manifest as a peak-like anomaly peak close to the transition to superconductivity which we will call the \emph{fluctuation peak} in the following. The fluctuation peak is a rarely observed phenomenon in low-$T_c$ superconductors. So far, to the best of our knowledge, only a few observations of the fluctuation peak in low-$T_c$ superconductors are known in the literature. Besides the observation in the recent works of Lortz \emph{et al.}~\cite{Lortz2006September,Lortz2007April}, a similar peak in the heat capacity below $T_c(H)$ was first observed in $1967$ by Barnes and Hake \cite{Barnes1967} for a type-II superconductor in the extreme dirty limit, although they were unable to study the behavior in the critical region because of additional sample broadening of the transition. Effects of fluctuations have also been observed in the specific heat of extremely dirty films by Zally and Mochel \cite{Zally1971December}. Later in $1975$, an enhanced heat capacity above and also below $T_c(H)$ was also observed in measurements on Nb by Farrant and Gough \cite{Farrant1975}; their sample was in the clean limit, a very pure crystal with negligible broadening due to impurities. The high resolution and data-point density of the DTA method used by us benefitted us in the investigation of the suspected fluctuation peak in Nb$_3$Sn. In this paper we present high resolution specific-heat data on a homogeneous single crystal of Nb$_3$Sn. We confirm the observation of the fluctuation peak close to $T_c(H)$ as observed in an earlier work by Lortz \emph{et al.}~\cite{Lortz2006September}.\\
\indent In the so-called \emph{peak effect region} close to $T_c$ the rise of the critical current at the onset of the peak effect has been attributed to an abrupt softening of the shear modulus of the vortex lattice as $H$ approaches the upper critical field $H_{c2}(T)$ \cite{Pippard1969,Larkin1979,Brandt1977,Brandt1977a,Brandt1977b,Brandt1977c,Brandt1993September}, which may has its origin in thermal fluctuations. When the vortex lattice becomes less rigid, the vortices can bend and adjust better to the pinning sites. In line with the collective-pinning description of Larkin and Ovchinnikov \cite{Larkin1970,Larkin1973,Larkin1979} the peak effect points to a transition of the vortex lattice from an ordered phase to a disordered phase. Whether this transition is a thermodynamic phase transition is still a matter of dispute to date. In case it is a true thermodynamic phase transition one expects to find an anomaly in the specific heat and the equilibrium magnetization. The associated metastability of an underlying first-order vortex-lattice melting transition has been proposed as the origin of the peak effect. One assumes that the melting transition is hindered by the strong pinning in the peak-effect region. In order to be able to observe the proposed melting transitions one can try to equilibrate the vortex lattice by the application of a small magnetic so-called \emph{vortex-shaking field} $h_{ac}$ as was done by Willemin \emph{et al.}~\cite{Willemin1998September,Willemin1998November}. According to theories \cite{Brandt1986November,Brandt1994April,Brandt1996March,Brandt1996August,Mikitik2000September,Mikitik2001August,Brandt2002July,Brandt2004,Brandt2004a,Brandt2004b,Brandt2007}, transversal and also longitudinal vortex shaking with a small oscillating magnetic field $h_{ac}$ can cause magnetic vortices to "walk" through a superconductor, which leads to a reduction of non-equilibrium current distributions and thereby to an annealing of the vortex lattice. This technique was applied, for example, for resistivity measurements \cite{Cape1968February}, torque magnetometry \cite{Willemin1998September,Willemin1998November,Weyeneth2009}, local magnetization measurements \cite{Avraham2001May}, and also for specific-heat investigations \cite{Lortz2006September}.
Lortz \emph{et al.}~\cite{Lortz2006September} applied such a shaking field parallel to the main magnetic field ($h_{ac} \parallel H$) to a Nb$_3$Sn single crystal and they claim to have observed a vortex-lattice melting transition in the peak-effect region in high resolution specific-heat data. We also applied a shaking field to the same Nb$_3$Sn sample during some of our high resolution specific-heat measurements, but with a different field configuration was different. However, according to our interpretation we did not observe a vortex-lattice melting transition, in contrast to the work by Lortz \emph{et al.}~\cite{Lortz2006September}.
\section{EXPERIMENT}
The Nb$_3$Sn single crystal (mass $m \approx 11.9\, $mg, thickness $d \approx 0.4\, $mm, and cross section $A \approx 0.44\, $mm$^2$) used for this study was characterized by Toyota \emph{et al.}~\cite{Toyota1988September} and was further investigated by Lortz \emph{et al.}~\cite{Lortz2006September,Lortz2007March,Lortz2007April} in calorimetric measurements and in resistivity measurements in our group \cite{Reibelt2010March}. The crystal exhibits a pronounced peak effect near $H_{c2}$ \cite{Lortz2007March,Reibelt2010March}, and its transition to superconductivity in zero magnetic field as determined by a resistivity measurement occurs at $T_{c} \approx 18.0\, $K.\\
\indent The specific heat was measured with a home-made differential-thermal analysis calorimeter, which achieves a very high sensitivity and data-point density \cite{Schilling2007March,Reibelt2008}. For some specific-heat measurements we applied a small shaking field $h_{ac}$ perpendicular to the main magnetic field $H$ continuously during the measurement. The shaking field was oriented perpendicular to the longest dimension of the sample, that is perpendicular to the critical currents inside the sample, a so-called \emph{transversal vortex-shaking configuration} ($h_{ac} \perp j_c$ and $h_{ac} \perp H$) \cite{Brandt2002July,Brandt2004}. We varied the amplitude of the shaking field to maximal $\mu_0h_{ac} \approx 1.3\, $mT and the frequency varied between $5$ and $100\, $Hz. The vortex lattice preparation prior to the measurement was a field cooled (FC) procedure. For most measurements the FC procedure was very slow and took several hours. The measuring cell was thermally decoupled from the rest of the insert very well, heat exchange took place only via its nylon wire suspension. For a few measurements we used a cooling clamp which decreased the cooling time during the FC procedure to about $15$ minutes.
\section{RESULTS AND DISCUSSION}
\subsection{A. Fluctuation peak}
\begin{figure}
\includegraphics[width=75mm,totalheight=200mm,keepaspectratio]{PRB_2012_all_fields_pdf}
\caption{Specific heat of a Nb$_3$Sn single crystal for different fixed magnetic fields from $0\, $T to $8\, $T. Here $1\, $gat (gram-atom) is $1/4\, $mole.\label{fig.PRB_2012_all_fields_pdf}}
\end{figure}
In Fig.~\ref{fig.PRB_2012_all_fields_pdf} we plotted the temperature dependence of the specific heat at different fixed magnetic fields, note the high data-point density. A peak-like anomaly near $T_c(H)$ is discernible for all shown nonzero magnetic fields, the fluctuation peak. This fluctuation peak was also seen by Lortz \emph{et al.}~\cite{Lortz2006September,Lortz2007April} in the same crystal. With increasing magnetic field, the transition widths at $T_c$ increases and the fluctuation peak broadens. The existence of the fluctuation peak and the increase of the transition width can be attributed to the presence of critical fluctuations of the superconductivity order-parameter and a one-dimensional ($1$D) character in the presence of a magnetic field \cite{Eilenberger1967,Thouless1975April}. Since the nature of the fluctuation is changed rather drastically when a magnetic field is applied, one expects that the critical region will be broadened \cite{Lee1972April}, which in turn leads to a broadening of the fluctuation peak. Figure \ref{fig.PRB_2012_fluctuation_peak_development_pdf} shows the development of the fluctuation peak at low magnetic fields. It can be discerned already at $\mu_0H=0.2\, $T and its magnitude increases with increasing magnetic field until it saturates at about $\mu_0H=1\, $T where its size reaches $\sim 5\%$ of the total jump of the specific heat at $T_c(H)$. The Ginzburg temperature $\tau_G = 0.5\, k^2_B\, T^3_c(0)/(H^2_c(0)\, \xi^3_0)^2$ determines the temperature range around $T_c$ where the contribution of fluctuations to the specific heat are of the same order of magnitude as the mean-field jump at $T_c$. Using $T_c \approx 18\, $K, $H_c(0) = 5200\, $Oe and $\xi_0 = 30\, ${\AA} \cite{Guritanu2004November}, we obtain $\tau_G \approx 10^{-4}\, $K. Contributions of a few percent of the jump might therefore be observable in a range of $10^{-2}\, $K around $T_c$. As can be seen from Fig.~\ref{fig.PRB_2012_fluctuation_peak_development_pdf}, the fluctuation peak near $T_c(H)$ develops over a temperature range of several $10^{-2}\, $K.\\
\FloatBarrier
\begin{figure}
\includegraphics[width=75mm,totalheight=200mm,keepaspectratio]{PRB_2012_fluctuation_peak_development_pdf}
\caption{Development of the fluctuation peak in the specific heat of Nb$_3$Sn at low magnetic fields.\label{fig.PRB_2012_fluctuation_peak_development_pdf}}
\end{figure}
\FloatBarrier
\indent According to Lortz \emph{et al.}~\cite{Lortz2007April} for their data the width of the transition to superconductivity and also the width of the fluctuation peak in Nb$_3$Sn scaled according to the $3$D-LLL scaling model for $\mu_0 H \geq 4\, $T and to the $3$D-XY scaling model for smaller magnetic fields. The problem which arises here is the fact that according to the $3$D-XY scaling model, the fluctuation peak should be most pronounced for zero magnetic field. Due to the divergence of the coherence length $\xi$ at $T_c$, the peak should also diverge in an ideal case, but finite size effects due to sample dimensions, sample inhomogeneities lead to a broadening of the peak; in addition the normal conducting cores of the vortices in the presence of a magnetic field lead to a broadening of the peak with increasing magnetic field. In the high-$T_c$ superconductors YBa$_2$Cu$_3$O$_{7-\delta}$ \cite{Lortz2003November} and NdBa$_2$Cu$_3$O$_7$ \cite{Plackowski2005} one has observed this diverging behavior for zero magnetic field. Therefore within the $3$D-XY model it is hard to understand, why in Nb$_3$Sn the peak is absent in zero magnetic field and develops only with increasing magnetic field. However, in the picture of lowest Landau levels (LLL) one may understand the observations. A magnetic field confines the quasiparticles to low Landau levels, thereby reducing the dimensionality of the fluctuations. For very low magnetic fields $\mu_0 H \leq 0.2\, $T, the confinement is not strong enough and no fluctuation peak is formed in the specific heat. With increasing magnetic field the confinement to low-lying Landau Levels gets stronger and as a result only one degree of freedom remains, namely that along the $z$ direction. The fluctuation specific heat is then proportional to the field and becomes one-dimensional in nature, diverging within the mean field theory, like $|T/T_{c}(H)-1|^{-3/2}$ as compared with a $|T/T_{c0}-1|^{-1/2}$ divergence in the absence of a magnetic field. This results in a substantial enhancement of the specific heat close to the transition temperature \cite{Lee1972April} and may explain the observed development of a fluctuation peak in the Nb$_3$Sn specific-heat data above $0.2\, $T.\\
\FloatBarrier
\begin{figure}
\includegraphics[width=75mm,totalheight=200mm,keepaspectratio]{PRB_2012_2T_vs_8T_pdf}
\caption{Broadening of the fluctuation peak in Nb$_3$Sn: \textbf{a)} Specific heat at $\mu_0H=2\, $T. \textbf{b)} Specific heat at $\mu_0H=8\, $T.\label{fig.PRB_2012_2T_vs_8T_pdf}}
\end{figure}
\FloatBarrier
\indent Besides the initial increase in magnitude of the fluctuation peak, one can also observe a broadening with increasing magnetic field. This is expected, if one keeps in mind that also $\tau_G$ increases in a magnetic field due to a reduction of the effective dimensionality arising from the confinement of the excitations to a few low Landau orbits \cite{Lee1972April}. In order to illustrate the increase of the broadening of the fluctuation peak with increasing magnetic field we compare in Fig.~\ref{fig.PRB_2012_2T_vs_8T_pdf} a measurements at $\mu_0H=2\, $T with a measurement at $\mu_0H=8\, $T. Note that the axes of Figs.~\ref{fig.PRB_2012_2T_vs_8T_pdf}a and \ref{fig.PRB_2012_2T_vs_8T_pdf}b are scaled to the same size.
\subsection{B. Absence of vortex-lattice melting}
We next applied a small shaking field at different amplitudes $h_{ac}$ and frequencies $f$ perpendicular to the main magnetic field $H$ continuously during the specific-heat measurements. For low magnetic fields (Figs.~\ref{fig.PRB_2012_1T_5Hz_pdf} and \ref{fig.PRB_2012_2T_pdf}), no peak-like deviation from the data without a shaking field can be noticed in our data.
\FloatBarrier
\begin{figure}
\includegraphics[width=75mm,totalheight=200mm,keepaspectratio]{PRB_2012_1T_5Hz_pdf}
\caption{Specific heat of Nb$_3$Sn at $\mu_0H=1\, $T. A small shaking field with $h_{ac} \approx 0.5\, $mT and $f = 5\, $Hz was applied perpendicular to the main magnetic field $H$ for the red curve.\label{fig.PRB_2012_1T_5Hz_pdf}}
\end{figure}
\begin{figure}
\includegraphics[width=75mm,totalheight=200mm,keepaspectratio]{PRB_2012_2T_pdf}
\caption{Specific heat of Nb$_3$Sn at $\mu_0H=2\, $T. A small shaking field with different amplitudes $h_{ac}$ and frequencies $f$ was applied perpendicular to the main magnetic field $H$.\label{fig.PRB_2012_2T_pdf}}
\end{figure}
\indent According to their interpretation, Lortz \emph{et al.}~\cite{Lortz2006September} observed the first sign of vortex-lattice melting at $\mu_0H=3\, $T and they investigated and observed the feature up to $\mu_0H=6\, $T (see also their magnetic phase diagram in Fig.~$6$ of Ref.~\cite{Lortz2007March}). Their observed sharp peak-like feature increased in magnitude with increasing magnetic field as one would expect it for a true vortex-lattice melting transition. In Fig.~\ref{fig.PRB_2012_3T_pdf} and Fig.~\ref{fig.PRB_2012_4T_50Hz_pdf} we present our data at $\mu_0H=3\, $T and $\mu_0H=4\, $T for amplitudes up to $h_{ac} = 0.76\, $mT and frequencies up to $f= 50\, $Hz. We did not note any sign of a sharp peak which may be connected to a latent heat of a first-order melting transition. However, Lortz \emph{et al.}~\cite{Lortz2006September} used a higher amplitude of about $h_{ac} \approx 1\, $mT and a higher frequency of $f=1\, $kHz. Our shaking coil was not able to reach these parameters. However, Lortz \emph{et al.}~\cite{Lortz2006September} applied the shaking field not perpendicular but parallel to the main magnetic field.
\FloatBarrier
\begin{figure}
\includegraphics[width=75mm,totalheight=200mm,keepaspectratio]{PRB_2012_3T_pdf}
\caption{Specific heat of Nb$_3$Sn at $\mu_0H=3\, $T. A small shaking field with different amplitudes $h_{ac}$ and frequencies $f$ was applied perpendicular to the main magnetic field $H$.\label{fig.PRB_2012_3T_pdf}}
\end{figure}
\begin{figure}
\includegraphics[width=75mm,totalheight=200mm,keepaspectratio]{PRB_2012_4T_50Hz_pdf}
\caption{Specific heat of Nb$_3$Sn at $\mu_0H=4\, $T. A small shaking field at frequency $f=50\, $Hz with different amplitudes $h_{ac}$ was applied perpendicular to the main magnetic field $H$.\label{fig.PRB_2012_4T_50Hz_pdf}}
\end{figure}
\FloatBarrier
\noindent According to Brandt and Mikitik, application of a small shaking field perpendicular (transversal vortex-shaking \cite{Brandt2002July,Brandt2004}) or parallel (longitudinal vortex-shaking \cite{Mikitik2003March}) to the critical current inside the sample is able to cause the critical currents and the irreversible magnetic moment of the sample to relax completely, however application of a small shaking field parallel to the main magnetic field $H$ and perpendicular to the critical currents inside the sample plane is only able to do so if a transport current is applied to the sample simultaneously \cite{Mikitik2001August}. The latter configuration was successfully used by us in resistivity measurements \cite{Reibelt2010March}. Lortz \emph{et al.}~\cite{Lortz2006September} applied a small shaking field $h_{ac} \approx 1\, $mT parallel to the main magnetic field $H$ but they did not apply the necessary transport current to the sample. Therefore, from the current theoretical point of view it is questionable whether their shaking field was able to relax the irreversible currents inside the sample as claimed by Lortz \emph{et al.}~\cite{Lortz2006September}. However, Lortz \emph{et al.}~\cite{Lortz2006September} placed the sample not inside their shaking coil but above it, therefore only the weak external stray field of the shaking coil reached the sample. The external stray field of a shaking coil was not homogeneous and not all its components were parallel to the main magnetic field, some of its field components were perpendicular to the main magnetic field and inside the sample plain. These perpendicular components may have been able to conduct transversal and longitudinal vortex-shaking inside the sample \cite{Brandt2004,Mikitik2003March}. According to Brandt \emph{et al.}~\cite{Brandt1986November,Brandt1994April,Brandt1996March,Brandt1996August,Mikitik2000September,Mikitik2001August,Brandt2002July,Brandt2004,Brandt2004a,Brandt2004b,Brandt2007} there exists a threshold amplitude below which no movement of the vortices is generated. In order to be able to cause the vortices to \emph{walk}, the shaking amplitude has to fulfill the condition $h_{ac} \geq J_c/2$, where $J_c$ is the critical sheet current density, which is the integration of the critical current density $j_c$ over the thickness $d$ of the sample. $J_c$ is proportional to the pinning strength. However, it is questionable whether Lortz \emph{et al.}~\cite{Lortz2006September} were able to reach this necessary threshold shaking-field amplitude $h_{ac} \geq J_c/2$ to cause the "walking" of the vortices \cite{Brandt2002July}. The field components of their shaking field inside the plane of the sample are most likely smaller than the amplitude of the shaking field which was applied by us in the transversal shaking configuration. Regarding the lower frequency used by us, Brandt \emph{et al.}~\cite{Brandt2002July} mention only a threshold amplitude but not a threshold frequency for the transversal shaking configuration. Therefore we assume that the measurements conducted by us should have been able to reveal the vortex-lattice melting transition, if present, even at frequencies $f<1\, $kHz.\\
\indent The presence of the peak effect near the upper critical field in this Nb$_3$Sn sample has been observed by Lortz \emph{et al.}~\cite{Lortz2006September} and was confirmed in our recent work \cite{Reibelt2010March}. Lortz \emph{et al.}~\cite{Lortz2007March} have shown that the peak effect coincides in their magnetic phase diagram with their claimed observation of a vortex-lattice melting transition. However, the presence of the peak effect can complicate the interpretation of data. From the description of the conduction of the experiment in Ref.~\cite{Lortz2006September} it appears that Lortz \emph{et al.}~continuously applied a shaking field during the measurement and did not turn it off while taking the data at each data point. Therefore it might be possible that the sample got heated up due to vortex motion. In the following we will call the heating up of the sample which is not due to the cell heater but due to vortex motion inside the sample itself as \emph{self heating}. Directly at the peak effect the vortex lattice becomes stronger pinned which leads to an increased critical current. In this region of increased pinning the self heating of the sample due to vortex motion should decrease. Depending on the background subtraction procedure and possible adjustments to gain absolute values, this decrease in self heating might have been misinterpreted as a peak-like increase in the specific heat.\\
\begin{figure}
\includegraphics[width=75mm,totalheight=200mm,keepaspectratio]{PRB_2012_4T_5Hz_pdf}
\caption{Distorted specific heat of Nb$_3$Sn at $\mu_0H=4\, $T, for different shaking amplitudes $h_{ac}$ at fixed frequency $f=5\, $Hz.\label{fig.PRB_2012_4T_5Hz_pdf}}
\end{figure}
\indent In Fig.~\ref{fig.PRB_2012_4T_5Hz_pdf} we set the shaking frequency to $f=5\, $Hz at $\mu_0H=4\, $T. Close to $T_c$ self heating occurs which increases with increasing amplitude of the shaking field $h_{ac}$. The surplus heat created inside the sample causes the DTA calorimeter to underestimate the specific heat of the sample which leads to the formation of a dip-like pattern in the data, which we will call \emph{self-heating dip} in the following. Therefore, for a nonzero shaking field the curves in Fig.~\ref{fig.PRB_2012_4T_5Hz_pdf} do not represent the specific heat in the regions where self heating occurs. However, these data of a distorted specific heat are still of value since a sharp first-order phase transition should still be discernible as a sharp peak since the self-heating causes merely a smooth distortion. However, no superimposed peak can be discerned in our data which we plotted in Fig.~\ref{fig.PRB_2012_4T_5Hz_pdf}.\\
\begin{figure}
\includegraphics[width=75mm,totalheight=200mm,keepaspectratio]{PRB_2012_6T_100Hz_pdf}
\caption{Distorted specific heat of Nb$_3$Sn at $\mu_0H=6\, $T. A small shaking field with frequency $f=100\, $Hz and different amplitudes $h_{ac}$ was applied perpendicular to the main magnetic field $H$. Self heating of the sample sets in for $h_{ac}>0$ which distorts the specific heat in a region close to $T_c$.\label{fig.PRB_2012_6T_100Hz_pdf}}
\end{figure}
\begin{figure}
\includegraphics[width=75mm,totalheight=200mm,keepaspectratio]{PRB_2012_6T_10Hz_pdf}
\caption{Distorted specific heat of Nb$_3$Sn at $\mu_0H=6\, $T. A small shaking field was applied perpendicular to the main magnetic field $H$ at different amplitudes $h_{ac}$. \textbf{a)} $f=10\, $Hz; \textbf{b)} $f=5\, $Hz.\label{fig.PRB_2012_6T_10Hz_pdf}}
\end{figure}
\indent The highest magnetic field investigated by Lortz \emph{et al.}~\cite{Lortz2006September} with the shaking technique was $\mu_0H=6\, $T where they observed the largest latent heat for their assumed first-order phase transition. In Fig.~\ref{fig.PRB_2012_6T_100Hz_pdf} we present our data at $\mu_0H=6\, $T and a shaking-field frequency $f=100\, $Hz. Again, no superimposed peak can be discerned in our data which we plotted in Fig.~\ref{fig.PRB_2012_6T_100Hz_pdf}. The same holds true for our $10\, $Hz measurements presented in Fig.~\ref{fig.PRB_2012_6T_10Hz_pdf}. For the measurement with $h_{ac}=0.52\, $mT in Fig.~\ref{fig.PRB_2012_6T_10Hz_pdf}a, the self-heating already sets in at about $14.1\, $K and generates a broad dip structure in the data in the region where Lortz \emph{et al.}~\cite{Lortz2006September} observed a small sharp peak in the specific heat. However, no such peak is discernible in our data in Figs.~\ref{fig.PRB_2012_6T_10Hz_pdf}a or \ref{fig.PRB_2012_6T_10Hz_pdf}b. As we further increase the shaking amplitude $h_{ac}$ at a fixed frequency $f=10\, $Hz in Fig.~\ref{fig.PRB_2012_6T_10Hz_both_pdf}, a broad dome-like feature is "carved out" by the shaking field. This dome-like feature seems to be superimposed to an extended self-heating dip. We drew a pink dashed line into Fig.~\ref{fig.PRB_2012_6T_10Hz_both_pdf}b in order to sketch the suspected shape of the self-heating dip how it would appear without the dome-like feature. The self-heating dip has expanded on the temperature scale to lower temperatures and its depth increased compared to the orange curve due to the increased shaking amplitude. According to the collective pinning theory by Larkin and Ovchinnikov \cite{Larkin1979}, the pinning strength decreases with increasing temperature (the peak effect is an exception). Decreasing the temperature at a constant magnetic field increases the pinning strength. A stronger shaking amplitude can overcome the stronger pinning at lower temperatures and make the vortices also \emph{walk} at lower temperatures leading to the observed expansion of the self-heating dip towards lower temperatures for higher $h_{ac}$. We marked in Fig.~\ref{fig.PRB_2012_6T_10Hz_both_pdf}b the onset of the self-heating dip as $T_{sdo}$ (\emph{sdo} stands for self-heating-dip onset) and the first sharp step-like appearance of self heating on increasing the temperature as $T_{so}$ (\emph{so} stands for self-heating onset). The maximum at the dome-like feature, which we identified with the center of the peak-effect region, we marked with $T_p$. As it appears, there are two different regions of self heating. On heating up during the measurement, the first sign of self heating sets in at $T_{so}$, where the curve suddenly drops very sharply about $5\, $mJ gat$^{-1}$ K$^{-2}$ below the shaking free $C/T$-curve and runs from $T_{so}$ on more or less parallel to the shaking free $C/T$-curve until it reaches the second region of self heating where at $T_{sdo}$ the self-heating dip sets in. With increasing shaking amplitude $h_{ac}$, the sharp onset of the first sign of self heating $T_{so}$ is shifted to lower temperatures in line with the collective pinning theory where for lower temperatures the pinning strength is stronger. In the regions where self heating is present, the magnetic vortices continuously enter the sample on one side of the sample, "walk" through the sample and leave it on the opposite side of the sample, thereby continuously generating heat inside the sample.\\
\begin{figure}
\includegraphics[width=75mm,totalheight=200mm,keepaspectratio]{PRB_2012_6T_10Hz_both_pdf}
\caption{\textbf{a)} Distorted specific heat of Nb$_3$Sn at $\mu_0H=6\, $T for different shaking amplitudes $h_{ac}$ at fixed frequency $f = 10\, $Hz. \textbf{b)} Closeup of the region near $T_c$. \label{fig.PRB_2012_6T_10Hz_both_pdf}}
\end{figure}
\begin{figure}
\includegraphics[width=75mm,totalheight=200mm,keepaspectratio]{PRB_2012_6T_50Hz_pdf}
\caption{Distorted specific heat of Nb$_3$Sn at $\mu_0H=6\, $T. A small shaking field with frequency $f=50\, $Hz was applied perpendicular to the main magnetic field $H$ at different amplitudes $h_{ac}$. The FC process prior to the measurement was very slow.\label{fig.PRB_2012_6T_50Hz_pdf}}
\end{figure}
In Fig.~\ref{fig.PRB_2012_6T_50Hz_pdf} we show data at $\mu_0H=6\, T$ and $f=50\, $Hz for different shaking amplitudes $h_{ac}$. We zoomed in on the peak-effect region near $T_c$, the sharp step-like onset of the self heating takes place at lower temperatures outside the zoomed in region. Again, a dome-like feature is "carved out" by the shaking field at the peak-effect region which has lower self heating due to the increased pinning strength in this region. This dome-like feature is broader than the peak-like feature observed by Lortz \emph{et al.}~\cite{Lortz2006September} at their supposed melting transition. In addition, the temperature of the center of the dome seems to be lower than the center of the supposed melting transition observed by Lortz \emph{et al.}~\cite{Lortz2006September}. Therefore it appears not very likely that the reduced self heating at the peak effect might have led to the peak observed by Lortz \emph{et al.}~\cite{Lortz2006September} at their supposed melting transition. However, one has to keep in mind, that different specific-heat methods and different shaking methods were used by the two groups. Interestingly, in Fig.~\ref{fig.PRB_2012_6T_50Hz_pdf} for higher amplitudes the dome-like feature is embedded in regions of increased self heating which appear as dips. According to an explanation of the peak effect by Pippard \cite{Pippard1969}, the vortex lattice softens in the peak-effect region, i.e., the shear modulus $C_{66}$ is vastly reduced, the vortex lattice becomes less rigid, and the vortices can bend and adjust better to the pinning sites. One may interpret our data in the way that the onset of the self-heating dip at $T_{sod}$ is the onset of the reduction of the shear modulus and the resulting less rigid vortex lattice is more prone to accelerations by the shaking field thereby leading to more self heating which causes the observed self-heating dip. It is known that the presence of pinning causes a friction during the motion of the vortices which leads to the heat dissipation. The bending of the vortices is crucial in this mechanism. When the shear modulus $C_{66}$ softens, the vortices can bend more easily due to the alternating forces exerted by the shaking field. On further increasing the temperature the shear modulus continues to reduce and the mechanism of better adjustment to the pinning sites sets in which increases the pinning strength and thereby hinders vortex motion in the region of the peak at $T_p$, at least in a part of the sample. The dome-like feature at $T_p$ inside the peak-effect region, superimposed to the self-heating dip, is in this picture the consequence of the increased pinning strength in the peak-effect region, since a hindered vortex motion also reduces the self heating.\\
\begin{figure}
\includegraphics[width=75mm,totalheight=200mm,keepaspectratio]{PRB_2012_0T_5Hz_pdf}
\caption{Specific heat of Nb$_3$Sn at $\mu_0H=0\, $T. A small shaking field with $h_{ac} \approx 1.3\, $mT and $f=5\, $Hz was applied perpendicular to the main magnetic field $H$.\label{fig.PRB_2012_0T_5Hz_pdf}}
\end{figure}
\indent We next want to decide the question wether the self-heating dip is really caused by the motion of vortices or caused by different means like, e.g., the increase of quasiparticles with increasing temperature in the picture of the two-fluid model, where eddy currents made up of quasiparticles may cause the self heating. If the self-heating dip is mainly caused by vortex motion, it should disappear or at least diminish strongly in zero magnetic field despite the presence of a shaking field. We present in Fig.~\ref{fig.PRB_2012_0T_5Hz_pdf} a measurement at $\mu_0 H = 0\, $T; the shaking field had the frequency $f = 5\, $Hz and the rather high amplitude $h_{ac} \approx 1.3\, $mT. No sign of self-heating can be observed near $T_c$. We therefore conclude that the above observed dip in our data close to $T_c$ for $\mu_0H>3\, $T is most likely caused by self heating due to the motion of magnetic vortices.\\
\begin{figure}
\includegraphics[width=75mm,totalheight=200mm,keepaspectratio]{PRB_2012_7T_5Hz_pdf}
\caption{Distorted specific heat of Nb$_3$Sn at $\mu_0H=7\, $T. A small shaking field with frequency $f=5\, $Hz was applied perpendicular to the main magnetic field $H$ at different amplitudes $h_{ac}$. The data were shifted for clarity.\label{fig.PRB_2012_7T_5Hz_pdf}}
\end{figure}
\indent If our interpretation of the dome-like feature as a manifestation of the increased pinning strength in the peak-effect region is correct, then further increasing the shaking amplitude $h_{ac}$ should be able to overcome the pinning strength even in the peak-effect region. Increasing the shaking amplitude was not possible for us due to the limitations of our shaking coil. However, we were able to increase the main magnetic field to $\mu_0H=7\, $T. The effectiveness of the shaking field increases with increasing main magnetic field, because the pinning strength decreases with increasing magnetic field as can be seen from the hysteresis in the magnetization loops of Nb$_3$Sn (see Refs.~\cite{Lortz2007March,Reibelt2010March} for magnetization measurements of this Nb$_3$Sn sample). In Fig.~\ref{fig.PRB_2012_7T_5Hz_pdf} we show data at $\mu_0H=7\, T$ and $f=5\, $Hz for different shaking amplitudes $h_{ac}$, the data was shifted for clarity. For medium shaking-field amplitudes $h_{ac}$ again the dome-like feature embedded in a self-heating dip emerges. Strikingly, for a large shaking-field amplitude $h_{ac}=0.65\, $mT (green curve) the dome-like feature diminishes and for the largest amplitude $h_{ac}=0.72\, $mT (purple curve) the dome-like feature even vanishes completely. If the dome-like feature origins from a first-order vortex-lattice melting transition, this vanishing for high shaking-filed amplitudes $h_{ac}$ would be unexpected. However, in the picture of our interpretation where the dome-like feature is a manifestation of the increased pinning strength in the peak-effect region, the vanishing of the dome-like feature with increasing $h_{ac}$ is expected.\\
\begin{figure}
\includegraphics[width=75mm,totalheight=200mm,keepaspectratio]{PRB_2012_8T_5Hz_pdf}
\caption{Distorted specific heat of Nb$_3$Sn at $\mu_0H=8\, $T. A small shaking field with frequency $f=5\, $Hz and $h_{ac} = 0.54\, $mT was applied perpendicular to the main magnetic field $H$.\label{fig.PRB_2012_8T_5Hz_pdf}}
\end{figure}
\begin{figure}
\includegraphics[width=75mm,totalheight=200mm,keepaspectratio]{PRB_2012_up_down_comparison_pdf}
\caption{Raw magnetocaloric data $\Delta T$ for increasing and decreasing temperature. The data were shifted for clarity.\label{fig.PRB_2012_up_down_comparison_pdf}}
\end{figure}
\indent We next present data at $\mu_0 H = 8\, $T. In Fig.~\ref{fig.PRB_2012_8T_5Hz_pdf} the typical self-heating pattern emerges again. In Fig.~\ref{fig.PRB_2012_up_down_comparison_pdf} we plotted the raw magnetocaloric data $\Delta T=T_r-T_s$, where $T_r$ is the temperature of the reference thermometer and $T_s$ is the temperature of the sample thermometer where the sample is mounted. We plotted data for a measurement on increasing the temperature and for a measurement on decreasing the temperature. Strikingly, a similar self-heating pattern emerges for both measurements in the peak-effect region. If the dome-like feature would have been due to a first-order melting transition, the related latent heat should have been released in one measurement and absorbed in the other measurement. However, we observe for both measurements a dome-like feature where the sample gets colder. The occurrence of self heating is rather cumbersome when it comes to the identification of phase transitions. The broad and huge shape of the dome-like feature already disqualify it as a manifestation of a melting transition to some degree. However, due to the comparison of a measurement on increasing with a measurement on decreasing the temperature, we were able to definitely exclude a first-order melting transition as the origin for the dome-like feature.\\
\begin{figure}
\includegraphics[width=75mm,totalheight=200mm,keepaspectratio]{PRB_2012_comparison_slow_fast_pdf}
\caption{Comparison of a measurement with slow cooled FC procedure preparation with a measurement with fast cooled FC procedure preparation.\label{fig.PRB_2012_comparison_slow_fast_pdf}}
\end{figure}
\indent Finally, we want to report on the influence of the cooling rate during the FC procedure on the self-heating pattern. In Fig.~\ref{fig.PRB_2012_comparison_slow_fast_pdf} we plotted two measurements at $\mu_0H=5\, $T, $f=10\, $Hz, and $\mu_0h_{ac}=0.6\, $mT. For the green curve we used the usual slow cooled FC procedure and the usual sharp step-like onset of the self heating occurs in the data, which we marked with a green arrow. For the red curve we used a very high cooling rate by the use of a cooling clamp which connected the measuring cell directly with the cold insert. It appears as if the formerly very sharp step-like onset of self heating turned into a smeared kink, which we marked with a red arrow in the figure. The faster cooling lead to a fast passing of the peak-effect region, giving the vortex lattice not enough time to adjust to the strongest pinning centers while the vortex lattice is easy bendable in the peak-effect region where the shear modulus is reduced. The ordered weakly pinned vortex lattice gets quenched, it has not enough time to pick up much disorder and strong pinning before it becomes rigid at low temperatures where it cannot adjust to the strongest pinning centers anymore. On heating up the sample again during the measurement, the less strong pinned vortex lattice allows the shaking field already at lower temperatures to cause vortex motion. The broadening of the onset may be caused by the circumstance that some parts of the sample are stronger pinned than others for the fast cooled FC procedure. In contrast, for the slow cooled FC procedure the sample stays for a long enough time in the peak-effect region on cooling the sample, allowing the vortices throughout the whole sample to adjust to the strongest pinning centers, which leads to a homogeneous strong pinning throughout the whole sample and as a result to the sharper self-heating onset at higher temperatures on shaking during the measurement.\\
\indent This finding is likely not a new one. However, we want to mention its practical relevance. It appears to be beneficial to cool down type-II superconductors very slowly across their peak-effect region (if present) in the presence of a magnetic field, in order to achieve (freeze in) the strongest pinned and most homogeneous phase throughout the whole superconductor.
\section{CONCLUSION}
To conclude, we confirm the observation of the fluctuation peak in the specific heat of Nb$_3$Sn near $T_c(H)$, which was previously reported by Lortz \emph{et al.}~\cite{Lortz2006September,Lortz2007April} for the same crystal. Along with work done by other authors our data support the view that thermal fluctuations manifest not only in high-$T_c$ superconductors but also in low-$T_c$ superconductors like Nb$_3$Sn \cite{Lortz2006September,Lortz2007April}, Nb \cite{Farrant1975}, and dirty Bi$_{0.4}$Sb$_{0.6}$ films \cite{Zally1971December}.\\
\indent However, we were not able to observe the by Lortz \emph{et al.}~\cite{Lortz2006September} reported melting of the vortex lattice in this Nb$_3$Sn crystal. We observed no small sharp peak at the by the measurements of Lortz \emph{et al.}~\cite{Lortz2006September} indicated position in the magnetic phase diagram in any of our measurements. We want to point out that our data at $\mu_0H=3\, $T and even some of our data at $\mu_0H=4\, $T are virtually free of self heating and still we do not observe any sign of a vortex melting. The by us observed broad dome-like feature in the measurements at higher magnetic fields where self heating is present, is interpreted by us as a manifestation of the increased pinning in the peak-effect region. We want to emphasize again that the discrepancy between the data of Lortz \emph{et al.}~\cite{Lortz2006September} and our data presented in this work may originate from the different specific-heat methods and different shaking methods and parameters used by the two groups. However, it would be interesting to repeat the measurements of Lortz \emph{et al.}~\cite{Lortz2006September} with their ac technique but with a truly transversal or longitudinal vortex-shaking configuration. In addition, higher shaking-field amplitudes may be used with their already existing setup in order to exclude a vanishing of the supposed vortex-lattice melting peak as it was the case for the dome-like feature in our data.
\section{ACKNOWLEDGMENT}
We thank the group of A.~Schilling for their support. This work was supported by the Schweizerische Nationalfonds zur F\"{o}rderung der Wissenschaftlichen Forschung, Grants No.~$20$-$111653$ and No.~$20$-$119793$.
|
{
"timestamp": "2012-05-22T02:02:48",
"yymm": "1203",
"arxiv_id": "1203.1848",
"language": "en",
"url": "https://arxiv.org/abs/1203.1848"
}
|
\section{Introduction}
The theory of open quantum systems has
become an increasingly important topic in, e.g., quantum information
science, quantum measurement, and quantum optics. Traditionally, the
dynamics of an open quantum system was often investigated using a
Markov master equation derived by invoking the Born-Markov
appoximation. However, this formalism fails for many solid-state
systems (see, e.g., \cite{jqyou1}) where the system-environment
coupling is strong and the environment is structured. Thus a
non-Markovian master equation is required when considering the
memory effect and back action of the environment. It is known that
the derivation of an exact non-Markovian master equation has long
been a challenging task. One of the breakthroughs is the exact
non-Markovian master equation for quantum Brownian motion model derived
by Hu {\it et al.}~\cite{Hu} using the Feynman-Vernon influence
functional path-integral method~\cite{Feynman}.
Of all the theoretical strategies used to deal with open quantum
systems, a non-Markovian quantum trajectory theory known as
non-Markovian quantum state diffusion (NMQSD)~\cite{Diosi,Strunz1}
provides a powerful approach to the dynamics of an open quantum
system in bosonic environments. In this approach, when the so-called
$O$-operator is obtained, the quantum dynamics of an open system is
determined by solving the NMQSD equation (i.e., a diffusive
stochastic Schr\"{o}dinger equation) and the non-Markovian master
equation can also be derived~\cite{Yu1,Strunz2}. In contrast to the
conventional master equation under the Born approximation, this
non-Markovian mater equation is derived non-perturbatively, so it
applies even for a strong system-environment coupling. Indeed, some
exact $O$-operators have been found in a variety of quantum
models~\cite{Diosi,Strunz1,Yu1,Strunz2,Jing1,Eberly}, including
multilevel models~\cite{Jing2}.
In addition to bosonic baths, fermionic baths are also involved in
many physical systems, particularly in solid-state systems. The
Feynman-Vernon influence functional path-integral method can also be
used to study the quantum dynamics of an open system in a fermionic
environment (see, e.g., \cite{Hedegard}). Recently, this
path-integral method was extended to derive non-Markovian master
equations for nanodevices~\cite{Zhang1,Zhang2}. Also, there were
other studies on quantum dynamics and transport of nanostructured
systems, including the rate-equation approach~\cite{Gurvitz,Stoof},
and the Markovian (see, e.g., \cite{Haug1,Cottet}) and non-Markovian
(see, e.g., \cite{Goan}) master equation approaches. In addition,
the nonequilibrium Green's function method can be used to study the
quantum transport through the nanostructures (see,
e.g.,~\cite{Haug2}). Nevertheless, the extension of NMQSD method to
an open quantum system in fermionic baths has been a long-standing
unsolved problem because this open system involves anticommutative
noise functions (i.e., Grassmann variables) that are intrinsically
different from the noise functions of bosonic baths.
In this paper,
we develop an NMQSD method to study the open quantum system in
fermionic baths. This NMQSD approach is formulated in a
non-perturbative manner and it applies for both weak and strong
system-environment couplings. We not only obtain the NMQSD equation
for quantum states of the system, but also derive the non-Markovian
master equation. Moreover, as interesting examples, we apply this
NMQSD method to single and double quantum-dot systems.
Note that our NMQSD method belongs to the quantum trajectory approach involving continuous time evolution. There is another kind of quantum trajectory approach which involves discontinuous time evolution, i.e., quantum jumps (see, e.g., \cite{Dali}). The non-Markovian quantum trajectory approach in \cite{Piilo} generalizes the Markovian quantum jump method and can also apply to both bosonic and fermionic baths.
\section{Quantum state diffusion equation}
We consider a quantum
system coupled to two fermionic baths: $H=H_{\rm sys}+H_{\rm
env}+H_{\rm int}$, with (we set $\hbar=1$)
\begin{align}
&H_{\rm env}= \sum_{k}(\omega_{Lk}a_{Lk}^\dagger a_{Lk}+
\omega_{Rk}a_{Rk}^\dagger a_{Rk}), \label{bath} \\
&H_{\rm int}=
\sum_{k}(g_{Lk}c_{L}^\dagger a_{Lk}+ g_{Rk}c_{R}^\dagger a_{Rk}+
{\rm H.c.}). \label{int}
\end{align}
Here $H_{\rm sys}$ denotes the Hamiltonian of the system, $H_{\rm
env}$ is the Hamiltonian of the two electric leads acting as
fermionic baths, and $H_{\rm int}$ models the interactions between
the system and the two baths. The spectral density function of each
bath is
\begin{equation}
J_{\lambda}(\omega)=\sum_{k}\lvert g_{\lambda
k}\rvert^2\delta(\omega-\omega_{\lambda k}),
\end{equation}
where $\lambda=L$ or
$R$. In Eq.~(\ref{bath}), $a_{\lambda k}^\dagger$ ($a_{\lambda k}$)
is the fermionic creation (annihilation) operator for a quantum
state with wave vector $k$ in the left or right lead. We assume that
the system of interest couples to the two leads via single channels
characterized by the fermionic creation (annihilation) operators
$c_{\lambda k}^\dagger$ ($c_{\lambda k}$) [see Eq.~(\ref{int})].
Extension to a multi-channel case is straightforward.
In an NMQSD approach, environments are required to be initially at
zero temperature, so as to conveniently represent the environmental
degrees of freedom with the coherent state basis. As for
environments initially with a nonzero temperature, one can map the
nonzero-temperature density operator to a zero-temperature density
operator using a Bogoliubov transformation~\cite{Yu2}. In the case
of fermionic baths, this requires to add $\sum_{\lambda
k}\omega_{\lambda k}b_{\lambda k} b_{\lambda k}^\dagger$ to
Eq.~(\ref{bath}), corresponding to the part involving holes in the
electric leads. The Bogoliubov transformation for fermionic
operators can be introduced as
\begin{align}
& a_{\lambda k}= \sqrt{1- \bar{n}_{\lambda k}}
d_{\lambda k}- \sqrt{\bar{n}_{\lambda k}}e_{\lambda k}^\dagger \nonumber\\
& b_{\lambda k}= \sqrt{1- \bar{n}_{\lambda k}}e_{\lambda k}+
\sqrt{\bar{n}_{\lambda k}}d_{\lambda k}^\dagger, \label{Bog}
\end{align}
where $\bar{n}_{\lambda k}=[e^{(\omega_{\lambda
k}-\mu_{\lambda})/k_{B}T}+1]^{-1}$ is the average number of
electrons in the $k$th state of the left (right) electric lead with
chemical potential $\mu_{\lambda}$. In Eq.~(\ref{Bog}), the
coefficients are determined by the requirement that the derived
master equation reduces to a Lindblad form in the Markovian limit.
The transformed Hamiltonian $\mathcal{H}$ is written as
\begin{eqnarray}
\mathcal{H}\!&\!=\!&\! H_{\rm sys}+\sum_{\lambda
k}\left[\omega_{\lambda k}(d_{\lambda k}^\dagger d_{\lambda
k}+e_{\lambda k}e_{\lambda
k}^\dagger)\right.\nonumber\\
&\!&\!+\left. (\sqrt{\bar{n}_{\lambda k}} g_{\lambda k}^*
c_{\lambda}e_{\lambda k}+\sqrt{1-\bar{n}_{\lambda k}} g_{\lambda
k}c_{\lambda}^\dagger d_{\lambda k}+ {\rm H.c.})\right],~~~~
\end{eqnarray}
where the new fermionic operator $d_{\lambda k}$ ($e_{\lambda
k}^\dagger$) corresponds to the annihilation of electrons (holes) in
the virtual fermionic baths. Note that the effects of temperature
are incorporated into the transformed Hamiltonian and the fermionic
baths with {\it nonzero} initial temperatures are mapped to virtual
fermionic baths with {\it zero} initial temperature.
In the interaction picture with respect to the environmental
Hamiltonian $\mathcal{H}_{\rm env}= \sum_{\lambda k} \omega_{\lambda
k}(d_{\lambda k}^\dagger d_{\lambda k}+e_{\lambda k}e_{\lambda
k}^\dagger)$, the total Hamiltonian reads
\begin{eqnarray}
\mathcal{H}(t)\!&\!=\!&\! H_{\rm sys}+ \sum_{\lambda
k}(\sqrt{\bar{n}_{\lambda k}} g_{\lambda k}^* c_{\lambda}e_{\lambda k}e^{i\omega_{\lambda k}t}\nonumber\\
&&\!+\sqrt{1- \bar{n}_{\lambda k}} g_{\lambda k}c_{\lambda}^\dagger
d_{\lambda k}e^{-i\omega_{\lambda k}t}+ {\rm H.c.}),
\end{eqnarray}
and the quantum state of the total system satisfies the equation of
motion
\begin{equation}
\partial_{t}\lvert\Psi_{t}\rangle=
-i\mathcal{H}(t)\lvert\Psi_{t}\rangle. \label{EOM}
\end{equation}
We assume that the quantum state of the total system is factorized
at the initial time $t=0$, so that $\lvert\Psi_{0}\rangle=\lvert
\varphi_{0}\rangle\otimes\lvert 0\rangle$, with the virtual
fermionic baths initially in the ground state (i.e., at zero
temperature): $\lvert 0\rangle=\bigotimes_{\lambda}\lvert
0\rangle_{\lambda d}\otimes\lvert 0\rangle_{\lambda e}$, where
$d_{\lambda k}\lvert 0\rangle=0$, and $e_{\lambda k}\lvert
0\rangle=0$.
Define a femionic coherent-state basis for the environmental degrees
of freedom:
\begin{equation}
\lvert zw\rangle=\bigotimes_{\lambda}\lvert
z\rangle_{\lambda}\otimes\lvert w\rangle_{\lambda},
\end{equation}
with
\begin{eqnarray}
\lvert z\rangle_{\lambda}\!&\!=\!&\! \bigotimes_{k}\lvert
z_{k}\rangle_{\lambda}
= e^{-\sum_{k}z_{\lambda k}d_{\lambda k}^\dagger}\lvert 0\rangle \nonumber\\
\lvert w\rangle_{\lambda}\!&\!=\!&\! \bigotimes_{k}\lvert
w_{k}\rangle_{\lambda}= e^{-\sum_{k}w_{\lambda k}e_{\lambda
k}^\dagger}\lvert 0\rangle,
\end{eqnarray}
where $z_{k}$ and $w_{k}$ are Grassmann variables that obey the
anticommutation relation. With the completeness relation for
coherent states
$\int e^{-z^*z-w^*w}\lvert zw \rangle\langle zw \rvert d^2 z d^2
w=1$,
the state $\lvert\Psi_{t}\rangle$ can be expressed as
\begin{equation}
\label{state2}\lvert\Psi_{t}\rangle= \int e^{-z^*z-w^*w}\lvert
zw\rangle\otimes\lvert \psi_{t}(z^*,w^*)\rangle d^2 z d^2 w,
\end{equation}
where
\begin{eqnarray}
&&z^*z\equiv\sum_{\lambda k}z^*_{\lambda k}z_{\lambda k},
~~~w^*w\equiv\sum_{\lambda k}w^*_{\lambda k}w_{\lambda k},\nonumber\\
&&d^2z\equiv\prod_{\lambda k}dz_{\lambda k}^*dz_{\lambda k},
~~~d^2w\equiv\prod_{\lambda k}dw_{\lambda k}^*dw_{\lambda k}.~~~
\end{eqnarray}
The
actions of annihilation (creation) operators $d_{\lambda k}$ and
$e_{\lambda k}$ ($d_{\lambda k}^{\dagger}$ and $e_{\lambda
k}^{\dagger}$) on fermionic coherent states satisfy the
relations~\cite{Atland}:
\begin{eqnarray}
&&d_{\lambda
k}|z\rangle_{\lambda}=z_{\lambda k}|z\rangle_{\lambda},~~~ d_{\lambda
k}^{\dagger}|z\rangle_{\lambda}=-\frac{\partial}{\partial z_{\lambda
k}}|z\rangle_{\lambda},\nonumber\\
&&e_{\lambda k}|w\rangle_{\lambda}=w_{\lambda k}|w\rangle_{\lambda},
~~~e_{\lambda k}^{\dagger}|w\rangle_{\lambda}=-\frac{\partial}{\partial w_{\lambda
k}}|w\rangle_{\lambda}.~~~~~~
\end{eqnarray}
When projecting onto the coherent-state
basis, the equation of motion (\ref{EOM}) can be reduced to the
NMQSD equation for a pure state of the system
$\lvert\psi_{t}(z^*,w^*)\rangle\equiv \langle
zw\rvert\Psi_{t}\rangle$:
\begin{eqnarray}
\frac{\partial}{\partial t}\lvert\psi_{t}\rangle\!&\!=\!&\!-iH_{\rm
sys}\lvert\psi_{t}\rangle
-\sum_{\lambda}\left[c_{\lambda}z_{\lambda}^*(t)\lvert\psi_{t}\rangle
+c_{\lambda}^\dagger w_{\lambda}^*(t)\lvert\psi_{t}\rangle\right]
\nonumber\\
&&\!-\sum_{\lambda}c_{\lambda}^\dagger\int_{0}^t\alpha_{\lambda
1}(t-s) \frac{\delta}{\delta
z_{\lambda}^*(s)}\lvert\psi_{t}\rangle ds \nonumber\\
&&\!-\sum_{\lambda}c_{\lambda}\int_{0}^t\alpha_{\lambda 2}(t-s)
\frac{\delta}{\delta w_{\lambda}^*(s)}\lvert\psi_{t}\rangle ds,
\label{QSD-1}
\end{eqnarray}
which initiates from $\lvert\psi_{t=0}(z^*,w^*)\rangle=\lvert
\varphi_{0}\rangle$. Here the noise functions $z_{\lambda}^*(t)$ and
$w_{\lambda}^*(t)$ are defined as
\begin{align}
&z_{\lambda}^*(t)=
-i\sum_{k}\sqrt{1-\bar{n}_{\lambda k}}
g_{\lambda k}^*z_{\lambda k}^*e^{i\omega_{\lambda k}t}, \nonumber\\
&w_{\lambda}^*(t)= -i\sum_{k}\sqrt{\bar{n}_{\lambda k}}g_{\lambda k}w_{\lambda
k}^*e^{-i\omega_{\lambda k}t}.
\end{align}
The temperature-dependent environment correlation functions are
\begin{eqnarray}
\alpha_{\lambda1}(t-s)\!&\!\equiv\!&\!\mathcal{M}\{z_{\lambda}(t)z_{\lambda}^*(s)\}\nonumber\\
\!&\!=\!&\!\int
d\omega[1-\bar{n}_{\lambda}(\omega)]J_{\lambda}(\omega)e^{-i\omega(t-s)},\nonumber\\
\alpha_{\lambda2}(t-s)\!&\!\equiv\!&\!\mathcal{M}\{w_{\lambda}(t)w_{\lambda}^*(s)\}\nonumber\\
\!&\!=\!&\!\int
d\omega\bar{n}_{\lambda}(\omega)J_{\lambda}(\omega)e^{i\omega(t-s)},
\end{eqnarray}
where $\mathcal{M}\{\cdot\}$ denotes the statistical mean over all
noise variables: $\mathcal{M}\{\cdot\}\equiv\int
e^{-z^*z-w^*w}\{\cdot\} d^2z d^2w$.
Introducing $O$-operators by
\begin{eqnarray}
\frac{\delta}{\delta
z_{\lambda}^*(s)}\lvert\psi_{t}(z^*,w^*)\rangle\!&\!=\!&\!O_{\lambda
1}(t,s,z^*,w^*)\lvert\psi_{t}(z^*,w^*)\rangle, \nonumber\\
\frac{\delta}{\delta
w_{\lambda}^*(s)}\lvert\psi_{t}(z^*,w^*)\rangle\!&\!=\!&\!O_{\lambda
2}(t,s,z^*,w^*)\lvert\psi_{t}(z^*,w^*)\rangle,~~~
\end{eqnarray}
we can write the
NMQSD equation in a time-local form:
\begin{eqnarray}
\frac{\partial}{\partial t}\lvert\psi_{t}\rangle\!&\!=\!&\!-iH_{\rm
sys}\lvert\psi_{t}\rangle-\sum_{\lambda}
\left[c_{\lambda}z_{\lambda}^*(t)+c_{\lambda}^\dagger
w_{\lambda}^*(t)\right.\nonumber\\
&&\!\left.+c_{\lambda}^\dagger\bar{O}_{\lambda
1}(t,z^*,w^*)+c_{\lambda}\bar{O}_{\lambda
2}(t,z^*,w^*)\right]\lvert\psi_{t}\rangle,~~~~~~\label{QSD-2}
\end{eqnarray}
where $\bar{O}_{\lambda n}\equiv\int_{0}^tds\alpha_{\lambda
n}(t-s)O_{\lambda n}(t,s,z^*,w^*)$, $n=1,2$.
With the consistency conditions
\begin{equation}
\frac{\partial}{\partial t}
\frac{\delta\lvert\psi_{t}\rangle}{\delta z_{\lambda}^*(s)}
=\frac{\delta}{\delta z_{\lambda}^*(s)}
\frac{\partial\lvert\psi_{t}\rangle}{\partial t},~~~
\frac{\partial}{\partial
t}\frac{\delta\lvert\psi_{t}\rangle}{\delta w_{\lambda}^*(s)}
=\frac{\delta}{\delta
w_{\lambda}^*(s)}\frac{\partial\lvert\psi_{t}\rangle}{\partial t},
\end{equation}
as well as the initial conditions $O_{\lambda
1}(t,s,z^*,w^*)\lvert_{t=s}=c_{\lambda}$ and $O_{\lambda
2}(t,s,z^*,w^*)\lvert_{t=s}= c_{\lambda}^\dagger$, we obtain the
equations of motion for the $O$-operators:
\begin{eqnarray}
\frac{\partial O_{\lambda n}}{\partial t}\!&\!=\!&\! [-iH_{\rm
sys}-\sum_{\lambda'}(c_{\lambda'}^\dagger\bar{O}_{\lambda' 1}
+c_{\lambda'}\bar{O}_{\lambda' 2}), O_{\lambda n}]+Q_n\nonumber\\
&&\!+\sum_{\lambda'}\left(\{c_{\lambda'},O_{\lambda n}\}
z_{\lambda'}^*(t)+\{c_{\lambda'}^\dagger,O_{\lambda n}\}
w_{\lambda'}^*(t)\right),\nonumber\\ \label{EOM-O}
\end{eqnarray}
where the square and curly brackets denote the commutator and
anticommutator, respectively, and
\begin{equation}
Q_n=c_{L}^\dagger\frac{\delta\bar{O}_{L
1}}{\delta\Lambda_n}+c_{R}^\dagger\frac{\delta\bar{O}_{R
1}}{\delta\Lambda_n}+c_{L}\frac{\delta\bar{O}_{L
2}}{\delta\Lambda_n}+c_{R}\frac{\delta\bar{O}_{R
2}}{\delta\Lambda_n},
\end{equation}
with $\Lambda_1=z_{\lambda}^*(s)$, and $\Lambda_2=w_{\lambda}^*(s)$.
\section{Master equation}
The reduced density operator of an
open quantum system by tracing over the environmental degrees of
freedom can be obtained by taking the statistical mean for a density
operator related to the state $\lvert\psi_{t}(z^*,w^*)\rangle$:
\begin{equation}
\rho_{t}={\rm Tr}_{\rm
env}\lvert\Psi_{t}\rangle\langle\Psi_{t}\rvert
=\mathcal{M}\{P_t\},
\end{equation}
where
$P_t\equiv\lvert\psi_{t}(z^*,w^*)\rangle\langle\psi_{t}(-z,-w)\rvert$.
Using the relation
\begin{eqnarray}
\frac{\partial P_{t}}{\partial t}\!&\!=\!&\!\frac{\partial
\lvert\psi_{t}(z^*,w^*)\rangle}{\partial
t}\langle\psi_{t}(-z,-w)\rvert \nonumber\\
&&\!+
\lvert\psi_{t}(z^*,w^*)\rangle\frac{\partial\langle\psi_{t}(-z,-w)\rvert}{\partial
t},
\end{eqnarray}
and Eq.~(\ref{QSD-2}), we derive the following non-Markovian
master equation:
\begin{eqnarray}
\frac{\partial\rho_{t}}{\partial t}\!&\!=\!&\! -i[H_{\rm
sys},\rho_{t}]+
\sum_{\lambda}\left([c_{\lambda},\mathcal{M}\{P_{t}\bar{O}_{\lambda 1}^\dagger(t,-z,-w)\}] \right.\nonumber\\
&&\!-[c_{\lambda}^\dagger,\mathcal{M}\{\bar{O}_{\lambda 1}(t,z^*,w^*)P_{t}\}]\nonumber\\
&&\!-[c_{\lambda},\mathcal{M}\{\bar{O}_{\lambda
2}(t,z^*,w^*)P_{t}\}] \nonumber\\
&&\!\left.+[c_{\lambda}^\dagger,\mathcal{M}\{P_{t}\bar{O}_{\lambda
2}^\dagger(t,-z,-w)\}]\right). \label{g-master}
\end{eqnarray}
This master equation is derived non-perturbatively, so it applies
even for a strong coupling between the system and the environments.
Moreover, in addition to trace preserving, it also preserves the
positivity and hermiticity.
In the Markovian limit, there are
\begin{eqnarray}
\alpha_{\lambda
1}(t-s)\!&\!\rightarrow\!&\!(1-\bar{n}_{\lambda})\Gamma_{\lambda}\delta(t-s),\nonumber\\
\alpha_{\lambda 2}(t-s)\!&\!\rightarrow\!&\!\bar{n}_{\lambda}\Gamma_{\lambda}\delta(t-s),
\end{eqnarray}
where $\Gamma_{\lambda}= 2\pi\rho_{\lambda}\lvert
g_{\lambda}\rvert^2$, with $\lambda=L$ ($R$), is the electron
tunneling rate between the system and the left (right) lead. Also,
the time-integrated $O$-operators become
\begin{eqnarray}
\bar{O}_{\lambda 1}\!&\!\rightarrow\!&\!\frac{1}{2}\Gamma_{\lambda}(1-\bar{n}_{\lambda})c_{\lambda},\nonumber\\
\bar{O}_{\lambda 2}\!&\!\rightarrow\!&\!
\frac{1}{2}\Gamma_{\lambda}\bar{n}_{\lambda}c_{\lambda}^\dagger.
\end{eqnarray}
Therefore, the master equation (\ref{g-master}) is reduced to
\begin{eqnarray}
\frac{\partial\rho_{t}}{\partial t}\!&\!=\!&\! -i[H_{\rm sys},\rho_{t}]\nonumber\\
&&\!+\sum_{\lambda}\frac{\Gamma_{\lambda}}{2}\left[\bar{n}_{\lambda}
(2c_{\lambda}^\dagger \rho_{t}c_{\lambda}
- c_{\lambda}c_{\lambda}^\dagger\rho_{t}- \rho_{t}c_{\lambda}c_{\lambda}^\dagger)\right.\nonumber\\
&&\!+\left.(1-\overline{n}_{\lambda})
(2c_{\lambda}\rho_{t}c_{\lambda}^\dagger- c_{\lambda}^\dagger
c_{\lambda}\rho_{t}- \rho_{t}c_{\lambda}^\dagger
c_{\lambda})\right].~~~ \label{m-master}
\end{eqnarray}
It is clear that this Markov master equation has a Lindblad form.
\section{Application to quantum-dot systems}
Below we apply our NMQSD approach to single and double quantun-dot systems.
\subsection{Single quantum dot}
Suppose that the single quantum dot is
in the strong Coulomb blockade regime, so that only one electron is
allowed therein. The Hamiltonian of the system is written as
\begin{equation}
H_{\rm sys}=\omega_{0}c^\dagger c,
\end{equation}
and $c_{L}=c_{R}=c$ for $H_{\rm int}$ in Eq. (\ref{int}).
The non-Markovian master equation is exactly derived as (see Appendix A)
\begin{align}
\frac{\partial\rho_{t}}{\partial t}=&-i[H_{\rm sys},\rho_{t}]
+\Gamma_{1}(t)[c,\rho_{t}c^\dagger]+
\Gamma_{2}(t)[c,c^\dagger\rho_{t}]\nonumber\\
&+ \Gamma_{1}^*(t)[c\rho_{t},c^\dagger]+
\Gamma_{2}^*(t)[\rho_{t}c,c^\dagger], \label{dot-master}
\end{align}
with time-dependent rates
\begin{equation}
\Gamma_{j}(t)=\int_{0}^t[\alpha_{1}(s-t)A_j(t,s)-\alpha_{2}(t-s)B_j(t,s)]ds,
\label{coefficient a}
\end{equation}
where $\alpha_{j}(t)=\alpha_{Lj}(t)+\alpha_{Rj}(t)$; $A_j(t,s)$ and
$B_j(t,s)$ are determined by the integro-differential equations:
\begin{align}
&(\frac{\partial}{\partial s}-i\omega_{0})A_j(t,s)+
\int_{0}^s\beta(s-s')A_j(t,s')ds'= U(t,s)\nonumber\\
&(\frac{\partial}{\partial s}-i\omega_{0})B_j(t,s)+
\int_{0}^s\beta(s-s')B_j(t,s')ds'= V(t,s),
\end{align}
with
\begin{eqnarray}
&&\beta(s-s')\equiv\alpha_{1}(s'-s)+\alpha_{2}(s-s'),\nonumber\\
&&U(t,s)\equiv\int_{0}^t\alpha_{2}(s-s')h(t,s')ds',\nonumber\\
&&V(t,s)\equiv\int_{0}^t\alpha_{1}(s'-s)h(t,s')ds',
\end{eqnarray}
and the final
conditions at $s=t$: $A_1(t,t)=B_1(t,t)=1$, and
$A_2(t,t)=B_2(t,t)=0$. Here $h(t,s)$ satisfies the equation
\begin{equation}
(\frac{\partial}{\partial s}-i\omega_{0})h(t,s)-
\int_{s}^t\beta(s-s')h(t,s')ds'= 0,
\end{equation}
with the final condition $h(t,t)=1$.
In the Markovian limit, there are
\begin{eqnarray}
\Gamma_{1}(t)\!&\!\rightarrow\!&\!\frac{1}{2}[1-\bar{n}_{L}(\omega_{0})]\Gamma_{L}
+\frac{1}{2}[1-\bar{n}_{R}(\omega_{0})]\Gamma_{R}, \nonumber\\
\Gamma_{2}(t)\!&\!\rightarrow\!&\!-\frac{1}{2}\bar{n}_{L}(\omega_{0})\Gamma_{L}
-\frac{1}{2}\bar{n}_{R}(\omega_{0})\Gamma_{R}.
\end{eqnarray}
Let us consider the
zero-temperature case with $\bar{n}_{\lambda}(\omega_{0})\rightarrow
\theta(\mu_{\lambda}- \omega_{0})$, were $\theta$ is the Heaviside
step function. If the single-dot level $\omega_0$ lies within the
energy window $\mu_{L}>\omega_{0}>\mu_{R}$, the master equation
(\ref{dot-master}) reduces to
\begin{eqnarray}
\frac{\partial}{\partial t}\rho_{t}\!&\!=\!&\!-i\omega_{0}[c^\dagger
c,\rho_{t}]+ \frac{1}{2}\Gamma_L(2c^\dagger\rho_{t}c-
cc^\dagger\rho_{t}- \rho_{t}cc^\dagger)\nonumber\\
&&\!+\frac{1}{2}\Gamma_R(2c\rho_{t}c^\dagger- c^\dagger c\rho_{t}-
\rho_{t}c^\dagger c).
\end{eqnarray}
With the basis state $|0\rangle$ ($|1\rangle$) which denotes an
empty (occupied) dot, it follows that the density matrix elements
satisfy
\begin{eqnarray}
&&\dot{\rho}_{00}= -\Gamma_{L}\rho_{00}+ \Gamma_{R}\rho_{11},\nonumber\\
&&\dot{\rho}_{11}= \Gamma_{L}\rho_{00}- \Gamma_{R}\rho_{11},\nonumber\\
&&\dot{\rho}_{10}= -(i\omega_{0}+ \Gamma_{L}+ \Gamma_{R})\rho_{10},
\end{eqnarray}
which are exactly the rate equations obtained by Gurvitz and
Prager~\cite{Gurvitz}.
\begin{figure}
\includegraphics[width=3.4in,
bbllx=52,bblly=335,bburx=536,bbury=619]{fig1.eps} \caption{(Color
online) (a)~Real and (b)~imaginary parts of the time-dependent
coefficient $\Gamma_{1}(t)$ in the non-Markovian master equation of
a single quantum dot system, where $\Gamma_{L}=\Gamma_{R}= 100\mu
eV$, $\omega_{0}=50\mu eV$, and $t_{0}=2\pi/\omega_{0}$. }
\label{fig1}
\end{figure}
Figure~\ref{fig1} presents both real and imaginary parts of the
time-dependent coefficients $\Gamma_{1}(t)$ in
Eq.~(\ref{dot-master}). For simplicity, we consider the large bias
regime (i.e., $\mu_{L}\gg\omega_{0}\gg\mu_{R}$) and a zero
temperature for fermionic environments. The tunneling rates are
chosen in the symmetric case of $\Gamma_{L}=\Gamma_{R}=\Gamma$, so
that $\Gamma_{2}(t)=- \Gamma_{1}(t)$. Moreover, the noise is modeled
as the Ornstein-Uhlenbeck process, and the related correlation
functions are then given by
\begin{eqnarray}
&&\alpha_{L2}(t,s)= \alpha_{R1}(t,s)=
\frac{\Gamma d}{2}e^{-d\lvert t-s\rvert},\nonumber\\
&&\alpha_{L1}(t,s)=\alpha_{R2}(t,s)= 0,
\end{eqnarray}
where $1/d$ characterizes the memory time of
each environment. In the Markovian limit with $d\rightarrow\infty$,
$\alpha_{L2}(t,s)=\alpha_{R1}(t,s)\rightarrow\Gamma\delta(t-s)$, and
$\Gamma_{1}(t)$ becomes time-dependent. Indeed, Fig.~\ref{fig1}
shows that $\Gamma_{1}$ oscillates with time $t$, but it quickly
decays to a constant for a large value of $d$.
\subsection{Double quantum dot (DQD)}
Suppose that the DQD is in the
strong Coulomb blockade regime, so that at most one electron is allowed in
each dot. The Hamiltonian of the DQD can be written as
\begin{equation}
H_{\rm sys}= \omega_{1}c_{1}^\dagger c_{1}+ \omega_{2}c_{2}^\dagger
c_{2}+ \Omega_{0}(c_{2}^\dagger c_{1}+ c_{1}^\dagger c_{2}),
\end{equation}
where $\Omega_{0}$ denotes the inderdot coupling. For $H_{\rm int}$
in Eq.~(\ref{int}), $c_{L}= c_{1}$, and $c_{R}= c_{2}$. The exact
non-Markovian master equation is given by
\begin{align}
\frac{\partial\rho_{t}}{\partial t} =&-i[H_{\rm sys},\rho_{t}]
+\left\{\left(\Gamma_{L1}(t)[c_{1},\rho_{t}c_{1}^\dagger]
+\Gamma_{L2}(t)[c_{1},c_{1}^\dagger\rho_{t}]\right.\right.\nonumber\\
&+\Gamma_{L3}(t)[c_{1},\rho_{t}c_{2}^\dagger]
+\Gamma_{L4}(t)[c_{1},c_{2}^\dagger\rho_{t}]\nonumber\\
&+\Gamma_{R1}(t)[c_{2},\rho_{t}c_{1}^\dagger]
+\Gamma_{R2}(t)[c_{2},c_{1}^\dagger\rho_{t}]\nonumber\\
&+\left.\left.\Gamma_{R3}(t)[c_{2},\rho_{t}c_{2}^\dagger]
+\Gamma_{R4}(t)[c_{2},c_{2}^\dagger\rho_{t}]\right)+ {\rm H.c.}\right\}, \label{DQD-master}
\end{align}
with time-dependent coefficients
\begin{equation}
\Gamma_{\lambda j}(t)= \int_{0}^t[\alpha_{\lambda 1}(s-t)A_{\lambda
j}(t,s)-\alpha_{\lambda 2}(t-s)B_{\lambda j}(t,s)]ds,
\end{equation}
where $A_{\lambda j}(t,s)$ and $B_{\lambda j}(t,s)$ satisfy a set of
integro-differential equations (see Appendix B), with the final
conditions: $A_{L1}(t,t)=A_{R3}(t,t)=B_{L2}(t,t)=B_{R4}(t,t)=1$, and
$A_{\lambda j}(t,t)=0$, $B_{\lambda j}(t,t)=0$ for other $\lambda$
and $j$. A similar non-Markovian master equation was also obtained
using the Feynman-Vernon influence functional path-integral
method~\cite{Zhang1}.
In the Markovian limit, there are
\begin{eqnarray}
\Gamma_{L1}(t)\!&\!\rightarrow\!&\! \frac{1}{2}
[1-\bar{n}_{L}(\omega_{1})]\Gamma_{L},~~ \Gamma_{R1}\rightarrow
0; \nonumber\\
\Gamma_{L2}(t)\!&\!\rightarrow\!&\! -\frac{1}{2}\bar{n}_{L}(\omega_{1})
\Gamma_{L},~~ \Gamma_{R2}\rightarrow 0; \nonumber\\
\Gamma_{L3}(t)\!&\!\rightarrow\!&\! 0,~~ \Gamma_{R3}\rightarrow \frac{1}{2}[1-
\bar{n}_{R}(\omega_{2})]\Gamma_{R}; \\
\Gamma_{L4}(t)\!&\!\rightarrow\!&\! 0,~~ \Gamma_{R4}\rightarrow
-\frac{1}{2}\bar{n}_{R}(\omega_{2})]\Gamma_{R}. \nonumber
\end{eqnarray}
We also consider
the zero-temperature case with
$\bar{n}_{\lambda}(\omega_n)\rightarrow \theta(\mu_{\lambda}-
\omega_n)$, and the two single-dot levels of the DQD all lie within
the energy window $\mu_{L}>\omega_n>\mu_{R}$, where $n=1,2$. We use
$|l\rangle$, $l=0,1,2$ and 3, to denote the states with both dots
empty, the left dot occupied, the right dot occupied, and both dots
occupied, respectively. From Eq.~(\ref{DQD-master}), it follows that
the master equations for density matrix elements are reduced to
\begin{align}
&\dot{\rho}_{00}=-\Gamma_{L}\rho_{00}+ \Gamma_{R}\rho_{22},~~~\dot{\rho}_{33}=\Gamma_{L}\rho_{22}-\Gamma_{R}\rho_{33},\nonumber\\
&\dot{\rho}_{11}=\Gamma_{L}\rho_{00}+\Gamma_{R}\rho_{33}+i\Omega_{0}(\rho_{12}- \rho_{21}),\nonumber\\
&\dot{\rho}_{22}=-(\Gamma_{L}+\Gamma_{R})\rho_{22}- i\Omega_{0}(\rho_{12}- \rho_{21}),\\
&\dot{\rho}_{12}=-i(\omega_{1}- \omega_{2})\rho_{12}+
i\Omega_{0}(\rho_{11}- \rho_{22})-
\frac{\Gamma_L+\Gamma_R}{2}\rho_{12},\nonumber
\end{align}
which are identical to the rate equations obtained in
\cite{Gurvitz}. For a DQD, both intradot and interdot Coulomb
repulsions can play an important role in the Coulomb-blockade effect
(see, e.g., \cite{jqyou2}). Thus, if both intradot and interdot
Coulomb repulsions are so strong that only one electron is allowed
in the whole DQD, the master equations for density matrix elements
are reduced to
\begin{align}
&\dot{\rho}_{00}=-\Gamma_{L}\rho_{00}+ \Gamma_{R}\rho_{22}\nonumber\\
&\dot{\rho}_{11}=\Gamma_{L}\rho_{00}+ i\Omega_{0}(\rho_{12}- \rho_{21})\nonumber\\
&\dot{\rho}_{22}=-\Gamma_{R}\rho_{22}- i\Omega_{0}(\rho_{12}- \rho_{21})\\
&\dot{\rho}_{12}= -i(\omega_{1}-\omega_{2})\rho_{12}+
i\Omega_{0}(\rho_{11}- \rho_{22})-
\frac{\Gamma_{R}}{2}\rho_{12},\nonumber
\end{align}
which are exactly the rate equations obtained in \cite{Stoof}.
\section{Conclusion}
We have developed an NMQSD method to study
the dynamics of an open quantum system in fermionic baths. We not
only obtain the NMQSD equation for quantum states of the system, but
also derive the non-Markovian master equation. This non-Markovian
approach is formulated in a non-perturbative manner and it applies
even for a strong coupling between the system and the fermionic
baths. Moreover, as useful examples, we have applied this NMQSD
method to single and double quantum-dot systems.
{\it Note added}: After finishing this work, we were aware of a
closely related work in~\cite{Yu3}, which also investigated an open
quantum system in fermionic baths by using a quantum state diffusion
approach.
\begin{acknowledgments}
This work was supported by the National Basic Research Program of
China Grant No. 2009CB929302 and the National Natural Science
Foundation of China Grant No. 91121015.
\end{acknowledgments}
|
{
"timestamp": "2013-05-10T02:01:06",
"yymm": "1203",
"arxiv_id": "1203.2217",
"language": "en",
"url": "https://arxiv.org/abs/1203.2217"
}
|
\section{Introduction}
In current and future cellular networks, handling interference in
the network is one of the most critical problems. Among the many
ways of handling interference, MIMO antenna techniques and base
station cooperation are considered as the key technologies to the
interference problem. Indeed, the 3GPP Long-Term
Evolution-Advanced considers the base station cooperation and
MIMO techniques to mitigate inter-cell interference under the name
of Coordinated Multipoint (CoMP) \cite{3GPP:10,
Sawahashietal:10WCM}. Mathematically, when each mobile station has
a single receive antenna and data is not shared among base
stations, the system is modelled as a MISO interference channel
(IC), and extensive research has been conducted on beam design for
this MISO IC, especially under the assumption of practical linear
beamforming treating interference as noise. First, Jorswieck {\em
et al.} investigated the structure of optimal beam vectors
achieving Pareto boundary points of the achievable rate region of
the MISO IC with linear beamforming
\cite{Jorswieck&Larsson&Danev:08SP} and showed that any
Pareto-optimal beam vector at each transmitter is a normalized
convex combination of the ZF beam vector and matched-filtering
(MF) (i.e., maximal ratio transmission) beam vector in the case of
two users and a linear combination of the channel vectors from the
transmitter to all receivers in the general case of an arbitrary
number of users. The result is extended in
\cite{Mochaourab&Jorswieck:11SP} to general MISO interference
networks with arbitrary utility functions having monotonic
property. Moreover, the parameterization for the Pareto-optimal
beam vector is compressed from $K(K-1)$ complex numbers
\cite{Jorswieck&Larsson&Danev:08SP} to $K(K-1)$ real numbers. In
addition to these results, other interesting works for MISO ICs
include the consideration of imperfect CSI
\cite{Lindblom&Larsson&Jorswieck:10WCOM}, shared data
\cite{Bjornson&etal:10SP}, second-order cone programming
\cite{Qiu&Zhang&Luo&Cui:11SP}, etc. Although these works provide
significant theoretical insights into the optimal beam structure
and parameterization of Pareto-optimal beam vectors, it is not
easy to use these results to design an optimal beam vector in the
real-world systems, and the beam design problem in the general
case still remains as a non-trivial problem practically.
With a sufficient number of transmit antennas, the simplest beam design method for base station coordination is ZF, which perfectly eliminates interference leakage to undesired receivers. However, it is well known that the ZF method is not optimal in the sense of sum data rate or Pareto-boundary achievability, and there have been several ideas to enhance the ZF beam design method. In the case of multi-user MISO/MIMO broadcast channels, the regularized channel inversion (RCI) \cite{Peel&Hochwald&Swindlehurst:05Com} and the signal-to-leakage-plus-noise (SLNR) method \cite{Sadek&Tarighat&Sayed:07TW} were proposed for this purpose. In particular, the SLNR method maximizes the ratio of signal power (to the desired receiver) to leakage (to undesired receivers) plus noise power, and its solution is given by solving a generalized
eigenvalue problem. The SLNR method can easily be adapted to the
MISO/MIMO IC. Recently, Zakhour and Gesbert rediscovered this method in the context of MISO IC under the name of the virtual signal-to-interference-plus-noise (SINR) method, and have further (and more importantly) shown that this method can achieve any point on the Pareto boundary theoretically, but practically can achieve one uncontrolled point
on the Pareto boundary of the achievable rate region in the case
of two\footnote{It can be shown that the virtual SINR (or SLNR)
method can theoretically achieve any Pareto-optimal point in the
general MISO IC case, too. See the appendix of \cite{Park&Lee&Sung&Yukawa:12Arxiv}.} users
\cite{Zakhour&Gesbert:09WSA},\cite{Zakhour&Gesbert:10WC}.
Another way of generalizing ZF in MISO IC was proposed by relaxing
the ZF leakage constraints to undesired users in
\cite{Shang&Chen&Poor:11IT}, \cite{Zhang&Cui:10SP},
\cite{Lee&Park&Sung:11WCSP}. First, Shang {\em et al.} showed that
all boundary points of the achievable rate region of MISO IC with
single-user decoding can be obtained by linear beamforming
\cite{Shang&Chen&Poor:11IT}, by converting the non-convex weighted
sum rate maximizing precoder design problem into a set of separate
convex problems by taking a lower bound on the achievable rate of
each user under the relaxed ZF (RZF) framework. This method was
further investigated by Zhang and Cui \cite{Zhang&Cui:10SP}, who
showed that separate rate optimization under the RZF framework
with a set of well-chosen interference leakage levels to undesired
users is Pareto-optimal for MISO ICs in addition to being sum-rate
optimal. In \cite{Lee&Park&Sung:11WCSP}, Lee {\em et al.} extended
the RZF framework to the case of MIMO IC. In this RZF beamforming
framework, each transmitter maximizes its own rate under
interference leakage constraints to undesired receivers. The idea
is based on the simple observation that the ZF beam design method
overreacts to inter-cell interference by completely nulling
out the interference. Most receivers (i.e., mobile stations) that
are affected by inter-cell interference are cell-edge users, and
thus, thermal noise remains even if the inter-cell interference is
completely removed. Thus, it is unnecessary to completely
eliminate the inter-cell interference and it is sufficient to
limit the inter-cell interference to a certain level comparable to
that of the thermal noise. By relaxing ZF interference
constraints, we do not need the condition that the number of
transmit antenna is larger than or equal to that of receivers and
have a larger feasible set yielding a larger rate than that of the
ZF scheme. In this paper, we explore and develop this RZF idea
fully in several aspects to provide a useful design paradigm for
coordinated beamforming (CB) for current and future cellular
networks. The contributions of the paper is summarized as follows:
\noindent $\bullet$ In the MISO IC case, a new structural representation of optimal beam vector for RZF coordinate beamforming is derived.
\noindent $\bullet$ In the MISO IC case, based on the new structural
representation, the {\it sequential
orthogonal projection combining (SOPC) method} for the RZF beam
design is proposed. In the case of $K=3$, an approximate
closed-form solution is provided.
\noindent $\bullet$ In the RZF framework, the allowed interference leakage
levels to undesired receivers at each transmitter are design
parameters, and the rate-tuple is controlled by controlling these
interference leakage levels. A centralized algorithm and a fully distributed heuristic algorithm are provided to control the location of the designed
rate-tuple (roughly) along the Pareto boundary of the achievable rate
region.
The controllability of rate is a desirable feature in network
operation since the required data rate of each
transmitter-receiver pair may be different from those of others in
practice, as in an example that one user is a voice user and the
others are high rate data users.
\noindent $\bullet$ Finally, the RZF CB (RZFCB) is extended to the MIMO IC case.
In the MIMO case, a new lower bound on
each user's rate is derived to decompose the beam design
problem into separate problems at different transmitters, and the
projected gradient method \cite{goldstein64} is adopted to solve
the MIMO RZFCB problem.
\noindent \textbf{Notations and Organization}
In this paper, we will make use of standard notational
conventions. Vectors and matrices are written in boldface with
matrices in capitals. All vectors are column vectors. For a matrix
${\bf A}$, ${\bf A}^H$, $\|{\bf A}\|$, $\|{\bf A}\|_F$, $\mbox{tr}({\bf A})$, and
$|{\bf A}|$ indicate the Hermitian transpose, 2-norm, Frobenius
norm, trace, and determinant of ${\bf A}$, respectively, and
${\cal C}({\bf A})$ denotes the column space of
${\bf A}$. ${\bf I}_n$ stands for the identity matrix of size $n$
(the subscript is omitted when unnecessary).
${\bf \Pi}_{{\bf A}}={\bf A}({\bf A}^H{\bf A})^{-1}{\bf A}^H$ represents the
orthogonal projection onto ${\cal C}({\bf A})$ and
${\bf \Pi}_{{\bf A}}^\perp={\bf I} - {\bf \Pi}_{{\bf A}}$. For matrices ${\bf A}$ and
${\bf B}$, ${\bf A} \ge {\bf B}$ means that
${\bf A}-{\bf B}$ is positive semi-definite. $[{\bf a}_1,\cdots,{\bf a}_L]$ or
$[{\bf a}_i]_{i=1}^L$ denotes the matrix composed of vectors
${\bf a}_1,\cdots,{\bf a}_L$. ${\bf x}\sim\mathcal{CN}(\hbox{\boldmath$\mu$\unboldmath},\hbox{$\bf \Sigma$})$ means that
${\bf x}$ is circular-symmetric complex Gaussian-distributed with mean vector $\hbox{\boldmath$\mu$\unboldmath}$ and covariance matrix $\hbox{$\bf \Sigma$}$. ${\mathbb{R}}$,
${\mathbb{R}}_+$, and ${\mathbb{C}}$ denote the sets of real
numbers, non-negative real numbers, and complex numbers,
respectively. For a set $A$, $|A|$ represents the cardinality
of the set.
The remainder of this paper is organized as follows.
The system model and the preliminaries are provided in Section \ref{sec:systemmodel}. In
Section \ref{sec:MISO_IC_RZF}, the RZFCB in MISO ICs is formulated,
and its solution structure and a fast algorithm for RZFCB are
provided. In Section \ref{sec:rate_control}, the rate-tuple control
problem under the RZFCB framework is considered and two approaches
are proposed to control the designed rate-tuple. The RZFCB problem
in MIMO ICs is considered in Section \ref{sec:MIMO_IC}, followed by
conclusions in Section \ref{sec:conclusion}.
\section{System Model and Preliminaries}
\label{sec:systemmodel}
In this paper, we consider a multi-user interference channel with
$K$ transmitter-receiver pairs. In the first part of the paper, we restrict
ourselves to the case that the transmitters are equipped with $N$
antennas and each receiver is equipped with one receive antenna
only. In this case, the received signal at receiver $i$ is given by
\begin{equation} \label{eq:rec_signal}
y_{i} = {\bf h}_{ii}^H{\bf v}_{i} s_i + \sum\limits_{j=1, j \neq i}^K
{\bf h}_{ij}^H{\bf v}_j s_j + n_i,
\end{equation}
where ${\bf h}_{ij}$ denotes the $N\times 1$ (conjugated) channel
vector from transmitter $j$ to receiver $i$, and ${\bf v}_j$ and
$s_j$ are the $N \times 1$ beamforming vector and the scalar
transmit symbol at transmitter $j$, respectively.
We assume that the transmit symbols are from a Gaussian code book with unit
variance, the additive noise $n_i$ is from ${\mathcal{CN}}(0, \sigma_i^2)$, and each transmitter has a transmit power constraint, $\|{\bf v}_i\|^2 \leq P_i$, $i = 1, \cdots, K$.
The first term on the right-hand side (RHS) of
\eqref{eq:rec_signal} is the desired signal and the second term
represents the sum of interference from $K-1$ undesired
transmitters. Under single-user decoding at each receiver treating interference as noise, for a given set of
beamforming vectors $\{{\bf v}_1,\cdots,{\bf v}_K\}$ and a channel
realization $\{{\bf h}_{ij}\}$, the rate of receiver $i$ is given by
\begin{equation} \label{eq:R_iOneUser}
R_i({\bf v}_1,\cdots,{\bf v}_K)
=
\log\left(1+\frac{|{\bf h}_{ii}^H{\bf v}_i|^2}
{\sigma_i^2+ \sum_{j\neq i} |{\bf h}_{ij}^H{\bf v}_j|^2} \right).
\end{equation}
Then, for the given channel realization, the achievable rate
region of the MISO IC with transmit beamforming and single-user decoding is defined as
the union of the rate-tuples that can be achieved by all possible
combinations of beamforming vectors under the power constraints:
\begin{equation} \label{eq:rate_region}
{\mathcal{R}} :=
\hspace{-1.4em}
\bigcup_{\left\{\substack{ {\bf v}_i:{\bf v}_i\in{\mathbb{C}}^N, \\
\|{\bf v}_i\|^2\leq P_i,\ 1\leq i\leq K }\right\}}
\hspace{-1.4em}
(R_1({\bf v}_1, \cdots, {\bf v}_K),\ \cdots,\ R_K({\bf v}_1, \cdots, {\bf v}_K)).
\end{equation}
The outer boundary of the rate region ${\mathcal{R}}$ is called the
$\textit{Pareto boundary}$ of ${\cal R}$ and
it consists of the rate-tuples for which the rate of any one user cannot
be increased without decreasing the rate of at least one other user \cite{Jorswieck&Larsson&Danev:08SP}.
At each transmitter, the interference to undesired receivers can be
eliminated completely by ZF CB (ZFCB).
Due to its simplicity and fully distributed nature, there has been extensive research on ZFCB, e.g., \cite{Spencer&Swindlehurst&Haardt:04SP,
Shim&Kwak&Heath&Andrews:08WC,
Somekh&Simeone&BarNess&Haimovich&Shamai:09IT}. The best ZF
beamforming vector at transmitter $i$ can be obtained by solving the
following optimization problem:
\begin{align}
{\bf v}_i^* =& \mathop{\arg\max}_{{\bf v}_i \in\ {\mathbb{C}}^{N} }\ \
\log \left( 1+\frac{|{\bf h}_{ii}^H{\bf v}_{i}|^2}{\sigma_i^2} \right)
\label{eq:ZFCB} \\
&\mbox{subject to\ \ \ } |{\bf h}_{ji}^H {\bf v}_{i}| =
0, ~\forall~ j\ne i ~~~~\mbox{and} \quad \|{{\bf v}}_{i}\|^2 \le
P_i.\nonumber
\end{align}
Here, $|{\bf h}_{ji}^H {\bf v}_{i}| = 0$ is the ZF leakage constraint at
transmitter $i$ for receiver $j$. If $N \ge K$, the problem
\eqref{eq:ZFCB} has a non-trivial solution and the solution is
given by ${\bf v}_i^{ZF}=c{\bf \Pi}_{[{\bf h}_{1i},\cdots,{\bf h}_{i-1,i},{\bf h}_{i+1,i},\cdots,{\bf h}_{Ki}]}^\perp
{\bf h}_{ii}$ for some scalar $c$ satisfying the transmit power
constraint. In this paper, however, we do not assume that $N
\ge K$ necessarily as in the ZF beamforming, but assume that
{\em (A.1)} In the case of $N \ge K$, $\{{\bf h}_{ji}, j=1,\cdots,K\}$ are linearly independent for each $i$.
In the case of $N < K$, the element vectors of any subset of $\{{\bf h}_{ji}, j=1,\cdots,K\}$ with cardinality $N$ are linearly independent for each $i$.
\noindent Assumption {\em (A.1)} is almost surely satisfied for randomly realized channel vectors.
\section{RZF Coordinated Beamforming in MISO Interference Channels}
\label{sec:MISO_IC_RZF}
\subsection{Formulation}
Although the ZFCB provides an effective way to handling inter-cell
interference, the ZFCB is not optimal from the perspective of Pareto optimality, i.e., the rate tuples achieved by ZFCB are in the interior of the achievable rate region \cite{Larsson&Jorswieck:08JSAC}. and requires the condition $N \ge K$. As mentioned before, even with such complete interference nulling, there exists thermal noise at each receiver, and thus, a certain level of interference leakage comparable to the power of thermal noise can be allowed for better performance. In the MISO IC case, the RZF leakage constraint at transmitter $i$ for receiver $j$ is formulated as follows:
\begin{equation}\label{eq:RZF}
|{\bf h}_{ji}^H {\bf v}_{i} |^2 \le \alpha_{ji}\sigma_j^2, \quad \forall
i, j \neq i,
\end{equation}
where $\alpha_{ji} \ge 0$ is a constant\footnote{In the RZF
scheme, $\{\alpha_{ji}, j,i=1,\cdots,K, j\neq i\}$ are system design
parameters that should be designed properly for optimal
performance. The practical significance of the parameterization in terms of the interference leakage levels will be clear in Section \ref{subsec:distributedControl}.} that controls the allowed level of interference
leakage from transmitter $i$ to receiver $j$ relative to the
thermal noise level $\sigma_j^2$ at receiver $j$. When $\alpha_{ji}=0$ for all $j\neq i$, the RZF constraints reduce to the conventional ZF constraints. When $\alpha_{ji} >0$, on the other hand, the ZF constraints are relaxed to yield a larger feasible set for ${\bf v}_i$ than that associated with the ZF constraints and due to this relaxation the condition $N \ge K$ is not necessary anymore.
Under the RZF framework, the power of interference from undesired transmitters at receiver $i$ is upper bounded as
\begin{equation}\label{eq:interf_power}
\textstyle
\sum_{j=1,j\neq i}^K |{\bf h}_{ij}^H{\bf v}_{j}|^2 \le \sum_{j\neq i}
\alpha_{ij}\sigma_i^2 =: \epsilon_i\sigma_i^2.
\end{equation}
Therefore, a lower bound on the rate of user $i$
under RZF is obtained by using \eqref{eq:interf_power} as
\begin{equation}\label{eq:lower_sum_rate}
\log\bigg( 1+\frac{|{\bf h}_{ii}^H{\bf v}_i|^2}{\sigma_i^2+\sum_{j\neq i}
|{\bf h}_{ij}^H{\bf v}_j|^2} \bigg) \ge \log\bigg( 1+\frac{|
{\bf h}_{ii}^H{\bf v}_i |^2}{(1+\epsilon_i)\sigma_i^2} \bigg).
\end{equation}
The lower bound on the rate at each receiver does not depend on the
beamforming vectors of undesired transmitters and thus,
exploiting the RZF constraints, we can convert the intertwined coordinated beam design problem into a set of separate problems for different users based on the lower bound \cite{Shang&Chen&Poor:11IT}. The separate problem for each transmitter based on RZF is given as follows \cite{Shang&Chen&Poor:11IT,Zhang&Cui:10SP}:
\begin{problem}\label{prob:MISO_RZF_formulation1}
For each transmitter $i \in \{1,\cdots,K\}$,
\begin{eqnarray}
&\underset{{\bf v}_i}{\mbox{maximize}} ~ &
\log\bigg(1+\frac{|{\bf h}_{ii}^H{\bf v}_i|^2}{(1+\epsilon_i)\sigma_i^2}\bigg) \\
&\mbox{subject to} & |{\bf h}_{ji}^H{\bf v}_i|^2 \leq
\alpha_{ji}\sigma_j^2,
\qquad \forall j\neq i, \\
& & \|{\bf v}_i\|^2 \leq P_i.
\end{eqnarray}
\end{problem}
\vspace{0.5em} \noindent Then, due to the monotonicity of the
logarithm, Problem \ref{prob:MISO_RZF_formulation1} is equivalent
to the following problem:
\begin{problem}[The MISO RZFCB problem]
\label{prob:MISO_RZF_formulation2} For each transmitter $i \in
\{1,\cdots,K\}$,
\begin{eqnarray}
&\underset{{\bf v}_i}{\mbox{maximize}}~ &|{\bf h}_{ii}^H{\bf v}_i|^2 \\
&\mbox{subject to} & |{\bf h}_{ji}^H{\bf v}_i|^2 \leq \alpha_{ji}\sigma_j^2, \label{eq:Problem3RZFconst}
\qquad \forall j\neq i, \\
& & \|{\bf v}_i\|^2 \leq P_i.
\end{eqnarray}
\end{problem}
\noindent From now on, we will consider Problem
\ref{prob:MISO_RZF_formulation2} (the RZFCB problem) and refer to
the solution to Problem \ref{prob:MISO_RZF_formulation2} as the RZF
beamforming vector.
\subsection{The Optimality and Solution Structure of RZFCB in MISO Interference Channels}
\label{subsec:solution_structure}
In this subsection, we will investigate the optimality and
structure of the solution to Problem
\ref{prob:MISO_RZF_formulation2}. We start with the optimality of
the RZFCB scheme. Without inter-cell interference, it is optimal
for the transmitter to use the MF beam vector with
full transmit power. However, with inter-cell interference,
such a selfish strategy leads to poor performance due to large mutual interference \cite{Larsson&Jorswieck:08JSAC}. Thus, to enhance
the overall rate performance in the network, the beamforming
vector should be designed to be as close as possible to the MF
beam vector without giving too much interference to undesired
receivers, and this strategy is the RZFCB in Problem
\ref{prob:MISO_RZF_formulation2} (or Problem
\ref{prob:MISO_RZF_formulation1} equivalently). The optimality of
the RZFCB is given in the following theorem of Shang \textit{et al.} \cite{Shang&Chen&Poor:11IT} or Zhang and Cui \cite{Zhang&Cui:10SP}.
\begin{theorem}\cite{Zhang&Cui:10SP}\label{theo:pareto_achievability}
Any rate-tuple $(R_1,\cdots, R_K)$ on the Pareto boundary of the
achievable rate region defined in \eqref{eq:rate_region} can be
achieved by the RZFCB if the levels $\{\alpha_{ij}\sigma_i^2,\
\forall i,j\neq i\}$ of interference leakage
are properly
chosen.
\end{theorem}
\begin{proof}
See Proposition 3.2 in \cite{Zhang&Cui:10SP}.
\end{proof}
\noindent Surprisingly, the separate beam design based on the rate lower bound in Problem \ref{prob:MISO_RZF_formulation2} can
achieve any Pareto-optimal point of the achievable rate region if the interference relaxation parameters are well chosen.\footnote{The beamforming vectors from Problem \ref{prob:MISO_RZF_formulation2} are necessary to achieve any point on the Pareto boundary but not sufficient. Not any choice of parameters $\{\alpha_{ij}\}$ leads to a point on the Pareto boundary. }
It was also shown that Problem 2 and the approach in \cite{Mochaourab&Jorswieck:11SP} are two different approaches to the same multi-objective optimization problem \cite{Vazquez&Neira&Lagunas:12WSA}.
Due to Theorem \ref{theo:pareto_achievability}, in the MISO IC
case, the remaining problems for the RZFCB are {\it i)} {\em to
construct an efficient algorithm to solve the RZFCB problem for given
$\{\alpha_{ij}\sigma_i^2,\ \forall i,j\neq i\}$} and {\it ii)} {\em to
devise a method to design $\{\alpha_{ij}\sigma_i^2,\ \forall
i,j\neq i\}$ for controlling the location of the rate-tuple along
the Pareto boundary of the achievable rate region.} We will
consider Problem \ref{prob:MISO_RZF_formulation2} for given
$\{\alpha_{ij}\sigma_i^2,\ \forall i,j\neq i\}$ here and will
consider the rate control problem in the next section.
First, we will derive an
efficient algorithm for obtaining a good approximate solution to
Problem \ref{prob:MISO_RZF_formulation2} for given
$\{\alpha_{ij}\sigma_i^2,\ \forall i,j\neq i\}$. To do this, we
need to investigate the solution structure of the RZFCB problem.
Instead of solving Problem \ref{prob:MISO_RZF_formulation1} as in
\cite{Zhang&Cui:10SP} (this becomes complicated due to the
logarithm), we here solve Problem
\ref{prob:MISO_RZF_formulation2}, which is equivalent to Problem
\ref{prob:MISO_RZF_formulation1}.
Note that Problem \ref{prob:MISO_RZF_formulation2} is not a
convex optimization problem since it maximizes a convex cost function
under convex constraint sets instead of minimizing the cost. However,
Problem \ref{prob:MISO_RZF_formulation2} can be made an equivalent
convex problem by exploiting the phase ambiguity of the solution to
Problem \ref{prob:MISO_RZF_formulation2} and making ${\bf h}_{ii}^H{\bf v}_i$
real and nonnegative without affecting the value of
$|{\bf h}_{ii}^H{\bf v}_i|$ as follows \cite{Bengtsson&Ottersten:99}:
\begin{problem}\label{prob:MISO_RZF_formulation3}
For each transmitter $i \in \{1,\cdots,K\}$,
\begin{eqnarray}
&\underset{{\bf v}_i}{\mbox{maximize}} ~ &
{\bf h}_{ii}^H{\bf v}_i \\
&\mbox{subject to} & |{\bf h}_{ji}^H{\bf v}_i|^2 \leq
\alpha_{ji}\sigma_j^2,
\qquad \forall j\neq i, \\
& & \|{\bf v}_i\|^2 \leq P_i , \\
& & {\bf h}_{ii}^H{\bf v}_i \geq 0. \label{eq:Prob3imag}
\end{eqnarray}
\end{problem}
Here, the constraint \eqref{eq:Prob3imag} implies $\mbox{imag}({\bf h}_{ii}^H{\bf v}_i)=0$ and due to this constraint, maximizing $|{\bf h}_{ii}^H{\bf v}_i|^2$ is equivalent to maximizing ${\bf h}_{ii}^H{\bf v}_i$.
\begin{lemma} \label{lemma:combination}
Let ${\bf v}_i^{opt}$ be a solution of the RZFCB problem
(i.e., Problem \ref{prob:MISO_RZF_formulation2}) for transmitter $i$.
Then, ${\bf v}_i^{opt}$ is represented as follows:
\begin{equation} \label{eq:JorswieckParam}
{\bf v}_i^{opt} = c_{ii}{\bf h}_{ii}+\sum_{j \in \Gamma_i}c_{ji}{\bf h}_{ji}
\end{equation}
for some $\{c_{ji}\in{\mathbb{C}}: j\in\Gamma_i\cup\{i\}\}$, where
$\Gamma_i :=\{j: |{\bf h}_{ji}^H{\bf v}_i^{opt}|^2 =
\alpha_{ji}\sigma_j^2\}$, $\|{\bf v}_i^{opt}\|^2 = P_i$ for $N\ge K$, and $\|{\bf v}_i^{opt}\|^2\le P_i$ for $N < K$.
\end{lemma}
\begin{proof} Proof is based on the equivalent formulation in Problem \ref{prob:MISO_RZF_formulation3}.
Since Problem \ref{prob:MISO_RZF_formulation3} is a convex optimization problem, the optimal solution can be obtained by the Karush-Kuhn-Tucker (KKT) conditions. The Lagrangian of Problem \ref{prob:MISO_RZF_formulation3} for transmitter $i$ is given by
\begin{eqnarray}\label{eq:Lagrangian}
& &\hspace{-1.8em} {\mathcal{L}}({\bf v}_i, \hbox{\boldmath$\lambda$\unboldmath}, \mu, \nu)
= - {\bf h}_{ii}^H {\bf v}_i \\
& &\hspace{-1.7em}
+\sum_{j=1, j\neq i}^{K}
\lambda_j(|{\bf h}_{ji}^H{\bf v}_i|^2-\alpha_{ji}\sigma_j^2)
+\mu(\|{\bf v}_i\|^2-P_i)
-\nu{\bf h}_{ii}^H{\bf v}_i, \nonumber
\end{eqnarray}
where $\hbox{\boldmath$\lambda$\unboldmath}:=\{\lambda_j\ge0: j=1,\cdots,i-1,i+1,\cdots, K\}$ and $\mu, \nu\ge 0$ are real dual variables. With optimal dual variables $\hbox{\boldmath$\lambda$\unboldmath}^\star$, $\mu^\star$, and $\nu^\star$, the (complex) gradient of the Lagrangian should be zero at ${\bf v}_i^{opt}$, i.e.,
\begin{align}\label{eq:grad}
\mathbf{0} & = \nabla_{{\bf v}_i^*}{\mathcal{L}}({\bf v}_i, \hbox{\boldmath$\lambda$\unboldmath}^\star,
\mu^\star, \nu^\star)\big|_{{\bf v}_i={\bf v}_i^{opt}} \\
& =
- {\bf h}_{ii} + \sum_{j=1,j\neq i}^{K}
\lambda_j^\star{\bf h}_{ji}{\bf h}_{ji}^H{\bf v}_i^{opt}
+ \mu^\star{\bf v}_i^{opt} - \nu^\star{\bf h}_{ii} \nonumber \\
& = - {\bf h}_{ii} + \sum_{j\in\Gamma_i}
\lambda_j^\star{\bf h}_{ji}{\bf h}_{ji}^H{\bf v}_i^{opt} +
\mu^\star{\bf v}_i^{opt} - \nu^\star{\bf h}_{ii}, \nonumber
\end{align}
where $\Gamma_i := \{j: \lambda_j^\star>0 \}$ and $\nabla_{{\bf v}_i^*}$ is the conjugate Wirtinger gradient. From the
complementary slackness condition, $\lambda_j^\star>0$ only when
$|{\bf h}_{ji}^H{\bf v}_i|^2=\alpha_{ji}\sigma_i^2$. Also, from the complementary slackness, we have
$\nu^\star=0$. Otherwise, ${\bf h}_{ii}^H{\bf v}_i^{opt}=0$ and thus no rate is provided to user $i$.
Thus, the gradient of the Lagrangian becomes zero if and only if
\begin{equation} \label{eq:Lagrangian_deriv}
\textstyle
{\bf h}_{ii}
=\left( \mu^\star{\bf I}
+\sum_{j\in\Gamma_i}\lambda_j^\star{\bf h}_{ji}{\bf h}_{ji}^H\right)
{\bf v}_i^{opt}.
\end{equation}
If ${\bf Q}:=(\mu^\star{\bf I}
+\sum_{j\in\Gamma_i}\lambda_j^\star{\bf h}_{ji}{\bf h}_{ji}^H)$ is singular, then ${\bf v}_i^{opt}$ exists if and only if ${\bf h}_{ii}\in{\mathcal{C}}({\bf Q})$. However, the condition ${\bf h}_{ii}\in{\mathcal{C}}({\bf Q})$ does not
occur almost surely for randomly realized channel vectors, which is assumed here.
Therefore, ${\bf Q}$ should have full rank for the existence of ${\bf v}_i^{opt}$ and the corresponding
${\bf v}_i^{opt}$ has two different forms according to the optimal dual variable $\mu^\star$.
$i)\ \mu^\star > 0$: This corresponds to the case in
which the transmitter uses full power, i.e.,
$\|{\bf v}_i^{opt}\|^2=P_i$. In this case, the optimal solution is given by
\begin{eqnarray} \label{eq:lagrangian2}
\textstyle
{\bf v}_i^{opt} =
\left(
\mu^\star{\bf I}+\sum_{j\in\Gamma_i}\lambda_j^\star{\bf h}_{ji}{\bf h}_{ji}^H
\right)^{-1}{\bf h}_{ii}.
\end{eqnarray}
By applying the matrix inversion lemma recursively, it can be shown that
${\bf v}_i^{opt}$ is a linear combination of $\{{\bf h}_{ji}:
~j\in\Gamma_i^\prime:=\Gamma_i\cup\{i\}\}$. Thus, the solution is represented as \eqref{eq:JorswieckParam}.
$ii)\ \mu^\star = 0$: This case corresponds to the case in
which full power is not used at transmitter $i$. In this case,
${\bf Q}= \sum_{j\in{\Gamma}_i}\lambda_j^\star{\bf h}_{ji}{\bf h}_{ji}^H$. The
matrix ${\bf Q}$ in this case is non-singular if and only if
$|\Gamma_i|\ge N$ (i.e., $K > N$) under the assumption {\em (A.1)}, and
the corresponding solution is
given by
\begin{eqnarray} \label{eq:lagrangian3}
\textstyle
{\bf v}_i^{opt} =
\left(
\sum_{j\in\Gamma_i}\lambda_j^\star{\bf h}_{ji}{\bf h}_{ji}^H
\right)^{-1}{\bf h}_{ii}.
\end{eqnarray}
In this case, $\{{\bf h}_{ij}, j \in \Gamma_i\}$ alone span
${\mathbb{C}}^N$ fully and
it is therefore clear that the solution is represented as
\eqref{eq:JorswieckParam}.
Indeed, any subset of
$\{{\bf h}_{ji},j=1,\cdots,K\}$ with cardinality $N$ forms a full basis
for ${\mathbb{C}}^N$ under the assumption {\em (A.1)} in this case.
Furthermore, when $N \ge K$, ${\bf v}_i^{ZF}$ is feasible and thus, we
can always increase power and rate without causing interference to the
undesired receivers. Therefore, the optimal solution uses full power,
i.e., $||{\bf v}_i^{opt}||^2 =P_i$ when $N \ge K$. On the other hand,
when $N < K$, we can have either $\mu^\star > 0$ ($||{\bf v}_i^{opt}||^2 =
P_i$) or $\mu^\star = 0$ ($||{\bf v}_i^{opt}||^2 < P_i$).
\end{proof}
The solution to RZFCB for a given set of interference
relaxation levels is a linear combination of the desired channel
and a subset of interference channels for which the RZF constraint
\eqref{eq:Problem3RZFconst} is satisfied with equality.
Furthermore,
it was shown that the interference leakage levels should be designed to make the RZF interference leakage constraints be satisfied tightly in order to achieve a point on the Pareto boundary \cite{Zhang&Cui:10SP}. In this case, $\Gamma_i = \{1,\cdots,K\}\backslash \{i\}$ and thus, the RZF beam structure in Lemma \ref{lemma:combination} coincides with the Pareto-optimal beam structure derived by Jorswieck {\em et al.} in \cite{Jorswieck&Larsson&Danev:08SP}. Now, based on Lemma \ref{lemma:combination}, we present a new useful representation of ${\bf v}_i^{opt}$ that provides a clear insight into the RZFCB solution and a basis for fast algorithm construction.
\begin{theorem} \label{theo:SuccessiveZeroForcing}
For transmitter $i$, the RZFCB solution can also be expressed as
\begin{equation}\label{eq:solution_structure}
{\bf v}_i^{opt} = c_0
\frac{{\bf h}_{ii}}{\|{\bf h}_{ii}\|} + c_1
\frac{{\bf \Pi}_{{\bf A}_1}^\perp{\bf h}_{ii}}{\|{\bf \Pi}_{{\bf A}_1}^\perp{\bf h}_{ii}\|}
+ \cdots +c_{|\widetilde{\Gamma}_i|}
\frac{{\bf \Pi}_{{\bf A}_{|\widetilde{\Gamma}_i|}}^\perp{\bf h}_{ii}}{\|{\bf \Pi}_{{\bf A}_{|\widetilde{\Gamma}_i|}}^\perp{\bf h}_{ii}\|},
\end{equation}
where
$c_j\in{\mathbb{C}}, ~ j = 0, 1,\cdots , |\widetilde{\Gamma}_i|$ and
${\bf A}_j$ is constructed recursively as
\begin{equation} \label{theo:SuccessiveZeroForcing_ABFj}
{\bf A}_j:=[{\bf A}_{j-1},\ {\bf h}_{\widetilde{\Gamma}_i(j), i}], ~~~j=1,
\cdots, |\widetilde{\Gamma}_i|.
\end{equation}
Here for convenience we let ${\bf A}_0$
be an $N \times 0$ 'matrix'.
$\widetilde{\Gamma}_i$ is a
set made by permuting the elements of $\Gamma_i$ according to an arbitrary order, and $\widetilde{\Gamma}_i(j)$ denotes the $j$-th element of $\widetilde{\Gamma}_i$.
\end{theorem}
\begin{proof}
From Lemma \ref{lemma:combination}, we know that ${\bf v}_i^{opt}\in
{\cal C}([{\bf h}_{ji}]_{j\in\Gamma_i^\prime})$. Proof of the theorem is
given by showing the equivalence of the two subspaces
${\cal C}([{\bf h}_{ji}]_{j\in\Gamma_i^\prime})$ and
${\cal C}([{\bf h}_{ii},{\bf \Pi}_{{\bf A}_{1}}^\perp{\bf h}_{ii}, \cdots,$
${\bf \Pi}_{{\bf A}_{|\widetilde{\Gamma}_i|}}^\perp{\bf h}_{ii}])$.
{\em Case (i). $|\Gamma_i|(=|\tilde{\Gamma}_i|) \le N-1$: } In this case,
$\{{\bf h}_{ii},{\bf \Pi}_{{\bf A}_{1}}^\perp{\bf h}_{ii}, \cdots,
{\bf \Pi}_{{\bf A}_{|\widetilde{\Gamma}_i|}}^\perp{\bf h}_{ii}\}$ are linearly
independent. This is easily shown by replacing
${\bf \Pi}_{{\bf A}_j}^\bot$ with ${\bf I}-{\bf \Pi}_{{\bf A}_j}$ and by using the
linear independence of $\{{\bf h}_{ji}\}_{j\in\Gamma_i^\prime}$. Thus,
the dimension of ${\cal C}([{\bf h}_{ii}, {\bf \Pi}_{{\bf A}_{1}}^\perp{\bf h}_{ii},
\cdots,$ ${\bf \Pi}_{{\bf A}_{|\widetilde{\Gamma}_i|}}^\perp{\bf h}_{ii}])$ is
$(|\widetilde{\Gamma}_i|+1)$, which is the same as that of
${\cal C}([{\bf h}_{ji}]_{j\in\Gamma_i^\prime})$. Now, consider the
projection of any vector in ${\cal C}([{\bf h}_{ii},
{\bf \Pi}_{{\bf A}_1}^\perp{\bf h}_{ii}, \cdots,
{\bf \Pi}_{{\bf A}_{|\widetilde{\Gamma}_i|}}^\perp{\bf h}_{ii}])$ onto the
orthogonal complement of ${\cal C}([{\bf h}_{ji}]_{j\in\Gamma_i^\prime})$: {
\begin{eqnarray}
& & {\bf \Pi}_{[{\bf h}_{ji}]_{j\in\Gamma_i^\prime}}^\perp\big( c_0{\bf h}_{ii}
+c_1 {\bf \Pi}_{{\bf A}_1}^\perp{\bf h}_{ii} + \cdots
+c_{|\widetilde{\Gamma}_i|}
{\bf \Pi}_{{\bf A}_{|\widetilde{\Gamma}_i|}}^\perp{\bf h}_{ii}\big)
\nonumber \\
&=& {\bf \Pi}_{[{\bf h}_{ji}]_{j\in\Gamma_i^\prime}}^\perp\big( c_0{\bf h}_{ii}
+c_1 ({\bf I}-{\bf \Pi}_{{\bf A}_1}){\bf h}_{ii} + \cdots
+c_{|\widetilde{\Gamma}_i|}
({\bf I}-{\bf \Pi}_{{\bf A}_{|\widetilde{\Gamma}_i|}}){\bf h}_{ii}\big)
\nonumber \\
&=& {\bf \Pi}_{[{\bf h}_{ji}]_{j\in\Gamma_i^\prime}}^\perp\big( c_0{\bf h}_{ii}
+c_1 ({\bf I}-{\bf \Pi}_{{\bf h}_{\widetilde{\Gamma}_i(1),i}}){\bf h}_{ii} + \cdots
+c_{|\widetilde{\Gamma}_i|}
({\bf I}-{\bf \Pi}_{[{\bf h}_{\widetilde{\Gamma}_i(j),i}]_{j=1}^{|\widetilde{\Gamma}_i|}}
){\bf h}_{ii}\big) \nonumber \\
&=& \textstyle {\bf \Pi}_{[{\bf h}_{ji}]_{j\in\Gamma_i^\prime}}^\perp\Big(
\sum_{j=0}^{|\widetilde{\Gamma}_i|} c_j{\bf h}_{ii}
- c_1 {\bf \Pi}_{{\bf h}_{\widetilde{\Gamma}_i(1),i}}{\bf h}_{ii}
- \cdots
- c_{|\widetilde{\Gamma}_i|}
{\bf \Pi}_{[{\bf h}_{\widetilde{\Gamma}_i(j),i}]_{j=1}^{|\widetilde{\Gamma}_i|}}
{\bf h}_{ii}\Big) \nonumber \\
&=& 0. \label{eq:proj_onto_ortho_comple1}
\end{eqnarray}}
By \eqref{eq:proj_onto_ortho_comple1} the orthogonal complement of
${\cal C}(\big[{\bf h}_{ii}, {\bf \Pi}_{{\bf A}_1}^\perp{\bf h}_{ii}, \cdots,$
${\bf \Pi}_{{\bf A}_{|\widetilde{\Gamma}_i|}}^\bot{\bf h}_{ii}\big])$ is
included in that of ${\cal C}([{\bf h}_{ji}]_{j\in\Gamma_i^\prime})$, but
${\cal C}(\big[{\bf h}_{ii}, {\bf \Pi}_{{\bf A}_1}^\perp{\bf h}_{ii}, \cdots,$
${\bf \Pi}_{{\bf A}_{|\widetilde{\Gamma}_i|}}^\bot{\bf h}_{ii}\big])$ and
${\cal C}([{\bf h}_{ji}]_{j\in\Gamma_i^\prime})$ have the same dimensions.
Thus, the two orthogonal complements are the same, and hence, the
two subspaces themselves are the same. Consequently, for any
$c_{ii}{\bf h}_{ii}+\sum_{j \in \Gamma_i} c_{ji}{\bf h}_{ji}$ with
arbitrary $\{c_{ji} \in {\mathbb{C}}: j\in\Gamma_i^\prime\}$,
there exists some $\{c_j\in{\mathbb{C}}: 0\le j\le
|\widetilde{\Gamma}_i|\}$ s.t.
\[
{\bf v}_i^{opt}
= c_{ii} {\bf h}_{ii} + \sum_{j \in{\Gamma_i}} c_{ji} {\bf h}_{ji}
= c_0 \frac{{\bf h}_{ii}}{\|{\bf h}_{ii}\|}
+ \sum_{j=1}^{|\widetilde{\Gamma}_i|}
c_j\frac{{\bf \Pi}_{{\bf A}_j}^\bot{\bf h}_{ii}}{\|{\bf \Pi}_{{\bf A}_j}^\bot{\bf h}_{ii}\|}.
\]
{\em Case (ii). $|\Gamma_i|\ge N$:} In this case, both $\{{\bf h}_{ji}, j\in\Gamma_i^\prime\}$ and
$\{{\bf h}_{ii},{\bf \Pi}_{{\bf A}_{1}}^\perp{\bf h}_{ii}, \cdots, {\bf \Pi}_{{\bf A}_{|\widetilde{\Gamma}_i|}}^\perp{\bf h}_{ii}\}$ span the whole ${\mathbb{C}}^N$. Thus, the claim is trivially satisfied.
\end{proof}
\noindent Theorem \ref{theo:SuccessiveZeroForcing} states that the RZF
solution is a linear combination of vectors that are obtained by
projecting the desired channel vector onto the orthogonal
complements of a series of subspaces spanned by the channels from
the transmitter to the undesired receivers. Furthermore, the
series of subspaces are obtained by sequentially including one
additional interference channel vector at a time, as shown in
\eqref{theo:SuccessiveZeroForcing_ABFj}. Soon, it will be shown
that, to obtain the RZF solution to Problem
\ref{prob:MISO_RZF_formulation2}, the order of interference
channel inclusion for constructing ${\bf A}_j$s in Theorem
\ref{theo:SuccessiveZeroForcing} is determined by the set of
allowed interference levels and the channel realization.
\subsection{The Sequential Orthogonal Projection Combining Method
and Closed-Form Solutions}
In this subsection, we propose an efficient beam design method for
RZFCB that successively allocates the transmit power to certain
vectors obtained by sequential orthogonal projection of the
desired channel vector onto monotonically decreasing subspaces.
Furthermore, we provide the closed-form solution to the RZFCB
problem in the two-user case and an approximate closed-form
solution in the three-user case.
To obtain the RZF beamforming
vector under given interference relaxation constraints for a given
channel realization, Problem \ref{prob:MISO_RZF_formulation2}
should be solved. One can use a numerical method \cite{Bland&Goldfarb&Todd:81OR}, as in \cite{Zhang&Cui:10SP}. However, such a method
requires a numerical search for determining the Lagrange dual
variables satisfying the RZF constraints and the transmit power
constraint. To circumvent such difficulty and to increase the
practicality of the RZFCB, we exploit Theorem
\ref{theo:SuccessiveZeroForcing} to construct an efficient method
to find the RZFCB solution. Theorem
\ref{theo:SuccessiveZeroForcing} provides us with a very
convenient way of obtaining the RZFCB solution for given
interference leakage levels for a given channel realization; we
only need to find $\widetilde{\Gamma}_i$ and complex coefficients
$\{c_i\}$ in \eqref{eq:solution_structure} for each transmitter.
The idea is based on the fact that the RZF beamforming vector
should be designed to be as close as possible to the MF beam
vector under the interference leakage constraints for the maximum
rate under RZF, as described in Problem
\ref{prob:MISO_RZF_formulation2}. Hereafter, we will explain how
the coefficients $\{c_i\}$ and the matrices $\{{\bf A}_i\}$ in
Theorem \ref{theo:SuccessiveZeroForcing} can be obtained to
maximize the rate under the RZF interference and power
constraints. Consider transmitter $i$ without loss of generality.
For the given transmit power constraint $\|{\bf v}_i\|_2^2 \le P_i$,
it may not be possible to allocate all of the transmit power to the
MF direction ${\bf h}_{ii}$ because this allocation may violate the
RZF constraints. The rate greedy approach under the
RZF constraints for a given channel realization is explained as follows.
First, we should start to allocate
the transmit power to the direction of ${\bf h}_{ii}$ by increasing
$c_0$ with some phase until this allocation hits one of the RZF constraints with equality, i.e., the interference level to one of the undesired receivers reaches
the allowed maximum exactly. (In the case that the allowed interference
levels to all undesired receivers are the same, this
receiver is the receiver whose channel vector has the maximum inner product
with ${\bf h}_{ii}$.) The index of this receiver is
$\widetilde{\Gamma}_i(1)$. At this point, transmitter $i$ cannot allocate
the transmit power to the direction ${\bf h}_{ii}$ anymore since this
would violate the RZF constraint for receiver $\widetilde{\Gamma}_i(1)$. Since the RZF constraints for other
undesired receivers are still met with strict inequality,
transmitter $i$ can still cause interference to the remaining
receivers. Thus, for the maximum rate under the RZF constraints,
transmitter $i$ should now start to allocate the remaining power to the
direction of
${\bf \Pi}_{{\bf A}_1}^\perp
{\bf h}_{ii}$, where ${\bf A}_1=[{\bf h}_{\widetilde{\Gamma}_i(1),i}]$, until
this allocation hits another RZF constraint with equality. The
index of this receiver is $\widetilde{\Gamma}_i(2)$. (Note that
${\bf \Pi}_{{\bf A}_1}^\perp {\bf h}_{ii}$ is the direction of maximizing
the data rate without causing additional interference to receiver
$\widetilde{\Gamma}_i(1)$.) Now, transmitter $i$ cannot cause
interference to receiver $\widetilde{\Gamma}_i(2)$ in addition to
receiver $\widetilde{\Gamma}_i(1)$ anymore. Therefore, at this point,
transmitter $i$ should start to allocate its remaining power to
the next greedy direction ${\bf \Pi}_{{\bf A}_2}^\perp {\bf h}_{ii}$, where
${\bf A}_2=[{\bf h}_{\widetilde{\Gamma}_i(1),i},
{\bf h}_{\widetilde{\Gamma}_i(2),i} ]$.
This greedy power allocation
without violating the RZF constraints should be done until either all the
transmit power is used up ($\mu^\star > 0$ in Lemma \ref{lemma:combination}) or we cannot find a new direction that does not cause interference to the users that are already in the set $\tilde{\Gamma}_i$ ($\mu^\star = 0$ in Lemma \ref{lemma:combination}). When $N \ge K$ and transmit power still remains even after hitting all the $K-1$ interference leakage constraints with equality, from then on, all the remaining power should be
allocated to the ZF direction. This coincides with our intuition that ZF is optimal at a high signal-to-noise ratio (SNR) in the case of $N \ge K$. On the other hand,
when all the transmit power is used up before reaching the remaining interference constraints
with equality, the corresponding remaining interference channel vectors do not
appear in the solution. The final RZF solution is the sum of these
component vectors and has the form in
\eqref{eq:solution_structure}. In this way, the RZFCB solution can
be obtained by combining the sequential projections of the desired
channel vector ${\bf h}_{ii}$ onto the orthogonal complements of the
subspaces ${\cal C}({\bf A}_1) \subset \cdots \subset
{\cal C}({\bf A}_{|\widetilde{\Gamma}_i|})$. Thus, we refer to this beam
design method as the {\em sequential orthogonal projection combining (SOPC)
method.\footnote{The rate optimality of the SOPC strategy under the RZF constraints is straightforward to see. Suppose that we are given any beam vector that is a linear combination of $\{{\bf h}_{ji}\}$, satisfies the RZF interference and power constraints but is not the SOPC solution. Then, the vector can still be represented in terms of the SOPC basis in Theorem \ref{theo:SuccessiveZeroForcing} and some of the basis component vectors with larger inner product with the MF direction do not satisfy the RZF constraints with equality. Thus,
the rate can be increased by allocating power from the basis component vector with smaller inner product with the MF direction to the basis component vector with larger inner product with the MF direction until the RZF constraints are satisfied with equality.} } By Theorem \ref{theo:pareto_achievability}, {\em the SOPC strategy with a well chosen set of interference relaxation levels is Pareto-optimal for MISO $K$-pair interference channels with single-user decoding.}
An interesting interpretation of the SOPC strategy is in an analogy with the water-filling strategy. The water-filling strategy distributes power to resource bins according to the effectiveness of each bin, and the power fills into the bin with the lowest noise level (or the most effective bin) first.
Similarly, the SOPC strategy allocates power to the most effective
direction first and then the next most effective direction when
the first direction cannot accommodate power anymore. This procedure
continues until either the procedure uses up the power or it cannot find a new feasible direction.
So, the SOPC strategy can be viewed graphically as pouring water on top of a multi-tiered fountain, as
illustrated in Fig. \ref{fig:SOPC_strategy}.
The relationship of the RZFCB/SOPC design and the two-user result by Jorswieck {\em et al.} \cite{Jorswieck&Larsson&Danev:08SP} is explained in Fig.
\ref{fig:SOPC_interpret}.
In the two user case, Jorswieck {\em et al.} have shown that a Pareto-optimal beam vector is a convex combination of the MF beam ${\bf v}_i^{MF}$ and the ZF beam ${\bf v}_i^{ZF}$
satisfying the power constraint, i.e., ${\bf v}_i=\sqrt{P_i}\frac{\lambda_i{\bf v}_i^{MF}+(1-\lambda_i){\bf v}_i^{ZF}}{\|\lambda_i{\bf v}_i^{MF}+(1-\lambda_i){\bf v}_i^{ZF}\|}$,
where $0\le\lambda_i\le 1$.
Thus, the feasible set of optimal
beam vectors is the arc denoted by ${\cal F}$ in Fig. \ref{fig:SOPC_interpret}. All the points on this arc can be represented by the sum of
the two vectors in red, and the size of the component vector in
the MF direction is determined by its projection onto
${\cal C}({\bf h}_{21})$, i.e., the allowed interference level to the
other receiver in the RZF context. Thus, the two-user result by
Jorswieck {\em et al.} can be viewed as a special case of the SOPC
strategy when the number of users is two. The key difference is the parameterization; $\alpha_{12}$ and $\alpha_{21}$ are the parameters in the RZF framework whereas the linear combining coefficients $\lambda_1$ and $\lambda_2$ are the parameters in \cite{Jorswieck&Larsson&Danev:08SP}.
\begin{figure}[t]
\centerline{
\begin{psfrags}
\psfrag{a}[c]{${\bf h}_{ii}$} %
\psfrag{b}[c]{$\ {\bf \Pi}_{{\bf A}_1}^\bot{\bf h}_{ii}$} %
\psfrag{c}[c]{$\ \dots$} %
\psfrag{d}[c]{$\ \ {\bf \Pi}_{{\bf A}_K}^\bot {\bf h}_{ii}$} %
\psfrag{t1}[l]{$P_i^{(0)}$} %
\psfrag{t2}[l]{$P_i^{(1)}$} %
\psfrag{t3}[l]{$P_i^{(K-1)}$} %
\psfrag{bd}[c]{{\@setsize\small{9pt}\viiipt\@viiipt\let\@listi\@listI ZF}} %
\psfrag{mf}[c]{{\@setsize\small{9pt}\viiipt\@viiipt\let\@listi\@listI MF}} %
\scalefig{0.4}\epsfbox{figures/water_fall1.eps}
\end{psfrags}} \caption{The SOPC strategy in the case of $N \ge K$:
Water-pouring on a
multi-tiered fountain.} \label{fig:SOPC_strategy}
\end{figure}
\begin{figure}[t]
\centerline{
\begin{psfrags}
\psfrag{fc}[c]{${\mathcal{F}}$} %
\psfrag{coh21}[c]{{\@setsize\small{9pt}\viiipt\@viiipt\let\@listi\@listI ${\cal C}({\bf \Pi}_{{\bf h}_{21}}^\bot{\bf h}_{11} )$}} %
\psfrag{ch11}[c]{{\@setsize\small{9pt}\viiipt\@viiipt\let\@listi\@listI ${\cal C}({\bf h}_{11})$}} %
\psfrag{r}[c]{{\@setsize\small{9pt}\viiipt\@viiipt\let\@listi\@listI $\sqrt{P_1}$}} %
\psfrag{as2}[c]{{\@setsize\small{9pt}\viiipt\@viiipt\let\@listi\@listI $\sqrt{\alpha_{21}\sigma_2^2}$}} %
\psfrag{ch21}[l]{\!\!\!{\@setsize\small{9pt}\viiipt\@viiipt\let\@listi\@listI ${\cal C}({\bf h}_{21})$}} %
\psfrag{opt}[c]{${\bf v}_1^{opt}$} %
\scalefig{0.40}\epsfbox{figures/trajectory.eps}
\end{psfrags}} \caption{SOPC interpretation of the two-user result.} \label{fig:SOPC_interpret}
\end{figure}
Now consider the detailed implementation of the SOPC method.
Before considering the general case of an arbitrary number $K$ of
users, we consider simple two-user and three-user cases. Here, we
restrict the combining coefficients $\{c_i\}$ to the set of real
numbers. It will shortly be shown that the performance loss
caused by restricting $\{c_i\}$ to real numbers is negligible.
Furthermore, it is the optimal solution of the RZFCB when $K=2$.
For simplicity, we only provide the solution for transmitter 1.
The solutions for other transmitters can be obtained in a similar
way.
\begin{proposition} \label{prop:sopck2}
The closed-form SOPC solution in the two-pair MISO IC case is
given by
\begin{align} \label{eq:sopc_K2}
{\bf v}_1 =
\left\{
\begin{array}{ll}
\sqrt{P_1}{\bf v}_1^{MF}, ~~ & \text{if} ~~ P_1
\le \frac{\alpha_{21}\sigma_2^2}{|{\bf h}_{21}^H{\bf v}_1^{MF}|^2}, \\
\xi_0{\bf v}_1^{MF}+\xi_1{\bf v}_1^{ZF}, & \text{otherwise},
\end{array}
\right. \nonumber
\end{align}
where ${\bf v}_1^{MF}=\frac{{\bf h}_{11}}{\|{\bf h}_{11}\|}$,
${\bf v}_1^{ZF}=\frac{{\bf \Pi}_{{\bf h}_{21}}^{\perp}{\bf h}_{11}}{\|{\bf \Pi}_{{\bf h}_{21}}^{\perp}
{\bf h}_{11}\|}$, $\xi_0 =
\sqrt{\frac{\alpha_{21}\sigma_2^2}{|{\bf h}_{21}^H{\bf v}_1^{MF}|^2}}$,
and $\xi_1 = -\rho\xi_0+\sqrt{P_1-\xi_0^2(1-\rho^2)}$. Here, $\rho
=
({\bf v}_1^{MF})^H{\bf v}_1^{ZF}=\|{\bf \Pi}_{{\bf h}_{21}}^\bot{\bf h}_{11}\|/\|{\bf h}_{11}\|
\in {\mathbb{R}}_+$.
\end{proposition}
\vspace{0.3em}
\textit{Proof:}
Proof of Proposition \ref{prop:sopck2} can be found in \cite{Park&Lee&Sung&Yukawa:12Arxiv}.
$\hfill\blacksquare$
\noindent Now, we consider the case of $K=3$. This case is particularly
important when the hexagonal cell structure is used and three
cells are coordinating their beam vectors. In the case of $K=3$,
the solution can have six different forms depending on the
transmit power and channel realization. We will provide the
closed-form solution under the real coefficient restriction for transmitter 1 in the case that the
interference leakage to receiver $3$ reaches the allowed level
before the interference leakage to receiver $2$ reaches the
allowed level. (For this, we should first take inner products
$\langle{\bf h}_{21}, {\bf h}_{11}\rangle$ and $\langle{\bf h}_{31},{\bf h}_{11}\rangle$
and compare the ratio of their magnitudes with some threshold. The solutions
of the other case and of other users can be derived in the same
manner.)
\begin{proposition} \label{prop:sopck3}
For $K=3$ and
$\frac{|{\bf h}_{31}^H{\bf v}_1^{MF}|^2}{|{\bf h}_{21}^H{\bf v}_1^{MF}|^2} \geq
\frac{\alpha_{31}\sigma_3^2}{\alpha_{21}\sigma_2^2 }$, the
closed-form SOPC solution with the restriction to real coefficients at transmitter $1$ is given in \eqref{eq:sopc_k3}.
\begin{figure*}
\hspace{-12em}
\begin{equation} \label{eq:sopc_k3}
{\bf v}_1 =
\left\{
\begin{array}{ll}
\sqrt{P_1}{\bf v}_1^{MF}, \hspace{-4.5em} &
\text{if} ~~ P_1\in\Psi_1:=\{P_1 \in {\mathbb{R}}^+:\sqrt{P_1} \le \beta_0\}, \\
\beta_0{\bf v}_1^{MF} + \beta_1 \frac{{\bf \Pi}_{{\bf h}_{31}}^\perp{\bf h}_{11}}
{\|{\bf \Pi}_{{\bf h}_{31}}^\perp {\bf h}_{11}\|}, &
\mbox{if} ~~ P_1 \in \Psi_2:=
\Big\{P_1 \in {\mathbb{R}}^+: \sqrt{P_1} > \beta_0, \\
& \hspace{7.5em}
\left|{\bf h}_{21}^H\Big(\beta_0{\bf v}_1^{MF} +
\beta_1\frac{{\bf \Pi}_{{\bf h}_{31}}^\perp {\bf h}_{11}}{\|{\bf \Pi}_{{\bf h}_{31}}^\perp {\bf h}_{11}\|}\Big)\right|^2 \le \alpha_{21}\sigma_2^2\Big\}, \\
\beta_0{\bf v}_1^{MF} +
\beta_1^\prime\frac{{\bf \Pi}_{{\bf h}_{31}}^\perp{\bf h}_{11}}
{\|{\bf \Pi}_{{\bf h}_{31}}^\perp {\bf h}_{11}\|} + \beta_2 {\bf v}_1^{ZF}, &
\mbox{if} ~~
P_1 \in \Psi_3:=\Big\{P_1 \in {\mathbb{R}}^+: \\
& \hspace{7.5em}
\Big|{\bf h}_{21}^H\Big(\beta_0{\bf v}_1^{MF} +
\beta_1\frac{{\bf \Pi}_{{\bf h}_{31}}^\perp
{\bf h}_{11}}{\|{\bf \Pi}_{{\bf h}_{31}}^\perp {\bf h}_{11}\|}\Big)\Big|^2 >
\alpha_{21}\sigma_2^2 \Big\}.
\end{array}
\right.
\end{equation}
\end{figure*}
In \eqref{eq:sopc_k3}, $\beta_0$, $\beta_1$, $\beta_1^\prime$ and $\beta_2$ are given by
$\beta_0 = \frac{\sqrt{\alpha_{31}\sigma_3^2}}{|{\bf h}_{31}^H{\bf v}_1^{MF}|}$,
$\beta_1 = -a\beta_0 + \sqrt{P_1 - (1-a^2)\beta_0^2}$,
$\beta_1^\prime =
\frac{1}{c}
{(-d \beta_0+\sqrt{d^2\beta_0^2-c(b\beta_0^2-\alpha_{21}\sigma_2^2)})}$,
and
$\beta_2 = -(f\beta_0+e\beta_1^\prime) +
\sqrt{(f\beta_0+e\beta_1^\prime)^2 -
(2a\beta_0\beta_1^\prime+\beta_0^2+\beta_1^{\prime 2}-P_1)}$,
where $a:=\mbox{Re}\{\langle{\bf v}_1^{MF},{\bf \Pi}_{{\bf h}_{31}}^\perp
{\bf h}_{11}/||{\bf \Pi}_{{\bf h}_{31}}^\perp
{\bf h}_{11}||\rangle\}=||{\bf \Pi}_{{\bf h}_{31}}^\perp {\bf v}_1^{MF}||$,
$b =|{\bf h}_{21}^H {\bf v}_1^{MF}|^2$,
$c = \Big|{\bf h}_{21}^H
\frac{{\bf \Pi}_{{\bf h}_{31}}^\perp {\bf h}_{11}}{\|{\bf \Pi}_{{\bf h}_{31}}^\perp
{\bf h}_{11}\|}\Big|^2$,
$d=$ $\mbox{Re}\Big\{({\bf h}_{21}^H{\bf v}_1^{MF})^* \Big({\bf h}_{21}^H
\frac{{\bf \Pi}_{{\bf h}_{31}}^\perp{\bf h}_{11}}{\|{\bf \Pi}_{{\bf h}_{31}}^\perp
{\bf h}_{11}\|}\Big)\Big\}$,
$e=\frac{|{\bf h}_{21}^H{\bf h}_{11}|^2}{\|{\bf h}_{21}\|^2}$, and $f =$
$({\bf v}_1^{MF})^H{\bf v}_1^{ZF}$.\end{proposition}
\vspace{0.3em}
\textit{Proof:}
Proof of Proposition \ref{prop:sopck3} can be found in \cite{Park&Lee&Sung&Yukawa:12Arxiv}.
$\hfill\blacksquare$
\noindent In the case of $K > 3$, it is cumbersome to
distinguish all possible scenarios for deriving an explicit SOPC
solution. Thus, we propose an algorithm implementing the SOPC
strategy with real combining coefficients in Table \ref{table:SOPC}. In the general case of $K > 3$,
the implementation of the SOPC algorithm can be simplified by the known result in the Kalman filtering theory, provided in the following lemma.
\begin{lemma}[Sequential orthogonal projection \cite{Lototsky:book}]
\label{lemma:sequential_orthogonal} Let ${\cal H}$ be a Hilbert space
with norm $\| \cdot \|$ and inner product $\langle\cdot,\cdot\rangle$.
Consider ${\bf x}\in {\cal H}$ and a closed linear subspace ${\bf A}_j$ of
${\cal H}$. For some ${\bf y}\in {\cal H}$ but ${\bf y}\not\in{\bf A}_j$, the
following equality holds
\begin{align}
{\bf \Pi}_{{\bf A}_{j+1}}{\bf x}
&= {\bf \Pi}_{[{\bf A}_j, {\bf y}]}{\bf x} \\
&=
{\bf \Pi}_{{\bf A}_j}{\bf x} + \frac{\langle{\bf x}-{\bf \Pi}_{{\bf A}_j}{\bf x},\
{\bf y}-{\bf \Pi}_{{\bf A}_j} {\bf y}\rangle}
{\|{\bf y}-{\bf \Pi}_{{\bf A}_j}{\bf y}\|^2}({\bf y}-{\bf \Pi}_{{\bf A}_j}{\bf y}). \nonumber
\end{align}
\end{lemma}
\vspace{0.5em} \noindent Since we need to compute
${\bf \Pi}_{{\bf A}_j}^\perp {\bf h}_{ii} = ({\bf I} -{\bf \Pi}_{{\bf A}_j}) {\bf h}_{ii}$ in
the SOPC algorithm, Lemma \ref{lemma:sequential_orthogonal} can be
applied recursively by exploiting the fact ${\bf A}_{j} = [{\bf A}_{j-1},
{\bf h}_{\widetilde{\Gamma}(j),i}]$. Thus, we only need to compute
${\bf \Pi}_{{\bf A}_{j-1}}{\bf h}_{\tilde{\Gamma}(j),i}$ for each $j \in
\{1,2,\cdots, K\}$.
The proposed algorithm in Table \ref{table:SOPC} computes the direction and size of the component vector for SOPC directly in each step.
\begin{table}[t]
\centering\caption{The sequential orthogonal projection combining
algorithm.} \label{table:SOPC}
\begin{tabular}{p{330pt}}
\hline
Given channel realization $\{{\bf h}_{ji}, i, j=1,\cdots, K\}$, pre-determined interference levels $\{\alpha_{ji}\sigma_j^2:$ $i, j=1,\cdots, K, j\neq i\}$, and maximum transmit power $\{P_i:i=1,\cdots,K\}$, perform the following procedure at each transmitter $i \in \{1,\cdots,K\}$.
\textbf{Initialization:} ${\bf v}_i={\mathbf{0}}$,
${\bf A} = \emptyset$,
$\Phi_i= \{1, \cdots, i-1, i+1, \cdots, K\}$, and $k=1$.
\textbf{While} $\ k \le \min(N, K)$,
~~ 1. Let ${\bf u}:=\frac{{\bf \Pi}_A^\bot {\bf h}_{ii}}{\|{\bf \Pi}_A^\bot{\bf h}_{ii}\|}$.
~~ 2. $\mu_p$ is a positive solution of
$\| {\bf v}_i + \mu_p {\bf u} \|_2^2 = P_i$, i.e.,
\[
\mu_p := -\rho_p + \sqrt{\rho_p^2 - (\|{\bf v}_i\|_2^2-P_i)}
\]
~~~~~ where $\rho_p = Re({\bf u}^H{\bf v}_i)$.
~~ 3. $\mu_j$ is a positive solution of $|{\bf h}_{ji}^H({\bf v}_i+\mu_j {\bf u})|^2 = \alpha_{ji}\sigma_j^2$
~~~~~ for each $j\in \Phi_i$, i.e.,
\[
\mu_j := \tfrac{-\rho_j+\sqrt{\rho_j^2-|{\bf h}_{ji}^H{\bf u}|^2\cdot
(|{\bf h}_{ji}^H{\bf v}_i|^2-\alpha_{ji}\sigma_j^2) }}{|{\bf h}_{ji}^H{\bf u}|^2}
\]
~~~~~ where $\rho_j=Re({\bf v}_i^H{\bf h}_{ji}{\bf h}_{ji}^H{\bf u})$.
~~ 4. Obtain $\mu_j^* = \min\limits_{j\in\Phi_i} \{\mu_j\}$ and
$j^* = \mathop{\arg\min}\limits_{j\in\Phi_i} \{\mu_j\}$.
~~ 5. If $\mu_p > \mu_j^*$,
${\bf v}_i = {\bf v}_i + \mu_j^* {\bf u}$, ${\bf A} = [{\bf A}, {\bf h}_{j^* i}]$,
$\Phi_i=\Phi_i \backslash\{j^*\}$,
~~~~~ $k=k+1$, and go to step 1.
~~~~~ If $\mu_p \le \mu_j^*$,
${\bf v}_i = {\bf v}_i + \mu_p {\bf u}$. Terminate iteration.
\textbf{end}
\\
\hline
\end{tabular}
\end{table}
\begin{figure}[t]
\centerline{ \scalefig{0.5}
\epsfbox{figures/apsm_wf_2to6user.eps} } \vspace{-1.5em} \caption{Average sum
rates of the exact RZFCB solution and the proposed SOPC algorithm with real coefficients.
(Here, $N=K$ and the average sum rate is obtained over $50$ i.i.d.
channel realizations.)} \label{fig:apsm_wf}
\end{figure}
The proposed SOPC solution based on real coefficients is a sub-optimal solution to the RZFCB problem in the case of $K\ge 3$. However, the performance loss between the optimal RZFCB (or exact SOPC) beamforming vector and the proposed SOPC solution based on real coefficients is insignificant for a wide range of meaningful SNR values, as seen in Fig. \ref{fig:apsm_wf}. Thus, practically, the proposed SOPC solution can be used with negligible performance loss. Note that the
necessary computations for the proposed SOPC solution are a few inner product
and square root operations and the complexity of the SOPC method is
simply $O(N)$, where $N$ is the number of transmit antennas at the
transmitter. The proposed SOPC method reduces computational complexity to obtain an RZF solution by order of hundreds
when compared to
the ellipsoid method for the RZFCB solution used in \cite{Zhang&Cui:10SP}, as shown in Fig. \ref{fig:complexity}, and the solution procedure can easily be programmed in a real hardware.
\begin{figure}[htbp]
\centering
\scalefig{0.5}\epsfbox{figures/complexity.eps}
\caption{Computational complexity for RZFCB beam design: Ellipsoid method \cite{Bland&Goldfarb&Todd:81OR,Zhang&Cui:10SP} versus SOPC ($N=K$ and SNR=5 dB)}.
\label{fig:complexity}
\end{figure}
\section{Rate-tuple Control}
\label{sec:rate_control}
In the previous section, we provided an $O(N)$-complexity algorithm
to solve the RZFCB problem for a given set $\{\alpha_{ji}\}$ of
interference relaxation parameters. Now, we consider how to
design these parameters. We first provide a centralized approach
to determine $\{\alpha_{ji}\}$ with the aim of controlling the
rate-tuple along the Pareto boundary of the achievable rate region
and then a fully-distributed heuristic approach that exploits the
parameterization in terms of interference relaxation levels in
RZFCB and is able to control the rate-tuple location roughly along
the Pareto boundary of the achievable rate region.
\subsection{A Centralized Approach}
By Theorem \ref{theo:pareto_achievability}, with a set of well
chosen allowed interference leakage levels, the RZFCB can achieve
any Pareto-optimal point of the rate region. However, the problem
of designing the interference leakage levels $\{\alpha_{ji}\}$ in
the network remains. Under the RZFCB framework, in
\cite{Zhang&Cui:10SP}, a necessary condition for the interference
relaxation parameters at each receiver to achieve a Pareto-optimal point
was derived. Based on the necessary condition, the authors proposed an
iterative algorithm that updates the interference relaxation
parameters. Although the algorithm in \cite{Zhang&Cui:10SP} is
applicable to general $K$-user MISO interference channels, it
cannot control the rate-tuple location on the Pareto boundary
to which the algorithm converges. To control the rate-tuple to an arbitrary point along the Pareto-boundary of the achievable rate region, we here apply the utility function based approach in \cite{Jorswieck&Larsson:08ICASSP} to the RZF parameterization in terms of interference leakage levels.
Exploiting the fact that the RZFCB can achieve any Pareto-boundary point by adjusting $\{\alpha_{ji}\}$,
we convert the problem of finding a desired point on the
Pareto boundary of the achievable rate region into that of finding
an optimal point of the following optimization problem: {\@setsize\small{9pt}\viiipt\@viiipt\let\@listi\@listI
\begin{eqnarray}
&\max\limits_{\{\alpha_{ji}\}}\ ~~& u \big(R_1(\{{\bf v}_i^{RZF}(\{\alpha_{ji}\})\}), \cdots, R_K(\{{\bf v}_i^{RZF}(\{\alpha_{ji}\})\}) \big), \nonumber \\
&\mbox{subject to}~~& |{\bf h}_{ji}^H{\bf v}_i|^2\le \alpha_{ji}\sigma_j^2, \quad \forall i, ~j\neq i, \label{eq:newOptProbCentral} \\
& & \|{\bf v}_i\|^2\le P_i, \quad\quad\quad~\forall i, \nonumber
\end{eqnarray}}
where $u(R_1, \cdots, R_K)$ is the desired utility function and several
examples include the weighted sum rate $u(R_1,\cdots,R_K)$ $= \sum w_i
R_i$,
where $w_i\ge 0$ and $\sum w_i = 1$, the Nash bargaining point $u(R_1,\cdots,R_K)$
$=\prod_{i=1}^{K} (R_i-R_i^{NE})$, where
$R_i^{NE}=\log_2\left(1+\frac{|{\bf h}_{ii}^H {\bf v}_i^{MF}|^2}{\sigma_i^2+\sum_{j\neq
i}|{\bf h}_{ij}^H {\bf v}_j^{MF} |^2} \right)$, and the egalitarian point
$u(R_1,\cdots,R_K) = \min(R_1,\cdots, R_K)$
\cite{Jorswieck&Larsson:08ICASSP}.
The optimization \eqref{eq:newOptProbCentral} can be solved
by an alternating optimization technique. That is, we fix all other
$\alpha_{ji}$'s except one interference relaxation parameter and
update the unfixed parameter so that the utility function is
maximized. After this update, the next $\alpha_{ji}$ is picked for update. This procedure continues until converges. The
proposed algorithm is described in detail in Table
\ref{table:alpha_algorithm}. For a given utility function $u(R_1,\cdots,R_K)$, the RZF beam vectors $\{{\bf v}_i^{RZF}\}$ can be obtained as functions of $\{\alpha_{ji}\}$ by the SOPC method, the rate-tuple can be computed as a function of $\{{\bf v}_i^{RZF}\}$ by \eqref{eq:R_iOneUser}, and finally the utility function value can be computed as a function of $(R_1,\cdots,R_K)$. Thus, the utility value as a function of $\{\alpha_{ji}\}$ can be computed very efficiently by the SOPC method for the proposed centralized algorithm, and this fact makes it easy to apply a numerical optimization method such as the interior point method to the per-iteration optimization in Table \ref{table:alpha_algorithm}.
\begin{figure}[ht]
\centering
\scalefig{0.5}\epsfbox{figures/10_convergence_new.eps}
\caption{ Convergence of the proposed centralized approach ($K=N=2$, $P_i=\sigma_i=1$ for $i=1,2$).}
\label{fig:10_convergence}
\end{figure}
Due to the non-convexity of utility functions w.r.t.
$\{\alpha_{ji}\}$, the convergence of the proposed algorithm to
the global optimum is not guaranteed, but the proposed algorithm
converges to a locally optimal point by the monotone convergence
theorem since the utility function is upper bounded and the
proposed algorithm yields a monotonically increasing sequence of
utility function values. Furthermore, the proposed algorithm is
also stable by the monotone convergence theorem.
Fig. \ref{fig:10_convergence} shows the convergence behavior of the proposed utility function based algorithm for 10 different channel realizations when $K=N=2$, $P_i=\sigma_i=1$ ($i=1,2$), and $u(R_1,R_2)=2R_1 + R_2$. It is seen in the figure that the algorithm converges in a few iterations in most cases. Fig. \ref{fig:learning_curves} shows the convergence behavior of several known rate control algorithms for the same setting as in Fig. \ref{fig:10_convergence} for one channel realization. The considered three algorithms converge to the same value eventually in this case.
It is also seen in Figs. \ref{fig:Pareto1} (a) and (b) that the proposed centralized algorithm yields desired points on the Pareto boundary although it is not theoretically guaranteed.
\begin{table}[!tp]
\caption{A centralized algorithm for determining
$\{\alpha_{ji}\}$.} \label{table:alpha_algorithm} \centering
\begin{tabular}{p{330pt}}
\hline \vspace{-0.8em}
For given channel realization $\{{\bf h}_{ji}, i, j=1,\cdots, K\}$,
noise power $\{\sigma_i^2, i=1,\cdots,K\}$, and a utility function
$u(\{R_i\})$, perform the following procedure to determine interference
leakage levels $\{\alpha_{ji}\}$.
\vspace{0.5em}
\textbf{Initialization:}
$\{\alpha_{ji}^1=0, ~ i,j = 1,\cdots,K, j\neq i\}$,
$\{R_i^0 = 0, ~i=1,\cdots, K\}$, $\epsilon>0$, and $l=1$.
\vspace{0.5em}
\textbf{while}
$\left|
u(\{\alpha_{ji}^l\} ) - u(\{\alpha_{ji}^{l-1}\})\right| > \epsilon $
\vspace{0.5em}
\begin{itemize}
\item[] $l = l+1;$
\item[] \textbf{for}~ $i=1,\cdots, K$,
\item[] \hspace{1em} \textbf{for}
~ $j=1,\cdots,i-1, i+1, \cdots, K$,
\item[] \hspace{2.6em}
$\alpha_{ji}^l ~~=
\underset{0\le \alpha_{ji} \le P_i|{\bf h}_{ji}^H{\bf v}_i^{MF}|^2, }
{\arg\max}\
u\Big(\big\{R_k^l(\{{\bf v}_i^{RZF}(\{\alpha_{ji}^l\})\})\big\}\Big)$
\item[] \hspace{1em} \textbf{end}
\item[] \textbf{end}
\end{itemize}
\textbf{end}
\\ \hline
\end{tabular}
\end{table}
\begin{figure}[H]
\centering
\scalefig{0.5}\epsfbox{figures/learning_curves.eps}
\caption{Convergence of several known algorithms ($K=N=2$,
$P_i=\sigma_i=1$ for $i=1,2$)}.
\label{fig:learning_curves}
\end{figure}
\subsection{A Distributed Heuristic Approach and Practical Considerations}
\label{subsec:distributedControl}
The proposed centralized algorithm in the previous subsection requires central processing with the knowledge of all
$\{{\bf h}_{ji}: i,j=1,2,\cdots,K\}$ and
$\{\sigma_{j}^2: j=1,2,\cdots,K\}$. This reduces the practicality
of the centralized approach when communication among the base stations is limited or experiences large delay as in real systems. Note that the RZFCB framework in Problem
\ref{prob:MISO_RZF_formulation2} itself is distributed.
Transmitter $i$ only needs to know $\{{\bf h}_{ji}, j=1,2,\cdots,K\}$
and $\{\sigma_j^2, j=1,2,\cdots,K\}$ and needs to control
$\{\alpha_{1i},\cdots,\alpha_{i-1,i},\alpha_{i+1,i},
\alpha_{Ki}\}$. In the RZF framework, heuristic rate control is possible with the
knowledge of $\{{\bf h}_{ji}, j=1,2,\cdots,K\}$ and $\{\sigma_j^2,
j=1,2,\cdots,K\}$ at transmitter $i$.
For fully distributed CB operation with limited inter-base station communication, instantaneous information such as the channel vectors should not be exchanged since inter-base station communication delay is typically larger than the channel coherence time for mobile users. One possible way to roughly control the rate-tuple in the network is to design a table composed of sets of interference relaxation parameters, as in the right side of Fig. \ref{fig:Pareto1}, based on the channel statistics.
When the transmitters form a
coordinating cluster, they can negotiate their rates based on
the requests from their receivers for a communication session. In this phase, one set of interference relaxation levels from the table is picked, shared among the base stations, and used during the communication session.
Heuristic guidelines to design the parameter table are based on the RZF parameterization itself.
Note that $\epsilon_i =
\sum_{j\ne i} \alpha_{ij}$ in \eqref{eq:interf_power} is the
additional interference power relative to thermal noise power
$\sigma_i^2$ at receiver $i$ and $\epsilon_i =1$ means
that the SINR of receiver $i$ is lower than the SNR of the same
receiver by 3dB. Thus, the designed interference level should not be too high compared to the thermal noise level.
Furthermore, to (roughly) obtain corner points
of the Pareto boundary of the rate region, another heuristic idea
works. One transmitter should use a nearly MF beam vector, and the
rest of the transmitters should use nearly ZF beam vectors.
More systematic ways based on vast computer simulation can be
considered to design the parameter table. One possible way is as
follows. We first generate a set of channel vectors randomly
according to the channel statistics. For this realized channel
set, we obtain graphs of interference relaxation parameters on the
Pareto boundary. The process is repeated over many different
channel realizations and the best fitting graphs are obtained from
the graphs of interference relaxation parameters of different
channel realizations by some regression model. Finally, the table
is constructed by selecting some points in the best fitting
graphs. The parameter table in the right side of Fig.
\ref{fig:Pareto1} is obtained in this manner for $K=N=2$
when each element of channel vector is i.i.d. zero-mean
complex Gaussian distributed with unit variance and the SNR is 0
dB. Figures \ref{fig:Pareto1} (a) and (b) show the rate control
performance of the parameter table designed in this manner for two
different channel realizations. It is seen that the heuristic
method performs well; the five rate points are all near the Pareto
boundary for each figure.
\begin{figure*}[t]
\begin{minipage}{0.8\textwidth}
\centerline{
\SetLabels
\L(0.25*-0.1) (a) \\
\L(0.76*-0.1) (b) \\
\endSetLabels
\leavevmode
\strut\AffixLabels{
\scalefig{0.5}\epsfbox{figures/Pareto_boundary_K2_convex_new.eps}
\hspace{-2em}
\scalefig{0.5}\epsfbox{figures/Pareto_boundary_K2_nonconvex_new2.eps}
} }
\vspace{2em}
\end{minipage}
\hspace{-2em}
\begin{minipage}{0.15\textwidth}
{\vspace{-2em}
\begin{tabular}{| c | c | c | }
\hline
& $\alpha_{21}$ & $\alpha_{12}$ \\ \hline
1 & $ 0 $ & $0.8511$ \\ \hline
2 & $0.0667$ & $0.5780$ \\ \hline
3 & $0.2444$ & $0.3268$ \\ \hline
4 & $1.0889$ & $0.2393$ \\ \hline
5 & $2.2$ & $0.0735$ \\ \hline
\end{tabular}
}
\end{minipage}
\caption{Performance of RZFCB with the proposed rate control
algorithms: The centralized rate control, marked with $*$,
searches for the weighted sum rate maximizing point. (The weight
vector ${\bf w}$ is shown in the figure.) The distributed rate
control scheme, marked with $+$, sets the interference leakage
levels as shown in the table. In Figs. \ref{fig:Pareto1} (a) and
(b), 'virtual SINR' denotes the rate-tuple obtained by the virtual SINR
(or SLNR) beamforming method in \cite{Zakhour&Gesbert:09WSA}. }
\label{fig:Pareto1}
\end{figure*}
Several advantages in the RZFCB are
summarized below.
\noindent $\bullet$ Real-time fully distributed operation is possible based on the proposed heuristic control approach. Transmitter $i$ only needs to know
$\{{\bf h}_{ji}: j=1,2,\cdots,K\}$ and $\{\sigma_j^2:
j=1,2,\cdots,K\}$.
\noindent $\bullet$ Once transmitter $i$ knows
$\{\alpha_{1i},\cdots,\alpha_{i-1,i},\alpha_{i+1,i},
\alpha_{Ki}\}$, there exists a very fast algorithm, the SOPC
algorithm, to design the RZFCB beam vector. Furthermore, in the
case of $K=3$, there is an approximate closed-form solution.
\noindent $\bullet$ Transmitter $i$ knows its SINR and achievable rate exactly,
and its achievable rate is given by
$R_i = \log\bigg( 1+\frac{| {\bf h}_{ii}^H{\bf v}_i
|^2}{(1+\epsilon_i)\sigma_i^2} \bigg)$.
So, transmission based on this rate will be successful with high
probability. This is true even when $\{\alpha_{ji}\}$ are designed
suboptimally, i.e., away from the Pareto boundary of the rate
region. Thus, the RZFCB scheme is robust.
\noindent $\bullet$ On the contrary to the ZF scheme, RZFCB does not require $N \ge K$.
\section{RZFCB for MIMO Interference Channels}
\label{sec:MIMO_IC}
In this section, we consider the case that both transmitters and
receivers are equipped with multiple antennas i.e., MIMO
interference channels. In the MIMO case, we consider the weighted sum rate maximization under the RZF framework and then propose a
solution to the MIMO RZFCB based on the projected gradient method
\cite{goldstein64}. The rate control idea in the MISO case can be
applied to the MIMO case too.
\subsection{Problem Formulation} \label{subsec:MIMOformulation}
We assume that each receiver has $M$ receive antennas and each
transmitter has $N$ transmit antennas. In this case, the received
signal at receiver $i$ is given by
\begin{equation}
{\bf y}_i = {\bf H}_{ii}{\bf V}_i{\bf s}_i + \sum_{j\neq i}{\bf H}_{ij}{\bf V}_j{\bf s}_j + {\bf n}_i,
\end{equation}
where ${\bf H}_{ij}$ is the $M \times N$ channel matrix from
transmitter $j$ to receiver $i$, ${\bf V}_i$ is the $N \times d_i$
beamforming matrix, ${\bf s}_i$ is the $d_i \times 1$ transmit
symbol vector at transmitter $i$ from a Gaussian codebook with
${\bf s}_i \sim {\mathcal{CN}}(0,{\bf I}_{d_i})$, and
${\bf n}_i\sim{\mathcal{CN}}(0,\sigma_i^2{\bf I})$ is the additive noise.
As in the MISO case, we have a transmit power constraint,
$\|{\bf V}_j\|_F^2 \le P_j$, for transmitter $j$. The proposed RZF constraint
in the MIMO case is given by an inequality with the Frobenius norm
as
\begin{equation}\label{eq:MIMO_RZF}
\|{\bf H}_{ji} {\bf V}_{i} \|_F^2 \le \alpha_{ji}\sigma_j^2,
\quad \forall i, \ j \neq i
\end{equation}
for some constant $\alpha_{ji} \ge 0$. As in the MISO case, the RZF
constraints reduce to ZF constraints when $\alpha_{ji}=0$ for all
$i, j\neq i$. With the MIMO RZF constraints, a cooperative beam
design problem that maximizes the weighted sum rate is formulated as follows:
\vspace{0.5em}
\begin{problem}[RZF cooperative beamforming problem]
\label{prob:co_MIMO}
\begin{eqnarray}
&\underset{\{{\bf V}_i\}}{\mbox{max}} & ~\sum_{i=1}^{K} w_i \log\bigg|
{\bf I}_M+(\sigma_i^2{\bf I}+{\bf B}_i)^{-1}{\bf H}_{ii}{\bf V}_{i}{\bf V}_{i}^H{\bf H}_{ii}^H\bigg|,
\label{eq:MIMO_Problem1} \nonumber\\
&\mbox{subject to}&
{\mathrm{(C. 1)}} \quad \| {{\bf H}}_{ji} {{\bf V}}_i \|_F^2 \le
\alpha_{ji}\sigma_j^2,
\quad \forall i, j \neq i, \label{eq:MIMO_RZFprobConstr1} \nonumber \\
& &
{\mathrm{(C. 2)}} \quad \|{\bf V}_{i}\|_F^2\le P_i, \quad \forall i,
\label{eq:MIMO_RZFprobConstr2}
\end{eqnarray}
where $w_i \ge 0$, $\sum_i w_i =1$, and ${\bf B}_i = \sum_{j\neq i}
{\bf H}_{ij}{\bf V}_{j}{\bf V}_{j}^H{\bf H}_{ij}^H$ is the interference
covariance matrix at receiver $i$.
\end{problem}
\noindent Note that, in Problem \ref{prob:co_MIMO}, the interference
from other transmitters is incorporated in the rate formula through
the interference covariance matrix ${\bf B}_i$ capturing the residual
inter-cell interference under the RZF constraints. As in the MISO
case, we will derive a lower bound on the rate of each user by exploiting the
RZF constraints to convert the joint design problem into a set of separate design problems. Note that, under the RZF constraints, the total
power of interference from undesired transmitters is upper bounded
as
\begin{equation}\label{eq:MIMO_interf_power}
\mbox{tr}({\bf B}_i)
= \sum_{j\neq i} \|{\bf H}_{ij}{\bf V}_{j}\|_F^2
\le \sigma_i^2\sum_{j\neq i} \alpha_{ij}
=: \epsilon_i\sigma_i^2,
\end{equation}
which implies ${\bf B}_i \le \epsilon_i\sigma_i^2 {\bf I}$.
Hassibi and Hochwald derived a lower bound on the ergodic rate of
a MIMO channel with interference \cite{Hassibi&Hochwald:03IT}.
However, their result is not directly applicable
here since the rate here is for an instantaneous channel
realization. Thus, we present a new lower bound under the RZF
interference constraints in the following Lemma.
\begin{lemma} \label{lemma:MIMO_LB}
A lower bound on the rate of receiver $i$ under the RZF constraints
is given by
\begin{eqnarray}\label{eq:MIMO_lower_sum_rate}
& &\hspace{-3em} \log\bigg| {\bf I}_M+(\sigma_i^2{\bf I}+{\bf B}_i)^{-1}
{\bf H}_{ii}{\bf V}_i{\bf V}_i^H{\bf H}_{ii}^H \bigg| \nonumber \\
& &\hspace{1em} \ge
\log\bigg|{\bf I}_M+\frac{1}{\sigma_i^2(1+\epsilon_i)}
{\bf H}_{ii}{\bf V}_i{\bf V}_i^H{\bf H}_{ii}^H \bigg|,
\end{eqnarray}
where $\mbox{tr}({\bf B}_i) = \sum_{j\neq i}\|{\bf H}_{ij}{\bf V}_j\|_F^2 \le \epsilon_i\sigma_i^2$ for all $i$.
\end{lemma}
\begin{proof}
The rate at receiver $i$ is given by
\begin{equation}
\log|{\bf I}+\hbox{$\bf \Phi$}_i^{-1}{\bf A}_i| = \log\prod_{k=1}^M
(1+\lambda_k(\hbox{$\bf \Phi$}_i^{-1}{\bf A}_i))
\end{equation}
where $\hbox{$\bf \Phi$}_i=\sigma_i^2{\bf I}+{\bf B}_i$,
${\bf A}_i={\bf H}_{ii}{\bf V}_i{\bf V}_i^H{\bf H}_{ii}^H$, and $\lambda_k({\bf X})$
denotes the $k$-th largest eigenvalue of ${\bf X}$. By the
Rayleigh-Ritz theorem \cite[p.176]{Horn&Johnson:book}, we have
\begin{equation}
\lambda_M(\hbox{$\bf \Phi$}_i^{-1}{\bf A}_i) \le
\frac{{\bf x}^H{\bf A}_i{\bf x}}{{\bf x}^H\hbox{$\bf \Phi$}_i{\bf x}} =
\frac{{\bf p}^H\hbox{$\bf \Phi$}_i^{-\frac{1}{2}}{\bf A}_i\hbox{$\bf \Phi$}_i^{-\frac{H}{2}}{\bf p}}{{\bf p}^H{\bf p}}
\le \lambda_1(\hbox{$\bf \Phi$}_i^{-1}{\bf A}_i)
\end{equation}
for any non-zero vector ${\bf x}\in{\mathbb{C}}^M$ and
${\bf p}:=\hbox{$\bf \Phi$}_i^{\frac{H}{2}}{\bf x}$. From the Courant-Fischer
theorem \cite[p.179]{Horn&Johnson:book}, the $k$-th largest
generalized eigenvalue of $\hbox{$\bf \Phi$}_i^{-1}{\bf A}_i$, $k=1,\cdots, M$
is given by
\begin{eqnarray}
\lambda_k(\hbox{$\bf \Phi$}_i^{-1}{\bf A}_i) ~~~~ =
\underset{\substack{{\bf p}\neq 0,~ {\bf p}\in{\mathbb{C}}^M,\\
{\bf p}\bot{\bf p}_1,\cdots,{\bf p}_{k-1}}}{\max}
\frac{{\bf p}^H\hbox{$\bf \Phi$}_i^{-\frac{1}{2}}{\bf A}_i\hbox{$\bf \Phi$}_i^{-\frac{H}{2}}{\bf p}}{{\bf p}^H{\bf p}}
\end{eqnarray}
where ${\bf p}_i$ is the eigenvector associated with the $i$-th
largest eigenvalue of
$\hbox{$\bf \Phi$}_i^{-\frac{1}{2}}{\bf A}_i\hbox{$\bf \Phi$}_i^{-\frac{H}{2}}$. Let
${\bf A}_i={\bf U}_i\hbox{$\bf \Sigma$}_i{\bf U}_i^H$ be the eigen-decomposition of
${\bf A}_i$, where $\hbox{$\bf \Sigma$}_i =
\mbox{diag}(\lambda_1({\bf A}_i),\cdots,\lambda_M({\bf A}_i))$. Then,
for all $k$
{\@setsize\small{9pt}\viiipt\@viiipt\let\@listi\@listI
\begin{eqnarray} \lambda_k(\hbox{$\bf \Phi$}_i^{-1}{\bf A}_i) &=&
~~~~
\underset{\substack{{\bf p}\neq 0,~ {\bf p}\in{\mathbb{C}}^M,\\
{\bf p}\bot{\bf p}_1,\cdots,{\bf p}_{k-1}}}{\max}
\frac{{\bf p}^H\hbox{$\bf \Phi$}_i^{-\frac{1}{2}}{\bf A}_i\hbox{$\bf \Phi$}_i^{-\frac{H}{2}}{\bf p}}{{\bf p}^H{\bf p}}, \nonumber \\
&=& ~~~~ \underset{\substack{{\bf p}\neq 0,~ {\bf p}\in{\mathbb{C}}^M,\\
{\bf p}\bot{\bf p}_1,\cdots,{\bf p}_{k-1}}}{\max} \frac{
{\bf p}^H\hbox{$\bf \Phi$}_i^{-\frac{1}{2}}{\bf U}_i
\hbox{$\bf \Sigma$}_i
{\bf U}_i^H\hbox{$\bf \Phi$}_i^{-\frac{H}{2}}{\bf p} }
{{\bf p}^H{\bf p}}, \nonumber \\
&=& \underset{\substack{{\bf z}\neq 0,~ {\bf z}\in{\mathbb{C}}^M,\\
{\tiny \hbox{$\bf \Phi$}_i^{H/2}}{\bf U}_i{\bf z}\bot {\bf p}_1,\cdots,{\bf p}_{k-1}
}}{\max}
\frac{{\bf z}^H\hbox{$\bf \Sigma$}_i{\bf z}}{{\bf z}^H{\bf U}_i^H\hbox{$\bf \Phi$}_i{\bf U}_i{\bf z}}, \nonumber \\
& & \qquad\quad ({\bf z}:= {\bf U}_i^H\hbox{$\bf \Phi$}_i^{-\frac{H}{2}}{\bf p}), \nonumber \\
&\stackrel{(a)}{\ge}&
\underset{\substack{{\bf z} \ne 0,~ ||{\bf z}||=1,~ {\bf z}\in{\mathbb{C}}^M,\\
z_{k+1} = z_{k+2} = \cdots = z_M = 0, \\
{\tiny \hbox{$\bf \Phi$}_i^{H/2}}{\bf U}_i{\bf z}\bot
{\bf p}_1,\cdots,{\bf p}_{k-1},}}{\max}
\frac{{\bf z}^H\hbox{$\bf \Sigma$}_i{\bf z}}{{\bf z}^H{\bf U}_i^H\hbox{$\bf \Phi$}_i{\bf U}_i{\bf z}}, \nonumber \\
& & \qquad\quad ({\bf z}=[z_1,z_2,\cdots,z_M]^T)\nonumber \\
&=&
\underset{\substack{{\bf z} \ne 0,~ ||{\bf z}||=1,~ {\bf z}\in{\mathbb{C}}^M,\\
z_{k+1} = z_{k+2} = \cdots = z_M = 0, \\
{\tiny \hbox{$\bf \Phi$}_i^{H/2}}{\bf U}_i{\bf z}\bot
{\bf p}_1,\cdots,{\bf p}_{k-1},}}{\max}
\frac{\sum_{j=1}^k\lambda_j({\bf A}_i)|z_j|^2}
{{\bf z}^H{\bf U}_i^H\hbox{$\bf \Phi$}_i{\bf U}_i{\bf z} }, \nonumber \\
&\stackrel{(b)}{\ge}&
\frac{\lambda_k({\bf A}_i)}{\lambda_1(\hbox{$\bf \Phi$}_i)},
\label{eq:MIMOeigenBound}
\end{eqnarray}
}
where (a) is satisfied since the feasible set for ${\bf z}$ is reduced and (b) is satisfied
since $||{\bf z}||^2=|z_1|^2+\cdots+|z_k|^2=1$, $\lambda_1({\bf A}_i)\ge \cdots \ge
\lambda_k({\bf A}_i)$, and ${\bf z}^H{\bf U}_i^H\hbox{$\bf \Phi$}_i{\bf U}_i{\bf z} \le
\lambda_1(\hbox{$\bf \Phi$}_i)$ by Rayleigh-Ritz theorem.
Based on \eqref{eq:MIMOeigenBound}, a lower bound on the
rate is given by
\begin{align}
\log|{\bf I}+\hbox{$\bf \Phi$}_i^{-1}{\bf A}_i|
\ge
\left(1+\frac{\lambda_k({\bf A}_i)}{\lambda_1(\hbox{$\bf \Phi$}_i)}\right).
\end{align}
Since $\hbox{$\bf \Phi$}_i = \sigma_i^2{\bf I} + \sum_{j\neq i} {\bf H}_{ij}{\bf V}_j{\bf V}_j^H{\bf H}_{ij}^H$, we have
$\lambda_1(\hbox{$\bf \Phi$}_i) = \sigma_i^2 + \lambda_1\left(\sum_{j\neq i}
{\bf H}_{ij}{\bf V}_j{\bf V}_j^H{\bf H}_{ij}^H\right)$, where the maximum
eigenvalue of the interference covariance matrix is upper bounded
by
$\lambda_1\Big(\sum_{j\neq i}
{\bf H}_{ij}{\bf V}_j{\bf V}_j^H{\bf H}_{ij}^H\Big)
\le \mbox{tr}\Big(\sum_{j\neq i} {\bf H}_{ij}{\bf V}_j{\bf V}_j^H{\bf H}_{ij}^H\Big)
= \sum_{j\neq i}\alpha_{ij}\sigma_i^2=\epsilon_i\sigma_i^2$.
Thus, a lower bound of rate at receiver $i$ is given by
\[
|{\bf I}+\hbox{$\bf \Phi$}_i^{-1}{\bf A}_i|
\ge
\Big|{\bf I}_M+\frac{1}{\sigma_i^2(1+\epsilon_{i})}{\bf A}_i\Big|.
\]
\end{proof}
\noindent Note that in \eqref{eq:MIMO_lower_sum_rate} the
inter-user dependency is removed and the beam design can be
performed at each transmitter in a distributed manner. Based on
the lower bound \eqref{eq:MIMO_lower_sum_rate}, the RZFCB problem
is now formulated as a distributed problem:
\begin{problem}[The MIMO RZFCB problem] \label{prob:MIMORZFCBProblem}
\begin{eqnarray} \label{eq:MIMOProblem}
&\underset{{\bf V}_i}{\mbox{max}}& \phi_i({\bf V}_i) :=
\log\left|{\bf I}_M+\frac{1}{\sigma_i^2(1+\epsilon_i)}{\bf H}_{ii}{\bf V}_{i}{\bf V}_{i}^H{\bf H}_{ii}^H
\right|, \nonumber \\
&\mbox{subject to}& {\mathrm{(C.1)}} \quad \|{\bf H}_{ji} {\bf V}_{i}\|_F^2 \le
\alpha_{ji}\sigma_j^2, \quad \forall j\neq i, \nonumber \\
& & {\mathrm{(C.2)}} \quad \|{\bf V}_{i}\|_F^2\le P_i,
\end{eqnarray}
for each transmitter $i = 1, 2, \cdots, K$.
\end{problem}
\noindent Note that Problem \ref{prob:MIMORZFCBProblem} is now fully distributed.
One of several known algorithms for constrained optimization
can be used to solve Problem \ref{prob:MIMORZFCBProblem} for given
$\{\alpha_{ji}\}$. In particular, we choose to use the projected
gradient method (PGM) by Goldstein \cite{goldstein64}. The proposed PGM-based beam design algorithm for MIMO ICs is provided in
Table \ref{table:algorithm}. Detailed explanation of the beam design with PGM algorithm is
provided in \cite{Park&Lee&Sung&Yukawa:12Arxiv}.
\begin{figure*}[t]
\centerline{
\SetLabels
\L(0.16*-0.1) (a) \\
\L(0.49*-0.1) (b) \\
\L(0.82*-0.1) (c) \\
\endSetLabels
\leavevmode
\strut\AffixLabels{
\scalefig{0.33}\epsfbox{figures/mimo_K3N6M2_sumrate.eps}
\scalefig{0.33}\epsfbox{figures/mimo_K3N8M2_sumrate.eps}
\scalefig{0.33}\epsfbox{figures/mimo_K4N6M2_sumrate.eps} } }
\vspace{1.5em} \caption{Sum rate of RZFCB: (a) (K,M,N)=(3, 2, 6),
(b) (K,M,N)=(3, 2, 8), and (c) (K,M,N)=(4, 2, 6).} \label{fig:sumrate}
\end{figure*}
\begin{table}[ht]
\caption{Beam design algorithm for MIMO IC using PGM.}
\label{table:algorithm} \centering
\begin{tabular}{p{310pt}}
\hline \vspace{-0.8em}
\normalsize{ For each transmitter $i\in\{1,\cdots, K\}$,
\textit{0.} Initialize ${\bf V}_i$ as the ZF beamforming matrix.
\textit{1.} Compute gradient of $\phi({\bf V}_i)$.
\textit{2.} Perform a steepest descent shift of ${\bf V}_i$.
\textit{3.} Perform successive metric projections of ${\bf V}_i$ onto
constraint sets.
\textit{4.} Go to Step 1 and repeat until the relative difference
of $\phi_i({\bf V}_i)$ is
~~ less than a pre-determined threshold. }
\\
\hline
\end{tabular}
\end{table}
\vspace{-2em}
\subsection{Numerical Results}
In this section, we provide some numerical results for the
performance of RZFCB in the MIMO case.
We consider three MIMO
interference channels with system parameters $(K,M,N)=(3, 2, 6)$,
$(3, 2, 8)$, and $(4, 2, 6)$. In each case, we set $\alpha_{ji}=0.01, 0.1$ and $0.2$ for all $i$ and $j$. The step size parameter
for the PGM is chosen to be $0.01$ for all iterations.
Figures \ref{fig:sumrate} (a), (b), and (c) show the sum rate
performance of the ZFCB and RZFCB averaged over 30 independent
channel realizations. In Fig. \ref{fig:sumrate} (a) it is seen that the RZFCB
outperforms the ZFCB at all SNR and the gain of the RZFCB over the
ZFCB at low SNR is large when $N=KM$. This large gain at low SNR is especially
important because most cell-edge receivers operate in the low SNR
regime. In Fig. \ref{fig:sumrate} (b) it is seen that the ZF scheme performs well when the number of TX antenna is more than enough and the dimension of ZF beams is large, as expected. In the case of $N < KM$ as in Fig. (c), the ZFCB is infeasible but the RZFCB still works well.
\section{Conclusion}
\label{sec:conclusion}
We have considered coordinated beamforming for MISO and MIMO
interference channels under the RZF framework. In the MISO case, we
have shown that the SOPC strategy with a set of well chosen
interference relaxation levels is Pareto-optimal. We have provided (approximate) closed-form solutions for the
SOPC strategy in the cases of two and three users and the SOPC
algorithm in the general case for a given set of
interference relaxation levels. In the MIMO case, we have
formulated the RZFCB problem as a distributed optimization
problem based on a newly derived rate lower bound and have
provided an algorithm based on the PGM to solve the MIMO RZFCB
beam design problem. We have also considered the rate control
problem under the RZFCB framework and have provided a centralized
approach and a fully-distributed heuristic approach to control the
rate-tuple location roughly along the Pareto boundary of the achievable
rate region. Numerical results validate the RZFCB paradigm.
\vspace{-0.5em}
|
{
"timestamp": "2012-10-03T02:02:11",
"yymm": "1203",
"arxiv_id": "1203.1758",
"language": "en",
"url": "https://arxiv.org/abs/1203.1758"
}
|
\section{Introduction and preliminary results}
The inequality of Ostrowski gives us an estimate for the deviation of the
values of a smooth function from its mean value. More precisely, if
f:[a,b]\rightarrow \mathbb{R}$ is a differentiable function with bounded
derivative, the
\begin{equation*}
\left\vert f(x)-\frac{1}{b-a}\int\limits_{a}^{b}f(t)dt\right\vert \leq \left[
\frac{1}{4}+\frac{(x-\frac{a+b}{2})^{2}}{(b-a)^{2}}\right] (b-a)\left\Vert
f^{\prime }\right\Vert _{\infty }
\end{equation*
for every $x\in \lbrack a,b]$. Moreover the constant $1/4$ is the best
possible.
For some generalizations of this classic fact see the book \cite[p.468-484
{[13]} by Mitrinovic, Pecaric and Fink. A simple proof of this fact can be \
done by using the following identity \cite{[13]}:
If $f:[a,b]\rightarrow \mathbb{R}$ is differentiable on $[a,b]$ with the
first derivative $f^{\prime }$ integrable on $[a,b],$ then Montgomery
identity holds
\begin{equation*}
f(x)=\frac{1}{b-a}\int\limits_{a}^{b}f(t)dt+\int\limits_{a}^{b}P_{1}(x,t)f^
\prime }(t)dt,
\end{equation*
where $P_{1}(x,t)$ is the Peano kernel defined b
\begin{equation*}
P_{1}(x,t):=\left\{
\begin{array}{ll}
\dfrac{t-a}{b-a}, & a\leq t<x \\
& \\
\dfrac{t-b}{b-a}, & x\leq t\leq b
\end{array
\right.
\end{equation*}
This inequality provides an upper bound for the approximation of integral
mean of a function $f$ by the functional value $f(x)$ at $x\in \lbrack a,b].$
In 2001, , Cheng \cite{[1]} proved the following Ostrowski-Gr\"{u}ss type
integral inequality.
\begin{theorem}
\label{t1} Let $I\subset
\mathbb{R}
$ be an open interval, $a,b\in I,a<b$. If $f:I\rightarrow
\mathbb{R}
$ is a differentiable function such that there exist constants $\gamma
,\Gamma \in
\mathbb{R}
$, with $\gamma \leq f^{^{\prime }}\left( x\right) \leq \Gamma $, $x\in
\left[ a,b\right] $. Then hav
\begin{eqnarray}
&&\left\vert \frac{1}{2}f\left( x\right) -\frac{\left( x-b\right) f\left(
b\right) -\left( x-a\right) f\left( a\right) }{2\left( b-a\right) }-\frac{1}
b-a}\int\limits_{a}^{b}f\left( t\right) dt\right\vert \label{hh} \\
&& \notag \\
&\leq &\frac{\left( x-a\right) ^{2}+\left( b-x\right) ^{2}}{8\left(
b-a\right) }\left( \Gamma -\gamma \right) \text{, for all }x\in \left[ a,
\right] . \notag
\end{eqnarray}
\end{theorem}
Theorem \ref{t1} is a generalization of the following Ostrowski-Gr\"{u}ss
type integral inequality, which was firstly by Dragomir and Wang in \cit
{[2]} and further improved by Matic et al. in \cite{[5]}.
\begin{theorem}
\label{t2} Let the assumptions of Theorem \ref{t1} hold. Then for all $x\in
\left[ a,b\right] $, we have
\end{theorem}
\begin{eqnarray}
&&\left\vert f\left( x\right) -\frac{f\left( b\right) -f\left( a\right) }{b-
}\left( x-\frac{a+b}{2}\right) -\frac{1}{b-a}\int\limits_{a}^{b}f\left(
t\right) dt\right\vert \label{hhh} \\
&& \notag \\
&\leq &\frac{1}{4}\left( b-a\right) \left( \Gamma -\gamma \right) . \notag
\end{eqnarray}
The above two inequalities are both connections between the Ostrowski
inequality \cite{[3]}\ and the Gr\"{u}ss inequality \cite{[4]}\ and can be
applied to bound some special mean and some numerical quadrature rules.
During the past few years many researchers have given considerable attention
to the above inequalities and various generalizations, extensions and
variants of these inequalities have appeared in the literature, see \cit
{[1]}, \cite{[2]}, \cite{[5]}, \cite{[24]} and the references cited therein.
For recent results and generalizations concerning Ostrowski and Gr\"{u}ss
inequalities, we refer the reader to the recent papers \cite{[1]}-\cite{[5]
, \cite{[13]}-\cite{[16]}, \cite{[18]}-\cite{[24]}.
The theory of fractional calculus has known an intensive development over
the last few decades. It is shown that derivatives and\ integrals of
fractional type provide an adequate mathematical modelling of real objects\
and processes see (\cite{[6]}, \cite{[7]}, \cite{[9]}, \cite{[10]}, \cit
{[21]}, \cite{[22]}). Therefore, the study of fractional differential
equations need more developmental of inequalities of fractional type. The
main aim of this work is to develop new weighted Montgomery identity for
Riemann-Liouville fractional integrals that will be used to establish new
weighted Ostrowski inequalities. Let us begin by introducing this type of
inequality.
In \cite{[6]} and \cite{[21]}, the authors established some inequalities for
differentiable mappings which are connected with Ostrowski type inequality
by used the Riemann-Liouville fractional integrals, and they used the
following lemma to prove their results:
\begin{lemma}
\label{l} Let $f:I\subset \mathbb{R}\rightarrow \mathbb{R}$ be
differentiable function on $I^{\circ }$ with $a,b\in I$ ($a<b$) and
f^{\prime }\in L_{1}[a,b]$, the
\begin{equation}
f(x)=\frac{\Gamma (\alpha )}{b-a}(b-x)^{1-\alpha }{\Large J}_{a}^{\alpha
}f(b)-{\Large J}_{a}^{\alpha -1}(P_{2}(x,b)f(b))+{\Large J}_{a}^{\alpha
}(P_{2}(x,b)f^{^{\prime }}(b)),\ \ \ \alpha \geq 1, \label{z}
\end{equation
where $P_{2}(x,t)$ is the fractional Peano kernel defined b
\begin{equation*}
P_{2}(x,t)=\left\{
\begin{array}{ll}
\dfrac{t-a}{b-a}(b-x)^{1-\alpha }\Gamma (\alpha ), & a\leq t<x \\
& \\
\dfrac{t-b}{b-a}(b-x)^{1-\alpha }\Gamma (\alpha ), & x\leq t\leq b
\end{array
\right.
\end{equation*}
\end{lemma}
In this article, we use the Riemann-Liouville fractional integrals to
establish some new integral inequalities of Ostrowski-Gr\"{u}ss type. From
our results, the classical Ostrowski-Gr\"{u}ss type inequalities can be
deduced as some special cases.
Firstly, we give some necessary definitions and mathematical preliminaries
of fractional calculus theory which are used further in this paper. More
details, one can consult \cite{[12]}, \cite{[17]}.
\begin{definition}
The Riemann-Liouville fractional integral operator of order $\alpha \geq 0$
with $a\geq 0$ is defined a
\begin{eqnarray*}
J_{a}^{\alpha }f(x) &=&\frac{1}{\Gamma (\alpha )}\dint\limits_{a}^{x}(x-t)^
\alpha -1}f(t)dt, \\
J_{a}^{0}f(x) &=&f(x).
\end{eqnarray*}
\end{definition}
Recently, many authors have studied a number of inequalities by used the
Riemann-Liouville fractional integrals, see (\cite{[6]}, \cite{[7]}, \cit
{[9]}, \cite{[10]}, \cite{[21]}, \cite{[22]}) and the references cited
therein.
\section{Main Results}
\begin{lemma}
\label{lm} Let $f:I\subset \mathbb{R}\rightarrow \mathbb{R}$ be a
differentiable function on $I^{\circ }$ with $a,b\in I$ ($a<b$)$,$ $\alpha
\geq 1$ and $f^{\prime }\in L_{1}[a,b]$, then the generalization of the
Montgomery identity for fractional integrals holds
\begin{eqnarray}
f\left( x\right) &=&(\alpha +1)\Gamma (\alpha )\frac{\left( b-x\right)
^{1-\alpha }}{(b-a)}{\Large J}_{a}^{\alpha }f(b)-{\Large J}_{a}^{\alpha
-1}(P_{2}(x,b)f(b))-\frac{\left( b-x\right) ^{2-\alpha }}{(b-a)}\Gamma
(\alpha )J_{a}^{\alpha -1}f\left( b\right) \notag \\
&& \label{9} \\
&&-\frac{\left( b-x\right) ^{1-\alpha }(x-a)}{\left( b-a\right) ^{2-\alpha }
f(a)+2J_{a}^{\alpha }\left( K_{1}\left( x,b\right) f^{^{\prime }}\left(
b\right) \right) , \notag
\end{eqnarray
where $K_{1}\left( x,t\right) $ is the fractional Peano kernel defined b
\begin{equation}
K_{1}\left( x,t\right) :=\left\{
\begin{array}{ll}
\left( t-\dfrac{a+x}{2}\right) \dfrac{\left( b-x\right) ^{1-\alpha }}{b-a
\Gamma \left( \alpha \right) , & t\in \lbrack a,x) \\
\left( t-\dfrac{b+x}{2}\right) \dfrac{\left( b-x\right) ^{1-\alpha }}{b-a
\Gamma \left( \alpha \right) , & t\in \lbrack x,b]
\end{array
\right. \label{91}
\end{equation}
\end{lemma}
\begin{proof}
By definition of $K_{1}\left( x,t\right) $, we hav
\begin{eqnarray*}
&&J_{a}^{\alpha }\left( K_{1}\left( x,b\right) f^{^{\prime }}\left( b\right)
\right) \\
&& \\
&=&\frac{1}{\Gamma \left( \alpha \right) }\dint\limits_{a}^{b}\left(
b-t\right) ^{\alpha -1}K_{1}\left( x,t\right) f^{\prime }\left( t\right) dt
\\
&& \\
&=&\frac{\left( b-x\right) ^{1-\alpha }}{b-a}\left[ \dint\limits_{a}^{x
\left( b-t\right) ^{\alpha -1}\left( t-\frac{a+x}{2}\right) f^{^{\prime
}}\left( t\right) dt+\dint\limits_{x}^{b}\left( b-t\right) ^{\alpha
-1}\left( t-\frac{b+x}{2}\right) f^{^{\prime }}\left( t\right) dt\right]
\end{eqnarray*
that can be written a
\begin{equation}
J_{a}^{\alpha }\left( K_{1}\left( x,b\right) f^{^{\prime }}\left( b\right)
\right) =\frac{1}{2}{\Large J}_{a}^{\alpha }(P_{2}(x,b)f^{^{\prime }}(b))
\frac{\left( b-x\right) ^{1-\alpha }}{2(b-a)}\dint\limits_{a}^{b}\left(
b-t\right) ^{\alpha -1}\left( t-x\right) f^{^{\prime }}\left( t\right) dt.
\label{10}
\end{equation
For term in the right hand side of (\ref{10}) integrating by parts implies
that
\begin{eqnarray}
&&\dint\limits_{a}^{b}\left( b-t\right) ^{\alpha -1}\left( t-x\right)
f^{^{\prime }}\left( t\right) dt \notag \\
&=&\left( b-x\right) \dint\limits_{a}^{b}\left( b-t\right) ^{\alpha
-1}f^{^{\prime }}\left( t\right) dt-\dint\limits_{a}^{b}\left( b-t\right)
^{\alpha }f^{^{\prime }}\left( t\right) dt \label{11} \\
&& \notag \\
&=&(x-a)\left( b-a\right) ^{\alpha -1}f(a)+\left( b-x\right) \Gamma (\alpha
)J_{a}^{\alpha -1}f\left( b\right) -\Gamma (\alpha +1)J_{a}^{\alpha }f\left(
b\right) \notag
\end{eqnarray}
Substituting ${\Large J}_{a}^{\alpha }(P_{2}(x,b)f^{^{\prime }}(b))$ in the
Lemma \ref{l} and (\ref{11}) in (\ref{10}), we obtain tha
\begin{eqnarray*}
&&J_{a}^{\alpha }\left( K_{1}\left( x,b\right) f^{^{\prime }}\left( b\right)
\right) \\
&& \\
&=&\frac{1}{2}f(x)-(\alpha +1)\Gamma (\alpha )\frac{\left( b-x\right)
^{1-\alpha }}{2(b-a)}{\Large J}_{a}^{\alpha }f(b)+\frac{1}{2}{\Large J
_{a}^{\alpha -1}(P_{2}(x,b)f(b)) \\
&& \\
&&+\frac{\left( b-x\right) ^{1-\alpha }(x-a)}{2}\left( b-a\right) ^{\alpha
-2}f(a)+\frac{\left( b-x\right) ^{2-\alpha }}{2(b-a)}\Gamma (\alpha
)J_{a}^{\alpha -1}f\left( b\right) .
\end{eqnarray*
The proof is completed.
\end{proof}
\begin{remark}
Letting $\alpha =1,$ formula (\ref{9}) reduces the following identit
\begin{equation*}
\frac{1}{2}f\left( x\right) =\frac{1}{(b-a)}\dint\limits_{a}^{b}f(t)dt+\frac
\left( x-b\right) f\left( b\right) -(x-a)f(a)}{2(b-a)}+\dint\limits_{a}^{b}
\left( x,t\right) f^{^{\prime }}(t)dt
\end{equation*
wher
\begin{equation*}
K\left( x,t\right) :=\left\{
\begin{array}{ll}
\left( t-\dfrac{a+x}{2}\right) , & t\in \lbrack a,x) \\
\left( t-\dfrac{b+x}{2}\right) , & t\in \lbrack x,b
\end{array
\right.
\end{equation*
which was given by Tong and Guan in \cite{[24]}. Using the above identity,
the authors proved another simple proof of Theorem \ref{t1}.
\end{remark}
Now using the new Montgomery identity for fractional integrals (\ref{9}) and
the corresponding fractional Peano kernel (\ref{91}), we derive a new
Ostrowski-Gr\"{u}ss type inequality of fractional type.
\begin{theorem}
Let\ $f$ be a differentiable \ function on $\left[ a,b\right] $ and
\left\vert f^{\prime }\left( x\right) \right\vert \leq M$ for any $x\in
\left[ a,b\right] $. Then the following fractional inequality holds
\begin{eqnarray}
&&\left\vert \frac{1}{2}f\left( x\right) -(\alpha +1)\Gamma (\alpha )\frac
\left( b-x\right) ^{1-\alpha }}{2(b-a)}{\Large J}_{a}^{\alpha }f(b)+\frac{1}
2}{\Large J}_{a}^{\alpha -1}(P_{2}(x,b)f(b))\right. \notag \\
&& \label{s} \\
&&\left. +\frac{\left( b-x\right) ^{2-\alpha }}{2(b-a)}\Gamma (\alpha
)J_{a}^{\alpha -1}f\left( b\right) +\frac{\left( b-x\right) ^{1-\alpha }(x-a
}{2\left( b-a\right) ^{2-\alpha }}f(a)\right\vert \notag \\
&& \notag \\
&\leq &\frac{M\left( b-x\right) ^{1-\alpha }}{\left( b-a\right) }\left[
\frac{\left( b-a\right) ^{\alpha }(x-a)+\left( b-x\right) ^{\alpha }(a+b-2x
}{2\alpha }\right] \notag
\end{eqnarray
for $\alpha \geq 1.$
\end{theorem}
\begin{proof}
From Lemma \ref{lm}, we hav
\begin{eqnarray*}
&&\left\vert \frac{1}{2}f\left( x\right) -(\alpha +1)\Gamma (\alpha )\frac
\left( b-x\right) ^{1-\alpha }}{2(b-a)}{\Large J}_{a}^{\alpha }f(b)+\frac{1}
2}{\Large J}_{a}^{\alpha -1}(P_{2}(x,b)f(b))\right. \\
&& \\
&&\left. +\frac{\left( b-x\right) ^{2-\alpha }}{2(b-a)}\Gamma (\alpha
)J_{a}^{\alpha -1}f\left( b\right) +\frac{\left( b-x\right) ^{1-\alpha }(x-a
}{2\left( b-a\right) ^{2-\alpha }}f(a)\right\vert \\
&& \\
&=&\frac{1}{\Gamma \left( \alpha \right) }\left\vert \int_{a}^{b}\left(
b-t\right) ^{\alpha -1}K_{1}\left( x,t\right) f^{\prime }\left( t\right)
dt\right\vert .
\end{eqnarray*
Taking into account the assumptions on the function $f$, it yield
\begin{eqnarray}
&&\left\vert \frac{1}{2}f\left( x\right) -(\alpha +1)\Gamma (\alpha )\frac
\left( b-x\right) ^{1-\alpha }}{2(b-a)}{\Large J}_{a}^{\alpha }f(b)+\frac{1}
2}{\Large J}_{a}^{\alpha -1}(P_{2}(x,b)f(b))\right. \notag \\
&& \notag \\
&&\left. +\frac{\left( b-x\right) ^{2-\alpha }}{2(b-a)}\Gamma (\alpha
)J_{a}^{\alpha -1}f\left( b\right) +\frac{\left( b-x\right) ^{1-\alpha }(x-a
}{2\left( b-a\right) ^{2-\alpha }}f(a)\right\vert \notag \\
&& \label{14} \\
&\leq &\frac{M\left( b-x\right) ^{1-\alpha }}{\left( b-a\right) }\left[
\dint\limits_{a}^{x}\left( b-t\right) ^{\alpha -1}\left\vert t-\frac{a+x}{2
\right\vert dt\right. \notag \\
&& \notag \\
&&\left. +\dint\limits_{x}^{b}\left( b-t\right) ^{\alpha -1}\left\vert t
\frac{b+x}{2}\right\vert dt\right] . \notag
\end{eqnarray
Noting the left hand side of (\ref{14}) by $I$ then integrating by parts the
right hand side of (\ref{14}), we obtai
\begin{equation*}
I\leq \frac{M\left( b-x\right) ^{1-\alpha }}{2\left( b-a\right) }\left[
\frac{\left( b-a\right) ^{\alpha }-\left( b-x\right) ^{\alpha }}{\alpha
\left( x-a\right) +\frac{\left( b-x\right) ^{\alpha +1}}{\alpha }\right] .
\end{equation*
Consequentl
\begin{equation*}
I\leq \frac{M\left( b-x\right) ^{1-\alpha }}{\left( b-a\right) }\left[ \frac
\left( b-a\right) ^{\alpha }(x-a)+\left( b-x\right) ^{\alpha }(a+b-2x)}
2\alpha }\right] .
\end{equation*
This completes the proof.
\end{proof}
\begin{remark}
Letting $\alpha =1,$ formula (\ref{s}) reduces the following inequality
\begin{eqnarray}
&&\left\vert \frac{1}{2}f\left( x\right) -\frac{\left( x-b\right) f\left(
b\right) -(x-a)f(a)}{2(b-a)}-\frac{1}{(b-a)}\dint\limits_{a}^{b}f(t)dt\righ
\vert \notag \\
&& \notag \\
&\leq &\frac{M}{\left( b-a\right) }\left[ \frac{(x-a)^{2}+\left( b-x\right)
^{2}}{2}\right] . \notag
\end{eqnarray
which connected with Ostrowski-Gr\"{u}ss type integral inequality. If we
take $x=\frac{a+b}{2}$ in this inequality, it follows tha
\begin{eqnarray}
&&\left\vert \frac{1}{2}f\left( \frac{a+b}{2}\right) +\frac{f\left( b\right)
+f(a)}{4}-\frac{1}{(b-a)}\dint\limits_{a}^{b}f(t)dt\right\vert \notag \\
&& \notag \\
&\leq &\frac{M\left( b-a\right) }{4}. \notag
\end{eqnarray}
\end{remark}
|
{
"timestamp": "2012-03-13T01:01:32",
"yymm": "1203",
"arxiv_id": "1203.2283",
"language": "en",
"url": "https://arxiv.org/abs/1203.2283"
}
|
\section{Introduction}
Frustrated magnets have attracted much attention in recent years.
Exotic spin liquid phases are of special interest which have been
found in some of them \cite{Balents}. A chiral spin liquid phase
is an example of such an exotic state of matter in which there are
neither quasi long-range nor long-range magnetic orders but a
chiral order parameter $\langle {\bf S}_i\times{\bf S}_j\rangle$
is nonzero. Existence of such a phase is discussed in context of
one-dimensional frustrated quantum magnetic systems\cite{Chain},
and it is found experimentally in Ref.~\cite{ChainExp}.
In larger dimensions, one of the systems in which the chiral spin
liquid phase can be found at finite temperature is a classical
planar (XY) helimagnet with $\mathbb{Z}_2\otimes SO(2)$ symmetry
in which the helical structure results from a competition of
exchange interactions between localized spins. Critical behavior
of spin systems from this class is described by two order
parameters. Besides the conventional magnetization with $SO(2)$
symmetry, one has to take into account also the chiral order
parameter that is an Ising variable with $\mathbb{Z}_2$ symmetry.
This parameter characterizes the direction of the helix twist and
distinguishes left-handed and right-handed helical structures.
In three-dimensional helimagnets, the phase transitions on the
magnetic and the chiral order parameters occur simultaneously. It
was found numerically that the transition is of the weak first
order or of the "almost second order"\cite{Zumbach, DMT} type in
helical antiferromagnets on a body-centered tetragonal lattice
\cite{Helix} and on a simple cubic lattice with an extra competing
exchange coupling along one axis \cite{Sorokin}. These systems
belong to the same (pseudo)universality class as, e.g., the model
on a stacked-triangular lattice \cite{Kaw1} and $V_{2,2}$ Stiefel
model \cite{KunZumb}. The possibility of existence and
stabilization of the chiral spin liquid phase by, e.g.,
Dzyaloshinsky-Moria interaction in 3D helimagnets is discussed
recently in Refs.~\cite{Onoda}.
In two dimensions, the situation is rather different
\cite{Olsson1}. Two successive transitions were observed with the
temperature decreasing. The chiral order appears as a result of
the first transition that is of the Ising type. Another one is the
Berezinskii-Kosterlitz-Thouless (BKT) transition driven by the
unbinding of vortex-antivortex pairs \cite{BKT}. Then, the chiral
spin liquid phase arises between these transitions with the chiral
order and without a magnetic one. Various 2D systems from the
class $\mathbb{Z}_2\otimes SO(2)$ were investigated numerically
(see Ref. \cite{Hasenbush} for review): triangular antiferromagnet
\cite{Triangular, TriangNNN}, J$_1$-J$_2$ model \cite{JJ}, the
Coulomb gas system of half-integer charges \cite{Gas}, two coupled
XY models \cite{XY-XY}, Ising-XY model \cite{Ising-XY, Hasenbush,
CFT} and the generalized fully frustrated XY model \cite{GFFXY}.
And surely, the most famous of them is the fully frustrated XY
model (FFXY) introduced by Villain \cite{Villain}. This model is
of great interest because it describes a superconducting array of
Josephson junctions under an external transverse magnetic field
\cite{Teitel}. It was found that the temperature of the Ising
transition $T_I$ is $1\mbox{-}3\%$ larger than that of the BKT
transition for the most of above-named systems \cite{Teitel,
Olsson1, FFXY, Hasenbush}.
Korshunov argued \cite{Korshunov} that a phase transition, driven
by unbinding of kink-antikink pairs on the domain walls associated
with the $\mathbb{Z}_2$ symmetry, can take place in models similar
to 2D FFXY one at temperatures appreciably smaller than $T_{BKT}$
(see also Ref.~\cite{Kinks}). Such a transition could lead to a
decoupling of phase coherence across domain boundaries, producing
in this way two separate bulk transitions with $T_{BKT}<T_{I}$
\cite{Tw}. It was pointed out however in Ref.~\cite{Korshunov}
that these two continuous transitions can merge into a single
first order one. These conclusions do not depend on the particular
form of interactions in system as soon as the ground state
degeneracy remains the same. They are confirmed by numerical
studies of the models mentioned above \cite{Teitel, Olsson1, FFXY,
Hasenbush, JJ, Triangular, TriangNNN, Gas}.
Nevertheless the situation remains contradictory in 2D helimagnets
belonging to the same $\mathbb{Z}_2\otimes SO(2)$ class as FFXY
model and the antiferromagnet on the triangular lattice. Garel and
Doniach \cite{Garel} (see also \cite{Okwamoto}) considered the
simplest helimagnet on a square lattice with an extra competing
exchange coupling along one axis that is describing by the
Hamiltonian
\begin{equation}
H=\sum\limits_{\mathbf{x}}\Bigl(J_1\cos(\fix-\fixa)
+J_2\cos(\fix-\fixaa)
-J_b\cos(\fix-\fixb)\Bigr),
\label{ham}
\end{equation}
where the sum runs over sites $\mathbf{x}=(x_a,x_b)$ of the
lattice, $\mathbf{a}=(1,0)$ and $\mathbf{b}=(0,1)$ are unit
vectors of the lattice, the coupling constants $J_{1,2}$ are
positive. Using arguments of Ref.~\cite{Einhorn}, they concluded
\cite{Garel} that at low temperatures the vertices are bound by
strings, which would inhibit the BKT transition and make the Ising
transition occur first with the temperature increasing. Kolezhuk
noticed \cite{Kolezhuk} that those arguments are not valid for a
helimagnet, and showed that the Ising transition temperature is
larger than the BKT one at least near the Lifshitz point
$J_2=J_1/4$. It was found by Monte Carlo simulations in the recent
paper \cite{Cinti} that $T_{BKT}>T_{I}$ at $J_2=0.3$ and
$J_1=J_b=1$ (i.e., very near the Lifshitz point) in accordance
with Ref.~\cite{Garel} and in contrast to Ref.~\cite{Kolezhuk}.
To account for the discordance between results for helical magnets
and the general arguments for $\mathbb{Z}_2\otimes SO(2)$ class,
we perform extensive Monte Carlo simulations of the model (1) for
different values of $J_2$. We obtain reliable results at
$J_2>0.4J_1$ which show that $T_{BKT}<T_I$. On the other hand the
value of $T_I$ close to the Lifshitz point is hiding among effects
of the finite size scaling and is not accessible for ordinary
estimation methods. We obtain the Ising transition temperature
from the chiral order parameter distribution and find that
$T_{BKT}<T_I$ near the Lifshitz point too. At the same time we
find in accordance with results of Ref.~\cite{Cinti} that the
specific heat and susceptibilities have subsidiary peaks at low
$T<T_{BKT}$ near the Lifshitz point. These are anomalies which are
attributed in Ref.~\cite{Cinti} to the Ising phase transition.
However, we demonstrate that these anomalies do not signify a
continuous phase transition. Apparently, their origin is in
metastable states which lead also to a peculiar distribution of
the chiral order parameter. We find no such features in the
specific heat and susceptibilities far from the Lifshitz point (at
$J_2>0.4J_1$). As a result we obtain the phase diagram shown in
Fig.~\ref{phase}.
\begin{figure}
\center
\vspace{-7mm}
\includegraphics[height=80mm]{fig1.eps}
\vspace{-7mm}
\caption{{\small Phase diagram of the model \eqref{ham} that is found in the present paper.}}
\label{phase}
\end{figure}
The rest of the present paper is organized as follows. We discuss
in Sec.~\ref{model} the model \eqref{ham} in more detail and
introduce quantities to be found in our calculations. Numerical
results are discussed in Sec.~\ref{res}. In particular, the Ising
and the BKT transitions are considered in Secs.~\ref{ising} and
\ref{bkt}, respectively. The neighborhood of the Lifshitz point
and the phase diagram are discussed in Sec.~\ref{phsec}.
Sec.~\ref{conc} contains our conclusions.
\section{Model and methods}
\label{model}
We consider the model (\ref{ham}) of the classical XY magnet on a
square lattice. We set $J_1=J_b=1$ for simplicity and the value of
the extra exchange interaction $J_2$ is a variable. The Lifshitz
point corresponds to $J_2=1/4$ in this notation. The system has a
collinear antiferromagnetic ground state at $J_2<1/4$. To discuss
the phase transition from the (quasi-)antiferromagnetic phase to
the paramagnetic one we consider $J_2=0$ and $J_2=0.1$ (see
Fig.~\ref{phase}). The ground state has a helical ordering at
$J_2>1/4$. The turn angle $\theta_0$ between two neighboring spins
along $\mathbf{a}$ axis is given by $\cos\theta_0=-J_1/4J_2$ at
zero temperature.
To discuss the number and the sequence of phase transitions from
the (quasi-)helical phase to the paramagnetic one we consider
$J_2\approx 0.309$, 0.5 and $1.76$ corresponding at $T=0$ to
angles of commensurate helices $\theta_0=4\pi/5$, $2\pi/3$ and
$6\pi/11$, respectively. We use lattices with $L^2$ cites, where
$L$ is divisible by the size of the helix pitch and it lies in the
range from 20 to 120. We apply the periodic (toric) boundary
conditions as well as the cylindrical ones (i.e., with the
periodic condition along the $\mathbf{b}$ axis and the free one
along the $\mathbf{a}$ axis). We have found that both conditions
lead to the same values of transition temperatures and indexes. In
contrast values of Binder's cumulants and the chiral order
parameter distribution at $J_2\approx 0.309$ depend on boundary
conditions as we discuss below in detail. Standard Metropolis
algorithm \cite{Metropolis} has been used. The thermalization was
maintained within $4\cdot10^5$ Monte Carlo steps in each
simulation. Averages have been calculated within $3.6\cdot10^6$
steps for ordinary points and $6\cdot10^6$ for points close to the
critical ones. We have used also the histogram analysis technique
in which the range of each quantity has been divided into
$6.4\cdot10^5$ bins.
\subsection{Order parameters}
The BKT transition is driven by the magnetic order parameter for
which we use two definitions. Similar to the triangular lattice
\cite{Triangular}, one can introduce a number of sublattices in
the case of helix pitches which are divisible by the lattice
constant. Then, one can write for the magnetic order parameter
\begin{equation}
\mathbf{m}_i=\frac{n_{sl}}{L^2}\sum_{\mathbf{x}_i}\mathbf{S}_{\mathbf{x}_i},\quad
\overline{m}=\sqrt{\frac1{n_{sl}}\sum\limits_{i}\left<\mathbf{m}_i^2\right>},
\label{m}
\end{equation}
where index $i$ enumerates $n_{sl}$ sublattices, the sum over
$\mathbf{x}_i$ runs over sites of the $i$-th sublattice, spin
$\mathbf{S}_{\mathbf{x}_i}=(\cos\phi_{\mathbf{x}_i},\sin\phi_{\mathbf{x}_i})$
is a classical two-component unit vector, and
$\langle\ldots\rangle$ denotes the thermal average. The second
definition of the order parameter is valid both for commensurate
and incommensurate helices
\begin{equation}
\mathbf{M}_j=\frac1{L}\sum_i\mathbf{S}_{j\mathbf{a}+i\mathbf{b}},\quad
\overline{M}=\sqrt{\frac1{L}\sum_{j}\left<\mathbf{M}_j^2\right>}.
\label{ml}
\end{equation}
Our calculations show that definitions \eqref{m} and \eqref{ml}
lead to the same results away from the Lifshitz point. We have
found that $\bf m$ shows an anomalous behavior at $J_2\approx1/4$
and we use definition \eqref{ml} in this case. Thus, we
demonstrate that it is useful in numerical discussion of
helimagnets to choose parameters of the Hamiltonian so that the
helix pitch at $T=0$ to be commensurate.
The Ising transition is driven by the
chiral order parameter defined as
\begin{equation}
k=\frac{1}{L^2\sin\theta_0}\sum\limits_\mathbf{x} \sin(\fix-\fixa),
\quad \overline{k}=\sqrt{\left<k^2\right>}.
\label{kiral}
\end{equation}
\subsection{Susceptibilities and cumulants}
We introduce corresponding susceptibilities for all order
parameters \cite{Binder}
\begin{equation}
\chi_p=\left\{
\begin{array}{ll}
\displaystyle \frac{L^2}{T}\left(\left<p^2\right>-\left<|p|\right>^2\right), \quad & T<T_c, \\
&\\
\displaystyle \frac{L^2}{T}\left<p^2\right>, \quad & T\ge
T_c.
\end{array}
\right.
\label{suscept}
\end{equation}
The second line in this definition is used below for estimation of critical
exponents. Binder's cumulants \cite{Binder} are define as
\begin{equation}
U_p=1-\frac{\langle p^4 \rangle}{3\langle p^2 \rangle^2}.
\label{up}
\end{equation}
We discuss also the cumulant
\begin{equation}
V_k=\frac{\partial}{\partial(1/T)}\ln
\langle k^2\rangle=L^2\left(\frac{\left<k^2E\right>}{\left<k^2\right>}-\langle E\rangle\right),
\label{Vp}
\end{equation}
using which the critical exponent $\nu_k$ can be found by
finite-size scaling analysis \cite{Ferren}.
\subsection{Helicity modulus}
\label{helmod}
It is useful to introduce the helicity modulus (or the spin
stiffness) \cite{Helicity} to discuss the BKT transition that is
defined by the increase in the free energy density $F$ due to a
small twist $\Delta_\mu$ across the system in one direction ($\bf
a$ or $\bf b$)
\begin{equation}
\Upsilon_{\mu}=\left.\frac{\partial^2 F}{\partial{\Delta_\mu}^2}\right|_{\Delta_\mu=0},
\label{Y=dF po dd}
\end{equation}
where $\mu=a,b$ denotes the direction. Important universal
properties of a BKT transition predicted by Kosterlitz and Nelson
\cite{Jump} are the jump of the helicity modulus (\ref{Y=dF po
dd}) from zero at $T>T_{BKT}$ to the value of $2T_{BKT}/\pi$ at
$T=T_{BKT}$ and the value of the exponent $\eta(T=T_{BKT})=1/4$.
These properties have become standard methods of finding the
transition temperature.
As a result of the fact that the exchange couplings along
$\mathbf{a}$ and $\mathbf{b}$ axes are different, the helicity
moduli in these directions differ too. Thus, at zero temperature
$\Upsilon_a(0)=4J_2-J_1^2/(4J_2)$, while $\Upsilon_b(0)=J_b$.
Nevertheless, both $\Upsilon_a$ and $\Upsilon_b$ must vanish at
the same temperature with the identical value of the jump.
One finds after trivial calculations using Eqs.~\eqref{ham} and
\eqref{Y=dF po dd} that the helicity modulus $\Upsilon_b$ is
expressed via correlation functions and has a common view
\cite{Ohta}
\begin{equation}
\Upsilon_b = \left<E''_b\right>-\frac{L^2}{T}\left<(E'_b)^2\right>,
\label{Yy}
\end{equation}
where $T$ is measured in units of $J_1=J_b=1$, we set $k_B=1$,
$E'_b = L^{-2}\sum_\mathbf{x}\sin(\fix-\fixb)$ and $E''_b =
L^{-2}\sum_\mathbf{x}\cos(\fix-\fixb)$. Similar calculations give
for the helicity modulus in the $\mathbf{a}$ direction
\begin{equation}
\Upsilon_a=\left<E''_a\right>-\frac{L^2}{T}\left<(E'_a)^2\right>+
\frac{L^2}{T}\left<E'_a\right>^2,
\label{Yx}
\end{equation}
where $E'_a = L^{-2}\sum_\mathbf{x}
(\sin(\fix-\fixa)+2J_2\sin(\fix-\fixaa) )$ and $E''_a =
-L^{-2}\sum_\mathbf{x}(\cos(\fix-\fixa)+4J_2\cos(\fix-\fixaa))$.
It may seem that the last term in Eq.~\eqref{Yx} can be discarded
as it is done with the corresponding term in Eq.~\eqref{Yy} which
is equal to zero. However it is not so because
$\left<E'_b\right>=0$ at all $T$ whereas $\left<E'_a\right>=0$ at
$T\ge T_I$ and $\left<E'_a\right>\ne0$ at $T<T_I$. To demonstrate
this let us apply an infinitesimal twist $\Delta_a$ across the
system in the $\bf a$ direction, i.e., let us replace in
Eq.~\eqref{ham} $\fix-\fixa$ by $\fix-\fixa+\Delta_a$. One writes
in the first order in $\Delta_a$
\begin{eqnarray}
&&\sum_{\bf x}\Bigl(\cos(\fix-\fixa+\Delta_a)+J_2\cos(\fix-\fixaa+2\Delta_a)\Bigr)\nonumber\\
&&\approx
\sum_{\bf x}\Bigl(\cos(\fix-\fixa)+J_2\cos(\fix-\fixaa)\Bigr)
\nonumber\\ &&-\Delta_a \sum_{\bf
x}\Bigl(\sin(\fix-\fixa)+2J_2\sin(\fix-\fixaa)\Bigr).
\label{exp}
\end{eqnarray}
Comparing the last term in Eq.~\eqref{exp} with the chiral order
parameter $k$ definition \eqref{kiral} and noting that one can use
an equivalent definition $\tilde k=L^{-2}\sum_\mathbf{x}
\sin(\fix-\fixaa)$ we conclude that the last term in
Eq.~\eqref{exp} is a linear combination of $k$ and $\tilde k$.
However, $k$ and $\tilde k$ have opposite signs in the case
considered. In particular, their combination $k+2J_2\tilde k$ in
the last term in Eq.~\eqref{exp} is equal to zero at $T=0$.
Nevertheless our numerical results presented below show that this
combination is not equal to zero at $T\ne0$ and it can be
considered as the Ising order parameter at $T\sim T_I$. Then, one
see from Eq.~\eqref{exp} that $\Delta_a$ plays the role of the
"chiral" field and, consequently, $\partial
F/\partial\Delta_a|_{\Delta_a=0}$ (that is equal in our notation
to $\left<E'_a\right>$) is proportional to the chiral order
parameter which is equal to zero at $T\ge T_I$ and which is finite
at $T<T_I$.
\begin{figure}[t]
\vspace{-7mm}
\parbox{0.48\textwidth}{
\centering
\includegraphics[height=70mm]{fig2.eps}
\vspace{-7mm}
\caption{{\small Distribution of the value $E_b'$ defined in Eq.~\eqref{Yy} for $J_2=0.5$, $L=$ and three $T$ values: $T>T_I$, $T<T_{BKT}$, and $T_{BKT}<T<T_I$.}}
\label{fig11}}
\hspace{0.02\textwidth}
\parbox{0.48\textwidth}{
\center
\vspace{-4mm}
\includegraphics[height=70mm]{fig3.eps}
\vspace{-7mm}
\caption{{\small Distribution of the value $E_a'$ defined in Eq.~\eqref{Yx} for $J_2=0.5$, $T=0.67<T_I$ and different $L$.}}
\label{fig12}}
\end{figure}
To illustrate this consideration we present in Fig.~\ref{fig11}
the distribution of $E'_b$ for $J_2=0.5$ and for various
temperatures below $T_{BKT}\approx0.671$, between $T_{BKT}$ and
$T_{I}\approx0.69$, and above $T_I$ (the values of $T_I$ and
$T_{BKT}$ are obtained below). The distribution has a Gaussian
form with the zero expected value $\langle E'_b\rangle=0$.
Fig.~\ref{fig12} shows the distribution of $E'_a$ for $T=0.67<T_I$
and various lattice sizes. A non-zero expected value of $\langle
E'_a\rangle$ is seen. One can observe a double-peak structure of
$E'_a$ distribution (see Fig.~\ref{fig12}) for small lattices or
at $T$ that is close enough to $T_I$, when the system can tunnel
to a configuration with opposite chirality. The probability of
such tunneling is estimated as
\begin{equation}
p(\overline{k}_+\to\overline{k}_-)=\exp\left(-\frac{2Lf_{dw}}{T}\right),
\end{equation}
where $f_{dw}$ is a domain wall tension \cite{domainwall} that is
positive at $T<T_I$ and it vanishes at $T=T_I$. That is why we
observe two peaks for small lattices and only one peak for large
ones. Quite expectedly, we find the single-peak distribution of
$E'_a$ at $T>T_I$ demonstrated in Fig.~\ref{fig13}.
\begin{figure}[t]
\vspace{-7mm}
\centering
\includegraphics[height=70mm]{fig4.eps}
\vspace{-7mm}
\caption{{\small Same as in Fig.~\ref{fig12} for $T>T_I$.}}
\label{fig13}
\end{figure}
It should be stressed that the disappearance of the double-peak
structure at the critical temperature $T_I(L)$ is a signature of
the transition on a lattice with size $L$. The value of $T_I(L)$
is close to the correct value of the transition temperature for
large lattices. We use this circumstance below in our analysis of
the Lifshitz point neighborhood.
It should be noted also that we replace in our numerical
calculations $\left<E'_a\right>$ by $\left<|E'_a|\right>$ at
$T<T_I$ in the last term in Eq.~\eqref{Yx} as it is usually done
in considerations of order parameters \cite{Binder}. It is done
because the order parameter distribution has tails in both
positive and negative regions even below the transition
temperature. The value $\left<p\right>$ is replaced by
$\left<|p|\right>$ in Eq.~\eqref{suscept} by the same reason.
\section{Numerical results}
\label{res}
We discuss in this section in detail our results for the special
case of $J_2=0.5$ that corresponds at $T=0$ to 120$^\circ$ helical
structure with three sublattices. Then, we discuss the phase
diagram. We consider lattices with $L=24$, 30, 36, 42, 48, 60, 72,
90, 120. The lattice with $L=18$ is also used for estimation of
some quantities.
\subsection{Ising transition}
\label{ising}
To obtain the Ising transition temperature $T_I$ we use the Binder
cumulant crossing method \cite{Binder}. We find for $L=24$ and 30
the temperature $T_{L'}$ as a function of
$\ln^{-1}\left(L'/L\right)$ at which curves $U_k(L)$ intersect for
different lattice sizes $L'>L$. Extrapolation to the thermodynamic
limit $L'\to\infty$ gives the following transition temperature
(see Figs.~\ref{fig1} and \ref{fig2})
\begin{figure}[t]
\vspace{-7mm}
\parbox{0.48\textwidth}{
\centering
\includegraphics[height=65mm]{fig5.eps}
\vspace{-9mm}
\caption{{\small Estimation of the transitions temperature by
the Binder cumulant crossing method.}}
\label{fig1}}
\hspace{0.02\textwidth}
\parbox{0.48\textwidth}{
\center
\includegraphics[height=65mm]{fig6.eps}
\vspace{-8mm}
\caption{{\small Binder's cumulant $U_k(L)$ defined by Eq.~\eqref{up} as a function of
temperature, for $L=24,\ldots,90$.}}
\label{fig2}}
\end{figure}
\begin{equation}
\label{ti}
T_I=0.689(1).
\end{equation}
The dispersion in the value of $T_I$ obtained for the different
lattice sizes $L$ gives the error of the transition temperature
estimation. Notice that error bars are not shown in
Figs.~\ref{fig1} and \ref{fig2} and in all figures below if they
are smaller than or comparable with symbols size.
A peak in the specific heat shown in Fig.~\ref{fig3} is found
approximately at the same temperature. For the largest lattices
the peak is located in the range of temperature from $0.690$ to
$0.695$. One expects that this peak corresponds to the logarithmic
divergence of the specific heat that is a characteristic of the 2D
Ising model in which the critical exponent $\alpha$ is equal to
zero. Insufficient accuracy of our data for the specific heat
prevents us from the immediate estimation of $\alpha$.
Critical exponents $\nu_k$, $\beta_k$ and $\gamma_k$ are obtained
by the finite-size scaling theory. To estimate the exponent $\nu_k$,
we find a maximum of the quantity $V_k$ given by Eq.~(\ref{Vp}) as a
function of lattice size $L$ \cite{Ferren}
\begin{equation}
\left(V_k^{(L)}\right)_{\mbox{\scriptsize max}}\sim
L^{1/\nu_k}.
\end{equation}
The fitting presented in Fig.~\ref{fig4} gives
\begin{equation}
\nu_k=0.97(4)
\end{equation}
that coincides within the computational error with the exact value of
$\nu=1$ for the 2D Ising model.
\begin{figure}[h]
\vspace{-7mm}
\parbox{0.48\textwidth}{
\centering
\includegraphics[height=65mm]{fig7.eps}
\vspace{-7mm}
\caption{{\small Specific heat $C(T)$.}}
\label{fig3}}
\hspace{0.02\textwidth}
\parbox{0.48\textwidth}{
\center
\includegraphics[height=65mm]{fig8.eps}
\vspace{-7mm}
\caption{{\small Estimation of the exponent $\nu_k$ using the cumulant $V_k$ defined by Eq.~\eqref{Vp}.}}
\label{fig4}}
\end{figure}
\begin{figure}[t]
\vspace{-7mm}
\parbox{0.48\textwidth}{
\centering
\includegraphics[height=65mm]{fig9.eps}
\vspace{-7mm}
\caption{{\small Estimation of the exponents $\beta_k$ and $\gamma_k$.}}
\label{fig5}}
\hspace{0.02\textwidth}
\parbox{0.48\textwidth}{
\center
\includegraphics[height=65mm]{fig10.eps}
\vspace{-7mm}
\caption{{\small Binder's cumulant $U_m(L)$ as a function of
temperature, for $L=24,\ldots,90$.}}
\label{fig6}}
\end{figure}
Exponents $\beta_k$ and $\gamma_k$ are found from scaling
properties of the order parameter $\overline{k}$ and the susceptibility
$\chi_k$ at the critical point
\begin{equation}
\left(\bar{k}^{(L)}\right)_{T=T_I}\sim L^{\beta_k/\nu_k},\quad
\left(\chi_k^{(L)}\right)_{T=T_I}\sim L^{\gamma_k/\nu_k},
\end{equation}
with the following result (see Fig.~\ref{fig5}):
\begin{equation}
\beta_k=0.118(8), \quad \gamma_k=1.70(6).
\end{equation}
These values coincide within computational errors with exact
values of $1/8$ and $7/4$, correspondingly, of the 2D Ising model.
Using the scaling relations we find other exponents
\begin{equation}
\alpha=2-2\nu_k=0.06(8),\quad
\eta_k=2-\gamma_k/\nu_k=0.25(5).
\end{equation}
Note that the scaling relation
$\alpha+2\beta_k+\gamma_k=2.00(8)\approx2$ is satisfied within the computational error.
We have found also the universal value of the Binder cumulant
$U^*=0.615(6)$ at the critical temperature (see Fig.~\ref{fig2})
that is in agreement with the value $U^*\approx0.611$ observed in
the 2D Ising model \cite{2DIsing} with periodic boundary
conditions.
\subsection{BKT transition}
\label{bkt}
According to the Mermin-Wagner theorem \cite{Mermin} there is no
spontaneous magnetization at finite temperature in 2D magnets with
short range interactions and a continuous symmetry. But a
quasi-long-range order appears at non-zero temperature due to the
Berezinskii-Kosterlitz-Thouless mechanism \cite{BKT} in XY magnets
with $SO(2)$ symmetry.
It is important that as long as we measure the temperature in
units of $J_1$, the universal value of the jump $2T_{BKT}/\pi$
perturbs by the factor of $J_1/J_{\mathrm{eff}}$, where
$J_{\mathrm{eff}}$ is an effective coupling constant. The
competing exchange coupling $J_2$ gives rise to this factor. To
obtain it we considered the Coloumb-gas representation of the
model (\ref{ham}) using standard duality transformations
\cite{BKT, Duality}. As a result we obtain
\begin{equation}
J_{\mathrm{eff}}=\sqrt{J_b\left(J_1-4J_2\right)}
\label{jeffc}
\end{equation}
for the antiferromagnetic phase ($J_2<J_1/4$), and
\begin{equation}
J_{\mathrm{eff}}=\sqrt{\frac12J_b\left(4J_2-\frac{J_1^2}{4J_2}\right)}
\end{equation}
for the helimagnetic one ($J_2>J_1/4$). Eq.~\eqref{jeffc} is in
accordance with results of Refs.~\cite{Garel, Kaplan}. In
particular, $J_{\mathrm{eff}}=\sqrt{3}/2$ for $J_2=0.5$.
A few authors have investigated the properties of the Binder
cumulant for magnetization as an alternative method of the BKT
transition temperature estimation \cite{Loison1}. Due to
finite-size corrections this method gives a value for the
transition temperature $T_{B}$ that is slightly larger than the
true value $T_{BKT}$. Nevertheless this method is useful as it
provides an estimation of the BKT transition temperature and an
extra evidence of separated transitions (if one finds that
$T_{B}<T_{I}$). Using the Binder cumulant crossing method
described above we find $T_{B}=0.679(2)$ (see Figs.~\ref{fig1} and
\ref{fig6}) that is $1.8\%$ smaller than $T_{I}$ given by
Eq.~\eqref{ti}.
\begin{figure}[t]
\vspace{-7mm}
\parbox{0.48\textwidth}{
\centering
\includegraphics[height=65mm]{fig11.eps}
\vspace{-7mm}
\caption{{\small Helicity modulus $\Upsilon_b$ in the $\mathbf{b}$ direction and its
intersection with the line $2T/\pi J_{\mathrm{eff}}$.}}
\label{fig7}}
\hspace{0.02\textwidth}
\parbox{0.48\textwidth}{
\center
\includegraphics[height=60mm]{fig12.eps}
\vspace{-9mm}
\caption{{\small Root-mean-square fit error $\Delta_c$ of the helicity
modulus $\Upsilon_b$ to the Weber-Minnhagen scaling equation \eqref{Y=1+lnC}.}}
\label{fig8}}
\end{figure}
\begin{figure}
\vspace{-7mm}
\parbox{0.48\textwidth}{
\centering
\includegraphics[height=65mm]{fig13.eps}
\vspace{-7mm}
\caption{{\small Same as in Fig.~\ref{fig7} for $\Upsilon_a$.}}
\label{fig14}}
\hspace{0.02\textwidth}
\vspace{-7mm}
\parbox{0.48\textwidth}{
\centering
\includegraphics[height=60mm]{fig14.eps}
\vspace{-7mm}
\caption{{\small Same as in Fig.~\ref{fig8} for $\Upsilon_a$.}}
\label{fig142}}
\vspace{5mm}
\end{figure}
To obtain $T_{BKT}$ precisely we use the Weber-Minnhagen
finite-size-scaling analysis \cite{weber} that is based on
consideration of logarithmic corrections to the value of the
helicity modulus at temperature close to $T_{BKT}$ having the form
\begin{equation}
\Upsilon(T,L)=\frac{2T}{\pi J_{\mathrm{eff}}} \left(1+\frac1{2\ln L+c}\right),
\label{Y=1+lnC}
\end{equation}
where $c$ is a fitting parameter. Fixing $T$ we find the
root-mean-square error $\Delta_c$ of the least-square fit of our
numerical data for $\Upsilon(T,L)$ with different $L\leq60$ that
is based on Eq.~(\ref{Y=1+lnC}). The minimum of $\Delta_c$
as a function of $T$ gives the value of the transition temperature
\cite{weber}. We obtain for the helicity modulus in the $\bf b$
direction (see Figs.~\ref{fig7} and \ref{fig8})
\begin{equation}
T_{BKT}^{(\Upsilon_b)}=0.671(1).
\label{tbktb}
\end{equation}
Corresponding results for $\Upsilon_a$ are shown in Figs.~\ref{fig14} and \ref{fig142}. For the
largest lattice of $L\geq48$ the Weber-Minnhagen finite-size scaling
analysis estimates the BKT transition temperature as
\begin{equation}
T_{BKT}^{(\Upsilon_a)}=0.673(2)
\end{equation}
that is in agreement with Eq.~\eqref{tbktb}. It should be noted
that inaccuracy in estimation of $\Upsilon_a$ is larger than of
$\Upsilon_b$. That is why we use below the more precise value
\eqref{tbktb} for comparison between different methods.
To verify our results, we use also cylindric boundary conditions
with the periodic condition along the $\mathbf{b}$ axis and with
the free one along the $\mathbf{a}$ axis. We obtain results
consistent with those for periodic conditions. In particular,
transitions temperatures $T_{BKT}=0.671(2)$ and $T_I=0.6907(6)$
were estimated by the Weber-Minnhagen analysis and the Binder
cumulant crossing method (see Fig.~\ref{fig161}). The universal
value of the Binder cumulant at the critical temperature is
$U^*=0.496(7)$ that is close to the value expected for the 2D Ising
model \cite{2DIsing} with mixed (cylindric) boundary conditions.
\begin{figure}
\parbox{0.48\textwidth}{
\centering
\includegraphics[height=65mm]{fig15.eps}
\vspace{-11mm}
\caption{{\small Intersection of the exponent $\eta(T)$ with the bound $\eta=0.25$.}}
\label{fig9}}
\hspace{0.02\textwidth}
\parbox{0.48\textwidth}{
\centering
\vspace{1mm}
\includegraphics[height=60mm]{fig16.eps}
\vspace{-6mm}
\caption{{\small Same as in Fig. \ref{fig2} for cylindric boundary conditions.}}
\label{fig161}}
\end{figure}
Another indication of the BKT transition is the equality to $0.25$
of the exponent $\eta(T)$. Below the transition temperature the
susceptibility diverges with the size of the system as
\begin{equation}
\chi_m(T,L)\sim L^{2-\eta(T)}.
\label{ch}
\end{equation}
The exponent $\eta$ as a function of temperature found using
Eq.~\eqref{ch} is shown in Fig.~\ref{fig9}. The intersection of
$\eta(T)$ with the bound $\eta=0.25$ gives
\begin{equation}
T_{BKT}^{(\eta)}=0.676(2).
\label{tbkte}
\end{equation}
Comparing Eqs.~\eqref{tbktb} and \eqref{tbkte} one notes that
$T_{BKT}^{(\eta)}>T_{BKT}^{(\Upsilon)}$. Because \cite{Jump}
$\eta(T_{BKT})=$ $T_{BKT} J_\mathrm{eff}/ 2\pi J_1
\Upsilon(T_{BKT})$, we can not exclude that at the
true transition temperature the exponent $\eta$ and the jump of
the helicity modulus have non-universal values. Such a possibility
has been considered for other models from the class
$\mathbb{Z}_2\otimes SO(2)$. \cite{Teitel, FFXY, Triangular, Gas,
nonuniv} If it is so $\eta$ has a value smaller than 0.25 and the
jump is greater than $2J_1/\pi J_\mathrm{eff}$. Thus, our data
show that $\eta(T_{BKT}^{(\Upsilon)})\approx0.22$.
\subsection{Neighborhood of the Lifshitz point and the phase diagram}
\label{phsec}
Besides the case of $J_2=0.5$ considered above in detail we have
carried out similar discussions of $J_2=0$, 0.1, 0.309 and 1.76 to
obtain the phase diagram shown in Fig.~\ref{phase}. Some results
of this consideration are summarized in Table~\ref{table}. The
case of $J_2=0$ corresponds to the well known XY model on a square
lattice and we find $T_{BKT}=0.891(2)$ that is consistent with the
previous results \cite{MCforKT}.
\begin{figure}[t]
\centering
\includegraphics[height=90mm]{fig17.eps}
\vspace{-7mm}
\caption{{\small Specific heat $C(T)$, helicity modulus $\Upsilon_b$ and susceptibilities $\chi_{k,m}$ for $J_2\approx 0.309$.}}
\label{fig163}
\end{figure}
It should be stressed that
we obtain $T_I>T_{BKT}$ at $J_2>1/4$ in accordance with
conclusions of Refs.~\cite{Korshunov,Kolezhuk} and in contrast to
Refs.~\cite{Garel,Cinti}. Because this our finding is at odds with
that of the similar numerical consideration of the same model
carried out in Ref.~\cite{Cinti}, our special interest is to
consider the case of $J_2\approx 0.309$ that is close to $J_2=0.3$
discussed in Ref.~\cite{Cinti}.
\begin{table}
\caption{Some results of our discussion of the model \eqref{ham}. Here $L_{max}$ is the maximum value of $L$ considered.}
\centering
\begin{tabular}{ccccc}
\hline
\hline
$J_2/J_1$ & $\theta_0$ & $L_{max}$ & $T_{BKT}/J_1$ & $T_I/J_1$\\
\hline
$0$& 0 &40& 0.891(2) & - \\
$0.1$& 0 &30& 0.781(3) & - \\
$\approx0.309$& $4\pi/5$ &100& 0.443(5) & 0.48(1)\\
0.5& $2\pi/3$ &150& 0.671(1) & 0.690(1)\\
$\approx1.76$ & $6\pi/11$ &66& 1.24(1) & 1.285(7)\\
\hline
\hline
\end{tabular}
\label{table}
\end{table}
The case of $J_2\approx 0.309$ corresponds at $T=0$ to
$\theta_0=4\pi/5$. In particular, we find $T_{BKT}=0.443(5)$ that
is very close to the value reported in Ref.~\cite{Cinti}.
Estimating the temperature of the Ising critical point by the
Binder cumulant crossing method, we encounter the anomalous
behavior of the chiral order parameter and do not obtain a
reliable result. Apparently, it is the reason why the authors of
Ref.~\cite{Cinti} base their conclusion about the Ising transition
on the behavior of the specific heat $C(T)$ and susceptibilities
$\chi_{k,m}$. Our results for these quantities are shown in
Fig.~\ref{fig163} and they are consistent with those of
Ref.~\cite{Cinti}. It is seen from Fig. \ref{fig163} that the
chiral susceptibility has a high peak at $T\approx 0.4$, while the
specific heat and the magnetic susceptibility have subsidiary
peaks at $T\approx 0.4$ which grow with the lattice size
increasing. These anomalies at $T\approx 0.4<T_{BKT}$ are
attributed in Ref.~\cite{Cinti} to the Ising transition.
However, we observe that for other $J_2>1/4$ the specific heat has
only one peak corresponding to the logarithmic divergence which
characterizes the Ising transition (see, e.g., Fig.~\ref{fig3} for
$J_2=0.5$). Therefore, the behavior of $C(T)$ shown in
Fig.~\ref{fig163} is not normal for the model discussed and it is
characteristic of the Lifshitz point neighborhood. To account for
this anomaly we examine the behavior of the chiral order parameter
in detail.
Fig. \ref{fig17} shows the chiral order parameter distribution for
$J_2=0.5$ and $L=45$. It looks like a customary order parameter
distribution in a system with the second order transition. In
particular, the distribution has a Gaussian form below a critical
temperature with a peak at $\bar{k}$ (see the curve for $T=0.6$).
In approaching to the critical temperature the distribution
acquires an appreciable tail (see curves for $T=0.68$ and
$T=0.7$). Such a broad distribution leads to a peak in the
susceptibility. The distribution has a peak at $k=0$ above the
critical point (see curves for $T=0.72$ and $T=0.8$).
\begin{figure}[t]
\vspace{-7mm}
\parbox{0.48\textwidth}{
\center
\includegraphics[height=65mm]{fig18.eps}
\vspace{-7mm}
\caption{{\small The chiral order parameter distribution for $J_2=0.5$ and $L=42$.}}
\label{fig17}}
\hspace{0.02\textwidth}
\parbox{0.48\textwidth}{
\centering
\includegraphics[height=65mm]{fig19.eps}
\vspace{-7mm}
\caption{{\small The chiral order parameter distribution for $J_2=0.309$, $L=45$ at different
temperatures.}}
\label{fig18}}
\end{figure}
\begin{figure}[t]
\vspace{-7mm}
\parbox{0.48\textwidth}{
\centering
\includegraphics[height=65mm]{fig20.eps}
\vspace{-7mm}
\caption{{\small The chiral order parameter distribution for $J_2=0.309$, $L=25$ and $L=35$
at $T/J_1=0.43$.}}
\label{fig20}}
\hspace{0.02\textwidth}
\parbox{0.48\textwidth}{
\center
\includegraphics[height=65mm]{fig21.eps}
\vspace{-7mm}
\caption{{\small The chiral order parameter distribution for $J_2=0.309$, $L=65$ and $L=100$
at $T/J_1=0.43$.}}
\label{fig21}}
\end{figure}
\begin{figure}[t]
\vspace{-7mm}
\parbox{0.48\textwidth}{
\centering
\includegraphics[height=65mm]{fig22.eps}
\vspace{-7mm}
\caption{{\small Distribution of $E_a'$ defined in Eq.~\eqref{Yx} for $L=65$.}}
\label{fig22}}
\hspace{0.02\textwidth}
\parbox{0.48\textwidth}{
\center
\includegraphics[height=60mm]{fig23.eps}
\vspace{-7mm}
\caption{{\small Estimation of $T_I$ for $J_2=0.309$ that is based on Eq.~\eqref{extr}.}}
\label{fig23}}
\end{figure}
However the picture for the chiral order parameter distribution is
quite different close to the Lifshitz point, e.g., at $J_2\approx
0.309$. We show in Fig.~\ref{fig18} the chiral order parameter
distribution for $L=45$ and different temperatures. One can see
one Gaussian peak at $T<0.35$ in agreement with the common picture
described above. But a few additional peaks arise at $T>0.35$.
Then, the number and the breadth of peaks depend on the lattice
size $L$ (see Figs.~\ref{fig20} and \ref{fig21}) and, what is much
more important, on the boundary conditions. For the cylindrical
boundary conditions, these peaks are broader and they are
accompanied by great number of accessory peaks. Such a
distribution of the chiral order parameter is characteristic in
the case of $L=45$ to the range of temperature from $T\approx0.35$
to $T\approx0.55$. One can see from Fig.~\ref{fig18} that at
$T=0.57$ the distribution has a form that is typical for a
disordered phase with the peak at zero value of the order
parameter. Then, it is clear that the order parameter distribution
at $T\approx0.4$ shown in Fig.~\ref{fig18} does not correspond to
a critical point of a continuous transition.
Apparently, the origin of such a peculiar behavior at $J_2\approx
0.309$ are metastable states with different values of the chiral
order parameter which prevent the system investigation
considerably. The multi-peak structure of the order parameter
distribution leads to a sudden jump of the susceptibility and
states intermediate between metastable configurations give rise to
the specific heat anomaly. It should be noted that energy values
of the metastable states are close since we observe in our
simulations that the energy distribution has a Gaussian form even
for the largest lattice size.
To estimate the Ising transition temperature at $J_2\approx 0.309$
we analyze $E'_a$ distribution defined in Eq.~\eqref{Yx} and
discussed in Sec.~\ref{helmod}. As it is pointed out above,
$\langle E'_a\rangle\ne0$ whenever a helical ordering exists.
Fig.~\ref{fig22} shows that the distribution is not symmetric
relative to zero at $T=0.52$ and $L=65$, while it is definitely
symmetric at $T\ge0.53$. Therefore, the critical temperature for
$L=65$ can be roughly estimated as $T_I(L)=0.53(1)$. The
transition temperature can be estimated using the relation
\begin{equation}
\label{extr}
T_I=T_I(L)-\frac{A}{L^\nu},
\end{equation}
where $A$ is constant and $\nu=1$ as it is expected for an Ising
transition. An extrapolation to the thermodynamic limit using
Eq.~\eqref{extr} gives $T_I=0.477(12)$ (see Fig.~\ref{fig23}).
Then, we obtain that $T_{BKT}<T_I$ even in the neighborhood of the
Lifshitz point.
\section{Conclusion}
\label{conc}
We discuss critical properties of the 2D helimagnet described by
the Hamiltonian \eqref{ham}. It belongs at $J_2/J_1>1/4$ to the
same class universality as the fully frustrated XY model and the
antiferromagnet on triangular lattice which have two successive
phase transitions upon the temperature decreasing: the first one
is associated with breaking of the discrete $\mathbb{Z}_2$
symmetry and the second one is of the BKT type at which the
$SO(2)$ symmetry breaks. We confirm that this scenario is realized
also in the model \eqref{ham} at $J_2/J_1>1/4$ and obtain the
phase diagram shown in Fig.~\ref{phase}. A narrow region exists on
this phase diagram between lines of the Ising and the BKT
transitions that corresponds to the chiral spin liquid.
In particular, we demonstrate that the number and sequence of
transitions do not depend on the turn angle $\theta_0$ of the
helix twist at $T=0$. Then, this quantity is not a critical
parameter that has been already found \cite{Helix, Kawamura2} in
three-dimensional helimagnets. We find that it is useful in
numerical discussion of helimagnets to choose parameters of the
Hamiltonian so that the helix pitch at $T=0$ to be commensurate.
It allows to use definition \eqref{m} of the magnetic order
parameter in which the summation over sublattices is involved.
We find in accordance with results of Ref.~\cite{Cinti} that the
specific heat and susceptibilities have subsidiary peaks at low
$T$ near the Lifshitz point $J_2/J_1=1/4$ (see Fig.~\ref{fig163}).
However, in contrast to the conclusion of Ref.~\cite{Cinti} we
demonstrate that these anomalies do not signify a continuous phase
transition. Apparently, their origin is in metastable states near
the Lifshitz point which lead also to a peculiar distribution of
the chiral order parameter shown in Fig.~\ref{fig18}.\bigskip
This work was supported by the RF President (grant MD-274.2012.2),
the RFBR grant 12-02-01234, and the Program "Neutron Research of
Solids".
|
{
"timestamp": "2012-03-13T01:00:56",
"yymm": "1203",
"arxiv_id": "1203.2235",
"language": "en",
"url": "https://arxiv.org/abs/1203.2235"
}
|
\section{Introduction}
Presumably next year the Multi Unit Spectroscopic Explorer \emph{MUSE}\ \citep{2010SPIE.7735E...7B} will see its first light
as a second-generation instrument for \emph{ESO}'s \emph{Very Large Telescope} at Paranal, Chile. The instrument is an highly efficient AO-supported integral field unit (IFU)
and its outstanding combination of a large field of view ($1 \times 1\,\mathrm{arcmin}^2$) and high spatial sampling
of 0.2" with a spectral resolution of R=2000-4000 over the optical range of 4650-9300\,\AA\
will allow us to obtain unprecedented observations.
In our group, we intent to use it for the analysis of galactic globular clusters, which due to the heavy
crowding towards the center are only accessible through their giants by other instruments. With \emph{MUSE},
we will be able to point directly in the center of the clusters and obtain thousands of stellar spectra
even from stars well below the main-sequence turnoff point with one single exposure.
For the analysis of those spectra, we need a grid of model spectra that matches both the wavelength range
and resolution of \emph{MUSE}\ as well as our requirements for an extensive parameter space (as given by
previous observations of globular clusters) and being able to adjust it as needed. Therefore a decision
was made to create a new grid of model atmospheres and synthetic spectra with \emph{PHOENIX}\ \citep{1999JCoAM.109...41H}.
We will present this new library together with a short description of the methods used for
analyzing \emph{MUSE}\ spectra and some preliminary results on a simulated data cube.
A paper \citep{husser2012} about our new synthetic stellar library with more detailed descriptions
and informations for downloading the spectra is in preparation and will be published
soon.
\section{Globular Clusters}
There are about 150 known globular clusters in our galaxy with masses of $10^5-10^6\,M_\mathrm{sun}$, which consist
of very old stellar populations with an age of $\geq 10\,\mathrm{Gyr}$. A couple of years ago those populations were
assumed to be very simple with a single isochrone.
Unexpectedly, latest observations showed evidence for the existence of multiple main-se\-quen\-ces within globular
clusters, e.\,g.\ \cite{2004ApJ...605L.125B} for Omega Centauri, and also for a split
in the (sub)giant branch \citep{1999Natur.402...55L}. This seems to be caused by multiple stellar populations with different abundances of
helium and iron, at least for massive clusters. A variation
of some lighter elements seems to be more ubiquitos, like the Na-O anti-correlation observed by Carretta (2009). Those
observations indicate a complex enrichment history with multiple epochs of star formation.
In globular clusters, we observe a velocity dispersion of $5-20\,\mathrm{km/s}$ and high central stellar densities
of $\sim$10$^6\,\mathrm{M}_\mathrm{sun}/\mathrm{pc}^3$. Given these high stellar densities, scenarios have been proposed
for the formation of intermediate-mass black holes in the cluster centres \citep{2002ApJ...576..899P}. An extrapolation of the tight relation between total mass and the
mass of the black hole in the bulges of galaxies (Marconi and Hunt 2003) towards typical masses of globular clusters predicts black
holes with a mass of $\sim$10$^3\,\mathrm{M}_\mathrm{sun}$.
In contrast to field stars, where we observe a binary fraction of about 50\%, in globular clusters we
find a value of only 30\% or even less. Monte Carlo simulations \citep{2005MNRAS.358..572I} showed that
the number of binaries in the core decreases rapidly over time. \cite{1987degc.book.....S} already showed
that a depletion of binaries in the core is necessary for it to collapse, so core-collapsed clusters
like M15 seem to be dynamically more evolved than others. A core-collapse could also result in the formation
of new binaries. Therefore, studying the binary fraction in globular clusters is an important task in order to understand their evolution.
\section{The new \emph{PHOENIX}\ grid}
The \emph{PHOENIX}\ version 16 that we are using for calculating the grid uses a new equation of state called ACES,
which is a state-of-the-art treatment of the chemical equilibrium in each layer of a stellar atmosphere.
The element abundances we used for the atmospheres were taken from \cite{2009ARA&A..47..481A}.
\renewcommand{\tabcolsep}{0.5mm}
\begin{table}
\caption{Parameter space of the grid. An extension in $T_{\mathrm{eff}}$ up to $12\,000\,\mathrm{K}$ is work in progress and
up to $25\,000\,\mathrm{K}$ in planning.
Alpha element abundances $[\alpha/Fe] \neq 0$ are only available for $3\,500\,\mathrm{K} \leq T_{\mathrm{eff}} \leq 8\,000\,\mathrm{K}$
and $-2 \leq [Fe/H] \leq 0$.}
\label{table:paramspace}
\centering %
\begin{tabular}{rrrlc}
\hline \hline
& & \multicolumn{2}{c}{Range} & Step size \\ \hline
$T_{\mathrm{eff}}$ [K] & & 2\,300 & -- 7\,000 & 100 \\
& & 7\,000 & -- 8\,000 & 200 \\
$\log(g)$ & & 0.0 & -- +6.0 & 0.5 \\
$[Fe/H]$ & & -4.0 & -- -2.0 & 1.0 \\
& & -2.0 & -- +1.0 & 0.5 \\
$[\alpha/Fe]$ & & -0.3 & -- +0.8 & 0.1 \\ \hline
\end{tabular}
\end{table}
\renewcommand{\tabcolsep}{2mm}
The parameter space of the new \emph{PHOENIX}\ grid that we are presenting in this paper is given in Table~\ref{table:paramspace}.
An extension towards hotter stars including NLTE treatment of important elements is both work-in-progress (up to 12\,000\,K) and intended
(up to 25\,000\,K). The grid is complete in its first three dimensions effective temperature $T_{\mathrm{eff}}$, surface
gravity $\log(g)$ and metallicity $[Fe/H]$. Different alpha element (including O, Ne, Mg, Si, S, Ar, Ca and Ti) abundances are provided for
$3\,500\,\mathrm{K} \leq T_{\mathrm{eff}} \leq 8\,000\,\mathrm{K}$ and $-2 \leq [Fe/H] \leq 0$ only.
\renewcommand{\tabcolsep}{0.5mm}
\begin{table}
\caption{Spectral resolution of the grid.}
\label{table:resolution}
\centering %
\begin{tabular}{rlrl}
\hline \hline
\multicolumn{2}{c}{Range [\AA]} & \multicolumn{2}{c}{Resolution} \\ \hline
500 & -- 3\,000 & $\Delta\lambda$ & = $0.1$\AA \\
3\,000 & -- 25\,000 & $R$ & $\approx 500\,000$ \\
25\,000 & -- 55\,000 & $R$ & $\approx 100\,000$ \\ \hline
\end{tabular}
\end{table}
\renewcommand{\tabcolsep}{2mm}
Although the primary intent for creating the grid was the analysis of \emph{MUSE}\ spectra, we decided to increase both the
wavelength range and the resolution (see Table~\ref{table:resolution}), so that teams working on other existing and upcoming
instruments will be able to use them for their purposes. Due to to this, our spectra are applicable for the analysis of
e.\,g.\ CRIRES \citep{2004SPIE.5492.1218K} and X-Shooter \citep{2011A&A...536A.105V} data.
\begin{figure}
\includegraphics[width=\textwidth]{mass_bw.eps}
\caption{Distribution of stellar masses for that part of the grid with solar abundances for different
effective temperatures $T_{\mathrm{eff}}$ and surface gavities $\log(g)$. Color-coded is the
stellar mass in units of solar mass from $0M_\odot$ (black) to $9M_\odot$ (white).}
\label{figure:mass}
\end{figure}
In order to define a spherical symmetric atmosphere as it is used in \emph{PHOENIX}, we need to define an
effective temperature $T_{\mathrm{eff}}$, a surface gravity $\log(g)$ and either a radius $r_0$ or a mass
$M_\star$. We decided to use the mass by taking a mass-luminosity relation $L_\star/L_\odot = (M_\star/M_\odot)^3$
for main-sequence stars and letting it tend towards higher values for giants and super giants:
\begin{equation}
M_{\star} = c \cdot M_{\mathrm{sun} } \cdot \left( \frac{T_{\mathrm{eff}}}{5\,770\,\mathrm{K}} \right)^2,
\label{eq:math}
\end{equation}
with values for the coefficient as given in the following table:
\begin{center}
\begin{tabular}{r|ccccccc}
\hline \hline
log(g) & $>4$ & $>3$ & $>2$ & $>1.6$ & $>0.9$ & $>0$ & $\leq0$ \\
c & 1 & 1.2 & 1.4 & 2 & 3 & 4 & 5 \\ \hline
\end{tabular}
\end{center}
Figure~\ref{figure:mass} shows the distribution of masses in our grid for solar abundances.
\emph{PHOENIX}\ uses the mixing length theory \citep{Prandtl,1953ZA.....32..135V} for describing convection within the atmosphere.
For our spectra, we used the formula provided by \citet{1999A&A...346..111L}, which has been calibrated
using 3D RHD models:
\begin{equation}
\alpha = a_0 + (a_1 + (a_3 + a_5 T_s + a_6 g_s) T_s + a_4 g_s) T_s + a_2 g_s,
\end{equation}
with
\begin{equation}
T_s = \frac{T_{\mathrm{eff}} - 5\,770\,\mathrm{K}}{1\,000} \quad\mathrm{and}\quad g_s = \log \left( \frac{10^{\log(g)}}{27\,500} \right),
\end{equation}
and coefficients given by:
\begin{center}
\begin{tabular}{c|c|c|c|c|c|c}
$a_0$ & $a_1$ & $a_2$ & $a_3$ & $a_4$ & $a_5$ & $a_6$ \\ \hline
1.587 & -0.054 & 0.045 & -0.039 & 0.176 & -0.067 & 0.107
\end{tabular}
\end{center}
\begin{figure}
\centering
\includegraphics[width=\textwidth]{microturb_bw.eps}
\caption{Distribution of micro-turbulences for that part of the grid with solar abundances for different
effective temperatures $T_{\mathrm{eff}}$ and surface gavities $\log(g)$. Color-coded is the
micro-turbulence from $0\textrm{km/s}$ (black) to $6\textrm{km/s}$ (white).}
\label{figure:microTurb}
\end{figure}
For matching synthetic spectra with observed ones, we need micro-turbulence as an additional adhoc parameter.
Our definition of this is that a large scale (macro-) turbulent motion triggers a small scale
(micro-) turbulent motion on length scales below the photon main free path length, which affects the strength of spectral lines \citep{gray2005observation}.
In this picture, the micro-turbulence is strongly related to macro-turbulent motion, therefore we use
$v_{\mathrm micro} = 0.5 \cdot \left< v_{\mathrm conv} \right>$ as an experimental
formula that follows from 3D radiative hydrodynamic investigations of cool M-stars \citep{2009A&A...508.1429W}.
So we first calculate the model atmosphere and then synthesize a spectrum from it using a micro-turbulence,
which is assumed to be half the mean convective velocity in the photosphere. Fig.~\ref{figure:microTurb}
shows the distribution of micro-turbulences in our grid for solar abundances.
Unfortunately we had a problem with \emph{PHOENIX}\ concerning convection for giants around 7\,000\,K, so we had to disable convection
for those models. Therefore there is no micro-turbulence included as well.
\section{Fitting stellar parameters}
The spectroscopic analysis of crowded stellar fields, such as star clusters or nearby galaxies has been limited to
relatively small samples of stars thus far. The main problem in this respect is that traditionally used techniques
like multi-object spectroscopy are restricted to the brighter, isolated stars in the field. We have developed a
new method to overcome this limitation using integral field spectroscopy. Taking advantage of the combined spatial
and spectral coverage provided by an integral field spectrograph, we developed a new analysis approach which we
call ``crowded field spectroscopy'' \citep{2004AN....325..155B,Kamann}. Via PSF fitting techniques, single object spectra for the stars above the confusion
limit are extracted. This deblending technique works so well that we obtain clean stellar spectra for a significantly
higher number of stars than hitherto possible.
For the extracted spectra we determine the stellar parameters using a weighted constrained
non-linear least-squares minimization (Levenberg-Marquardt), similar to the ULySS package by
\cite{2009A&A...501.1269K}, which has been used as well e.\,g.\ by \cite{2011RAA....11..924W}. Usually six different parameters are fitted: effective
temperature $T_{\mathrm{eff}}$, surface gravity $\log(g)$, metallicity $[Fe/H]$, $\alpha$-element abundance $[\alpha/Fe]$,
radial velocity $v_\mathrm{rad}$ and line broadening $\sigma$.
In every iteration of the Levenberg-Marquardt algorithm, a model spectrum is extracted from the \emph{PHOENIX}\ grid
using an N-dimensional spline interpolator. Then a line of sight velocity distribution (LOSVD) is applied to the
model, which adds line broadening $\sigma$ and a shift caused by e.\,g.\ radial velocity $v_{\mathrm{rad}}$. Finally an N-dimensional Legendre
polynomial (i.\,e.\ the continuum difference between model and observation) is determined using a linear fit in a way that
the model multiplied with this polynomial matches the observation as best as possible. Due to this we are independent
of the continuum and the fit is done on spectral lines only.
\begin{figure}
\centering
\includegraphics[height=5.3cm]{47tuc_field1_cmd.eps}
\quad
\includegraphics[height=5.3cm]{fitVrad.eps}
\caption{The color-magnitude diagram for a simulated \emph{MUSE}\ data cube is shown on the left. For the
analysis we only used 1/9 of the cube, for which the stars are marked in red. On the right the
errors in fitted radial velocities are plotted.}
\label{figure:dryRun}
\end{figure}
Figure~\ref{figure:dryRun} shows some first preliminary results for a simulated \emph{MUSE}\ data cube based
on real HST observations of 47 Tuc, obtained by \cite{2007AJ....133.1658S}. As one can see, at the main-sequence
turnoff point at $\sim$17mag we still can fit the radial velocity with an accuracy of about 5km/s. The systematic
offset in the results is caused by a known error in the creation of the simulated data cube. At that magnitude,
the corresponding error in the fitted effective temperature is of the order of 100\,K.
\cite{2008ApJ...682.1217K} showed that even with medium resolution spectra it is possible to determine the abundance of single
elements, in their case it was some of the alpha elements (Mg, Si, Ca, Ti). For this analysis, they created a mask in order to fit only those parts
of a spectrum against a grid of synthetic spectra, where it changes most when varying the analysed element. They
created the mask by looking for regions, where two spectra with a given $T_{\mathrm{eff}}$, a fixed $\log(g)$,
a metallicity of $[Fe/H]=-1.5$ and element abundances of $[X/Fe]=\pm0.3$ differ by more then $0.5\%$. Masks for several
different temperatures where combined into a single mask that was used for the analysis.
The main advantage of this method was that they did not need new dimensions in the grid for
every new element, but could use an existing one (here $[\alpha/Fe]$), since the masks did not overlap.
With our new \emph{PHOENIX}\ grid, we can go one step further and create a mask specifically
for every single spectrum that we want to analyse. Therefore we use the same method as described by
\cite{2008ApJ...682.1217K}, but use the previously fitted values for $T_{\mathrm{eff}}$ and $\log(g)$ of
the observed spectrum. Using the individual mask for each star, alpha element abundances can then be
determined with an uncertainty of typically 0.1-0.2dex.
Of course our intention is to extent this method even further in order to fit the
abundance of other elements.
When observing the same field multiple times, we will have radial velocities for several epochs, so that
we can determine orbital parameters for binaries that we find. Furthermore we can extend our method described
above to fit simultaneously the two components of a binary star and henceforth derive the atmospheric
parameters of both stars.
\section{Conclusion}
We presented a new extensive grid of synthetic stellar spectra from \emph{PHOENIX}\ atmospheres with
a wavelength range and resolution that should cover all existing and upcoming instruments. Currently
its parameter range is optimized for the analysis of globular clusters, but we intend to extend
it to higher temperatures. We also introduced a new self-consistent way of describing the
micro-turbulence in model atmospheres.
Furthermore we presented a first view on the methods for analyzing globular clusters with data
obtained with the \emph{VLT} \emph{MUSE}\ together with preliminary results on simulated data. We showed that we
will be able to examine both the kinematics as well as the binary fraction. In addition we will have
accurate stellar parameters for most of the stars in the field.
|
{
"timestamp": "2012-03-12T01:00:16",
"yymm": "1203",
"arxiv_id": "1203.1941",
"language": "en",
"url": "https://arxiv.org/abs/1203.1941"
}
|
\section{Introduction}
The central exclusive production (CEP) process $pp\to p + X + p$,
where $X$ stands for a centrally produced system separated from the
two very forward protons by large rapidity gaps, has been proposed
in Refs.~\cite{SNS90,Bialas:1991wj} as an alternative way of searching for
the neutral Higgs boson (see Ref.~\cite{Albrow:2010yb} for a review).
If momenta of the outgoing protons are measured by forward
proton detectors placed at 220 m and 420 m
from the ATLAS/CMS interaction point \cite{FP420},
the mass of the $X$ system may be reconstructed~\cite{Albrow:2000na}
with very precise resolution.
The exclusive reaction $pp\to pHp$ has been
intensively studied by the Durham group \cite{Durham} in the last
decade. This study was motivated by the clean environment and
largely reduced background due to a suppression of $b\bar b$ production
as a consequence of the spin-parity conservation in the forward limit.
However, very recent precise calculations of Refs.~\cite{MPS_bbbar}
have shown that the situation with Higgs CEP background in the $b\bar b$
channel is more complicated and the signal is to a large extent shadowed by the
exclusive non-reducible continuum $b{\bar b}$ production.
In addition, reducible backgrounds from a misidentification of gluonic
jets as $b$-quark jets can be very difficult to separate
\cite{MPS2011_gg}. Since the total cross section for the Higgs CEP
is quite small and rather uncertain, the issue with the Higgs CEP is
still far from its final resolution, from both theoretical and
experimental point of view.
The final system $X$ in the midrapidity region is predominantly
produced in the $J_z=0$ state as dictated by the well-known $J_z=0$
selection rule \cite{Durham}. However, corrections to this rule due
to slightly off-forward protons can be important for lower (a few GeV)
mass central systems and may lead to sizeable contributions in the observable
signals, in particular, in the $\chi_c$ mesons \cite{chic,LKRS10},
$b\bar b$ \cite{MPS_bbbar} and $gg$ \cite{Cudell:2008gv,MPS2011_gg} CEP.
The emission of gluons from the "screening" gluon
could also violate the $J_z=0$ selection rule as has
recently been emphasized in Ref.~\cite{Cudell:2008gv}.
In order to reduce the theoretical uncertainties
of the CEP mechanism, coming from both the hard subprocess
(Sudakov form factor \cite{Cudell:2008gv,Dechambre:2011py},
next-to-leading order QCD corrections \cite{Khoze:2006um})
and the soft interactions
(color screening effects at extremely small gluon $x$ \cite{soft-col},
rapidity gap survival factor \cite{SF},
poorly known unintegrated gluon distribution functions (UGDFs) at small gluon
$q_\perp$ and $x$ \cite{chic,LKRS10}),
new experimental data
on various exclusive production channels are certainly required
and expected to come soon from ongoing LHC measurements.
In particular, as it was stressed e.g. in Ref.~\cite{Dechambre:2011py}
the measurements of the exclusive dijets production at the LHC could
largely reduce the theoretical uncertainty in the Higgs boson CEP.
Other measurements, e.g.
heavy quarkonia \cite{chic,LKRS10},
$\gamma\gamma$ \cite{LKRS10},
high-$p_\perp$ light mesons \cite{Szczurek:2006bn,HarlandLang:2011qd},
associated charged Higgs $H^+W^-$ \cite{EP2011} CEP, etc.,
are also important in this context.
Some of these results have been compared to
experimental data from the Tevatron~\cite{CDF,CDF_gamgam},
and a rough quantitative agreement between them has been achieved.
In this paper, we focus on exclusive production of $W^+W^-$ pairs
in high-energy proton-proton collisions.
It was found recently \cite{royon,piotrzkowski} that the reaction is an
ideal case to study experimentally $\gamma W^+ W^-$ and
$\gamma \gamma W^+ W^-$ couplings
\footnote{Some more subtle aspects of the beyond Standard Model
anomalous couplings were discussed e.g. in \cite{MMN08}.}.
The $\gamma \gamma \to W^+ W^-$ process is interesting reaction to
test the Standard Model and any other theory beyond the Standard Model.
The linear collider would be a good option to study the couplings of
gauge bosons in the distant future.
For instance in Ref.\cite{NNPU} the anomalous coupling in locally
SU(2) $\times$ U(1) invariant effective Lagrangian was studied.
Other models also lead to anomalous gauge boson coupling.
The photon-photon contribution for the purely exclusive production
of $W^+ W^-$ was considered so far in the literature.
The diffractive production and decay of Higgs boson into the $W^+W^-$ pair
was discussed in Ref.~\cite{WWKhoze}, and the corresponding cross
section turned out to be significantly smaller than that for
the $\gamma\gamma$-contribution.
Provided this is the case, the $W^+W^-$ pair production signal would
be particularly sensitive to New Physics contributions in
the $\gamma \gamma \to W^+ W^-$ subprocess \cite{royon,piotrzkowski}.
Similar analysis has been considered recently
for $\gamma \gamma \to Z Z$ \cite{Gupta:2011be}.
These previous analyses strongly motivate our present detailed study
on a competitive diffractive contribution.
The $pp\to pW^+W^-p$ process going through the diffractive QCD
mechanism with the $gg \to W^+W^-$ subprocess naturally constitutes
a background for the exclusive electromagnetic
$pp\to p(\gamma\gamma\to W^+W^-)p$ process.
We consider not only the mechanism with intermediate Higgs boson but
also quark box contributions never estimated in exclusive processes.
Both the Higgs and box contribution may interfere together.
We discuss here the interference effects.
Corresponding measurements will be possible to perform at the ATLAS
detector with the use of very forward proton detectors \cite{royon}.
In order to quantify to what extent the QCD mechanism competes with the
``signal'' from the $\gamma\gamma$ fusion, we calculate both
contributions and compare them differentially
as a function of several relevant kinematical variables.
Since the box contribution of exclusive diffractive $p p \to p p W^+ W^-$
process is very similar to the $p \bar{p} \to p \bar{p} \gamma \gamma$ process
which has been measured recently \cite{CDF_gamgam}, we discuss the
latter one and compare corresponding results with the recent CDF data.
\section{Diffractive mechanism of exclusive $W^+W^-$ pair production}
A schematic diagram for central exclusive production of
$W^{\pm}W^{\mp}$ pairs in proton-proton scattering $pp\to
pW^{\pm}W^{\mp}p$ is shown in Fig.~\ref{fig:WWCEP}. Similar
mechanisms have been considered in inclusive production of $W^+ W^-$
pairs (see e.g. Refs.~\cite{inclusive,gamgam_WW,DDS95}). In what
follows, we use the standard theoretical description of CEP
processes developed by Khoze, Martin and Ryskin for the exclusive
production of Higgs boson \cite{Durham}.
\begin{figure}[h!]
\centerline{\epsfig{file=WW_CEP.eps,width=6.0cm}} \caption{Generic
diagram for the central exclusive $WW$ pair production in $pp$
collisions. Momenta of incident particles are shown explicitly.}
\label{fig:WWCEP}
\end{figure}
The momenta of intermediate gluons are given by Sudakov
decompositions in terms of the incoming proton four-momenta
$p_{1,2}$
\begin{eqnarray}\nonumber
&&q_1=x_1p_1+q_{1\perp},\quad q_2=x_2p_2+q_{2\perp},\quad 0<x_{1,2}<1,\\
&&q_0=x'p_1-x'p_2+q_{0\perp}\simeq q_{0\perp},\quad x'\ll x_{1,2},
\label{moms}
\end{eqnarray}
where $x_{1,2},x'$ are the longitudinal momentum fractions for
active (fusing) and color screening gluons, respectively.
In the forward proton scattering limit, we have
\begin{eqnarray}\nonumber
&&t_{1,2}=(p_{1,2}-p'_{1,2})^2\simeq{p'}^2_{1,2\perp}\to0 \,,\\
&&q_{\perp} \equiv q_{0\perp} \simeq -q_{1\perp} = q_{2\perp} \,.
\label{forward}
\end{eqnarray}
The QCD factorisation of the process at the hard scale $\mu_F$ is
provided by the large invariant mass of the $WW$ pair $M_{WW}$, i.e.
\begin{eqnarray}\label{sx1x2}
\mu_F^2\equiv s\,x_1x_2\simeq M_{WW}^2\,.
\end{eqnarray}
It is convenient to introduce the Sudakov expansion for $W^{\pm}$
boson momenta
\begin{eqnarray}
k_+=x_1^+ p_1+x_2^+ p_2+k_{+\perp},\quad
k_-=x_1^- p_1+x_2^- p_2+k_{-\perp}
\end{eqnarray}
leading to
\begin{eqnarray}\label{xqq}
x_{1,2}=x_{1,2}^+ + x_{1,2}^-,\quad
x_{1,2}^+=\frac{m_{+\perp}}{\sqrt{s}}e^{\pm y_+},\quad
x_{1,2}^-=\frac{m_{-\perp}}{\sqrt{s}}e^{\pm y_-},\quad
m_{\pm\perp}^2=m_W^2+|{\mbox{\boldmath $k$}}_{\pm\perp}|^2\,,
\end{eqnarray}
in terms of $W^{\pm}$ rapidities $y_{\pm}$ and transverse masses
$m_{\pm\perp}$. For simplicity, in actual calculations we work in
the forward limit given by Eq.~(\ref{forward}), which implies that
${\mbox{\boldmath $k$}}_{+\perp}=-{\mbox{\boldmath $k$}}_{-\perp}$.
In actual calculations below, $W^{\pm}$ bosons are assumed to be
on-mass-shell, whereas particular contributions to the observables
can then be estimated in the narrow-width approximation. For
example, in the leptonic channel we have the following observable
cross section
\begin{eqnarray}
\sigma_{l^+\nu l^-\nu}\simeq \sigma_{WW}\times \mbox{BR}(W^+\to
l^+\nu)\,\mbox{BR}(W^-\to l^-\nu) \, ,
\end{eqnarray}
where $\mbox{BR}(W^+ \to l^+ \nu) = (10.80 \pm 0.09) \times 10^{-2}$
\cite{PDG} for a given lepton flavor. Both electrons and muons can
be used in practice \cite{royon}.
We write the amplitude of the diffractive process, which at high energy
is dominated by its imaginary part, as
\begin{eqnarray} \label{ampl}
{\cal M}_{\lambda_+\lambda_-}(s,t_1,t_2) &\simeq&is\frac{\pi^2}{2}
\int d^2 {\mbox{\boldmath $q$}}_{0\perp} V_{\lambda_+\lambda_-}(q_1,q_2,k_{+},k_{-})
\frac{f_g(q_0,q_1;t_1)f_g(q_0,q_2;t_2)}
{{\mbox{\boldmath $q$}}_{0\perp}^2\,{\mbox{\boldmath $q$}}_{1\perp}^2\,{\mbox{\boldmath $q$}}_{2\perp}^2}\,,
\end{eqnarray}
where $\lambda_{\pm}=\pm 1,\,0$ are the polarisation states
of the produced $W^{\pm}$ bosons, respectively, $f_g(r_1,r_2;t)$ is
the off-diagonal unintegrated gluon distribution function (UGDF),
which depends on the longitudinal
and transverse components of both gluons momenta.
The gauge-invariant $gg\to W_{\lambda_+}^+ W_{\lambda_-}^-$
hard subprocess amplitude
$V_{\lambda_+\lambda_-}(q_1,q_2,k_{+},k_{-})$
is given by the light cone projection
\begin{eqnarray}\label{GIproj}
V_{\lambda_+\lambda_-}=
n^+_{\mu}n^-_{\nu}V_{\lambda_+\lambda_- , \mu\nu}=
\frac{4}{s}
\frac{q^{\mu}_{1\perp}}{x_1}
\frac{q^{\nu}_{2\perp}}{x_2}
V_{\lambda_+\lambda_-,\mu\nu},\quad
q_1^{\mu}V_{\lambda_+\lambda_-,\mu\nu}=
q_2^{\nu}V_{\lambda_+\lambda_-,\mu\nu}=0\,,
\end{eqnarray}
where $n_{\mu}^{\pm} = p_{1,2}^{\mu}/E_{p,cms}$ and the
center-of-mass proton energy $E_{p,cms} = \sqrt{s}/2$. We adopt the
definition of gluon transverse polarisation vectors proportional to
the transverse gluon momenta $q_{1,2 \perp}$, i.e. $\epsilon_{1,2} \sim q_{1,2 \perp} /
x_{1,2}$. The helicity matrix element in the previous expression reads
\begin{eqnarray}
V_{\lambda_+\lambda_-}^{\mu\nu}(q_1,q_2,k_{+},k_{-})=
\epsilon^{*,\rho}(k_+,\lambda_+)
\epsilon^{*,\sigma}(k_-,\lambda_-)V_{\rho\sigma}^{\mu\nu}\,,
\label{Vepsilon}
\end{eqnarray}
in terms of the Lorentz and gauge invariant $2\to2$ amplitude
$V_{\rho\sigma}^{\mu\nu}$ and $W$ boson polarisation vectors
$\epsilon(k,\lambda)$. Below we will analyze the exclusive
production with polarized $W^+$ or $W^-$. In Eq.~(\ref{Vepsilon})
$\epsilon_{\mu}(k_+,\lambda_+)$ and
$\epsilon_{\nu}(k_-,\lambda_-)$ can be defined easily in the
proton-proton center-of-mass frame
as
\begin{eqnarray}
\epsilon(k,0) &=& \frac{E_{W}}{m_{W}}
\left(\frac{k}{E_{W}},\,\cos\phi \sin\theta, \,
\sin\phi \sin\theta,\,\cos\theta\right) \,,\nonumber \\
\epsilon(k,\pm 1) &=& \frac{1}{\sqrt{2}} \left(0,\, i\sin\phi \mp
\cos\theta\cos\phi,\, -i\cos\phi \mp \cos\theta\sin\phi,\, \pm
\sin\theta\right)\,,
\label{vectors}
\end{eqnarray}
where $\phi$ is the azimuthal angle of a produced boson, and satisfy
$\epsilon^{\mu}(\lambda)\epsilon^*_{\mu}(\lambda)=-1$ and
$\epsilon^*_{\mu}(k_+,\lambda_+)k_+^{\mu}=\epsilon^*_{\nu}(k_-,\lambda_-)k_-^{\nu}=0$.
In the forward limit, provided by Eq.~(\ref{forward}), the azimuthal
angles of the $W^+$ and $W^-$ bosons are related as $\phi_{-} = \phi_{+} + \pi$.
The diffractive amplitude given by Eq.~(\ref{ampl}) is averaged over
the color indices and over the two transverse polarizations of the
incoming gluons. The relevant color factor which includes summing
over colors of quarks in the loop (triangle or box) and averaging
over fusing gluon colors (according to the definition of
unintegrated gluon distribution function) is the same as in the
previously studied Higgs CEP (for more details on derivation of the
generic $pp\to pXp$ amplitude, see e.g. Ref.~\cite{Albrow:2010yb}).
The matrix element $V_{\lambda_{+},\lambda_{-}}$ contains twice the
strong coupling constant $g_s^2 = 4 \pi \alpha_{s}$. In our
calculation here we take the running coupling constant
$\alpha_s(\mu_{hard}^2=M_{WW}^2)$ which depends on the invariant
mass of $WW$ pair as a hard renormalisation scale of the process.
The choice of the scale approximately introduces roughly a factor of
two model uncertainties when varying the hard scale $\mu_{hard}$
between $2M_{WW}$ and $M_{WW}/2$ values.
The bare amplitude above is subjected to absorption corrections that
depend on the collision energy and typical proton transverse
momenta. As in the original KMR calculations \cite{Durham}, the bare
production cross section is usually multiplied by a rapidity gap survival
factor which we take the same as for the Higgs boson and $b \bar b$
production to be $S_{g} = 0.03$ at the LHC energy (see e.g.
Ref.~\cite{MPS2011_gg}).
\subsection{The hard subprocess}
The typical contributions to the $gg\to W^+W^-$ subprocess are shown
in Fig.~\ref{fig:WWhard}. The total number of topologically
different loop diagrams amounts to two triangles, and six boxes. In
the central exclusive $W^+W^-$ production, triangle diagrams with
$\gamma$ and $Z$ bosons in the intermediate state are suppressed due
to the $J_z$ = 0 and parity selection rule for singlet gluon-gluon to
(virtual) photon transition strictly valid in the on-shell limit of
fusing gluons and Landau-Yang theorem for intermediate $Z$ boson.
Then the only non-zeroth contribution comes from the Higgs resonant
diagram, and in the next subsection we will discuss it in detail.
However, this can only lead to a sizeable enhancement of the cross
section close to its threshold $m_{h^0}\simeq M_{WW}\gtrsim 2 m_W$
\cite{WWKhoze}. The Standard Model Higgs bosons with such large
masses have been recently excluded by the Tevatron
\cite{CDF_exclusion} and LHC \cite{ATLAS_exclusion,CMS_exclusion}
measurements. For yet allowed values of Higgs mass 115 GeV $\lesssim
m_{h^0} \lesssim $ 130 GeV, corresponding contribution to the
$W^{+}W^{-}$ channel is far from the Higgs boson resonance and turned
out to be suppressed compared to box contributions at low invariant masses.
However, due to interference effects at rather large
invariant masses $M_{WW}$ the resonant (triangles) contribution
could become comparable to the non-resonant (boxes) one. Below, for
comparison we have calculated box and triangle (through the
$s$-channel SM Higgs boson exchange) contributions in different phase
space regions
\footnote{Close to the $WW$-threshold instability
of $W$ bosons \cite{Wsmear} should be included.}
which could be interesting for future measurements
with forward detectors at ATLAS or CMS.
\begin{figure}[h!]
\centerline{\epsfig{file=gg_WW.eps,width=10.5cm}}
\caption{Representative diagrams of the hard subprocess
$gg\to W^{\pm}W^{\mp}$,
which contribute to the exclusive $WW$ pair production.}
\label{fig:WWhard}
\end{figure}
\subsubsection{Higgs contribution}
The matrix element for the $gg \to h^0 \to W^+ W^-$ transition with
intermediate $s$-channel Higgs boson exchange
(see first two diagrams in Fig.~\ref{fig:WWhard})
can be written in the narrow-width approximation as
\begin{eqnarray}\nonumber
&&V_{gg\to h^0\to
W^+W^-}(q_1,q_2,k_+,k_-)=\delta^{(4)}(q_1+q_2-k_+-k_-)\times \\
&&\qquad V_{gg \to h^0}(q_1,q_2,p_{h^0}) \, \frac{i}{M_{WW}^2 -
m_{h^0}^2 + iM_{WW}\Gamma_{\rm{tot}}^h} \, V_{h^0 \to
W^+W^-}(k_+, k_-,\lambda_+,\lambda_-),
\label{triangle}
\end{eqnarray}
where the Higgs boson momentum is $p_{h^0}=q_1+q_2$, and the
$\delta$-function reflects the momentum conservation in the process.
In order to get a correct resonant invariant mass distribution, the
standard Breit-Wigner Higgs propagator with the total Higgs decay
width $\Gamma^{h}_{\rm{tot}}$, which can be found e.g. in
Ref.~\cite{Passarino}, is used.
In Eq.~(\ref{triangle}), first the $gg\to h^0$ amplitude of the Higgs
boson production through the top-quark triangle in the $k_t$-factorisation approach
can be written as (see e.g. Ref.~\cite{incl-Higgs})
\begin{equation}
V_{gg \to h^0}\simeq
\frac{i\delta^{ab}}{v}\,\frac{\alpha_s(\mu_F^2)}{\pi}\,(\mbox{\boldmath $q$}_{1\perp}\cdot
\mbox{\boldmath $q$}_{2\perp})\,\frac{2}{3}\left(1+\frac{7}{120}\frac{M_{WW}^2}{m_{\rm{top}}^2}\right)\,,\qquad
v=\left(G_F\sqrt{2}\right)^{-1/2}.
\end{equation}
The second tree-level $h^0 \to W^+ W^-$ ``decay'' amplitude reads:
\begin{equation}
V_{h^0 \to W^+W^-}\simeq i m_W\,\frac{e}{\sin\theta_W}\,
\epsilon^*(k_+,\lambda_+) \epsilon^*(k_-,\lambda_-)\,,
\end{equation}
where the polarisation vectors in the direction of motion of $W^+$
and $W^-$ bosons in the proton-proton center-of-mass frame are used
in practical calculations.
Potentially interesting contribution could come from the Higgs
resonance if the Higgs mass was close to the $WW$ production
threshold. Similar resonance effects have been considered recently
in inclusive \cite{Enberg:2011ae} and exclusive associated
\cite{EP2011} charged Higgs boson production, and large
contributions beyond the Standard Model were found. However, the SM
Higgs mass $\sim$160 GeV has been recently excluded in inclusive
searches by the CDF Collaboration at Tevatron \cite{CDF_exclusion}
and by the ATLAS and CMS Collaborations at LHC
\cite{ATLAS_exclusion,CMS_exclusion}, so yet realistic SM Higgs
boson mass interval $m_{h^0}\sim 115-130$ GeV leads to a suppressed triangles'
contribution to exclusive $W^{+} W^{-}$ pair production.
In the calculation presented here we take $m_{h^{0}}$ = 120 GeV.
Since the Higgs mass is certainly much smaller than the threshold value
a precise value of the Higgs boson mass is not very important.
A contribution from an extended Higgs sector
beyond the Standard Model \cite{Enberg:2011ae} could be
interesting, but we postpone this issue for a later study.
In this work, we are primarily interested in estimation of dominant
box contributions as well as in possible box-triangle interference
effects within the Standard Model as potentially important irreducible
background for the $\gamma\gamma\to W^+W^-$ signal relevant for a
precision study of anomalous couplings. Thus, our numerical
estimates provide minimal limit for the central exclusive $WW$
production signal.
\subsubsection{Contribution of box diagrams}
The box contributions to the $gg\to W^+W^-$ parton level subprocess
amplitude (see diagrams No.~(3-8) in Fig.~\ref{fig:WWhard}) for
on-shell fusing gluons were calculated analytically by using the
Mathematica-based {\tt FormCalc} (FC) \cite{FC} package. The
complete matrix element was generated automatically by the FC tools in
terms of one-loop Passarino-Veltman two-, three- and four-point
functions and other internally-defined functions (e.g. gluon and
vector bosons polarisation vectors) and kinematical variables.
At the next step, the Fortran code for the matrix element was
generated, and then used as an external subroutine in our numerical
calculations together with other FC routines setting up the Standard
Model parameters, coupling constants and kinematics. Instead of
built-in FC polarisation vectors we have used transverse gluon
polarisation vectors which enter the projection in
Eq.~(\ref{GIproj}), and the standard $W^{\pm}$ polarisation vectors
defined in Eq.~(\ref{vectors}), giving us an access to individual
polarisation states of the $W$ bosons. In accordance with the
$k_t$-factorisation technique, the gauge invariance of the resulting
amplitudes for the on-mass-shell initial gluons is ensured by a
projection onto the gluon transverse polarisation vectors
proportional to the transverse gluon momenta $q_{1,2\perp}$
according to Eq.~(\ref{GIproj}).
For the evaluation of the scalar master tree- and four-point
integrals in the gluon-gluon fusion subprocess we have used the {\tt
LoopTools} library \cite{FC}. The result is summed up over all
possible quark flavors in loops and over distinct loop topologies.
We have also checked that the sum of relevant diagrams is explicitly
finite and obeys correct asymptotical properties and energy
dependence. It is worth to mention that a large cancelation between
separate box contributions in the total sum of diagrams takes place,
which is expected from the general Standard Model symmetry
principles
\footnote{We are thankful to Prof. O. Nachtmann for
an enlightening discussion on this matter.}.
As soon as the hard subprocess matrix element (denoted above as
$V_{\lambda_+\lambda_-}$) has been defined as a function of relevant
kinematical variables (four-momenta of incoming/outgoing particles),
the loop integration over $q_{0\perp}$ in Eq.~(\ref{ampl}) was
performed to obtain the diffractive amplitude, which then has been
used to calculate the differential distributions for (un)polarised
$W$ bosons.
As we will demonstrate below, in the Standard Model the total box
contribution is somewhat larger than the triangle one, for the realistic
Higgs boson masses. We, however, keep both the triangle and box
contributions and investigate a possible interference between them,
which, in fact, is quite important, especially at rather large $W^+W^-$-pair
invariant masses, i.e. in the region we are interested in.
\subsection{Exclusive $p p \to p p \gamma \gamma$ process}
The same formalism as described above is used to calculate the amplitude
for the $p p \to p p \gamma \gamma$ process.
We write the amplitude of the diffractive
$p p \to p p \gamma \gamma$ process as
\begin{eqnarray} \label{ampl_gamgam}
{\cal M}_{\lambda_+\lambda_-}(s,t_1,t_2) &\simeq&is\frac{\pi^2}{2}
\int d^2 {\mbox{\boldmath $q$}}_{0\perp}
V_{\lambda_+\lambda_-}^{gg \to \gamma \gamma}(q_1,q_2,k_{+},k_{-})
\frac{f_g(q_0,q_1;t_1)f_g(q_0,q_2;t_2)}
{{\mbox{\boldmath $q$}}_{0\perp}^2\,{\mbox{\boldmath $q$}}_{1\perp}^2\,{\mbox{\boldmath $q$}}_{2\perp}^2}\,,
\end{eqnarray}
where now $\lambda_{\pm}=\pm 1$ are the helicity polarisation states
of the produced photons
and corresponding polarisation vectors are defined
easily in the $pp$ center-of-mass frame
\begin{eqnarray}
\epsilon(k,\pm 1) &=& \frac{1}{\sqrt{2}} \left(0,\, i\sin\phi \mp
\cos\theta\cos\phi,\, -i\cos\phi \mp \cos\theta\sin\phi,\, \pm
\sin\theta\right)\, .
\label{vectors_gamgam}
\end{eqnarray}
The typical contributions to the leading order $gg\to \gamma\gamma$
subprocess are shown in Fig.~\ref{fig:GGhard}. The total number of
topologically different loop diagrams in the Standard Model amounts
to twelve boxes. So the $\gamma\gamma$ does not exhibit resonant
features, and can potentially serve as a probe for New Physics
resonant contributions.
\begin{figure}[h!]
\centerline{\epsfig{file=gg_gam_gam.eps,width=6.5cm}}
\vspace{-2cm}
\caption{Representative diagrams of the hard subprocess $gg\to
\gamma\gamma$, which contribute to the exclusive $\gamma\gamma$ pair
production.} \label{fig:GGhard}
\end{figure}
The box contributions to the $gg\to \gamma\gamma$ parton level
subprocess amplitude in Fig.~\ref{fig:GGhard} for on-shell fusing
gluons were calculated analytically by using the Mathematica-based
{\tt FormCalc} (FC) \cite{FC} package. The complete matrix element
was automatically generated by FC tools in terms of one-loop
Passarino-Veltman two-, three- and four-point functions and other
internally-defined functions (e.g. gluon and vector bosons
polarisation vectors) and kinematical variables.
Other details of the calculation are very much the same as those
for the $W^+ W^-$ production. We will not repeat here the details.
\subsection{Gluon $k_{\perp}$-dependent densities in
the forward limit}
In the $k_t$-factorisation approach, the density of gluons in the
proton is described in terms of the off-diagonal unintegrated gluon
distribution functions (UGDFs)
$f_g(q_0,q_{1,2};t_{1,2})=f^{\mathrm{off}}_g(x',x_{1,2},{\mbox{\boldmath $q$}}_{0\perp}^2,{\mbox{\boldmath $q$}}_{1/2\perp}^2,\mu_F^2;t_{1,2})$
at the factorization scale $\mu_F \sim M_{WW}\gg |{\mbox{\boldmath $q$}}_{0\perp}|$.
In the forward scattering (see Eq.~(\ref{forward})) and asymmetric
limit of $x'\ll x_{1,2}$, the off-diagonal UGDF is written as
a skewedness factor $R_g(x')$ multiplied by the diagonal UGDF, which
describes the coupling of gluons with longitudinal momentum
fractions $x_{1,2}$ to the proton (see Refs.~\cite{Kimber:2001sc,MR}
for details). The skewedness parameter $R_g$ is
expected to be roughly constant at LHC energies and gives only a
small contribution to the overall normalization uncertainty.
We take $R_{g} = 1.3$ in practical calculations.
In the kinematics considered here, the unintegrated gluon density can be
written in terms of the conventional integrated gluon distribution
$g(x,{\mbox{\boldmath $q$}}_{\perp}^2)$ as~\cite{MR}
\begin{eqnarray}\nonumber
f_g(q_0,q_{1,2};t_{1,2})&\simeq&
R_g f_g(x_{1,2},{\mbox{\boldmath $q$}}_{\perp}^2,\mu_F^2)\exp(bt_{1,2}/2)=\\
&&R_g\frac{\partial}{\partial\ln {\mbox{\boldmath $q$}}_{\perp}^2}
\Big[x_{1,2} g(x_{1,2},{\mbox{\boldmath $q$}}_{\perp}^2)\sqrt{T_g({\mbox{\boldmath $q$}}_{\perp}^2,\mu_F^2)}\Big]
\exp(bt_{1,2}/2)\,, \label{ugdfkmr}
\end{eqnarray}
where the diffractive slope is taken to be $b=4$ GeV$^{-2}$.
$T_g$ is the Sudakov form factor which suppresses real emissions
from active gluons during the evolution,
so that the rapidity gaps are not populated by gluons.
It is given by~\cite{MR}
\begin{eqnarray}
T_g({\mbox{\boldmath $q$}}_{\perp}^2,\mu_F^2)&=&{\rm exp}
\bigg(-\int_{{\mbox{\boldmath $q$}}_{\perp}^2}^{\mu_F^2} \frac{d
{\mbox{\boldmath $k$}}_{\perp}^2}{{\mbox{\boldmath $k$}}_{\perp}^2}\frac{\alpha_s({\mbox{\boldmath $k$}}_{\perp}^2)}{2\pi}\int_{0}^{1-\Delta}
\!\bigg[ z P_{gg}(z) + \sum_{q} P_{qg}(z) \bigg]dz \!\bigg)\,,
\label{Sudakov}
\end{eqnarray}
where $\Delta$ in the upper limit is taken to be
\cite{Coughlin:2009tr}
\begin{equation}\label{delta}
\Delta=\frac{|{\mbox{\boldmath $k$}}_{\perp}|}{|{\mbox{\boldmath $k$}}_{\perp}|+M_{WW}} \,.
\end{equation}
In our calculations we take $\mu_F^2 = M_{WW}^2$.
The choice of the scale introduces uncertainties roughly
of about factor two. Since in the present calculations we need
values of $T_g({\mbox{\boldmath $q$}}_{\perp}^2,\mu_F^2)$ for extremely large scales
$\mu_F^2$ the integration in Eq.~(\ref{Sudakov}) is performed rather
in $\log_{10}(k^2/k_0^2)$, where $k_{0} = 1$ GeV was introduced for convenience.
\section{Four-body phase space in the forward limit}
The diffractive $WW$ CEP amplitude (\ref{ampl}) described above is
used now to calculate the corresponding cross section including
certain limitations of the phase space. The cross section for the
two-boson production can be obtained by integration over the
four-body phase space given by
\begin{eqnarray}
\sigma=\frac{(2 \pi)^4}{2s}\int\overline{ |{\cal M}|^2}\delta^4 (p_1
+ p_2 - p'_1 - p'_2 - k_+ - k_-) \frac{d^3 p'_1}{(2 \pi)^3 2 E'_1}
\frac{d^3 p'_2}{(2 \pi)^3 2 E'_2} \frac{d^3 k_+}{(2 \pi)^3 2 E_+}
\frac{d^3 k_-}{(2 \pi)^3 2 E_-} , \nonumber\\
\label{full_phase_space}
\end{eqnarray}
where $E'_{1,2}$ and $E_{\pm}$ are the energies of the final-state
protons and produced $W^{\pm}$ bosons, respectively, $\overline{
|{\cal M}|^2} = \sum_{\lambda_{+},\lambda_{-}}
{\cal M}_{\lambda_{+}\lambda_{-}} {\cal M}^{*}_{\lambda_{+}\lambda_{-}}$
assuming, as usual, that the helicities of both protons are
unchanged in the considered process. In order to calculate the
total cross section one has to take the eight-dimensional integral
numerically (for details see e.g. Ref.~\cite{LS2010}). However, the
evaluation of the corresponding hard subprocess amplitude
$V_{\lambda_{+}\lambda_{-}}$, its subsequent convolution with the
gluon UPDFs in the diffractive amplitude (\ref{ampl}) and the full
phase space integration (\ref{full_phase_space}) is extremely time
consuming. Clearly the calculation of diffractive mechanism must be
simplified to be feasible. Such a simplification seems possible for
the diffractive process considered here. We start from the choice of
integration variables as in Ref.~\cite{LS2010}. Then
\begin{eqnarray}
d\sigma=\frac{1}{2s}\overline{ |{\cal
M}|^2}\,\frac{1}{2^4}\frac{1}{(2\pi)^8}\frac{1}{E'_1E'_2}\,\frac14\,
dt_1 dt_2 d{\phi_1} d{\phi_2}\,\frac{p_{m\perp}}{4}{\cal
J}^{-1}\,dy_+ dy_- dp_{m\perp} d\phi_m \, ,
\label{redPS0}
\end{eqnarray}
where $p_{m\perp}=|\mbox{\boldmath $k$}_{+\perp}-\mbox{\boldmath $k$}_{-\perp}|$ is the
difference between transverse momenta of $W^+$ and $W^-$,
$\mbox{\boldmath $k$}_{+\perp}$ and $\mbox{\boldmath $k$}_{-\perp}$, respectively, and
$\phi_m$ is the corresponding azimuthal angle.
For the sake of simplicity, assuming an exponential slope of
$t_1 / t_2$-dependence of the KMR UGDFs (see Eq.~(\ref{ugdfkmr})), and as a
consequence of the approximately exponential dependence of the cross
section on $t_1$ and $t_2$ (proportional to $\exp(b t_1)$ and
$\exp(b t_2)$), the four-body phase space can be calculated as follows
\begin{eqnarray}
d \sigma \approx && \frac{1}{2s}\overline{
|{\cal M}|^2}\Big|_{t_{1,2}=0}\,\frac{1}{2^4}
\frac{1}{(2\pi)^8}\frac{1}{E'_1E'_2}\,
\frac14\,
\frac{1}{b^2}\,(2\pi)^2\,
\frac{p_{m\perp}}{4}{\cal J}^{-1}\,dy_+
dy_- dp_{m\perp} d\phi_m \,.
\label{redPS}
\end{eqnarray}
Since in this approximation we have assumed no correlations between
outgoing protons (which is expected here and is practically true for
the production of $b \bar b$ \cite{MPS_bbbar} or $g g$
\cite{MPS2011_gg} dijets) there is no dependence of the integrand in
Eq.~(\ref{redPS}) on $\phi_m$, which means that the phase space
integration can be further reduced to three-dimensional one. The
Jacobian ${\cal J}$ in Eq.~(\ref{redPS0}) is given in Ref.~\cite{LS2010}
\begin{eqnarray}
{\cal J}=\Bigg| \frac{p_{1z}'}{\sqrt{m_p^2+{p'}_{1z}^2}} -
\frac{p_{2z}'}{\sqrt{m_p^2+{p'}_{2z}^2}} \Bigg| \, .
\end{eqnarray}
In actual calculations below we shall use the reduced form of the
four-body phase space Eq.~(\ref{redPS}), and it is checked to give
correct numerical results against the full phase space calculation
for some simple reactions.
Different representations of the phase space depending on a
particular kinematical distributions needed can be found in Ref.~\cite{LS2010}.
\section{$\gamma \gamma \to W^+ W^-$ mechanism}
In this section, we briefly discuss the $\gamma \gamma \to W^+ W^-$
mechanism, considered already in the literature (see
Refs.~\cite{royon,piotrzkowski}).
The relevant subprocess diagrams are shown in Fig.~\ref{fig:gamgam_WW}.
Let us start from the reminder about the $\gamma \gamma \to W^+ W^-$
coupling within the Standard Model.
The three-boson $WW \gamma$ and four-boson $WW \gamma\gamma$ couplings,
which contribute to the $\gamma \gamma \to W^+ W^-$ process in
the leading order read
\begin{eqnarray}
\mbox{$\mathcal{L}$}_{WW\gamma} & = &
-ie( A_\mu W^-_\nu \twosidep{\mu} W^{+\nu}
+ W_\mu^- W^+_\nu \twosidep{\mu} A^\nu
+ W^+_\mu A_\nu \twosidep{\mu} W^{-\nu}) \, ,
\label{eq:anom:lagrww1}\\
\mbox{$\mathcal{L}$}_{WW\gamma\gamma} & = &
-e^2(W^{-}_\mu W^{+\mu}A_\nu A^\nu-W_\mu^-A^\mu W^+_\nu A^\nu) \, ,
\label{eq:anom:lagrww2}
\end{eqnarray}
where the asymmetric derivative has the form
$X\twosidep{\mu}Y=X\partial^{\mu}Y-Y\partial^{\mu}X$.
\begin{figure}[h!]
\centerline{\epsfig{file=gamgam_WW.eps,width=8.0cm}}
\caption{The Born diagrams for the $\gamma \gamma \to
W^{\pm}W^{\mp}$ subprocess.}
\label{fig:gamgam_WW}
\end{figure}
Then within the Standard Model, the elementary tree-level cross
section for the $\gamma \gamma \to W^+ W^-$ subprocess can be written in the
compact form in terms of the Mandelstam variables (see e.g. Ref.~\cite{DDS95})
\footnote{This formula does not include the
process with virtual Higgs boson $\gamma \gamma \to H \to W^+ W^-$
\cite{gammagamma_H_WW}. For heavy Higgs boson, this would lead to
clear Higgs boson signal modifying the cross section (typical
resonance $+$ background effect) \cite{DDS95}, however, with the
present limits for Higgs boson mass
\cite{ATLAS_exclusion,CMS_exclusion} only deeply off-shell Higgs
boson contribution could be possible. Also, the diagram with an
intermediate Higgs boson is, of course, of a higher order compared
to the contributions considered here. This automatically means
rather small effect on the measured cross section, in particular, on
the $W^+ W^-$ invariant mass distribution in our case of the
four-body $p p \to p W^+ W^- p$ reaction.}
\begin{equation}
\frac{d\hat{\sigma}}{d \Omega} = \frac{3 \alpha^2 \beta}{2\hat{s}} \left(
1 - \frac{2 \hat{s} (2\hat{s}+3m_W^2)}{3 (m_W^2 - \hat{t}) (m_W^2 -
\hat{u})} + \frac{2 \hat{s}^2(\hat{s}^2+ 3m_W^4)}{3 (m_W^2 -
\hat{t})^2(m_W^2 - \hat{u})^2} \right) \,,
\label{gamgam_WW}
\end{equation}
where $\beta=\sqrt{1-4m_W^2/\hat{s}}$ is the velocity of the $W$
bosons in their center-of-mass frame and the electromagnetic
fine-structure constant $\alpha=e^{2}/(4\pi) \simeq 1/137$ for the
on-shell photon. The total elementary cross section can be obtained
by integration of the differential cross section above.
In the Weizs\"acker-Williams approximation,
the total cross section for the $pp \to pp (\gamma \gamma) \to W^+ W^-$
can be written as in the parton model
\begin{equation}
\sigma = \int d x_1 d x_2 \, f_1^{WW}(x_1) \, f_2^{WW}(x_2) \,
\hat{\sigma}_{\gamma \gamma \to W^+ W^-}(\hat s) \, .
\label{EPA}
\end{equation}
We take the Weizs\"acker-Williams equivalent photon fluxes of protons from Ref.~\cite{DZ}.
To calculate differential distributions the following parton formula
can be conveniently used
\begin{equation}
\frac{d\sigma}{d y_+ d y_- d^2 p_{W\perp}} = \frac{1}{16 \pi^2 {\hat s}^2}
\, x_1 f_1^{WW}(x_1) \, x_2 f_2^{WW}(x_2) \,
\overline{ | {\cal M}_{\gamma \gamma \to W^+ W^-}(\hat s, \hat t, \hat u)
|^2} \, ,
\label{EPA_differential}
\end{equation}
where momentum fractions of the fusing gluons $x_{1,2}$ are defined in
Eq.~(\ref{xqq}). We shall not discuss here any approach beyond the Standard Model.
A potentially interesting Higgsless scenario of the
$WW$-pair production has previously been discussed e.g. in
Refs.~\cite{royon,piotrzkowski}.
In Fig.~\ref{fig:gamma-gamma} we show distribution in $\xi_1 =
\log_{10}(x_1)$ and $\xi_2 = \log_{10}(x_2)$ at $\sqrt{s}$ = 14 TeV.
We observe a maximum of the cross section at $\xi_1, \xi_2 \approx
-2$ which means that corresponding longitudinal momentum fractions
carried by photons are typically 10$^{-2}$.
\begin{figure}[!h]
\includegraphics[width=5cm]{dsig_dxi.eps}
\includegraphics[width=5cm]{map_xi1xi2.eps}
\caption{
\small Summary of the $\gamma \gamma \to W^+ W^-$ contribution. The
lines were calculated within EPA approximation as described in the
text with photon fluxes obtained in Ref.~\cite{DZ}. Here, $\xi_{1.2}
= \log_{10}(x_{1,2})$, where $x_{1,2}$ are photon longitudinal
fractions with respect to parent protons.}
\label{fig:gamma-gamma}
\end{figure}
\section{Inclusive production of $W^+W^-$ pairs}
For a test and for a comparison we also consider a gluon-gluon contribution
to the inclusive cross section.
We are not interested in the quark-antiquark component
which is simple and well known. We also omit $pp \to t\bar{t}X \to W^{+}W^{-}b\bar{b}X$
process very important at high energy.
In the lowest order of pQCD the inclusive cross section
for the gluon-gluon fusion can be written as
\begin{equation}
\frac{d \sigma^{gg}}{d y_+ d y_- d^2 p_{W\perp}} = \frac{1}{16 \pi^2 {\hat s}^2}
x_1 g(x_1, \mu_F^2) x_2 g(x_2,\mu_F^2)
\overline{| {\cal M}_{gg \to W^+ W^-}(\lambda_1, \lambda_2, \lambda_+,
\lambda_-) |^2} \, .
\label{inclusive_cs}
\end{equation}
The corresponding matrix elements have been discussed in the
literature in detail \cite{inclusive}. The distributions in rapidity
of $W^+$ ($y_+$), rapidity of $W^-$ ($y_-$) and transverse momentum
of one of them $p_{W\perp}$ can be calculated in a straightforward way from
Eq.~(\ref{inclusive_cs}). The distribution in invariant mass can be
then obtained by an appropriate binning.
Our inclusive $d \sigma/ d M_{WW}$ distribution seems consistent with
similar distributions presented in the past in the literature.
The total cross section can be obtained from a simpler formula:
\begin{equation}
\sigma_{pp \to W^{+}W^{-}X}^{gg} =
\int d x_1 d x_2 \, g(x_1,\mu_{F}^{2}) \, g(x_2,\mu_{F}^{2}) \,
\hat{\sigma}_{gg \to W^+ W^-}(\hat s) \, .
\label{gg_WW}
\end{equation}
Let us concentrate for a while a the elementary $g g \to W^+ W^-$ cross
section shown in Fig.\ref{fig:gg_WW}.
In this calculation we have assumed $m_{h^{0}}$ = 125 GeV \cite{Higgs}.
We also show a vertical line at the $t \bar t$ threshold.
The figure demonstrates a cancellation
pattern between box and triangle contributions. We will discuss similar
cancellation for the $p p \to p p W^+ W^-$ reaction in the next section.
We wish to notice that
$\hat{\sigma}_{gg \to W^{+}W^{-}} \ll \hat{\sigma}_{\gamma \gamma \to W^{+}W^{-}}
\, {}^{\underrightarrow{\hat{s} \to \infty}} \,\thicksim 10^{2}$ pb.
This shows a potential role of photon-photon induced processes
of $W^+ W^-$ production not discussed
so far in the context of inclusive process.
\begin{figure}[!h]
\includegraphics[width=7cm]{gg_WW_results.eps}
\caption{
\small
The integrated elementary cross section for the $gg \to W^{+}W^{-}$ reaction.
The solid line represents the coherent sum of all contributions.
We show separate contributions of boxes (dashed line) and triangles (dotted line).
}
\label{fig:gg_WW}
\end{figure}
As discussed before, in the case of exclusive scattering
the $J_z =0$ contribution is the dominant one.
In the case of inclusive process the situation is
slightly different.
In Fig.\ref{fig:ggWW_Jz} we present the $J_{z} = 0$
and $|J_{z}| = 2$ components to angular distributions.
The $J_{z} = 0$ contribution is generally larger than the $|J_{z}| = 2$ one.
As in the exclusive case, at forward scattering ($cos \, \theta = \pm 1$)
we observe the dominance of the $J_z = 0$ contribution.
At $\sqrt{\hat s}$ = 500 GeV it happens very close
to $\cos \, \theta \approx \pm$ 1.
\begin{figure}[!h]
\includegraphics[width=7cm]{dsig_dz_ggWW_200_Jzsum.eps}
\includegraphics[width=7cm]{dsig_dz_ggWW_500_Jzsum.eps}
\caption{
\small
Centre-of-mass scattering angle dependence of
the hard subprocess $gg \to W^{+}W^{-}$ cross section
averaged over incoming gluon polarizations.
The solid line represents the coherent sum of all contributions
The $J_{z} = 0$ (dashed line) and the $|J_{z}| = 2$ (dotted line))
contributions are shown separately.
}
\label{fig:ggWW_Jz}
\end{figure}
For completeness in Fig.\ref{fig:pp_WWX}
we show corresponding contributions to
the rapidity distribution of one of $W$'s
in the $p p \to W^+ W^- X$ process.
Here the $J_z$ = 0 contribution is larger
in the whole range of rapidities.
\begin{figure}[!h]
\includegraphics[width=7cm]{dsig_dy_incl.eps}
\caption{
\small
The $J_z = 0$ (dashed line) and $|J_z| = 2$ (dotted line)
contributions to the inclusive $p p \to W^+ W^- X$
rapidity distribution.
}
\label{fig:pp_WWX}
\end{figure}
\section{Results}
Before we go to the presentation of results for the
$p p \to p p W^+ W^-$ reaction we wish to show results for
the $p \bar{p} \to p \bar{p} \gamma \gamma$ reaction.
The latter reaction was
studied experimentally in Ref.\cite{CDF_gamgam}.
\subsection{$p p \to p p \gamma \gamma$}
The $p \bar{p} \to p \bar{p} \gamma \gamma$ process was discussed recently in \cite{HKRS2012}.
No differential distributions have been discussed there.
The CDF Collaboration has measured photons in the interval
$|\eta(\gamma)| <$ 1.0, $E_T >$ 2.5 GeV and with the condition of no
other particles detected in -7.4 $< \eta <$ 7.4. They have obtained
$\sigma_{\gamma \gamma}$ = 2.48 pb with about quarter of relative
uncertainty. We obtain 2.99 pb for the GJR NLO gluon distribution
\cite{GJR}, 2.46 pb for the MSTW08 NLO gluon distribution \cite{MSTW_PDF}
and 2.1 pb for the CT12 NLO gluon distribution \cite{CT_PDF}. Our results
very well agree with the CDF experimental data. In this calculation we
have assumed averaged soft gap survival factor $S_g$ = 0.05 and the scale
of the Sudakov form factor was taken as $\mu^2 = M_{\gamma \gamma}^2$.
Cuts on the gluon transverse momenta $q_{\perp,cut}^{2} = 0.5$ GeV$^{2}$
were imposed.
In Fig.\ref{fig:dsig_dM34_dp1t_gamgam} (left panel) we show distribution of photon-photon
invariant mass with experimental CDF cuts.
We show results for three different gluon
distributions \cite{GJR,MSTW_PDF,CT_PDF}. We obtain very good description
of the CDF experimental data \cite{CDF_gamgam}, both in shape and absolute
normalization.
In the right panel we show corresponding distribution
in photon transverse momentum.
\begin{figure}[!h]
\includegraphics[width=7cm]{dsig_dM34_tev.eps}
\includegraphics[width=7cm]{dsig_dp1t_tev.eps}
\caption{
\small
Left panel: Photon-photon invariant mass distribution.
We show results for three different gluon distributions
specified in the figure.
The experimental data are taken from Ref.\cite{CDF_gamgam}.
Right panel:
}
\label{fig:dsig_dM34_dp1t_gamgam}
\end{figure}
Finally in Fig.\ref{fig:dsig_dy_gamgam} we show corresponding
distribution in photon pseudorapidity in the left panel, again for three
different gluon distributions. In the right panel we present
decomposition into different $pp$ center-of-mass photon helicity components.
\begin{figure}[!h]
\includegraphics[width=7cm]{dsig_dy_tev.eps}
\includegraphics[width=7cm]{dsig_dy_tev_deco.eps}
\caption{
\small Distribution in photon pseudorapidity for three different
gluon distributions (left panel) and the decomposition into different
$pp$ center-of-mass photon helicity components.
}
\label{fig:dsig_dy_gamgam}
\end{figure}
Having shown that the results of the approach used in the present paper
nicely describe the CDF experimental data \cite{CDF_gamgam} we can
confidentially present our predictions for the $p p \to p p W^+ W^-$ reaction.
\subsection{$p p \to p p W^+ W^-$}
Let us present now our results for the central exclusive $W^+W^-$ pair production.
In Fig.~\ref{fig:dsig_dy} we compare rapidity distribution of $W^+$
(or $W^-$) for the electromagnetic $\gamma \gamma \to W^+ W^-$ and
diffractive $gg \to W^+ W^-$ mechanisms. The two-photon induced
contribution is almost three orders of magnitude larger than the
diffractive contribution, in which all polarization components for
$W^+$ and $W^-$ have been included. For a reference, we show also
inclusive cross section ($g g \to W^+ W^-$ contribution only) which
is roughly two more orders of magnitude bigger than the exclusive
$\gamma \gamma \to W^+ W^-$ contribution. We see, therefore, that the
exclusive diffractive component is five orders of magnitude smaller
for its inclusive counterpart.
The diffractive contribution was calculated with the GJR NLO
\cite{GJR} collinear gluon distribution, in order to generate the
off-diagonal UGDFs given by Eq.~(\ref{ugdfkmr}). This collinear PDF allows us to
use quite small values of gluon transverse momenta ($q_{\perp,cut}^{2}$ = 0.5 GeV$^2$).
A much smaller diffractive contribution compared to the two-photon
one requires a special comment as it is rather exceptional.
For example, it is completely opposite than for $p p \to p p H$ \cite{MPS_bbbar},
$p p \to p p M$ (e.g. light/heavy quarkonia production \cite{chic,LKRS10}) or
$p p \to p p Q\bar Q$ \cite{MPS_bbbar,MPS_ccbar} CEP processes.
The standard relative suppression,
present also in the latter cases, is due to soft gap survival probability factor
($S_g \sim$ 0.03 for diffractive contribution versus $S_g
\sim$ 1 for two-photon contribution), and due to a suppression by
the Sudakov form factor calculated at very large scales, here at
$\mu_{hard} = M_{WW}$. The main difference compared to other cases is that
in the diffractive case the leading contribution comes from loop
diagrams while in the two-photon case already from tree level
diagrams.
\begin{figure}[!h]
\includegraphics[width=7cm]{dsig_dy.eps}
\caption{
\small Rapidity distribution of $W$ bosons. The diffractive
contribution is shown by the bottom line while the $\gamma \gamma
\to W^+ W^-$ contribution by the middle line. For comparison, we
also show the cross section for the inclusive ($gg$-fusion only)
production case (upper line).
}
\label{fig:dsig_dy}
\end{figure}
In Fig.~\ref{fig:dsig_dy_deco} we present, in addition, individual
polarization components for the diffractive mechanism, along with
the unpolarized cross section. The calculation of the helicity
contributions is performed in the $pp$ center-of-mass frame (in
which all the experimental studies of the exclusive production
processes are usually performed).
As can be seen from the figure,
the contribution of $(\lambda_{+},\lambda_{-}) = (\pm 1, \mp 1)$ is
bigger than other contributions and
the contribution of $(\lambda_{+},\lambda_{-}) = (\pm 1, \pm 1)$
concentrated mostly at midrapidities. Since we use
$p p$ center of mass helicities there is on simple relation
to the often used in a qualitative discussion $J_z$ = 0 dominance rule.
Discussion of the $J_z$ = 0 rule would require complicated
transformations between different reference frames and
going beyond approximations made here.
This clearly goes beyond the scope of this paper.
In particular, as it is seen from
Fig.~\ref{fig:dsig_dy_deco} the helicity contributions obey
the following relation
\begin{eqnarray}
\frac{d\sigma_{\lambda\lambda'}(y_+)}{dy_+}=\frac{d\sigma_{\lambda'\lambda}(y_-)}{dy_-} \,,
\end{eqnarray}
where $y_{\pm}$ are rapidities of $W^{\pm}$ bosons, respectively.
The unpolarized cross section does not show up any peculiarities in
$y$-dependence and is symmetric with respect to $y=0$ for both $W^+$
and $W^-$ bosons.
\begin{figure}[!h]
\includegraphics[width=7cm]{dsig_dy_deco_yplus.eps}
\includegraphics[width=7cm]{dsig_dy_deco_yminus.eps}
\caption{
\small Rapidity distribution of separate polarisation components to
the diffractive $W$ bosons production. The individual contributions
are marked in the figure.
}
\label{fig:dsig_dy_deco}
\end{figure}
In Fig.~\ref{fig:dsig_dpt} we show distribution in $W^+$ ($W^-$) transverse momentum.
The distribution for exclusive diffractive production is much
steeper than that for the electromagnetic contribution.
A side remark is in order here.
The diffractive contribution peaks at $p_{t,W} \sim$ 25 GeV.
This is somewhat smaller than for the $\gamma \gamma \to W^+ W^-$ mechanism
where the maximum is at $p_{t,W} \sim$ 40 GeV.
The exclusive cross section for photon-photon
contribution is at large transverse momenta $\sim$ 1 TeV smaller
only by one order of magnitude than the inclusive $gg \to W^{+}W^{-}$ component.
The situation could be even more
favorable if New Physics would be at the game \cite{royon}.
\begin{figure}[!h]
\includegraphics[width=7cm]{dsig_dp1t.eps}
\caption{
\small Distribution in transverse momentum of one of the $W$ bosons.
The diffractive contribution is shown by the bottom solid line while
the $\gamma \gamma \to W^+ W^-$ contribution by the middle solid
line. The top solid line corresponds to the inclusive
two-gluon initiated $pp \to W^+W^-X$ component. Separate contributions
of boxes (dashed) and triangles (dotted) are shown in addition for
illustrating the cancellation effect.
}
\label{fig:dsig_dpt}
\end{figure}
Fig.~\ref{fig:dsig_dMWW} shows distribution in the $W^+ W^-$ invariant
mass which is particularly important for the New Physics searches at
the LHC \cite{royon}. The distribution for the diffractive component
drops quickly with the $M_{WW}$ invariant mass. For reference and
illustration, we show also distribution when the Sudakov form
factors in Eq.~(\ref{ugdfkmr}) is set to one. As can be seen from
the figure, the Sudakov form factor lowers the cross section by a
large factor. The damping is $M_{WW}$-dependent as can be seen by
comparison of the two curves. The larger $M_{WW}$ the larger the
damping. We show the full result (boxes + triangles) and the result
with boxes only which would be complete if the Higgs boson
does not exist. At high invariant masses, the interference of boxes
and triangles decreases the cross section. The distribution for the
photon-photon component drops very slowly with $M_{WW}$ and at
$M_{WW} >$ 1 TeV the corresponding cross section is even bigger than
the $gg \to W^+ W^-$ component to inclusive production of $W^+ W^-$
pairs.
\begin{figure}[!h]
\includegraphics[width=7cm]{dsig_dmWW.eps}
\caption{
\small Distribution in $W^+W^-$ invariant mass. We show both the QCD
diffractive contribution and the electromagnetic $\gamma \gamma \to
W^+ W^-$ contribution. The result when the Sudakov form factor is
put to one is shown for illustration of its role. The most upper
curve is for the inclusive gluon-initiated $pp \to W^+ W^-X$ component.}
\label{fig:dsig_dMWW}
\end{figure}
Finally, in Fig.~\ref{fig:dsig_dy1dy2} we show for completeness the
two-dimensional distributions in rapidities of $W^+$/$W^-$ bosons in
both electromagnetic and QCD mechanisms.
We see a typical correlation pattern characteristic for 2 $\to$ 2 subprocesses.
This distribution does not show any specific behavior which could be
used to differentiate the diffractive and the two-photon contributions.
\begin{figure}[!h]
\includegraphics[width=6cm]{dsig_dy1dy2_via_gg.eps}
\includegraphics[width=6cm]{dsig_dy1dy2_via_gamgam.eps}
\caption{
\small
Two-dimensional distribution in rapidity of $W^+$ and $W^-$ bosons
for the diffractive mechanism (left panel) and two-photon mechanism
(right panel).
}
\label{fig:dsig_dy1dy2}
\end{figure}
\section{Conclusions}
We have calculated the QCD diffractive contribution to the exclusive
$p p \to p W^+ W^- p$ process for the first time in the literature
with the full one-loop $gg\to W^+W^-$ matrix element. Two mechanisms
have been considered. First mechanism is a virtual (highly off-shell)
Higgs boson production and its subsequent transformation into real
$W^+ W^-$ pair. Second mechanism relies on the formation of
intermediate quark boxes, very much similar to ones in the exclusive
two photon production mechanism.
We have calculated corresponding amplitudes using computer
program package {\tt FormCalc}. We have made a first estimate of
the cross section using amplitudes in the forward limit ``corrected''
off-forward via a simple exponential (slope dependent) extrapolation.
In order to gain confidence to our calculations and the formalism used we
consider also the $p \bar{p} \to p \bar{p} \gamma \gamma$ process
which was measured recently by the CDF Collaboration.
Here the formalism of calculating
quark box diagrams is essentially the same
as for the exclusive production of
$W^+ W^-$ pairs. We have obtained very nice agreement with experimental
diphoton invariant mass distribution.
Having verified the formalism for diphoton production we have performed
similar calculation for $W^+ W^-$ production.
Differential distributions in the $W^{\pm}$ transverse momentum,
rapidity and $W^+ W^-$ pair invariant mass have been calculated and
compared with corresponding distributions for discussed in the
literature $\gamma \gamma \to W^+ W^-$ mechanism.
The contribution of triangles with the intermediate Higgs boson
turned out to be smaller
than the contribution of boxes taking into account recent
very stringent limitations on Higgs boson mass from Tevatron and LHC data.
We have found that, in contrast to exclusive production of
Higgs boson or dijets, the two-photon fusion dominates over the
diffractive mechanism for small four-momentum transfers squared in
the proton lines ($t_1, t_2$) as well as in a broad range of $W^+
W^-$-pair invariant masses, in particular, for large $M_{WW}$.
Estimated theoretical uncertainties cannot disfavor this statement.
The large $M_{WW}$ region is damped in the diffractive model via
scale dependence of the Sudakov form factor.
One could focus on the diffractive contribution by imposing lower
cuts on $t_1$ and/or $t_2$ using very forward detectors
on both sides of the interaction point at distances of
220 m and 420 m as planned for future studies at ATLAS and CMS. The
corresponding cross section is, however, expected to be extremely low.
Compared to the previous studies in the effective field theory
approach, in this work we have included the complete one-loop
(leading order) $gg\to W^+W^-$ matrix element, and have shown that
extra box diagrams, even though they are larger than the resonant
(s-channel Higgs) diagrams, constitute a negligibly small background for
a precision study of anomalous couplings.
The unique situation of the dominance of the $\gamma \gamma \to W^+
W^-$ contribution over the diffractive one opens a possibility of
independent tests of the Standard Model as far as the triple-boson
$\gamma W W$ and quartic-boson $\gamma \gamma W W$ coupling
is considered. It allows also for stringent tests of some Higgsless
models as discussed already in the literature (see e.g.
Ref.~\cite{royon}).
\section{Acknowledgments}
Useful discussions with Rikard Enberg, Gunnar Ingelman,
Valery Khoze, Otto Nachtmann, Christophe Royon, Torbj\"{o}rn Sj\"{o}strand
and Marek Tasevsky are gratefully acknowledged. This study was partially
supported by the MNiSW grants No. DEC-2011/01/N/ST2/04116 and
DEC-2011/01/B/ST2/04535. Authors are grateful to the European Center
for Theoretical Studies in Nuclear Physics and Related Areas
(ECT$^*$, Trento, Italy) for warm hospitality
during their stay when this work was completed.
Piotr Lebiedowicz is thankful to the THEP Group at
Lund University (LU), Sweden, for hospitality during his
collaboration visit at LU.
|
{
"timestamp": "2013-02-19T02:06:10",
"yymm": "1203",
"arxiv_id": "1203.1832",
"language": "en",
"url": "https://arxiv.org/abs/1203.1832"
}
|
\section{Introduction}
Two of the hallmark features of graphene are its linear Dirac-cone quasiparticle dispersion and the finite minimum conductivity.\cite{Geim07} In non-suspended graphene (NSG) the minimum conductivity $\sigma_{\text{min}}$, i.e., the minimal value of the dc conductivity $\sigma_{\text{dc}}$ with respect to variations of the electron density, is weakly dependent on temperature $T$ but varies considerably from sample to sample.\cite{Geim07,Tan07} In the attempts to approach the ballistic limit of Dirac fermions without scattering, realizations of suspended graphene (SG) sheets have been prepared, which enable unprecedented electron mobilities \cite{Morozov08,Bolotin08mob} and show a stronger increase of $\sigma_{\text{min}}$ upon increasing $T$.\cite{Du08,Bolotin08} For low electron densities when the Fermi energy is slightly off-center of the Dirac cone, $\sigma_{\text{dc}}$ decreases with $T$ below a crossover temperature, while the resistivity increases linearly with $T$ at high densities.\cite{Bolotin08}
Free electrons on the half-filled honeycomb lattice constitute a perfect conductor with a non-zero Drude weight at finite temperature and a finite dc conductivity $\sigma_{0}=\pi e^2/2h$ at zero temperature.\cite{Ziegler07,Stauber08,Lewkowicz09,Neto09review} This is a direct consequence of the Dirac-cone structure of the electronic dispersion. For finite temperatures also the optical conductivity is of order $\sigma_{0}$ in the visible frequency range. This theoretical result, neglecting disorder effects, is indeed observed in optical absorption experiments on charge-neutral graphene.\cite{Nair08}
Graphene samples are not pristine, however.\cite{Novoselov05,Zhang05} Scanning-tunneling microscopy images of NSG samples show inhomogeneous patterns;\cite{Martin08} their origin was traced to the presence of impurities at the substrate-graphene interface.\cite{Zhang09} Impurities may nucleate electron- or hole-rich puddles,\cite{Guinea08} which obscure the intrinsic Dirac fermion physics of pristine graphene. In suspended graphene (SG) scattering can occur due to microscopic corrugations of the otherwise unstable two-dimensional crystal, so-called ripples, which were observed by transmission electron microscopy \cite{Meyer07} and theoretically analyzed as one possible source for electron scattering \cite{Fasolino07,Katsnelson08} or charge inhomogeneity.\cite{Brey08} Impurity effects are quantitatively less important for the optical conductivity, but play a major role in determining the dc transport properties of graphene; the latter are the topic of this paper.
$\sigma_{\text{dc}}$ of graphene is minimal at the charge neutrality point, which corresponds to a honeycomb lattice at half-filling, i.e., with one electron per lattice site. Upon applying a gate voltage of either sign, $\sigma_{\text{dc}}$ increases,\cite{Bolotin08} hence the name ``minimum conductivity''. Disorder-induced charge-density modulations imply a spatially varying chemical potential and thereby conceal clean Dirac fermion physics. Also the dc transport measurement process itself may introduce a bias, e.g., due to a charge transfer at metal contacts.\cite{Blake09}
\begin{figure}[htp!]
\begin{center}
\includegraphics[width=\hsize,angle=0]{sketch_pristine.eps} \\
\end{center}
\vspace{-5mm}
\caption[Transport in pristine graphene]{
Transport at the charge neutrality point in pristine graphene. (a) At $T=0$ the valence band (states below the chemical potential $\mu$), shown here for a single Dirac cone, is filled (indicated by the dark blue shading) and the conduction band is empty. At $T>0$ electrons are thermally excited to the conduction band (lighter blue shading). (b) Contributions to the conductivity: Only interband transitions are allowed at $T=0$ due to Fermi blocking. Intraband transitions contribute to dc transport at finite temperatures. (c) Dynamical conductivity of pristine graphene within the Dirac-cone approximation at $T=0$ and $T>0$. At $T=0$ interband transitions lead to a universal finite conductivity of $\sigma_{0}$ $=$ $\pi e^2/2h$. For $T>0$ a Drude peak emerges due to the intraband transitions with a Drude weight $D$ $\propto$ $T$. For visual frequencies $\omega_{\text{vis}}$ the optical conductivity still is of order $\sigma_{0}$.}
\label{fig:0}
\vspace{-3mm}
\end{figure}
Theoretical work on transport in graphene comprises studies of charged-impurity scattering, as reviewed in Ref.\ \onlinecite{Adam09review}, different sources of disorder,\cite{Adam10review} and also the crossover between low- and high-density regimes.\cite{Adam09rapcomm} More recent studies focused on ballistic \cite{Mueller09} or diffusive transport,\cite{Adam09preprint} and also the effects of finite-range scattering at finite densities.\cite{Ferreira11}
Unresolved problems remain in particular in the low-density regime, which is relevant for the minimum conductivity at zero bias.
Several predictions exist for a minimum conductivity of
$4e^2/\pi h$ in the absence of disorder.\cite{theories1,theories2,theories3,theories4,theories5} This limiting value at zero temperature is obtained if the dc limit is taken first and the zero-disorder limit afterwards.\cite{Ziegler07,Lewkowicz09} Experiments on both NSG and SG samples \cite{Geim07,Bolotin08,Du08} with non-universal values of the minimum conductivity were reported, with the trend that the $T$ dependence of the minimum conductivity is enhanced in clean SG samples as opposed to dirty SG \cite{Bolotin08} or NSG samples.\cite{Bolotin08,Du08} In fact, the minimum conductivity increases with increasing temperature, i.e., as in a semiconductor. In contrast, at sufficiently large gate voltages a metallic $T$ dependence of the conductivity is observed, with a resistivity increasing linearly with $T$ and a slope that decreases upon an increase in the gate voltage.\cite{Bolotin08}
Here we evaluate the Kubo formula for the dc conductivity of electrons with a linear Dirac cone dispersion. Disorder effects are included by a random chemical potential, which is treated within the coherent-potential approximation (CPA).\cite{Taylor67,Soven67} The associated disorder energy scale $\Gamma$ may itself depend on temperature. We specifically investigate the case of a Lorentzian disorder distribution of width $\Gamma$ (``Lloyd model''), for which the Kubo formula can be evaluated exactly within CPA. As a result $\sigma_{\text{dc}}$ at half-filling depends only on the type of disorder distribution and the dimensionless ratio $T/\Gamma$. For $T/\Gamma$ $\gg$ 1 the minimum conductivity increases linearly with $T/\Gamma$. For a temperature dependent $\Gamma$ $=$ $\Gamma_0 + \alpha_1 T$ the minimum conductivity thus saturates at high temperatures. With this simple ansatz and a choice of typical meV energy scales for $\Gamma$, the $T$ dependence of $\sigma_{\text{dc}}$ changes from semiconducting at half-filling to metallic at sufficiently large band filling. At intermediate densities $\sigma_{\text{dc}}$ evolves from metallic to semiconducting behavior in the temperature range between 0 and 200 K. Moreover, the experimentally observed linearly increasing resistivity at high temperatures in the metallic regime as well as the decreasing slope upon increasing the density are reproduced in this ansatz.
\section{Model and Method}
An infinite sheet of pristine graphene is modeled by a tight-binding Hamiltonian with an effective next nearest neighbor hopping on a honeycomb lattice without impurity scattering and electron-electron interactions. In the absence of current-vertex corrections the dc conductivity follows from
\begin{equation}
\sigma_{\text{dc}} = \frac{2\pi e^2}{\hbar^2} \int\limits_{-\infty}^{\infty} \hspace{-1mm} \text{d}\nu \hspace{-1mm} \int\limits_{-\infty}^{\infty} \hspace{-1mm} \text{d}\epsilon \; \tilde{\rho}(\epsilon)
\left[ A_{\epsilon}(\nu)+A_{-\epsilon}(\nu) \right]
A_{\epsilon}(\nu)\frac{-\text{d} f_{\nu-\mu}}{\text{d} \nu},
\label{Kubo}
\end{equation}
where $f_{x}$ $=$ $1/(1+\exp(x/T))$ is the Fermi-Dirac distribution function (with $k_B$ $=$ $1$) and $\tilde{\rho}(\epsilon)$ $=$ $L^{-1}\sum_{\bm{k}} (\partial\epsilon_{\bm{k}}/\partial k_x)^2 \delta(\epsilon-\epsilon_{\bm{k}})$; $L$ is the number of unit cells of the lattice. $\mu$ is the chemical potential which vanishes at half-filling. In Eq.~(\ref{Kubo}) a prefactor of 4 has been incorporated; it accounts for the spin and valley degeneracies of graphene.
For free electrons the spectral functions simply reduce to $A_{\epsilon}(\nu)$ $=$ $\delta(\epsilon-\nu)$. We use the Dirac cone approximation $\tilde{\rho}(\epsilon)$ $=$ $\hbar|\epsilon|/2\pi$ for $|\epsilon|$ $<$ $\epsilon_{\text{max}}$, where $\epsilon_{\text{max}}$ is a cutoff energy chosen as the half-bandwidth of graphene. Indeed, the Dirac cone approximation for $\tilde{\rho}(\epsilon)$ gives the correct result for $\sigma_{\text{dc}}$ and serves as a good approximation even in the visual frequency range,\cite{Stauber08} where the band dispersion leads to only weak quadratic corrections to $\sigma(\omega)$ at low frequencies.
The term in Eq.\ (\ref{Kubo}) which involves $A_{\epsilon}(\nu)^2$ in the integrand leads to the usual intraband conductivity as in single-band models. It gives rise to a Drude-like contribution, hence an infinite dc conductivity in a perfect conductor. Also in the presence of electron-electron interactions this expression for the intraband conductivity remains correct, if the self-energy $\Sigma(\nu)$ is local and $A_{\epsilon}(\nu)$ $=$ $-\text{Im}(\epsilon-\nu-\Sigma(\nu))^{-1}/\pi$.\cite{opticaldmft1,opticaldmft2} The second term in Eq.\ (\ref{Kubo}), involving $A_{\epsilon}(\nu) A_{-\epsilon}(\nu)$, describes excitations with a particle at energy $\epsilon$ and a hole at $-\epsilon$ and accounts for interband transitions. This contribution accounts for the visual transparency of graphene in the dc limit of the optical conductivity, $\sigma_{0}$ $=$ $\pi e^2/2h$ (see Fig.\ \ref{fig:1}c).\cite{Nair08}
The presence of disorder complicates the situation considerably. Discrete translational invariance is broken, rendering microscopic theoretical approaches much more difficult than in the homogeneous case. One standard approach is the Anderson model \cite{Anderson58} with local potential impurities,
\begin{equation}
H = H_0 + \sum_i V_i n_i,
\end{equation}
where $H_0$ is the tight-binding Hamiltonian for the clean system, $n_i$ is the local density operator on site $i$ of the lattice and $V_i$ is a random variable determined from a probability distribution $P(V_i)$. Here we do not aim at a full microscopic description of disorder, e.g., in the spirit of a self-consistent diagrammatic treatment of impurity scattering effects,\cite{Vollhardt80} and recall that weak localization is suppressed by long-range scattering in graphene.\cite{Morozov06, Wakabayashi07, Ziegler08} Instead we apply the coherent-potential approximation (CPA)\cite{Taylor67,Soven67,Elliott74} to determine an effective random medium described by a local self-energy $\Sigma(\omega)$, which is determined by a self-consistent solution of the CPA equations
\begin{eqnarray}
\bar{G}(\omega) &=& G_0(\omega-\Sigma(\omega)),\nonumber\\
\bar{G}(\omega) &=& \tilde{D}\left[\mathcal{G}^{-1}(\omega)\right] = \int \text{d}V\; \frac{P(V)}{\mathcal{G}^{-1}(\omega)-V},\nonumber\\
\mathcal{G}^{-1}(\omega) &=& \bar{G}^{-1}(\omega) + \Sigma(\omega).
\label{CPA}
\end{eqnarray}
In Eq.~(\ref{CPA}) $G_0(z)=\int \text{d}\omega\; \rho_{\text{DOS}}(\omega)/(z-\omega)$ is the local Green function of the clean system described by $H_0$, $\mathcal{G}(\omega)$ is a dynamical Weiss field and $\tilde{D}[z]$ is the Hilbert transform with respect to the disorder distribution function $P(V_i)$. The CPA expression for the conductivity \cite{Elliott74} agrees with the Kubo formula Eq.\ (\ref{Kubo}) with $A_{\epsilon}(\nu)$ $=$
$-\text{Im}(\epsilon-\nu-\Sigma(\nu))^{-1}/\pi$.
The CPA equations (\ref{CPA}) can be solved, at least numerically, for an arbitrary disorder distribution. In order to keep the subsequent analysis as simple and transparent as possible, we focus on the specific case of a Lorentzian disorder distribution of width $\Gamma$,
\begin{equation}
P(x) = \frac1\pi \frac{\Gamma}{\Gamma^2+x^2}.
\end{equation}
This is the so-called Lloyd model, for which the CPA equations are exactly solvable using $\tilde{D}[z]$ $=$ $(z+i\Gamma)^{-1}$, which yields $\Sigma(\nu)$ $=$ $-\text{i}\Gamma$. Hence we obtain for the Lloyd model
\begin{equation}
A_{\epsilon}(\nu) = \frac1\pi \frac{\Gamma}{\Gamma^2+(\epsilon-\nu)^2}
\end{equation}
as the input quantity for Eq.\ (\ref{Kubo}). $A_{\epsilon}(\nu)$ is of the form $a((\epsilon-\nu)/\Gamma)/\Gamma$, implying that the dc conductivity is only a function of the ratio $T/\Gamma$ for $\mu=0$ and of $\mu/\Gamma$ for $T=0$; the latter holds only, if $\mu$ $\ll$ $\epsilon_{\text{max}}$, which is fulfilled in the experiments cited above.
The scaling behavior of $\sigma_{\text{min}}$ has two reasons: (a) the cutoff energy $\epsilon_{\text{max}}$ can be replaced by infinity in the $\epsilon$-integral in Eq.\ (\ref{Kubo}) and thus does not appear as an additional energy scale, and (b) for dimensional reasons the dc conductivity is universal in the sense that it does not depend on the hopping matrix element $t$ of the underlying two-dimensional tight-binding Hamiltonian and, consequently, not on the Fermi velocity $v_F$. Both reasons are directly related to the linearity of the dispersion in graphene up to energies much larger than the relevant temperatures. For finite densities this universality no longer holds, since the density variations are determined by the chemical potential which thereby depends on the hopping matrix element $t$.
Experimentally it is the gate voltage which controls the electronic density $n$, measured relative to half-filling. For given temperature $T$, disorder strength $\Gamma$, and chemical potential $\mu$ the density is given by
\begin{equation}
n = \int_{-\infty}^{\infty} \text{d}\omega\; \rho_{\text{DOS}}(\omega) \left(f_{\omega-\mu}-f_{\omega}\right),
\end{equation}
where
\begin{equation}
\rho_{\text{DOS}}(\omega) = \frac{4}{\sqrt{3}\pi t^2 A_u} \int_{-\epsilon_{\text{max}}}^{\epsilon_{\text{max}}}\text{d}\epsilon\; |\epsilon| A_{\epsilon}(\omega)
\end{equation}
is the density of states (summed over both spin projections) for the disordered system, $t$ $=$ 2.7 eV the hopping matrix element of the tight-binding model, $A_u$ $=$ $3\sqrt{3}a_0^2/2$ the size of the unit cell, and $a_0$ $=$ 1.42 $\times$ $10^{-10}$ m the interatomic distance on the honeycomb lattice.\cite{Neto09review}
\section{Results}
We first keep the disorder strength $\Gamma$ fixed and discuss basic properties of the minimum conductivity and the conductivity at finite chemical potential and zero temperature. In a second step we evaluate the density dependence of the conductivity for a typical disorder strength (on the order of meV\cite{Du08,Martin08}) and for temperatures in the range from 0 K to 200 K. Especially we consider the temperature dependence of the resistivity $\sigma_{\text{dc}}^{-1}$ at fixed densities and show that a temperature-independent $\Gamma$ at high densities is insufficient to explain the experimentally observed linear $T$ dependence of the resistivity. Adding a phenomenological linear $T$ dependent contribution to $\Gamma$, the experimental observation is matched by our ansatz. Moreover, the observed density dependence of the slope in the $T$-linear regime of the resistivity follows naturally without further assumptions. For a selected $T$ dependent $\Gamma$ $=$ $\Gamma_0 + \alpha_1 T$, the minimum conductivity increases with temperature but with a decreasing slope; $\sigma_{\text{min}}$ saturates at high temperatures closely similar to the experiments. Finally we show the $T$ dependence of the conductivity with a density dependent crossover from metallic ($\text{d}\sigma_{\text{dc}}/\text{d}T$ $<$ 0) at low $T$ to semiconducting behavior ($\text{d}\sigma_{\text{dc}}/\text{d}T$ $>$ 0) at high $T$.
\begin{figure}[htp!]
\includegraphics[width=\hsize,angle=0]{sgmin_lloyd.eps} \\
\vspace{5mm}
\includegraphics[width=\hsize,angle=0]{sgnu_lloyd.eps}
\caption[DC conductivity for the Lloyd model]{
DC conductivity in units of $e^2/h$ for the Lloyd model in CPA. Top panel: Minimum conductivity ($\mu$ $=$ 0) as a function of $T/\Gamma$. Bottom panel: DC conductivity at $T$ $=$ 0 as a function of $\mu/\Gamma$.}
\label{fig:1}
\end{figure}
The temperature dependence of the minimum conductivity $\sigma_{\text{min}}$ for the Lloyd model with disorder strength $\Gamma$ is shown in the top panel of Fig.\ \ref{fig:1}. For $T/\Gamma$ $\rightarrow$ 0, $\sigma_{\text{min}}$ tends to the limiting value $4e^2/\pi h$, which coincides with the clean limit discussed in Refs.\ \onlinecite{theories1,theories2,theories3,theories4,theories5}. However, $4e^2/\pi h$ should not be considered a universal value, but rather a particular result of the Lloyd model. Other non-Lorentzian disorder distributions are likely to lead to other values of the minimum conductivity. At high temperatures $\sigma_{\text{min}}$ increases linearly with $T/\Gamma$ for $T/\Gamma$ $\gg$ 1,
\begin{equation}
\sigma_{\text{min}} = \frac{e^2}{h}
\bigg(2 \ln(2)\,\frac{T}{\Gamma}
+{\cal O}\bigg(\frac{\Gamma}{T}\bigg)
\bigg)
\,.\label{hightemp}
\end{equation}
To understand the physical processes involved we discuss the relevant contributions to $\sigma_{\text{min}}$. First we note that the clean case at zero temperature is not recovered by our theory for $\sigma_{\text{dc}}$. However, this is not a shortcoming but rather a generic feature of the conductivity as a function of disorder strength, temperature, and frequency. We recall that at zero temperature intraband excitations are prohibited. Thus only the interband excitations are responsible for $\text{Re}\;\sigma(\omega \rightarrow 0)$ $=$ $\sigma_{0}$ \cite{Stauber08} (see Fig.\ \ref{fig:0}), i.e. when the dc limit is taken \emph{after} the limits of zero temperature and zero disorder strength. The theory presented here instead aims at describing dc measurements, for which the dc limit must be taken \emph{first}. In the latter case, both interband and intraband excitations are relevant and both contribute equally ($2e^2/\pi h$ for the Lloyd model) to the $T/\Gamma$ $\rightarrow$ 0 limit. The discrepancy between $\sigma_{0}$ and $\sigma_{\text{min}}(T \rightarrow 0)$ may also be understood by noting that for $\sigma_{0}$ the largest energy scale in the system is the frequency (taken to zero last), while the largest energy scale for $\sigma_{\text{min}}$ is the disorder strength $\Gamma$.
At finite temperatures thermally excited particles in the conduction band render intraband particle-hole excitations possible, leading to a non-zero Drude weight and thus an infinite $\sigma_{\text{dc}}$ (for $\Gamma$ $\rightarrow$ 0 and therefore $T/\Gamma$ $\rightarrow$ $\infty$), while interband low-energy excitations are blocked by thermally occupied states in the conduction band. Viewed as a function of temperature at fixed $\Gamma$ the conductivity is semiconducting, i.e., $\text{d}\sigma_{\text{dc}}/\text{d}T$ $>$ 0.
At finite chemical potentials the interband excitations become less important, and the behavior is determined mostly by intraband excitations. The situation for higher densities thus resembles more and more the case of a single partially filled band, where metallic behavior sets in for sufficiently low temperatures, i.e., $\text{d}\sigma_{\text{dc}}/\text{d}T$ $<$ 0.
The dc conductivity at $T$ $=$ 0 as a function of $\mu/\Gamma$ is shown in the lower panel of Fig.\ \ref{fig:1}. It tends to the limiting value $4e^2/\pi h$ for $|\mu/\Gamma|$ $\rightarrow$ 0 and increases linearly for $|\mu/\Gamma|$ $\gg$ 1 but well below the cutoff $\epsilon_{\text{max}}/\Gamma$. This linear dependence on the chemical potential for large $|\mu/\Gamma|$ is analytically obtained from the Kubo formula, taking into account intraband excitations only,
\begin{eqnarray}
\sigma_{\text{dc}}(\mu,T=0) &\approx& \frac{2e^2}{\pi h} \int_{-\infty}^{\infty} \text{d}\epsilon |\epsilon| \frac{\Gamma^2}{\left(\Gamma^2+(\epsilon-\mu)^2\right)^2}
\nonumber
\\
&=&
\frac{2e^2}{\pi h} \left( 1 + \frac{\mu}{\Gamma} \arctan\frac{\mu}{\Gamma} \right)
\sim
\left|\frac{\mu}{\Gamma}\right| \frac{e^2}{h},
\end{eqnarray}
where the last asymptotic expression is valid for $|\mu/\Gamma|$ $\gg$ 1.
\begin{figure}[htp!]
\begin{center}
\includegraphics[width=\hsize,angle=0]{sgn_lloyd.eps} \\
\vspace{5mm}
\includegraphics[width=\hsize,angle=0]{rhon_lloyd.eps}
\end{center}
\vspace{-5mm}
\caption[Density-dependent conductivity and resistivity]{
Upper panel: Conductivity as a function of density at different temperatures for a temperature independent $\Gamma$. Lower panel: The data for the resistivity $\rho$ $=$ $\sigma_{\text{dc}}^{-1}$.}
\label{fig:2}
\vspace{-3mm}
\end{figure}
For fixed value of the disorder strength $\Gamma$ $=$ 1 meV we show in Fig.\ \ref{fig:2} the dc conductivity and its inverse, the resistivity $\rho$, as a function of density for selected temperatures. Since the density depends quadratically on the chemical potential for $\mu/\Gamma$ $\gg$ 1 and $\sigma_{\text{dc}}$ depends linearly on $|\mu/\Gamma|$ in this limit at zero temperature, the low-temperature conductivity increases like $\sqrt{n}$ at high densities. Experimentally a sublinear density dependence of the conductivity was also reported in Ref.\ \onlinecite{Du08}.
In Ref.\ \onlinecite{Bolotin08} a linear increase of the resistivity as a function of temperature was observed for high densities at elevated temperatures. For a temperature independent $\Gamma$, $\sigma_{\text{dc}}$ $=$ $|\mu/\Gamma|$ $e^2/h$ for large $|\mu/\Gamma|$, and the temperature dependence of the chemical potential at fixed densities follows $\mu$ $=$ $\mu(T=0)+\mathcal{O}(T^2)$. Hence, $\sigma_{\text{dc}}(T)$ $=$ $\sigma_{\text{dc}}(T=0)+\mathcal{O}(T^2)$ for fixed densities and thus $\rho$ $=$ $\rho(T=0)+\mathcal{O}(T^2)$. A linear temperature dependence of $\rho$ therefore requires a temperature dependent $\Gamma$ within our ansatz. In fact, a linearly increasing resistivity at high densities naturally follows from $\Gamma$ $=$ $\Gamma_0+\alpha_1 T$,
\begin{equation}
\rho \approx \frac\Gamma\mu \frac{h}{e^2} = \frac{\Gamma_0}{\mu}\frac{h}{e^2} + \frac{\alpha_1 T}{\mu}\frac{h}{e^2}.
\label{Tdep}
\end{equation}
$\mu$ thereby depends not only explicitly on temperature, but also implicitly via the $T$ dependent $\Gamma$. This implicit $T$ dependence is, however, negligible for large fillings, when also the explicit $T$ dependence is very weak since it scales like temperature over Fermi energy.
\begin{figure}[htp!]
\begin{center}
\includegraphics[width=\hsize,angle=0]{rhon.eps} \\
\vspace{5mm}
\includegraphics[width=\hsize,angle=0]{rhot3.eps}
\end{center}
\vspace{-5mm}
\caption[Density- and temperature-dependent resistivity]{
Upper panel: Resistivity as a function of density for different temperatures.
Lower panel: Resistivity increase $\Delta \rho$ $=$ $\rho(T)-\rho(\text{10 K})$ above 10 K as a function of temperature for different densities. The dashed lines are linear fits to the data points between 100 K and 200 K.}
\label{fig:3}
\vspace{-3mm}
\end{figure}
In the following we adopt the $T$ dependent disorder strength $\Gamma(T)$ $=$ $\Gamma_0$ $+$ $\alpha_1$ $T$ and fix the parameters $\Gamma_0$ $=$ 1 meV and $\alpha_1$ $=$ 3.25 meV/200 K such that the temperature dependence of the minimum conductivity (see lower panel of Fig.\ \ref{fig:4}) approximately matches the experimental data of Ref.\ \onlinecite{Du08}. The density dependence of the resistivity for the selected $T$ dependent disorder strength is shown in the upper panel of Fig.\ \ref{fig:3}. For moderate temperatures below 200 K the temperature dependence of the resistivity is presented in the lower panel of Fig.\ \ref{fig:3}. Eq.\ (\ref{Tdep}) implies that the slope in the linear regime is proportional to $\alpha_1/\mu$, and since $\mu$ $\propto$ $\sqrt n$ the slope decreases like $1/\sqrt{n}$. For high temperatures well above 200 K and sufficiently large densities, or below 200 K for moderate densities, there is a deviation from linear behavior, and the resistivity decreases again due to interband excitations.
\begin{figure}[htp!]
\vspace{6mm}
\begin{center}
\includegraphics[width=\hsize,angle=0]{sgt.eps} \\
\vspace{5mm}
\includegraphics[width=\hsize,angle=0]{sgmin.eps}
\end{center}
\vspace{-5mm}
\caption[Temperature-dependent conductivity]{
Upper panel: Conductivity as a function of temperature for different densities with a $T$ dependent $\Gamma$ in a double-logarithmic scale. The density is measured with respect to half-filling. Lower panel: Temperature dependence of the minimum conductivity.}
\label{fig:4}
\vspace{-3mm}
\end{figure}
The crossover from metallic behavior at finite densities and low temperatures to semiconducting behavior at elevated temperatures is shown in the upper panel of Fig.\ \ref{fig:4}. The crossover temperature vanishes at zero density ($\mu$ $=$ 0), since $d\sigma_{\text{min}}/dT$ $>$ 0 for all temperatures, and also increases with increasing density. In fact, for the selected temperature dependence of $\Gamma$ the conductivity is metallic below 200 K for densities larger than 8 $\times$ 10$^{10}/\text{cm}^2$. The temperature dependence of the minimum conductivity for the same $T$ dependent $\Gamma$ is shown in the lower panel of Fig.\ \ref{fig:4}. Here a sublinear $T$ dependence of $\sigma_{\text{min}}$ is observed for elevated temperatures. Indeed, the curvature of $\sigma_{\text{dc}}$ changes sign at an intermediate temperature depending on the relative sizes of $\Gamma_0$ and $\alpha_1$. An increasing $\sigma_{\text{min}}$ as a function of temperature with a sublinear behavior at elevated $T$, yet below 200 K, is similarly observed in experiments.\cite{Bolotin08,Du08}
\section{Summary and Discussion}
We have presented a phenomenological theory for the temperature dependence of the dc conductivity of graphene at zero and finite particle densities including potential disorder in a coherent-potential approximation (CPA). Specifically we have chosen a Lorentzian disorder distribution (``Lloyd model''), for which the CPA equations are exactly solvable. This approach recovers well-established limits in the clean case and at the same time provides a phenomenological context for the remarkable transport properties of graphene in the presence of impurity scattering. For the Lloyd model the minimum conductivity is $4e^2/\pi h$, which coincides with previous predictions for the dc limit in the clean system provided that the zero frequency limit is taken before the clean limit at zero temperature. At finite temperatures, the enhanced $T$ dependence of the minimum conductivity in cleaner SG samples is explained, and we find $\sigma_{\text{min}}$ $\propto$ $T/\Gamma$ for $T/\Gamma$ $\gg$ 1. As a consequence we expect a very steep increase of $\sigma_{\text{min}}$ with temperature in even cleaner samples. Moreover we have shown that the $T$ linear resistivity at high densities and the density dependence of its slope follow naturally from a temperature dependent $\Gamma$ $=$ $\Gamma_0 + \alpha_1 T$. This phenomenologically determined $T$ dependence of $\Gamma$ suggests the existence of at least two sources for scattering in suspended graphene. The constant $\Gamma_0$ points to static potential disorder, whereas the $T$-linear part $\alpha_1 T$ may arise from scattering off a thermally excited perturbation. One obvious possibility are thermally excited ripples, since even the linear $T$ dependence of the scattering rate could be explained within the ripple scenario \cite{Katsnelson08}.
Here we have investigated the role of disorder, as described by an Anderson impurity model with a phenomenological disorder strength, as a source for scattering in graphene. As pointed out in Ref.\ \onlinecite{Blake09}, it is important to understand which additional extrinsic effects may mask the intrinsic properties of graphene, especially the sensitive Dirac fermion physics at the neutrality point. Possible extrinsic perturbations are contact resistances, spurious chemical doping into the contact regions, or macroscopic charge inhomogeneity on length scales comparable to the sample size. Such effects need to be incorporated in order to understand the unusual transport properties of graphene in particular at the charge-neutrality point. Also improved doping techniques using organic molecules \cite{Coletti10} may help to unveil the intrinsic transport properties of grapheme. Further theoretical and experimental activity should clarify these aspects and the promising prospects of graphene as a basis of future electronic devices.
We acknowledge discussions with Prabuddha Chakraborty, Krzysztof Byczuk, Holger Fehske, Andreas Sinner, and Wolfgang H\"ausler. This work was supported by the Deutsche Forschungsgemeinschaft through TRR 80.
|
{
"timestamp": "2012-03-13T01:00:37",
"yymm": "1203",
"arxiv_id": "1203.2216",
"language": "en",
"url": "https://arxiv.org/abs/1203.2216"
}
|
\section{Background}
\label{sec:background}
In this section, we review the problem of detecting stepping stones and then review both the passive and active
approaches to the problem. We compare the advantages and disadvantages of the two techniques,
motivating our approach.
\subsection{Stepping Stone Detection}
A stepping stone is a host that is used to relay traffic through an enterprise network to another remote destination.
Stepping stones are used to disguise the true origin of an attack. Detecting stepping stones can help trace
attacks back
to their true source. Also, stepping stones are often indicative of a compromised machine.
Thus detecting stepping stones is a useful part of enterprise security monitoring.
Generally, stepping stones are detected by noticing that an outgoing
flow from an enterprise matches an incoming flow.
Since the relayed connections are often encrypted (using SSH~\cite{ssh:rfc4251}, for
example), only characteristics such as packet sizes, counts, and
timings are available for such detection. And even these are not
perfectly replicated from an incoming flow to an outgoing flow, as
they are changed by padding schemes, retransmissions, and jitter.
As a result, statistical methods are used to detect correlations
among the incoming and outgoing flows. We next review the passive
and active approaches.
\subsection{Passive Traffic Analysis}
In general, passive traffic analysis techniques operate by recording characteristics of incoming streams
and then correlating them with outgoing ones. The right place to do this is often at the border router
of an enterprise, so the overhead of this technique is the space used to store the stream characteristics
long enough to check against correlated relayed streams, and the CPU time needed to perform the correlations.
In a complex enterprise with many interconnected networks, a connection relayed through a stepping stone may
enter and leave the enterprise through different points; in such cases, there is additional communications
overhead for transmitting traffic statistics between border routers.
The passive schemes have explored using various characteristics for correlating streams. Zhang and
Paxson~\cite{zhang:sec00} model interactive flows as on--off processes and detect linked flows by
matching up their on--off behavior. Wang et al.~\cite{wang:esorics02} focus on inter-packet delays, and consider
several different metrics for correlation. More recently,
He and Tong used packet counts for stepping stone detection~\cite{he:tosp07}.
Donoho et al. were the first to consider intruder evasion techniques~\cite{donoho:raid02}. They defined
a \emph{maximum-tolerable-delay} (MTD) model of attacker evasion and suggested wavelet methods to detect stepping
stones while being robust to adversarial action. Blum et al. used a Poisson model of flows to create a technique
with provable upper bounds on false positive rates~\cite{blum:raid04}, given the MTD model. However, for
realistic settings, their techniques require thousands of packets to be observed to achieve reasonable
rates of false errors.
\subsection{Watermarks}
To address some of the efficiency concerns of passive traffic
analysis, Wang et al. proposed the use of
watermarks~\cite{wang:ccs03}. In this scenario, a border router
will modify the traffic timings of the incoming flows to contain a
particular pattern---the watermark. If the same pattern is present
in an outgoing flow, a stepping stone is detected.
Watermarks improve upon passive traffic analysis in two ways. First, by inserting a pattern that is
uncorrelated with any other flows, they can improve the detection efficiency, requiring smaller numbers of
packets to be observed (hundreds instead of thousands) and providing lower false-positive rates
($10^{-4}$ or lower, as compared to $10^{-2}$ with passive watermarks). Second, they can operate in a
\emph{blind} fashion: after an incoming flow is watermarked, there is no need to record or communicate the
flow characteristics, since the presence of a watermark can be detected independently. The detection is also
potentially faster, as here is no need to compare each outgoing flow to all the incoming flows within the same
time frame.
Watermarking techniques for network flows have been based on existing techniques for multi-media watermarking.
For example, Wang et al. based their scheme on QIM
watermarks~\cite{chen:vlsisp01}. Two other watermark schemes~\cite{pyun:infocom07,wang:oakland07} are
based on patchwork watermarking~\cite{bender:ibmsys96}, and Yu et al.~\cite{yu:oakland07} developed one
based on spread-spectrum techniques~\cite{cox:toip97}. Some of the schemes target anonymous communication
rather than stepping stones as the application area (both involve the problem of linking flows), but
the techniques for both are comparable.
\subsection{Watermark Properties}
To motivate our design, we first propose some desirable properties of network flow watermarks. First of all,
a watermark should be \emph{robust} to modifications of the traffic characteristics that will occur inside an
enterprise network, such as jitter. Watermarks should also be resilient to an adversary who actively tries
to remove them from the flow, a property we call \emph{active robustness}. The watermarks should also introduce
little \emph{distortion}, in that they should not significantly impact the performance of the flows. This
is important because in a stepping-stone scenario, most watermarked flows will be benign. Finally,
watermarks should be \emph{invisible} even to attackers who specifically try to test for their presence.
Looking at previous designs, all of them fail to be invisible: the watermarks introduce large delays, on
the order of hundreds of milliseconds, on some packets, which can be easily detected by an
attacker~\cite{peng:oakland06}. In fact, they cannot even be considered low-distortion, as such large delays
are easily noticeable and bothersome to legitimate users. The watermarks are also not actively robust,
as demonstrated by recent attacks~\cite{peng:oakland06,kiyavash:sec08}.
We also observe that active robustness and invisibility are likely
to be impossible to achieve at the same time. This is because to be
invisible, the watermark can only introduce minute changes to the
packet stream. In particular, it cannot introduce jitter of more
than a few milliseconds, since otherwise it will be possible to tell
it apart from the natural network jitter. However, an active attacker will be
willing to introduce large delays to the network; for example, the
maximum tolerable delay suggested in previous work is 500ms. As
such, he will be able to destroy any low-order effects that will be
introduced by the watermark.
Further, it is easy to imagine an attacker determined to hide his tracks using even more drastic measures,
such as using dummy packets to generate a completely independent Poisson process~\cite{blum:raid04}, which
will render any linking techniques ineffective. As such, we decided to design a watermark scheme that is
robust to normal network interference, though not actively robust, and is invisible. This will serve to
detect stepping stones where attackers are unwilling (or unable) to actively distort their stream as it
crosses a stepping stone. Further, as the watermark will be invisible, attackers will not be able to tell
if they are being traced and thus will be less likely to try to apply costly watermark countermeasures.
\section{Conclusions\label{sec:conclusionsFuturework}}
In this paper, we introduce the first non-blind active traffic analysis scheme, RAINBOW.
Using the tools from the detection and estimation theory, we find the optimum passive and (non-blind) active traffic analysis schemes for different types of the network flows.
We show that, for different traffic models, the optimum active detectors outperform the optimum passive detectors. This advantage is more significant for the more correlated network traffic, e.g., the web browsing traffic. Considering the fact that both passive and non-blind active approaches of traffic analysis are constrained by similar scalability issues, this finding motivated the use of non-blind active approaches over the passive approaches.
\section{Introduction}
Internet attackers commonly relay their traffic through a number of
(usually compromised) hosts in order to hide their identity. Detecting
such hosts, called stepping stones, is therefore an important problem in
computer security. The detection proceeds by finding correlated flows
entering and leaving the network. Traditional
approaches have used patterns inherent
in traffic flows, such as packet timings, sizes, and counts, to link
an incoming flow to an outgoing one~\cite{staniford-chen:oakland95,zhang:sec00,donoho:raid02,wang:esorics02,blum:raid04}. More recently, an active
approach called \emph{watermarking} has been
considered~\cite{wang:ccs03,pyun:infocom07}. In this approach,
traffic characteristics of an incoming flow are actively perturbed
as they traverse some router to create a distinct pattern, which can
later be recognized in outgoing flows. These techniques also
have relevance to anonymous communication, as linking two flows can
be used to break anonymity, and both
passive traffic
analysis~\cite{levine:fc04,danezis:pet04} and active
watermarking~\cite{wang:ccs05,wang:oakland07,yu:oakland07} have been
studied in that domain as well.
The choice between passive and active techniques for traffic analysis exhibits a tradeoff.
Passive approaches require observing relatively long-lived network flows, and storing or
transmitting large amounts of traffic characteristics. Watermarking approaches are more efficient,
with shorter observation periods necessary. They are also \emph{blind}: rather than storing or
communicating traffic patterns, all the necessary information is embedded in the flow itself.
This, however, comes at a cost: to ensure robustness, the watermarks introduce large delays (hundreds of milliseconds)
to the flows, interfering with the activity of benign users, and making them subject to
attacks~\cite{peng:oakland06,kiyavash:sec08}.
Motivated by this, we propose a new category for network flow watermarks, the \textit{non-blind flow watermark}s. Non-blind watermarking lies in the middle of passive techniques and (blind) watermarking techniques: similar to passive techniques (and unlike blind watermarks),
non-blind watermarks will record traffic pattern of incoming flows and correlate them with outgoing flows.
On the other side, similar to blind watermarks (and unlike passive techniques), non-blind watermarking aids traffic analysis by applying some modifications to the communication patterns of the intercepted flows.
We develop and prototype the first non-blind flow watermark, called RAINBOW.
RAINBOW records the \textit{timing} pattern of incoming flows and correlate them with the timing pattern of the outgoing flows. On each incoming flow,
RAINBOW also inserts a watermark by delaying some packets, after recording the received timings.
As such a watermark is generated independently of the
flows, this will diminish the effect of natural similarities between two unrelated flows, and allow a flow
linking decision to be made over a much shorter time period. RAINBOW uses spread-spectrum techniques to
make the delays much smaller than previous work. RAINBOW uses delays that are on the order of only a few milliseconds;
this means that RAINBOW watermarks not only do not interfere with traffic patterns of normal users, they are
also virtually \emph{invisible}, since the delays are of the same magnitude as natural network jitter.
In \cite{houmansadr:ndss09} we use different information theoretical tools to verify the invisibility of RAINBOW, and demonstrate its high performance in linking network flows through a prototype implementation over the PlanetLab~\cite{bavier:nsdi04} infrastructure.
In this paper, we thoroughly analyze the detection performance of RAINBOW non-blind watermark, and compare it with that of passive traffic analysis schemes. By using \textit{hypothesis testing} mechanisms from the detection and estimation theory \cite{poor88}, we find the optimum detection schemes for RAINBOW as well as the optimum passive detectors under different models for network traffic. Modeling real-world network traffic is a complicated problem as it depends on many different parameters; as a result, we only consider two extreme models of the network traffic: (1) independent flows where each flow is modeled as a Poisson process (traffic model A), and, (2) completely correlated flows where all flows are considered to have similar timing patterns (traffic model B). We assume that any real-world traffic model lies in the middle of these two extreme models.
Our analysis leads to the following important conclusions:
i) Non-blind watermarking \emph{always} performs a better detection than passive traffic analysis. This is an essential result in motivating the use of non-blind watermarks over passive traffic analysis, since both have similar scalability constraints, i.e., both approaches have $O(n)$ communication overheads and $O(n^2)$ computation overheads \cite{houmansadr:ndss09}. Not that this point is not necessary (nor is always true) to motivate the use of traditional (blind) watermarks over passive traffic analysis, since blind watermarks provide much better scalability (i.e., $O(1)$ communication overhead and $O(n)$ computation overhead \cite{houmansadr:ndss09} ).
ii) Our analysis shows that the performance advantage of non-blind watermarking (over passive schemes) is only marginal for uncorrelated network traffic, while it is very significant for correlated network traffic. This knowledge can be used to decide the best traffic analysis approach in various applications.
We validate our analysis through simulating the detection schemes on real network traces. In particular, we show that for highly correlated traffic, e.g., same webpage downloads, passive traffic analysis performs very poorly while a RAINBOW watermark is highly effective.
iii) We also show (through both analysis and experiments) that the optimum watermark detector derived for correlated traffic (namely $SLCorr$) also performs very good for uncorrelated traffic (while the optimum watermark detector for uncorrelated traffic does not do well for correlated traffic). This allows one to use $SLCorr$ as the sole watermark detector regardless of the type of traffic being observed. This is especially useful in real-world applications where the observed traffic is a mixture of different flow types.
Note that in this paper we do not discuss the performance advantage of non-blind watermarks over traditional blind watermarks, as this has been justified in \cite{houmansadr:ndss09}.
The rest of this paper is organized as follows: we review the problem of stepping stone detection and existing schemes in Section~\ref{sec:background}. Our RAINBOW scheme is presented in Section~\ref{sec:NBSS}. In Section~\ref{sec:detection}, we use hypothesis testing to find and analyze the optimum likelihood ratio detectors for passive and non-blind active (watermark) approaches under different traffic models, and analyze their false error rates. In Section~\ref{sec:implementation}, we validate the analysis results through simulation of the detection schemes over real network traces. Finally, the paper is concluded in Section~\ref{sec:conclusionsFuturework}.
|
{
"timestamp": "2012-03-13T01:01:25",
"yymm": "1203",
"arxiv_id": "1203.2273",
"language": "en",
"url": "https://arxiv.org/abs/1203.2273"
}
|
\section{Conclusion \& Discussion}
In this paper, we studied the problem of scheduling in wireless networks with interference
constraints where the capacity of links changes over time. We have analyzed the
performance of a well-known algorithm, Greedy-Maximal Scheduling (GMS), to
the case of general wireless networks with fading structure. We
defined Fading-Local pooling factor for graphs with fading and showed
that it characterizes the fraction of throughput that can be achieved by
GMS. We have derived useful yet easily computable bounds on F-LPF through alternate formulations.
By analyzing F-LPF, we have studied the effect of fading on the performance of GMS. It is a priori not clear whether
fading can enhance/degrade the relative performance of GMS. In this work, we have showed that fading can in fact
exhibit both behaviors through two simple examples, one in which fading increases the efficiency
ratio of GMS and other in which fading decreases the efficiency ratio as compared to non-fading case.
\bibliographystyle{plain}
\section{Extensions to Multiple Fading States}
We now extend our results for 'ON/OFF' channels to channel models
where each link capacity is time-varying and takes values from a
finite state space. Let us denote the set of values in the state space
by $\{0,c_1, c_2,.....,c_m\}$. The global state $GS(t)$ of the system
now refers to the exact channel state of each link. Let $\pi(X_1,
X_2,...,X_K)$ denote the fraction of time the network is in global
channel state $(X_1, X_2, X_3,....X_K).$ Let us denote the state
$(X_1, X_2, X_3,....,X_K)$ by ${\bf X}$.
Let $M_{\bf X}$ denote the matrix consisting of $K$ rows one for each
link. Each column now represents a possible maximal independent set on
the set of links with non-zero channel states. For a given column,
the entries of a given row is set to zero if link $l$ (corresponding to row)
does not belong to independent set, or is set to equal to channel value $X_l$ if
it belongs to independent set. For example, consider the Interference
graph in Figure \ref{fig:intgraph} with each link taking 3 channel
states $\{0,1,2\}$. Then $M_{(1,2,1,0)}$ is given by,
\[ M_{(1,2,1,0)} = \left( \begin{array}{cc}
1 & 0 \\
0 & 2 \\
1 & 0 \\
0 & 0 \end{array} \right) \]
The throughput region $\Lambda_f$ for the above general network model
with fading pattern $\pi({\bf X})$ is given by:
\begin{eqnarray*}
\Lambda_f^g = \{\vec{\lambda}: \vec{\lambda} > 0&,&\vec{\lambda}
\,\leq \, \sum_{{\bf X}} \, \pi({\bf X}) \vec{\eta}_{{\bf X}} \,
\textrm{where}\\
& & \, \, \vec{\eta}_{{\bf X}} \in {\cal CH}(M_{{\bf X}}) \}.
\end{eqnarray*}
We now define the F-LPF for a set of links $L$ as follows:
\begin{equation}
\label{eq: sigmaLgen}
\sigma_L^* (\boldsymbol{\pi})= \textrm{inf} \{\sigma : \exists \,
\vec{\phi_1}, \vec{\phi_2} \in \Phi^g(L) \textrm{ such that } \sigma
\vec{\phi_1} \geq \vec{\phi_2} \},
\end{equation}
where,
\begin{equation}
\Phi^g(L) = \{\vec{\phi}: \vec{\phi}= \sum_{{\bf X}}{\pi({\bf X})
\vec{\eta}_{{\bf X}}} \textrm{ where } \vec{\eta}_{{\bf X}} \in
{\cal CH}(M_{{\bf X_L}})\},
\end{equation}
${\bf X_L}$ is constructed from ${\bf X}$ by setting the values of
links that do not belong to set $L$ in ${\bf X}$ to zero.
Theorem 1 can be shown to hold for the general model with the above
modified definition of F-LPF. The proof of Theorem 1 for the 'ON/OFF'
channels can be easily modified to above system with general channels
and is therefore omitted.
\section{Introduction}
This paper analytically investigates the effect of fading on the
throughput performance of a natural and popular scheduling algorithm:
Greedy Maximal Scheduling (GMS)
\cite{Mck_95,LeoNeeLea_06,DimWal_06,JooLinShr_08}. As with any
scheduling algorithm, GMS is a way to determine which wireless links
can transmit at any given time, based on their mutual interference
characteristics and their current level of fading. In particular, GMS
involves first associating a weight with each link -- which depends on
the load of the link and its channel condition. Then, GMS involves
iteratively turning on the heaviest link that does not interfere with
links already turned on. This is repeated every time slot.
GMS has empirically shown to have very good throughput and delay
performance; recent theoretical advances
\cite{LecNiSri_09,JooLinShr_08,ZusBrxMod_08,BirMarBer_10,LonEytCha_10,LiRoh_10}
characterize its throughput. All of these works assume that there is
no fading; ie that the rate a link can support is invariant as long as
all the links that interfere with it are not simultaneously on. Our
work investigates what happens to this performance in the more
realistic setting with intrinsic channel fading as well. In
particular, we compare the relative throughput of GMS as compared to
that of an optimal scheduler.
Our results demonstrate that the effect of fading is quite subtle; in
particular, in some instances fading can degrade the relative
performance of GMS, while in other cases it can improve it. The former
reflects the fact that fading provides an extra degree of freedom and
complexity in the system, which GMS may not be able to handle as well
as in a system without this fading. The latter reflects the, perhaps more
subtle, fact that the sub-optimality of GMS (even without fading) is
tied to the existence of special global system configurations that
result in poor performance. The presence of fading ``breaks up" these
global configurations -- not allowing them to occur too often -- allowing GMS to
perform relatively better.
Specifically, our contributions are as follows: For a given wireless
network with fading channels,
\begin{enumerate}
\item We define a new quantity, called Fading-Local Pooling Factor
(F-LPF), analogous to LPF defined in \cite{JooLinShr_08} that
characterizes the performance of Greedy Maximal Scheduling (GMS) in
wireless networks with fading channels. Furthermore, we show that
Fading-LPF is a lower bound on the fraction of throughput that can
be stabilizable by the GMS when the arrivals and channels are
independent and identically distributed over time.
\item With arbitrary arrival and channel state process, we show that
Fading-LPF is an upper bound on the fraction of throughput that can
be stabilizable by the greedy schedule. More specifically, we
construct an adversarial arrival and channel process with long term
averages that lie outside the scaled throughput region and show that
GMS policy cannot stabilize the queues.
\item We further provide lower and upper bounds on Fading-LPF that are
easy to evaluate. We provide two example networks with specific
fading structure and use the derived bounds to demonstrate that
fading can either enhance or degrade the relative performance of GMS
as compared to the non-fading scenario.
{
\item With fading, we can represent the channel model as a collection
of global channel-states, where each state is associated with an
independent set and an occurance probability. A natural question
that arises is the following: Is the acheivable rate-region with
fading simply the (channel-probability weighted) average of the
per-state {\em scaled} rate regions, with the scaling parameter
simply being the conventional LPF for each state? We show that this
is in general not true. However, we derive a region that {\em can} be
stabilized by the GMS in wireless networks with fading
channels. This region is characterized based on the interference
degree of the subgraphs (generated from original network) and the
fading distribution.
}
\end{enumerate}
\subsection{Related Work:}
Transmission scheduling has been a key challenge in modern wireless
systems. The MaxWeight algorithm, proposed in \cite{TasEph_92}, has
been the inspiration for many approaches to address this in various
wireless systems (see \cite{LeoNeeLea_06} for several
variants). However, this algorithm suffers from centralization as well
as computational complexity.
Thus, there has been significant research in finding sub-optimal
(i.e., achieving a subset of the throughput region) distributed
scheduling algorithms with low complexity. The authors in
\cite{Mck_95} propose one such policy called Greedy Maximal
Scheduling, whose time complexity is linear in the number of links, and has
a distributed implementation \cite{LecNiSri_09}. There are other
sub-optimal, randomized algorithms that have been proposed with similar
performance as GMS \cite{LinRas_06,JooShr_07}.
The authors in \cite{DimWal_06} have been the first to study the
performance of GMS under a general interference model. They have
identified conditions (so called 'Local Pooling') under which there is
no loss in the network throughput region with GMS. The notion of
Local Pooling has been extended to a multi-hop regime by
\cite{ZusBrxMod_08}.
This condition being identified as too restrictive, the authors in
\cite{JooLinShr_08} have defined a new quantity called Local Pooling
Factor (LPF) that exactly characterizes the fraction of throughput
region achieved by GMS, and show that over tree networks with a
$K-$hop model for interference, GMS achieves the entire throughput
region.
Additional characterizations, including a
per-link LPF \cite{LiBoyXia_09} and bounds to characterize the
stability region \cite{LiRoh_10}, have been proposed in literature.
The authors in \cite{BirMarBer_10} exactly characterize, using graph
theoretic methods, the set of network graphs (with only the primary
interference constraints) where GMS is optimal (LPF $= 1$).
Finally, the authors in
\cite{LonEytCha_10} have studied the performance of GMS with the SINR
interference model, and have shown that GMS exhibits zero LPF in the
worst case.
All the above results assume that there are no channel variations
(fading). In this paper, we study the effect of channel variation on
the performance of GMS.
\section{Proofs of Results}
\label{sec:main}
\begin{theorem*}[\textbf{1}]
a) (Upper Bound) Under a given network topology and channel state
distribution with Assumption A1 on the arrivals and fading channels,
the efficiency ratio of GMS ($\gamma^*$) is less than or equal to
$\sigma_G^*(\boldsymbol{\pi})$.
b) (Achievability) Under a given network topology and channel state
distribution $\boldsymbol{\pi}$ with Assumptions A1 and A2 on the
arrivals and fading channels, the efficiency ratio of GMS
($\gamma^*$) is greater than or equal to
$\sigma_G^*(\boldsymbol{\pi})$.
\end{theorem*}
\begin{IEEEproof}
{\em The proof follows the method developed by the authors in
\cite{JooLinShr_08, DimWal_06} for the non-fading case; however we
have extended it to take in to account the fading
structure. First, for the converse (to show instability for
arrivals outside the stability region), we explicitly construct an
adversarial channel variations pattern that satisfies the
time-averages imposed by the fading assumption, and this is used
in conjunction with the adversarial arrival process. The
achievability part is more straightforward -- we augment the
analysis in \cite{DimWal_06,JooLinShr_08} to include the fluid
limit of the channel fading process.} We now provide the proof
more detail:
\textbf{Proof (Theorem 1. a)}: The result follows from the following general
lemma.
\begin{lemma}
\label{lemma: ub}
If there exists a subset of links $L (\subseteq {\cal K})$, a positive
number $\sigma$ and two vectors $\vec{\mu}, \vec{\nu} \in \Phi(L)$
such that $\sigma \vec{\mu} > \vec{\nu}$, then for arbitrary small
$\epsilon > 0 $, there exists a traffic pattern with offered load
$\vec{\nu} + \epsilon \vec{e}_L$ and a fading pattern, such that
system is unstable under greedy maximal schedule.
\end{lemma}
\textbf{Proof (Lemma \ref{lemma: ub})}: The idea of the proof is as
follows -- we construct a traffic pattern and channel variations
pattern with offered load $\vec{\nu} + \epsilon \vec{e}_L $ and show
that under this traffic/channel fading pattern, the queue lengths go
to infinity under GMS, thus making the system unstable.
\emph{As remarked earlier, this proof technique was introduced in
\cite{JooLinShr_08}, where authors only needed to construct
adversarial arrival process that makes the queues in the system to
overflow. However, in our setting, we need to account for the fading
process and construct both arrival and channel fading pattern that
makes the network unstable.}
Since $\vec{\nu} \in \Phi(L)$, there exist vectors $\vec{w}^J$ such
that $\vec{\nu}$ can be expressed as,
\begin{equation}
\vec{\nu} = \sum_{J \subseteq L} \pi_L(J) \Big(M_{J,L} \vec{w}^J \Big).
\end{equation}
Fix $\delta > 0$, we then find a vector $\vec{r}^J$ in the set of
rational numbers, $\mathbb{Q}$, such that $\lVert \vec{r}^J -\vec{w}^J
\rVert < \delta.$
Assume packets arrive to a link at beginning of the time slot. Let
the queues of all the links in $L$ are empty at $t =0$. Let $T_J$ be
the smallest integer such that for all $i$, $r_i^J T_J $ is an
integer. Let $t_i^J = r_i^J T_J$. Also, there exists integers
$n_1,n_2,...n_{2^L}$ such that
\begin{equation}
\big | \frac{n_J T_J}{\sum_{S: S\subseteq L} n_S T_S} -
\pi_L(J)\big | \leq \frac{\delta}{2^L}.
\end{equation}
Let us define $\tilde{\pi}_L(J) \in \mathbb{Q}$ as follows,
\begin{equation}
\tilde{\pi}_L(J) := \frac{n_J T_J}{\sum_{S\subseteq L} n_S T_S}.
\end{equation}
Using the rational quantities $\tilde{\pi}_L(J)$ and $\vec{r}^J$, we
define $\vec{\nu}^r$ as follows,
\begin{equation}
\vec{\nu}^r = \sum_{J : J \subseteq L} \tilde{\pi}_L (J) \big( M_{J,L} \vec{r}^J \big).
\end{equation}
Consider a total time period of $\sum_J n_J T_J $. We assume that
channel state remains in $J$ state for $T_J$ time slots (denoted as a
time frame). It is easy to observe that with the above described
fading pattern, we achieve the same channel state distribution as
$\tilde{\pi}_L(J)$ on links of set $L$. We now describe the arrival
pattern for $T_J$ time slots when the channel is in state $J$.
Assume that all the queue lengths (of links in $L$) are equal at the
beginning of $T_J$ time slots. We now construct arrival pattern that
keeps the queue lengths of all links in set $L$ equal at the end of
$T_J$ time slots under the GMS policy. The arrival process is as
follows:
\begin{enumerate}
\item The time frame of $T_J$ slots is further divided in to $t_1^J,
t_2^J,....t_{|IS^J|}$ time slots, where $t_i^J = r_i^J T_J$ and
$|IS^J|$ denotes the number of columns in $M_J$.
\item During the $t_i^J , i \neq |IS^J| $ time slots, apply one packet
to each link that is 'ON' in the $i^{th}$ column of $M_J$. For the
last $t_{|IS^J|}^J$ time slots, apply one packet to each link that
is ON in the last column of $M_J$ at the beginning of the time slot
except for the last one time slot. For the last one time slot, with
probability $1-\epsilon$ we do the same as described before and with
probability $\epsilon$, we apply two packets to each link that is ON
in the last column of $M_J$ and 1 packet to rest of links in $L$.
\end{enumerate}
\emph{Note that the arrival process is modified compared to one
proposed in \cite{JooLinShr_08} so as to ensure that all queues
remain equal after $T_J$ time slots.}
It is now easy to see that at the end of $T_J$ time slots, all the
queue lengths are equal and increase by 1 with probability $\epsilon.$
Thus the above arrival and channel variation pattern make the system
unstable under GMS schedule. We now show that the arrival rate is same
as $\vec{\nu} + \epsilon \vec{e}_L$.
Let $\vec{e}_i$ denote the vector of all zeros except for $i$ th
position which is set to one. Let $\sum_J = \sum_{J \subseteq L}$ for
the remaining part of the proof. For the constructed adversarial
arrival process, the arrival rate is given by the following,
\begin{equation}
\vec{\lambda}_{\textrm{adv}} = \frac{\sum_{J} n_J (\sum_{i=1}^{|IS^J|}
t_i^J M_J \vec{e}_i+ \epsilon \vec{e})}{\sum_{J} n_J
(\sum_{i=1}^{|IS^J|} t_i^J)}
\end{equation}
Rewriting the above expression in terms of $\tilde{\pi}_L(J)$, we have
that
\begin{equation}
\vec{\lambda}_{\textrm{adv}} = \sum_{J} \tilde{\pi}_L(J)
(\sum_{i=1}^{|IS^J|} r_i^J M_J \vec{e}_i)+ \epsilon \big(\sum_J
\frac{\tilde{\pi}_L(J)}{T_J}\big) \vec{e}
\end{equation}
Thus we have,
\begin{equation}
\vec{\lambda}_{\textrm{adv}} = \sum_{J} \tilde{\pi}_L(J) \big(M_{J,L}
\vec{r}^J \big) + \epsilon (\sum_J \frac{\tilde{\pi}_L(J)}{T_J})
\vec{e}
\end{equation}
We choose small enough $\delta$ so that the arrival rate is strictly
less than $\vec{\nu} + \epsilon \vec{e}_L.$
\textbf{Proof (Theorem 1. b)}: {\em This proof is a simple extension of that
in \cite{JooLinShr_08,DimWal_06}, however modified to include the
fluid limit arising due to the channel fading process. Thus, we have
provided a detailed sketch and refer to
\cite{JooLinShr_08,DimWal_06} for full details.}
We consider the fluid limit of the queuing process and we provide a
Lyapunov function and show negative drift under GMS schedule whenever
arrival rate $\vec{\lambda} \in (\sigma_G^*(\pi)-\epsilon) \Lambda_f$.
Consider a sequence of systems $ \frac{1}{n} \vec{Q}^{n}(nt)$ (scaled
in time and space by a factor of $n$), where $\vec{Q}^n(.)$ denotes
the queue lengths of original system, satisfying $\sum Q_l^n(0) \leq
n$ at time $t = 0.$ Let us index the sequence of systems by $n =
\{1,2,....\}$. We apply the same arrival processes to all the above
defined systems (i.e $\vec{A}^n(.) = \vec{A}(.)$) and assume that
queues are served according to greedy maximal schedule. Let
$\vec{A}^n(t)$ and $\vec{D}^n(t)$ denote the cumulative arrival and
departure process of system $n$ up to time $t$.
Using the results from \cite{Dai_95}, it can be shown that the
sequence of processes $(\vec{Q}^n(.), \vec{A}^n(.), \vec{D}^n(.))$ as
$n \to \infty$ converges to a fluid limit almost surely along a
subsequence $\{n_k\}$ in the topology of uniform convergence over
compact sets,
\begin{eqnarray}
\frac{1}{n_k} A_l^{n_k}(n_k t) & \to & \lambda_l t ,\\
\frac{1}{n_k} D_l^{n_k}(n_k t) & \to & \sum_J \pi(J) \big( \int_0^t
\mu_l^J(s) ds \big) , \\
\frac{1}{n_k} Q_l^{n_k}(n_k t) & \to & q_l(t).
\end{eqnarray}
Also, the fluid limits $(q_l(t), \mu_l^J(t))$ satisfy the following equality:
\begin{equation}
q_l(t) = q_l(0) + \lambda_l t - \sum_J \pi(J) \big( \int_0^t \mu_l^J(s) ds \big).
\end{equation}
Moreover, fluid limits are absolutely continuous, and at regular times
$t$ (i.e., those points in time where the derivatives exist) we have
the following condition satisfied:
\begin{displaymath}
\frac{d}{dt} q_l(t) = \left \{ \begin{array}{ll}
\lambda_l - \mu_l(t) & \textrm{if}\quad q_l(t) > 0 \\
(\lambda_l - \mu_l(t))^+& \textrm{if}\quad q_l(t) = 0, \end{array} \right.
\end{displaymath}
where $\mu_l(t) = \sum_J \pi(J) \mu_l^J(t)$ satisfies the GMS
properties. Let $L_0$ denote the set of links with the longest queues
at time $t$,
\begin{equation}
\label{eq:Lo}
L_0(t) = \big\{ i \in K | q_i(t) = \textrm{max}_{j \in K} q_j(t) \big\}
\end{equation}
Let $L(t)$ denote the set of links with the largest derivative of
queue length among the links in $L_0(t),$
\begin{equation}
\label{eq:L}
L(t) = \big\{ i \in L_0(t) | \frac{d}{dt} q_i(t) = \textrm{max}_{i \in
L_0(t)} \frac{d}{dt}q_i(t)\big\}
\end{equation}
{
\begin{lemma}
\label{GMS}
Under the greedy maximal schedule, the service rate satisfies $\vec{\mu}(t)|_{L(t)} \in
\Phi(L(t))$, where $\vec{u}|_{L}$ denotes the projection of vector on $u$ on to set of links $L$.
\end{lemma}
The proof of the above lemma is similar to one in \cite{DimWal_06,JooLinShr_08} and is presented in appendix. The idea, roughly is that, queues in the set $L(t)$ will remain the longest for small enough amount of time past $t$ and GMS
picks the maximal schedule restricted to links in $L(t)$ that are in
'ON' state.
}
Since the arrival rates are strictly with in
$\sigma_L^*(\boldsymbol{\pi}) \Lambda_f$, there exists a service
vector $\vec{\nu} \in \Phi(L)$ and $\vec{\nu} <
\sigma_L^*(\boldsymbol{\pi}) \Lambda_f$ such that $\vec{\lambda}(L)
\leq \vec{\nu}$, where $\vec{\lambda}(L)$ is projection of arrival
vector on to the set $L$. Given any two vectors in set $\Phi(L)$, note
that one vector never dominates the other one in all the dimensions by
a factor more than $\sigma_L^*(\boldsymbol{\pi})$. Therefore we have
that $\frac{d}{dt} \textrm{max}_{i \in L(t)} q_i(t) $ is strictly
negative when ever $\textrm{max} \, q_i(t) > 0.$
Let $V(t) = \textrm{max} \, q_l(t)$ denote the Lyapunov function used
for the fluid system. Since we have a negative drift for the Lyapunov
function, using the results from \cite{Dai_95}, we have that fluid
system is stable (i.e there exists $t_0 > 0$ such that $q_l(t) = 0 \,
\forall t > t_0$). Therefore from \cite{Dai_95}, we have that the queues in
the original queuing system are stable.
\end{IEEEproof}
\begin{theorem*}[\textbf{2}]
For every $J \subseteq {\cal K}$ and any $(\vec{\mu}_J, \vec{\nu}_J,
H_J)$ such that $\vec{\mu}_J, \vec{\nu}_J \in {\cal CH}(M_J)$,
$\vec{\nu}_J \leq H_J \vec{\mu}_J$, we have that
\begin{equation*}
\sigma_G^*(\boldsymbol{\pi}) \leq \textrm{max}_l \, \frac{\sum_{J \subseteq
{\cal K}} \pi(J) H_J \mu_J(l)}{\sum_{J \subseteq {\cal K}}
\pi(J) \mu_J(l)},
\end{equation*}
where $\mu_J(l) = 0 $ if $l \notin J.$
\end{theorem*}
\begin{IEEEproof}
Since $(\vec{\mu_J}, \vec{\nu_J}, H_J)$ satisfy the inequality,
\begin{equation}
\vec{\nu}_J \leq H_J \vec{\mu}_J
\end{equation}
Summing over all subsets with positive scaling constants $\pi(J)$,
\begin{equation}
\sum_J \pi(J) \nu_J(l) \leq \sum_J \pi(J) \big( H_J \mu_J(l) \big)
\end{equation}
Using the maximum constant over all the inequalities, we have the following,
\begin{equation}
\sum_J \pi(J) \vec{\nu}_J \leq \Big( \textrm{max}_l \frac{\sum_J \pi(J) H_J \mu_J(l)}{\sum_J \pi(J) \mu_J(l)} \Big) \sum_J \pi(J) \vec{\mu}_J
\end{equation}
By observing the fact that $(\sum_J \pi(J) \vec{\nu}_J, \sum_J \pi(J) \vec{\mu}_J)$ belong to the $\Phi({\cal K})$, we have the result.
\end{IEEEproof}
\begin{theorem*}[\textbf{3}]
\begin{equation}
\sigma_L^*(\boldsymbol{\pi}) \geq \frac{\sum_{J \subseteq L} \pi_L(J)
n(M_J)}{\sum_{J \subseteq L} \pi_L(J) N(M_J)},
\end{equation}
where $n(M) = \textrm{min}_j \sum_i M_{ij}, N(M) = \textrm{max}_j
\sum_i M_{ij} $ and $\boldsymbol{\pi}_L$ denotes the marginal distribution on set
of links $L$ induced by $\boldsymbol{\pi}$.
\end{theorem*}
\begin{IEEEproof} We first state a lemma that describes the dual
problem that finds the fading Local Pooling Factor as the optimal
solution. The dual characterization of Local Pooling Factor was
presented previously in \cite{DimWal_06,LiBoyXia_09}. We now
provide such characterization for F-LPF in Lemma~\ref{lemma: dual}
by generalizing the arguments in \cite{LiBoyXia_09}. In particular,
the multiple global channel states due to fading each induce a
different constraint -- combining all of these appropriately while
satisfying the long-term average fractions $\{\pi_L(J)\}$ results in
a $\max \min$ problem, as detailed below. This result is used to
derive the lower bound.
\begin{lemma}
\label{lemma: dual}
The following optimization problem characterizes $\sigma_L^*(\boldsymbol{\pi}):$
\begin{align*}
&\sigma_L^*(\boldsymbol{\pi}) = \max \displaystyle \sum_{J : J
\subseteq L} \pi_L(J) a(J) \label{eq: Dual}\\
\hbox{s.t : }
&\displaystyle x' M_{J,L} \geq a(J) e' \quad \forall J \subseteq L \nonumber \\
&\displaystyle x' M_{J,L} \leq b(J) e' \quad \forall J \subseteq L \nonumber \\
&\displaystyle\sum_{J \subseteq L} \pi_L(J) b(J) = 1
\end{align*}
\end{lemma}
\begin{IEEEproof}
Consider the definition of $\sigma_L^*(\boldsymbol{\pi})$ in (\ref{eq:
sigmaL}). The corresponding optimization problem is given by:
\begin{align*}
&\inf \quad \quad \quad \displaystyle \sigma \\
\hbox{s.t : }
&\displaystyle \sigma \sum_{J \subseteq L}\pi_L(J) M_{J,L}
\vec{\alpha}(J) \geq \sum_{J \subseteq L}\pi_L(J) M_{J,L}
\vec{\beta}(J) \nonumber \\
& \lVert \vec{\alpha}(J) \rVert = 1 \quad \quad \forall \quad J \subseteq L \nonumber \\
& \lVert \vec{\beta}(J) \rVert = 1 \quad \quad \forall \quad J \subseteq L \nonumber \\
& \vec{\alpha}(J), \vec{\beta}(J) \geq 0
\end{align*}
where $\lVert . \rVert$ is defined as the sum of all the elements of
the vector. Let us define a new variable $\vec{\gamma}(J) = \sigma
\vec{\alpha}(J)$. Thus, we have:
\begin{align*}
&\inf \quad \quad \quad \displaystyle \sigma \\
\hbox{s.t : }
&\displaystyle \sum_{J \subseteq L}\pi_L(J) M_{J,L} (\vec{\beta}(J)-
\vec{\gamma}(J)) \leq 0 \nonumber \\
& \lVert \vec{\gamma}(J) \rVert = \sigma \quad \quad \forall \quad J
\subseteq L \nonumber \\
& \lVert \vec{\beta}(J) \rVert = 1 \quad \quad \forall \quad J
\subseteq L \nonumber \\
& \vec{\gamma}(J), \vec{\beta}(J) \geq 0
\end{align*}
For the above LP, let $(\vec{x}, \{y(J)\}, \{z(J)\})$ denote the dual
variables associated with the constraints. The dual is given by
\begin{align*}
\max_{\vec{x}, \{y(J)\}, \{z(J)\} } & \min_{\sigma, \vec{\alpha}(J),
\vec{\beta}(J)} \sigma + \\
& \sum_{i =1}^L x_i \Big( \sum_{J \subseteq L} \pi_L(J)
[\sum_{j=1}^{|IS_J|} M_{ij}^J (\beta_j^J- \gamma_j^J)] \Big) + \\
& \sum_{J \subset L} y(J) \big(\vec{\gamma}(J)'e - \sigma\big) + \\
&\sum_{J \subset L} z(J) \big(\vec{\beta}(J)'e - 1\big) \\
\hbox{s.t:} \vec{\gamma}(J), \vec{\beta}(J) \geq 0
\end{align*}
Rewriting the above dual optimization problem, we have
\begin{align*}
\max_{\vec{x}, \{y(J)\}, \{z(J)\} } \min_{\sigma, \vec{\alpha}(J),
\vec{\beta}(J)} & -\sum_{J} z(J) + \sigma (1- \sum_{J}y(J)) +\\
& \sum_{j =1}^{|IS_J|} \beta_j^J \big[\pi_L(J) \sum_{i=1}^L x_i M_{ij}^J + z(J) \big] + \\
& \sum_{j =1}^{|IS_J|} - \gamma_j^J \big[\pi_L(J) \sum_{i=1}^L x_i M_{ij}^J + y(J) \big] \\
\hbox{s.t:} \vec{\gamma}(J), \vec{\beta}(J) \geq 0
\end{align*}
Equivalently, the above program can be reduced to
\begin{align*}
&\max \displaystyle \sum_{J : J \subseteq L} - z(J) \\
\hbox{s.t : }
&\displaystyle \pi_L(J) x' M_{J,L} + z(J) e' \geq 0 \quad \forall J
\subseteq L \nonumber \\
&\displaystyle - \pi_L(J) x' M_{J,L} + y(J) e' \geq 0 \quad \forall
J \subseteq L \nonumber \\
&\displaystyle \sum_{J \subseteq L} y(J) = 1
\end{align*}
Denoting $\frac{-z(J)}{\pi(J)}$ by $a(J)$ and $\frac{y(J)}{\pi(J)}$ by
$b(J)$ we have the desired result.
\end{IEEEproof}
From the above Lemma \ref{lemma: dual}, we have that
$\sigma_L^*(\boldsymbol{\pi}) $ is equal to,
\begin{align*}
&\max_{x,a(J),b(J)} \displaystyle \sum_{J : J \subseteq L} \pi_L(J) a(J) \\
\hbox{s.t : }
&\displaystyle x' M_{J,L} \geq a(J) e' \quad \forall J \subseteq L \nonumber \\
&\displaystyle x' M_{J,L} \leq b(J) e' \quad \forall J \subseteq L \nonumber \\
&\displaystyle\sum_{J \subseteq L} \pi_L(J) b(J) = 1
\end{align*}
Observe that $(\frac{1}{\sum \pi_L(J) N(M_J)} e, \frac{n(M_J)}{\sum
\pi_L(J) N(M_J)}, 1)$ is a valid point in the search
space. Substituting the point in the above function, we have the
desired inequality.
\end{IEEEproof}
\emph{Corollary 1: } $\sigma_G^*(\boldsymbol{\pi}) \geq \frac{1}{d_I(G)} $
\begin{IEEEproof}
Observing the fact that $n(M_J) \geq \frac{1}{d_I(G)} N(M_J)$ and
using the above lemma, we have the desired inequality.
\end{IEEEproof}
{
\begin{theorem*}[\textbf{4}]
Under a given network topology and channel state distribution with Assumption A1 on the arrivals and fading channels, GMS can stabilize the network if the arrival rates are inside the region $\Lambda_f(\vec{x})$, where $x(S) = \frac{1}{d_I(S)}$.
\end{theorem*}
\begin{IEEEproof}
We consider a continuous model similar to the one described in the proof of Theorem 1b. In this model, the queuing system evolves according to the following equation,
\begin{displaymath}
\frac{d}{dt} q_l(t) = \left \{ \begin{array}{ll}
\lambda_l - \mu_l(t) & \textrm{if}\quad q_l(t) > 0 \\
(\lambda_l - \mu_l(t))^+& \textrm{if}\quad q_l(t) = 0, \end{array} \right.
\end{displaymath}
where $\mu_l(t) = \sum_J \pi(J) \mu_l^J(t)$ satisfies the GMS properties. In the original system with fading channels note that the weight of GMS schedule is always greater than $\frac{1}{d_I(S)}$ of the weight of the max-weight schedule where $S$ is the set of links that are in 'ON' state. Therefore in the fluid model, we can show that $\mu_l^J(t)$ satisfies the following condition
\begin{displaymath}
\sum_{l} q_l(t) \mu_l^J(t) \geq \frac{1}{d_I(J)} \max_{\vec{\eta}_J \in {\cal CH}(M_{J,{\cal K}})} \sum_{l} q_l(t) \eta_J(l).
\end{displaymath}
Let us the consider the following Lyapunov function,
\begin{equation}
V(\vec{q}(t)) = \sum_l q_l^2(t).
\end{equation}
Taking the derivate of the Lyapunov function, we have that
\begin{equation}
\dot{V}(\vec{q}(t)) \leq 2 \sum_l q_l(t) (\lambda_l) - \mu_l(t)).
\end{equation}
Using the GMS properties of $\mu_l(t),$ we have
\begin{equation}
\begin{split}
\dot{V}(\vec{q}(t)) \leq & \left( 2 \sum_l q_l(t) \lambda_l - \sum _J \frac{2}{d_I(J)} \pi(J) \max_{\vec{\eta}_J \in {\cal CH}(M_{J,{\cal K}})} \sum_{l} q_l(t) \eta_J(l) \right)
\end{split}
\end{equation}
As $\vec{\lambda}$ is assumed to lie inside the region $\Lambda_f(\vec{x})$, there exists $\vec{\eta}_J \in {\cal CH}(M_{J,{\cal K}})$ such that
\begin{equation}
\lambda_l < \sum_J \frac{1}{d_I(J)} \pi(J) \eta_J(l) .
\end{equation}
Using the above inequality, we have that
\begin{equation}
\begin{split}
\dot{V}(\vec{q}(t)) < & \left( 2 \sum_l q_l(t) \sum_J \frac{1}{d_I(J)} \pi(J) \eta_J(l) - \sum _J \frac{2}{d_I(J)} \pi(J) \max_{\vec{\eta}_J \in {\cal CH}(M_{J,{\cal K}})} \sum_{l} q_l(t) \eta_J(l) \right)
\end{split}
\end{equation}
Thus from the above inequality we have that $\dot{V}(q(t)) < 0$ whenever $q(t) > 0$.
We can now use the results from \cite{Dai_95} to argue that the original system is stable under the assumed arrival process as the fluid model is stable.
\end{IEEEproof}
}
\section{Main Results}
\label{sec:mainresults}
In this paper, we characterize the performance of GMS algorithm for
wireless networks with time-varying channels. We define the fading
local pooling factor, $\sigma_L^* (\boldsymbol{\pi})$, for a set of links $L
(\subseteq {\cal K})$ with fading structure $\boldsymbol{\pi}$ as follows:
\begin{equation}
\label{eq: sigmaL}
\sigma_L^* (\boldsymbol{\pi})= \textrm{inf} \{\sigma : \exists \,
\vec{\phi_1}, \vec{\phi_2} \in \Phi(L) \textrm{ such that } \sigma
\vec{\phi_1} \geq \vec{\phi_2} \},
\end{equation}
where,
\begin{equation}
\Phi(L) = \{\vec{\phi}: \vec{\phi}= \sum_{J: J \subseteq {\cal
K}}{\pi(J) \vec{\eta}_J} \textrm{ where } \vec{\eta}_J \in {\cal
CH}(M_{J \cap L,L})\},
\end{equation}
\noindent and \emph{Fading-Local Pooling Factor (F-LPF)} for a network
$G$, $\sigma_G^* (\boldsymbol{\pi})$, with fading structure $\boldsymbol{\pi}$ as follows:
\begin{equation}
\sigma_G^*(\boldsymbol{\pi}) = \displaystyle \textrm{min}_{L : L \subseteq
{\cal K}} \sigma_L^*(\boldsymbol{\pi}),
\end{equation}
Note that the above definition reduces to the known definition of LPF
for a graph \cite{JooLinShr_08} when there is no fading, i.e, when
$\pi({\cal K}) = 1$.
The F-LPF can be understood as follows: Consider arrivals only to links of
set $L$ (assume arrivals to other links are 0); when the links in set
$J$ are 'ON' (others are 'OFF'), GMS will pick a maximal schedule among
the 'ON' links, i.e. a column of $M_{J \cap L,L}.$ Thus vector
$\vec{\eta}_J$ is the long run average of these maximal schedules when
system is in state $J$; so $\vec{\eta}_J \in {\cal CH}(M_{J \cap
L,L}).$ Thus $\Phi(L)$ is the set of all long-run average service
vectors that could appear due to GMS when the arrivals are restricted
only to set of links in $L$. For any two vectors $\vec{\phi}_1,
\vec{\phi}_2 \in \Phi(L)$, it may thus happen that GMS results in
$\vec{\phi}_2$ service vector, when it should have been $\vec{\phi}_1$
(for the optimal case). Thus $\sigma_L^*(\boldsymbol{\pi})$ is the worst possible
ratio difference among all the possible service vectors of $\Phi(L)$.
\noindent \textbf{Dual Characterization and Implications:} In
the same spirit as \cite{DimWal_06,LiBoyXia_09}, the {\em Fading}-
Local Pooling Factor has a dual characterization, as noted in
Lemma~\ref{lemma: dual}, and displayed below. The F-LPF, $\sigma_L^*(\boldsymbol{\pi}),$ is given by the solution to the following optimization problem:
\begin{align}
&\sigma_L^*(\boldsymbol{\pi}) = \max_{x,a(J), b(J)} \displaystyle \sum_{J: J \subseteq L}
\pi_{L}(J) a(J) \label{eqn: dualchar}\\
\hbox{s.t : }
&\displaystyle x' M_{J,L} \geq a(J) e' \quad \forall J \subseteq L \nonumber \\
&\displaystyle x' M_{J,L} \leq b(J) e' \quad \forall J \subseteq L \nonumber \\
&\displaystyle\sum_{J: J \subseteq L} \pi_{L}(J) b(J) = 1 ,
\end{align}
where $e$ is a column vector of all ones, $(\cdot)'$ is the vector
transposition operation and $\boldsymbol{\pi}_L$ denotes the marginal distribution on set
of links $L$ induced by $\boldsymbol{\pi}$.
Observe that each fading state $J$ {\em induces} a network defined by
ON edges (i.e., all OFF links are removed from the network). Thus, one
could ask if with fading channels, the F-LPF can be determined simply
by computing the ``standard'' LPF (denoted by $\sigma^*(J)$) for each
of these induced networks, and then averaging these quantities
(weighted by the steady-state fractions of times for each of the
fading states) over all possible fading states? In other words,
\textit{is the following true}?
\begin{align*}
\sigma_L^*(\boldsymbol{\pi}) \stackrel{?}{=} \sum_{J: J \subseteq L} \pi_{L}(J) \sigma^*(J)
\end{align*}
where $\sigma^*(J)$ is the standard LPF \cite{JooLinShr_08} for the
network that is induced by state $J.$
An important insight that emerges from the dual characterization is
that such \textbf{averaging does necessarily not hold}, in particular
because the possibly adversarial nature of the fading channel does not
permit averaging. Note that the adversary {\em cannot} change the
long-term fractions of the global states -- it can merely change the
temporal correlations. Inspite of this, averaging does not hold, as
clearly shown in Example~B in Section~\ref{sec: appl}).
In a tree network with fading as in Example~B (see Section~\ref{sec:
appl}), while the LPF for each state is '1', the F-LPF is less than
$4/5$ which is lower than {\em any} convex averaging of the states!
This discussion implies that the regular LPF does not immediately
extend to the case with fading. This motivates us to explicitly
develop the local pooling factor in the presence fading, and
understand its implications.
\noindent \textbf{Contributions:}
\subsection{Characterization in terms of F-LPF:}
Our first contribution, Theorem~\ref{thm: flpf}, characterizes the efficiency ratio of GMS algorithm in the
presence of fading.
\begin{theorem}
\label{thm: flpf}
a) (Upper Bound) Under a given network topology and channel state
distribution with Assumption A1 on the arrivals and fading channels,
the efficiency ratio of GMS ($\gamma^*$) is less than or equal to
$\sigma_G^*(\boldsymbol{\pi})$.
b) (Achievability) Under a given network topology and channel state
distribution $\boldsymbol{\pi}$ { with Assumption A2} on the arrivals and
fading channels, the efficiency ratio of GMS ($\gamma^*$) is greater
than or equal to $\sigma_G^*(\boldsymbol{\pi})$.
\end{theorem}
\emph{Implications:} The above result enables us to understand the
performance of GMS compared to the optimal scheduler in the presence
of fading. In particular, computing bounds on $\sigma_G^*(\boldsymbol{\pi})$
leads to insights on the positive and negative aspects of fading
(discussed further in Theorems~\ref{thm: ub} and \ref{thm:
lb}). Observe first that as long as the long-term averages on the
arrivals and channels are satisfied (Assumption A1), we can construct
an arrival and channel process that ensures that the efficiency {\em
cannot} exceed the F-LPF $\sigma_G^*(\boldsymbol{\pi}).$ Further, for {\em
typical} arrival and channel processes with sufficient
randomness (in this paper i.i.d. assumptions have been imposed,
however this can be weakened), the converse holds wherein
$\sigma_G^*(\boldsymbol{\pi})$ is achievable.
\emph{Proof Discussion:} For the first part, we extend the ideas in
\cite{JooLinShr_08}, to construct an adversarial arrival and {\em
fading} process pattern when arrival rates are outside the
$(\sigma_G^*(\boldsymbol{\pi})+\epsilon) \Lambda_f$ and show that a set of
queues are unstable under GMS policy. For the second part, we use the
approach in \cite{DimWal_06,JooLinShr_08} as follows: we show that
if $\vec{\lambda}$ is inside $(\sigma_G^*(\boldsymbol{\pi})-\epsilon)
\Lambda_f$ then GMS policy can stabilize all the queues in the
network. We look at the deterministic fluid limit of the system and
exhibit a Lyapunov function whose drift is negative under the GMS
policy. We have that fluid model is stable and therefore that the
original system is stable.
\begin{theorem}[Upper Bound]
\label{thm: ub}
For every $J \subseteq {\cal K}$ and any $(\vec{\mu}_J, \vec{\nu}_J,
H_J)$ such that $\vec{\mu}_J, \vec{\nu}_J \in {\cal CH}(M_J)$,
$\vec{\nu}_J \leq H_J \vec{\mu}_J$, we have that
\begin{equation*}
\sigma_G^*(\boldsymbol{\pi}) \leq \textrm{max}_l \, \frac{\sum_{J \subseteq
{\cal K}} \pi(J) H_J \mu_J(l)}{\sum_{J \subseteq {\cal K}}
\pi(J) \mu_J(l)},
\end{equation*}
where $\mu_J(l) = 0 $ if $l \notin J.$
\end{theorem}
\emph{Implications:} While $\sigma_G^*(\boldsymbol{\pi})$ is defined only
though an optimization problem, the upper bound permits an explicit
solution. This bound is useful, as evidenced in Example~B provided in
Section~\ref{sec: appl}. In particular this upper bound is useful to
illustrate that the F-LPF is not a simple convex combination of the
standard LPF averaged over the fading states, and that adversarial
fading can indeed worsen the performance of GMS.
\emph{Proof Discussion:} Though the proof follows from straightforward
algebraic computations, the value of the theorem lies in the smart
selection of $(\vec{\mu}_J, \vec{\nu}_J, H_J)$ vectors that
satisfy the inequality stated in the above theorem. In the worst case
the bound yields 1; however we can use the existing results in
literature \cite{BirMarBer_10} to get good bounds. Thus, the tightness
of the upper bound depend up on the ability to identify good vectors
that satisfy the above constraints.
\begin{theorem}[Lower Bound]
\label{thm: lb}
\begin{equation}
\sigma_L^*(\boldsymbol{\pi}) \geq \frac{\sum_{J \subseteq L} \pi_L(J)
n(M_J)}{\sum_{J \subseteq L} \pi_L(J) N(M_J)},
\end{equation}
where $n(M) = \min_j \sum_i M_{ij}, N(M) = \max_j
\sum_i M_{ij} $. $\boldsymbol{\pi}_L$ denotes the marginal distribution on set
of links $L$ induced by $\boldsymbol{\pi}$ { and can be computed as follows,
$$ \boldsymbol{\pi}_L(J) = \sum_{I:I \subseteq {\cal K}, I \cap L = J} \boldsymbol{\pi}(I) $$. }
\end{theorem}
\emph{Implications:} The ability to compute a lower bound leads to the interesting
observation that fading can help \emph{improve} efficiency. This is because,
by turning links 'OFF', fading ``breaks up'' some of the bad global
states that can lead to poor GMS performance. This is explicitly
brought out in Example~A in the context of a six-link network.
\emph{Proof Discussion:} The lower bound is derived using the dual
formulation of the F-LPF, see (\ref{eqn: dualchar}). We find a point
in the dual search space that satisfies all the constraints in the
dual characterization, thus yielding a lower bound on the primal
problem. Observe that $n(M_J)$ corresponds to the minimum number of
links that needs to be 'ON' in any maximal schedule on set of $J$
links and $N(M_J)$ denotes the maximum number of links that could be
'ON' among all the maximal schedules on set of $J$ links. Thus, the
lower bound can be computed easily and can be shown to be tight for
some wireless networks. As an interesting aside, note that the lower
bound provided is always better than the inverse of the interference
degree of graph $G$ (see Corollary~1).
We now present two examples: A and B, one in which fading reduces the
relative performance of GMS and the other in which fading enhances the
relative performance of GMS respectively to illustrate the value of
the above results.
\subsection{Examples: Benefit and Detriment with Fading}
\label{sec: appl}
\begin{figure}[!]
\centering
\includegraphics[scale = 0.9]{examplemod1.eps}
\hspace{1.5in}\parbox{6in}{\caption{Interference graphs for the two example networks} }
\label{fig:examples}
\end{figure}
\noindent {\bf Example A: A network where fading structure improves
the relative performance of GMS:} Consider a graph with six links
${\cal K} = \{a,b,c,d,e,f\}$. The interference graph for the six links
is shown in the { Figure \ref{fig:examples}}. Each link is either is state 'ON' or
'OFF'. We consider the following fading structure, $\boldsymbol{\pi}$, for $J
\subseteq {\cal K}$
\begin{equation*}
\pi(J) = p^{|J|} (1-p)^{6-|J|},
\end{equation*}
where $|J|$ denotes the size of set $J$. Note that $p = 1$ corresponds
to the no-fading case.
Using our results, we compute the lower bound and upper bounds on
local pooling factor $\sigma_G^*(\boldsymbol{\pi})$ {and is plotted in Figure \ref{fig:bounds}}.
\begin{figure}[h!]
\centering
\includegraphics[width = 134mm, height = 88mm]{plot3.eps}
\centering{\hspace{1.5in}\parbox{6in}{\caption{Bounds on the fading local pooling factor for the Hexagon network}}}
\label{fig:bounds}
\end{figure}
It is known \cite{JooLinShr_08} that the non-fading LPF for the above
example is equal to 2/3. From the graph, we observe that for smaller
values of $p$, F-LPF for above hexagon network
with fading is greater than LPF with out fading
structure. As p tends to zero, the fraction of time network remains a
cycle also tends to be small and it is known that GMS is optimal for
tree networks. Therefore, it fits well with intuition to see that
fading enhances the F-LPF for graphs with cycles.
\noindent {\bf Example B: A network where fading structure worsens
the relative performance of GMS:}
Consider the graph with 3 links $a,b,c$ as shown above. The
interference sets for each link is: ${\cal I}_a = \{b\}, {\cal I}_b =
\{a,c\} \textrm{and } {\cal I}_c = \{b\}.$ We assume each link is
either in state 'ON'(1) or 'OFF'(0). So the global channel state
$'110'$ denotes that link $a$ and $b$ are in 'ON' state and link $c$
is in 'OFF' state. The fading structure is defined as follows:
$\pi('110') = \pi('011') = \pi('111') = 1/3.$
For each global channel state, the possible maximal independent sets are as follows:
\[ M_{ab, abc} = \left( \begin{array}{cc}
1 & 0 \\
0 & 1 \\
0 & 0 \end{array} \right) \]
and
\[ M_{bc, abc} = \left( \begin{array}{cc}
0 & 0 \\
1 & 0 \\
0 & 1 \end{array} \right) \]
and
\[ M_{abc} = \left( \begin{array}{cc}
1 & 0 \\
0 & 1 \\
1 & 0 \end{array} \right) \]
Any vector that belongs to $\Phi(\{abc\})$ can be represented as follows,
\begin{equation}
\vec{\phi} = \frac{1}{3} M_{ab} [\alpha \, 1-\alpha]' + \frac{1}{3} M_{bc} [\beta \, 1-\beta]' + \frac{1}{3} M_{abc}[\gamma \, 1- \gamma]'.
\end{equation}
Let $\vec{\phi_1}$ be obtained using $(\alpha, \beta, \gamma) =
(1,0,0)$ and $\vec{\phi_2}$ be obtained using $(\alpha, \beta,
\gamma) = (1/2,1/2,3/4)$. Evaluating the above expression using the
above values, we have $\vec{\phi_1} = \frac{1}{3} [1 \,1 \, 1]'$ and
$\vec{\phi_2} = \frac{5}{12} [1 \, 1 \,1]'$. Observing the fact that
$\frac{4}{5} \vec{\phi_2} = \vec{\phi_1}$, using Theorem~\ref{thm: ub}, we have that local pooling factor for the wireless network with the above fading structure is less than or equal to $\frac{4}{5}$. But, it is known that the local pooling factor of GMS for tree networks (with no fading) is 1.
This result though sounds counter-intuitive, stems from the fact that we allow the fading to be arbitrary. Thus fading can act as adversary and as demonstrated, can degrade the performance of GMS algorithm.
{
\subsection{Characterization in terms of Interference degree}
So far, we have characterized the performance of GMS through a single
scaling factor of the entire throughput region. Note that each fading
state $J$ induces a network defined on the set of edges that are in
'ON' state and GMS can stabilize the network if arrivals are inside
the region $\sigma^*(J)\Lambda_J$. It is natural to ask for the fading
scenario, i.e. network with distribution $\pi(J),$ \emph{if GMS could
stabilize the region} $\sum_J \pi(J) \sigma^*(J) \Lambda_J?$ We
answer the above question in two parts.
In the first part, we show the interesting result that \emph{GMS
cannot stabilize} the above averaged region. In other words, there
exists an arrival process with rate outside the region
$\Lambda_f(\vec{x})$ for $x(J) = \sigma^*(J)$ (standard LPF) that can
make the network unstable under GMS algorithm. We illustrate this
using a simple example described below.
\noindent {\bf Counter Example:} Consider the network with 3 nodes as
in Example B. Note that the standard LPF \cite{BirMarBer_10} for all
the three fading states is $1$. Thus the region
$\Lambda_f(\sigma^*(J))$ is exactly same as the actual throughput
region $\Lambda_f$. However, we have shown earlier that F-LPF is
strictly less than $0.8$. Thus there exists an arrival process with
rates outside the region $0.8 \Lambda_f$ that cannot be stabilized by
the greedy maximal schedule.
Given the previous negative result, in the second part we show that
GMS can stabilize the region $\Lambda_f(\frac{1}{d_I(J)})$. Note that
this region is strictly inside the region $\Lambda_f(\vec{x})$ with
$x(J) = \sigma^*(J)$. More formally, our result is as follows:
\begin{theorem}
Under a given network topology and channel state distribution with
Assumption A1 on the arrivals and fading channels, GMS can stabilize
the network if the arrival rates are inside the region
$\Lambda_f(\vec{x})$, where $x(S) = \frac{1}{d_I(S)}$.
\end{theorem}
\emph{Implications:} The above theorem provides an elegant
characterization of the rate region that can be stabilizable by the
GMS algorithm. Also, we find that that the above region is \emph{not a
subset} of the achievable region stated in Theorem 1b (i.e
$\sigma_G^*(\boldsymbol{\pi}) \Lambda_f$). We illustrate the above
observation through a simple example described below.
Consider the wireless network with 3 nodes and fading distribution
similar to example B. Note that the interference degree for fading
state $'110'$ is $d_I('110') = 1$, for state $'011'$ is $d_I('011') =
1$ and for the fading state $'111'$ is $d_I('111') = 0.5.$ Any arrival
rate vector that belongs to the new region defined using the
interference degree can be expressed as below,
\begin{equation}
\vec{\lambda} = \frac{1}{3} M_{ab} [\alpha \, 1-\alpha]' + \frac{1}{3} M_{bc} [\beta \, 1-\beta]' + \frac{1}{3} \frac{1}{2} M_{abc}[\gamma \, 1- \gamma]',
\end{equation}
where $\alpha, \beta \, \textrm{and} \, \gamma$ are positive constants
that are bounded by $1$. Using $(\alpha, \beta, \gamma) = (0,1,0)$, we
have that rate vector $(0,\frac{5}{6},0)$ is inside the new region
characterized by the interference degree. However, note that we have
shown the F-LPF is upper bounded by $\frac{4}{5}$ for example B
network. Thus, all arrival rates that are inside the region
$\frac{4}{5} \Lambda_f$ satisfy the constraint that $\lambda_2 <
\frac{4}{5}$ and hence rate vector $(0, \frac{5}{6}, 0)$ belongs to
the new region and not the region characterized by F-LPF.
\emph{Proof Discussion:} We consider the continuous time model with
deterministic arrival and channel state processes. We then exhibit a
Lyapunov function, sum of squares of queue lengths, whose derivative
is strictly less than zero under the GMS policy whenever the arrival
rate is strictly inside the new region. Therefore, the fluid model is
stable and thus using the results from \cite{Dai_95} we conclude that
the original network model is stable. }
\section{System Model and Back Ground}
\label{sec:model}
We consider a wireless network consisting of $K$ links labeled as
$\{1,2,3,...,K\}$. Let ${\cal K} $ denote the set of links in the
network. Each link $l$ consists of a transmitter and receiver. We
assume time to be slotted. Each time slot is composed of two parts.
The first (control) part is reserved for making the transmission
decision and second part for transmitting the packet. At time slot
$t$, we denote the channel capacity of link by $C_l[t].$ We assume
that the capacity varies from slot to slot, and is constant during a
time slot. We consider collision interference/protocol model and
denote the set of links that interfere with link $l$ by ${\cal I}_l$. We
say that the transmission on link $l$ at time $t$ is successful, if no
link in the ${\cal_I}_l$ transmits during the same time $t$. The
maximum number of packets that can be successfully transmitted in time
slot $t$ on link $l$ is bounded by $C_l[t]$.
We assume single hop flows in the network. Let $A_l[t]$ denote the
number of packets that arrive at transmitter of link $l$ at time slot
$t$. We assume that arrival processes is bounded and average rate of
arrivals for link $l$ is denoted by $\lambda_l.$
For simplicity we first consider ON/OFF channels (i.e $C_l[t] = 0
\,\textrm{or}\, 1 $) and later show that our results can be extended
to channels with finite number of channel states. For the ON/OFF
setting, global state (GS) refers to specifying the set of links that
are in 'ON' state. Let $GS(t)$ denote the set of links that are in
'ON' state in time slot $t$. Let $\pi(J)$ denote the fraction of time
the network is in global channel state $J$, where links in set $J$ are
'ON' and links in the set ${\cal K} \backslash J$ are in 'OFF' state.
Let $\boldsymbol{\pi}$ := $\{\pi(J), J \subset {\cal K}\}$ denote the \emph{fading
structure}.
\begin{assumption}
: \\
\emph{A1 (Long-term Averages):} We assume that the long-term time
averages of arrivals and channel states satisfy the following:
\begin{equation} \frac{1}{T} \sum_{t = 0}^T A_l[t] \to \lambda_l
\quad \textrm{as} \quad T \to \infty. \end{equation} and
\begin{equation} \frac{1}{T} \sum_{t =0}^T {\bf 1}_{GS(t) = J} \to
\pi(J) \quad \textrm{as} \quad T \to \infty. \end{equation}
\emph{ A2 (Randomness):} We assume that arrivals are mutually
independent i.i.d processes with $\lambda_l = E[A_l[t]]$. Similarly
the channels are independent across time and form a stationary
process with $\pi(J) = E[1_{GS(t) = J}]$.
\end{assumption}
While both assumptions A1 and A2 specify the same long-term averages,
we note that assumptions in A1 allow for arrival and channel state
processes to be \emph{dependent across time and across links} in a
deterministic, and possibly {\em adversarial manner}. The necessity
for the above sets of assumptions will be clear as we state our main
results in Section \ref{sec:mainresults}.
\subsection{Preliminaries}
As discussed earlier, there is a rich history of analysis of GMS
algorithms for the non-fading case
\cite{DimWal_06,JooLinShr_08,LiBoyXia_09,LiRoh_10,BirMarBer_10,LonEytCha_10}. In
this section we build on this notation in literature to allow for
time-varying (fading) channels.
We define Interference graph ${\cal IG}$ for a set of links as
follows: Each link is represented by a node and an edge is drawn
between two nodes if transmissions on the corresponding links in the
original graph interfere with each other. This model captures many
existing wireless models and is quite general. We define the
Independent set on this graph as set of nodes with no edges between
them. Let $Q_l[t]$ denote the number of packets present at the
transmitter at time $t$ waiting to get scheduled on link $l$. Let
$S_l[t] \in \{0,1\}$ denote the schedule decision for link $l$ at time
$t$. At each time $t$, a schedule $\vec{S}[t]$ is determined based on
the global queue state and channel state information at time $t$, that
is $(\vec{Q}[t]), \vec{C}[t])$. We also assume that arrivals occur at
the end of time slot, thus we have the following queue dynamics:
\begin{equation}
Q_l[t+1] = (Q_l[t] - C_l[t] S_l[t])^+ + A_l[t],
\end{equation}
\noindent where $a^+ = \textrm{max}(0, a)$.
Given the arrival traffic rate $\{\lambda_l\}_{l\in \cal{L}}$ and a
scheduling policy, we say that the network is \emph{stable} under
scheduling policy if the mean of the sum of queue lengths is bounded.
We say that an arrival rate vector $\{\lambda_l\}_{l\in \cal{L}}$ is
\emph{supportable} if there exists any scheduling policy that can make
the network stable. We call the set of all arrival vectors that are
supportable by $\emph{throughput region}$ and denote it as
$\Lambda_f$, where $f$ denotes that the channels are fading.
We say that a scheduling policy is throughput optimal if it can
stabilize the network for all arrival rates inside the throughput
region.
\emph{Definition 1:} (\cite{JooLinShr_08}) The interference degree
$d_I(l)$ of link $l$ is the maximum number of links in the set $\{l
\cup {\cal I}_l\} $that can be active at the same time with out
interfering with each other. The interference degree $d_I(G)$ of a
graph $G = \{V, E\}$ is the maximum interference degree across all its
links in $E$
Consider a wireless system with 4 links. Let ${\cal I}_1 = \{2\}$,
${\cal I}_2 = \{1,3,4\}$, ${\cal I}_3 = \{2,4\}$ and ${\cal I}_4 =
\{2,3\}$. The interference graph is shown in the {\color{black}Figure \ref{fig:intgraph} }with the
corresponding $d_I(l)$. The interference degree of this example graph
is 2.
\begin{figure}[h!]
\centering
\includegraphics[scale = 1]{figexam2.eps}
\hspace{1.5in}\caption{Interference Graph where nodes denote the links and edges
denote the interference constraints.}
\label{fig:intgraph}
\end{figure}
\emph{Definition 2:} Given an interference graph, an independent set
corresponds to set of nodes (links in the original graph) such that
there is no edge between any two nodes in the set (no two links
interfere in the original graph). Further, it is maximal if it is not
a subset of any other independent set. For a set of links $L$, define
a matrix $M_L$ whose columns represent the maximal independent sets on
the set $L$, with $|L|$ rows one for each link. We assume links are
naturally ordered and rows in $M_L$ are assigned according to the
defined order. For $J \subset L$, let $M_{J,L}$ denote the matrix with
$|L|$ rows and is constructed from $M_J$ as follows: columns from
$M_J$ are used and zero row vectors are added for links which do not
belong to set $J.$ Let ${\cal CH}(M_{J,L})$ denote the convex hull of
all column vectors of matrix $M_{J,L}.$
For the above example with 4 links, let $J = \{1,2,3\}$ and $L = \{1,2,3,4\}$, we have
\[ M_{J} = \left( \begin{array}{cc}
1 & 0 \\
0 & 1 \\
1 & 0 \end{array} \right) \]
and
\[ M_{J,L} = \left( \begin{array}{cc}
1 & 0 \\
0 & 1 \\
1 & 0 \\
0 & 0 \end{array} \right) \]
Note that the set $ \Lambda_L := \{\vec{\lambda}: \vec{\lambda} < \vec{\mu}; \vec{\mu} \in {\cal CH}(M_L)\}$ characterizes the throughput region of set of $L$ links if no fading were present. We now define the throughput region with the \emph{fading structure},
\emph{Definition 3:} The throughput region $\Lambda_f$ for a given
network with fading pattern $\pi(J)$ is described as follows,
\begin{eqnarray*}
\Lambda_f = \Big\{ \vec{\lambda}: \vec{\lambda} > 0, \vec{\lambda} \,\leq \, \sum_J \, \pi(J) \vec{\eta}_J \, \textrm{where} \, \, \vec{\eta}_J \in {\cal CH}(M_{J,{\cal K}}) \Big\}.
\end{eqnarray*}
\emph{Definition 4:} (\cite{JooLinShr_08}) The efficiency ratio
$\gamma_{pol}^*$ under a given scheduling policy is defined as follows,
\begin{eqnarray*}
\gamma_{pol}^* = \textrm{sup} \Big\{\gamma : \textrm{the policy can stabilize for all} \textrm{the arrival rate vectors} \, \lambda \in \gamma \Lambda_f \Big\}.
\end{eqnarray*}
{
\emph{Definition 5:} Given $x(J) \in [0,1]$, we define a new region $\Lambda_f(\vec{x})$ as follows,
\begin{eqnarray*}
\Lambda_f(\vec{x}) = \Big\{\vec{\lambda}: \vec{\lambda} > 0, \vec{\lambda} \,\leq \, \sum_J \, x(J) \pi(J) \vec{\eta}_J \, \textrm{where}
\, \, \vec{\eta}_J \in {\cal CH}(M_{J,{\cal K}}) \Big\}.
\end{eqnarray*}
Note that throughput region is same as $\Lambda_f(1)$.
}
\subsection{GMS Algorithm \cite{Mck_95}}
We now describe the Greedy Maximal Scheduling(GMS) Algorithm. GMS
essentially finds a maximal schedule in a greedy fashion. Each node in
the interference graph is assigned weight equal to $f(Q_l(t) C_l(t))$,
where $f(.)$ is a strictly increasing function that is zero at $0$ and
tends to infinity as $Q_l(t)C_l(t) \to \infty.$ It then proceeds as
follows: it finds the node with maximum weight in the whole network
and adds it to GMS schedule (ties are broken arbitrarily), it further discards all the neighboring
nodes along with the selected node and repeats the above procedure on
the reduced graph, till there are no more nodes left in the
interference graph.
|
{
"timestamp": "2012-03-12T01:00:45",
"yymm": "1203",
"arxiv_id": "1203.1997",
"language": "en",
"url": "https://arxiv.org/abs/1203.1997"
}
|
\section{Introduction} \label{intro}
Currently there is a large body of data coming from cosmological and astrophysical observations that is mostly consistent with the existence of dark matter. Such observations also suggest that the hypothesized particles that constitute dark matter have very small cross section and typically travel much slower than light. These lead to the cold dark matter (CDM) framework, which is one of the pillars of the current standard cosmological model $\Lambda$CDM.
It is not only tempting, but mandatory to check if such dark matter particles exist (by detecting them in laboratory based experiments, for instance) and also to check if the gravitational effects that lead to the dark matter hypothesis could follow from a more detailed and complete approach to gravity. The effects of pure classical General Relativity at galaxies have been studied for a long time and, considering galaxy kinematics, the differences between General Relativity and Newtonian gravity are negligible (for a dispute on the latter, see however Ref. \cite{Cooperstock:2006dt}).
One of the subjects which attract significant attention currently is modifications of General Relativity in order to replace either dark matter or dark energy. When all the astrophysical and cosmological data are considered, there is currently no particular model in this class that was proved to be as successful as the $\Lambda$CDM model, however diverse nontrivial achievements were accomplished by such new approaches to gravity; whilst the $\Lambda$CDM model has its own difficulties and problems, particularly at the galactic scale\cite{Gentile:2004tb, 2009NJPh...11j5029P, 2010Natur.465..565P, Kroupa:2012qj}. Probably the most well known success of the modified gravity approach relies on galaxy kinematics, where MOND\cite{1983ApJ...270..365M,1983ApJ...270..371M,Milgrom:2003ui} is largely the most cited example and achieved success in some areas where the $\Lambda$CDM results are at least unclear \cite{2012AJ....143...40M, 2007A&A...472L..25G}. There are examples in the cosmological realm as well. In particular, considering the evolution of linear perturbations and its comparison to the CMB and the LSS data, results either close or identical to $\Lambda$CDM results can be found in different scenarios \cite{Banados:2008fj, Batista:2011nu, Moffat:2009cv, Skordis:2005xk}. The Bullet Cluster \cite{Clowe:2006eq} was some times cited as a definitive proof on the existence of dark matter, however this system can also be modeled from a modified gravity perspective \cite{Brownstein:2007sr, Dai:2008sf}. These phenomenological results motivate the use of gravitational theories that are different from General Relativity in its standard form.
Independent on whether gravity should be or should not be quantized, we know that the matter fields should. It is well known that the renormalization group
can be extended to quantum field theory (QFT) on curved space time (e.g., Refs. \cite{Buchbinder:1992rb,Shapiro:2008sf,Shapiro:2009dh}). In particular, concerning
the high energy (UV) behavior, there is hope that the running of $G$ may converge to a non-Gaussian fixed point in accordance with the asymptotic safety approach \cite{Niedermaier:2006wt, Weinberg:2009wa}. Our present concern is, however, not about the UV completeness, but with the behavior of $G$ in the far infrared (IR) regime.
In the far IR regime of quantum electrodynamics, one finds classical electromagnetism, and hence no renormalization group running of its coupling constant. This behavior is in accordance with the Appelquist-Carazzone decoupling \cite{Appelquist:1974tg} (see Ref.\cite{Goncalves:2009sk} for a recent derivation). In the case of gravity (in the context of QFT in curved space time) the same effect of decoupling has been obtained for the higher derivative terms in the gravitational action \cite{Gorbar:2002pw,Gorbar:2003yt}. Currently, it remains unclear whether the Einstein-Hilbert action coupling parameter behaves as a constant at the far IR or not. Since in pure theoretical grounds with no additional hypothesis it is hard to advance in this direction, this possibility of General Relativity deviation has been developed on different grounds a number of times before, e.g. \cite{Goldman:1992qs, Bertolami:1993mh, Dalvit:1994gf, Bertolami:1995rt, Shapiro:2004ch, Reuter:2004nx, Reuter:2007de}.
In \cite{Rodrigues:2009vf} we presented new results on the application of renormalization group corrections to General Relativity in the astrophysical domain. Previous attempts to apply this picture to galaxies have considered for simplicity point-like galaxies (e.g., \cite{Shapiro:2004ch, Reuter:2004nx}). We extended previous considerations by identifying a proper renormalization group energy scale $\mu$ and by evaluating the consequences considering the observational data of disk galaxies. We proposed the existence of a relation between $\mu$ and the local value of the Newtonian potential (this relation was reinforced afterwards from a different approach, by using a scale setting formalism \cite{Domazet:2010bk}). With this choice, the renormalization group-based approach (RGGR) was capable to mimic dark matter effects with great precision. Also, it is remarkable that this picture induces a very small variation on the gravitational coupling parameter $G$, namely a variation of about $10^{-7}$ of its value across a galaxy (depending on the matter distribution). We call our model RGGR, in reference to renormalization group effects in General Relativity.
The main purpose of this work is to extend the analysis of \cite{Rodrigues:2009vf} (see also \cite{Shapiro:2004ch, Farina:2011me, Rodrigues:2011cq},\cite{Fabris:2012wg}) towards elliptical galaxies, testing the RGGR approach in this context. This extension is also of value to future RGGR applications, for instance in cluster of galaxies. Many of the elliptical galaxies behave as spherically symmetric stable systems which are mainly supported by velocity dispersions, in sharp contrast with the disk galaxies, which are axially symmetric and are mainly supported by rotation velocity. To this end, we deduce the additional effective mass introduced by RGGR, detail some aspects of its general behavior on elliptical galaxies and present a detailed fitting using recent observational data of the galaxies NGC 4374 (giant elliptical) and NGC 4494 (ordinary elliptical). Moreover we use exactly the same data of NGC 4374 to evaluate MOND and compare its results with the RGGR ones.\footnote{We do not do the MOND analysis for NGC 4494 since it was previously analyzed in Ref. \cite{Milgrom:2003ui} and this galaxy does not constitute a hard test for MOND.} The numerical evaluation of the velocity dispersion (VD) curves and related procedures use a program made by us on Wolfram Mathematica.
\section{A brief review on RGGR}
The gravitational coupling parameter $G$ may behave as a true constant in the far IR limit, leading to standard General Relativity in such limit. Nevertheless, in the context of QFT in cuved space time, there is no proof on that. According to Refs. \cite{Shapiro:2004ch, Farina:2011me}, a certain logarithmic running of $G$ is a direct consequence of covariance and must hold in all loop orders. Hence the situation is as follows: either there is no new gravitational effect induced by the renormalization group in the far infrared, or there are such deviations and the gravitational coupling runs as
\begin{equation}
\beta_{G^{-1}} \equiv \mu \frac{dG^{-1}}{d \mu} = 2 \nu \, \frac{M_{\mbox{\tiny Planck}}^{2}}{c \, \hbar} = 2 \nu G_0^{-1}.
\label{betaG}
\end{equation}
Equation (\ref{betaG}) leads to the logarithmically varying $G(\mu)$ function,
\begin{equation}
\label{gmu}
G(\mu) = \frac {G_0}{ 1 + \nu \,\mbox{ln}\,(\mu^2/\mu_0^2)},
\end{equation}
where $\mu_0$ is a reference scale introduced such that $G(\mu_0) =G_0 $. The constant $G_0$ is the gravitational constant as measured in the Solar System (actually, there is no need to be very precise on where $G$ assumes the value of $G_0$, due to the smallness of the variation of $G$). The dimensionless constant $\nu$ is a phenomenological parameter which depends on the details of the quantum theory leading to eq. (\ref{gmu}). Since we have no means to compute the latter from first principles, its value should be fixed from observations. Even a small $\nu$ of about $\sim 10^{-7}$ can lead to observational consequences at galactic scales. Note that the first possibility, namely of no new gravitational effects in the far infrared, corresponds to $\nu=0$.
The action for this model is simply the Einstein-Hilbert one in which $G$ appears inside the integral, namely,\footnote{We use the $(- + + +)$ space-time signature.}
\begin{equation}
S_{\mbox{\tiny RGGR}}[g] = \frac {c^3}{16 \pi }\int \frac {R } G \, \sqrt{-g} \, d^4x.
\label{rggraction}
\end{equation}
In the above, $G$ is an external scalar field, it satisfies (\ref{gmu}). For a complete cosmological picture, $\Lambda$ is necessary and it also runs covariantly with the RG flow of $G$ \cite{Shapiro:2004ch, Reuter:2007de, Koch:2010nn}. In the above, the $\Lambda$ term was not written since its role is expected to be negligible for the galaxy internal dynamics, and it will not be used through the main part of this paper. In the Appendix \ref{appendixA} we numerically solve the variation of $\Lambda$ inside the galaxies that are analyzed in this paper (NGC 4494 and NGC 4374) and show that albeit the value of $\Lambda$ can increase many times inside a galaxy, it is far from sufficient to lead to any significative observational effect. To our knowledge, this is the first time that the variation of $\Lambda$ in galaxies was directly evaluated in the context of renormalization group effects.
There is a simple procedure to map the solutions from the Einstein equations with the gravitational constant $G_0$ into RGGR solutions. In this review, we will proceed to find RGGR solutions via a conformal transformation of the Einstein-Hilbert action, and to this end first we write
\begin{equation}
G = G_0 + \delta G,
\end{equation}
and we assume $\delta G / G_0 \ll 1$, which will be justified latter. Introducing the conformally related metric
\begin{equation}
\bar g_{\mu \nu} \equiv \frac {G_0}{G} g_{\mu \nu},
\label{ct}
\end{equation}
the RGGR action can be written as
\begin{equation}
S_{\mbox{\tiny RGGR}}[g] = S_{\mbox{\tiny EH}}[\bar g] + O(\delta G^2),
\end{equation}
where $S_{\mbox{\tiny EH}}$ is the Einstein-Hilbert action with $G_0$ as the gravitational constant. The above suggest that the RGGR solutions can be generated from the Einstein equations solutions via the conformal transformation (\ref{ct}). Indeed, within a good approximation, one can check that this relation persists when comparing the RGGR equations of motion to the Einstein equations even in the presence of matter \cite{Rodrigues:2009vf}.
In the context of galaxy kinematics, standard General Relativity gives essentially the same predictions of Newtonian gravity. In the weak field limit and for velocities much lower than that of light, the gravitational dynamics can be derived from the Newtonian potential, which is related to the metric by
\begin{equation}
\bar g_{00} = - \left ( 1 + \frac {2 \Phi_N}{c^2} \right ).
\end{equation}
Hence, using eq. (\ref{ct}), the effective RGGR potential $\Phi$ is given by
\begin{equation}
\Phi = \Phi_N + \frac {c^2}2 \frac{\delta G}{G_0}.
\label{PhiRGGR}
\end{equation}
An equivalent result can also be found from the geodesics of a test particle\cite{Rodrigues:2009vf}. For weak gravitational fields $\Phi_N/ c^2 \ll 1$ (with $\Phi_N = 0$ at spatial infinity), hence even if $\delta G/G_0 \ll 1$ eq. (\ref{PhiRGGR}) can lead to a significant departure from Newtonian gravity.
In order to derive a test particle acceleration, we have to specify the proper energy scale $\mu$ for the problem setting in question, which is a time-independent gravitational phenomena in the weak field limit. This is a recent area of exploration of the renormalization group application, where the usual procedures for high energy scattering of particles cannot be applied straightforwardly. Previously to \cite{Rodrigues:2009vf} the selection of $\mu \propto 1/r$, where $r$ is the distance from a massive point, was repeatedly used, e.g. \cite{Reuter:2004nv,Dalvit:1994gf,Bertolami:1993mh,Goldman:1992qs, Shapiro:2004ch}. This identification adds a constant velocity proportional to $\nu$ to any rotation curve. Although it was pointed as an advantage due to the generation of ``flat rotation curves'' for galaxies, it introduced difficulties with the Tully-Fisher law \cite{Tully:1977fu}, the Newtonian limit, and the behavior of the galaxy rotation curve close to the galactic center, since there the behavior is closer to the expected one without dark matter. In \cite{Rodrigues:2009vf} we introduced a $\mu$ identification that seems better justified both from the theoretical and observational points of view. The characteristic weak-field gravitational energy scale does not comes from the geometric scaling $1/r$, but should be found from the Newtonian potential $\Phi_N$, the latter is the field that characterizes gravity in such limit. Therefore,
\begin{equation}
\frac{\mu}{\mu_0} = f\( \frac{\Phi_N}{\Phi_0}\).
\label{murggr}
\end{equation}
If $f$ would be a complicated function with dependence on diverse constants, that would lead to a theory with small (or null) prediction power. The simplest assumption, $ \mu \propto \Phi_N$, leads to $\mu \propto 1/r$ in the large $r$ limit; which is unsatisfactory on observational grounds (bad Newtonian limit and correspondence to the Tully-Fisher law). One way to recover the Newtonian limit is to impose a suitable cut-off, but this rough procedure does not solves the Tully-Fisher issues \cite{Shapiro:2004ch}. Another one is to use \cite{Rodrigues:2009vf}
\begin{equation}
\frac {\mu}{\mu_0} =\left( \frac{\Phi_N}{\Phi_0} \right)^\alpha,
\label{muphi}
\end{equation}
where $\Phi_0$ and $\alpha$ are constants. Apart from the condition $ \Phi_0 < 0$, in order to guarantee $\delta G/G_0 \ll 1$, the precise value of $\Phi_0$ is largely irrelevant for the dynamics, since $\Phi'(r)$ does not depends on $\Phi_0$. The relevant parameter is $\alpha$, which will be commented below. The above energy scale setting (\ref{muphi}) was recently re-obtained from a more fundamental perspective \cite{Domazet:2010bk}, where a renormalization group scale-setting formalism is employed.
The parameter $\alpha$ is a phenomenological parameter that needs to depend on the mass of the system, and it must go to zero when the mass of the system goes to zero. This is necessary to have a good Newtonian limit. From the Tully-Fisher law, it is expected to increase monotonically with the increase of the mass of disk galaxies. In a recent paper, an upper bound on $\nu \alpha$ in the Solar System was derived \cite{Farina:2011me}. In galaxy systems, $\nu \alpha|_{\mbox{\tiny Galaxy}} \sim 10^{-7}$, while for the Solar System, whose mass is about $10^{-10}$ of that of a galaxy, $\nu \alpha|_{\mbox{\tiny Solar System}} \lesssim 10^{-17}$. It shows that a linear increase on $ \alpha$ with the mass (ignoring possible dependences on the mass distribution) is sufficient to satisfy both the current upper bound from the Solar System and the results from galaxies. Actually, in Sec. \ref{Sec.RoleOfAlpha} it is shown that a close-to-linear dependence on the mass can also be found for elliptical galaxies by using the fundamental plane.
Once the $\mu$ identification is set, it is straightforward to find the rotation velocity for a static gravitational system sustained by its centripetal acceleration \cite{Rodrigues:2009vf},
\begin{equation}
V^2_{\mbox{\tiny RGGR}} \approx V^2_N \left ( 1 - \frac {\nu \, \alpha \, c^2} {\Phi_N} \right ).
\label{v2rggr}
\end{equation}
Contrary to Newtonian gravity, the value of the Newtonian
potential at a given point does play a significant role
in this approach. This sounds odd from the perspective of
Newtonian gravity, but this is not so
from the General Relativity viewpoint, since the latter has no
free zero point of energy. In particular, the Schwarzschild
solution is not invariant under a constant shift of the
potential.
Equation (\ref{v2rggr}) was essential for the derivation of galaxy rotation curves. Since elliptical galaxies are mainly supported by velocity dispersions (VD), the main equation for galaxy kinematics in this case will not be eq.(\ref{v2rggr}), but eq.(\ref{sigma_pK}) with the mass $M(r)$ given by eq. (\ref{MRGGR}).
\section{Mass modeling of elliptical galaxies}
While disk galaxies are extended gravitational systems mainly supported by rotation, elliptical galaxies are mainly supported by velocity dispersions (VD). There is good evidence that the elliptical galaxies dealt in this paper (and many others) are close to spherical systems, and we will consider this approximation in this paper.
For stationary spherical systems without rotation, the mean velocity at a small cell centered at position $\mathbf{r}$ is $\< \mathbf{v} \>(\mathbf{r}) = \mathbf{0}$, while the mean square velocity satisfies $\< v^2_\theta\> = \< v^2_\varphi\> $, where $\theta$ and $\varphi$ refer to the angular components of spherical coordinates. For such systems, from the colissionless Boltzman equation one derives the following Jeans equation \cite{0691084459},
\begin{equation}
\frac1{\ell} \frac{{\partial}}{{\partial} r} \( \ell \sigma^2_r\) + \frac 2 r \beta \sigma^2_r = -\frac{{\partial} \Phi}{{\partial} r}.
\label{jeans}
\end{equation}
In the above, $\sigma^2_r (r)\equiv \<v^2_r \>(r) - \< v_r\>^2(r) = \<v^2_r \>(r) $ is the VD radial component at the radius r, $\ell$ is the local luminosity density, $\beta(r) \equiv 1 - \sigma^2_\theta(r) / \sigma^2_r(r)$ is the local anisotropy and $\Phi$ is the potential of the total force per mass that acts in the cell centered at $\mathbf{r}$ (i.e., $- \mathbf{\nabla} \Phi(r) = \ddot{\mathbf{r}}$). In the absence of dark matter, and considering Newtonian gravity, $\Phi$ would be the gravitational potential generated by the stars alone (we consider only elliptical galaxies with negligible amount of gas).
It is a well known theorem of Newtonian gravity that for spherical systems
\begin{equation}
\frac{{\partial} \Phi}{{\partial} r} = \frac{G_0 M(r)}{r^2},
\end{equation}
where $G_0$ is the Newton's constant and $M(r)$ the total mass inside the radius $r$. Therefore the effect of a spherical dark matter profile is to replace the total mass from the stellar one $M_*(r)$ to $M(r) = M_*(r) + M_{dm}(r)$. Hence, from the knowledge of $M_*(r)$, $M_{dm}(r)$, $\ell(r)$ and $\beta(r)$, one can solve the Jeans eq. (\ref{jeans}) to find the radial VD contribution induced by the stellar mass ($\sigma^2_{r *}$), the radial VD contribution from the dark matter ($\sigma^2_{r \,dm}$), and the total radial VD, which satisfies
\begin{equation}
\sigma^2_{r } = \sigma^2_{r *}+\sigma^2_{r \, dm}.
\label{sigmardecomp}
\end{equation}
The above decomposition of $\sigma_r$ is a consequence of $\Phi$ being linear on the mass and the left hand side of eq. (\ref{jeans}) being linear on $\sigma_r^2$. The case of non-Newtonian gravity will be commented latter.
Astrophysical observations cannot measure $\sigma_r^2$ since the VD seen from the Solar System is not the radial, but the line of sight VD (which we will also call ``projected VD''). From straightforward geometrical considerations, the observed projected VD can be computed from \cite{0691084459}
\begin{eqnarray}
I(R) &=& 2 \int_R^\infty \frac{\ell(r) \, r \, dr}{\sqrt{r^2 - R^2}},\\[.1in]
\label{IandEll}
\sigma_p^2 (R)&=& \frac 2 {I(R)} \int_R^\infty \( 1 - \beta(r) \frac{R^2}{r^2}\)\frac{ \ell(r) \, \sigma_r^2(r) \, r \, dr }{\sqrt{r^2 - R^2}}.
\label{sigma_p}
\end{eqnarray}
The procedure described above to first evaluate $\sigma_r^2$ to then find $\sigma_p^2$ is computationally demanding (specially for models with constant but free $\beta$). A significant computational time improvement is achieved by bypassing the computation of $\sigma_r^2$ \cite{Mamon:2004xk},
\begin{equation}
\sigma_p^2(R) = \frac {2 G_0}{I(R)} \int_R^\infty K\(\frac r R\) \frac{\ell(r) M(r) } r dr,
\label{sigma_pK}
\end{equation}
where
{\small
\begin{equation}
K(u) \equiv \frac 12 u^{2 \beta - 1}\[ \(\frac 32 - \beta \) \sqrt \pi \frac{\Gamma(\beta - 1/2)}{\Gamma(\beta)} + \beta B\(\frac 1{u^2}, \beta + \frac 12, \frac 12 \) - B\(\frac 1{u^2}, \beta - \frac 12, \frac 12 \)\],
\end{equation}}
$B(x,a,b) = \int_0^x t^{a-1} (1-t^{b-1}) dt$ is the incomplete beta function and $\Gamma$ is the Gamma function.
\bigskip
The above concludes a short review on how to relate mass distribution to observed velocity dispersions for Newtonian gravity. Nevertheless, almost all the above can be applied to non-Newtonian gravity as well, the single exception being the relation between $\Phi$ and $M(r)$. A useful procedure to generalize the picture above to other gravity theories, is to introduce the concept of total effective mass $M_{E}(r)$ as follows,
\begin{equation}
M_{E}(r) \equiv \frac{\Phi'(r) \, r^2}{ G_0} = \frac{g(r) \, r^2}{G_0},
\end{equation}
where $g(r)$ is the norm of the local acceleration.
As above discussed, the dark matter effect is to enhance the total mass inside a radius r from that computed from stars alone ($M_*(r)$) to $M(r) = M_*(r) + M_{dm}(r)$. In the case of RGGR and MOND (without dark matter), a similar phenomenology is achieved since the potential $\Phi$ is enhanced due to a change on the gravitational theory. The latter can be interpreted as the effect of an additional effective mass such that the total effective mass is
\begin{equation}
M_{E}(r) = M_*(r) + M_{\mbox{\tiny RGGR}}(r),
\label{MeRGGR}
\end{equation}
or, in the case of MOND, $M_{E}(r) = M_*(r) + M_{\mbox{\tiny MOND}}(r)$.
\bigskip
For MOND, the acceleration $ g$ felt by a test particle due to the gravitational force of a spherical mass distribution is only equal to the Newtonian one for sufficiently large accelerations ($ g \gg a_0$). In this framework, the physical acceleration $g$ is related to the Newtonian one ($g_N$) by
\begin{equation}
g \, \mu\(\frac{g}{a_0} \) = g_N.
\label{gMOND}
\end{equation}
The function $\mu$ (MOND's interpolating function) is such that $\mu(x) =1 $ for $x \ll 1$ and $\mu(x) =x $ for $x \gg 1$ \cite{1983ApJ...270..371M,1983ApJ...270..365M}. This introduces a certain ambiguity to this framework, since each choice of the function $\mu$ leads to physically different results. Fortunately, for many sounding choices of $\mu$, and for the purposes of some physical tests, the differences are negligible. For concreteness, we will fix $\mu$ as one of the most cited interpolating functions, namely the one proposed in Ref. \cite{Famaey:2005fd} which reads
\begin{equation}
\mu(x) = \frac{x}{1 + x}.
\label{simplemu}
\end{equation}
With the above, the eq. (\ref{gMOND}) can be explicitly solved and it leads to the following additional effective mass contribution,
\begin{equation}
M_{\mbox{\tiny MOND}}(r) \equiv M_{E}(r) - M_*(r) = \frac 12 M_*(r) \sqrt{1 + \frac{4 \, a_0 \, r^2} {G_0 M_*(r)} } - \frac{M_*(r)}2.
\label{MMOND}
\end{equation}
The constant $a_0$ is assumed to be an universal constant for MOND (i.e., all galaxies should be subjected to the physical effects induced by the same value of $a_0$). From a best fit to the kinematics of diverse disk galaxies, and considering the $\mu(x)$ function above, its value was found to be $a_0 = 1.35 \times 10^{-8} cm/s^2$\cite{Famaey:2006iq}. --- If we had adopted the original $\mu$ function proposed in \cite{1983ApJ...270..371M,1983ApJ...270..365M}, whose $a_0$ value is slightly lower according to disk galaxies data \cite{Famaey:2006iq}, a worse concordance with the NGC 4374 galaxy would be found.
\bigskip
In the RGGR case, in order to find its effective additional mass, we start from the expression for the acceleration of a test particle in the weak field and low velocity limits, which was found in \cite{Rodrigues:2009vf},\footnote{In the presence of matter as a fluid this relation is not valid as an equality, but within a good approximation (since the $G$ variation is very small), the relation (\ref{PhiRGGR2}) holds. See Ref. \cite{Rodrigues:2009vf} for further details. }
\begin{equation}
\Phi'_{\mbox{\tiny RGGR}} \approx \Phi'_N \( 1 - \frac {c^2 \alpha \nu}{\Phi_N} \),
\label{PhiRGGR2}
\end{equation}
where $\Phi_N$ stands for the Newtonian potential of all the relevant matter. In the case of elliptical galaxies, assuming no dark matter, it is essentially the Newtonian potential from the stars (i.e., $\Phi_N = \Phi_*$).
Equation (\ref{PhiRGGR2}) is the analogous of the eqs. (\ref{gMOND}, \ref{simplemu}), since it establishes the relation between the physical acceleration (considering its theoretical framework) to the Newtonian one. Contrary to MOND, eq. (\ref{PhiRGGR2}) poses no fundamental acceleration scale. In particular, there is no MOND's $\mu(x)$ interpolating function that is capable to lead to the relation given in eq.(\ref{PhiRGGR2}).
For spherical systems,
\begin{eqnarray}
\Phi'_{\mbox{\tiny RGGR}}(r) &=& \frac{G_0 M_*(r)}{r^2} \( 1 + \frac{c^2 \alpha \nu}{\frac{G_0 M_*(r)}{r} + 4 \pi G_0 \int_r^{\infty } \rho_*(r') r' dr' }\) \\
&=& \frac{G_0 M_*(r)}{r^2} + \frac{c^2 \alpha \nu}{r + \frac{4 \pi r^2}{M_*(r)}\int_r^{\infty } \rho_*(r') r' dr' }.
\end{eqnarray}
Hence, the effective additional mass of RGGR for spherical systems is given by
\begin{equation}
M_{\mbox{\tiny RGGR}} (r) \equiv (\Phi'_{\mbox{\tiny RGGR}} - \Phi'_N) \frac {r^2}{G_0} = \frac{\alpha \nu c^2}{G_0} \frac{r}{1 + \frac{4 \pi r }{M_*(r)} \int_{r}^{\infty} \rho_*(r') r' dr'}.
\label{MRGGR}
\end{equation}
\section{Observational data and numerical procedures}
\subsection{Observational data}
In order to disclose the RGGR consequences to real elliptical galaxies, we selected two elliptical galaxies, an ordinary (NGC 4494) and a giant one (NGC 4374). The analysis of both of them are based on recent data, which in particular include extended line-of-sight velocity dispersion (VD) data obtained by both long-slit observations (inner radii) and by the VD of Planetary Nebulae (PNe).
\begin{table}[htdp]
\begin{center}
{\footnotesize
\begin{tabular}{l c c c c c c c}
\multicolumn{8}{c}{\emph{ \normalsize Distance, luminosity, expected $\Upsilon_*$ and S\'ersic parameters}}\\
\hline \hline
Galaxy $^{(1)}$ & Distance$^{(2)}$ &$L_{\mbox{\tiny V}}$$^{(3)}$ &$\< \Upsilon_*\>^{\; (4)}$ &$R_e$$^{\; (5)}$ &$n^{(6)}$ &S\'ersic Ext.$^{(7)}$ & Main Ref.'s$^{(8)}$ \\
& (Mpc) &($10^{10} L_{\odot_V}$) & ($M_\odot / L_{\odot_V}$) & & & & \\ \cline{1-8}
NGC 4374 & 17.1 &7.64 &$ 4.5 $ &113.5'' &6.11 &290'' & \cite{2009ApJS..182..216K, 2009MNRAS.394.1249C, 2011MNRAS.411.2035N}\\
NGC 4494 & 15.8 & 2.64 &$ 3.8 $ & 48.2'' &3.30 & 273'' & \cite{ 2009MNRAS.394.1249C,2009MNRAS.393..329N} \\
\hline \hline
\end{tabular}
\caption {\label{DLS} \footnotesize 3) Luminosity in the $V$ band. 4) The expected V-band stellar mass-to-light ratio considering a Kroupa IMF, see comments in text. 5) The effective radius. 6) The S\'ersic index. The last two values were found from S\'ersic profile fits to the observational surface brightness along the projected intermediate radius (for details see \cite{2011MNRAS.411.2035N, 2009MNRAS.393..329N}). 7) The radius at which the S\'ersic extension to the observational surface brightness is implemented here.}}
\end{center}
\end{table}%
In Table \ref{DLS} we list the main global properties of each galaxy which were relevant for the evaluation of their mass models. The expected stellar mass-to-light ratio is based on previous analysis that use the Kroupa IMF \cite{2001MNRAS.322..231K, 2002Sci...295...82K}, see also Ref. \cite{2012ApJ...748....2D}. We have not added to the table above, but $\beta$ estimates can be found in Ref. \cite{2012ApJ...748....2D}, where it is found that both of the galaxies are close to isotropic ($-0.5 <\beta < 0.5$). Converting to the V band the $\Upsilon_*$'s derived in Refs. \cite{Gerhard:2000ck, 2009MNRAS.396.1132T}, it is fair to assume that, for NGC 4374, $\<\Upsilon_*\>_{\mbox{\tiny Kroupa IMF}} \sim (4.5 \pm 1.0) M_\odot/L_{\odot, V}$ (see also Ref. \cite{2011MNRAS.411.2035N}). For NGC 4494, we use $\<\Upsilon_*\>_{\mbox{\tiny Kroupa IMF}} \sim (3.8 \pm 1.0) M_\odot/L_{\odot, V}$ \cite{2009MNRAS.393..329N} (it should be noted that this galaxy kinematics is not compatible with the Salpeter IMF \cite{2012ApJ...748....2D}). It is not our purpose to be ultimately precise on the Kroupa IMF, the essential issue is to verify that RGGR has a tendency of being in conformity with IMF's that lead to significantly lower $\Upsilon_*$'s than the Salpeter one. This tendency could thus provide another important physical test for our proposal once stelar population models become more precise.
For each of the above galaxies, we use the observational surface brightness data up to a the largest radius where the observation is trustworthy, and then extend it with its S\'ersic profile (analogously to the case of disk galaxies, where the extension is implemented with a simple exponential profile). The observational surface brightness profile, which we call $\mu_{\mbox{\tiny obs}}(R)$, is determined from observations up to a certain radius. It is currently known that the surface brightness of many galaxies can be well approximated by S\'ersic profiles \cite{1968adga.book.....S}. From $\mu_{\mbox{\tiny obs}}$ one can fit the parameters $R_e$ and $n$ to find the corresponding S\'ersic surface brightness of each galaxy, which we call $\mu_{\mbox{\tiny S}}(R)$. The latter can be used to generate an extended surface brightness profile $\mu_{\mbox{\tiny ext}}$, which is composed by $\mu_{\mbox{\tiny obs}}$ from the center up to a given radius and $\mu_{\mbox{\tiny S}}$ from that point out to a larger radius.
The intensity associated with the S\'ersic profile reads (see Ref. \cite{2005PASA...22..118G} for a review),
\begin{equation}
I(R) = I_0 \, e^{- \beta_n \( \frac R {R_e}\)^{1/n}},
\label{sersicprof}
\end{equation}
where $R$ is the projected radius, $n$ is the S\'ersic index and $R_e$ is the effective radius (which is defined as the radius that encloses half of the total luminosity). Since the total luminosity is given by $L = 2 \pi \int_0^\infty I(R) dR^2$, the constant $\beta_n$ must satisfy $\Gamma(2 n) = 2 \gamma(2 n , \beta_n)$, where $\Gamma$ and $\gamma$ stand for the gamma function and the (generalized) incomplete gamma function (i.e., $\gamma(a,b) = \int_0^b t^{a - 1} e^{-t} dt$). Solutions for $\beta_n$ can be found numerically.
Since most photometric observations are stated in unities of mag/arcsec$^2$, the following relation between surface brightness $\mu(R)$ and the intensity $I(R)$ is useful,
\begin{equation}
\mu(R) = - 2.5 \, \mbox{log}_{10} \[ I(R) \frac{ \mbox{kpc}^2}{L_\odot } \( \frac { \pi 10^{-2}}{180 \times 60 \times 60}\)^2 \] + {\cal M}_\odot,
\end{equation}
where ${\cal M}_\odot$ is the magnitude of the sun in the appropriate band.
\bigskip
\subsection{The stellar contribution to the observed VD} \label{stelarVD.subsec}
To find the stellar contribution to the total projected VD, we proceed as follows: \\
\noindent
$i$) Find the luminosity density, which comes from the inversion of eq. (\ref{IandEll}) \cite{0691084459}, namely
\begin{equation}
\ell (r)= - \frac{1} {\pi} \int_r^\infty \frac{I'(R)}{\sqrt{R^2 - r^2}} dR.
\end{equation}
From the above, a vector $\{\ell(r_i)\}$ is built. This vector is numerically interpolated to generate a numerical function for $\ell(r)$ (this step is necessary to improve computational performance). At this and similar procedures, the $r_i$ resolution is chosen such that the errors between the original function and the one build from the interpolation are no more than $1 \%$. \\
\noindent
$ii$) Using a linear relation between stellar mass density and luminosity density, $\rho_* (r)= \Upsilon_* \, \ell(r)$, it is straightforward to build the total stellar mass inside the radius $r$, $M_*(r)$. \\
\noindent
$iii$) We use $\ell$, $M_*$ and the eq. (\ref{sigma_pK}) to build the numerical table $\{\sigma^2_{*} (R_i, \beta_j)/ \Upsilon_*\}$, with $\beta_j \in [-1,1]$. In general, $\beta \in (- \infty, 1]$, but physical considerations (based on galaxy formation or reasonable restrictions on the distribution function) disfavor large negative (tangential) anisotropy, while favors either isotropy or radial anisotropy \cite{2011MNRAS.411.2035N,2009MNRAS.393..329N}.
\bigskip
The three steps above determines $\sigma_{p*}$, apart from $\Upsilon_*$ and $\beta$, which will be analyzed latter.
\bigskip
\subsection{The non-Newtonian contributions to the observed VD}
To find the RGGR additional contribution to the projected VD, which we call $\sigma_{p \, {\mbox{\tiny RGGR}}}(R)$ and satisfies
\begin{equation}
\sigma_p^2 = \sigma^2_{p *} + \sigma^2_{p\, {\mbox{\tiny RGGR}}},
\label{sigmap}
\end{equation}
we re-do the steps $ii$ and $iii$ from above with $M_{\mbox{\tiny RGGR}}$ (see eq. (\ref{MRGGR})) in place of $M_*$. With this procedure, the total projected VD for RGGR is completely determined except for $\beta$ and two constants that appear linearly inside $\sigma_{p *}^2$ and $\sigma_{p \, {\mbox{\tiny RGGR}}}^2$, namely: the stellar mass-to-light ratio $\Upsilon_*$ and the constant $\nu \alpha$.
\bigskip
The effective additional mass provided by MOND (eq. \ref{MMOND}) has a simple form, but it depends on different powers of $\Upsilon_*$. Hence, in the step $iii$ described previously, the numerical function $\sigma_{ p \, {\mbox{\tiny MOND}}}$ is found from the three dimensional interpolation of the following table, $\{ \sigma^2_{ p\, {\mbox{\tiny MOND}}} (\Upsilon_{* i}, R_j, \beta_k ) \}$, where $\Upsilon_*$ was constrained to be in the range\footnote{No physical issues are expected due to this constrain for this galaxy in this band.} $1\le \Upsilon_*/(M_\odot/L_{\odot_V}) \le 10$, while $R$ and $\beta$ have the same resolutions and range as in the RGGR model described above.
Before proceeding, we remark that the stellar component modeling used for MOND is identical to the modeling used for RGGR and Newtonian gravity. In Ref.\cite{2011A&A...531A.100R} it is argued in favor of the use of Jaffe profiles in the outer parts of the baryonic mass profiles of ellipticals. Here this route is not pursued.
\section{Elliptical galaxies within RGGR without dark matter: general aspects}
\subsection{S\'ersic profiles} \label{generalaspects}
\begin{figure}[thbp]
\begin{center}
\includegraphics[width=100mm]{MRGGRsersic.pdf}
\caption{The RGGR additional mass contribution $M_{\mbox{\tiny RGGR}}$, as a function of the deprojected radius $r$, for two S\'ersic profiles. One of the profiles has S\'ersic index $n = 6.1$ (solid line) and the other with $n = 3.3$ (dashed line). Many of the elliptical galaxies have intermediate S\'ersic indices. For $\alpha \nu \sim 10^{-7}$, the additional mass $M_{\mbox{\tiny RGGR}}$ is about the same order of the baryonic mass in a galaxy.}
\label{MRGGRsersic}
\end{center}
\end{figure}
\begin{figure}[thbp]
\begin{center}
\includegraphics[width=100mm]{SigmapRGGRSersicN4374.pdf}
\caption{The projected VD contribution from RGGR ($\sigma_{p \, {\mbox{\tiny RGGR}}}(R)$) for a matter distribution given by the S\'ersic profile (\ref{sersicprof}) with $n = 6.1$. The thickest (green) curve corresponds the isotropic case $\beta =0 $. The others are associated to the following $\beta$ values: -1.00, -0.75, - 0.50, - 0.25, 0.25, 0.50, 0.75, 1.00 (from light gray to black, respectively). }
\label{SigmapRGGRSersicN4374}
\end{center}
\end{figure}
\begin{figure}[thbp]
\begin{center}
\includegraphics[width=100mm]{SigmapRGGRSersicN4494.pdf}
\caption{The same of Fig. \ref{SigmapRGGRSersicN4374}, but with S\'ersic index $n= 3.3$.}
\label{SigmapRGGRSersicN4494}
\end{center}
\end{figure}
For spherical stationary systems, the main differences between Newtonian gravity and the RGGR one are encoded in the expression for $M_{\mbox{\tiny RGGR}}$ (\ref{MRGGR}). From a given matter density $\rho(r)$, it is straightforward to derive the additional effective mass posed by RGGR. In the end, the single uncertainty relies on the constant $\alpha$, on which further comments on its behavior will be presented in the next section.
Since the matter density of diverse elliptical galaxies are fairly proportional to a deprojected S\'ersic profile (which, for $n=4$, coincides with the de Vaucouleurs profile), here we will consider the consequences of RGGR for matter profiles deduced from deprojected S\'ersic profiles (for all radii). Two cases will be explicitly evaluated: the case $n=3.3$ and $n = 6.1$ (the same S\'ersic indices of NGC 4374 and NGC 4494). Such values are useful to the purposes of this section, since many ellipticals have S\'ersic indices that lie between these values.
The RGGR mass profile $M_{\mbox{\tiny RGGR}}(r)$ is shown in Fig. \ref{MRGGRsersic}. The contributions of such profiles to the projected VD (see eq. (\ref{sigmap})), including different assumptions for the anisotropic parameter $\beta$, are depicted in Figs. \ref{SigmapRGGRSersicN4374}, \ref{SigmapRGGRSersicN4494}.
Apart from the $\alpha$ role, from the eq. (\ref{MRGGR}) one sees that $M_{{\mbox{\tiny RGGR}}}$ is not sensible to either the constant $I_0$ in eq. (\ref{sersicprof}) or the value of any constant mass-to-light ratio; it only depends on the effective radius $R_e$ and the S\'ersic index $n$.
\subsection{The role of $\alpha$ and the fundamental plane} \label{Sec.RoleOfAlpha}
Here we point and illustrate that future analysis on the relation between RGGR and the fundamental plane \cite{1987ApJ...313...59D,1987ApJ...313...42D} can provide significant constraints on the correlation between $\alpha$ and the mass distribution. It is beyond the scope of this work to do a solid analysis on the RGGR consequences for the fundamental plane interpretation (interpretation which is currently an open problem within any framework). In particular, the status of such task for MOND can be checked in Ref. \cite{2011MNRAS.412.2617C} and references therein.
The existence of a plane in the space $(\< I\>_e, R_e, \sigma_0)$ (the fundamental plane) is usually interpreted as a consequence of the virial theorem (in the context of Newtonian gravity), see Ref. \cite{0521857937} for a review. Also, the data scattering can be further reduced around the theoretical surface if the S\'ersic index $n$ is included as a new axis, a ``fundamental hyperplane'' \cite{2002MNRAS.334..859G}. It is admissible to use the same relation in context of RGGR, except that one should replace the virial mass $M_v$ by the corresponding effective mass ${M_E}_v$ (\ref{MeRGGR}) (considering stationary and spherically symmetric systems). Note that the total effective mass is not finite, but such feature is not a novelty, since well known dark matter profiles do display the same feature. One may define, for instance, the virial radius at the radius at which the local effective density is about 200 times the cosmological critical density. The details of such definition will not be relevant to the approach below.
The theoretical plane in the $(\< I\>_e, R_e, \sigma_0)$ space, which is derived from Newtonian gravity, together with the virial theorem and with constant $M/L$, is tilted in regard to the observed one. The tilt can be interpreted as an evidence of dark matter (within the Newtonian gravity context), since the necessary $M/L$ variation to match theory and observation is larger than the expected from stellar population variations (e.g., \cite{2006MNRAS.366.1126C}). Hence, one way to explain the tilt of the fundamental plane is from the assumption that
\begin{equation}
\frac M L \propto L^A \< I\>_e^B,
\label{MLAB}
\end{equation}
where the values of $A$ and $B$ depends on the values of the parameters that determine the fundamental plane (from observations). For instance, using the values of Ref. \cite{1996MNRAS.280..167J} one finds $A = 0.31$ and $B=0.02$ (see also Ref.\cite{0521857937}). Part of the above scaling relation would be due to systematic stellar mass-to-light ratios ($M_*/L$) variations. For simplicity, we will here assume no systematic variation of $ M_*/L $, hence
\begin{equation}
\frac M L = \frac {M_*} L \frac M {M_*} \propto 1 + \frac{M_{\mbox{\tiny RGGR}}}{M_*}.
\label{MLMRGGR}
\end{equation}
At the virial radius, it is fair to use $M_* \ll M_{\mbox{\tiny RGGR}}$. Combining the last comment with eqs. (\ref{MLAB},\ref{MLMRGGR}),
\begin{equation}
\frac{M_{\mbox{\tiny RGGR}}}{M_*} \propto I_0^{A+ B} R_e^{2A} g^{A+ B}(n),
\label{MRGGRMstarsI0}
\end{equation}
since $\<I\>_e = I_0 \, g(n)/ (2 \pi)$ and $L = I_0 \,R_e^2 \,g(n)$, where
\begin{equation}
g(n) \equiv 2 \pi \frac{n e^{2 \beta_n}}{{\beta_n}^{2n}} \Gamma(2 n) .
\end{equation}
In Sec. \ref{generalaspects} it is shown that $M_{\mbox{\tiny RGGR}}(r)/\alpha$ does not depend on $I_0$, hence from eq. (\ref{MRGGRMstarsI0}) a rough estimate on the variation of $\alpha$ with $M_*$ can be found as
\begin{equation}
\alpha \propto M_*^{1 + A + B} f(R_e,n).
\end{equation}
That is, for elliptical galaxies of about the same shape (i.e., with similar $R_e$ and $n$), $\alpha$ should increase with the stellar mass of the the galaxy faster than linear, but slower than quadratic. This is a quick procedure to evaluate the dependence of $\alpha$ with the galaxy parameters. The best procedure would be to use a large sample of elliptical galaxies to deduce that, and also unveil the function $f(R_e, n)$.
\section{NGC 4374 and NGC 4494 results}
\subsection{Introduction}
Here a detailed numerical analysis of NGC 4374 and NGC 4494 is done. The main results are shown in Tables \ref{N4374resultsTable}, \ref{N4374RGGRmassTable}, \ref{N4494resultsTable}, \ref{N4494RGGRmassTable} and Figs. \ref{N4374MONDFig},\ref{N437RGGRPhotoFig}, \ref{N4494RGGRPhoto3}. The main purpose of this section is to show that recent data on elliptical galaxies is in conformity with the RGGR model, even assuming a negligible amount of dark matter. Moreover, we compare our results to MOND (which also constitute an original work presented in this paper).
Many elliptical galaxies display a certain degree of rotation. Such rotation do not appear directly in the velocity dispersion (VD) data, but it also traces the galaxy effective mass. In order to mass model such galaxies without simply neglecting the rotational (sub-dominant) component, one can insert a compensation in the VD data. The data that is here called ``line of sight VD'' has already a compensation for rotation, whose computation was done in Refs.\cite{2009MNRAS.393..329N, 2011MNRAS.411.2035N}. We use the same data that these references use for mass modeling these galaxies.
\begin{table}[htdp]
\begin{center}
{\footnotesize
\begin{tabular}{l c c c c c c}
\multicolumn{6}{c}{\emph{ \normalsize NGC 4374}}\\
\hline \hline
\multicolumn{6}{c}{\emph{Newtonian gravity without dark matter} } \\
Stellar model $^{(1)}$ & --- & $\beta^{\; (2)}$ &$\Upsilon_*^V/ (\frac{M_\odot}{L_{\odot,\mbox{\tiny V}}})$ $^{\; (3)}$ &${\chi^{2}}^{\; (4)}$ &${\chir}^{\; (5)}$\\ \cline{1-6}
$\beta_{[0]}$ & & 0 & 7.59$\pm$0.15 & 140 & 6.4 \\
$\beta_{[-1,1]}$ & &$-1.00^{+0.11}_{-0.00}$ & 7.82$\pm$0.23 & 104 & 4.9 \\
K.IMF+$\beta_{[0]}$ & & 0 & 6.66$\pm$0.11 & 292 & 13 \\
K.IMF+$\beta_{[-1,1]}$ & &$-1.00^{+0.37}_{-0.00}$ & 6.78$\pm$0.17 & 276 & 13 \\
& & & & &\\
\multicolumn{6}{c}{\emph{RGGR without dark matter} } \\
Stellar model & $\alpha\,\nu \times 10^{7}$$^{\;(6)}$& $\beta$ &$\Upsilon_*^V/ (\frac{M_\odot}{L_{\odot,\mbox{\tiny V}}})$ &${\chi^{2}}$ &${\chir}$\\ \cline{1-6}
$\beta_{[0]}$ &$14.9\pm2.4$ &0 &$4.14\pm0.57$ &21.1 &1.0\\
$\beta_{[-1,1]}$ &$15.3^{+5.5}_{-4.7}$ &$0.12^{+0.66}_{-1.12}$ &$3.9^{+1.7}_{-2.4}$ &21.0 &1.1\\
K.IMF+$\beta_{[0]}$ &$13.8\pm1.4$ &0 &$4.41\pm0.28$ &21.8 &0.99\\
K.IMF+$\beta_{[-1,1]}$ &$14.0\pm1.9$ &$-0.18^{+0.43}_{-0.78}$ &$4.48\pm0.38$ &21.3 &1.0\\
& & & & &\\
\multicolumn{6}{c}{\emph{MOND without dark matter} } \\
Stellar model & --- & $\beta$ &$\Upsilon_*^V/ (\frac{M_\odot}{L_{\odot,\mbox{\tiny V}}})$ &${\chi^{2}}$ &${\chir}$\\ \cline{1-6}
$\beta_{[0]}$ & &0 &$6.8\pm0.1$ &34.2 &1.6 \\
$\beta_{[-1,1]}$ & &$-1.0^{+0.4}_{-0.0}$ &$7.2\pm0.2$ &24.9 &1.2\\
K.IMF+$\beta_{[0]}$ & &0 &$6.1\pm0.1$ &117 &5.1\\
K.IMF+$\beta_{[-1,1]}$ & &$0.4^{+0.2}_{-0.3}$ &$6.0\pm0.2$ &113 &5.2\\
\hline \hline
\end{tabular}
\caption{\label{N4374resultsTable} \footnotesize NGC 4374 results. (1) $\beta_{[0]}$ indicates isotropic VD, $\beta_{[-1,1]}$ indicates constant anisotropy with $\beta \in [-1,1]$, K.IMF is a reference to Kroupa IMF, and it means that the expected value of $\Upsilon_*$ was used, see Table \ref{DLS} and eq. (\ref{chiUps}). (4) and (5) are the chi squared and reduced chi squared parameters, for details see Sec.\ref{Sec.chi2}. }}
\end{center}
\end{table}
\begin{table}[htdp]
\begin{center}
{\footnotesize
\begin{tabular}{l c }
\hline \hline
Stellar Model &$M_*/(10^{10} M_\odot)$ \\
\footnotesize (1) & (2) \\
\hline
$\beta_{[0]}$ &$31.7\pm8.5$\\
$\beta_{[-1,1]}$ &$30^{+17}_{-22}$\\
K.IMF+$\beta_{[0]}$ &$33.7\pm6.5$\\
K.IMF+$\beta_{[-1,1]}$ &$34.2\pm7.4$\\[0.2cm]
\hline \hline
\end{tabular}
\caption{\label{N4374RGGRmassTable} \footnotesize RGGR mass results for NGC 4374.}}
\end{center}
\end{table}%
\begin{table}[htdp]
\begin{center}
{\footnotesize
\begin{tabular}{l c c c c c c}
\multicolumn{6}{c}{\emph{ \normalsize NGC 4494}}\\
\hline \hline
\multicolumn{6}{c}{\emph{Newtonian gravity without dark matter} } \\
Stellar model $^{(1)}$ & --- & $\beta^{\; (2)}$ &$\Upsilon_*^V/ (\frac{M_\odot}{L_{\odot,\mbox{\tiny V}}})$ $^{\; (3)}$ &${\chi^{2}}^{\; (4)}$ &${\chir}^{\; (5)}$\\ \cline{1-6}
$\beta_{[0]}$ & & 0 & 4.206$\pm$0.044 & 20.6 & 0.82 \\
$\beta_{[-1,1]}$ & &$-0.55^{+0.26}_{-0.32}$ &$4.164\pm0.067$ &6.25 &0.26\\
K.IMF+$\beta_{[0]}$ & & 0 & 4.190$\pm$0.043 & 23.6 & 0.91 \\
K.IMF+$\beta_{[-1,1]}$ & &$-0.56^{+0.25}_{-0.32}$ &$4.150\pm0.065$ &8.54 &0.34\\[.1in]
& & & & &\\
\multicolumn{6}{c}{\emph{RGGR without dark matter} } \\
Stellar model & $\alpha\,\nu/10^{-7}$$^{\;(6)}$& $\beta$ &$\Upsilon_*^V/ (\frac{M_\odot}{L_{\odot,\mbox{\tiny V}}})$ &${\chi^{2}}$ &${\chir}$\\ \cline{1-6}
$\beta_{[0]}$ &$1.55^{+0.60}_{-0.58}$ &0 &$3.49\pm0.28$ &3.19 &0.13\\
$\beta_{[-1,1]}$ &$1.2^{+1.6}_{-1.2}$ &$-0.15^{+0.43}_{-0.55}$ &$3.67^{+0.51}_{-0.73}$ &2.81 &0.12\\
K.IMF+$\beta_{[0]}$ &$1.21\pm0.44$ &0 &$3.66\pm0.20$ &5.07 &0.20\\
K.IMF+$\beta_{[-1,1]}$ &$0.86^{+0.72}_{-0.69}$ &$-0.24^{+0.31}_{-0.42}$ &$3.80^{+0.29}_{-0.31}$ &3.08 &0.13\\[.1in]
& & & & &\\
\hline \hline
\end{tabular}
\caption{\label{N4494resultsTable} \footnotesize NGC 4494 results. See Table \ref{N4374resultsTable} for details. }}
\end{center}
\end{table}%
\begin{table}[htdp]
\begin{center}
{\footnotesize
\begin{tabular}{l c }
\hline \hline
Stellar Model &$M_*/(10^{10} M_\odot)$ \\
\footnotesize (1) & (2) \\
\hline
$\beta_{[0]}$ &$9.2 \pm 1.5$\\
$\beta_{[-1,1]}$ &$9.7^{+2.1}_{-2.7}$ \\
K.IMF + $\beta_{[0]}$ &$9.7\pm1.3$ \\
K.IMF + $\beta_{[-1,1]}$ &$10.0\pm1.6$ \\
\hline \hline
\end{tabular}
\caption{\label{N4494RGGRmassTable} \footnotesize RGGR mass results for NGC 4494.}}
\end{center}
\end{table}%
\begin{figure}[thbp]
\begin{center}
\includegraphics[width=110mm]{N4374StellarPhoto2.pdf}
\caption{\footnotesize NGC 4374 mass models with Newtonian gravity (NG). In the larger plots at the left, the circles and stars with error bars refer to the observational VD, either from long-slit observations (the stars) or from planetary nebulae (the circles). The curves refer to mass models composed by a stellar component with Newtonian gravity (NG). The first model (i) assumes isotropy ($\beta = 0$), while the second (ii) assumes $\beta \in [-1,1]$. Models iii and iv use the expected mass-to-light ratio from the Kroupa IMF (K.IMF) as part of the data to be fitted (see eq. (\ref{chiUps}) and Table \ref{DLS} for details). The four thin lines that appear in the lower left plot depict the theoretical error bars on the stellar velocity dispersion (the solid thin line refers to the model iii, while the dashed thin line to the model iv, which differ since they have different $\beta$ values). The vertical dashed line signs the radius above which the observational data is considered for the fitting procedure (10 arcsec) \cite{2011MNRAS.411.2035N}. The four small plots at the right show the $1 \sigma$ and $2 \sigma$ confidence levels for each of the models. } \label{N4374StarsPhotoFig}
\end{center}
\end{figure}
\begin{figure}[thbp]
\begin{center}
\includegraphics[width=140mm]{N4374RGGRPhoto08.pdf}
\caption{NGC 4374 mass models with RGGR. The curves refer to mass models composed by the stellar component (inferred from photometric data with S\'ersic extrapolation) and RGGR gravity. The black solid line in each of the four large plots is the resulting VD for each model, the yellow dashed and blue dotted lines are respectively the stellar Newtonian and non-Newtonian contributions to the total VD, and the thin dark gray lines in models iii and iv are the effective error bars for $\Upsilon_*$ (see text for details). The first model (i) assumes isotropy ($\beta = 0$), the second (ii) assumes $\beta \in [-1,1]$, while models iii and iv also consider the expected mass-to-light ratio ($\Upsilon_*$) in the fitting procedure (see eq. (\ref{chiUps}) and Table \ref{DLS} for details). The vertical dashed line signs the radius above which the observational data is considered for the fitting procedure (10 arcsec). The smaller plots at the right show the $1 \sigma$ and $2 \sigma$ confidence levels for each of the four models.}
\label{N437RGGRPhotoFig}
\end{center}
\end{figure}
\begin{figure}[thbp]
\begin{center}
\includegraphics[width=125mm]{N4374MONDPhoto.pdf}
\caption{ NGC 4374 mass models with MOND. See Fig. \ref{N437RGGRPhotoFig} for details.}
\label{N4374MONDFig}
\end{center}
\end{figure}
\begin{figure}[thbp]
\begin{center}
\includegraphics[width= 110mm]{N4494StellarPhoto.pdf}
\caption{NGC 4494 mass models with Newtonian gravity (NG). See Fig. \ref{N4374StarsPhotoFig} for details. }
\label{N4494StarsPhotoFig}
\end{center}
\end{figure}
\begin{figure}[thbp]
\begin{center}
\includegraphics[width=140mm]{N4494RGGRPhoto3.pdf}
\caption{ NGC 4494 mass models with RGGR. See Fig. \ref{N437RGGRPhotoFig} for details.}
\label{N4494RGGRPhoto3}
\end{center}
\end{figure}
In the Newtonian gravity without dark matter framework, there are two parameters that the procedures described in Sec.\ref{stelarVD.subsec} do not fix, namely: the mass-to-light raio $\Upsilon_*$ and the anisotropy parameter $\beta$. There are theoretical expectations for $\Upsilon_*$, but they are considerably more uncertain than other astrophysical data.
Contrary to MOND, the RGGR approach was not developed in order to remove the necessity of dark matter, but to explore a possible QFT effect on large scales that may have a significant impact on astrophysics and cosmology. In this paper, likewise in Ref.\cite{Rodrigues:2009vf}, we explore the extremum possibility of having no dark matter in galaxies and only standard gravitation with the running of $G$ as given by eq. (\ref{gmu}). It would be a leap of faith the assume that this single running of $G$ could solve all the dark matter issues at all scales, and we do not expect that. On the other hand, the galactic picture we are describing can be more than a mere starting point, since it does work as reasonable approximation of a universe with a lower amount of dark matter, and of a warmer type. Considering this picture, dark matter itself would have a lesser role in galaxies, while the RGGR role would be the dominant one.
Albeit the results here presented for NGC 4374 favor RGGR over MOND, a natural criticism would be that with RGGR one more parameter is being fitted than with MOND. The comparison done here is worthwhile since: $i$) had we systematically found for RGGR a worse observational concordance than for MOND, that would sign that RGGR had deep problems and that it should be dismissed; $ii$) in case MOND systematically deviates from the observational data (e.g., data from giant elliptical galaxies), it should be dismissed or modified. $iii$) MOND and RGGR seem to display different tendencies for the stellar mass-to-light ratios, and thus developments on stellar population models may favor one over the other, in spite of any differences on the number of parameters.
\subsection{On the $\chi^2$ minimization} \label{Sec.chi2}
In order to find the best fit for each model, we proceed with a standard $\chi^2$ minimization. We numerically search for the global minimum of
\begin{equation}
\chi^2(\Upsilon_*, \Theta_1, \Theta_2) = \sum_{i =1}^N \( \frac{\sigma_{\mbox{\tiny model}}(\Upsilon_*,\Theta_1, \Theta_2, R_i) - \sigma_{\mbox{\tiny obs}\, i}}{\zeta_i} \) ^2.
\label{chi2}
\end{equation}
In the above, $N$ is the total number of observational VD points, $\Theta_1$ and $\Theta_2$ stand for additional parameters (e.g., $\beta$ and $\alpha \nu$), $\sigma_{\mbox{\tiny obs}\, i}$ is the VD observational value at radius $R_i$, while $\zeta_i$ is its corresponding uncertainty.
The procedure above only evaluates the shape of the VD curve, hence it may lead to small values of $\chi^2$ when the inferred mass-to-light ratios are very far from the theoretical expectations. On the other hand, to simply fix the value of $\Upsilon_*$ according to a certain estimate seems (at least in many cases) unrealistic, since such estimates have significantly large error bars. Here, besides using the standard minimization procedure as above, we also use a novel procedure in which the expected value of $\Upsilon_*$ is inserted as an additional data to be fitted, as follows,
\begin{equation}
\chi^2_\Upsilon (\Upsilon_*, \Theta_1, \Theta_2) = \sum_{i =1}^N \( \frac{\sigma_{\mbox{\tiny model}}(\Upsilon_*,\Theta_1, \Theta_2, R_i) - \sigma_{\mbox{\tiny obs}\, i}}{\zeta_i} \) ^2 + N \(\frac{\Upsilon_* - \bar \Upsilon_*}{\zeta_\Upsilon}\)^2,
\label{chiUps}
\end{equation}
where $\bar \Upsilon_*$ is the theoretically expected stellar mass-to-light ratio, $\zeta_\Upsilon$ is its error or dispersion, and the $N$ multiplying the last term imposes the same weigh for both the curve shape and the mass-to-light ratio.
One way to evaluate the goodness of the fit, which is probably the most straightforward, is by the comparison of the $\chir$ values. In the tables \ref{N4374resultsTable} and \ref{N4494resultsTable} we list the values of the minimum of $\chi^2$ and its corresponding $\chir$. The last is given by the minimum value of $\chi^2$ divided by the difference between $N$ and the number of parameters being fitted. When using the $\chi^2$ given by eq. (\ref{chiUps}), the number of observational data, $N$, is added by one.\footnote{Another reasonable option would be to add by $N$, but either choice would not change our conclusions.}
Besides determining the parameters values for the best fit, we also find the confidence level curves for each model. The region inside the $n \; \sigma$'s confidence level curve is such that $\chi^2 (p_1,...,p_k) \le \chi^2_{\mbox{\tiny min}} + \Delta \chi^2(n,k)$, where $\{p_i\}$ are the model's free parameters and the values of $\Delta \chi^2(n,k)$ can be found in standard references on statistics. This is important for testing the sensibility of each model and for disclosing correlations between its parameters.
As a final remark, likewise was done in Refs. \cite{2011MNRAS.411.2035N,2009MNRAS.393..329N}, the galaxy kinematics inside its 10'' radius is not considered at the fitting procedure. This is done since it is unlikely that the Jeans model employed is a reasonable approximation for the galaxy dynamics at a range so close to the galaxy center. In all of the VD plots the observational data is present from 5'', and the fitted curves are accordingly extended.
\subsection{Specific comments on the NGC 4374 and NGC 4494 fits}
\noindent
{\it NGC 4374}
\noindent
For Newtonian gravity and isotropic VD ($\beta = 0$), a mass model composed only of stars cannot fit the observational data of the galaxy NGC 4374, as it it can be seen in Fig.\ref{N4374StarsPhotoFig} (see also Ref. \cite{2011MNRAS.411.2035N}). Besides being a poor fit considering the shape of the VD curve, it was achieved by using a stellar mass-to-light ratio $\Upsilon_*$ which is considerably above the theoretical expectations. If the isotropy condition is dropped, the fit for the shape of the curve can be slightly improved, but the anisotropy parameter goes toward the disfavored negative values $\beta \lesssim - 1$ and $\Upsilon_*$ increases even more. This feature is common to other giant elliptical galaxies.
From Fig. \ref{N4374MONDFig} and Table \ref{N4374resultsTable}, it is clear that MOND fits better the NGC 4374 observational data than Newtonian gravity without dark matter. However, it is still a poor fit, in particular since: $i$) There is a significant tendency towards a lower VD curve at large radii (the region of PNe data), tendency which is strongly enhanced once the fits considers the expected $\Upsilon_*$ (the models iii and iv in Fig. \ref{N4374MONDFig}); $ii$) if the expected $\Upsilon_*$ is not used, the best fit is achieved for tangential anisotropy with $\beta \le -1$. Also, besides these two points, MOND is incompatible with the Kroupa IMF expectations for this galaxy, since even for the model iv the derived $\Upsilon_*$ is (far) outside the $2 \sigma$ confidence level (Fig. \ref{N4374MONDFig}). Other issues of MOND with the giant ellipticals can be found for instance in Ref.\cite{Gerhard:2000ck, Memola:2011vn}.
From Fig. \ref{N437RGGRPhotoFig} and Table \ref{N4374resultsTable}, it can be seen that RGGR fit to the data is a satisfactory one and outperforms MOND in all the points above. It seems that the single issue that MOND does better than RGGR is on it's VD curve continuation towards the galactic center (i.e., the extension from 10'' (0.83 kpc) to 5'' (0.41 kpc)). Since the application of this Jeans modeling to a region so close to the galactic center is probably meaningless \cite{2011MNRAS.411.2035N}, that would constitute no true advantage to MOND.
\bigskip
\noindent
{\it NGC 4494}
\noindent
There is a wiggle at large radius in our analysis (Fig.\ref{N4494StarsPhotoFig}) that is not presented in Ref. \cite{2009MNRAS.393..329N}. This wiggle is innocuous to the fits, and it appears due to a different convention on the S\'ersic extension starting radius.
Similarly to other ordinary elliptical galaxies, mass models derived from Newtonian gravity and a stellar component provide reasonable fits to the observational VD data \cite{Romanowsky:2003qv, 2009MNRAS.393..329N}, se Table \ref{N4494resultsTable} and Fig. \ref{N4494StarsPhotoFig}. Actually, the fits with Newtonian gravity without dark matter are so good that they have lead to a conflict with CDM expectations \cite{Romanowsky:2003qv}. Such lack of dark matter-like effects was explained in the context of MOND shortly after \cite{Milgrom:2003ui}, since the typical internal accelerations of such galaxies are higher than MOND's $a_0$ acceleration. In the case of RGGR, considering that NGC 4374 is a much brighter elliptical, the expectation would be that NGC 4494's corresponding value of $\alpha$ would be significantly lower than that of the giant elliptical, while probably (considering the disk galaxy fits) above $0.1/\nu$. Sharper expectations on the $\alpha$ of NGC 4494 can be drawn after similar galaxies have been analyzed in the RGGR framework. The resulting $\alpha$ value in Table \ref{N4494resultsTable} lies well inside that region.
Likewise in the NGC 4374 case, the RGGR derived $\Upsilon_*$ in models i and ii (Fig.\ref{N4494RGGRPhoto3}) is naturally compatible with the Kroupa IMF expectations (Table \ref{DLS}). Once such value enters as part of the data to be fitted, as in eq. (\ref{chiUps}) and models iii and iv, the parameters degeneracies are significantly lowered, and sharper predictions are done.
\section{Conclusions}
In summary, in this paper the RGGR \cite{Shapiro:2004ch, Rodrigues:2009vf} effective mass for stationary spherical systems was deduced, we presented general considerations on elliptical galaxies within RGGR, and evaluated two specific ellipticals (an ordinary and a giant one), both with extended observed VD by the use of recent planetary nebulae (PNe) data \cite{2011MNRAS.411.2035N,2009MNRAS.393..329N}. Good agreement between the RGGR VD curve and the observations was found. Considering the galaxies here analyzed, no strong tendency towards tangential anisotropy\footnote{Which is usually disfavored, see also Ref.\cite{2012ApJ...748....2D}.} was found for RGGR, while this behavior appeared in both Newtonian gravity without dark matter and MOND. The stellar mass-to-light ratios $\Upsilon_*$ found within RGGR are compatible with the expectations of the Kroupa IMF (or other similar IMF that lead to lower $\Upsilon_*$ than the Salpeter one), also in accordance with the findings of Ref. \cite{Rodrigues:2009vf}.
In the case of NGC 4374, the RGGR fit was clearly better than the MOND's one due to a number of features. The discrepancies between MOND and the observational data are significantly enhanced when a $\Upsilon_*$ compatible with the Kroupa IMF is assumed.
In additional to the standard procedure of using $\Upsilon_*$ as a free parameter, and only evaluating a posteriori whether the fitting procedure yielded reasonable results, additional fits were generated in which the expected value of $\Upsilon_*$ was part of the data do be fitted. This procedure, in comparison with simply fixing the value of $\Upsilon_*$ a priori, is useful to avoid being unphysically precise on the value of $\Upsilon_*$.
The best way to compare different models is to use the same assumptions and procedures whenever possible. This methodology was used here for Newtonian gravity, MOND and RGGR. In Refs. \cite{2009MNRAS.393..329N, 2011MNRAS.411.2035N} the NFW dark matter halo was analyzed for the same galaxies, but the higher order Jeans equations together with the kurtosis data were used as part of the data to be fitted. In particular, the observational kurtosis can put constraints on the anisotropy parameter $\beta$, once the distribution function is assumed to satisfy $f(E,L) = f_0(E) L^{- 2 \beta}$ for constant $\beta$ (for further details, see the Appendix B of Ref. \cite{2009MNRAS.393..329N} and references therein). In those references, the NFW halo fits were found to be compatible with the N-body simulations expectations and the Kroupa IMF, considering the effects of adiabatic contraction together with a crescent anisotropy along the radius (albeit with a low concentration value for NGC 4494). Comparing the shape of the VD curves, the NFW halo could achieve about the same or a better concordance with the observational data than RGGR (remember that NFW halos use one more free parameter than RGGR). It is not easy to draw a straight comparison between the anisotropy parameters, but RGGR has shown a tendency towards isotropy while being only mildly dependent on the precise value of $\beta$ (since the 1 $\sigma$ uncertainties for RGGR are about or larger than 0.5). Had the kurtosis been evaluated as part of the data to be fitted, that would probably insert a tendency towards higher radial anisotropy. Finally, considering the stellar mass-to-light ratios ($\Upsilon_*$), albeit both NFW and RGGR are compatible with the Kroupa IMF expectations, RGGR shows a tendency in these galaxies towards lower $\Upsilon_*$ than NFW. And that may eventually constitute a physical test for RGGR, once the stellar populations models become more precise.
Another well known gravitational theory, $f(R)$ gravity, had its consequences for elliptical galaxies recently evaluated in Ref. \cite{2012ApJ...748...87N}. The sample of data that we use is similar to theirs. The VD curve that they find is similar to the one derived for RGGR, while two clear differences are the $\Upsilon_*$ of NGC 4374 (RGGR has a lower value, more in the middle of the Kroupa IMF expectation) and the $\beta$ of NGC 4494 (since $f(R)$ shows a significative tendency towards radial anisotropy). It would be interesting to compare these models using a larger sample.
There is a number of natural developments for RGGR, both from the theoretical side and the phenomenological one. In particular, within the context of galaxy kinematics, it is desirable to better constrain the $\alpha$ variation. This can be achieved from the analysis of a larger sample of galaxies. The $\alpha$ variation with the galaxy parameters is not currently well known, but it should be stressed that RGGR without dark matter (or with a relatively small amount of it inside galaxies) imply the existence of correlations between dark matter-like effects and baryonic matter. The existence of some correlations are well known for a long time, in particular here we briefly explored the correlations implied by the fundamental plane. It might result that no sufficiently strong correlation compatible with an $\alpha$ as a reasonable function of $M_*, R_e, n$ to be found, hence galaxy kinematics alone (even if all the individual fits of galaxies are satisfactory) may refute, or significantly corroborate, this approach of RGGR without dark matter. Other developments at cluster and cosmological scales and on lensing effects are being done.
\vspace{.2in}
\acknowledgments
I am grateful to Ilya Shapiro for useful discussions on the renormalization group and for commenting on a preliminary version of this paper; to Nicola Napolitano for useful discussions on the analysis of elliptical galaxies and for providing data on NGC 4374, NGC 4494; to Oliver Piattella and J\'ulio Fabris for important discussions on conformal transformations; and I also thank CNPq and FAPES for partial financial support.
|
{
"timestamp": "2012-09-05T02:00:38",
"yymm": "1203",
"arxiv_id": "1203.2286",
"language": "en",
"url": "https://arxiv.org/abs/1203.2286"
}
|
\section{Introduction}
\label{s1}
The thermal history of nonbaryonic dark-matter (DM) species is highly relevant
to the shaping of the universe as we find it today. The existence of DM is based
on evidence at many length scales. At the scale of galactic halos, for example,
DM explains the observed flatness of the rotation curves of spiral galaxies
\cite{R1}. According to observations over the past twelve years, 23\% of the
energy of the universe consists of DM. This number has been obtained by best-fit
analyses of astrophysical data to the Standard Cosmological Model, which is a
Friedmann-Robertson-Walker cosmology involving cold DM as the dominant DM
species. The modern data is based on observations of type-Ia supernovae
\cite{R2}, the cosmic microwave background \cite{R3,R4}, baryon oscillations
\cite{R5}, and weak-lensing data \cite{R6}. It should be stressed that estimates
of the DM abundance depend crucially on the theoretical model that is
considered.
In the absence of dilaton effects from string theory, the evolution of the
appropriately normalized number density $Y(x)$ of a DM species $X$ of mass $m_X$
is governed by the Boltzmann equation
\begin{equation}
Y'(x)=-\lambda x^{-n-2}\left[Y^2(x)-Y_{\rm eq}^2(x)\right],
\label{e1}
\end{equation}
which is a Riccati equation in the dimensionless independent variable $x\equiv
m_X/T$, where $T$ is the temperature. The parameter $\lambda$ is a dimensionless
measure of the scattering of DM particles and is regarded as a large number
$\lambda\gg1$. The integer $n=0,\,1,\,2,\,\ldots$ comes from a partial-wave
analysis of the scattering of DM particles; $n=0$ refers to $S$-wave scattering.
For bosonic remnants the function $Y_{\rm eq}(x)$ is the distribution \cite{R7}
\begin{equation}
Y_{\rm eq}(x)=A\int_0^\infty ds\frac{s^2}{e^{\sqrt{s^2+x^2}}-1},
\label{e2}
\end{equation}
where $A=0.145 g/g_*$, $g$ is the degeneracy factor for the DM species, and
$g_*$ counts the total number of massless degrees of freedom \cite{R8}.
As the universe cools and $x$ increases, the nature of the solution $Y(x)$ to
(\ref{e1}) changes rapidly in the vicinity of a value $x=x_f$, the so-called
{\it freeze-out} point, and as $x\to\infty$ the solution $Y(x)$ approaches the
constant $Y_\infty$, called the {\it relic abundance}. Because a closed-form
analytical solution to this Riccati equation is unavailable, an approximate
heuristic approach is customarily used to treat this Riccati equation: One
approximation is made for $x<x_f$ and another is made for $x>x_f$. The solutions
in the two regions are then patched at $x=x_f$. This approach gives an intuitive
and reasonably accurate determination of $Y_\infty$ and it is widely adopted
\cite{R8}.
However, this splitting into two regions is only a mathematical convenience and
there is really no precise value $x_f$. Because the differential equation
(\ref{e1}) is first order, its solution is completely determined by one initial
condition, namely $Y(0)$. The usual method of splitting (\ref{e1}) into two
approximate first-order equations, which are valid in each of two regions, leads
to two conditions, an initial condition and a patching condition. We feel that
this gives rise to an unsatisfactory mathematical discussion that is prevalent
in the literature. The value of $x_f$, for example, becomes explicitly involved
in the determination of $Y_{\infty}$ when there is no reason for this.
Equation (\ref{e1}) is valid in a general-relativistic framework. However, given
the importance of understanding the current thermal-relic abundance of DM in
theories beyond the standard model of particle physics, we also reexamine here
the modifications of (\ref{e1}) due to string cosmology \cite{R9}. String theory
is widely accepted as a leading candidate for physics beyond the standard model,
and it places a constraint on the types of time-dependent backgrounds in
conformally invariant critical theories. As before, we are interested in eras in
which the temperature $T$ satisfies $m_X>T>T_0$, where $T_0$ is the current
temperature of the universe. String cosmology leads to a rolling dilaton source
in the Boltzmann equation \cite{R10} that describes DM species. Including this
source gives an additional linear term in the Boltzmann equation:
\begin{equation}
Y'(x)=-\lambda x^{-n-2}\left[Y^2(x)-Y_{\rm eq}^2(x)\right]+\Phi_0 Y(x)/x,
\label{e3}
\end{equation}
where $\Phi_0$ is a negative dimensionless constant of order 1.
The purpose of this paper is to study analytically the two Riccati equations
(\ref{e1}) and (\ref{e3}). These equations do not have exact closed-form
solutions. However, because $\lambda$ is a large parameter, one can attempt to
find asymptotic approximations to the solutions. The most direct approach is to
convert these Riccati equations into equations of Schr\"odinger type. When this
transformation is applied to (\ref{e1}), we obtain
\begin{equation}
v''(x)-\frac{n(n+2)}{4x^2}v(x)-\lambda^2x^{-2n-4}Y_{\rm eq}^2(x)v=0.
\label{e4}
\end{equation}
Now, if we set $n=0$, we obtain the standard time-independent Schr\"odinger
equation in which $1/\lambda$ plays the role of $\hbar$. While it is possible to
perform a local analysis of this equation for small $x$ and for large $x$, it
is not easy to use WKB analysis to find a global asymptotic approximation
because the equation is singular at $x=0$ and there is a turning point at $x=
\infty$.
Thus, in this paper we will use two other powerful asymptotic methods from which
we can extract global information. The first method is boundary-layer analysis.
This asymptotic technique, which has been used to solve approximately the
equations of fluid mechanics, gives very accurate results, and it has the
physical advantage of treating freeze-out as a boundary-layer region, very much
like the boundary between two fluids. The second technique, known as the {\it
delta expansion} \cite{R11}, is particularly well-suited to study the transition
from the equilibrium region to the large-$x$ behavior of the solutions without
the necessity of finding approximations to the Boltzmann equation in different
epochs. We will see that the presence of a source in (\ref{e3}) gives a solution
for $Y(x)$ in (\ref{e3}), whose qualitative behavior is significantly different
from the solution for $Y(x)$ in (\ref{e1}).
This paper is organized as follows. In Sec.~\ref{s2} we summarize the derivation
of the Boltzmann equations (\ref{e1}) and (\ref{e3}). In Sec.~\ref{s3} we apply
boundary-layer analysis to study (\ref{e1}) and (\ref{e3}). Next, in
Sec.~\ref{s4} we describe the delta expansion and then use it to study the
approximate behaviors of (\ref{e1}) and (\ref{e3}). Finally, in Sec.~\ref{s5} we
give some brief concluding remarks.
\section{Derivation of the Boltzmann equations}
\label{s2}
In this section we review the derivation of the two Boltzmann equations
(\ref{e1}) and (\ref{e3}).
\subsection{Derivation of (\ref{e1})}
\label{ss2a}
In the hot early universe DM particles interact with themselves and with other
particles. Particle species are assumed to react rapidly enough to maintain
equilibrium. However, the universe expands and cools throughout its history. The
timescale associated with this expansion is determined by the Hubble rate $H$.
There is also a timescale $\Gamma$ associated with the scattering cross-section
(that is, an interaction rate per particle). The dynamics of DM particles
depends on the ratio $\Gamma/H$. When $\Gamma/H\gg1$, conditions for equilibrium
hold and $Y(x)$ follows the canonical distribution obtained from equilibrium
statistical mechanics. However, for $\Gamma/H\ll1$ the DM particles are unable
to maintain equilibrium. There is a crossover to freeze-out behavior in which
$Y(x)$ is asymptotically a constant.
Let us consider a two-body scattering process in which particles of species 1
and 2 scatter reversibly into particles of species 3 and 4. The phase-space
distribution function $f_i\left(\vec{r},\vec{p},t\right)$ for the species $i$
gives the number of particles in an infinitesimal region of phase space around
the position $\vec{r}$ and momentum $\vec{p}$: $f_i\left(\vec{r},\vec{p},t\right
)d^3r\,d^3p$. The main bulk quantity of interest is the number density $n_i
\left(\vec{r},t\right)$, which is given by \cite{R8}
\begin{equation}
n_i\left(\vec{r},t\right)=g_i\int\frac{d^3 p}{(2\pi)^3}\,f_i\left(\vec{r},
\vec{p},t\right),
\label{e5}
\end{equation}
where $g_i$ is the degeneracy factor for the $i$th DM species. The evolution of
such a bulk quantity in the universe is given by the Liouville equation (in the
absence of collisions, for simplicity)
\begin{equation}
\frac{df_i}{dt}=L[f_i]\equiv\left(\frac{\partial}{\partial t}+\frac{d\vec{p}}{dt
}\cdot\nabla_{\vec{p}}+\frac{d\vec{r}}{dt}\cdot\nabla_{\vec{r}}\right)f_i=0.
\label{e6}
\end{equation}
The standard Robertson-Walker metric for an isotropic and expanding flat
universe is given by
\begin{equation}
ds^2=-dt^2+a^2(t)\left(dx^2+dy^2+dz^2\right),
\label{e7}
\end{equation}
where $a(t)$ is the scale factor \cite{R12}. The covariant generalization of
(\ref{e6}) is \cite{R8},
\begin{equation}
L\left[f_i\right]=\left(p^\mu\frac{\partial}{\partial x^\mu}-\Gamma_{\nu
\rho}^\mu p^\nu p^\rho\frac{\partial}{\partial p^\mu}\right)f_i=0,
\label{e8}
\end{equation}
where the Christoffel symbol is given by
$$\Gamma_{\nu\rho}^\mu\equiv g^{\alpha\mu}\left(g_{\alpha\nu,\rho}+
g_{\alpha\rho,\nu}-g_{\nu\rho,\alpha}\right)/2.$$
For the metric in (\ref{e7}), isotropy further implies that $f_i\left(\vec{p},\,
t\right)=f_i\left(\left|\vec{p}\right|,t\right)$. For the isotropic case
(\ref{e8}) takes the form
$$L[f(E,t)]=E\frac{\partial f}{\partial t}-\frac{\dot{a}}{a}\left|\vec{p}
\right|^2\frac{\partial f}{\partial E},$$
where $E=\sqrt{\vec{p}^2+m^2}$.
For a two-body collision process the Liouville equation (\ref{e8}) no longer has
a vanishing right side. This equation can then be used to describe the change in
the number density of a given species. For species 1, for example, one gets
\cite{R8}
\begin{eqnarray}
a^{-3}\frac{d\left(n_1a^3\right)}{dt} &=& \int\frac{d^3p_1}{(2\pi)^3 2E_1}\int
\frac{d^3 p_2}{(2\pi)^3 2E_2}\int\frac{d^3 p_3}{(2\pi)^3 2E_3}\int\frac{d^3p_4}
{(2\pi)^3 2E_4}\nonumber\\
&&\quad\times(2\pi)^4\delta^3\left(p_1+p_2-p_3-p_4\right)\delta\left(E_1+E_2-E_3
-E_4\right)|\mathcal{A}|^2\nonumber\\
&&\quad\times\left[f_3 f_4\left(1\pm f_1\right)\left(1\pm f_2\right)-f_1 f_2
\left(1\pm f_3\right)\left(1\pm f_4\right)\right],
\label{e9}
\end{eqnarray}
where the plus sign is used for a bosonic species and the minus sign is used for
a fermionic species. The symbol $\mathcal{A}$ represents the scattering
amplitude for the process $1+2\leftrightarrow 3+4$ and it is a function of the
$p_i$.
If the scattering process is sufficiently fast, $f_i$ can be parametrized by
canonical Fermi-Dirac or Bose-Einstein distributions. For temperatures $T\ll E-
\mu$ the Bose-Einstein and Fermi-Dirac distributions both take the form
\begin{equation}
f(E)\sim e^{\mu/T}e^{-E/T},
\label{e10}
\end{equation}
which implies that quantum statistics are not important. Hence, the
Pauli-blocking and Bose-enhancement are negligible ($f_i\ll 1$), and the
third line of (\ref{e9}) simplifies:
$$f_3 f_4\left(1\pm f_1\right)\left(1\pm f_2\right)-f_1 f_2\left(1\pm f_3\right)
\left(1\pm f_4\right)\sim e^{-(E_1+E_2)/T}\left[e^{(\mu_3+\mu_4)/T}-
e^{(\mu_1+\mu_2)/T}\right],$$
where the relation $E_1+E_2=E_3+E_4$ has been used. Also, combining (\ref{e5})
and (\ref{e10}), we get
$$n_i=g_i e^{\mu_i/T}\int\frac{d^3 p}{(2\pi)^3}e^{-E_i/T}.$$
The equilibrium number density in the absence of a chemical potential is denoted
by $n_i^{(0)}$. Thus,
\begin{equation}
a^{-3}\frac{d}{dt}\left(n_1 a^3\right)=n_1^{(0)}n_2^{(0)}\langle\sigma v\rangle
\left\{\frac{n_3 n_4}{n_3^{(0)}n_4^{(0)}}-\frac{n_1 n_2}{n_1^{(0)}n_2^{(0)}}
\right\},
\label{e11}
\end{equation}
where the thermally averaged annihilation cross-section $\langle\sigma v\rangle$
is given by
\begin{eqnarray}
\langle\sigma v\rangle &\equiv& \frac{1}{n_1^{(0)}n_2^{(0)}}\int\frac{d^3 p_1}
{(2\pi)^3 2E_1}\int\frac{d^3 p_2}{(2\pi)^3 2E_2}\int\frac{d^3 p_3}{(2\pi)^3 2E_3
}\int\frac{d^3 p_4}{(2\pi)^3 2E_4}e^{-\left(E_1+E_2\right)/T}\nonumber\\
&&\quad\times(2\pi)^4\delta^3\left(p_1+p_2-p_3-p_4\right)\delta\left(E_1+E_2-E_3
-E_4\right)|\mathcal{A}|^2.
\label{e12}
\end{eqnarray}
We now make the standard assumption \cite{R8} that the predominant interaction
of the cold DM species $X$ of mass $m_X$ is $XX\leftrightarrow ll$, where $l$ is
a light particle in equilibrium. As a consequence, in (\ref{e11}) we can replace
$n_1$ and $n_2$ by $n_X$, where $n_X$ is the number density of the species $X$.
Also, we replace and $n_3$ and $n_4$ by $n_l^{(0)}$. The resulting equation is
\begin{equation}
a^{-3}\frac{d}{dt}\left(n_X a^3\right)=
\langle\sigma v\rangle\left[\left(n_X^{(0)}\right)^2-n_X^2\right].
\label{e13}
\end{equation}
We now define $x\equiv m/T$ and note that $dx/x=-dT/T=da/a$ because $T$ scales
as $1/a$. Thus, $\frac{dx}{dt}=Hx$, where the Hubble rate $H\equiv\frac{d}{dt}
\log(a)$. Since the cosmological era for DM production is radiation dominated,
$a(t)\propto\sqrt{t}$. This translates into $H=H_m/x^2$ with $H_m=1.67g_*^{1/2}
m_X^2/m_{\rm Planck}$. It is known theoretically \cite{R8} that $\sigma v\propto
v^{2n}$ with $n=0$ for $s$-wave annihilation and $n=1$ for $p$-wave
annihilation. Since $\langle v\rangle\propto\sqrt{T}$, we have the
parametrization $\langle\sigma v\rangle=\sigma_0x^{-n}$ for $x\geq3$ \cite{R8}.
Finally, we introduce the dependent variable $Y\equiv n_X/T^3\propto n_Xa^3$.
Similarly, we define $Y_{\rm eq}\equiv n_X^{(0)}/T^3$. We then obtain the
Boltzmann equation in (\ref{e1}), where $\lambda\equiv\sigma_0 m_X^3/H_m\propto
m_{\rm Planck}/m_X$, and this explains why $\lambda$ is a large dimensionless
parameter \cite{R13}.
\subsection{Derivation of (\ref{e3})}
\label{ss2b}
String theory can be formulated in nonflat backgrounds, which is necessary when
considering cosmology. Here, we consider the world-sheet sigma-model approach
for dilaton-based cosmologies \cite{R9}. In superstring theory the bosonic part
of the supermultiplet with lowest energy consists of the following massless
states: the graviton $g_{MN}$, the spinless dilaton $\Phi$, and the
antisymmetric spin-one tensor $B_{MN}$. For expanding universes $\Phi$ provides
consistent time-dependent backgrounds. In such backgrounds the string sigma
model on the world sheet $\Sigma$ is given by \cite{R14}
\begin{equation}
S_\sigma=\int_\Sigma\frac{d^2\sigma}{4\pi\alpha'}\left[\sqrt{\gamma}\,
\gamma^{\alpha\beta}g_{MN}(X)\partial_\alpha X^M\partial_\beta X^N+B_{MN}(X)
\epsilon^{\alpha\beta}\partial_\alpha X^M\partial_\beta X^N+\alpha'\sqrt{\gamma}
\Phi(X)R^{(2)}/2\right],
\label{e14}
\end{equation}
where $X^M$ are target space-time coordinates with $M,N=0,\,1,\,\ldots,\,9$,
$\sigma^\alpha$ are the world-sheet coordinates with $\alpha,\beta=0,\,1$,
$\gamma^{\alpha\beta}$ is the world-sheet metric, $\gamma=\left|\det\left(\gamma
^{\alpha\beta}\right)\right|$, $R^{(2)}$ is the Ricci scalar associated with
$\gamma^{\alpha\beta}$, and $\alpha'$ is the string slope. Expanding around a
conformal flat background with the action $S^*$, we can write $S_\sigma$ as
\begin{equation}
S_\sigma=S^{*}+h^i\int_\Sigma d^2\sigma\,V_i,
\label{e15}
\end{equation}
where $h^i$ denotes the background fields $\left\{g_{MN},B_{MN},\Phi \right\}$
and $V_i$ are associated vertex operators \cite{R14}.
Short-distance singularities of the quantum field theory on the world sheet lead
to renormalized couplings $\left\{h_R^i\right\}$ and to dependence on the
renormalization-group scale $\mu$ \cite{R15}. Usually, this results in
nonvanishing $\beta$ functions: $\beta^i\equiv dh_R^i/d\log\mu$. To restore
conformal invariance, these $\beta$ functions must vanish. This leads to
equations of motion satisfied by the background fields. The usual procedure is
to consider an effective target-space action in the string frame that reproduces
the equations of motion:
\begin{equation}
S=-\frac{1}{2\alpha'^4}\int d^{10}x\,\sqrt{G}\,e^{-\Phi}\left[R+(\nabla\Phi)^2+
2\alpha'^4 U(\Phi)-\tilde{H}^2/12\right],
\label{e16}
\end{equation}
where $\tilde{H}^2=H_{\mu\nu\alpha}H^{\mu\nu\alpha}$, $H_{\mu\nu\alpha}\equiv
\partial_\mu B_{\nu\alpha}+\partial_\nu B_{\alpha\mu}+\partial_\alpha B_{\mu\nu
}$, and the potential $U(\Phi)$ has been introduced. With the help of duality
symmetries it is possible to find analytic solutions for the time dependence of
the dilaton field \cite{R9}.
From (\ref{e16}) it can be shown \cite{R10} that in three spatial dimensions the
energy density $\rho$ of the DM species $X$ satisfies
\begin{equation}
\frac{d\rho}{dt}+3H(\rho+p)-\frac{d\Phi}{dt}(\rho-3p)=0.
\label{e17}
\end{equation}
We then assume that the thermal DM species $X$ behaves like dust (that is, $p=
0$) and that the energy density of the DM is given by the simple formula $\rho=
m_X n_X$. Next, in place of the $0$ on the right side of (\ref{e17}), we include
a collision term, which is just the right side of (\ref{e13}):
\begin{equation}
\frac{d}{dt}n_X+3Hn_X-\frac{d\Phi}{dt}n_X=
\langle\sigma v\rangle\left[\left(n_X^{(0)}\right)^2-n_X^2\right].
\label{e18}
\end{equation}
Assuming that matter sources are perfect fluids and requiring scale-factor
duality symmetry, one can show \cite{R9} that up to an additive constant, $\Phi
(t)=\Phi_0\log a(t)$ where $\Phi_0={\rm O}(1)$ and $\Phi_0<0$.
Finally, we make the assumption that the behavior of the DM species is dominated
by radiation so that $a(t)\propto t^{1/2}$ \cite{R8}. As in Subsec.~\ref{ss2a},
we introduce the variables $Y(x)$ and $x$ and obtain the Boltzmann equation
(\ref{e3}).
\section{Boundary-layer solution to (\ref{e1}) and (\ref{e3})}
\label{s3}
In this section we show how to perform a boundary-layer asymptotic analysis of
(\ref{e1}) and (\ref{e3}). The advantage of this analysis is that it provides a
global picture of the cosmological development from the initial time to the
present as described by the Boltzmann equation, and not just the physics of the
equilibrium epoch or of the post-equilibrium epoch alone. It also establishes a
framework to describe in a clear and natural way the region of rapid transition
between these two epochs. A satisfactory description of this crucial transition
is lacking in earlier treatments in the literature because the earlier analysis
used {\it patching} (joining together two solutions to a differential equation
at an arbitrary and fictitious point, which produces an elbow in the solution)
rather than {\it asymptotic matching} \cite{R16}.
One may wonder why an asymptotic procedure as powerful as boundary-layer theory
should be used to solve a first-order ordinary differential equation as simple
as a Riccati equation. A general Riccati equation
$$y'(x)=a(x)y^2(x)+b(x)y(x)+c(x)$$
can be recast as a linear second-order equation,
\begin{equation}
w''(x)-\left[\frac{a'(x)}{a(x)}+b(x)\right]w'(x)+a(x)c(x)w(x)=0,
\label{e19}
\end{equation}
where $y(x)=-\frac{w'(x)}{a(x)w(x)}$.
Furthermore, (\ref{e19}) can be recast as a Schr\"odinger equation
\begin{equation}
v''(x)+\left\{p''(x)/p(x)-[b(x)+a'(x)/a(x)]^2/2+a(x)c(x)\right\}v(x)=0
\label{e20}
\end{equation}
by introducing $v(x)=w(x)/p(x)$, where $p'(x)/p(x)=[b(x)+a'(x)/a(x)]/2$.
The form (\ref{e20}) is often useful for asymptotic WKB analysis but the problem
of freeze-out poses mathematical difficulties. If we apply these transformations
to (\ref{e1}) for the case $n=0$ and use the leading asymptotic forms for
$Y_{\rm eq}(x)$ in Appendices A and B, we obtain the Schr\"odinger equations
\begin{equation}
v''(x)-\lambda^2 x^{-4}\eta^2v(x)=0,
\label{e21}
\end{equation}
where $\eta\equiv 2A\zeta(3)$ for $x\ll1$, and
\begin{equation}
v''(x)-\lambda^2A^2 x^{-1}e^{-2x}v(x)=0
\label{e22}
\end{equation}
for $x\gg1$. The role of $\hbar$ in these equations is played by $1/\lambda$
because $\lambda$ is treated as a large parameter.
The exact general solution of (\ref{e21}) is
\begin{equation}
v(x)=x\left(v_+e^{\lambda\eta/x}+v_-e^{-\lambda\eta/x}\right),
\label{e23}
\end{equation}
where $v_+$and $v_-$ are constants. The approximate general solution to
(\ref{e22}) can be found by using a standard application of WKB \cite{R16}. [A
detailed analysis of (\ref{e22}) for large $x$ is given in Appendix C.] However,
because (\ref{e22}) has a turning point at $x=\infty$ and because (\ref{e21})
has a singularity at $x=0$, it is very difficult to construct a uniform
asymptotic expansion that is valid for all $x$. We show below that
boundary-layer theory overcomes these difficulties.
\subsection{Boundary-layer analysis of (\ref{e1})}
\label{ss3a}
Whenever the highest-derivative term in a differential equation is multiplied by
a small parameter, one can attempt a boundary-layer analysis \cite{R16}. In such
an analysis one identifies an {\it outer} region (or regions) in which the
solution is slowly varying and an {\it inner} or {\it boundary-layer} region (or
regions) in which the solution is rapidly varying. If these regions have an
overlap, one tries to construct a global asymptotic approximation to the
differential equation by performing an asymptotic match of the outer solutions
to the inner solutions.
In boundary-layer form the derivative term in (\ref{e1}) is multiplied by $1/
\lambda$, which is regarded as small ($1/\lambda\ll1$). Thus, we begin by
looking for an outer solution; that is, a solution whose derivative is not
large. To leading order such a solution in the outer region satisfies a {\it
distinguished limit} (an asymptotic balance between two of the three terms in
the differential equation) in which we neglect the derivative term as $\lambda
\to\infty$:
\begin{equation}
Y(x)\sim Y_{\rm eq}(x)\quad(\lambda\to\infty).
\label{e24}
\end{equation}
Since $Y(x)$ is well approximated by $Y_{\rm eq}(x)$ in this region, we call
this outer region the {\it thermal-equilibrium} region.
To higher order, we seek a series expansion of this thermal-equilibrium outer
solution as a formal power series in inverse powers of $\lambda$:
\begin{equation}
Y^{\rm thermal-equilibrium}(x)\sim\sum_{k=0}^\infty\lambda^{-k}Y_k^{\rm
thermal-equilibrium}(x).
\label{e25}
\end{equation}
Substituting this series into (\ref{e1}) and collecting powers of $1/\lambda$
yields the higher-order terms in the outer series. For example, to first order
we get
\begin{equation}
Y_1^{\rm thermal-equilibrium}(x)=-\frac{1}{2}x^{n+2}\frac{d}{dx}\log\left[Y_{\rm
eq}(x)\right].
\label{e26}
\end{equation}
We must now determine the extent of the thermal-equilibrium region. We know from
Appendix A that for large $x$, $x\gg1$, the asymptotic behavior of $Y_{\rm eq}
(x)$ is given by
\begin{equation}
Y_{\rm eq}(x)\sim Ae^{-x}x^{3/2}\qquad(x\to\infty).
\label{e27}
\end{equation}
Thus, for large $x$ in the outer region
\begin{equation}
Y_0^{\rm thermal-equilibrium}(x)\sim Ae^{-x}x^{3/2}\quad{\rm and}\quad Y_1^{\rm
thermal-equilibrium}(x)\sim\frac{1}{2}x^{n+2}.
\label{e28}
\end{equation}
Hence, the second term in the outer series is no longer small compared with the
first term when
\begin{equation}
x\sim\log(2A\lambda)-(n+1/2)\log(x).
\label{e29}
\end{equation}
We will call the solution to this asymptotic relation the so-called {\it
freeze-out} value $x_{\rm f}$:
\begin{equation}
x_{\rm f}\sim\log(2A\lambda)-(n+1/2)\log\left(x_{\rm f}\right).
\label{e30}
\end{equation}
Note that if we take $\lambda\approx 10^{14}$ and $A\approx0.00145$, we see that
the outer asymptotic approximation ceases to be valid when $x$ exceeds the
approximate numerical value
\begin{equation}
x_{\rm f}\approx25.
\label{e31}
\end{equation}
Equation (\ref{e29}) defines the upper asymptotic limit of the
thermal-equilibrium region. However, it is important to emphasize here that {\it
freeze-out does not occur at a point}; $x_{\rm f}$ should not be viewed as a
number but rather as a large range of values of $x$ all satisfying the
asymptotic relation (\ref{e29}):
\begin{equation}
x\sim x_{\rm f}\quad(\lambda\to\infty).
\label{e32}
\end{equation}
A second possible distinguished limit of (\ref{e1}) could in principle consist
of an asymptotic balance between the left side and the second term on the right
side. However, this distinguished limit is inconsistent and must be rejected
because we are led to a contradiction: If we solve the resulting equation, we
find that for large $\lambda$ the first term on the right side is in fact {\it
not negligible} compared with the second term .
A third distinguished limit of (\ref{e1}) occurs when $x$ is so large that the
contribution of the equilibrium term $Y_{\rm eq}^2(x)$ is negligible. In this
case, the left side is asymptotic to the first term on the right side:
\begin{equation}
Y'(x)\sim -\lambda x^{-n-2}Y^2(x)\qquad(x\gg1).
\label{e33}
\end{equation}
In this second outer region, which we will call the {\it post-freeze-out}
region, the solution $Y^{\rm post-freeze-out}(x)$ to (\ref{e33}) is
\begin{equation}
Y^{\rm post-freeze-out}(x)\sim\frac{1}{1/C-\lambda x^{-n-1}/(n+1)},
\label{e34}
\end{equation}
where $C$ is a constant of integration to be determined. Note that this solution
is consistent and valid when $x\gg1$ because $Y_{\rm eq}(x)$ is exponentially
small when $x\gg1$. Note also that as $x\to\infty$, $Y^{\rm post-freeze-out}(x)$
approaches the limiting value $C$. Thus, $C$ represents the long-time limiting
value of the relic abundance.
The physical process of freeze-out can be recast in mathematical terms as a
process that occurs in an inner region (or boundary layer), which we treat as a
time interval that is comparatively short relative to the time intervals of
the two outer regions, the thermal-equilibrium region and the post-freeze-out
region. We begin the analysis of the freeze-out boundary layer by determining
the size of this region. To do so, we introduce the {\it inner variable} $X$:
\begin{equation}
x=x_{\rm f}+\kappa X.
\label{e35}
\end{equation}
We regard $|X|$ as a variable that may get large compared to $1$, say as large
as $X_{\rm max}$, but $X$ is still small compared with $\lambda$. Thus, since
$\kappa$ is expected to be a small parameter roughly of order $1/\lambda$, the
boundary layer is narrow because it extends roughly from $x_{\rm f}-\kappa
X_{\rm max}$ to $x_{\rm f}+\kappa X_{\rm max}$.
Making the change of variables (\ref{e35}), from which we get
\begin{equation}
\frac{d}{dx}=\frac{1}{\kappa}\frac{d}{dX},
\label{e36}
\end{equation}
and treating $\kappa X$ as small compared with $x_{\rm f}$, we find that
(\ref{e1}) becomes
\begin{equation}
\frac{1}{\kappa}\mathcal{Y}'(X)=-\lambda x_{\rm f}^{-n-2}\left[\mathcal{Y}^2(X)-
A^2x_{\rm f}^3e^{-2x_{\rm f}}\right],
\label{e37}
\end{equation}
where $\mathcal{Y}(X)=Y(x)$. A consistent dominant balance in this equation is
achieved if we take
\begin{equation}
\kappa=x_{\rm f}^{n+2}/\lambda,
\label{e38}
\end{equation}
and if we make this choice, we must neglect the second term on the right side
because it is of order $\lambda^{-2}$ compared with the first term on the right
side. This gives the simple inner differential equation
\begin{equation}
\mathcal{Y}'(X)=-\mathcal{Y}^2(X),
\label{e39}
\end{equation}
whose solution is
\begin{equation}
\mathcal{Y}(X)=\frac{1}{X+D},
\label{e40}
\end{equation}
where $D$ is an integration constant.
To complete the boundary-layer analysis, we must match the two outer solutions
to this boundary-layer solution. In order to perform the asymptotic match, we
re-express the outer solutions in terms of the inner variable $X$ and then
carry out an asymptotic approximation valid for small $\kappa$ to these
asymptotic approximations.
Let us look first at the outer solution in the post-freeze-out region:
\begin{equation}
Y^{\rm post-freeze-out}(X)\sim\frac{1}{1/C-\lambda \left(x_{\rm f}+\kappa X
\right)^{-n-1}/(n+1)},
\label{e41}
\end{equation}
which simplifies to
\begin{equation}
Y^{\rm post-freeze-out}(X)\sim\frac{1}{X+\frac{1}{C}-\frac{\lambda}{(n+1)\left(
x_{\rm f}\right)^{n+1}}}.
\label{e42}
\end{equation}
The coefficient of $X$ in the denominator is $1$, which agrees exactly with the
coefficient of $X$ in the inner solution (\ref{e40}). Thus, we have achieved an
asymptotic match, and the matching condition relates the constants $C$ and $D$:
\begin{equation}
D=\frac{1}{C}-\frac{\lambda}{(n+1)x_{\rm f}^{n+1}}.
\label{e43}
\end{equation}
Next, we match the boundary-layer solution in (\ref{e40}) to the outer solution
(\ref{e25}) in the thermal-equilibrium region. To do so, we must re-express the
outer solution in (\ref{e25}) in terms of the inner variable $X$. Although we
are matching to just one term of the inner freeze-out solution, it is essential
that we take the first {\it two} terms in the outer thermal-equilibrium series,
and not just the first term, because we have shown that as we approach the
freeze-out region, the first two terms in the outer solution become comparable
in size. Thus, we include a factor of two in the asymptotic behavior
\begin{eqnarray}
Y^{\rm thermal-equilibrium}(x)&\sim&2Ax^{3/2}e^{-x}\nonumber\\
&\sim&2A\left(x_{\rm f}+\kappa X\right)^{3/2}e^{-x_{\rm f}}e^{-\kappa X}
\nonumber\\
&\sim& \frac{1}{X+\frac{\lambda}{x_{\rm f}^{n+2}}}.
\label{e44}
\end{eqnarray}
Because the coefficient of $X$ in this behavior is $1$, we obtain once again a
perfect asymptotic match to the inner freeze-out solution in (\ref{e40}).
This allows us to determine the value of the constant $D$:
\begin{equation}
D=\lambda x_{\rm f}^{-n-2}.
\label{e45}
\end{equation}
Finally, combining this result with (\ref{e43}), we obtain the value of $C$:
\begin{equation}
C=\frac{(n+1)x_{\rm f}^{n+2}}{\lambda\left(n+1+x_{\rm f}\right)},
\label{e46}
\end{equation}
which is our result for the thermal-relic abundance. For $x_{\rm f}$ large
compared with $n+1$ this is in close agreement with the value $(n+1)x_{\rm f}^{
n+1}/\lambda$ given in Ref.~\cite{R8}.
\subsection{Boundary-layer analysis of (\ref{e3})}
\label{ss3b}
The arguments given in Subsec.~\ref{ss3a} apply to a modified version of
(\ref{e3}). We modify (\ref{e3}) as follows. If we let $\varphi=\left|\Phi_0
\right|$, then the substitution
$$Z(x)=Y(x)x^\varphi$$
reduces (\ref{e3}) to the simpler Riccati equation
\begin{equation}
Z'(x)=-\lambda x^{-n-2}\left[x^{-\varphi}Z^2(x)-x^\varphi Y_{\rm eq}^2(x)\right]
.
\label{e47}
\end{equation}
The advantage of this equation over (\ref{e3}) is that there are only three
rather than four terms, and thus it is easier to identify a dominant balance.
We can now analyze (\ref{e47}) using the procedure adopted in the previous
subsection. In the left outer region (the thermal-equilibrium region) we have
\begin{equation}
Z_0^{\rm thermal-equilibrium}(x)\sim Ae^{-x}x^{\varphi+3/2}\quad{\rm and}\quad
Z_1^{\rm thermal-equilibrium}(x)\sim\frac{1}{2}x^{\varphi+n+2}.
\label{e48}
\end{equation}
From this result we deduce that the freeze-out value $x_{\rm f}$ is given by
$$x_{\rm f}\sim\log(2A\lambda)-(n+1/2)\log\left(x_{\rm f}\right),$$
which is identical to the result in (\ref{e30}). This result shows that to
leading order in $1/\lambda$ the freeze-out temperature is independent of
$\Phi_0$; that is, the location of the freeze-out region is only weakly affected
by the presence of a dilaton.
Next we discuss the right outer region (post-freeze region). The analog of
(\ref{e34}) is
\begin{equation}
Z^{\rm post-freeze-out}(x)\sim\frac{1}{1/C-\lambda x^{-n-1-\varphi}/(n+1+
\varphi)},
\label{e49}
\end{equation}
where $C$ is a constant of integration to be determined by asymptotic matching.
As before, $C$ describes the long-term-abundance behavior. However, when $x$ is
large compared with the freeze-out temperature $\left(x\gg x_{\rm f}\right)$,
$Y(x)$ does not approach a constant. Rather,
\begin{equation}
Y(x)\sim x^{-\varphi}Z(x)\sim x^{-\varphi}C\quad(x\to\infty).
\label{e50}
\end{equation}
In the freeze-out boundary-layer region we again make the change of
variable in (\ref{e35}),
$$x=x_{\rm f}+\kappa X,$$
where the inner variable $X$ may become large compared to $1$, but it is still
small compared with $\lambda$. Thus, since $\kappa$ is expected to be a small
parameter of order $1/\lambda$, the boundary layer is narrow as before. A
consistent dominant-balance gives the value
\begin{equation}
\kappa=x_{\rm f}^{n+2+\varphi}/\lambda.
\label{e51}
\end{equation}
The inner differential equation then has the form
\begin{equation}
\mathcal{Z}'(X)=-\mathcal{Z}^2(X),
\label{e52}
\end{equation}
where $\mathcal{Z}(X)=Z(x)$. The solution to (\ref{e52}) is
\begin{equation}
\mathcal{Z}(X)=\frac{1}{X+D},
\label{e53}
\end{equation}
where $D$ is an integration constant. This is the analog of (\ref{e40}).
An asymptotic match of the right outer solution to the boundary-layer
solution produces the relation between the constants $C$ and $D$,
\begin{equation}
D=\frac{1}{C}-\frac{\lambda}{(n+1+\varphi)x_{\rm f}^{n+1+\varphi}},
\label{e54}
\end{equation}
which is the analog of (\ref{e43}). Finally, by matching the left outer
solution to the boundary-layer solution, we obtain the value of $C$:
\begin{equation}
C=\frac{(n+1+\varphi)x_{\rm f}^{n+2+\varphi}}{\lambda\left(n+1+\varphi+x_{\rm f}
\right)}.
\label{e55}
\end{equation}
In conclusion, we find that, due to the presence of a dilation, the
thermal-relic abundance in (\ref{e50}) remains {\it time dependent}; it vanishes
as $x\to\infty$ and does not approach a constant. Note also that if we eliminate
the effect of the dilaton by allowing $\Phi_0$ to approach $0$, the results in
(\ref{e50}) and (\ref{e55}) smoothly reduce to that in Subsec.~\ref{ss3a}).
\section{Application of the Delta expansion to (\ref{e1}) and (\ref{e3})}
\label{s4}
In this section we show how to apply the delta expansion to (\ref{e1}) and
(\ref{e3}). We begin with a brief summary of the delta-expansion technique.
\subsection{Summary of the delta expansion}
\label{ss4a}
The delta expansion is an unconventional perturbative technique for solving
nonlinear problems. It was first introduced to treat nonlinear aspects of
quantum field theory \cite{R17}. To prepare for applying it to the Boltzmann
equations (\ref{e1}) and (\ref{e3}), in this subsection we give a brief review
of the delta expansion.
The theme of the delta expansion is to introduce a parameter $\delta$ as a
measure of the nonlinearity of a problem; that is, the departure of the problem
from a corresponding linear problem. We then treat $\delta$ as small ($\delta
\ll1$), and solve the problem perturbatively by expanding about the linear
problem obtained by setting $\delta=0$. The basic ideas of the delta expansion
are explained in Ref.~\cite{R11}.
To illustrate the delta expansion, we consider the Thomas-Fermi nonlinear
boundary-value problem
\begin{equation}
y''(x)=[y(x)]^{3/2}/\sqrt{x},\qquad y(0)=1,~y(+\infty)=0.
\label{e56}
\end{equation}
This problem is extremely difficult and no closed-form analytical solution is
known. We introduce the parameter $\delta$ in the exponent of the nonlinear term
of the differential equation and consider the one-parameter family of problems
\begin{equation}
y''(x)=y(x)[y(x)/x]^\delta,\qquad y(0)=1,~y(+\infty)=0,
\label{e57}
\end{equation}
where we treat $\delta$ as a small perturbation parameter. The solution to the
unperturbed ($\delta=0$) linear problem is $y_0(x)=e^{-x}$, and we use $y_0(x)$
as the first term in the delta expansion of the solution to the nonlinear
problem (\ref{e57}):
\begin{equation}
y(x)=\sum_{k=0}^\infty\delta^k y_k(x).
\label{e58}
\end{equation}
Finally, we recover the solution to the original Thomas-Fermi problem by
setting $\delta=1/2$. Typically, only very few terms are needed in the delta
expansion to recover accurate numerical results. Furthermore, the accuracy
of the delta expansion can by accelerated by using Pad\'e techniques to sum
the delta expansion. In the case of the Thomas-Fermi problem a $(2,1)$-Pad\'e
approximant has a numerical error of about 1\%.
As a second example, consider the quintic polynomial equation
$$x^5+x-1=0,$$
which cannot be solved by quadrature. The real root of this equation is
$x=0.75487767\ldots$. Introducing the perturbation parameter $\delta$, we obtain
the equation
$$x^{1+\delta}+x=1.$$
We then seek a perturbation series of the form
\begin{equation}
x(\delta)=c_0+c_1\delta+c_2\delta^2+c_3\delta^3+\ldots
\label{e59}
\end{equation}
whose first term is $c_0=1/2$. The radius of convergence of the delta series
(\ref{e59}) is $1$, and therefore it diverges at $\delta=4$. However, a $(3,
3)$-Pad\'e approximant has a numerical error of $0.05\%$ and a $(6,6)$-Pad\'e
approximant has a numerical error of $0.00015\%$.
\subsection{Delta expansion for (\ref{e1})}
\label{ss4b}
To apply the delta expansion to (\ref{e1}), we insert the parameter $\delta$
in such a way that when $\delta=1$ we recover (\ref{e1}):
\begin{equation}
Y'(x)=-\lambda x^{-n-2}\left(Y-Y_{\rm eq}\right)(Y+Y_{\rm eq})^\delta.
\label{e60}
\end{equation}
There are, of course, many ways to insert the parameter $\delta$, but the
advantage of (\ref{e60}) is that the solution to the unperturbed linear problem
obtained by setting $\delta=0$ is qualitatively similar to the solution to
(\ref{e1}), which we have already investigated in Sec.~\ref{s3}. In particular,
when $\delta=0$, $Y(x)$ behaves like $Y_{\rm eq}(x)$ for small $x$, undergoes a
transition as $x$ increases, and then approaches a constant as $x\to\infty$.
Following the usual delta-expansion procedure, we represent $Y(x)$ as a series
in powers of $\delta$,
$$Y(x)=\sum_{k=0}^\infty y_k(x)\delta^k,$$
and then substitute this series into (\ref{e60}). Comparing powers of $\delta$,
we obtain a sequence of {\it inhomogeneous} differential equations for $y_k$:
\begin{equation}
y_k'(x)+\lambda x^{-n-2}y_k(x)=h_k(x)\quad(k=0,\,1,\,2,\,\ldots),
\label{e61}
\end{equation}
where
\begin{eqnarray}
h_0(x)&=&\lambda x^{-n-2}Y_{\rm eq}(x),\nonumber\\
h_1(x)&=&\lambda x^{-n-2}\left[Y_{\rm eq}(x)-y_0(x)\right]\log\left[Y_{\rm eq}
(x)+y_0(x)\right],\nonumber\\
h_2(x)&=&\lambda x^{-n-2}\left\{
y_1(x)\frac{Y_{\rm eq}(x)-y_0(x)}{Y_{\rm eq}(x)+y_0(x)}
+\frac{Y_{\rm eq}(x)-y_0(x)}{2}\log^2\left[Y_{\rm eq}(x)+y_0(x)\right]
-y_1(x)\log\left[Y_{\rm eq}(x)+y_0(x)\right]\right\},
\label{e62}
\end{eqnarray}
and so on.
The solution to (\ref{e61}), which is obtained by using the integrating factor
$\exp\left[-\lambda x^{-n-1}/(n+1)\right]$, has the quadrature form
\begin{equation}
y_k(x)=e^{\lambda x^{-n-1}/(n+1)}\int_0^x ds\,e^{-\lambda x^{-n-1}/(n+1)}h_k(s).
\label{e63}
\end{equation}
Because (\ref{e61}) is a first-order equation, its solution contains one
arbitrary constant for each $k$ and this constant is determined by the
requirement that $y_k(0)$ be finite. This requirement fixes the lower endpoint
of integration to be $0$ for all $k$. Note that if we evaluate the integral in
(\ref{e63}), we obtain the results $y_0(0)=Y_{\rm eq}(0)=2A\zeta(3)$ (see
Appendix B), $y_1(0)=y_2(0)=\ldots=0$. As $x$ increases, $y_0(x)$ remains close
to $Y_{\rm eq}(x)$ until $x$ is of order $\lambda$.
We can now express the freeze-out value $Y(\infty)$ as a series in powers of
$\delta$ and then evaluate this series at $\delta=1$. Here, we just calculate
the first term in ths series:
\begin{equation}
y_0(x)=\lambda e^{\lambda x^{-n-1}/(n+1)}\int_0^\infty ds\,s^{-n-2}
e^{-\lambda s^{-n-1}/(n+1)}Y_{\rm eq}(s).
\label{e64}
\end{equation}
Let us evaluate this integral assuming that the parameter $\lambda$ is large.
Since the integrand is exponentially small for small $s$, we may assume that
the only contribution to the integral comes from the region $s\gg1$, and in
this region we may replace $Y_{\rm eq}(s)$ by its asymptotic behavior $As^{3/2}
e^{-s}$ (see Appendix A). We thus obtain
\begin{equation}
y_0(\infty)\sim A\lambda\int_0^\infty ds\,s^{-n-1/2}e^{\phi(s)}\quad(\lambda\to
\infty),
\label{e65}
\end{equation}
where
$$\phi(s)=-s-\frac{\lambda}{n+1}s^{-n-1}.$$
To evaluate (\ref{e65}) we use Laplace's method with a moving maximum
\cite{R16}. We note that the maximum of $\phi(s)$, which occurs when $\phi'(s)=0
$, is at $s_0=\lambda^{1/(n+2)}$. Hence, we introduce the rescaled variable $t$:
$$s=t\lambda^{1/(n+2)}.$$
This gives the integral
\begin{equation}
y_0(\infty)\sim A\lambda^{5/(2n+4)}\int_0^\infty dt\,t^{-n-1/2}e^{\lambda^{
1/(n+2)}\theta(t)}\quad(\lambda\to\infty),
\label{e66}
\end{equation}
where
$$\theta(t)=-t-\frac{1}{n+1}t^{-n-1}.$$
The maximum of $\theta(t)$ occurs at $t=1$, and near this point we have
the quadratic approximation
$$\theta(t)\sim-\frac{n+2}{n+1}-\frac{n+2}{2}(t-1)^2.$$
Thus, evaluating the Gaussian integral, we obtain the result
\begin{equation}
y_0(\infty)\sim\frac{A\sqrt{2\pi}}{\sqrt{n+2}}\lambda^{2/(n+2)}
\exp\left[-\frac{n+2}{n+1}\lambda^{1/(n+2)}\right],
\label{e67}
\end{equation}
which reduces to
\begin{equation}
y_0(\infty)\sim A\lambda\sqrt{\pi}e^{-2\sqrt{\lambda}}
\label{e68}
\end{equation}
when $n=0$. Thus, the delta expansion predicts that at $x=\infty$ the freeze-out
value of $Y(x)$ is exponentially small.
We see from this calculation that the delta expansion gives a simple and
qualitatively accurate picture of the solution to the Boltzmann equation
(\ref{e1}). However, the prediction in (\ref{e67}) of the relic abundance
$Y_\infty$ is clearly too small and, of course, this is because we have only
kept the leading-order term in the delta expansion. We will see in the next
subsection that if we retain higher powers of $\delta$, the qualitative features
of the solution do not change but the quantitative prediction for the long-time
behavior of $Y(x)$ is improved.
\subsection{Delta expansion for (\ref{e3})}
\label{ss4c}
The delta expansion treatment of (\ref{e3}) parallels that for (\ref{e1}). We
insert the parameter $\delta$ into (\ref{e3}) as follows:
\begin{equation}
Y'(x)=-\frac{\lambda}{x^{n+2}}[Y(x)-Y_{\rm eq}(x)][Y(x)+Y_{\rm eq}(x)]^\delta
-\frac{\phi}{x}Y(x),
\label{e69}
\end{equation}
where $\phi=\left|\Phi_0\right|$. The analog of (\ref{e61}) is then
\begin{equation}
y_k'(x)+\left(\frac{\lambda}{x^{n+2}}+\frac{\phi}{x}\right)y_k(x)=h_k(x)\quad
(k=0,\,1,\,2,\,\ldots).
\label{e70}
\end{equation}
The solution to (\ref{e70}), which is obtained by using the integrating factor
$x^\phi\exp\left[-\lambda x^{-n-1}/(n+1)\right]$, has the quadrature form
\begin{equation}
y_k(x)=x^{-\phi}\exp\left[\lambda x^{-n-1}/(n+1)\right]\int_0^x ds\,s^\phi
\exp\left[-\lambda s^{-n-1}/(n+1)\right]h_k(s).
\label{e71}
\end{equation}
Using the modified Laplace method again, we obtain for $x\to\infty$ and large
$\lambda$ the asymptotic approximation
\begin{equation}
y_0(x)\sim x^{-\phi}B(\lambda),
\label{e72}
\end{equation}
where the constant $B(\lambda)$ is given by
$$B(\lambda)=A\lambda^{(\phi+2)/(n+2)}\sqrt{\frac{2\pi}{n+2}}\exp\left[-
\lambda^{1/(n+2)}(n+2)/(n+1)\right].$$
This shows that the dilatonic correction to the Boltzmann equation gives a
significant qualitative change in the freeze-out behavior of DM. The magnitude
of the DM abundance is era dependent because its leading behavior for large $x$
is an algebraic decay of the form $x^{-\phi}$. The delta expansion is
qualitatively in agreement with boundary layer theory.
The result in (\ref{e72}) is the analog of (\ref{e67}), and again we see that
while the delta expansion in leading-order gives a good qualitative description
of the solution to the Boltzmann equation, the quantitative prediction for the
coefficient $B(\lambda)$ of $x^{-\phi}$ in the large-$x$ behavior is much too
small. Thus, we extend the result in (\ref{e72}) to first order in $\delta$. The
calculation is a straightforward generalization of the zeroth-order calculation
and the result is
\begin{equation}
y_0(x)+\delta y_1(x)\sim x^{-\phi}B(\lambda)\left\{1-\delta\log[B(\lambda)]+
\delta\frac{\phi}{n+1}\left[\gamma+\log\left(\frac{\lambda}{n+1}\right)\right]
\right\},
\label{e73}
\end{equation}
where $\gamma=0.5772\ldots$ is Euler's constant.
For large $\lambda$, we can ignore all but the $\log[B(\lambda)]$ term, and we
obtain a rough asymptotic behavior, which is a simplified version of
(\ref{e73}):
\begin{equation}
y_0(x)+\delta y_1(x)\sim x^{-\phi}B(\lambda)\left\{1-\delta\log[B(\lambda)]
\right\}.
\label{e74}
\end{equation}
Not surprisingly, the second-order contribution contains a logarithm squared:
\begin{equation}
y_0(x)+\delta y_1(x)\sim x^{-\phi}B(\lambda)\left\{1-\delta\log[B(\lambda)]
+\frac{1}{2}\delta^2\log^2[B(\lambda)]\right\}.
\label{e75}
\end{equation}
In general, the dominant contribution to the coefficient of $\delta^k$ in the
delta expansion is $(-1)^k\log^k[B(\lambda)]/k!$. Thus, if we sum the
approximate delta series to all orders in $\delta$ and set $\delta=1$, the
multiplicative coefficient $B(\lambda)$, which is numerically incorrect because
it is much too small, is exactly canceled. This explains the mechanism by
which the delta expansion and the matched asymptotic analysis become compatible.
\section{Brief concluding remarks}
\label{s5}
We have applied two powerful perturbative techniques, boundary-layer theory and
the delta expansion, to find globally accurate solutions to two different
Boltzmann equations that describe dark-matter abundances in the early universe.
The first Boltzmann equation is based on the standard model of particle physics
and general relativity; the second includes additional effects due to dilatonic
contributions that arise in string theory. The boundary-layer solution consists
of contributions from three distinct eras, a thermal-equilibrium epoch, a
freeze-out region, and a nonequilibrium relic-abundance epoch, and the global
solution is obtained by the use of asymptotic matching. The delta-expansion
solution does not require the use of asymptotic matching and gives a good
qualitative picture of the behavior in these three epochs, but the results
to low orders in $\delta$ are not as accurate for long times.
We have shown that when dilatonic effects are not included, the
dark-matter-relic abundance approaches a constant for long times, but when
dilatonic effects are included, the relic abundance has a power-law decay
determined by the dilaton coupling.
\acknowledgments
We are grateful to N.~E.~Mavromatos for enlightening discussions. CMB thanks the
U.K.~Leverhulme Foundation and the U.S.~Department of Energy and SS thanks the
U.K.~Science and Technology Facilities Council for financial support.
|
{
"timestamp": "2012-03-09T02:03:16",
"yymm": "1203",
"arxiv_id": "1203.1822",
"language": "en",
"url": "https://arxiv.org/abs/1203.1822"
}
|
\section{Introduction}
In a consensus problem, a set of agents aim to make an agreement on
some quantities of interest via distributive decision making. The
information interactions are based upon local neighboring structure.
A consensus is said to be achieved if all agents in the system tend
to agree on the quantities of interest as time approaches infinity,
cf. survey papers \cite{3,5} and references therein. Especially, the
speed of synchronization is investigated in \cite{13}.
Ali Jadbabaie et al. \cite{1} studied a simple model of flocking
introduced by Vicsek et al. \cite{2} showing that all agents will
reach consensus as time goes on, provided the communication graph
switching deterministically over time is periodically jointly
connected. Some researchers have also treated random situations, see
e.g. \cite{6,8}. Recently, a new model called moving neighborhood
network is introduced in \cite{10}. In this model, each agent
carries an oscillator and diffuses in the environment. The computer
simulation shows that synchronization is possible even when the
communication network is spatially disconnected in general at any
given time instant. Subsequently, several researchers have derived
analytical results on the moving neighborhood networks, see e.g.
\cite{18,9,11,4,12}.
The aim of this paper is to implement consensus on moving
neighborhood network modeled by the famous Peterson graph \cite{7}.
See Fig. 1. There are many interesting characteristics of Peterson
graph in mathematics. For example, it is traceable but not
Hamiltonian. That is, it has a Hamiltonian path but doesn't have a
Hamiltonian cycle. It is also the canonical example of a
hypohamiltonian graph. In this paper, we show that it is possible to
reach consensus on them by using moving neighborhood model.
\begin{figure}[hbt]
\centering
\includegraphics[80pt,653pt][214pt,771pt]{peterson.eps}
\caption{An example of Peterson graph, which has 10 vertices and 15
edges.}
\end{figure}
\section{Preliminaries}
Let $G=(V,E,W)$ be a weighted graph with vertex set $V$. $E$ is a
set of pairs of elements of $V$ called edges. $W=(w_{ij})$ is the
weight matrix, in which $w_{ij}>0$ if $(i,j)\in E$, and $w_{ij}=0$
otherwise. Consider $n$ identical agents $\{v_1,v_2,\cdots,v_n\}$ as
random walkers on $G$, moving randomly to a neighbor of their
current location in $G$ at any given time. For each agent, the
random neighbor that is chosen is not affected by the agent's
previous trajectory. The $n$ random walk processes are independent
to each other. If $v_i$ and $v_j$ meet at the same node
simultaneously, then they can interact with each other by sending
information.
Let $X_i(t)\in\mathbb{R}$ be the state of agent $v_i$ at time t. We
use the following consensus protocol
\begin{equation}
X_i(t+1)=X_i(t)+\varepsilon\sum_{j\in
N_i(t)}b_{ij}(t)(X_j(t)-X_i(t))\label{1}
\end{equation}
where $\varepsilon>0$ and $N_i(t)$ is the index set of neighbors of
agent $v_i$ at time $t$. The factor $b_{ij}>0$ for $i\not=j$, and
$b_{ii}=0$ for $1\le i\le n$. Let $A(t)=(a_{ij}(t))$ be the
adjacency matrix of the moving neighborhood network, whose entries
are given by,
$$
a_{ij}(t)=\left\{
\begin{array}{cc}
b_{ij}(t),& (v_i,v_j)\in E(t)\\
0,& otherwise
\end{array}\right.
$$
for $1\le i,j\le n$. Suppose that $\triangle:=\max_{1\le i\le
n}(\sum_{j=1}^nb_{ij}(t))$, and we further assume
$\varepsilon\in(0,1/\triangle)$ for all $t$. We will show that the
states of all agents walking on a Peterson graph reach consensus as
time goes on.
\section{Numerical examples}
For Peterson graph represented in Fig. 1, we take the weight matrix
as the adjacency matrix. In addition, we take the $b_{ij}(t)$
randomly from a set of basic functions such as $e^t$, $\sin(t)$,
$\cos(t)$ and so on. In Fig. 2,3,4,5, we show that the consensus can
be achieved asymptotically.
\begin{figure}[htb]
\begin{center}
\scalebox{0.5}{\includegraphics{fig2.eps}}\caption{The consensus
over moving neighborhood network modeled by a Peterson graph.}
\end{center}
\end{figure}
\begin{figure}[htb]
\begin{center}
\scalebox{0.5}{\includegraphics{fig3.eps}}\caption{The consensus
over moving neighborhood network modeled by a Peterson graph.}
\end{center}
\end{figure}
\begin{figure}[htb]
\begin{center}
\scalebox{0.5}{\includegraphics{fig4.eps}}\caption{The consensus
over moving neighborhood network modeled by a Peterson graph.}
\end{center}
\end{figure}
\begin{figure}[htb]
\begin{center}
\scalebox{0.5}{\includegraphics{fig5.eps}}\caption{The consensus
over moving neighborhood network modeled by a Peterson graph.}
\end{center}
\end{figure}
\bigskip
|
{
"timestamp": "2012-03-09T02:04:37",
"yymm": "1203",
"arxiv_id": "1203.1900",
"language": "en",
"url": "https://arxiv.org/abs/1203.1900"
}
|
\section{Introduction}
Ballistic electron transport~\cite{Wharam88,vanWees88} in two-terminal graphene systems is in the focus
of intensive studies ever since the pioneering experiments on
single-layer carbon.~\cite{CastroNeto06,Geim07}
The Dirac Hamiltonian~\cite{CastroNeto06,Dirac28}
describes charge transport close to the charge-neutrality point and
leads to a linear dispersion relation $\epsilon=\hbar v_F k$. This
allows to observe several relativistic phenomena in solid-state
system, such as Klein tunneling~\cite{Klein28,Dombey99,Katsnelson07,Shytov08,Young09,Sonin09}
or the Zitterbewegung.~\cite{Katsnelson06,Schliemann08b,Rusin07b}
In the very early works on graphene the minimal
conductivity~\cite{Novoselov05,Zhang05,Katsnelson06,Katsnelson06a,Ziegler07}
$G\approx e^2/h$ per valley and pseudo-spin at the charge-neutrality
point has been found and stimulated the research on current and noise
properties. The current-current correlations around the minimal
conductivity lead to a zero frequency sub-Poissonion Fano factor with
a maximal value of $F=1/3$,~\cite{Tworzydlo06,ZhuGuo07,Groth08,Danneau08,DiCarlo08}
remarkably similar to diffusive systems as disordered metals.~\cite{Beenakker92,Nagaev92,Lewenkopf08}
The suppression of the Fano factor below the Poissonian value originates from noiseless,
open quantum channels that are found at all conductance minima in
graphene-based two-terminal structures~\cite{ZhuGuo07}, and can be
explained as an interplay between Klein tunneling, resonant tunneling
and pseudo-spin matching. This pseudo-diffusive behavior~\cite{Rycerz09} is due to
the special band-structure of graphene. Without impurity scattering,
coherent transport through such a graphene sheet~\cite{Sonin08} gives
rise to the same shot noise as in classical diffusive systems.
The opening of a gap~\cite{LopezRodriguez08} in the quasiparticle spectrum
leads to an enhanced Fano factor.~\cite{ZhuGuo07}
Such a gap can be opened for example by photon-assisted tunneling, as
shown recently for the case of a graphene p-n
junction~\cite{Williams07} with a linear potential drop across the
interface.~\cite{Fistul07,Syzranov08}
There, Landau-Zener like transitions stimulated by photon emission or absorption via resonant
interaction of propagating quasiparticles in graphene with an
irradiating electric field lead to hopping between different trajectories.\\
\begin{figure}[tb]
\centering
\includegraphics[width=0.8\columnwidth]{fig1.pdf}\\
\caption{Wide graphene strip ($W\gg L$) sandwiched between two heavily-doped,
metallic graphene leads. The Fermi-level of the sheet
can be tuned by a center gate voltage $V_g$.
Electron- and hole states are injected via the time-dependent
bias-voltages in left- and right leads $\mu_{L/R}(t)$.}
\label{fig1}
\end{figure}
The scattering approach as put forward by Landauer and
B\"uttiker~\cite{BlanterButtiker00} has been applied to ac-driven
charge transport~\cite{TienGordon62,Grifoni98,Platero04,Kohler05} through a metal-graphene interface with an abrupt
potential change.~\cite{Trauzettel07} The metal can be formed by a
graphene lead strongly electrostatically doped by a gate potential, thus
shifting the Dirac point far away from the Fermi energy. In this
work we adopt the formalism and parameterization
introduced in Refs.~\cite{Beenakker06,Trauzettel07} and calculate the
finite-frequency current-current correlations at finite dc- and
ac-bias voltages in the system depicted in Fig.~\ref{fig1}.
We complement recent results on ac-transport in
Fabry-P\'erot graphene devices of Ref.~\cite{Rocha09}, in which the
influence of different boundary conditions, i.e. zigzag or armchair
configurations, on the Fabry-P\'erot patterns in a combined
Tien-Gordon/tight-binding approach has been investigated.
The influence on transmission properties of a time-dependent potential barrier in
a graphene monolayer has been investigated.~\cite{Zeb08}
In our work the transverse boundary effects
are central and we assume so-called infinite mass
boundaries~\cite{Tworzydlo06,Berry87} describing a
short but wide ($L\ll W$) graphene strip.
We focus on the interplay between the Dirac-spectrum with the Fabry-Perot interferences.
Interestingly, the well-known oscillations as function of gate voltage on a
scale of the return frequency $\hbar v_F/L$, related to the length $L$ of the
graphene sheet, can be seen as a reminiscence of Zitterbewegung.~\cite{Katsnelson06}
The role of the complex reflection amplitude and the onset of contributions of
scattering states coming from terminal $\alpha$ and being scattered
into terminal $\beta$ will be the key characteristics in our
discussion of the results for the noise as function of bias
voltage and frequency. As a consequence of these onsets the
oscillations add up de- or constructively depending on
the precise values of voltage and frequency. In our setup,
the separation of oscillations caused by the
Fabry-P\'erot reflections and effects caused by the band-structure of
the Dirac Hamiltonian is a priori not obvious. In both cases
phase-coherent transport is essential. However, for charge injection
either into the conduction or the valence band only, effects like
Zitterbewegung should not be present and all oscillating features of
the noise spectra have to be of Fabry-P\'erot nature.
\section{Dirac equation and scattering formalism}
The ballistic graphene~\cite{Miao07,Laakso08,Schuessler09} sheet considered in the following can be
described by the two-dimensional Dirac equation for the two-component
spinor $\hat{\Psi}=(\hat{\Psi}_1, \hat{\Psi}_2)^T$ with indices
referring to the two pseudo-spins of the carbon sub-lattices.
Throughout this work we will neglect inter-valley scattering and Coulomb interactions.
We only consider the interaction of the electrons with the radiation field in the form of
photon-assisted transitions. With Fermi velocity $v_F$ the Dirac
equation can be cast into the form
\begin{align}
& \left[-i v_F \hbar
\left(\begin{array}{cc}
0 & \partial_x - i \partial_y\\
\partial_x + i \partial_y & 0
\end{array}\right)
-\mu({\mathbf x},t)\right]{\hat{\Psi}}({\mathbf{x}},t) \nonumber \\
& = i \hbar \partial_t {\hat{\Psi}}({\mathbf{x}},t) \,.
\label{diraceq}
\end{align}
The electrochemical potential $\mu({\mathbf{x}},t)$
includes static and harmonically driven potentials in the leads plus a
static gate voltage in the graphene sheet.
\begin{align}
\mu({\mathbf x},t)=\left\{
\begin{array}{ccc}
\mu_L + eV_{\mathrm{ac,L}} \cos(\omega t) & \mathrm{ if } & x<0\\
eV_g & \mathrm{ if } & 0<x<L\\
\mu_R + eV_{\mathrm{ac,R}}\cos(\omega t) & \mathrm{ if } & x>L
\end{array}\right..
\label{potentials}
\end{align}
Making use of the Tien-Gordon ansatz, we write the solution to the
time-dependent Dirac equation as a sum over
PAT modes:
\begin{align}
{\hat{\Psi}}({\mathbf{x}},t) = & {\hat{\Psi}}_{\mathrm{0}}({\mathbf{x}},t) e^{-i(eV_{ac}/\hbar \omega)\sin(\omega t)} \\
= & \sum\limits_{m=-\infty}^{\infty} J_m \left(\frac{eV_{ac}}{\hbar \omega } \right) {\hat{\Psi}}_{\mathrm{0}}({\mathbf{x}},t) e^{-i m\omega t}\\
\mathrm{ where } \qquad & {\hat{\Psi}}_{\mathrm{0}}({\mathbf{x}},t)={\hat{\Psi}}_{\mathrm{0}}({\mathbf{x}}) e^{- i \epsilon t}
\label{states}
\end{align}
The advantage of this ansatz is that the scattering problem has to be
solved for the time-independent case only. Therefor, in
terminals $\gamma=L,R$ we define stationary solutions
${\hat{\Psi}}_{\mathrm{0}}({\bf{x}},t)={\hat{\Psi}}_{\epsilon}({\bf{x}})
e^{- i \epsilon t}$ by the equation
\begin{align}&
\left[-i v_F \hbar
\left(\begin{array}{cc}
0 & \partial_x - i \partial_y\\
\partial_x + i \partial_y & 0
\end{array}\right)
-\mu_{\gamma}\right]{\hat{\Psi}}_{\mathrm{0}}({\mathbf{x}}) \\
&= \epsilon {\hat{\Psi}}_{\mathrm{0}}({\mathbf{x}}) \,.
\label{diraceq2}
\end{align}
\begin{figure}[tb]
\centering
\includegraphics[width=0.6\columnwidth]{fig2.pdf}
\caption{(color online) Transmission probability $T(\epsilon)=\left|t(\epsilon) \right|^2$
as a function of energy and transverse momentum q.
}
\label{fig2}
\end{figure}
The basis states in graphene
can be constructed as a superposition of left- and right movers,
\begin{equation}
\hat{\Psi}_0({\mathbf x})= \sum\limits_{k,q}^{} \left[{{\Psi}}_{\mathrm{0,+}}^{k,q} \hat{a}_{k,q} + {{\Psi}}_{\mathrm{0,-}}^{k,q} \hat{a}_{-k,q} \right] \, .
\end{equation}
$\alpha(\epsilon)$ describes the angle
between the momentum of a quasiparticle
and it's y-component $q$ in region $x=0 \ldots L$ of the graphene sheet.
Then the pseudo-spinors can be parametrized as
\begin{align}
{{\Psi}}_{\mathrm{0,+}}^{k,q} =&\frac{e^{i q y +i k(\epsilon)x}}{\sqrt{\cos\alpha(\epsilon)}}
\left(\begin{array}{c}
e^{-i \alpha(\epsilon)/2} \\
e^{i \alpha(\epsilon)/2}
\end{array}\right)
\\
{{\Psi}}_{\mathrm{0,-}}^{k,q} =&\frac{e^{i q y-i k(\epsilon)x}}{\sqrt{\cos\alpha(\epsilon)}}
\left(\begin{array}{c}
e^{i \alpha(\epsilon)/2} \\
-e^{-i \alpha(\epsilon)/2}
\end{array}\right) \,.
\end{align}
Here the dispersion is given by $\epsilon=\hbar v_F \sqrt{q^2+k^2}$.
The wave vector $k(\epsilon)$ and the angle $\alpha(\epsilon)$
are defined as
\begin{align}
\alpha(\epsilon)=&\arcsin\left(\frac{\hbar v_F q}{\epsilon +eV_g} \right)\\
k(\epsilon)=&\frac{\epsilon +eV_g}{\hbar v_F } \cos\left(\alpha(\epsilon)\right)\,.
\label{angel}
\end{align}
Therewith, and neglecting transverse momentum due to high doping, we have the basis states
\begin{align}
{{\Psi}}_{\mathrm{0,+}}^{k,0}&=\frac{e^{i k(\epsilon)x}}{\sqrt{2}}
\left(\begin{array}{c}
1 \\
1
\end{array}\right)\\
{{\Psi}}_{\mathrm{0,-}}^{k,0}&=\frac{e^{-i k(\epsilon)x}}{\sqrt{2}}
\left(\begin{array}{c}
1\\
-1
\end{array}\right)
\end{align}
in the leads. Additionally shifting the Fermi surface of the graphene sheet away
from the Dirac point, and thus changing the concentration of carriers,
is incorporated into the gate voltage $eV_g$.
For $\left|\epsilon+eV_g\right| < \left|\hbar v_F q\right|$
we have evanescent modes,~\cite{KatsnelsonGuinea07} with imaginary $\alpha(\epsilon)$ and $k(\epsilon)$.
Otherwise we have propagating modes and scattering is only at $x=0,L$.\\
Irradiating the two-terminal structure with a laser~\cite{Erbe06} can be
described by a harmonic ac-bias voltage with driving strength
$\alpha=eV_{ac}/\hbar \omega$ as discovered in the pioneering paper by
Tien and Gordon~\cite{TienGordon62}. Their theory can be incorporated
into the scattering formalism~\cite{ButtikerPretre93,PedersenButtiker98} and we are
applying it here to the two-terminal graphene structure.
We take the two valleys and two pseudo-spin states of the carbon lattice
into account in the pre-factor of the current operator of reservoir $\eta$, which reads
\begin{align}
& \hat{I}_{\eta}(t)=\frac{2 e W}{\pi \hbar} \sum\limits_{\gamma,\delta=L,R}^{} \sum\limits_{l,k=-\infty}^{\infty} \int\limits_{-\infty}^{\infty} d\epsilon d\epsilon' \int\limits_{0}^{\infty}dq J_{l}\left(\alpha_{\gamma}\right) J_{k}\left(\alpha_{\delta}\right) \nonumber\\
&\times \hat{a}_{\gamma}^{\dagger}(\epsilon - l \hbar \omega) A_{\gamma \delta}(\eta,\epsilon,\epsilon') \hat{a}_{\delta}(\epsilon' - k \hbar \omega)e^{i(\epsilon-\epsilon')t/\hbar} \,.
\label{currentop}
\end{align}
Indices $\gamma,\delta$ run over reservoirs $L,R$. Summation over all modes of y-momentum
is replaced by an integral since $W\gg L$. Scattering is contained within the current matrix
${A}_{ \gamma \delta}(\eta, \epsilon, \epsilon')=\delta_{\eta \gamma} \delta_{\eta \delta} - s^*_{\eta \gamma}(\epsilon) s_{\eta \delta}(\epsilon')$
of a current between leads $\gamma$ and $\delta$ measured in lead $\eta$ via the energy-dependent
scattering-matrix
\begin{figure}[tb]
\centering
\includegraphics[width=0.72\columnwidth]{fig3a.pdf}\\
\includegraphics[width=0.72\columnwidth]{fig3b.pdf}
\caption{(color online) Left: Conductivity $\sigma(\omega,\alpha)=(L/W) G(\omega,\alpha)$
as a function of
dc-voltage applied across the two-terminal setup.
We show curves for various
ac-driving strengths $\alpha$ applied to a) both reservoirs ($a=0$)
and b) to the left reservoir only ($a=1$) with $\omega L/v_f=5$.
}
\label{fig3}
\end{figure}
\begin{equation}
s(\epsilon)=\left(
\begin{array}{cc}
r(\epsilon) & t'(\epsilon) \\
t(\epsilon) & r'(\epsilon)
\end{array}
\right) \,.
\end{equation}
The scattering matrix connects in- and outgoing scattering states at the two barriers and
is calculated in Appendix \ref{app:bounds} by matching the wave functions at $x=0,L$.
Here we write the results for reflection and transmission amplitudes in an alternative version:
\begin{align}
t(\epsilon)&=\frac{2e^{ i k(\epsilon) L} \left(1+e^{2 i \alpha(\epsilon) }\right)}{e^{2 i k(\epsilon) L}\left(1-e^{i \alpha(\epsilon)} \right)^2+\left(1+ e^{i \alpha(\epsilon)} \right)^2}\\
r(\epsilon)&= \frac{\left(e^{2 i k(\epsilon) L}-1\right) \left(e^{2 i \alpha(\epsilon) }-1\right)}{e^{2 i k(\epsilon) L} \left(1-e^{i \alpha(\epsilon) }\right)^2+\left(1+e^{ i \alpha(\epsilon) }\right)^2}\,.
\label{transmission_exp}
\end{align}
We assume identical scattering for quasi-particles incident from left and right,
so $t(\epsilon)=t'(\epsilon)$ and $r(\epsilon)=r'(\epsilon)$.
$r(\epsilon)$ vanishes if $k(\epsilon)= \pi n/L$, with integer $n$.
The corresponding modes in $y$-direction are determined by
\begin{equation}
q=\left[\left(\frac{\epsilon }{\hbar v_F}\right)^2-\left(\frac{\pi n}{L} \right)^2\right]^{1/2}\, ,
\label{qjumps}
\end{equation}
giving rise to special features of the current fluctuations, going along with the phase
jumps of $\pi L/\hbar v_F$ in $r(\epsilon)$ we discuss later on.
At the Dirac point transmitted quasi-particles at perpendicular
incidence perform Klein tunneling via evanescent modes,
leading to finite transmission probability $T(\epsilon)=t^{\dagger}(\epsilon)t(\epsilon)$
at small transverse momentum, see Fig.~\ref{fig2}.
\section{Differential conductance \label{sec:conductance}}
\begin{figure}[tb]
\centering
\includegraphics[width=0.97\columnwidth]{fig4.pdf}\\
\caption{(color online) Schematic view of the different regions which occur in
the integrands of the correlators contributing to the
finite-frequency shot noise spectrum.}
\label{fig4}
\end{figure}
Since the average current has only a zero-frequency component,
PAT events in the conductance\cite{Snyman08,Peres06} can only be studied
by inducing photon-exchange via a time-dependent voltage as it is,
for example, generated by irradiating the setup with a laser beam.
Different polarizations of the coupled light field lead to different ac-driving in left and right leads.
Such an asymmetry can be described by a parameter $a \in \left[-1,1\right]$ which varies the
driving in the leads via $\alpha_{L/R}=\frac{a \pm 1}{2}\alpha \equiv \frac{V_{\mathrm{ac,L/R}}}{\hbar \omega}$.
We call the driving symmetric (in the amplitudes $V_{\mathrm{ac,L/R}}$) if $a=0$ and asymmetric if $a= \pm 1$.
For clearness we will only discuss $a=0,\pm 1$ since intermediate values are just a
mixture of those limiting cases.
For arbitrary $a$ the differential conductance can be derived from
Eqn.(\ref{currentop}) by taking the statistical- and time average and differentiating
with respect to voltage. At $k_BT=0$ it reads
\begin{align}
G(\omega,\alpha) =&\frac{2e^2 W}{\hbar} \int\limits_{0}^{\infty}dq \sum\limits_{m=-\infty}^{\infty} \left( J_{m}^2\left(\alpha_L\right) \left|t (m\hbar \omega+\frac{eV}{2})\right|^2 \right. \nonumber\\
&\left.+ J_{m}^2\left(\alpha_R\right)\left|t (m\hbar \omega-\frac{eV}{2})\right|^2 \right) \,.
\end{align}
Different orders $m$ of PAT do not mix but have to be summed up resulting
in independent contributions $G_m(\omega,\alpha)$ to differential conductance.
Since $G(\omega,\alpha)$ only depends on the Besselfunctions squared,
these pre-factors will always be positive. The influence of the driving
strength $\alpha$ on conductivity $\sigma(\omega,\alpha)= (L/W) G(\omega,\alpha)$
as a function of dc-bias is plotted in Fig.~\ref{fig3}.
PAT events lead to a substantial enhancement of the conductivity around zero dc-bias,
because more channels are available in comparison to the case without time-dependent voltages.
At large dc-bias voltages this effect gets negligible since the transmission probability
of the graphene sheet, see Fig.~\ref{fig2}, is not vanishing at large energies.
Thus, those contributions built a dominant background.
Conductance at arbitrary dc- and ac-bias is a sum of two integrated
transmission probabilities, where the integrand exhibits crossings of the two independent interference patterns,
as in region III$b$ in Fig.~\ref{fig4} a).
Each $G_m(\hbar \omega) $ shows a transition from a region with an oscillating, but in average not increasing
contribution to conductance for dc-bias voltages $|eV/2| < |m\hbar \omega|$,
to a regime with a linear increasing background at larger dc-bias voltages.
The photon-energy $m\hbar \omega$ introduces a phase shift in the oscillations of $G_m(\omega,\alpha)$
as a function of dc-bias voltage,
so for different $m$ we can have local minima or maxima at $eV=0$.
After summation, conductivity can also show a local minimum or maximum at $eV=0$,
as it can be observed for the various values of $\alpha$ in Fig.~\ref{fig3} a).
If $|a|$ tends to one this effect is hidden behind the contribution from the terminal where
driving gets small, as in Fig.~\ref{fig3} b) with $a=1$.
From the oscillations with period proportional to $L$, we expect no measurable effect on conductivity
or shot-noise~\cite{DiCarlo08,Danneau08}, as in the case without ac-driving and for the zero-frequency Fano factor.
In the scattering approach they are simply because the transmission function oscillates as a function of energy.
But imperfections of real samples, as impurities~\cite{Titov07} or lattice-mismatch, lead to scattering events.
Due to this randomizing effect on the path-lengths for propagating quasi-particles the
calculated oscillations are averaged out in experiment~\cite{DiCarlo08,Danneau08}.
\section{Frequency-dependent shot noise \label{sec:shotnoise}}
To get full informations on current-current correlations we study the
non-symmetrized noise-spectrum as it can be detected by an
appropriate measurement device in the quantum regime.~\cite{Schoelkopf97,Lesovik97,GavishImry00,GavishImry02,Beenakker01,Reydellet03,Nazarov08,Gabelli08,Gabelli09,Reulet03,Zakka-Bajjani09,EngelLoss04,EntinWohlman07,Rothstein08,Schonenberger03,Aguado00,Brandes04}\\
We allow harmonic ac-driving $eV_{ac} \cos(\omega t)$ in the leads,
so in Fourier space the current-current correlations are defined as
\begin{equation}
S_{\alpha \beta}(\Omega, \Omega', \omega)= \int\limits_{-\infty}^{\infty} dt dt'
S_{\alpha \beta}(t,t',\omega)
e^{i \Omega t + i \Omega' t'} \, .
\label{noisedef}
\end{equation}
The non-symmetrized shot noise correlates currents at two times:
\begin{equation}
S_{\alpha \beta}(t,t',\omega)=\left\langle \Delta \hat{I}_{\alpha}(t) \Delta \hat{I}_{\beta}(t') \right\rangle
\end{equation}
with variance $\Delta \hat{I}_{\alpha}(t) = \hat{I}_{\alpha}(t) -
\langle\hat{I}_{\alpha}(t)\rangle$.
Of experimental interest are the fluctuations on timescales large
compared to the one defined by the driving frequency $\omega$. Thus,
as in~\cite{PedersenButtiker98}, we introduce Wigner coordinates
$t=T+\tau/2$ and $t'=T-\tau/2$ and average over a driving period
$2\pi/\omega$. Then, the noise spectrum is defined by the quantum
statistical expectation value of the Fourier-transformed
current-operator $\hat{I}_{\alpha}(\Omega)$ via $S_{\alpha
\beta}(\Omega, \Omega',\omega)=2\pi
S_{\alpha
\beta}(\Omega,\omega)\delta(\Omega+\Omega') =
\langle\hat{I}_{\alpha}(\Omega)\hat{I}_{\beta}(\Omega')\rangle$.
$S_{\alpha \beta}(\Omega,\omega)$ is nothing but the Fourier transform of $S_{\alpha \beta}(\tau,\omega)$.
Similarly, in the case without ac-driving the noise is only a function
of relative times $\tau=t-t'$. In order to keep notation short,
in the dc-limit we write $S_{\alpha \beta}(\Omega):=S_{\alpha
\beta}(\Omega,\omega=0)$.
To get a deeper insight into the underlying processes of charge-transfer
we split the noise into four possible correlators~\cite{GavishImry02}, defined by
\begin{equation}
S_{LL}(\Omega,\omega):=\underset{\alpha, \beta =L,R}\sum C_{\alpha \rightarrow \beta}(\Omega,\omega) \,.
\end{equation}
The correlators itself can be seen as the building-blocks of noise spectra where
different combinations describe noise detected by corresponding measurement setups.~\cite{GavishImry00,GavishImry02}
First we discuss $S_{LL}(\Omega):=S_{LL}(\Omega,\omega=0)$, the case when no ac-driving is present.
We also skip $\omega$ in the arguments of the correlators.
Then evaluation of Eqn. (\ref{noisedef})
at $k_BT=0$ leads to the expressions:
\begin{subequations}
\label{correlators}
\begin{align}
C_{L \rightarrow L}(\Omega)=&
\frac{ e^2 \Theta(\hbar \Omega)}{2 \pi \hbar} \underset{\mu_L-\hbar \Omega}{\overset{\mu_L}\int} d\epsilon \underset{-\infty}{\overset{\infty}\int} dq \,
\left|r^*(\epsilon)r(\epsilon+\hbar\Omega) - 1 \right|^2 \label{correlators:LtoL}\\
C_{R \rightarrow R}(\Omega)=&
\frac{e^2 \Theta(\hbar \Omega)}{2 \pi \hbar} \underset{\mu_R-\hbar \Omega}{\overset{\mu_R}\int} d\epsilon \underset{-\infty}{\overset{\infty}\int} dq \,
T(\epsilon)T(\epsilon+\hbar\Omega)\label{correlators:RtoR}\\
C_{L \rightarrow R}(\Omega)=&
\frac{e^2 \Theta(\hbar \Omega-eV)}{2 \pi \hbar} \underset{\mu_R-\hbar \Omega}{\overset{\mu_L}\int} d\epsilon\underset{-\infty}{\overset{\infty}\int} dq \,
R(\epsilon)T(\epsilon+\hbar\Omega)\label{correlatorsL:toR}\\
C_{R \rightarrow L}(\Omega)=&
\frac{e^2 \Theta(\hbar \Omega+eV)}{2 \pi \hbar} \underset{\mu_L-\hbar \Omega}{\overset{\mu_R}\int} d\epsilon \underset{-\infty}{\overset{\infty}\int} dq \,
T(\epsilon)R(\epsilon+\hbar\Omega)\, . \label{correlators:RtoL}
\end{align}
\end{subequations}
At finite dc-bias voltages correlations with initial and final state related to the
measurement terminal $L$ are special in the sense that they can not be written in terms of
probabilities at finite frequency. For symmetrized noise, B\"uttiker~\cite{Buttiker92} discussed the essential
role of the complex reflection amplitudes in elastic electron transport and how they determine the equilibrium current fluctuations.
In the quantum regime at $k_BT=0$, the equilibrium fluctuations are given by
\begin{align}
S_{LL}(\Omega)=& \frac{e^2}{2 \pi \hbar} \Theta(\hbar \Omega) \underset{-\hbar \Omega}{\overset{0}\int} d\epsilon\underset{-\infty}{\overset{\infty}\int} dq \nonumber\\
& \left(2- r^*(\epsilon)r(\epsilon+\hbar\Omega)-r^*(\epsilon+\hbar\Omega)r(\epsilon) \right) \,.
\label{equilibrium}
\end{align}
For finite dc-bias the reflection amplitudes entering $C_{L\rightarrow L}(\Omega)$
play the same essential role as in equilibrium,
in the sense that finite-frequency current fluctuations are non-zero even for vanishing transmission.
The combination of scattering-matrices of the correlators integrands which enter in the current-current cross-correlation spectrum
\begin{equation}
S_{LR}(\Omega,\omega):=\underset{\alpha, \beta =L,R}\sum C^{\mathrm{c}}_{\alpha \rightarrow \beta}(\Omega,\omega)
\end{equation}
are substantially different than in the ones for the auto-terminal noise. Most of all, at finite frequency none of the
complex correlators can
be written as an integral over transmission- or reflection probabilities:
\begin{subequations}
\label{correlatorscross}
\begin{align}
C^{\mathrm{c}}_{L \rightarrow L}(\Omega)=& \frac{e^2 \Theta(\hbar \Omega)}{2 \pi \hbar} \underset{\mu_L-\hbar \Omega}{\overset{\mu_L}\int} d\epsilon\underset{-\infty}{\overset{\infty}\int} dq \, \nonumber \\
& t^*(\epsilon+\hbar\Omega) t(\epsilon) \left[1- r^*(\epsilon)r(\epsilon+\hbar\Omega) \right]\label{correlatorscross:LtoL} \\
C^{\mathrm{c}}_{R \rightarrow R}(\Omega)=& \frac{e^2 \Theta(\hbar \Omega) }{2 \pi \hbar} \underset{\mu_R-\hbar \Omega}{\overset{\mu_R}\int} d\epsilon \underset{-\infty}{\overset{\infty}\int} dq \, \nonumber \\
&
t^*(\epsilon) t(\epsilon+\hbar\Omega) \left[1- r^*(\epsilon+\hbar\Omega)r(\epsilon) \right]\label{correlatorscross:RtoR}\\
C^{\mathrm{c}}_{L \rightarrow R}(\Omega)=& \frac{-e^2 \Theta(\hbar \Omega-eV)}{2 \pi \hbar} \underset{\mu_R-\hbar \Omega}{\overset{\mu_L}\int} d\epsilon \underset{-\infty}{\overset{\infty}\int} dq \,\nonumber \\
&
r^*(\epsilon)t(\epsilon) r^*(\epsilon+\hbar\Omega)t(\epsilon+\hbar\Omega)\label{correlatorscross:LtoR}\\
C^{\mathrm{c}}_{R \rightarrow L}(\Omega)=& \frac{-e^2 \Theta(\hbar \Omega+eV)}{2 \pi \hbar} \underset{\mu_L-\hbar \Omega}{\overset{\mu_R}\int} d\epsilon \underset{-\infty}{\overset{\infty}\int} dq \, \nonumber \\
&
t^*(\epsilon)r(\epsilon)t^*(\epsilon+\hbar\Omega)r(\epsilon+\hbar\Omega)\label{correlatorscross:RtoL}
\end{align}
\end{subequations}
Unlike for symmetrized noise, quantum noise~\cite{Nazarov08,Schonenberger03} spectra discriminate between
photon absorption ($\Omega > 0$) and emission ($\Omega < 0$) processes between
quasi-particles in graphene and a coupled electric field~\cite{GavishImry00,GavishImry02,Aguado00,Brandes04,Marcos11}.
Energy for photon emission has to be provided by the voltage source, so at $k_BT=0$ the Heaviside-Theta functions ensure that only
terms satisfying this condition contribute at negative frequencies. In the dc-limit, our choice of chemical potentials
$-\mu_L=\mu_R=eV/2 > 0$ and the fact that the measurement is performed at reservoir $L$, leaves only
$C^{\mathrm{c}}_{R \rightarrow L}(\Omega) \ne 0$ if $\Omega \le 0$. When additional ac-voltages are present none
of the correlators of Eqn.(\ref{thenoise}) is given in terms of probabilities and integration boundaries are
changed by $\pm m\hbar \omega$. Then all correlators can contribute at frequencies $\Omega<0$.
\begin{figure}[tbp]
\centering
\includegraphics[width=0.95\columnwidth]{fig5.pdf}
\caption{(color online) Real parts of integrands of the four correlators
(Eqn.~\ref{correlators})
contributing to the shot-noise, namely a) $0.25|1-r^*(\epsilon)r(\epsilon+\hbar \Omega)|^2$, b) $T(\epsilon)T(\epsilon+\hbar \Omega)$, c) $R(\epsilon)T(\epsilon+\hbar \Omega)$ and d) $R(\epsilon+\hbar \Omega)T(\epsilon)$. Here the energy is fixed
$\epsilon = 0$ corresponding to vanishing dc-bias. The correlator
in a) cannot be written in terms of a probabilities, except in the
zero frequency limit the integrand results in
$T^2(\epsilon)$. Correlator b) contains one transmission
probability at zero energy that is only non-zero at small $q$.
Since for small transversal momentum $R(\epsilon)$
decays as $q^{-2}$ the correlator c) tends to zero in this regime and
otherwise mimics the behavior of $T(\epsilon)$. Integrand d) is
also restricted to low transverse momentum because
$T(\epsilon)=0$ otherwise.\\
}
\label{fig5}
\end{figure}
\begin{figure}[tbp]
\centering
\includegraphics[width=0.9\columnwidth]{fig6.pdf}
\caption{(color online) Real parts of integrands of correlators
(Eqn.~\ref{correlators}, see also Fig.~\ref{fig5}) contributing to the shot-noise
for fixed energy $\epsilon L/\hbar v_F=20$. At finite $\epsilon$ there is an additional
interference pattern along $q$ if $\hbar v_F |q| < |\epsilon|$,
leading to phase jumps in the Integrand of correlator a), the one where initial and final
state belong to the measurement terminal $L$. When the integrand can be written as a product
of probabilities, see b)-d), the phase jumps are absent but
two independent interference patterns are found. }
\label{fig6}
\end{figure}
\begin{figure}[tbp]
\centering
\includegraphics[width=0.9\columnwidth]{fig7.pdf}
\caption{(color online) Real parts of integrands of the correlators (Eqn.~(\ref{correlators}), see also Fig.~\ref{fig5})
contributing to the shot-noise with fixed frequency $\Omega L/v_F=20$.
Analogous to Fig.\ref{fig6} but as a function of $(q,\epsilon)$.
Phase jumps occur in the intervall $-\hbar \Omega<\epsilon<0$ in integrand a),
region III$_b$ of Fig.\ref{fig4}a).
The interplay of the two interference patterns can also be observed at larger energies and
transverse momenta for $\hbar v_F |q|< |\epsilon|$, $\hbar v_F |q|< |\epsilon+ \hbar \Omega|$ in all integrands a)-d).
}
\label{fig7}
\end{figure}
\begin{figure}[tbp]
\centering
\includegraphics[width=0.9\columnwidth]{fig8.pdf}
\caption{(color online) Real parts of integrands which appear in Eqn.~(\ref{correlatorscross})
contributing to the cross-correlation shot-noise
for fixed energy $\epsilon L/\hbar v_F=20$ (top) and fixed frequency $\Omega L/v_F=20$ (bottom),
namely a),c) $\Re[t^*(\epsilon+\hbar\Omega)t(\epsilon)(r^*(\epsilon+\hbar\Omega)r(\epsilon)-1)]$ and
b),d) $4\Re[r^*(\epsilon)t(\epsilon+\hbar\Omega)r^*(\epsilon+\hbar\Omega)r(\epsilon)]$.
Due to symmetry reasons the integrands are identical when interchanging index labels $L,R$.
As a function of frequency integrand a) leads to strongly oscillating
contributions to the noise spectrum. These oscillations are reduced due to
the alternating behavior along $q$ in cross-terminal contributions b).
In c), d) the integrands are plotted as a function of $(q,\epsilon)$ where they
reveal a similar structural difference.
}
\label{fig8}
\end{figure}
\begin{figure}[tbp]
\centering
\includegraphics[width=0.9\columnwidth]{fig9.pdf}
\caption{ (color online) Real parts of integrands appearing in
the correlators of Eqn.~(\ref{thenoise}) contributing to
the shot-noise for driving frequency $\omega L/v_F=7.5$ and fixed frequency $\Omega L/v_F=20$.
The integrands are a) $0.25(1-r^*(\epsilon)r(\epsilon+\hbar\Omega))(1-r^*(\epsilon+\hbar\Omega+\hbar\omega)r(\epsilon+\hbar\omega))$,
b) $t^*(\epsilon)t(\epsilon+\hbar\Omega)t^*(\epsilon+\hbar\Omega+\hbar\omega)t(\epsilon+\hbar\omega)$,
c) $r^*(\epsilon)t(\epsilon+\hbar\Omega)t^*(\epsilon+\hbar\Omega+\hbar\omega)r(\epsilon+\hbar\omega)$ and
d) $t^*(\epsilon)r(\epsilon+\hbar\Omega)r^*(\epsilon+\hbar\Omega+\hbar\omega)t(\epsilon+\hbar\omega)$.
When two frequencies are present none of
the correlators can be written in terms of probabilities and additional
phase jumps come into play.
}
\label{fig9}
\end{figure}
\begin{figure}[tbp]
\centering
\includegraphics[width=0.9\columnwidth]{fig10.pdf}
\caption{ (color online) Real parts of integrands of the correlators contributing to the
shot-noise with driving frequency $\omega L/v_F=7.5$
and fixed energy $\epsilon L/\hbar v_F=20$. Analogous to Fig.~(\ref{fig9}),
but as a function of $(q,\Omega$). As a
consequence of PAT horizontal interference lines occur for transverse
momenta $\hbar v_F|q|<|\epsilon+m\hbar \omega|$ as in figures a), c).}
\label{fig10}
\end{figure}
\section{Qualitative discussion \label{sec:qualitative}}
A good starting point to interpret results for conductivity and
shot-noise spectra is to examine the involved integrands in Eqs.~(\ref{correlators}) and (\ref{correlatorscross}).
Figure~\ref{fig4} provides a schematic overview of the different
regions occurring in the 2D-plots of Figs.~\ref{fig5}-\ref{fig10}.
We show the real parts of integrands either as a function of $(q,\epsilon)$ as in scheme ~\ref{fig4} a)
or of $(q,\Omega)$ as in scheme~\ref{fig4} b).
The former is divided by the four envelopes $q=|\epsilon|$
and $\hbar v_F q=|\epsilon+\hbar \Omega|$ into six areas:
I, where the regimes II$_a$ and II$_b$ of evanescent modes are merging
and the areas III$_a$,III$_a$, III$_a$ of propagating modes.
Area III$_b$ is defined by the two lines with origins $(q=0,\epsilon=0$), ($q=0,\epsilon=-\hbar \Omega)$
and intersection $(\hbar v_Fq=\hbar \Omega/2,\epsilon=-\hbar \Omega/2)$.
Areas in scheme~\ref{fig4}b) are separated by $\hbar v_F q=|\epsilon + \hbar \Omega|$
and the dashed horizontal line $\hbar v_F q=|\epsilon|$.
The transmission probability fits into this scheme when the horizontal separation is absent
so we are left with areas 1$_a$ and 2$_{a/b}$. Then area 1$_a$
includes the black region of Fig.~\ref{fig2} where no transmission is possible, and the regime
of evanescent modes with finite transmission probability for small $|\epsilon| < \hbar v_F |q|$ around $\epsilon=0$ due to Klein tunneling.
In regimes 2$_{a/b}$ a hyperbolic shaped interference pattern with oscillations along $\epsilon$ is prominent,
where the period of oscillations is on the order of $\hbar v_f/L$ for small $\hbar v_F |q| \ll |\epsilon|$.
Figure~\ref{fig5} shows the relevant integrands of
the four correlators $C_{\alpha \rightarrow \beta}(\Omega)$
contributing to the finite frequency quantum noise, plotted as a function of $(q,\Omega)$ when $\epsilon=0$.
Then the imaginary part of $r^*(\epsilon)r(\epsilon+\hbar \Omega)$ leads to finite
contributions in the region I$_a$ and I$_b$ in figure~\ref{fig5}a).
$T(\epsilon=0)$ is only non-zero for small $q$,
so integrands b) and d) vanish for large $q$.
Since $R(\epsilon)=1-T(\epsilon)$, integrand c) vanishes when $q\rightarrow 0$
and otherwise resembles the shape of $T(\epsilon)$.\\
Finite $\epsilon$, as in Fig.~\ref{fig6}, introduces another
interference pattern for propagating modes.
In region 1$_a$ non-zero values are possible and in 2$_a$ and 2$_b$ the usual interferences occur.
For $q$-values below $\hbar v_F |q|=|\epsilon|$ this additional pattern can be seen in region 1$_b$.
The interplay of both patterns leads to phase jumps of $\pi L/\hbar v_F$ in regions 3$_a$ and 3$_b$.
These phase jumps can be determined by requiring $|r^*(\epsilon) r(\epsilon+\hbar \Omega)-1|^2=1$
in Eq.(\ref{correlators:LtoL}), Fig.~\ref{fig6}a).
Therefor $r^*(\epsilon) r(\epsilon+\hbar \Omega)$ has to vanish, what is fulfilled by the transversal momenta
of Eq.~(\ref{qjumps}).
The condition $|r^*(\epsilon) r(\epsilon+\hbar \Omega)-1|^2=4$ for a
maximum in the integrand leads to modes which experience Klein tunneling.
Actually, this correlator can be written as integral over
$1 + R(\epsilon)R(\epsilon+\hbar\Omega) - 2 [R(\epsilon)R(\epsilon+\hbar \Omega)]^{1/2} \cos(\Phi(\epsilon,\Omega))$
including a scattering-phase $\Phi(\epsilon,\Omega) = \text{Arg}\left[r^*(\epsilon)r(\epsilon+\Omega)\right]$.
Thus it describes events containing the scattering-phase between time-reversed paths of
electron-hole pairs separated by the photon energy $\hbar \Omega$ reflected back into the measurement terminal.
The effect of the phase shifts on the integrands interference patterns is also obvious
in the $(q,\epsilon)$-plot of Fig.~\ref{fig7} a), region III$_b$.
Figs.~\ref{fig7} b)-d) show a similar interference pattern
although the corresponding correlators are defined in terms of probabilities.\\
Concerning cross-correlation noise, the integrands occurring in Eqs.(\ref{correlatorscross}) show alternating patterns of
positive and negative values. The ones which describe auto-terminal
contributions to $S_{LR}(\Omega)$ (Eqs.(\ref{correlatorscross:LtoL}) and (\ref{correlatorscross:RtoR})),
as in Fig.~\ref{fig8}a),
have an alternating sign along $\Omega$.
In the cross-terminal ones (Eqs.(\ref{correlatorscross:LtoR}) and (\ref{correlatorscross:RtoL})),
as in Fig. ~\ref{fig8}b), the additional
interference pattern along $q$ introduces another change of sign.
Plots ~\ref{fig8} c) and d) show a similar behavior as functions of $(q,\epsilon)$.
When ac-bias voltages introduce the driving frequency $\omega$, the integrands structures become even richer but also less clear, as in Fig.~\ref{fig9} and Fig.~\ref{fig10}.
Then alternating signs in all contributions to auto-correlation noise are observed, except for the correlator with initial and final sates in the measurement terminal.
This results in peculiar oscillatory features in the interference patterns at combinations of all
involved energies $\epsilon,\hbar \Omega,m \hbar \omega$. Predicting the effect of such features on the noise
spectra from the plotted integrands is then almost impossible because one still has to average over all possible
energies and $q$-values by integration.
\section{Auto-correlation noise \label{sec:autonoise}}
In contrast to conductivity, the shot-noise spectrum in general couples
different orders of PAT events, expressed by the product of
four Besselfunctions of arbitrary order. But since the driving is fixed,
non-vanishing contributions exist only up to a certain order depending on the
precise value of $\alpha$. When time-dependent voltages are present, current fluctuations of Eq.~(\ref{thenoise}) contain
products of four scattering matrices, each with a different energy argument.
After performing the dc-bias limit only transitions between $\epsilon$ and $\epsilon + \hbar \Omega$ are left.
\begin{figure}[tb]
\centering
\vspace{0.2cm}
\includegraphics[width=0.97\columnwidth]{fig11a.pdf}\\
\vspace{0.6cm}
\includegraphics[width=0.97\columnwidth]{fig11b.pdf}
\caption{(color online) Real parts of auto-correlation noise spectrum in units of $2\pi\hbar/e^2$.
We compare a setup where dc-bias voltages are fixed symmetrically around
the Dirac point (top,) with the case when $eV_0L/\hbar v_F=2eV/\hbar v_F=10$ (bottom).
Thick lines: Shot-noise and correlators.
Thin lines: Derivatives with respect to frequency.
Contributions from $C_{L\rightarrow L}(\Omega)$
are dominant at positive frequencies.
Top: Special features in the derivatives are seen for frequencies $\hbar \Omega < eV$
in the $R \rightarrow R$ contribution, when the lower bound of the energy-integration interval
approaches the Dirac point (compare to Figs.~\ref{fig6},~\ref{fig7}).
Bottom: The distance to the Dirac point is increased by the offset voltage.
Therefore oscillatory features appear in a larger frequency interval and in all four correlators,
since integration boundaries in all contributions are crossing the
Dirac-point with increasing $\Omega$.}
\label{fig11}
\end{figure}
\begin{figure}[tb]
\centering
\vspace{0.2cm}
\includegraphics[width=0.97\columnwidth]{fig12a.pdf}\\
\vspace{0.6cm}
\includegraphics[width=0.95\columnwidth]{fig12b.pdf}
\vspace{0.01cm}
\caption{(color online) Real parts of auto-correlation excess-noise spectrum in units of $2\pi\hbar/e^2$ (thick lines, upper panels)
and derivatives (thin lines, lower panels) with dc-bias symmetrically applied around
the Dirac point (top) and for finite $eV_0=2eV$ (bottom).
By subtracting the noise at zero dc-bias the divergent background
is removed. The structure and especially
the oscillatory behavior are coined by auto-terminal contributions
of Eq.(\ref{correlatorscross:LtoL}) related to the measurement terminal $L$.
The jump in the derivative of $C_{L \rightarrow R}(\Omega)$ is present because
this correlator does not contribute for frequencies $\hbar \Omega <eV$.
By applying an offset voltage $eV_0L/\hbar v_F=10$ a complicated
structure emerges, best visible in the derivatives.}
\label{fig12}
\end{figure}
\subsection{Shot noise spectrum}
In the regime $eV, \hbar \Omega, \hbar \omega \ll \hbar v_F/L$, the scattering matrix can be treated
as energy-independent. Then, as for a single level quantum dot in the broad-band limit,
asymmetric quantum noise as function of frequency is the sum of four straight lines,
with kinks at $\hbar \Omega=0,\pm eV$.~\cite{GavishImry00,GavishImry02}
For vanishing dc-bias we have
$C_{R \rightarrow L}^{\mathrm{}}(\Omega)=C_{L \rightarrow R}^{\mathrm{}}(\Omega)$ and
$C_{R \rightarrow L}^{\mathrm{}}(\Omega)\approx C_{L \rightarrow R}^{\mathrm{}}(\Omega)$, as long as $ \Omega \ll v_F/L$.
The richer regime, when $eV, \hbar \Omega, \hbar \omega > \hbar v_F/L$,
additionally exhibits strongly oscillating integrands.
Those oscillations are purely due to propagating modes as it is also clear from interference patterns of
the integrands in Figs.~\ref{fig5}-\ref{fig9}, regions II$_{a,b}$ and III$_{a,b,c}$.
In the special case of perpendicular incidence ($q,\alpha(\epsilon)=0$) we have Klein tunneling, thus
the frequency-dependence of the correlators is linear for this mode.
Then $C_{\alpha \rightarrow \beta}(\Omega)=0$ if $\alpha \ne \beta$ since $R(\epsilon)=0$.
Otherwise the $C_{\alpha \rightarrow \beta}(\Omega)$ mirror the interference patterns of the integrands.
So the noise spectrum (Fig.~\ref{fig11} solid, thick curve) shows oscillations on the scale of $L/\hbar v_F$ in the
regime $eV, \hbar \Omega, \hbar \omega \gg \hbar v_F/L$,
similar to the shot-noise at zero-frequency as a function of gate voltage~\cite{Tworzydlo06}.
Although present in all four correlators, the oscillations show up in the noise spectrum
mainly via $C^\mathrm{}_{L\rightarrow L}(\Omega)$ of the terminal where the fluctuating currents are probed.
That is because the correlator itself as well as the amplitude of the oscillations are significantly larger than for other contributions.
Therefore, in comparison to the absorption-branch (positive frequencies) the emission-branch of
the spectrum (negative frequencies) shows only small shot-noise. Indeed all correlators except $C_{R\rightarrow L}(\Omega)$ vanish when $\Omega \le 0$
since the energy for the emission of a photon has to be provided by the voltage source.
Especially the contribution dominant at positive frequencies vanishes:
$C^\mathrm{}_{L\rightarrow L}(\Omega)=0$ if $\hbar \Omega \le 0$.\\
We are considering the limit $k_BT=0$ where the correlators integration windows are exactly determined by the chemical potentials.
At finite temperature this so-defined onsets of the four contributions as a function of frequency
are smeared out by the broadening of the Fermi-functions.
Clearly a gate voltage does not affect these onsets since it does not enter in the Fermi functions of the leads,
but it still changes the transmission function resulting in a modified spectrum.
\begin{figure}[tb]
\centering
\vspace{0.0cm}
\includegraphics[width=0.97\columnwidth]{fig13a.pdf}\\
\vspace{0.0cm}
\includegraphics[width=0.965\columnwidth]{fig13b.pdf}
\caption{(color online) Real parts of auto-correlation current-current
fluctuations in units of $2\pi\hbar/e^2$ as a function of dc-bias for fixed frequency
$\hbar \Omega$ (thick lines, upper panels). We compare the symmetric setup without
dc-bias offset (top) and when $eV_0L/\hbar v_F=10$ (bottom).
Thin lines (lower panels) are used for the derivatives with respect to voltage.
Top: Due to symmetrically applied bias voltage the noise and the auto-terminal contributions
are symmetric in the voltage dependence and
$\left.S_{\alpha \rightarrow \beta}(\Omega)\right|_{V}=\left.S_{\beta \rightarrow \alpha}(\Omega)\right|_{-V}$ if $\alpha\ne \beta$.
Bottom: By applying an offset voltage we are breaking the setups symmetry.
Auto-terminal terms are then symmetric with respect to $eV=\pm 2eV_0$ while the summed up noise is asymmetric. }
\label{fig13}
\end{figure}
Those limits of energy-integration, as well as their position relative to region III$_b$, result in features
in the noise spectra besides the discussed oscillations.
In order to clarify the role of the Dirac Hamiltonian in comparison to
the role of pure Fabry-P\'erot interferences, we compare results
when the charge injection is only in the conduction or
valence band by shifting the dc-bias voltages above
the Fermi energy of the graphene sheet via the offset
voltage $V_0$ in $\mu_{L/R}=\pm eV/2 + eV_0$.
$C^\mathrm{}_{L\rightarrow R}(\Omega)$ can
never see the regime $-\hbar \Omega<\epsilon<0$ when $eV_0=0$, as in the upper plot of Fig.~\ref{fig11}.
Thus the oscillations visible in the derivate
have a well defined period over the whole spectrum on top of a
linearly increasing background.
When an offset voltage $eV_0= 2eV$ is applied,
as done when calculating the spectra for the lower plot of Fig.~\ref{fig11},
$C_{R\rightarrow R}(\Omega)$ shows
a complicated frequency dependence for small $\Omega$.
Contribution $C_{L\rightarrow L}(\Omega)$ describes correlations of
scattering states emanating from the left reservoir reflected back into
the same reservoir. We will discuss this contribution now in detail:
Special features for small frequency are due to the interplay of
the integration boundaries with the various regions in Fig.~\ref{fig4}a)
occurring in the integrands $(q,\epsilon)$-dependence of Fig.~\ref{fig7}a).
Integration is over all $q$-modes and from $\epsilon=-eV/2+eV_0-\hbar \Omega$
to $\epsilon=eV_0-eV/2$. When $eV_0=0,eV=0$ this corresponds
to $-\hbar\Omega < \epsilon <0$, regions III$_b$ and partly II$_{a,b}$
of Fig.~\ref{fig4}a). Now at finite $eV,eV_0$ as in Fig.~\ref{fig11},
the integration window can include region III$_b$ completely, partly, or not at all,
resulting in variations of the spectrum. At small $\hbar \Omega$,
features in the integrands interference patterns have stronger impact.
This can be seen from strongly non-harmonic features of the noise spectrum,
e.g. in $C_{L\rightarrow L}(\Omega)$
and $C_{R\rightarrow L}(\Omega)$ for $eV_0=2eV$.
For large frequencies averaging leads to nearly harmonic oscillations
on top of the increasing background.
With the chosen parameters the distance
of the chemical potential $\mu_L$ to the charge-neutrality point is given by
$e(-V/2+V_0) L/(\hbar v_F)=7.5$. Around the corresponding frequency the oscillatory behavior
of the spectrum is modified and flattened due to a reduced fraction of propagating modes.
Raising the frequency further increases this fraction again and oscillations
are roughly harmonic with period
$\pi L/\hbar v_F$, best visible in the derivatives $dC_{L\rightarrow L}(\Omega)/d\Omega$ of Fig.~\ref{fig11}.
That is also the point where the lower bound of energy integration starts
to include the special interference pattern of the integrands around
the energy interval $-\Omega< \epsilon < 0$, region III$_b$.
$C_{R\rightarrow R}(\Omega)$ is not influenced
by the measurement terminal itself, but probes transmission probabilities
via scattering events which are related to the right terminal only.
An analogous behavior of the spectrum as before is found,
this time with a distance $e(V/2+V_0) L/(\hbar v_F)=12.5$ of the
lower integration boundary to the charge neutrality point when
$\hbar \Omega=0$. Now increasing frequency is going along with
a decreasing slope of the derivative with respect to frequency
until the Dirac point is reached.
There the slope increases again since more open channels become available.
The same interpretation also explains features in the interval $\hbar \Omega < eV$
of the auto-terminal correlators shown in Fig.~\ref{fig11}, when $V_0=0$.
E.g. the spectrum of the correlator Eq.(\ref{correlators:RtoR}), with initial and final state
in the right lead, exhibits a reducing slope until $\hbar \Omega=eV/2$
from where on the oscillations have a well defined period.
The $dC_{R\rightarrow R}(\Omega)/d\Omega$ curve has a maximal slope at
$\hbar \Omega=eV$ when positive and negative energies with same magnitude
are present. For higher frequencies oscillations have again a well-defined phase.\\
We also study the excess noise at finite frequencies:
$S_{\mathrm{exc}}(\Omega,\omega):=\left. S(\Omega,\omega)\right|_{eV}-\left. S(\Omega,\omega)\right|_{eV=0}$.
Subtracting the noise at zero bias-voltage removes the divergent contributions
from the noise spectrum. Then oscillating features due to bias-voltages are
more obvious since they are now also prominent in the noise spectra of Fig.~\ref{fig12},
not only in derivatives.
When $eV_0=0$ the excess noise (thick, black, solid curve) is purely positive for
$\hbar \Omega \ll eV$ while for $\hbar \Omega > eV$ it is oscillating around zero,
because then cross-terminal contributions $C_{\alpha \rightarrow \beta}(\Omega)$ cancel each other
up to a constant offset acquired at small $\Omega$.
This offset is compensated by the $L\rightarrow L$
contribution. Oscillations of this contribution have again a considerable impact on the excess noise spectrum.
In the lower plot of Fig.~\ref{fig12} the offset voltage is fixed to $eV_0=2eV$.
For low frequencies $\hbar \Omega < eV$, complicated oscillations occur in all contributions
to excess noise and are accompanied by a strongly increasing slope up to frequencies $\hbar \Omega>eV_0 +eV/2$.
As for the noise itself, the frequency of the oscillations is determined by $\hbar\Omega_Z=2eV$ and equals the frequency
expected from the Zitterbewegung of relativistic Dirac fermions~\cite{Katsnelson06}.
This frequency corresponds to a period of $T=\pi$ in our plots.
It would be interesting to test experimentally if those much more pronounced oscillation,
compared to the overall shot-noise, can be detected in spite of randomization effects
of imperfections on the quasi-particles path lengths.
In summary, i) the impact of the Dirac Hamiltonian on the frequency-dependence of auto-terminal current fluctuations
leads to peculiar oscillation for energies in the vicinity of the Dirac point as an interplay of Klein tunneling,
phase-jumps in the correlators and their energy-integration limits. And ii) oscillations due to the FP setup have a constant
phase for high energies when propagating modes are dominant. Then $dS^{\mathrm{exc}}(\Omega)/d\Omega$ oscillates
between positive and negative values with a period as it is expected from the effect of Zitterbewegung.
\subsection{Dc-bias dependence at finite frequency}
\begin{figure}[tbp]
\centering
\vspace{0.6cm}
\includegraphics[width=0.95\columnwidth]{fig14a.pdf}\\
\vspace{1.0cm}
\includegraphics[width=0.95\columnwidth]{fig14b.pdf}
\caption{(color online) Real parts of cross-correlation spectrum in units of $2\pi\hbar/e^2$
when $eV_0=0$ (top) and $eV_0L/\hbar v_F=3eVL/\hbar v_F=18$ (bottom).
Without an offset voltage ($eV_0=0$) auto-terminal contributions are identical,
as well as cross-terminal ones at large frequencies $\hbar \Omega \gg eV$.
At finite $eV_0$ the asymmetric bias voltage is reflected in the frequency-dependence of the
auto-terminal correlators by their different heights and the shift of the oscillations maxima.
}
\label{fig14}
\end{figure}
Analogous to the spectrum, the dc-bias dependence for fixed frequency is featureless in the regime
$eV, \hbar \Omega, \hbar \omega \ll \hbar v_F/L$,
except the pronounced onsets of the four correlators.
This is not surprising when looking at the derivatives with respect to voltage:
\begin{subequations}
\label{derivativesV}
\begin{align}
& \frac{dC_{L \rightarrow L}}{dV}=\frac{e^2 \Theta(\Omega) }{4\pi \hbar} \underset{-\infty}{\overset{\infty}\int} dq \left[\left|1-r^*(-eV/2)r(\hbar\Omega-eV/2) \right|^2 \right.\nonumber\\
&\left. -\left|1-r^*(-eV/2-\hbar \Omega)r(-eV/2) \right|^2 \right] \label{derivativesV:LtoL} \\
&\frac{dC_{R \rightarrow R}}{dV}= \frac{e^2 \Theta(\Omega) }{4\pi \hbar}\underset{-\infty}{\overset{\infty}\int} dq\nonumber\\
&\left[T(eV/2)T(eV/2 +\hbar \Omega) -T(eV/2-\hbar\Omega)T(eV/2)\right]\label{derivativesV:RtoR}\\
&\frac{dC_{L \rightarrow R}}{dV}=\frac{e^2 \Theta(\Omega-eV) }{4\pi \hbar}\underset{-\infty}{\overset{\infty}\int} dq\nonumber\\ &\left[ T(\hbar\Omega-eV/2)R(-eV/2)-T(eV/2)R(eV/2-\hbar\Omega) \right] \label{derivativesV:LtoR}\\
& \frac{dC_{R \rightarrow L}}{dV}=\frac{e^2 \Theta(\Omega+eV) }{2\pi \hbar} \underset{-\infty}{\overset{\infty}\int} dq\nonumber\\ &\left[ T(eV/2)R(eV/2+\hbar\Omega) -T(-eV/2-\hbar\Omega) R(-eV/2)\right] \label{derivativesV:RtoL}
\end{align}
\end{subequations}
Scattering amplitudes are roughly constant for a given q-mode in this regime,
then correlators are straight lines as a function of dc-bias voltage.
E.g. a special situation that could exhibit interesting physics is when some derivatives are zero.
But this is, due to symmetry arguments, only possible at $eV=0,\pm \hbar \Omega$,
proofing a zero slope of the correlators at their onsets but revealing no additional effect.
By this means, as in the shot-noise spectrum, the dependence on the bias voltage
reveals again the onsets of the four correlators. Since we have chosen
positive $\hbar \Omega$, the auto-terminal contributions are non-zero over the whole bias range.
As before, cross-terminal ones vanish if no energy is provided by the voltage source:
$C_{L \rightarrow R}\ne 0$ if $eV>-\hbar \Omega$
and $C_{R \rightarrow L}\ne 0$ if $eV<\hbar \Omega $.
As it is clear from the bottom plot of Fig.~\ref{fig13},
the oscillations of the components are not in phase,
thus adding up to complicated oscillations in $S_{LL}(\Omega)$.
But, as mentioned in the beginning, we doubt this could be a measurable effect.
The shot-noise and the auto-terminal correlators are symmetric in the voltage dependence if $V_0=0$,
whereas the cross terminal ones obey $C_{\alpha \rightarrow \beta}(\Omega,V)=C_{\beta \rightarrow \alpha}(\Omega,-V)$.
Here the charge-neutrality point and the width of the region III$_b$ are revealed
as a minima in the slope of the correlator $C_{L \rightarrow L}(\Omega)$ at $eV=\pm 2 \hbar \Omega$
and in the change of sign in $dC_{R \rightarrow R}(\Omega)/dV$ at $eV=0$.
\section{Cross-correlation noise \label{sec:crossnoise}}
The explicit expressions of Eq.(\ref{correlatorscross}) for the cross-correlation current noise
spectrum of Fig.~\ref{fig14} can be extracted from the general expression Eq.(\ref{noisedef}) in the
same way as we did when deriving Eq.(\ref{correlators}).
From Figs. ~\ref{fig8} a) and b) it is also clear that the spectrum of auto-terminal
correlators are oscillating as a function of $\Omega$ with larger amplitude than cross-terminal ones,
since they show an alternating behavior between positive and negative integrands.
Dependence on $q$ in the relevant frequency range is weak, as shown in Fig.~\ref{fig8} a).
Contrary, cross-terminal contributions as in Fig.~\ref{fig8}
b) show features with an alternating sign along both variables, $\Omega$ and $q$.
Thus, integration along y-momentum leads to averaging and therefore
significantly smaller oscillation amplitudes occur.
As discussed for the excess noise of the auto-correlation noise spectral function,
we find the oscillations have a frequency $\hbar \Omega_Z=2eV$ what is tantamount to a period $T=2\pi$ in the plots.
Complex conjugation corresponds to time-reversed states.
Again, as the product of scattering matrices of the integrands in Eq.(\ref{noisedef}) suggests,
it is probing transmission- and reflection amplitudes
of electron-hole pairs separated by an energy quanta $\hbar \Omega$.
So, for cross-terminal noise not only the reflection but also the complex
transmission amplitude is essential even without ac-bias voltages.
Again it would be interesting to test if the resulting oscillations could be detected
in the challenging task of a finite-frequency cross-correlations experiment.
Analyzing the integrands reveals the symmetry
$C^{\mathrm{c}}_{\alpha \rightarrow \alpha}(\Omega)=C^{\mathrm{c}}_{\beta \rightarrow \beta}(\Omega)$ if
$\mu_L=-\mu_R$ as we show in Fig.~\ref{fig15}. This symmetry is distorted by applying an offset voltage $V_0$.
The spectrum of the correlator $C^{\mathrm{c}}_{L\rightarrow L}(\Omega)$ shows a shift of the maxima and minima of
the oscillations with respect to $C^{\mathrm{c}}_{R \rightarrow R}(\Omega)$ for finite $V_0$.
This shift is due to the fact that the distance between neighboring maxima
of the integrand is not constant when varying $\hbar \Omega$ at given q-mode
(see the bending of the maxima towards higher frequencies for larger $q$ in the integrands, e.g. Fig.~\ref{fig6}).
Derivatives of the correlators $C^{\mathrm{c}}_{\alpha \rightarrow \beta}(\Omega)$
with respect to voltage show a sequence of pairs of different maxima.
This observation is traced down to the same origin as above,
and so the appearance of peculiar oscillations in the
summed up cross-correlation shot-noise $S_{LR}(\Omega)$ is explained.
At $\hbar \Omega=0$ current conservation and the unitarity of the s-matrix
require $S_{LR}(\Omega)=-S_{LL}(\Omega)$.
Therefore the correlator described by Eq.~\ref{correlatorscross:RtoL} is negative.
\begin{figure}[tbp]
\centering
\vspace{0.6cm}
\includegraphics[width=0.95\columnwidth]{fig15.pdf}
\caption{(color online) Real parts of current-current cross-correlations
in units of $2\pi\hbar/e^2$, as function of symmetrically
applied dc-bias voltage and for fixed frequency.
Jumps in the derivatives at $\pm \Omega L/v_F$ are
due to the onsets of the cross-terminal contributions.}
\label{fig15}
\end{figure}
\section{Finite-frequency noise at ac-bias \label{sec:finitefreq}}
By applying an ac-bias voltage at the leads one can inject
charge-carriers at positive and negative energies of the Dirac cone
without applying a dc-voltage. Analogous to the minimal conductivity,
in the non-driven case going along with a maximal Fano factor, the
shot-noise at zero frequency but finite ac-bias $S_{\alpha
\alpha}(\Omega=0;\omega)$ mirrors the behavior of the conductivity
in Fig.~\ref{fig3}.
\begin{figure}[tbp]
\centering
\vspace{0.0cm}
\includegraphics[width=0.95\columnwidth]{fig16.pdf}
\caption{(color online) Derivatives of current-current correlations real parts
in units of $2\pi\hbar/e^2$ with respect to frequency
as function of frequency. We have chosen a symmetrically
applied dc-bias voltage with additional harmonic ac-driving
($\omega L/v_F=4$, $\alpha=0.5$ and $a=1$) in lead $L$.
Dashed vertical lines mark the step-positions, coloring specifies the correlator which
shows the step at corresponding $\Omega L/v_F$.}
\label{fig16}
\end{figure}
The noise spectrum Fig.~\ref{fig16} for the driven setup ($a=1$) is
similar to the one without driving but with additional steps in the
derivatives. For arbitrary ac-bias these steps can appear at frequencies
$\hbar \Omega=\left(\mu_{\alpha}-\mu_{\beta}\right)\pm n \hbar \omega$
due to the onset of higher-order PAT events. Since we set $a=1$ in Fig.~\ref{fig16},
the correlator with states $R\rightarrow R$ shows no ac-induced steps in the derivative.
But when $|a|\ne 1$ all integrands (Fig.~\ref{fig9}) are not given in terms of probabilities
and can take negative values as mentioned in section \ref{sec:qualitative}.
For the shot-noise spectrum there are then two possible sources of contributions that could reduce noise:
Either a correlators integrand or the the product of Besselfunctions is negative.
When the driving voltage is applied symmetrically ($a=0$) more PAT-induced steps
in the derivatives of the noise spectrum are visible and finite contributions
at negative $\Omega$ are possible for all correlators.
As proposed by Trauzettel et. al.,~\cite{Trauzettel07} a time-dependent
voltage could be used to induce interference between states in
particle- and hole-like parts of the Dirac spectrum. This should
correspond to Zitterbewegung like in relativistic quantum mechanics,
but we are not aware of any unique feature caused by Zitterbewegung
that can be distinguished from other oscillations, especially of Fabry-P\'erot
nature.
\section{Conclusions \label{sec:conclude}}
We have analyzed conductivity and non-symmetrized finite-frequency current-current correlations
for a Fabry-P\'erot graphene structure.
Oscillations on the intrinsic energy scale $L/\hbar v_F$ are still present in the finite frequency noise.
Emission spectra are diverging for large frequencies,
whereas the absorption branch of the spectrum has to vanish at $\hbar \Omega=-eV$.
As expected from the integrands, the current-noise also diverges for voltages $|eV| \gg \hbar v_F/L $.
Since the onset of the different noise contributions is defined by the four
possible combinations of the chemical potentials, the noise built by
all correlators consists of contributions oscillating with the same period but
different phases. Although dominated by $C_{L\rightarrow L}(\Omega)$ when
correlating the currents in terminal $L$ at large frequencies, this interplay
is revealed in the spectra and voltage dependence of all correlators.
Each contribution can show peculiar oscillations at low enough frequencies
or voltages. In this regime features in the integrands
$(q,\epsilon)$-dependence can have a prominent impact whereas they tend to be averaged out at large frequencies.
Another aspect is the appearance
of a special region showing phase jumps
in the energy dependence of the integrands when $\hbar \Omega \le 2eV$.
This interplay of the Dirac spectrum and the Fabry-P\'erot physics~\cite{Liang01,Herrmann07}
can be probed purely by applying an appropriate combination of dc-bias and offset voltage $V_0=V/2$,
thus connecting electron- and hole part of the Dirac spectrum symmetrically when $eV=2 \hbar \Omega$.
The way the scattering amplitudes are combined in this approach spoils the
clear picture in terms of transmission- and reflection probabilities. Instead, in the dc-limit it gives
rise to the interpretation of the $L \rightarrow L$ contribution in terms of jumps in the scattering-phase
between time-reversed electron-hole states separated by the photon energy $\hbar \Omega$.
In the same way the complex correlators for cross-correlation noise or for the driven setup
exhibit phase jumps and can not be written in terms of probabilities.
Complex contributions of the scattering matrices lead to large oscillations between positive and
negative values of cross-correlation noise or in the derivatives
with respect to frequency of the auto-terminal noise spectral function.
These oscillations have a frequency of $\hbar\Omega_Z=2eV$, what corresponds to a period of $T=2\pi$ in our plots.
This frequency corresponds to the Zitterbewegung frequency as it is known for relativistic Dirac fermions.
Again, strongly non-harmonic features can occur when the transition between different regimes
is probed, especially when region III$_b$ around the Dirac point comes into play.
Additional ac-bias complicates the picture because
combinations of $q, \hbar \Omega, m\hbar \omega$ define additional phase jumps,
onsets of the correlators and therefore steps in the noise when higher-order PAT events occur.
Then the special role of the complex reflection and transmission amplitudes
is essential for all possible correlators.
\acknowledgments
We would like to acknowledge the financial
support by the DFG (Grant No. SFB 767) and
thank B. Trauzettel for validating and sharing
corrections to reference~\cite{Trauzettel07}.
|
{
"timestamp": "2012-03-12T01:00:54",
"yymm": "1203",
"arxiv_id": "1203.2010",
"language": "en",
"url": "https://arxiv.org/abs/1203.2010"
}
|
\section{Results}
Fig. \ref{fig:pca} shows the 3 -- 20 keV X-ray flux from the source as a function of the orbital and superorbital phase (as seen by RXTE/PCA). From Fig. \ref{fig:pca} one could see that the source exhibits on average one episode of increased X-ray activity per orbit. A regular increase of X-ray activity in the phase interval $\Delta \phi_X\simeq 0.2-0.4$ is accompanied by random variations of the source flux, with short flares appearing on the time scales $T_{flares,X}\ll \Delta\phi_XP_{orb}$. The average source flux in 3-20 keV energy range during the active/quiet part of the orbit indicated by the white dashed lines on Fig. \ref{fig:pca} is about 1 mCrab/0.5 mCrab. Typical error of the flux measurment, $\sim$ 0.1 mCrab, is dominated by the uncertainity of the PCA background. The figure also has the color scale expressed in mCrab units.
The average phase of the X-ray activity period $\phi_{X}$ varies on the superorbital time scale. From Fig. \ref{fig:pca} we find that $\phi_X$ exhibits a systematic drift from $\phi_X\simeq 0.35$ to $\phi_X\simeq 0.75$ within one superorbital cycle. Such a drift is similar to the systematic drift of the phase of the periodic radio flares from $\phi_R\simeq 0.5$ to $\phi_R\simeq 1$ \citep{gregory02}.
Evolution of the orbital variability of the source in hard X-rays (20 - 60 keV) on the superorbital time scale, observed by INTEGRAL, is shown in Fig. \ref{fig:int}. One could see that, similarly to the 3-20~keV range, the maximum flux happens during the orbital phase $0.25 <\phi < 0.5$ in the superorbital cycle phase $0.5<\Phi<1$. The maximum becomes wider and shifts toward $0.25 < \phi_X < 0.75$ in the superorbital phase $0<\Phi<0.5$.
\begin{figure}
\includegraphics[width=\columnwidth,bb=24 420 585 675,clip]{integral_super_v2}
\caption{Averaged orbital variability of the hard X-ray flux (20 - 60 keV) from LSI~+61~303\ for the $0<\Phi<0.5$ (black solid crosses) and $0.5<\Phi<1$ (red dotted crosses) superobital phases. The corresponding exposures (from left to right) are 525, 664 and 452 ksec for the $0<\Phi<0.5$ and 298, 132 and 207 ksec for the $0.5<\Phi<1$.}
\label{fig:int}
\end{figure}
Observations of the systematic drift of the phase of the radio flares from the source reported by \cite{gregory02} were performed several superorbital cycles before the X-ray monitoring campaign by RXTE. To verify the long-term stability of the range of the shifts of $\phi_R$ over many superorbital cycles we use the data of monitoring of the source in the radio band which are contemporaneous with the RXTE monitoring campaign. Fig. \ref{fig:radio} shows the radio flux of the source as a function of the orbital and superorbital phases. {Colorbar shows flux measured in mJy.}
Comparing Fig. \ref{fig:radio} with the equivalent figure from \citet{gregory02}, we find that the overall drift pattern of the phase of the radio flare remained stable over several superorbital cycles. The same drift from the phase $\phi_R\simeq 0.55$ to $\phi_R\simeq 0.95$ is observed also in the radio data contemporaneous with the RXTE monitoring campaign.
\begin{figure}
\includegraphics[width=\columnwidth,bb= 70 217 540 575,clip]{radio_lsi_ph_map}
\caption{Radio flux from LSI~+61~303\ as a function of the orbital and superorbital phases. The color scale is expressed in mJy units.}
\label{fig:radio}
\end{figure}
Comparison of X-ray and radio superorbital variability patterns is shown in Figure \ref{fig:radio}. The average phase of the X-ray activity period always preceeds the phase of the radio flare by $\Delta\phi_{X-R}\simeq 0.2$, which corresponds to the time delay $\Delta T_{X-R}=\Delta\phi_{X-R}P\simeq 5.3$~d.
Contrary to the X-ray and radio bands, the superorbital modulation pattern is not clearly visible in the $\gamma$-ray\ band. Fig. \ref{fig:fermi} shows the source flux in the 0.1-10~GeV energy band plotted as a function of the orbital and superorbital phase, similarly to Figs. \ref{fig:pca} and \ref{fig:radio}. {Colorbar shows flux measured in mCrabs, typical error is about 10\% of the flux.} Long-term source behaviour of the source in the GeV band is puzzling. Orbital modulation was clearly observable at the beginning of Fermi observations at the superorbital phase $6.8<\Phi<6.9$. The phase of the maximum orbital modulation of the GeV flux in this superorbital phase range was close to the phase of the X-ray activity. However, in the time period following the superorbital phase $\Phi\simeq 6.9$ a clear orbital modulation pattern disappeared (see also \cite{hadasch11}). Further study of the source on the time scale of several superorbital cycles is needed to clarify the repeatability of the observed appearance / disappearance of the orbital modulation pattern and its relation to the overall 4.6~yr activity cycle of the source.
\begin{figure}
\includegraphics[width=\columnwidth,bb= 60 220 555 575,clip]{fermi_lsi_ph_map}
\caption{Very high energy (E$>$100 MeV) flux from LSI~+61~303\ as a function of the orbital and superorbital phases. The color scale is expressed in mCrab units.}
\label{fig:fermi}
\end{figure}
\section{Discussion.}
A constant time delay between the drifting orbital phases of X-ray and radio flares could be naturally explained if one takes into account that radio and X-ray emission originate from different regions. The radio emission is produced at large distance from the binary, $D_R\gtrsim 5\times 10^{13}$~cm \citep{zdziarski10} while the X-ray flux is most probably produced at shorter distances of the order of the binary separation $3\times 10^{12}\mbox{ cm}<D_X <10^{13}\mbox{ cm}$. Assuming that injection of high-energy electrons responsible for the X-ray and radio flares happens in the same event in the binary, one could attribute the time delay between the X-ray and radio flares to the time-of-flight of the high-energy particle filled plasma to the radio emission region. {The cooling times of high energy particles are long enough to allow them to travel to the radio emission region.} This gives an estimate of the plasma outflow velocity $v_R\simeq D_R/\Delta T_{X-R}\simeq 10^8$~cm/s, which is in good agreement with the asymptotic velocity of the stellar wind from the massive star in both the polar and equatorial regions \citep{zdziarski10}.
The constant time delay between the X-ray and radio flare phases suggests the following scenario of production of the periodic radio flares. Once per orbit, an event of interaction of the compact object with the stellar wind leads to injection of high-energy particles into the stellar wind. The high-energy particles mixed into the stellar wind escape from the binary system with the stellar wind velocity. A radio flare occurs at the moment when the portion of the stellar wind filled with high-energy electrons reaches the distances $D\sim D_R$ at which the system becomes transparent to the radio waves.
High-energy electrons, responsible for the radio-to-X-ray emission, are held in the plasma outflow by the magnetic field. Electrons producing synchrotron emission in the radio band have energies $E_e\simeq 10 \left[B/1\mbox{ G}\right]^{-1/2}\left[\nu_R/10\mbox{ GHz} \right]^{1/2}$~MeV, where $\nu_R$ is the frequency of the radio synchrotron emission. The main cooling mechanisms for such electrons are synchrotron and/or inverse Compton emission with the characteristic cooling time scales $t_S\simeq 0.7 \left[B/1\mbox{ G}\right]^{-3/2}\left[\nu_R/10\mbox{ GHz}\right]^{-1/2}$~yr and $t_{IC}\simeq 6.5\left[D/10^{13}\mbox{ cm}\right]^{-2}\mbox{ d}$. Unless magnetic field in the system is much higher than $B\sim 10$~G, the synchrotron and inverse Compton cooling time scales are not shorter than the time-of-flight from inside the binary orbit to the radio emission region. Thus energy losses do not prevent electrons injected in the binary system from traveling to the radio emission region on a time scale of several days.
The X-ray emission is most probably produced via the synchrotron mechanism by electrons of the energies $E_e\simeq 100\left[B/10\mbox{ G}\right]^{-1/2}\left[E_X/5.4\mbox{ keV}\right]^{1/2}\mbox{ GeV}$. This conjecture is supported by the observations of fast variability of X-ray emission on the time scales $t\sim 10$~s \citep{smith09} which is comparable to the synchrotron cooling time of 100~GeV electrons in the $B\sim 10$~G magnetic field. The absence of any obvious break / cut-off features in the keV-GeV source spectrum with a maximum in the $\sim 10-100$~MeV range is in favour of interpretation of the entire keV-GeV bump as a single spectral component.
In such a model the GeV band emission is due to the synchrotron emission by the 10-100~TeV electrons.
The phase of the $\gamma$-ray\ flare is close to the phase of the X-ray flare (Fig. \ref{fig:pca} and \ref{fig:fermi}), at least during a part of the superorbital cycle $6.8<\Phi<6.9$. It is natural to identify the phase of the X-ray/$\gamma$-ray\ flare with the moment of formation of high-energy particle outflow inside the binary. Long term evolution of this phase could not be followed in the $\gamma$-ray\ band, because of the large width of the activity period after the superorbital phase $\Phi>6.9$. In the X-ray band the width remains finite all over the superorbital cycle. Difference in the superorbital modulation pattern in X-rays and $\gamma$-ray s is, most probably, related to different regimes of acceleration/propagation/cooling of 10 MeV and 10 TeV electrons.
The phase of the X-ray activity is close to the phase of the periastron of the binary orbit $\phi_{per}\simeq 0.3$ at the superorbital phase $\Phi\simeq 0.5$ (Fig. \ref{fig:pca}) when the duration of the activity period is shortest.
The phase of activity gradually shifts to the post-periastron interval $\phi_X>0.3$ over the superorbital period and almost reaches the phase of the apastron $\phi_{ap}\simeq 0.8$ toward the end of the superorbital cycle so that the X-ray emission is always delayed with respect to the apastron. Shift of $\phi_X$ is accompanied by the increase of the width of activity period.
A possible explanation for such behaviour could be found in a scenario in which the 4.6~yr superorbital cycle is interpreted as the cycle of gradual buildup and decay of the equatorial disk of the Be star. At $\Phi\simeq 0.5$ the equatorial disk is weak. The phase of the closest encounter between the disk and the compact object is the phase of the periastron. Interaction of the compact object with the disk perturbs the disk and strips away a part of the disk which escapes from the system to the radio emission region. Gradual buildup of the equatorial disk due to ejection of matter from the Be star leads to the increase of the disk density and/or disk size. {Such a scenario implies that extended radio emission shouldn't have a clear jet-like morphology, but rather have an irregular morphology varying over the orbital and superorbital cycle.Such a variable morphologu is indeed observed in the radio band \cite{dhawan06}.}
The shift of the phase $\phi_X$ of ejection of a portion of the disk could be explained by the increase of the time of accumulation of energy sufficient for ejection. The kinetic energy needed to strip away a part of the disk is comparable to the gravitational binding energy of the disk, $U\sim G_NM_*\rho_dR_dH_d\sim 10^{40}\left[\rho_d/10^{13} \mbox{g} \cdot \mbox{cm}^{-3}\right]\left[R_d/10^{12}\mbox{ cm}\right]\left[H_d/10^{12}\mbox{ cm}\right]\times$ $[M_*/10M_\odot]$~erg, where $\rho_d, R_d$ and $H_d$ are the density, radius and thickness of the equatorial disk. Such energy should be transmitted to the disk by the compact object at each disk-compact object interaction event. Suppose that the compact object injects energy in the disk at a constant rate $P$~erg/s at each interaction event. The energy sufficient for ejection of a part of the disk is then accumulated on a time scale $T_{ej}\sim U/P\sim 1\left[P/10^{35}\mbox{ erg/s}\right]^{-1}$~d. Accumulation of mass in the disk leads to the increase of $T_{ej}$ and, as a consequence, to the shift of $\phi_X$.
If the disk size reaches the size of the binary orbit, the compact object always moves inside the disk and continuously perturbs it. Studies of the high-mass X-ray binaries with Be stars show that in this case the compact object induced instabilities in the disk might lead to destruction and
complete loss of the disk. Our hypothesis is that the loss of the disk corresponds to the period in the superorbital phase range $\Phi\simeq 0.4-0.5$ when the strength of X-ray and radio flares decreases and no systematic periodic variability of the source is observed (see Fig. \ref{fig:pca}, \ref{fig:radio} and \citet{gregory02}). Ejection of matter from the equatorial regions of Be star leads to formation of a new disk at around $\Phi\simeq 0.5$ and a new cycle of disk growth/decay starts.
In such a scenario, regularity of the superorbital modulation in the system could be readily explained. The constant growth rate of the equatorial disk of Be star is determined by the stable rotation of the Be star. The period of superorbital modulation is determined by the fixed time scale on which the disk growth to the size comparable to the size of the binary orbit. This scenario for the origin of the orbital and superorbital modulations of the source flux could be tested using the H$\alpha$ data which provide a diagnostic of the state of the equatorial disk of Be star \citep{zamanov99,mcswain10}. Growth and decay cycles of the disk lead to the variations of the overall strength and shape of the H$\alpha$ line. Variability of the line intensity and profile on the time scale of the superorbital modulation was demonstrated by \citet{zamanov99}. Monitoring of the H$\alpha$ line on the time scale of several superorbital cycles would show if the the observed variability corresponds to the periodic buildup and decay of the disk. Periodic ejection of a part of the disk as a result of the compact object-disk interaction might be responsible for occurence of transient red or blue ``shoulders'' of H$\alpha$ line as observed by \citet{mcswain10}. Systematic re-observation of the repetition of occurence of the shoulders in many orbital cycles and correlation of the phase of occurence of the shoulders with the phases of X-ray flares would provide a direct test for our model.
\textit{Note added in proof:} During the process of publication of this
article, a similar study by \cite{li12} appeared in press.
\textbf{Acknowledgements}
The authors thank participants of the ISSI team ``Study of Gamma-ray Loud Binary Systems'' for useful discussions, and the International
Space Science Institute (ISSI, Bern) for support. The authors also wish to acknowledge the SFI/HEA Irish Centre for High-End Computing (ICHEC) for the provision of computational facilities and support. The work of D.M. is supported in part by the Cosmomicrophysics programme of the National Academy of Sciences of Ukraine and by the State Programme of Implementation of Grid
Technology in Ukraine. S.M. and A.L. acknowledged the support from the program ``Origin, Structure and Evolution of the Objects in the Universe'' by the Presidium of the Russian Academy of Sciences, grant no.NSh-5069.2010.2 from the President of Russia, Russian Foundation for Basic Research (grants 11-02-01328 and 11-02-12285-ofi-m-2011), State contract 14.740.11.0611
|
{
"timestamp": "2012-03-12T01:00:16",
"yymm": "1203",
"arxiv_id": "1203.1944",
"language": "en",
"url": "https://arxiv.org/abs/1203.1944"
}
|
\section{Introduction}
The physical properties of a system at thermal equilibrium are determined by an equation of state. For a fluid of particles in the grand canonical ensemble, the equation of state relates a thermodynamic quantity such as pressure, density or entropy to temperature and chemical potential. It can take a complicated expression when the particles interact {\it via} a two-body potential $V({\bf r}_1-{\bf r}_2)$ which has no simple expression as is usually the case in real systems. Quite remarkably however, the low-temperature equation of state of a $d$-dimensional dilute gas is universal, in the sense that it depends only on a small number of parameters, such as the mass $m$ of the particles and the $s$-wave scattering length $a_d$, and is otherwise insensitive to the details of the two-body potential $V({\bf r}_1-{\bf r}_2)$. A well-know example of universality is given by a three-dimensional dilute Bose gas at zero temperature, the pressure of which is given by the mean-field result $P(\mu)=\mu^2m/(8\pi a_3\hbar^2)$ to leading
order in the small parameter $ma_3^2\mu/\hbar^2$. The first quantum correction, known as the Lee-Huang-Yang correction, is also entirely determined by $m$ and $a_3$ (besides the chemical potential $\mu$)~\cite{Lee57a,*Lee57b}. (In the following we set $k_B=\hbar=1$.)
The universality property of the equation of state of a dilute Bose gas can be understood from the point of view of the theory of phase transitions~\cite{Sachdev_book,Fisher89,Zhou10,Hazzard11}. By varying the chemical potential from negative to positive values at zero temperature, one induces a quantum phase transition between a state with vanishing pressure and no particles (vacuum) and a superfluid state with a nonzero pressure. This identifies the point $\mu=T=0$ as a quantum critical point (QCP). Above two dimensions (the upper critical dimension of the $T=0$ quantum phase transition), {\it i.e.} for $d\geq 2$, the boson-boson interaction is irrelevant and the critical behavior at the transition is mean-field like with a correlation-length exponent $\nu=1/2$ and a dynamical exponent $z=2$. However, the boson-boson interaction cannot be completely ignored and enters the equation of state~\cite{note2}. In the critical regime near the QCP, defined by $ml^2\mu\ll1$ and $ml^2T\ll 1$ with $l$ the ``natural"
low-energy length scale~\cite{Braaten06,note15}, the pressure takes the form
\begin{equation}
P(\mu,T) = \left(\frac{m}{2\pi}\right)^{d/2} T^{d/2+1} \calF_d\left(\frac{\mu}{T},\tilde g(T)\right) ,
\label{pressure0}
\end{equation}
where $\calF_d$ is a universal scaling function characteristic of the $d$-dimensional dilute Bose gas universality class. The temperature-dependent dimensionless interaction constant $\tilde g(T)$ is a known function of $ma_d^2T$, so that $P(\mu,T)$ can also be written in terms of a universal function of $\mu/T$ and $ma_d^2T$. In two dimensions and in the weak-interaction limit, $\tilde g(T)\equiv \tilde g$ is approximately temperature independent and the universal scaling function $\calF_2(\mu/T,\tilde g)$ depends on $\mu/T$ with the interaction strength $\tilde g$ as a parameter; the equation of state then exhibits an approximate scale invariance (with no characteristic energy scales other than $\mu$ and $T$)~\cite{Prokofev01,Prokofev02}. Equation~(\ref{pressure0}) also holds in a one-dimensional Bose gas ({\it i.e.} below the upper critical dimension of the $T=0$ vacuum-superfluid transition) but with the universal function $\calF_1$ depending only on $\mu/T$.
While Eq.~(\ref{pressure0}) follows from general renormalization-group (RG) arguments (see Sec.~\ref{sec_univ}), the theoretical determination of the universal scaling function $\calF_d(x,y)$ requires an explicit computation of the pressure $P(\mu,T)$. A perturbative calculation order by order in $\tilde g(T)$ is possible only for $d>2$ (it nevertheless breaks down in the critical regime of the thermal phase transition between the normal and the superfluid phase, which is controlled by the Wilson-Fisher fixed point of the classical O(2) model). In two dimensions, perturbative theory is plagued with infrared divergences at finite temperatures, thus making the determination of $\calF_2$ difficult, in particular in the quantum critical regime $|\mu|\ll T$. The Berezinskii-Kosterlitz-Thouless (BKT) transition~\cite{Berezinskii70,*Berezinskii71,*Kosterlitz73,*Kosterlitz74} and the low-temperature phase with quasi-long-range order are also beyond a mere perturbative treatment~\cite{Fisher88}.
The advantage of the point of view based on phase transitions is two-fold. Firstly it gives a straightforward explanation of universality in a dilute Bose gas. Secondly it shows that the universal equation of state~(\ref{pressure0}) holds not only for a dilute Bose gas but for any system near a quantum phase transition belonging to the same universality class. For instance, a Bose gas in an optical lattice near the vacuum-superfluid transition exhibits the same thermodynamics as a dilute Bose gas, provided that $m$ and $a_d$ are understood as the effective mass and scattering length of the bosons moving in the lattice. The thermodynamics of a Bose gas near the superfluid--Mott-insulator transition is also described by the equation of state~(\ref{pressure0}), since this quantum phase transition (when it is induced by a density change) belongs to the dilute Bose gas universality class~\cite{Sachdev_book,Fisher89}. In this manuscript we focus on the vacuum-superfluid transition in a two-dimensional Bose gas.
On the experimental side, cold atomic gases provide us with highly controlled and tunable systems where universal thermodynamics can be experimentally demonstrated. Altough cold gases are inhomogeneous and of finite size due to the harmonic confining potential, using a local-density approximation it is possible to deduce the equation of state $P(\mu,T)$ of the infinite homogeneous gas (with uniform density)~\cite{Cheng07,Ho09}. A number of experiments on weakly-interacting two-dimensional Bose gases have been reported~\cite{Hadzibabic06,Clade09,Rath10,Hung11,Yefsah11}, and the scale invariance of the equation of state $P(\mu,T)$ has been observed~\cite{Rath10,Hung11,Yefsah11}. More recently, the equation of state of a Bose gas in an optical lattice has been measured near the vacuum-superfluid transition in a regime where the interaction constant is not weak~\cite{Zhang12}. These experiments allow us to determine both the $\mu/T$ and $\tilde g(T)$ dependence of the universal scaling function $\calF_2$ in
various limits and will be thoroughly discussed in the manuscript.
The outline of the paper is as follows. In Sec.~\ref{sec_model}, we introduce and motivate the low-energy effective Hamiltonians which enable to derive the universal thermodynamics of three- and two-dimensional dilute Bose gases. Section~\ref{sec_univ} is devoted to a discussion of the thermodynamics of a dilute Bose gas using the language and concepts familiar from the theory of phase transitions. A detailed derivation of Eq.~(\ref{pressure0}) is given. In Sec.~\ref{sec_F}, we discuss the universal scaling function $\calF_2$ obtained from a nonperturbative renormalization-group (NPRG) approach. These theoretical results are compared with the experimental data of Refs.~\cite{Yefsah11,Hung11,Zhang12} in Sec.~\ref{sec_exp}. In particular, we make quantitative comparisons between the experimental data obtained with a Bose gas in an optical lattice~\cite{Zhang12} and theoretical results obtained in the framework of the Bose-Hubbard model. Finally, in Sec.~\ref{sec_bkt}, we discuss the NPRG prediction for the BKT
transition temperature and compare it with the estimate deduced from a quantum Monte Carlo simulation of an effective classical field theory.
\section{Model Hamiltonians}
\label{sec_model}
\subsection{Three-dimensional Bose gas}
The interaction between ultracold atoms is governed by a potential $V({\bf r}_1-{\bf r}_2)$ which is repulsive at short distances and determined by the van der Waals attraction $-C_6|{\bf r}_1-{\bf r}_2|^{-6}$ at long distances~\cite{Bloch08,Braaten06}. The latter defines the microscopic length scale $l_{\rm vdW}\sim (mC_6)^{1/4}$ ($m$ denotes the atomic mass). For length scales larger than $l_{\rm vdW}$ and energies smaller than $1/ml_{\rm vdW}^2$, collisions between atoms occur only in the $s$-wave channel and the scattering amplitude is well approximated by
\begin{equation}
f_{\rm 3D}({\bf q}) = - \frac{a_3}{1+i|{\bf q}|a_3} ,
\label{scat3D}
\end{equation}
where the three-dimensional $s$-wave scattering length $a_3$ is typically of the order of $l_{\rm vdW}$. In this low-energy regime, the ultracold gas can be described by the effective Hamiltonian
\begin{align}
\hat H ={}& \intr \biggl\lbrace \hat\psi^\dagger({\bf r}) \left(-\frac{\boldsymbol{\nabla}^2}{2m}-\mu \right)\hat\psi({\bf r}) \nonumber \\ & + \frac{g}{2}\hat\psi^\dagger({\bf r})\hat\psi^\dagger({\bf r}) \hat\psi({\bf r})\hat\psi({\bf r}) \biggr\rbrace ,
\label{ham}
\end{align}
with an ultraviolet momentum cutoff $\Lambda\sim l_{\rm vdW}^{-1}$ ($d=3$ for a three-dimensional gas). Here $\hat\psi^{(\dagger)}({\bf r})$ is a bosonic operator, $\mu$ the chemical potential and $g$ the ``microscopic'' interaction constant. The scattering amplitude obtained from~(\ref{ham}) takes the form~(\ref{scat3D}) with a scattering length
\begin{equation}
a_3 = \frac{mg}{4\pi+\frac{2}{\pi}mg\Lambda}
\end{equation}
which is a function of $g$ and $\Lambda$. The low-energy effective description is valid only for momentum scales much smaller than $\Lambda$, which requires both temperature and density to be small enough: $T\ll \Lambda^2/2m$ and $D\ll \Lambda^3$.
\subsection{Two-dimensional Bose gas}
\label{subsec_Q2D}
A quasi-two-dimensional gas can be created by subjecting a three-dimensional gas to a confining harmonic potential along one direction. The scattering amplitude then vanishes in the low-energy limit ${\bf q}\to 0$,
\begin{equation}
f_{\rm 2D}({\bf q}) = - \frac{2\pi}{\ln\left(\frac{|{\bf q}|a_2}{2}\right)+C-i\frac{\pi}{2}}
\label{scat2D}
\end{equation}
($C$ is the Euler constant), as in a strictly two-dimensional system~\cite{Petrov00a,Lim08}. The effective two-dimensional $s$-wave scattering length $a_2$ is a function of the thickness $l_z$ of the gas in the confining direction, as well as the $s$-wave scattering length and microscopic interaction strength of the three-dimensional (unconfined) Bose gas. At sufficiently low temperatures, when $T$ is much smaller than the $\omega_z=1/ml_z^2$, only the lowest level of the confining potential is populated and the gas behaves as a two-dimensional system. The quasi-two-dimensional gas can be described by the effective Hamiltonian~(\ref{ham}) with $d=2$ and a ``microscopic'' interaction constant~\cite{Petrov00a,Lim08}
\begin{equation}
g = \sqrt{8\pi} \frac{a_3}{ml_z} ,
\end{equation}
which reproduces the scattering amplitude~(\ref{scat2D}) with the scattering length
\begin{equation}
a_2 = \frac{2}{\Lambda} \exp\left( -\frac{2\pi}{mg}-C \right) .
\label{a2}
\end{equation}
Here $\Lambda\sim l_z^{-1}$ is an ultraviolet momentum cutoff below which the two-dimensional description holds. In addition to the condition $T\ll\omega_z\sim \Lambda^2/2m$, we must require the density to satisfy $D\ll\Lambda^2$. The typical energy per particle $gD$ is then much smaller than $\omega_z$ as it should be for the two-dimensional description to be justified. Note that in all experiments realized so far, the dimensionless interaction constant $\tilde g=2mg$ is small.
\subsection{Bose gas in an optical lattice}
Bosons in an optical lattice are described by the Bose-Hubbard model~\cite{Jaksch98,Fisher89},
\begin{equation}
\hat H = - t \sum_{\mean{{\bf r},{\bf r}'}} \left( \hat\psi^\dagger_{\bf r} \hat\psi_{{\bf r}'} + \hc \right) + \sum_{\bf r} \left[ -\mu \hat n_{\bf r} + \frac{U}{2} \hat n_{\bf r} (\hat n_{\bf r}-1) \right] ,
\label{BH}
\end{equation}
where $t$ is the hopping amplitude between nearest-neighbor sites $\mean{{\bf r},{\bf r}'}$, and $U$ the onsite interaction. $\hat\psi_{\bf r}^{(\dagger)}$ is an annihilation (creation) operator for a boson at site ${\bf r}$ of the lattice and $\hat n_{\bf r}=\hat\psi^\dagger_{\bf r} \hat\psi_{\bf r}$. An effective single-band description is valid only if the optical potential is strong enough and at sufficiently low temperatures. For a $d$-dimensional hypercubic lattice, the dispersion of the free bosons is given by the Fourier transform $t_{\bf q} = -2t\sum_i \cos(q_il)$ of the intersite hopping matrix ($l$ denotes the lattice spacing). It is convenient to use a shifted dispersion law
\begin{equation}
\epsilon_{\bf q} = 2t d -2t \sum_{i=1}^d \cos(q_il)
\label{epsq}
\end{equation}
which vanishes for ${\bf q}=0$ and behaves as $\epsilon_{\bf q}\simeq tl^2{\bf q}^2$ for $|{\bf q}|\ll l^{-1}$.
If the density $D$ is low enough ($Dl^d\ll 1$), the ground state is always a superfluid and we do not have to worry about the physics of the Mott transition~\cite{Fisher89}. Furthermore, at low temperatures $T\ll t$, the lattice does not matter and one can take the continuum limit where the Hamiltonian takes the form~(\ref{ham}) with an effective mass $m=1/2tl^2$, an interaction constant $g=Ul^d$, and a chemical potential $\mu+2dt$. The ultraviolet momentum cutoff $\Lambda$ is of the order of the inverse lattice spacing $l^{-1}$; the conditions $T\ll t$ and $Dl^d\ll 1$ then become $T\ll\Lambda^2/2m$ and $D\ll\Lambda^d$.
Thus, in the low-energy limit, a Bose gas in an optical lattice behaves similarly to a homogeneous Bose gas with an effective mass $m$ and an effective interaction constant $g$. To ensure that the effective continuum model reproduces the same low-energy physics as the lattice model, we must choose the cutoff $\Lambda$ so that it yields the same scattering length. In the two-dimensional case, we require Eq.~(\ref{a2}) to reproduce the scattering length of the two-dimensional Bose-Hubbard model~\cite{note9},
\begin{equation}
a_2 = \frac{l}{2\sqrt{2}} \exp\left(-\frac{4\pi t}{U} - C\right) ,
\label{a2lat}
\end{equation}
which gives $\Lambda=4\sqrt{2}/l$.
\section{Universal thermodynamics}
\label{sec_univ}
In this section, we discuss the thermodynamics of a $d$-dimensional dilute Bose gas from the point of view of phase transitions, starting from the Hamiltonian~(\ref{ham}). This description provides us with a natural explanation of universality as well as a simple derivation of Eq.~(\ref{pressure0}). Altough we will mainly focus on two-dimensional systems in the following sections, for generality we consider an arbitrary dimension $d\geq 2$.
\subsection{Vacuum-superfluid transition}
Let us first consider the vacuum-superfluid quantum phase transition induced by a change of chemical potential at zero temperature. For $d$ larger than the upper critical dimension $d_c^+=2$, boson-boson interactions are irrelevant (in the RG sense) and the critical behavior is described by non-interacting bosons. At the QCP $\mu=0$, the ground state is the vacuum, and the single-particle Green function is given by
\begin{equation}
G({\bf q},i\omega) = \left( i\omega - \frac{{\bf q}^2}{2m} \right)^{-1} ,
\label{propa}
\end{equation}
with $\omega$ a (bosonic) Matsubara frequency. This result is exact and holds for any value of the (bare) interaction constant $g$~\cite{note1}. We deduce the dynamic exponent $z=2$ while the anomalous dimension $\eta$ vanishes. Similarly, for $\mu\leq 0$, we find $G({\bf q},i\omega)^{-1} = i\omega+\mu -{\bf q}^2/2m$ and the critical exponent associated with the correlation length $\xi=|2m\mu|^{-1/2}$ takes the value $\nu=1/2$. The value of the renormalized interaction $g_R$ at the QCP is given by the $T$ matrix in vacuum (using again the fact that the ground state is the vacuum). In the low-energy limit, it takes the value $4\pi a_3/m$ in three dimensions, but vanishes logarithmically in two dimensions (see Eq.~(\ref{gtilde}) below).
The same analysis holds for bosons moving in a lattice. In the vacuum, the single-particle Green function is given by
\begin{equation}
G({\bf q},i\omega) = (i\omega +\mu+2dt- \epsilon_{\bf q})^{-1} ,
\end{equation}
where $\epsilon_{\bf q}$ is the dispersion of the free bosons [Eq.~(\ref{epsq})]. The $T=0$ QCP between the vacuum and the superfluid phase is now located at $\mu_c=-2dt$ and the elementary excitations have an effective mass $m=1/2tl^2$. As in the continuum model, the renormalized value $g_R$ of the interaction ({\it i.e.} the $T$ matrix in vacuum) can be expressed in terms in of the scattering length $a_d$ of the bosons moving in the lattice~\cite{Rancon11b}.
\subsection{RG approach}
The preceding results can be formulated in the language of the RG. In the Wilson formulation, a RG transformation consists in integrating out ``fast'' modes with momenta between $\Lambda$ and $\Lambda/s$ ($s>1$), and rescaling fields, momenta and frequencies in order to restore the original value of the cutoff $\Lambda$. This yields an effective Hamiltonian for the ``slow'' modes with a renormalized interaction constant $g(s)$~\cite{note16}. At the QCP $\mu=T=0$, there is no renormalization of the quadratic part of the Hamiltonian, in agreement with the fact that Eq.~(\ref{propa}) is exact. The dimensionless interaction constant $\tilde g(s)=2m \Lambda^{d-2}g(s)$ satisfies the RG equation
\begin{equation}
s \frac{d\tilde g(s)}{ds} = (2-d)\tilde g(s) - \frac{K_d}{2} \tilde g(s)^2 ,
\label{rgeq1}
\end{equation}
where $K_d=[2^{d-1}\pi^{d/2}\Gamma(d/2)]^{-1}$. Above the upper critical dimension $d_c^+=2$, $\tilde g(s)$ vanishes for $s\to\infty$ and the only fixed point of Eq.~(\ref{rgeq1}) is the Gaussian fixed point $\tilde g=0$, which therefore governs the quantum phase transition between the vacuum and the superfluid phase. From~(\ref{rgeq1}), we obtain
\begin{equation}
\tilde g(s) = \left\lbrace
\begin{array}{lcc}
\dfrac{8\pi\Lambda a_3}{s} & \mbox{if} & d=3 , \\
- \dfrac{4\pi}{\ln\left(\frac{\Lambda a_2}{2s}\right)+C} & \mbox{if} & d=2 ,
\end{array}
\right.
\label{gtilde}
\end{equation}
where the result for $d=3$ holds for $s\gg 1$.
There are two relevant perturbations about the Gaussian fixed point $\mu=T=\tilde g=0$: the chemical potential and the temperature. In a RG transformation, they transform as
\begin{equation}
\tilde T(s) = s^z \tilde T , \qquad
\tilde\mu(s) = s^{1/\nu} \tilde\mu ,
\label{rgeq2}
\end{equation}
near the QCP ({\it i.e.} when $|{\tilde\mu}(s)|\lesssim 1$ and $\tilde T(s)\lesssim 1$). We have introduced the dimensionless variables~\cite{note3}
\begin{equation}
\tilde T = \frac{2mT}{\Lambda^z}, \qquad \tilde\mu = \frac{2m\mu}{\Lambda^{1/\nu}} .
\end{equation}
Note that in the low-temperature regime where this analysis based on the effective Hamiltonian~(\ref{ham}) is valid, $|{\tilde\mu}|\ll 1$ and $\tilde T\ll 1$ (see Sec.~\ref{sec_model}). When $\mu$ and $T$ are nonzero, the RG equation for $\tilde g(s)$ is well approximated by~(\ref{rgeq1}) or (\ref{gtilde}) as long as $|{\tilde\mu}(s)|\lesssim 1$ and $\tilde T(s)\lesssim 1$. We can obtain a rough sketch of the phase diagram by noting that the low-energy behavior of the system depends on which of the conditions $|{\tilde\mu}(s)|\sim 1$ and $\tilde T(s)\sim 1$ is reached first. This yields two crossover lines defined by $|{\tilde\mu}|\sim \tilde T$, {\it i.e.} $|\mu|\sim T$ using $z=1/\nu=2$, in agreement with the generic phase diagram of a system near a quantum critical point (see Fig.~\ref{fig_phase_dia})~\cite{Sachdev_book}. For $\mu<0$ and $|\mu|\gg T$, the system behaves as a dilute classical gas and we expect a classical Boltzmann description to apply (see Sec.~\ref{sec_F}). The condition $|\mu|\ll T$ defines the quantum critical
regime where the physics is controlled by the QCP $\mu=T=0$ and its thermal excitations~\cite{Sachdev_book}.
\begin{figure}
\centerline{\includegraphics[width=6.5cm]{phase_dia.pdf}}
\caption{(Color online) Phase diagram of the dilute Bose gas ($d\geq 2$). The dashed lines are defined by $|\mu|\sim T$ and the solid one corresponds to the superfluid transition (of BKT type when $d=2$). The shaded area corresponds to the high-energy region $|\mu|,T\gtrsim \Lambda^2/2m$ where the thermodynamics is not universal. The value of the ultraviolet momentum cutoff $\Lambda$ is discussed in Sec.~\ref{sec_model} for a three- and a quasi-two-dimensional Bose gas.}
\label{fig_phase_dia}
\end{figure}
\subsection{Universal thermodynamics}
Let us now consider the dimensionless pressure~\cite{note3}
\begin{equation}
\tilde P({\tilde\mu},{\tilde T},\tilde g) = \frac{2m}{\Lambda^{d+z}} P(\mu,T) ,
\end{equation}
expressed in terms of the dimensionless variables $\tilde T$, $\tilde\mu$ and $\tilde g$ (note that $\tilde P$ has no explicit dependence on the ultraviolet cutoff $\Lambda$). In a RG transformation, $\tilde P$ transforms as
\begin{equation}
\tilde P({\tilde\mu},{\tilde T},\tilde g) = s^{-d-z} \tilde P(s^{1/\nu}{\tilde\mu},s^z\tilde T,\tilde g(s)) ,
\label{pressure}
\end{equation}
provided that $\tilde T(s)\ll 1$ and $|{\tilde\mu}(s)|\ll 1$. Equation~(\ref{pressure}) holds for the full pressure since the vanishing of $P$ when $\mu\leq 0$ and $T=0$ implies that $P$ has no regular part at the transition. Only the two-body interaction constant $\tilde g$ is taken into account. Higher-order interactions (which are inevitably generated in the RG procedure), such as the three-body term, are not considered here since they are irrelevant and give rise to subleading contributions to the pressure~\cite{note4,note8}
Setting $s=\tilde T^{-1/z}$ in Eq.~(\ref{pressure}), we obtain
\begin{equation}
\tilde P({\tilde\mu},{\tilde T},\tilde g) = \frac{{\tilde T}^{d/z+1}}{(4\pi)^{d/2}} \calF_d\left(\frac{{\tilde\mu}}{{\tilde T}^{1/\nu z}},\tilde g({\tilde T})\right) ,
\label{pressure2}
\end{equation}
where we use the notation $\tilde g(\tilde T)$ for $\tilde g(s={\tilde T}^{-1/z})$. Going back to dimensionful variables and setting $z=1/\nu=2$, we finally obtain Eq.~(\ref{pressure0}) where the energy-dependent interaction constant
\begin{equation}
\tilde g(\epsilon) = \left\lbrace
\begin{array}{lcc}
8\pi \sqrt{2ma_3^2\epsilon} & \mbox{if} & d=3 , \\
- \dfrac{4\pi}{\ln\left(\frac{1}{2}\sqrt{2ma_2^2\epsilon}\right)+C} & \mbox{if} & d=2 ,
\end{array}
\right.
\label{gT}
\end{equation}
is obtained from~(\ref{gtilde}) with $s=\tilde\epsilon^{-1/2}$ and $\tilde\epsilon=2m\epsilon/\Lambda^2$. We stress that $\calF_d$ is a universal scaling function characteristic of the $d$-dimensional dilute Bose gas universality class (the factor $1/(4\pi)^{d/2}$ in~(\ref{pressure2}) is introduced for later convenience); it is independent of microscopic parameters such as the mass $m$ of the bosons or the scattering length $a_d$ which depend on the system considered.
Using Eq.~(\ref{pressure0}), we can write any thermodynamic quantity in a scaling form. For instance, the density $D=\partial P/\partial\mu$ and the entropy per unit volume $s=\partial P/\partial T$ read
\begin{equation}
D(\mu,T) = \left(\frac{mT}{2\pi}\right)^{d/2} \calF_d^{(1,0)}\left(\frac{\mu}{T},\tilde g(T)\right) ,
\end{equation}
and
\begin{multline}
s(\mu,T) = \left(\frac{mT}{2\pi}\right)^{d/2} \biggl[ \left(\frac{d}{2}+1\right) \calF_d\left(\frac{\mu}{T},\tilde g(T)\right) \\ - \frac{\mu}{T} \calF_d^{(1,0)}\left(\frac{\mu}{T},\tilde g(T)\right) + T \tilde g'(T) \calF_d^{(0,1)}\left(\frac{\mu}{T},\tilde g(T)\right) \biggr] ,
\label{sdef}
\end{multline}
where $\calF_d^{(1,0)}(x,y)\equiv\partial_x\calF_d(x,y)$, $\calF_d^{(0,1)}(x,y)\equiv\partial_y\calF_d(x,y)$, and $\tilde g'(T)=d\tilde g/dT$. Since $T\tilde g'(T)=\frac{d-2}{2}\tilde g(T)$ for $d>2$ and $T\tilde g'(T)=\tilde g(T)^2/8\pi$ for $d=2$, $s(\mu,T)$ is a function of $\mu/T$ and $\tilde g(T)$ (up to the factor $(mT)^{d/2}$).
One often introduces the so-called phase-space pressure and phase-space density,
\begin{equation}
\begin{split}
\calP(\mu,T) &= P(\mu,T) \frac{\lambda_{\rm dB}^d}{T} = \calF_d\left(\frac{\mu}{T},\tilde g(T)\right) ,\\
{\cal D}(\mu,T) &= D(\mu,T) \lambda_{\rm dB}^d = \calF_d^{(1,0)}\left(\frac{\mu}{T},\tilde g(T)\right) ,
\end{split}
\label{calPD}
\end{equation}
where $\lambda_{\rm dB}=\sqrt{2\pi/mT}$ is the thermal de Broglie wavelength. $\calP$ and ${\cal D}$ provides a direct measure of the scaling function $\calF_d$ and its derivative $\calF_d^{(1,0)}$. One can also consider the entropy per particle ${\cal S}=D^{-1}\partial P/\partial T$,
\begin{equation}
{\cal S}(\mu,T) = -\frac{\mu}{T} + \left(\frac{d}{2}+1\right) \frac{\calF_d}{\calF_d^{(1,0)}} + T\tilde g'(T) \frac{\calF_d^{(0,1)}}{\calF_d^{(1,0)}} ,
\label{calS}
\end{equation}
where we use the shorthand notation $\calF_d\equiv\calF_d(\mu/T,\tilde g(T))$, etc. Note that ${\cal S}(\mu,T)$ is also a universal function of $\mu/T$ and $\tilde g(T)$ (see the remark about $T\tilde g'(T)$ following Eq.~(\ref{sdef})).
At zero temperature, the scaling function $\calF_d$ can be computed in perturbation theory. The one-loop correction to the mean-field result gives
\begin{equation}
P(\mu,0) = \Theta(\mu) \frac{m\mu^2}{8\pi a_3} \left(1 - \frac{64}{15\pi} \sqrt{ma_3^2\mu} \right)
\end{equation}
in three dimensions (the one-loop correction is known as the Lee-Huang-Yang correction~\cite{Lee57a,*Lee57b}), and
\begin{equation}
P(\mu,0) = - \Theta(\mu) \frac{m\mu^2}{4\pi} \left[ \ln\left(\frac{1}{2}\sqrt{ma_2^2\mu}\right)+C+\frac{1}{4} \right]
\label{P2}
\end{equation}
in two dimensions, where $\Theta$ denotes the step function~\cite{Schick71,Popov72a,Popov_book_2}. These results can be cast in the scaling form
\begin{equation}
P(\mu,T) = \left(\frac{m}{2\pi}\right)^{d/2} \mu^{d/2+1} {\cal G}_d\left(\frac{T}{\mu},\tilde g(\mu)\right) ,
\label{calG}
\end{equation}
which is equivalent to Eq.~(\ref{pressure0}) but more appropriate to the zero-temperature limit~\cite{note5}.
At finite temperature, the determination of the scaling function $\calF_2$ (or ${\cal G}_2$) is difficult in two dimensions, in particular in the quantum critical regime $|\mu|\ll T$. In the following section, we discuss the scaling function $\calF_2$ obtained from the NPRG approach.
\section{Scaling function $\calF$ of a two-dimensional Bose gas}
\label{sec_F}
The NPRG approach has recently been used to understand the physics of a Bose gas beyond the Bogoliubov approximation~\cite{Andersen99,Wetterich08,Dupuis07,*Dupuis09a,*Dupuis09b,Floerchinger08,*Floerchinger09b,Floerchinger09a,Sinner09,*Sinner10,*Eichler09}, but the computation of the scaling function $\calF_d$ has not been carried out except for the Lee-Huang-correction in a zero-temperature three-dimensional Bose gas~\cite{Floerchinger08,*Floerchinger09b}. Here we discuss the NPRG results for the scaling function $\calF\equiv\calF_2$ which determines the thermodynamics of a two-dimensional Bose gas. We use both the standard version of the NPRG as well as its lattice version~\cite{Rancon11a,Rancon11b} to directly study the Bose-Hubbard model. The NPRG approach is briefly reviewed in Appendix~\ref{app_nprg}. Our results are based on the numerical solution of the NPRG equations as well as analytical results in some limits, in particular for $\mu=0$ (see Appendix~\ref{app_muzero}).
\subsection{$\calF(x,y)$ vs $x$ ($y$ fixed)}
\label{subsec_Fx}
We first discuss the $x$ dependence of $\calF(x,y)$ for fixed $y$. Figure~\ref{fig_scaling} shows the phase-space pressure $\calP=\calF$ [Eqs.~(\ref{calPD})] as a function of $\mu/T$ for $\tilde g(T)=0.22$ and $\tilde g(T)=5$. We can verify that the scaling form~(\ref{pressure0}) holds by computing $\calP$ for various sets of parameters $(T,g,m,\Lambda)$. For the case $\tilde g(T)=0.22$, we choose the value $\tilde g=0.22$ for the bare interaction constant, so that the system is in the weak-coupling limit and $\tilde g(T)\simeq 0.22$ nearly temperature independent (see the discussion below). We find that the three sets of parameters $(T,g,m,\Lambda)$, $(T/2,g,m,\Lambda)$ and $(T,2g,m/2,\Lambda)$ (with $T=0.1\Lambda^2/2m$ and $2m=1$) yield the same results for the phase-space pressure $\calP$ in agreement with the expected scale invariance at weak coupling: $\calF(\mu/T,\tilde g(T))=\calF(\mu/T,\tilde g)$. In the case $\tilde g(T)=5$, the results obtained for the three sets of parameters $(T,g,m,\Lambda)$, $(
T/2,g,m,\Lambda/\sqrt{2})$ and $(2T,2g,m/2,\Lambda)$ also collapse on a single curve corresponding to the scaling function $\calF(x,y)$ (with $y\equiv \tilde g(T)$ fixed). In this case one must change simultaneously at least two parameters to keep $\tilde g(T)$ unchanged. In table~\ref{table_Tkt} we indicate the value of $(\mu/T)_{\rm BKT}$ at the BKT transition for various values of $\tilde g(T)\lesssim 1$ as obtained from the NPRG (Sec.~\ref{sec_bkt}) and Monte Carlo simulations~\cite{Prokofev02}. Note that neither our method nor the Monte Carlo simulations gives a reliable estimate of $(\mu/T)_{\rm BKT}$ in the strong-coupling limit $\tilde g(T)\gtrsim 1$.
\begin{figure}
\centerline{\includegraphics[width=7cm]{inv_scal_P.pdf}}
\caption{(Color online) Phase-space pressure $\calP(\mu,T)$ vs $\mu/T$ at fixed $\tilde g(T)$. The upper symbols are obtained for $\tilde g(T)=0.22$ with $T=0.1\Lambda^2/2m$ and $2m=1$. (Red) squares: $(T,g,m,\Lambda)$, (black) dots: $(T/2,g,m,\Lambda)$, (orange) diamonds: $(T,2g,m/2,\Lambda)$. The lower symbols are obtained for $\tilde g(T)=5$. (Blue) triangles: $(T,g,m,\Lambda)$, (green) triangles: $(T/2,g,m,\Lambda/\sqrt{2})$, (purple) triangles: $(2T,2g,m/2,\Lambda)$. The solid lines are guides to the eyes. $(\mu/T)_{\rm BKT}$ at the BKT transition is given in table~\ref{table_Tkt}.}
\label{fig_scaling}
\end{figure}
\begin{table}
\renewcommand{\arraystretch}{1.5}
\begin{center}
\begin{tabular}{ccccc}
\hline \hline
$\tilde g(T)$ & 0.1 & 0.22 & 0.5 & 1 \\
\hline
$(\mu/T)_{\rm BKT} $ & 0.08 & 0.15 & 0.29 & 0.51 \\
\hline
$(\mu/T)_{\rm BKT}$ (Ref.~\cite{Prokofev02}) & 0.09 & 0.17 & 0.31 & 0.51 \\
\hline \hline
\end{tabular}
\end{center}
\caption{$(\mu/T)_{\rm BKT}$ at the BKT transition for various various values of $\tilde g(T)\lesssim 1$ as obtained from the NPRG (Sec.~\ref{sec_bkt}) and Monte Carlo simulations~\cite{Prokofev02}.}
\label{table_Tkt}
\end{table}
\begin{figure}
\centerline{\includegraphics[width=6.5cm]{adimP.pdf}}
\centerline{\includegraphics[width=6.5cm]{adimD.pdf}}
\centerline{\hspace{0.35cm}\includegraphics[width=6.15cm]{SoverN.pdf}}
\caption{(Color online) Phase-space pressure $\calP(\mu,T)$, phase-space density ${\cal D}(\mu,T)$ and entropy per particle ${\cal S}(\mu,T)$ vs $\mu/T$. From top to bottom, the curves correspond to $\tilde g(T)=0.1,0.5,1,3,5$ for $\calP$, $\tilde g(T)=0.5,1,3,5$ for ${\cal D}$, and $\tilde g(T)=0.1,0.5,1$ for ${\cal S}$. The dots show the limiting behavior~(\ref{MF}) valid for $\mu\gg T$ and the diamonds the classical gas result~(\ref{classical}). [For numerical reasons, it is difficult to compute the entropy when $\tilde g(T)\gtrsim 1$.]}
\label{fig_F1}
\end{figure}
Figure~\ref{fig_F1} shows the phase-space pressure $\calP=\calF$, the phase-space density ${\cal D}=\calF^{(1,0)}$ and the entropy per particle ${\cal S}(\mu,T)$ as a function of $\mu/T$ for various values of $\tilde g(T)$ [Eqs.~(\ref{calPD}-\ref{calS})]. At large and negative chemical potential ($|\mu|/T\gg 1$), we find that the system behaves as a classical dilute gas,
\begin{equation}
\begin{gathered}
\calP(\mu,T) = {\cal D}(\mu,T) = e^{-|\mu|/T} , \\
{\cal S}(\mu,T) = 2 - \frac{\mu}{T} ,
\end{gathered}
\label{classical}
\end{equation}
which implies
\begin{equation}
\calF(x,y)=e^x \quad \mbox{for} \quad x<0 \quad \mbox{and} \quad |x|\gg 1.
\label{classical1}
\end{equation}
In the opposite limit of a large positive chemical potential ($\mu/T\gg 1$), the pressure can be approximated by its zero-temperature limit. Using the expression~(\ref{P2}), we obtain
\begin{equation}
\begin{split}
\calP(\mu,T) &= \left(\frac{\mu}{T}\right)^2 \left( \frac{2\pi}{\tilde g(\mu)} + \frac{1}{4} \ln 2 - \frac{1}{8} \right), \\
{\cal D}(\mu,T) &= \frac{\mu}{T} \left( \frac{4\pi}{\tilde g(\mu)} + \frac{1}{2} \ln 2 - \frac{1}{2} \right) , \\
{\cal S}(\mu,T) &= 0 ,
\end{split}
\label{MF}
\end{equation}
{\it i.e.}
\begin{equation}
{\cal G}(x,y)=\left( \frac{2\pi}{y} + \frac{1}{4} \ln 2 - \frac{1}{8} \right) \quad \mbox{for} \quad x\ll 1 ,
\label{MF1}
\end{equation}
where ${\cal G}\equiv{\cal G}_2$ is the scaling function defined in Eq.~(\ref{calG}) ($x\equiv T/\mu$)~\cite{note5}. Without the additive constants $-1/8+(\ln 2)/4$ and $-1/2+(\ln2)/2$, Eqs.~(\ref{MF}) coincide with the mean-field result assuming an effective interaction constant $g(\mu)$. These constants can be omitted in the weak-coupling limit $\tilde g(\mu)\ll 1$. As pointed out in Refs.~\cite{Prokofev01,Prokofev02}, in the weak-coupling limit -- where the BKT transition temperature $T_{\rm BKT}$ can be easily determined (see Sec.~\ref{sec_bkt}) -- the approximation~(\ref{MF1}) remains remarkably accurate all the way down to the transition point $(\mu/T)_{\rm BKT}$. We also observe that the limiting behaviors~(\ref{classical},\ref{classical1}) and (\ref{MF},\ref{MF1}) are very well satisfied not only in the weak-coupling limit~\cite{Prokofev01,Prokofev02} but also in the strong-coupling limit where $\tilde g(T)\gtrsim 1$ (see Fig.~\ref{fig_F1}).
The crossover regime $|\mu|\ll T$ is more difficult to analyze in simple terms, and a full numerical solution of the RG equations is necessary (see however Sec.~\ref{subsec_Fy} and Appendix~\ref{app_muzero} for an analytical solution in the case $\mu=0$).
In the weak-coupling limit $\tilde g=2mg\ll 1$, the scattering length $a_2$ is exponentially small. This implies that the renormalized interaction constant $\tilde g(T) \simeq \tilde g$ is nearly temperature independent except for exponentially small temperatures (which are experimentally unreachable)~\cite{note10}. It follows that the phase-space pressure and density and the entropy per particle,
\begin{equation}
\begin{split}
\calP(\mu,T) &= \calF\left(\frac{\mu}{T},\tilde g\right), \\
{\cal D}(\mu,T) &= \calF^{(1,0)}\left(\frac{\mu}{T},\tilde g\right), \\
{\cal S}(\mu,T) &= 2 \frac{\calP(\mu,T)}{{\cal D}(\mu,T)} - \frac{\mu}{T} ,
\end{split}
\label{PDweak}
\end{equation}
can be considered as functions of $\mu/T$ only, with the microscopic interaction constant $\tilde g$ entering the scaling function $\calF$ as a parameter. The equation of state then exhibits an approximate scale invariance (with no characteristic energy scales other than $\mu$ and $T$)~\cite{Prokofev01,Prokofev02}.
\subsection{$\calF(0,y)$ vs $y$}
\label{subsec_Fy}
\begin{figure}
\centerline{\hspace{0.2cm}\includegraphics[width=6.5cm]{P_scaling.pdf}}
\centerline{\hspace{0.2cm}\includegraphics[width=6.5cm]{D_scaling.pdf}}
\centerline{\includegraphics[width=6.75cm]{entropy_scaling.pdf}}
\caption{(Color online) Phase-space pressure $\calP(0,T)$, phase-space density ${\cal D}(0,T)$ and entropy per particle ${\cal S}(0,T)$ vs $\tilde g(T)$. The symbols show the the data of the ENS, Chicago I and Chicago II (with $\mu=\mu_c$) experiments~\cite{Yefsah11,Hung11,Zhang12} (see Sec.~\ref{sec_exp}).}
\label{fig_F2}
\end{figure}
The limit $|x|\ll 1$ is particularly interesting as it corresponds to the quantum critical regime. In this section, we discuss the function $\calF(0,y)$. Figure~\ref{fig_F2} shows $\calP(0,T)$, ${\cal D}(0,T)$ and ${\cal S}(0,T)$ as a function of $\tilde g(T)$. We show in Appendix~\ref{app_muzero} that
\begin{equation}
\calP(0,T) \equiv \calF(0,\tilde g(T)) \simeq \frac{\pi^2}{6} - \frac{\tilde g(T)}{2\pi} \ln^2\left(\frac{2\pi}{\tilde g(T)}\right)
\end{equation}
for $\tilde g(T)\to 0$. The result $\lim_{T\to 0}\calP(0,T)=\pi^2/6$ is exact. Experimentally, however, this limiting behavior cannot be observed due to the logarithmic temperature dependence of $\tilde g(T)$. In the weak-coupling limit, $\tilde g(T)=\tilde g$ is nearly temperature independent, and the phase-space pressure takes the form
\begin{equation}
\calP(0,T) = \calF(0,\tilde g) ,
\end{equation}
where $\calF(0,\tilde g)\leq \lim_{y\to 0}\calF(0,y)=\pi^2/6$. In the strong-coupling limit, $\calP(0,T)$ exhibits a weak temperature dependence coming from that of $\tilde g(T)$, but again reaching the limiting value $\lim_{T\to 0}\calP(0,T)=\pi^2/6$ requires extremely small (unrealistic) temperatures.
\subsection{Thermodynamics of the Bose-Hubbard model}
\label{subsec_bh}
\begin{figure}
\centerline{\includegraphics[width=7.3cm]{gt_latt_cont_WC.pdf}}
\centerline{\hspace{0.435cm}\includegraphics[width=6.735cm]{gt_latt_cont.pdf}}
\caption{(Color online) (Red) solid lines: dimensionless interaction constant $\tilde g_{\rm BH}(T)$ in the two-dimensional Bose-Hubbard model for $U/t=0.22$ (top) and $U/t=6.25$ (bottom) (note the different scales on the vertical axes). The (blue) dash-dotted lines show the universal limit $\tilde g(T)$.}
\label{fig_gtlat}
\end{figure}
\begin{figure}
\centerline{\includegraphics[width=7cm]{P_lat_cont_SC.pdf}}
\caption{(Color online) Phase-space pressure $\calP(\mu_c,T)$ vs $T/t$ for $U/t=6.25$. The (green) dashed line shows the result obtained for a flat density of states (DOS) in the energy window $[0,8t]$. The (blue) dash-dotted line shows the universal limit $\calF(0,\tilde g(T))$.}
\label{fig_P_lat}
\centerline{\includegraphics[width=7cm]{Pdim_t0v16.pdf}}
\caption{(Color online) Pressure $P(\mu_c,T)$ vs $T/t$ for $U/t=6.25$. The (green) dashed line shows the result obtained for a flat density of states in the energy window $[0,8t]$. The (blue) dash-dotted line shows the universal limit $T^2/(4\pi tl^2)\calF(0,\tilde g(T))$.}
\label{fig_Ptot_lat}
\centerline{\includegraphics[width=7cm]{DS_lattice.pdf}}
\caption{(Color online) Phase-space density ${\cal D}(\mu_c,T)$ ((red) solid line) and entropy per particle ${\cal S}(\mu_c,T)$ ((blue) dash-dotted line) vs $T/t$ for $U/t=6.25$. The (green) dashed lines show the corresponding universal limits.}
\label{fig_DS_lat}
\end{figure}
\begin{figure}
\centerline{\includegraphics[width=7cm]{compressibility.pdf}}
\caption{(Color online) Compressibility $\kappa(\mu_c,T)$ vs $T/t$ for $U/t=6.25$. The (green) dashed line shows the corresponding universal limit. (The numerical noise in the NPRG result follows from taking the second-order derivative of the pressure with respect to $\mu$.)}
\label{fig_kappa}
\end{figure}
In this section we discuss the results obtained in the two-dimensional Bose-Hubbard model [Eq.~(\ref{BH})] using the lattice version of the NPRG~\cite{Rancon11a,Rancon11b}. The energy-dependent interaction constant is defined by
\begin{equation}
\gbh(\epsilon) = \frac{U}{1+U\Pi(\epsilon)} ,
\label{gbh}
\end{equation}
with
\begin{equation}
\Pi(\epsilon) = l^2 \int \frac{d^2q}{(2\pi)^2} \frac{1}{2(\epsilon_{\bf q}+\epsilon)} ,
\label{pibh}
\end{equation}
where $\epsilon_{\bf q}$ is the lattice dispersion of the boson~[Eq.~(\ref{epsq})]. This definition, which is also that used in Ref.~\cite{Zhang12}, is justified in Appendix~\ref{app_disp}. In the low-energy limit $\epsilon\to 0$, it coincides with the universal form $l^{-2}g(\epsilon)$ [Eq.~(\ref{gT})], obtained from the continuum model with boson mass $m=1/2tl^2$ and scattering length $a_2$ given by Eq.~(\ref{a2lat}). The dimensionless interaction constant $\tilde g_{\rm BH}(T)=2ml^2\gbh(T)=\gbh(T)/t$ is shown in Fig.~\ref{fig_gtlat} for $U/t=0.22$ and $U/t=6.25$. In both cases, $\tilde g_{\rm BH}(T)$ is well approximated by its universal limit $\tilde g(T)$ [Eq.~(\ref{gT})] for $T\lesssim 8t$ (see insets in Fig.~\ref{fig_gtlat}).
\begin{figure}
\centerline{\includegraphics[width=7.13cm]{lattice_WC.pdf}}
\centerline{\hspace{0.155cm}\includegraphics[width=7cm]{lattice_SC.pdf}}
\caption{(Color online) Phase-space pressure $\calP(\mu,T)$ vs $\delta\mu/T$ for $U/t=0.22$ (top) and $U/t=6.25$ (bottom). $T/t=1/100$ (circles), $1/10$ (squares), $1$ (diamonds) and $3$ (triangles). The solid lines correspond to $\calF(\delta\mu/T,\tilde g(T))$. For $t/U=0.22$, $\mu/T\simeq 0.15$ at the BKT transition (see table~\ref{table_Tkt} and Sec.~\ref{sec_bkt}).}
\label{fig_scaling_lat}
\centerline{\includegraphics[width=6.5cm]{PDS_scaling.pdf}}
\caption{(Color online) Phase-space pressure $\calP(\mu_c,T)$, phase-space density ${\cal D}(\mu_c,T)$, and entropy per particle ${\cal S}(\mu_c,T)$ vs $\tilde g(T)$ ($U/t=6.25$). The lines show the universal limit obtained from the scaling function $\calF(0,\tilde g(T))$ and its derivatives.}
\label{fig_PDS_scaling}
\end{figure}
Figure~\ref{fig_P_lat} shows the phase-space pressure $\calP(\mu_c,T)$ vs $T/t$ for $U/t=6.25$ and $\mu=\mu_c=-4t$. We observe a maximum around $T/t\sim 2.5$ due to the enhanced density of states of the square lattice near the band center~\cite{note11}. This maximum disappears if we consider a flat density of states in the energy window $[0,8t]$. Comparing $\calP(\mu_c,T)$ and $\calF(0,\tilde g(T))$ (with $\tilde g(T)$ the universal limit of $\tilde g_{\rm BH}(T)$ discussed above) we see that the universal limit, where $\calP(\mu_c,T)$ becomes a universal function of $ma_2^2T$, is reached only at very low temperatures $T\ll t$. The identification of $t$ as the crossover temperature scale for quantum critical behavior is confirmed by the $T$ dependence of the pressure. For $T\lesssim t$, one finds that $P(\mu_c,T)=T^2/(4\pi tl^2)\calP(\mu_c,T)$ is well approximated by the universal limit $T^2/(4\pi tl^2)\calF(0,\tilde g(T))$ (Fig.~\ref{fig_Ptot_lat}). The phase-space density ${\cal D}(\mu_c,T)$ and entropy per
particle ${\cal S}(\mu_c,T)$ are shown in Fig.~\ref{fig_DS_lat} (the low-temperature regime where ${\cal D}(\mu_c,T)$ and ${\cal S}(\mu_c,T)$ coincide with their universal limits is not shown).
The fact that the universal regime is reached only at low temperatures can also be seen in the temperature dependence of the compressibility $\kappa=\partial^2 P/\partial\mu^2$ (Fig.~\ref{fig_kappa}). Although it is difficult to numerically compute the second-order derivative of the pressure with respect to $\mu$, our results clearly show that $\kappa(\mu_c,T)$ is below the universal limit $(1/4\pi tl^2)\calF^{(2,0)}(0,\tilde g(T))$. We also note that while $\kappa(\mu_c,T)$ varies weakly with $T$ in the temperature range $[t,10t]$, it should eventually diverge as $T\to 0$ (see Eq.~(\ref{rgeq11}) in appendix~\ref{app_muzero}). We thus disagree with the conclusion of Ref.~\cite{Fang11} that quantum criticality is observed below a characteristic temperature of the order of the single-particle bandwidth $8t$~\cite{note17}.
Figure~\ref{fig_scaling_lat} shows the phase-space pressure $\calP(\mu,T)$ versus $\delta\mu/T$ for $U/t=0.22$ and $U/t=6.25$ ($\delta\mu=\mu-\mu_c$), and various temperatures ranging from $t/100$ to $3t$. In the weak-coupling limit $U/t=0.22$, $\tilde g_{\rm BH}(T)=U/t$ is nearly temperature independent in the temperature range $[t/100,3t]$ (see Fig.~\ref{fig_gtlat}). At very low temperatures, we obtain a perfect agreement between $\calP(\mu,T)$ and the universal scaling function $\calF(\delta\mu/T,\tilde g)$ (with $\tilde g=0.22$). At higher temperatures, when $T\sim t$, we observe that $\calP(\mu,T)$ slightly deviates from $\calF(\delta\mu/T,\tilde g)$, in particular for large values of $\mu$. This agrees with the previous observation that $\calP(\mu_c,T)$ reaches the universal limit only for $T\lesssim t$ (see Figs.~\ref{fig_P_lat} and \ref{fig_Ptot_lat}). For $U/t=6.25$, we again find a good agreement between $\calP(\mu,T)$ and $\calF(\delta\mu/T,\tilde g(T))$ at low temperatures and small chemical
potential $\delta\mu$, but deviations are clearly visible at higher temperatures or larger values of $\delta\mu/T$.
The phase-space pressure $\calP(\mu_c,T)$, phase-space density ${\cal D}(\mu_c,T)$ and entropy per particle ${\cal S}(\mu_c,T)$ vs $\tilde g(T)$ are shown in Fig.~\ref{fig_PDS_scaling} for $U/t=6.25$. The maximum around $\tilde g(T)\sim 3.5$ is due to the enhanced density of states of the square lattice near the band center~\cite{note11}. For $\tilde g(T)\lesssim 2.5$, we recover the universal limit where the thermodynamics is determined by the scaling function $\calF(\delta\mu/T,\tilde g(T))$.
\section{Comparison with experiments}
\label{sec_exp}
\begin{figure}
\centerline{\includegraphics[width=6.6cm]{ensP.pdf}}
\centerline{\hspace{0.05cm}\includegraphics[width=6.5cm]{ensD.pdf}}
\centerline{\hspace{0.1cm}\includegraphics[width=6.37cm]{ensS.pdf}}
\caption{(Color online) Phase-space pressure $\calP(\mu,T)$, phase-space density ${\cal D}(\mu,T)$ and entropy per particle ${\cal S}(\mu,T)$ vs $\mu/T$ in the ENS experiment~\cite{Yefsah11}. The (red) solid lines show the NPRG results.}
\label{fig_ens}
\end{figure}
In this section, we compare our theoretical results for the scaling function $\calF\equiv\calF_2$ with three recent experiments on two-dimensional Bose gases. The first experiment was realized with a gas of $^{87}$Rb atoms with scattering length $a_3=5.3$\,nm and a thickness $l_z=240$\,nm in the confining direction leading to a dimensionless interaction constant $\tilde g=2mg=0.22$~\cite{Yefsah11,note6}. The second one was performed with $^{133}$Cs atoms and a scattering length $a_3$ controlled by a Feshbach resonance and varying in the range $2-10$\,nm resulting in $\tilde g=0.1-0.52$~\cite{Hung11}. The last one was realized with a $^{133}$Cs atom gas in an optical lattice and can be described by the Bose-Hubbard model with $t=2.7$\,nK, $U=16.7$\,nK ({\it i.e.} $U/t=6.25$), and a temperature varying in the range $5.8-32$\,nK ({\it i.e.} $2.15t-32t$)~\cite{Zhang12,note7}. This leads to a temperature-dependent dimensionless interaction constant $\tilde g_{\rm BH}(T)$ varying between $3.95$ and $5.75$. We
refer to these experiments as the ``ENS", ``Chicago I" and ``Chicago II" experiments, respectively.
In Fig.~\ref{fig_ens}, we compare the NPRG results with the ENS experiment. For $\tilde g=0.22$, the temperature dependence of $\tilde g(T)$ is negligible so that we expect the scaling forms~(\ref{PDweak}), which express $\calP$, ${\cal D}$ and ${\cal S}$ as universal functions of $\mu/T$ and $\tilde g$, to be very well satisfied. We find a nearly perfect agreement between the experimental data and the NPRG calculation of the universal function $\calF(\mu/T,\tilde g)$ (without any fitting parameter).
\begin{figure}
\centerline{\includegraphics[width=7.25cm]{chin_fig3.pdf}}
\caption{(Color online) Temperature dependence of the phase-space density ${\cal D}(\mu_c,T)$ in the Chicago II experiment (triangles)~\cite{Zhang12}. The (red) solid line shows the NPRG result and the (green) dashed one the universal limit.}
\label{fig_chin3}
\end{figure}
\begin{figure}
\centerline{\includegraphics[width=4.4cm]{P_chin_T=2v5t.pdf}
\includegraphics[width=3.9cm]{P_chin_T=4v1t.pdf}}
\caption{(Color online) Phase-space pressure $\calP(\mu,T)$ vs $\delta\mu/T$ for $T=6.7\,$nK and $T=11\,$nK in the Chicago II experiment~\cite{Zhang12}. The solid lines show the NPRG results obtained in the Bose-Hubbard model and the dashed lines the universal limit.}
\label{fig_chin24a}
\centerline{\includegraphics[width=7.6cm]{chin_fig2.pdf}}
\centerline{\hspace{0.8cm}\includegraphics[width=6.55cm]{chin_fig4a.pdf}}
\caption{(Color online) Phase-space density ${\cal D}(\mu,T)$ and entropy per particle ${\cal S}(\mu,T)$ vs $\delta\mu/T$ for $T=6.7$ and $T=11\,$nK in the Chicago II experiment~\cite{Zhang12}. The solid and dashed lines show the NPRG results obtained in the Bose-Hubbard model.}
\label{fig_chin24b}
\end{figure}
In Sec.~\ref{subsec_bh}, we have shown that a Bose gas in an optical lattice, described by the Bose-Hubbard model with $U/t=6.25$, reaches the universal limit only at temperatures of the order of $t$. In the Chicago II experiment, the lowest temperature $T\sim 2.15t$ is above $t$, and we should therefore expect experimental data to agree only approximately with results obtained from the universal function $\calF$. The temperature dependence of the phase-space density ${\cal D}(\mu_c,T)$ is shown in Fig.~\ref{fig_chin3}. There is an overall agreement between the experimental data and the NPRG results but the existence of a plateau for $T\lesssim 8t$ followed by a strong suppression of ${\cal D}(\mu_c,T)$ at higher temperatures, as advocated in Ref.~\cite{Zhang12}, is not supported by the theory. In Fig.~\ref{fig_chin24a} we show the phase-space pressure $\calP(\mu,T)$ vs $\delta\mu/T$ for $T/t=2.5$ and $T/t=4.1$. As expected the NPRG results show deviations from the universal limit $\calF(\delta\mu,\tilde g(T))$ (
note that $\tilde g(T)$ is nearly temperature independent in the temperature range $[2.5t-4.1t]$). For large and negative chemical potential $\delta\mu$, the pressure is very well approximated by the classical dilute gas expression
\begin{equation}
\calP(\mu,T) = 4\pi te^{-|\delta\mu|/T} l^2\int_{\bf q} e^{-\epsilon_{\bf q}/T} .
\label{calP1}
\end{equation}
The difference with the universal limit $\calP=e^{-|\delta\mu|/T}$ (Sec.~\ref{subsec_Fx}) is entirely due to the difference between the lattice dispersion $\epsilon_{\bf q}$ [Eq.~(\ref{epsq})] and the free quadratic dispersion ${\bf q}^2/2m$ with $m=1/2tl^2$. For $T/t=4.1$, Eq.~(\ref{calP1}) gives $\calP\simeq 1.3 e^{-|\delta\mu|/T}$ when $\delta\mu/T\lesssim -2$. On the other hand the experimental data show a remarkable agreement between the phase space pressure $\calP$ and the universal scaling function $\calF$ with only a small difference for positive $\delta\mu$. Such an agreement is difficult to understand in the framework of the Bose-Hubbard model. In particular, one would expect $\calP$ to differ from $e^{-|\delta\mu|/T}$ for large and negative $\delta\mu$ and $T/t\simeq 2-4$ due to lattice effects (see the discussion above). The phase-space density ${\cal D}(\mu,T)$ and the entropy per particle ${\cal S}(\mu,T)$ vs $\delta\mu/T$ for $T/t=2.5$ and $T/t=4.1$ are shown in Fig~\ref{fig_chin24b}; there is a good agreement
between theory and experiment.
The ENS, Chicago I and Chicago II experiments can be used to obtain $\calP(\mu_c,T)$, ${\cal D}(\mu_c,T)$ and ${\cal S}(\mu_c,T)$ as a function of the effective interaction constant $\tilde g(T)$. The results are shown in Fig.~\ref{fig_F2}. For all three experiments, we obtain a very good agreement with the universal limit~(\ref{calPD},\ref{calS}). This confirms that both the ENS and Chicago I experiments deal with a weakly interacting Bose gas in the universal regime. As for the Chicago II experiment, such a good agreement is partially accidental since for $\tilde g_{\rm BH}(T)\simeq\tilde g(T) \simeq 4.3$ (the relevant value of $\tilde g_{\rm BH}(T)$ corresponding to the experimental data shown in Fig.~\ref{fig_F2}), ${\cal S}(\mu_c,T)$ turns out to be very close to the universal limit even though the system has not reached the universal regime yet (see Fig.~\ref{fig_PDS_scaling}). We also note that for this value of this interaction constant, $\calP(\mu_c,T)$ and ${\cal D}(\mu_c,T)$ are nearly equal, which implies
that $P(\mu_c,T)\simeq T D(\mu_c,T)$ as observed in the Chicago II experiment.
\section{BKT transition temperature}
\label{sec_bkt}
In this section we show how the BKT transition temperature $T_{\rm BKT}$ can be estimated from the NPRG approach. For the classical O(2) model, the NPRG reproduces most of the universal properties of the BKT transition~\cite{Graeter95,Gersdorff01}. In particular one finds a value $\tilde\rho_0^*$ of the dimensionless order parameter (the spin-wave ``stiffness'') such that the beta function $\beta( \tilde\rho_{0,k})=k\partial_k \tilde\rho_{0,k}$ nearly vanishes for $\tilde\rho_{0,k}\geq \tilde\rho_0^*$ (here $k$ denotes the RG momentum scale, see Appendix~\ref{app_nprg}). This implies the existence of a line of quasi-fixed points and enables to identify a low-temperature phase ($T<T_{\rm BKT}$) where the running of the stiffness $\tilde\rho_{0,k}$, after a transient regime, becomes very slow, implying a very large (although not strictly infinite as expected in the low-temperature phase of the BKT transition) correlation length $\xi$. In this low-temperature phase, the anomalous dimension $\eta_k$ depends on the (slowly varying)
stiffness $\tilde\rho_{0,k}$. It takes its largest value $\sim 1/4$ when the RG flow crosses over to the disordered (long-distance) regime (for $\tilde\rho_{0,k}\sim\tilde\rho_0^*$ and $k\sim\xi^{-1}$), and is then rapidly suppressed as $\tilde\rho_{0,k}$ further decreases. On the other hand, the beta function is well approximated by $\beta( \tilde\rho_{0,k})=\const\times(\tilde\rho_0^*-\tilde\rho_{0,k})^{3/2}$ for $\tilde\rho_{0,k}\leq\tilde\rho_0^*$, and the essential scaling $\xi\sim e^{\const/(T-T_{\rm BKT})^{1/2}}$ of the correlation length above the BKT transition temperature $T_{\rm BKT}$ is reproduced~\cite{Gersdorff01}. Thus, although the NPRG approach does not yield a low-temperature phase with an infinite correlation length, it nevertheless allows us to estimate the BKT transition temperature from the value of $\tilde\rho^*_0$. A reasonable estimate of the BKT transition in the two-dimensional XY model has been obtained using the lattice NPRG~\cite{Machado10}. Here we use the NPRG to determine the BKT
transition temperature in a two-dimensional Bose gas~\cite{note13}.
\begin{figure}
\centerline{\includegraphics[width=7.3cm]{eta_ns.pdf}}
\caption{(Color online) Flow trajectories in the plane $(n_s,\eta)$ for a two-dimensional Bose gas with a dimensionless interaction constant $\tilde g=0.22$. The vertical line indicates the value of $n_s^*$. The (red) solid line shows the line of quasi-fixed points for $n_s\geq n_s^*$. The critical trajectory (which joins the line of quasi-fixed points for $n_s=n_s^*$) corresponds to $\mu/T\simeq 0.154$.}
\label{fig_eta_ns}
\end{figure}
The flow trajectories in the plane $(n_s,\eta)$ are shown in Fig.~\ref{fig_eta_ns} for the continuum model with $\tilde g=0.22$. $n_s$ denotes the superfluid density and is analog to the dimensionless order parameter $\tilde\rho_0$ of the classical O(2) model. At sufficiently low temperatures, the trajectories join a line of quasi-fixed points where the RG flow is very slow, before eventually crossing over to the disordered phase ($n_s\to 0$). The value of $n_s$ at the merging point with the line of quasi-fixed points depends on the temperature and chemical potential of the Bose gas. We estimate the BKT transition temperature by the trajectory for which the merging point corresponds to the value $n_s^*$ (analog to $\tilde\rho_0^*$ in the classical O(2) model) of the superfluid density. A precise determination of the value of $n_s^*$ (which can be obtained by fitting the beta function $k\partial_k n_{s,k}$ by $\const\times(n_s^*-n_{s,k})^{3/2}$ for $n_{s,k}<n_s^*$) is however difficult as it requires the full $\
calO(\partial^2)$ expansion of the effective action while we solve the NPRG equation within a simple truncation of the effective potential [Eq.~(\ref{trunc})]. Nevertheless, since the BKT transition in the Bose gas model and the classical O(2) model is controlled by the same fixed point, we expect the ratio $n_s^*/n_s^{\rm max}$, where $n_s^{\rm max}$ is the value of $n_s$ for which $\eta$ is maximum (Fig.~\ref{fig_eta_ns}), to be equal to $\tilde\rho_0^*/\tilde\rho_0^{\rm max}$.
Using this method, we have verified that the ratio $\mu/T$ at the BKT transition is a universal function of $\tilde g(T)$, {\it i.e.}
\begin{equation}
\left( \frac{\mu}{T}\right)_{\rm BKT} = {\cal H}\bigl(\tilde g(T)\bigr) ,
\label{ktuniv}
\end{equation}
with ${\cal H}$ a universal function. Equivalently, since $\tilde g(T)$ is a function of $ma_2^2T$, $(\mu/T)_{\rm BKT}$ can be seen as a universal function of $ma_2^2T$ or $ma_2^2\mu$. Figure~\ref{fig_Tkt} shows $(\mu/T)_{\rm BKT}$ obtained for two different temperatures, $T=T_\Lambda/10$ and $T_\Lambda/50$ ($T_\Lambda=\Lambda^2/2m$), and a range of values of $\tilde g$. The universal form~(\ref{ktuniv}) is well satisfied in the weak-coupling limit ($\tilde g(T)\simeq \tilde g\lesssim 1$). In this limit, we find
\begin{equation}
\left( \frac{\mu}{T}\right)_{\rm BKT} \simeq \frac{0.982}{2\pi}\tilde g\ln\left(\frac{2\times 9.48}{\tilde g}\right) ,
\end{equation}
in good agreement with the weak-coupling result~\cite{Popov_book_2,Fisher88,Prokofev01,Prokofev02}
\begin{equation}
\left( \frac{\mu}{T}\right)_{\rm BKT} = \frac{1}{2\pi}\tilde g\ln\left(\frac{2\zeta}{\tilde g}\right),
\label{Tktmc}
\end{equation}
where $\zeta\simeq 13.2\pm 0.4$ has been obtained from Monte Carlo simulation~\cite{Prokofev01,Prokofev02}. We ascribe the violation of universality at strong coupling, as seen in Fig.~\ref{fig_Tkt}, to a poor description of the BKT transition by the NPRG when $\tilde g\gtrsim 1$~\cite{note14}. Although we can use the same method to determine the BKT transition temperature in the Bose-Hubbard model, we cannot compare with the experimental result of the Chicago II experiment~\cite{Zhang12} which corresponds to a strong-interaction regime ($\tilde g_{\rm BH}(T)\sim 4.3$) where this method is not reliable.
\begin{figure}
\centerline{\includegraphics[width=8.cm]{TKTcont.pdf}}
\caption{(Color online) Ratio $(\mu/T)_{\rm BKT}$ vs $\tilde g(T)$ for $T=T_\Lambda/10$ and $T=T_\Lambda/50$, where $T_\Lambda=\Lambda^2/2m$. The (green) dotted line corresponds to the expression~(\ref{Tktmc}) with the Monte Carlo result $\zeta=13.2$~\cite{Prokofev01,Prokofev02}.}
\label{fig_Tkt}
\end{figure}
\section{Conclusion}
The scale invariance of the equation of state of a weakly interacting Bose gas, {\it i.e.} the fact that the phase-space pressure $\calP(\mu,T)$ depends only on $\mu/T$ when the dimensionless interaction constant $\tilde g$ is small, is well understood both experimentally and theoretically. We have shown that, more generally, the phase-space pressure $\calP(\mu,T)$ is a universal function of $\mu/T$ and the temperature-dependent dimensionless interaction constant $\tilde g(T)$ [Eq.~(\ref{pressure0})]. Using the NPRG approach, we have computed the corresponding universal scaling function $\calF(x,y)$ for a two-dimensional gas from weak to strong coupling. Recent measurements of the pressure, density and entropy in a weakly two-dimensional Bose gas~\cite{Yefsah11,Hung11} allow us to determine both the $x$ and $y$ dependence of $\calF(x,y)$ in some limits, and the results are found to agree remarkably well with the NPRG predictions.
We have also compared our theoretical results in the Bose-Hubbard model with recent experimental data obtained in a two-dimensional Bose gas in an optical lattice near the vacuum-superfluid transition~\cite{Zhang12}. Our theoretical analysis shows that the lowest temperature ($T=2.5t$) reached in the experiment remains slightly above the crossover temperature $T\sim t$ to the quantum critical regime where the thermodynamics is fully determined by the universal scaling function $\calF$. However, somewhat surprisingly, the experimental data do not show the small deviations from (universal) quantum critical behavior that are expected for $T=2.5t$ (see the discussion in Sec.~\ref{sec_exp}).
The experiment reported in Ref.~\cite{Zhang12} shows that it is now possible to measure the thermodynamics of a two-dimensional Bose gas in an optical lattice near the superfluid--Mott-insulator transition (where the Mott insulating phase is not the vacuum). Since this transition (when it is induced by a density change) belongs to the dilute Bose gas universality class, the thermodynamics in the superfluid phase is also determined by the scaling function $\calF$ (the BKT transition temperature being determined by the scaling function ${\cal H}$, see Eq.~(\ref{ktuniv})). The nonuniversal parameters $m$ and $a_2$ should be understood as the effective mass and effective scattering length of the elementary excitations at the (nontrivial) QCP between the superfluid phase and the Mott insulator. We have recently shown that Eq.~(\ref{pressure0}) indeed holds for a three-dimensional Bose gas in an optical lattice near the Mott transition and computed the non-universal parameters $m$ and $a_3$ in the framework of the
Bose-Hubbard model~\cite{Rancon12a}. Measuring the thermodynamics near the superfluid--Mott-insulator transition of a two- or three-dimensional Bose gas would allow for a very strong test of universality in strongly interacting quantum fluids.
\begin{acknowledgments}
We would like to thank X. Zhang, C. Chin, R. Desbuquois and J. Dalibard for discussions and/or correspondence and providing us with the experimental data shown in the manuscript. We are especially grateful to X. Zhang and C. Chin for numerous correspondences about the experiment reported in Ref.~\cite{Zhang12}.
\end{acknowledgments}
|
{
"timestamp": "2012-06-26T02:08:22",
"yymm": "1203",
"arxiv_id": "1203.1788",
"language": "en",
"url": "https://arxiv.org/abs/1203.1788"
}
|
\section{}
\label{}
\section{INTRODUCTION}
The notion of bisimulation introduced by
\cite{milner1989communication} has been successfully used as a
behavior equivalence in model checking \citep{clarke1997model},
software verification \citep{chaki2004abstraction} and formal
analysis of continuous \citep{tabuada2004bisimilar}, hybrid
\citep{tabuada2004compositional} and discrete event systems
(DESs). What makes bisimulation appealing is its capability in
complexity mitigation and branching behavior preservation,
specially when we deal with large scale distributed and concurrent
systems such as multi-robot cooperative tasking, networked
embedded systems, and traffic management.
Therefore, recent years have seen increasing research activities
in employing bisimulation to DESs. References
\citep{barrett1998bisimulation}, \citep{komenda2005control} and
\citep{sumodel} used bisimulation for the control of deterministic
systems subject to language equivalence.
\cite{madhusudan2002branching} investigated the control for
bisimulation equivalence with respect to a partial specification,
in which the plant is taken to be deterministic and all events are
treated to be controllable. \cite{tabuada2008controller} solved
the controller synthesis problem for bisimulation equivalence in a
wide variety of scenarios including continuous system, hybrid
system and DESs, in which the bisimilarity controller is given as
a morphism in the framework of category theory.
\cite{zhou2006control} investigated the bisimilarity control for
nondeterministic plants and nondeterministic specifications. A
small model theorem was provided to show that a supervisor
enforcing the bisimulation equivalence between the supervised
system and the specification exists if and only if a state
controllable automaton exists over the Cartesian product of the
system and specification state spaces. This small model theorem
was also extended for partial observation in
\citep{zhou2007small}. In both these works, the existence of a
bisimilarity supervisor depends on the existence of a state
controllable automaton, which is hard to calculate in a systematic
way, and the complexity of checking the existence condition is
doubly exponential. To reduce the computational complexity,
\cite{zhoubisimilarity2011} specialized to deterministic
supervisors. The existence condition for a deterministic
bisimilarity supervisor considering nondeterministic plants and
nondeterministic specifications was identified. Moreover, the
synthesis of deterministic supervisors, feasible supspecifications
and infimal subspecifications were developed as well.
\cite{liu2011bisimilarity} introduced a simulation-based framework
upon which the bisimilarity control for nondeterministic plants
and nondeterministic specifications was studied. In particular, a
new scheme based on the simulation relation was proposed for
synchronization which is different from those commonly used
synchronization operators such as parallel composition and product
in the supervisory control literature.
This paper studies the supervisory control of nondeterministic
plants for bisimulation equivalence with respect to deterministic
specifications. Compared to the existing literature, the
contributions of this paper mainly lie on the following aspects.
First, a novel notion of synchronous simulation-based
controllability is introduced as a necessary and sufficient
condition for the existence of a bisimilarity enforcing
supervisor. Although it is equivalent to the conditions in
\citep{zhoubisimilarity2011} specialized to deterministic
specifications, it provides a great insight into what characters
should a deterministic specification possesses for bisimilarity
control. Second, a test algorithm is proposed to verify the
existence condition, which is shown to be polynomial complexity
(less than the complexity of the conditions in
\citep{zhoubisimilarity2011}). When the existence condition holds,
we further present a systematic way to construct bisimilarity
enforcing supervisors. Third, since a given specification does
always guarantee the existence of a bisimilarity enforcing
supervisor, a key question arises is how to find a maximal
permissive specification which enables the synthesis of
bisimilarity enforcing supervisors. To answer this question, we
investigate the calculation of supremal synchronously
simulation-based controllable sub-specifications by using two
different methods. One is based on a recursive algorithm and the
other directly computes such a sub-specification based on
formulas.
The rest of this paper is organized as follows. Section 2 gives
the preliminary and problem formulation. Section 3 presents the
synthesis of bisimilarity enforcing supervisors. Section 4
investigates the test algorithm for the existence of a
bisimilarity enforcing supervisor. Section 5 explores the
calculation of maximal permissive sub-specifications. This paper
concludes with section 6.
\section{Preliminary and Problem Formulation}
\subsection{Preliminary Results}
A DES is modeled as a nondeterministic automaton $G
=(X,\Sigma,x_{0},\alpha, X_{m})$, where $X$ is the set of states,
$\Sigma$ is the set of events, $\alpha : \!X \times \Sigma \!
\rightarrow 2^X$ is the transition function, $x_0$ is the initial
state and $X_m \subseteq X$ is the set of marked states. The event
set $\Sigma$ can be partitioned into $\Sigma$ = $\Sigma_{uc} \cup
\Sigma_{c}$, where $\Sigma_{uc}$ is the set of uncontrollable
events and $\Sigma_{c}$ is the set of controllable events. Let
$\Sigma^{*}$ be the set of all finite strings over $\Sigma$
including the empty string $\epsilon$. The transition function
$\alpha$ can be extended from events to traces, $\alpha : \!X
\times \Sigma^{*} \!\rightarrow 2^{X}$, which is defined
inductively as: for any $x \in X$, $\alpha(x, \epsilon)=x$; for
any $s\in \Sigma^{*}$ and $\sigma \in \Sigma$, $\alpha(x,
s\sigma)=\alpha(\alpha(x, s), \sigma)$. If the transition function
is a partial map $\alpha: \!X \times \Sigma \!\rightarrow X$, $G$
is said to be a deterministic automaton. For $X_1 \subseteq X$,
the notation $\alpha|_{X_1 \! \times \! \Sigma}$ means $\alpha$ is
restricted from a smaller domain $X_1 \! \times \!\Sigma$ to
$2^{X_1}$. Given $X_1 \subseteq X$, the subautomaton of $G$ with
respect to $X_1$, denoted by $F_{G}(X_1)$, is defined as: $
F_{G}(X_1)= (X_1,\Sigma,x_{0},\alpha_1,X_{m1})$, where $\alpha_1
\! = \!\alpha \!\mid_{X_1 \!\times \! \Sigma}$ and $X_{m1}$ = $X_1
\! \cap X_{m}$. The active event set at state $x$ is defined as
$E_{G}(x)=\{\sigma \in \Sigma~|~\alpha(x, \sigma)$ is defined\}.
Given a string $s \in \Sigma^{*}$, the length of the string $s$,
denoted as $|s|$, is the total numbers of events, and $s(i)$ is
the $i$-$th$ event of this string, where $1 \leq i \leq |s|$.
Given $\Sigma_1 \subseteq \Sigma$, a projection
$P_{\Sigma\!\rightarrow \Sigma_1}$: $\Sigma^{*}\! \rightarrow
\Sigma_{1}^{*}$ is used to filter a string of events from $\Sigma$
to $\Sigma_1$, and it is defined inductively as follows:
$P_{\Sigma\!\rightarrow \Sigma_1}(\epsilon)=\epsilon$; for any
$\sigma \in \Sigma$ and $s \in \Sigma^{*}$,
$P_{\Sigma\!\rightarrow \Sigma_1}(s\sigma)=P_{\Sigma\!\rightarrow
\Sigma_1}(s)\sigma$ if $\sigma \in \Sigma_1$, otherwise,
$P_{\Sigma\!\rightarrow \Sigma_1}(s\sigma)=P_{\Sigma\!\rightarrow
\Sigma_1}(s)$. The language generated by $G$ is defined as
$L(G)=\{s \in \Sigma^{*} \mid \alpha(x_0, s)$ is defined$\}$, and
the marked language generated by $G$ is defined as $L_{m}(G)=\{s
\in \Sigma^{*} \mid \alpha(x_0, s) \cap X_m \neq \emptyset$\}.
Consider three languages $K, K_1, K_2 \subseteq \Sigma^{*}$. The
Kleene closure of $K$, denoted as $K^{*}$, is the language
$K^*=\cup_{n \in \mathbb{N}}K^{n}$, where $K^{0}=\{\epsilon\}$ and
for any $n \geq 0$, $K^{n+1}=K^{n}K$. The prefix closure of $K$,
denoted as $\overline{K}$, is the language $\overline{K}=\{s \in
\Sigma^{*}~|~(\exists t \in \Sigma^{*})~st\in K\}$. The quotient
of $K_1$ with respect to $K_2$, denoted as $K_1/K_2$, is the
language $K_1/K_2=\{s \in \Sigma^{*}~|~(\exists t \in K_2)~st \in
K_1\}$. For two languages $K_1, K_2 \in \Sigma^{*}$ with $K_2
\subseteq K_1 \neq \emptyset$, let $G_{(K_1, K_2)}$ be a
deterministic automaton such that $L(G_{(K_1, K_2)})=K_1$ and
$L_{m}(G_{(K_1, K_2)})=K_2$. For a nondeterministic $G$, let
$det(G)$ be a minimal deterministic automaton such that
$L(det(G))=L(G)$ and $L_{m}(det(G))=L_{m}(G)$.
To model the interaction between automata, we introduce parallel
composition as below \citep{cassandras2008introduction}.
\begin{Definition}\label{parallel}
Given $G_1 =(X_1,\Sigma_1,x_{01},\alpha_1, X_{m1})$ and $G_2
=(X_2,\Sigma_2,x_{02},\alpha_2,X_{m2})$, the parallel composition
of $G_1$ and $G_2$ is an automaton
\[
G_1 || G_2 = ( X_1 \times X_2, \Sigma_1 \cup \Sigma_2,
\alpha_{1||2}, (x_{01}, x_{02}), X_{m1} \times X_{m2}),
\]
where for any $x_1 \in X_1$, $x_2 \in X_2$ and $\sigma \in
\Sigma$, the transition function is defined as:
\[
\alpha_{1||2}((x_1, x_2),\sigma) = \left\{ {\begin{array}{*{20}c}
\alpha_1(x_1, \sigma) \times \alpha_2(x_2, \sigma) & {\sigma \in E_{G_1}(x_1) \cap E_{G_2}(x_2) }; \\
\alpha_1(x_1, \sigma) \times \{x_2\} & {\sigma \in E_{G_1}(x_1) \cap \sigma \in E_1 \!\setminus E_2}; \\
\{x_1\} \times \alpha_2(x_2, \sigma) & {\sigma \in E_{G_2}(x_2) \cap \sigma \in E_2 \!\setminus E_1}; \\
\emptyset & {otherwise}. \\
\end{array}} \right.
\]
\end{Definition}
When $\Sigma_1=\Sigma_2$, parallel composition can be understood
as a form of control, where a supervisor is designed to restrict
the behavior of the plant.
Next we present the synchronized state map, which is used to find
the synchronized state pairs of two automata
\citep{zhou2006control}.
\begin{Definition}
Given $G_1 =(X_1,\Sigma_1,x_{01},\alpha_1, X_{m1})$ and $G_2
=(X_2,\Sigma_2,x_{02},\alpha_2,X_{m2})$, the synchronized state
map $X_{synG_1G_2}$: $X_1 \rightarrow 2^{X_2}$ from $G_1$ to $G_2$
is defined as
\[
X_{synG_1G_2}(x_1)=\{x_2 \in X_2~|~(\exists s \in \Sigma^{*})~ x_1
\in \alpha_1(x_{01}, s) \wedge x_2 \in \alpha_2(x_{01}, s)\}.
\]
\end{Definition}
Most literature on supervisory control aims to achieve language
equivalence between the supervised system and the specification.
The necessary and sufficient condition for the existence of a
language enforcing supervisor is captured by the notion of
language controllability as below \citep{ramadge1987supervisory}.
\begin{Definition}\label{langc}
Given $G =(X,\Sigma,x_{0},\alpha, X_{m})$, a language $K \subseteq
L(G)$ is said to be language controllable with respect to $L(G)$
and $\Sigma_{uc}$ if
\[
\overline{K}\Sigma_{uc} \cap L(G) \subseteq \overline{K}.
\]
\end{Definition}
As a stronger behavior equivalence than language equivalence,
bisimulation is stated as follows \citep{milner1989communication}.
It is known that bisimulation implies language equivalence and
marked language equivalence, but the converse does not hold.
\begin{Definition}
Given $G_{1} =(X_{1},\Sigma,x_{01},\alpha_{1},X_{m1})$ and $G_{2}
=(X_{2},\Sigma,x_{02},\alpha_{2},X_{m2})$, a simulation relation
$\phi$ is a binary relation $\phi \subseteq X_1 \times X_2$ such
that $(x_{1}, x_{2}) \in \phi$ implies:
\begin{enumerate}
\item[(1)] $(\forall \sigma \in \Sigma)[\forall x_{1}^{'} \in
\alpha_{1}(x_{1},\sigma)\Rightarrow \exists x_{2}^{'} \in
\alpha_{2}(x_{2},\sigma)$ such that $(x_{1}^{'},x_{2}^{'}) \in
\phi]$;
\item[(2)] $x_{1} \in X_{m1} \Rightarrow x_{2} \in X_{m2}$.
\end{enumerate}
\end{Definition}
If there is a simulation relation $\phi$ $\subseteq$ $X_{1} \times
X_{2}$ such that $(x_{01},x_{02}) \in \phi$, $G_{1}$ is said to be
simulated by $G_{2}$, denoted by $G_{1} \prec_{\phi} G_{2}$. For
$\phi \subseteq (X_1 \cup X_2)^{2}$, if $G_{1} \prec_{\phi}
G_{2}$, $G_{2} \prec_{\phi} G_{1}$ and $\phi$ is symmetric, $\phi$
is called a bisimulation relation between $G_{1}$ and $G_{2}$,
denoted by $G_{1} \cong_{\phi} G_{2}$. We sometimes omit the
subscript $\phi$ from $\prec_{\phi}$ or $\cong_{\phi}$ when it is
clear from the context. Then we present a motivating example of
this paper.
\subsection{A Motivating Example}
\begin{figure}[!htb]
\begin{center}
\includegraphics*[scale=.5]{plant2.eps}
\caption{ multi-robot system (MRS) (Left), $G_1$ (Middle) and
$G_2$(Right)} \label{plantdet}
\end{center}
\end{figure}
Consider a cooperative multi-robot system (MRS) configured in Fig.
\ref{plantdet} (Left). The MRS consists of two robots $R_1$ and
$R_2$. Both of them have the same communication, position,
pushing, scent-sensing and frequency-sensing capabilities.
Furthermore, $R_1$ has color-sensing capabilities, while $R_2$ has
shape-sensing capability. $R_1$ and $R_2$ can cooperatively search
and clear a dangerous object (the white cube) in the workspace.
Initially, $R_1$ and $R_2$ are positioned outside the workspace.
Let $i=1, 2$. When the work request announces (event $w_i$), $R_i$
is required to enter the workspace. Due to actuator limitations,
it nondeterministically goes along one of two pre-defined paths
(event $g$). In the first path, $R_1$ activates color-sensing
(event $c$) and scent-sensing (event $o$) capabilities to detect
the dangerous object; whereas in the second path, besides
color-sensing and scent-sensing capabilities, $R_1$ also activates
frequency-sensing (event $f$) for detection. Similarly, $R_2$
activates shape-sensing (event $s$), scent-sensing and
frequency-sensing capabilities in the first path, while in the
second path it activates shape-sensing and scent-sensing
capabilities. After detecting the dangerous object, $R_i$ pushes
the dangerous object outward the workspace (event $p$), and then
returns to the initial position (event $r$) for the next
implementation.
\begin{figure}[!htb]
\begin{center}
\includegraphics*[scale=.5]{moti.eps}
\caption{$G_1||G_2$ (First Left), $R$ (Second Left), $S_1$ (Second
Right) and $S_2$ (First Right)} \label{spec}
\end{center}
\end{figure}
The automaton model $G_i$ of $R_i$ with alphabet $\Sigma_{i}$ is
shown in Fig. \ref{plantdet}, where $\Sigma_1=\{w_1, g, c, o, f,
p, r\}$ and $\Sigma_2=\{w_2, g, s, o, f, p, r\}$. Since $R_i$ can
not disable the host computer to broadcast the work announcement,
the event $w_i$ is deemed uncontrollable, that is $w_i \in
\Sigma_{uci}$. The rest events are controllable. The cooperative
behavior of $R_1$ and $R_2$ can be represented as $G_1||G_2$ (Fig.
\ref{spec} (First Left)). The specification $R$, configured in
Fig. \ref{spec}, is given in order to restrict the cooperative
behavior $G_1||G_2$. According to the specification, after both
$R_1$ and $R_2$ receive the work command and go to the workspace,
two possible states may be reached by the MRS
nondeterministically. In the first state, the color sensor, the
shape sensor and the scent sensors can be adopted to confirm an
objective is dangerous. However, to save the energy, in the second
state only the color sensor and the shape sensor can be adopted
for dangerous object detection. After the detection, the dangerous
object is cleared from the workspace.
\begin{figure}[!htb]
\begin{center}
\includegraphics*[scale=.5]{motispec1.eps}
\caption{$||_{i \in \{1, 2\}} G_i||S_i$ (Left), $R_{s_1}$ (Middle)
and $R_{s_2}$ (Right)} \label{proj}
\end{center}
\end{figure}
For such a MRS, if we use language equivalence as behavior
equivalence, the control target is to design supervisors $S_1$ and
$S_2$ such that $L(\parallel_{i \in \{1, 2\}}G_i||S_i)=L(R)$.
According to the results in \citep{willner1991supervisory}, this
problem can be solved by designing $S_i$ such that
$L(G_i||S_i)=P_{\Sigma_1\!\cup\!\Sigma_2\rightarrow
\Sigma_{i}}(L(R))$. Since $P_{\Sigma_1\!\cup\!\Sigma_2\rightarrow
\Sigma_{i}}(L(R))$ is language controllable with respect to
$L(G_i)$ and $\Sigma_{uci}$, we can construct $S_i$ as shown in
Fig. \ref{spec}. So the supervised system $||_{i \in \{1,
2\}}G_i||S_i$ (Fig. \ref{proj} (Left)) is language equivalent to
$L(R)$. However, it can be seen that $||_{i \in \{1, 2\}}G_i||S_i$
enables all the color sensor, the shape sensor and the scent
sensors for dangerous object detection, which violates the energy
saving requirement in the specification. Hence langauge
equivalence is not adequate for this case, which calls for the use
of bisimulation as behavior equivalence. That is, we need design
supervisor $S_i'$ such that $||_{i \in \{1, 2\}}G_i||S_i' \cong
R$. For such a bisimilarity control problem, a promising method
\citep{karimadini2011guaranteed} is to decompose the global
specification $R$ into sub-specifications $R_{s_i}$ with alphabet
$\Sigma_{i}$ for $R_i$ (Fig. \ref{proj}) such that $||_{i \in \{1,
2\}}R_{s_i}\cong R$ . If we can design $S_i'$ such that $G_i||S_i'
\cong R_{s_i}$, then $||_{i \in \{1, 2\}}G_i||S_i' \cong R$. In
particular, $R_{s_2}$ is deterministic, which motivates us to
consider the bisimilarity control for deterministic specifications
in this paper.
\subsection{Problem Formulation}
In the rest of paper, unless otherwise stated we will use $G=(X,
\Sigma, \alpha, x_{0}, X_m)$, $R=(Q, \Sigma, \delta, q_{0}, Q_m)$
and $S=(Y, \Sigma, \beta, y_0, Y_m)$ to denote the
nondeterministic plant, the deterministic specification and the
supervisor (possibly nondeterministic) respectively. Next we
formalize the notion of bisimilarity enforcing supervisor, which
always enables all uncontrollable events and enforces bisimilarity
between the supervised system and the specification.
\begin{Definition}\label{bissup}
Given a plant $G$ and a specification $R$, a supervisor $S$ is
said to be a bisimilarity enforcing supervisor for $G$ and $R$ if:
(1) There is a bisimulation relation $\phi$ such that $G||S
\cong_{\phi} R$;
(2) $(\forall y \in Y)(\forall \sigma \in \Sigma_{uc}) ~\beta(y,
\sigma) \neq \emptyset$.
\end{Definition}
This paper aims to solve the following problems.
{\bf Problem 1:} Given a nondeterministic plant $G$ and a
deterministic specification $R$, what condition guarantees the
existence of a bisimilarity enforcing supervisor $S$ for $G$ and
$R$?
{\bf Problem 2:} How to check this condition effectively?
{\bf Problem 3:} If the condition is satisfied, how to construct a
bisimilarity enforcing supervisor $S$?
{\bf Problem 4:} If the condition is not satisfied, how to obtain
a maximal permissive sub-specification which enables the synthesis
of bisimilarity enforcing supervisors?
\section{ Supervisory Control for Bisimilarity}
This section investigates Problem 1 and Problem 3, also called the
bisimilarity enforcing supervisor synthesis problem. We begin with
the existence condition of a bisimilarity enforcing supervisor.
For sufficiency, since we need design a bisimilarity enforcing
supervisor, the following concept is introduced.
\begin{Definition}
Given $G_1 =(X_1,\Sigma,x_{01},\alpha_1, X_{m1})$, the
uncontrollable augment automaton $G_{1uc}$ of $G_1$ is defined as:
\[
G_{1uc} =(X_1 \cup \{D_d \},\Sigma,x_{01},\alpha_{uc},X_{m1}),
\]
where for any $x \in X_1 \cup \{D_d \}$ and $\sigma \in \Sigma$:
\[
\alpha_{uc}(x,\sigma) = \left\{ {\begin{array}{*{20}c}
\alpha_1(x, \sigma) & \sigma \in E_{G_1}(x); \\
\{D_d\} & { (\sigma \in \Sigma_{uc} \!\setminus E_{G_1}(x)) \vee (x=D_d \wedge \sigma \in \Sigma_{uc})} ; \\
\emptyset & {otherwise.}\\
\end{array}} \right.
\]
\end{Definition}
We can see that an uncontrollable augment automaton can be
employed in the construction of bisimilarity enforcing supervisors
because it naturally satisfies the condition (2) required for a
bisimilarity enforcing supervisor (Definition \ref{bissup}).
On the other side, for necessity we have $G||S \cong R$, which
implies $R \prec G||S \prec G$. Hence $R \prec G$ is a necessary
condition to guarantee the existence of a bisimilarity enforcing
supervisor. Moreover, $G||S \cong R$ implies $L(G||S)=L(R)$, thus
language controllability of the specification is also a necessary
condition for the existence of a bisimilarity enforcing
supervisor. To satisfy those necessary conditions, we will
introduce synchronous simulation-based controllability as a
property of the specification. Before that, we need the following
concept.
\begin{Definition}
Given $G_{1} =(X_{1},\Sigma,x_{01},\alpha_{1},X_{m1})$, $G_{2}
=(X_{2},\Sigma,x_{02},\alpha_{2},X_{m2})$ and a simulation
relation $\phi$ such that $G_{1} \prec_{\phi} G_{2}$, $\phi$ is
called a synchronous simulation relation from $G_1$ to $G_2$ if
$(x_1, x_2) \in \phi$ for any $x_1 \in X_1$ and $x_2 \in
X_{synG_1G_2}(x_1)$.
\end{Definition}
If there exists a synchronous simulation relation $\phi$ from
$G_1$ to $G_2$, $G_1$ is said to be synchronously simulated by
$G_2$, denoted as $G_1 \prec_{syn\phi} G_{2}$. For a deterministic
specification $R$, if $R$ is synchronously simulated by $G$, then
$G$ possesses the branches which are bisimilar to $R$ and the
branches which are outside $L(R)$. Hence it turns out that $G||R
\cong R$. If $R$ is further language controllable with respect to
$L(G)$ and $\Sigma_{uc}$, then $G||R=G||R_{uc}$, implying that
$R_{uc}$ is a candidate of bisimilarity enforcing supervisor. Base
on this observation, we provide the following concept.
\begin{Definition}\label{syncdef}
Given $G_1 =(X_1,\Sigma,x_{01},\alpha_1,X_{m1})$ and $G_2
=(X_2,\Sigma,x_{02},\alpha_2,X_{m2})$, $G_1$ is said to be
synchronously simulation-based controllable with respect to $G_2$
and $\Sigma_{uc}$ if it satisfies:
(1) There is a synchronous simulation relation $\phi$ such that
$G_1 \prec_{syn\phi} G_{2}$;
(2) $L(G_1)$ is language controllable with respect to $L(G_2)$ and
$\Sigma_{uc}$.
\end{Definition}
It is immediate to see that when $R$ is synchronously
simulation-based controllable with respect to $G$ and
$\Sigma_{uc}$, it not only satisfies the necessary conditions ($R
\prec G$ and language controllability of $L(R)$) for the existence
of a bisimilarity enforcing supervisor but also enables the
development of $R_{uc}$ as a bisimilarity enforcing supervisor to
accomplish the sufficiency of the existence condition.
Then we present a necessary and sufficient condition for the
existence of a bisimilarity enforcing supervisor.
\begin{Theorem}\label{t}
Given a plant $G$ and a deterministic specification $R$, there
exists a bisimilarity enforcing supervisor $S$ for $G$ and $R$ if
and only if $R$ is synchronously simulation-based controllable
with respect to $G$ and $\Sigma_{uc}$.
\end{Theorem}
\begin{proof}
For sufficiency, we choose $R_{uc}$ as the supervisor. Let
$G||R=(X_{||}, \Sigma, (x_0, q_0),$ $ \alpha_{||},X_{m||})$.
Consider a relation $\phi_1=\{((x, q), q) ~|~ (x, q) \in
X_{||}\}$. We show that $\phi_1 \cup \phi_1^{-1}$ is a
bisimulation relation from $G||R$ to $R$. First note that $((x_0,
q_0), q_0) \in \phi_1$. Pick $((x, q), q) \in \phi_1$ and $(x',
q') \in \alpha_{||}((x, q), \sigma)$, where $\sigma \in \Sigma$.
By the definition of parallel composition, we have $q' \in
\delta(q, \sigma)$, which implies $((x', q'), q') \in \phi_1$.
When $(x', q') \in X_{m||}$, then $q' \in Q_{m}$. On the other
side, pick $(q, (x, q)) \in \phi_1^{-1}$ and $q' \in \delta(q,
\sigma)$. Since $(x, q) \in X_{||}$ and there is a synchronous
simulation relation $\phi$ such that $R \prec_{syn\phi} G$, we
have $(q, x) \in \phi$. Then there is $x' \in \alpha(x, \sigma)$
such that $(q', x') \in \phi$, and if $q' \in Q_{m}$, then $x' \in
X_{m}$. It follows that $(x', q') \in \alpha_{||}((x, q), \sigma)$
and $(x', q') \in X_{m||}$ when $q' \in Q_{m}$. That is, $(q',
(x', q')) \in \phi_{1}^{-1}$. Hence $G || R \cong_{\phi_1 \cup
\phi_1^{-1}} R$. Moreover from determinism and language
controllability of $R$ and the fact that $R_{uc}$ adds every state
a transition to $D_d$ through undefined uncontrollable events does
not change the result of parallel composition, we have
$G||R_{uc}=G||R$. It implies that $G||R_{uc} \cong_{\phi_1 \cup
\phi_1^{-1}}R$.
For necessity, suppose there is a bisimilarity enforcing
supervisor $S$ for $G$ and $R$. Then, there is a bisimulation
relation $\phi'=\phi \cup \phi^{-1}$ such that $R \prec_{\phi}
G||S$ and $G||S \prec_{\phi^{-1}} R$. Let $G||S=(X_{G||S}, \Sigma,
(x_0, y_0), \alpha_{G||S},X_{mG||S})$. Consider a relation
$\phi_1=\{(q, x) \in Q \times X ~|~(\exists y \in Y)~(q, (x, y))
\in \phi \}$. We show that $\phi_1$ is a synchronous simulation
relation from $R$ to $G$. By the definition of parallel
composition, $\phi_1$ is a simulation relation from $R$ to $G$.
Assume there is $q \in Q$ and $x' \in X_{synRG}(q)$ such that $(q,
x') \notin \phi_1$. Hence there exists $s \in \Sigma^{*}$ such
that $q \in \delta(q_0, s)$ and $x' \in \alpha(x_0, s)$. Since $R
\prec_{\phi} G||S$, for $q \in \delta(q_0, s)$, there is $(x, y)
\in \alpha_{G||S}((x_0, y_0), s)$ such that $(q, (x, y)) \in
\phi$, which implies $y \in \beta(y_0, s)$ and in turn implies
$(x', y) \in \alpha_{G||S}((x_0, y_0),s)$. Because $G||S
\prec_{\phi^{-1}} R$, for $(x', y) \in \alpha_{G||S}((x_0,
y_0),s)$, there is $q' \in \delta(q_0, s)$ such that $((x', y),
q') \in \phi^{-1}$. Since $R$ is deterministic, we have $q=q'$.
Therefore, $(q, (x', y)) \in \phi$, which implies $(q, x') \in
\phi_1$. It introduces a contradiction. Then the assumption is not
correct. That is, for any $q \in Q$ and $x \in X_{synRG}(q)$, $(q,
x) \in \phi_1$. So $R \prec_{syn\phi_1} G$. Next we show language
controllability of $L(R)$. Since a bisimilarity enforcing
supervisor $S$ enables all uncontrollable events at each state,
$L(G||S)$ is language controllable with respect to $L(G)$ and
$\Sigma_{uc}$, further, $G||S \cong R$ implies $L(G||S)=L(R)$. It
follows that $L(R)$ is language controllable w.r.t. $L(G)$ and
$\Sigma_{uc}$. So $R$ is synchronously simulation-based
controllable w.r.t. $G$ and $\Sigma_{uc}$.
\end{proof}
\begin{Remark}
Theorem \ref{t} shows that if a deterministic $R$ is synchronously
simulation-based controllable with respect to $G$ and
$\Sigma_{uc}$, $R_{uc}$ is a bisimilarity enforcing supervisor for
$G$ and $R$. Here synchronous simulation-based controllability of
$R$ is equivalent to the conditions ($G||det(R)\cong R$ and
language controllability of $L(R)$) specialized to deterministic
specifications \citep{zhoubisimilarity2011} to ensure the
existence of a deterministic bisimilarity supervisor. However, the
notion of synchronous simulation-based controllability offers
computation advantages compared to the conditions in
\citep{zhoubisimilarity2011} (See section 4). Moreover, it enables
the calculation of maximal permissive sub-specification when the
existence condition for a bisimilarity enforcing supervisor does
not hold (See section 5).
\end{Remark}
\begin{figure}[!htb]
\begin{center}
\includegraphics*[scale=.5]{clsall.eps}
\caption{ $S_1'$ (First Left), $S_2'$ (Second Left), $G_1||S_1'$
(Second Right) and $G_2||S_2'$ (First Right)} \label{moticls}
\end{center}
\end{figure}
Now we revisit the motivating example.
\begin{Example}\label{lcbis}
Let $i\!=1,2$. We need design supervisor $S_{i}'$ such that
$G_i||S_i' \cong R_{s_i}$. Since $R_{s_2}$ is deterministic and
synchronously simulation-based controllable with respect to $G_2$
and $\Sigma_{uc2}\!=\{w_2\}$, from Theorem \ref{t} we can design
$(R_{s_2})_{uc}$ to be $S_2'$ (Fig. \ref{moticls} (Second Left)).
The supervised system $G_2||S_2'$ is shown in Fig. \ref{moticls}
(First Right) and it can be seen that $G_2||S_2'\! \cong_{\phi
\cup \phi^{-1}} R_{s_2}$, where $\phi\!=\{(q_0', (x_0', y_0')),
(q_1', (x_1', y_1')),(q_2', (x_2', y_2')), (q_2', $ $(x_3',
y_2')),(q_3', (x_4', y_3')), (q_4', (x_5', y_4'))\}$. In addition,
$S_1'$ for $G_1$ can be designed as shown in Fig. \ref{moticls}
(First Left) according to our results in
\citep{sun2012bisimilarityacc}. Then $G_1||S_1' \cong R_{s_1}$
(Fig. \ref{moticls} (Second Right)). As a result, $||_{i \in \{1,
2\}}G_i||S_i' \cong R$.
\end{Example}
\section{A Test Algorithm for the Existence of a Bisimilarity
Enforcing Supervisor}
To solve Problem 2, an algorithm is proposed in this section to
test the existence of a bisimilarity enforcing supervisor. We
start by introducing synchronously simulation-based controllable
product, which will be used in the test algorithm.
\begin{Definition}
Given $G_1 =(X_1,\Sigma,x_{01},\alpha_1, X_{m1})$ and $G_2
=(X_2,\Sigma,x_{02},\alpha_2,X_{m2})$, the synchronously
simulation-based controllable product of $G_1$ and $G_2$ is an
automaton
\[
G_1 ||_{sync} G_2 = ( (X_1 \times X_2) \cup \{q_d, q_d'\}, \Sigma,
\alpha_{12}, (x_{01}, x_{02}), X_{m1} \times X_{m2}),
\]
where for any $(x_1, x_2) \in X_1 \times X_2$ and $\sigma \in
\Sigma$, the transition function is defined as:
\[
\alpha_{12}\!((x_1\!, \!x_2),\!\sigma) = \left\{
{\begin{array}{*{20}c}
\!\alpha_1(\!x_1, \!\sigma) \! \times\! \alpha_2\!(x_2,\! \sigma) \! & \! {\sigma \in E_{G_1}(x_1)\cap E_{G_2}(x_2) }; \\
q_d & {\sigma \! \in \! E_{G_1}(x_1)\! \setminus \! E_{G_2}(x_2)}; \\
q_d' & {\sigma \! \in \! \Sigma_{uc}\cap(E_{G_2}(x_2)\! \setminus E_{G_1}(x_1))}; \\
\emptyset & {otherwise}. \\
\end{array}} \right.
\]
\end{Definition}
Since synchronous simulation-based controllability is a necessary
and sufficient condition for the existence of a bisimilarity
enforcing supervisor, the following algorithm for testing
synchronous simulation-based controllability of $R$ also verifies
the existence of a bisimilarity enforcing supervisor for $G$ and
$R$.
\begin{Algorithm}\label{algsync}
Given a plant $G$ and a deterministic specification $R$, the
algorithm for testing synchronous simulation-based controllability
of $R$ with respect to $G$ and $\Sigma_{uc}$ is described as
below.
Step 1: Obtain $ R ||_{sync} G=(X_{sync}, \Sigma, \alpha_{sync},
(q_0, x_0), X_{msync})$;
Step 2: $R$ is synchronously simulated-based controllable with
respect to $G$ and $\Sigma_{uc}$ if and only if $q_d$ and $q_d'$
are not reachable in $R||_{sync} G$ and $x \in X_{m}$ for any
reachable state $(q, x)$ in $R||_{sync} G$ with $q \in Q_{m}$.
\end{Algorithm}
\begin{Theorem}\label{alg1c}
Algorithm 1 is correct.
\end{Theorem}
\begin{proof}
From the definition of synchronously simulation-based controllable
product, it is obvious that any $(q, x)$ satisfying $x \in
X_{synRG}(q)$ is a state reachable in $R||_{sync}G$, and any $(q,
x) \in X_{sync}\!\setminus \!\{q_d, q_d'\}$ satisfies that $x \in
X_{synRG}(q)$. For synchronous simulation-based controllability to
hold, condition (1) and condition (2) of Definition \ref{syncdef}
should be satisfied. On the other hand, if condition (1) is
violated, there are two cases. Case 1: there exist $(q, x)$ and
$\sigma \in \Sigma$ such that $x \in X_{synRG}(q)$ and $\sigma \in
E_{R}(q) \!\setminus E_{G}(x)$. So $q_d \in \alpha_{sync}((q, x),
\sigma)$. Case 2: there is $(q, x)$ such that $x \in X_{synRG}(q)$
and $x \notin X_{m}$ when $q \in Q_{m}$. If condition (2) is
violated, i.e. there exist $(q, x)$ and $\sigma \in \Sigma_{uc}$
such that $x \in X_{synRG}(q)$ and $\sigma \in E_{G}(x)
\!\setminus E_{R}(q)$. So $q_d' \in \alpha_{sync}((q, x),
\sigma)$. It follows that $q_d$ and $q_d'$ are reachable in
$R||_{sync} G$ or $x \notin X_{m}$ for any reachable state $(q,
x)$ in $R||_{sync} G$ with $q \in Q_{m}$ iff $R$ is not
synchronously simulated-based controllable w.r.t. $G$ and
$\Sigma_{uc}$.
\end{proof}
\begin{Remark}
Algorithm 1 can be terminated because the state sets and the event
sets of $R$ and $G$ are finite. Since $G$ is nondeterministic and
$R$ is deterministic, their numbers of transitions are $O(|X|^{2}
|\Sigma|)$ and $O(|Q||\Sigma|)$ respectively. Then the complexity
of constructing $R||_{sync}G$ is $O(|X|^{2}|Q|^{2} |\Sigma|)$. In
addition, the complexity of checking the reachability of $q_d$ and
$q_d'$ in $R||_{sync}G$ is $O(log(|X||Q|))$
\citep{jones1975space}. So the complexity of Algorithm 1 is
$O(|X|^{2} |Q|^{2} |\Sigma|)$. That is, the algorithm for testing
the existence of a bisimilarity enforcing supervisor has
polynomial complexity. \cite{zhoubisimilarity2011} used the
conditions such as $G||det (R) \cong R$ and $L(R)$ is language
controllable with respect to $L(G)$ and $\Sigma_{uc}$ to guarantee
the existence of a deterministic supervisor that achieves
bisimulation equivalence. The complexity of verifying those
conditions with respect to deterministic specifications is
$O(|X|^{2}|Q|^{2}|\Sigma|^{3}log (|X||Q|^{2}))$ (Remark 2 in
\citep{zhoubisimilarity2011}). Hence, we argue that Algorithm 1 is
more effective.
\end{Remark}
We provide the following example to illustrate the algorithm for
checking synchronous simulation-based controllability.
\begin{figure}[!htb]
\begin{center}
\includegraphics*[scale=.5]{chk11.eps}
\caption{ Plant $G$ (Left), Specification $R$ (Middle) and
$R||_{sync} G$ (Right) of Example 2} \label{chk1}
\end{center}
\end{figure}
\begin{Example}\label{checksync}
Consider a plant $G$ and a specification $R$ with
$\Sigma_{uc}=\{b, e\}$ configured in Fig. \ref{chk1}. We can see
that $R$ is not synchronously simulation-based controllable with
respect to $G$ and $\Sigma_{uc}$ because for $f \in L(G)\cap L(R)$
and $e \in \Sigma_{uc}$, $fe \in L(G)\!\setminus L(R)$, and $e$ is
defined at $q_7$ but not $x_8 \in X_{synRG}(q_7)$.
Next we use Algorithm \ref{algsync} to test synchronously
simulation-based controllability of $R$. The synchronously
simulation-based controllable product $R||_{sync}G$ is shown in
Fig. \ref{chk1} (Right). It can be seen that $q_d$ and $q_d'$ are
reachable in $R||_{sync}G$. Hence $R$ is not synchronously
simulation-based controllable with respect to $G$ and
$\Sigma_{uc}$.
\end{Example}
\section{Supremal Synchronously Simulation-Based Controllable Sub-specifications}
This section studies Problem 4, i.e., the synthesis of supremal
synchronously simulation-based controllable sub-specifications,
because a synchronous simulation-based controllable
sub-specification ensures the existence of a bisimilarity
enforcing supervisor. First we introduce the notion of supremal.
Given $(A, \leq)$ and $A' \subseteq A$, where $\leq \subseteq A
\times A$ is a transitive and reflexive relation over $A$, $x \in
A$ is said to be a supremal of $A'$, denoted by $supA'$, if it
satisfies:
(1) $\forall y \in A'$: $y \leq x$;
(2) $\forall z \in A: [\forall y \in A': y \leq z] \Rightarrow [x
\leq z]$.
When we define the supremal of $A'$, a set $(A, \leq)$ should be
given with respect to the element of $A'$. If the elements of $A'$
are languages, the set $(2^{\Sigma^{*}}, \subseteq)$ should be
applied because $2^{\Sigma^{*}}$ includes all languages over
alphabet $\Sigma$ and language inclusion fully captures the
comparison between two languages. However, if the elements of $A'$
are automata, the set $(B, \prec)$ should be applied, where $B$ is
a full set of automata with alphabet $\Sigma$ and $\prec \subseteq
B \times B$ is the simulation relation, since $B$ includes all
automata over alphabet $\Sigma$ and the simulation relation is
adequate for automata (possibly nondeterministic) comparison.
We consider the class of sub-specifications that satisfies
synchronous simulation-based controllability as below.
\begin{eqnarray}
C_1 &:=& \{R' ~|~R'~ is ~deterministic, R' \prec R ~and ~R'~ is~ synchronous~ \nonumber\\
& & simulation-based ~controllable~ w.r.t.~ G ~and~
\Sigma_{uc}\} \nonumber
\end{eqnarray}
It can be seen that the supremal of $C_1$ with respect to $(B,
\prec)$ is a supremal synchronously simulation-based controllable
sub-specification. However, it is difficult to directly calculate
the supremal of $C_1$ because $C_1$ is not closed under the upper
bound (join) operator with respect to $(B, \prec)$
\citep{zhoubisimilarity2011}. To encounter this problem, we would
like to convert the automaton set $C_1$ into equivalently
expressed language sets which are closed under the upper bound
(set union) operator with respect to $(2^{\Sigma^{*}}, \subseteq)$
\citep{cassandras2008introduction}. Next we do this conversion
item by item. First, for two deterministic automata $R'$ and $R$,
the condition $R' \prec R$ is equivalent to the language condition
$L(R') \subseteq L(R)$ and $L_{m}(R') \subseteq L_{m}(R)$. Second,
language controllability required in synchronous simulation-based
controllability is naturally a language description. It remains to
convert synchronous simulation relation required in synchronous
simulation-based controllability to an equivalent language
condition. To complete the conversion, we need the following
concept.
\begin{Definition}
Given $G=(X, \Sigma, x_{0}, \alpha, X_{m})$, the synchronous state
merger operator on $G$ is defined as an automaton
\[
F_{syn}(G) = (X_{syn},\Sigma, \{x_{0}\},\alpha_{syn},X_{msyn}),
\]
where $X_{syn} = 2^{X}$, $X_{msyn}=\{ Y_1 ~|~Y_1 \subseteq X_{m}
\}$, and for any $A \in X_{syn}$ and $\sigma \in \Sigma$, the
transition function is defined as:
\[
\alpha_{syn}(A,\sigma) = \left\{ {\begin{array}{*{20}c}
\cup_{x \in A} \alpha(x, \sigma) & {\sigma \in \cap_{ x \in A}E_{G}(x) }; \\
undefined & {otherwise}. \\
\end{array}} \right.
\]
\end{Definition}
By using $F_{syn}(G)$, the synchronous simulation relation from a
deterministic automaton $G_1$ to a plant $G$ is equivalent to
language conditions $L(G_1) \subseteq L(F_{syn}(G))$ and
$L_{m}(G_1) \subseteq L_{m}(F_{syn}(G))$, which is illustrated by
the following proposition.
\begin{Proposition}\label{gsyn}
Given a plant $G$ and a deterministic automaton $G_1$, there is a
synchronous simulation relation $\phi$ such that $G_{1}
\prec_{syn\phi} G$ iff $L(G_1) \subseteq L(F_{syn}(G))$ and
$L_m(G_1) \subseteq L_{m}(F_{syn}(G))$.
\end{Proposition}
\begin{proof}
Let $F_{syn}(G) = (X_{f},\Sigma, \{x_{0}\},\alpha_{f},X_{mf})$,
$G_{1}=(X_1, \Sigma, x_{01}, \alpha_1, X_{m1})$ and
$G_{L}=G_{1}||G=(X_{L},\Sigma, (x_{01}, x_0),\alpha_{L},X_{mL})$.
For sufficiency, consider a relation $\phi=\{(x_1, x) \in X_1
\times X~|~x \in X_{synG_{1}G}(x_1) \}$. We show that $\phi$ is a
synchronous simulation relation from $G_{1}$ to $G$. First note
that $(x_{01}, x_0) \in \phi$. Pick $(x_1, x) \in \phi$ and $x_1'
\in \alpha_{1}(x_1, \sigma)$, where $\sigma \in \Sigma$. Since $x
\in X_{synG_{1}G}(x_1)$, there is $s \in \Sigma^{*}$ such that
$x_1 \in \alpha_1(x_{01}, s)$ and $x \in \alpha(x_0, s)$. Hence
$s, s\sigma \in L(G_1)$, moreover, $L(G_1)\subseteq
L(F_{syn}(G))$. It follows that $s, s\sigma \in L(F_{syn}(G))$.
Therefore there exist $A = \alpha_{f}(\{x_{0}\}, s)$ and $A_1 =
\alpha_{f}(A, \sigma)$. By the definition of $F_{syn}(G)$, we have
$x \in A$ and $\sigma \in \cap_{x''\in A}E_{G}(x'')$, which
implies there is $x' \in \alpha(x, \sigma)$ such that $x' \in
X_{synG_1G}(x_1')$, i.e. $(x_1', x') \in \phi$. Next we show that
$x_1 \in X_{m1}$ implies $x \in X_{m}$. Because $x_1 \in X_{m1}$,
we have $s \in L_m(G_1)$, in addition, $L_m(G_1) \subseteq
L_{m}(F_{syn}(G))$. It follows $s \in L_{m}(F_{syn}(G))$, that is
$A \subseteq X_{m}$, implying $x \in X_{m}$. So
$G_{1}\prec_{syn\phi}G$.
For necessity, the induction method is used to prove $s\in
L(F_{syn}(G))$ for any $s \in L(G_1)$, that is $L(G_1) \subseteq
L(F_{syn}(G))$. (1) $|s|=0$, then $s=\epsilon$. It is obvious that
$\epsilon \in L(F_{syn}(G))$. (2) Assume when $|s|=n$, we have $s
\in L(F_{syn}(G))$ for any $s \in L(G_1)$. (3) $|s|=n+1$. Let
$s=s_1\sigma$, where $\sigma \in \Sigma$. Because $s_1\sigma \in
L(G_1)$ and $G_{1}$ is deterministic, for any $x_2 \in
\alpha_{1}(x_{01}, s_1)$, we have $\sigma \in E_{G_{1}}(x_2)$.
Since $G_{1} \prec_{syn\phi} G$, for any $x'' \in \alpha(x_0,
s_1)$, we have $(x_2, x'') \in \phi$. It follows that $\sigma \in
\cap_{x'' \in \alpha(x_0, s_1)} E_{G}(x'')$. In addition,
$|s_1|=n$ implies $s_1 \in L(F_{syn}(G))$, which in turn implies
there is $A_1 = \alpha_{f}(\{x_0\}, s_1)$ such that $x'' \in A_1$.
Hence $A_2=\alpha_{f}(A_1, \sigma)=\cup_{x''\in A_1} \alpha(x'',
\sigma)$, that is, $s_1\sigma \in L(F_{syn}(G))$. Therefore for
any $s \in L(G_1)$, we have $s \in L(F_{syn}(G))$, i.e. $L(G_1)
\subseteq L(F_{syn}(G))$. Next we show $L_m(G_1) \subseteq
L_{m}(F_{syn}(G))$ by proving $s' \in L_{m}(F_{syn}(G))$ for any
$s' \in L_m(G_1)$. Since $s' \in L_m(G_1)$, there is $x_4
\in \alpha_1(x_{01}, s')$ such that $x_4 \in X_{m1}$. Because
$G_{1} \prec_{syn\phi} G$ implies $(x_4, x''') \in \phi$ for any
$x''' \in \alpha(x_0, s')$, we have $x''' \in X_{m}$. Definition
of $F_{syn}(G)$ implies $s' \in L_{m}(F_{syn}(G))$, i.e. $L_m(G_1)
\subseteq L_{m}(F_{syn}(G))$.
\end{proof}
Hence the automaton set $C_1$ can be converted into the following
langauge sets:
\begin{eqnarray}
C_2 &:=& \{L_1 \subseteq L(R) \cap L(F_{syn}(G))~|~
L_1=\overline{L_1}
~and~ L_1 ~is ~language ~controllable \nonumber \\
& & ~w.r.t.~ L(G) ~and~ \Sigma_{uc}\}; \nonumber \\
C_3 &:=& \{L_1 \cap L_{m}(R) \cap L_{m}(F_{syn}(G))~|~L_1 \in
C_2\}. \nonumber
\end{eqnarray}
The computation of supremal synchronously simulation-based
controllable sub-specification, i.e., $supC_1$, with respect to
$(B, \prec)$, can be achieved through the computation of the
supremal languages of $C_2$ and $C_3$ with respect to
$(2^{\Sigma^{*}}, \subseteq)$ as shown in the following theorem.
\begin{Theorem}\label{supeq}
Given a plant $G$ and a deterministic specification $R$, if
$supC_2 \neq \emptyset$, then $G_{(supC_2,\! supC_3)}$ $\in
supC_1$.
\end{Theorem}
\begin{proof}
Let $L_1\!=supC_2\!\neq \emptyset$ and $L_1'\!=supC_2 \!\cap
L_{m}(R) \!\cap L_{m}(F_{syn}(G))\!=supC_3$. First we show that
$G_{(L_1, L_1')}\!\in C_1$. Since $L_1 \!= supC_2$, we have $L_1
\!\in C_2$, which implies $L_1$ is language controllable w.r.t.
$L(G)$ and $\Sigma_{uc}$ and $L_1 \!\subseteq L(F_{syn}(G))$. In
addition, definition of $L_1'$ implies $L_1' \!\subseteq
L_{m}(F_{syn}(G))$. From Proposition \ref{gsyn}, it follows that
$G_{(L_1, L_1')}$ is synchronously simulation-based controllable
w.r.t. $G$ and $\Sigma_{uc}$. Since $L_1 \! \in C_2$ also implies
$L_1 \!\subseteq L(R)$ and $L_1' \!\subseteq L_{m}(R)$ and $R$ and
$G_{(L_1, L_1')}$ are deterministic, we have $G_{(L_1, L_1')}\!
\prec R$. Therefore, $G_{(L_1, L_1')}\! \in C_1$. Next we show
that $R_1 \!\prec G_{(L_1, L_1')}$ for any $R_1 \in C_1$. Suppose
there is $R_1 \! \in C_1$ such that $R_1 \! \nprec G_{(L_1,
L_1')}$. Since $R_1 \! \in C_1$, it implies $R_1 \! \prec R$,
moreover, $R_1$ and $R$ are deterministic. It follows that $L(R_1)
\! \subseteq L(R)$ and $L_{m}(R_1) \! \subseteq L_{m}(R)$. In
addition, $R_1 \! \in C_1$ also implies synchronous
simulation-based controllability of $R_1$. Hence $L(R_1)$ is
language controllable with respect to $L(G)$ and $\Sigma_{uc}$ and
there is a synchronous simulation relation $\phi$ such that $R_1
\! \prec_{syn\phi} G$ implying $L(R_1) \! \subseteq L(F_{syn}(G))$
and $L_{m}(R_1) \!\subseteq L_{m}(F_{syn}(G))$ according to
Proposition \ref{gsyn}. Hence $L(R_1)\! \in C_2$. Moreover,
$L_{m}(R_1) \! \subseteq L(R_1)$. By the definition of supremal,
we have $L(R_1) \! \subseteq supC_2\!= L_1$ and $L_{m}(R_1) \!
\subseteq supC_3 \!=L_1'$, further, $R_1$ and $G_{(L_1, L_1')}$
are deterministic. It follows that $R_1\!\prec G_{(L_1, L_1')}$,
which introduces a contradiction. Hence, the assumption is not
correct. That is, we have $R_1 \! \prec G_{(L_1, L_1')}$ for any
$R_1 \!\in C_1$. So $G_{(L_1, L_1')} \!=G_{(supC_2, supC_3)} \!
\in supC_1$.
\end{proof}
Next we present a recursive algorithm for computing the supremal
synchronously simulation-based controllable sub-specification.
\begin{Algorithm}\label{alg2}
Given a plant $G$ and a deterministic specification $R$, the
algorithm for computing the supremal synchronously
simulation-based controllable sub-specification with respect to
$G$ and $\Sigma_{uc}$ is described as follows:
Step 1: Obtain $det(G)=(X_{det}, \Sigma, x_{0det}, \alpha_{det},
X_{mdet})$, $G'=(F_{syn}(G)||R)_{uc}=(X', \Sigma, x_{0}', \alpha',
X_{m}')$ and $G''=G'||$ $det(G)=(X'', \Sigma, x_{0}'', \alpha'',
X_{m}'')$;
Step 2: $Z_0:=\{(x_1', x_2) \in X' \times X_{det}~|~x_1'=D_d\}$;
Step 3: $\forall k \geq 0$, $Z_{k+1}=Z_{k} \cup \{z \in
X''-Z_{k}~|~(\exists \sigma \in \Sigma_{uc}) ~ \alpha''(z, \sigma)
\in Z_{k} \}$;
Step 4: If $Z_{k+1}=Z_{k} \neq Z$, then the subautomaton
$F_{G''}(X''-Z_{k})$ of $G''$ is a supremal synchronously
simulation-based controllable sub-specification with respect to
$G$ and $\Sigma_{uc}$.
\end{Algorithm}
\begin{Theorem}\label{calz}
Algorithm \ref{alg2} is correct.
\end{Theorem}
\begin{proof}
Consider $R''\!=F_{G''}(X''\!-Z_{k})\!=(Q'', \Sigma, q_0'',
\delta'', Q_{m}'')$, where $Z_{k+1}\!=Z_{k}\!\neq Z$ with $k
\!\geq $ $0$. First we show that $L(R'') \!\in C_2$. Definition of
$Z_{k}$ implies $L(R'')$ is language controllable w.r.t. $L(G)$
and $\Sigma_{uc}$, and the fact that $L(det(G))\!=L(G)$ implies
$L(R'') \!\subseteq L(F_{syn}$ $ (G)) \!\cap L(R)$ and $L_{m}(R'')
\!\subseteq \!L_{m}(F_{syn}(G)) \!\cap L_{m}(R)$. It follows that
$L(R'') \!\in C_2$. Next we show that $L_2 \!\subseteq L(R'')$ for
any $L_2 \! \in C_2$. Suppose there is $L_2 \! \in C_2$ such that
$L_2 \!\nsubseteq L(R'')$, that is, there is $s \!\in \Sigma^{*}$
such that $s \!\in L_2 \!\setminus L(R'')$. Since $s \! \notin
L(R'')$, there exists $s_1 \!\in \overline{\{s\}}$ such that
$(x_1', x_1) \!\in Z_{k'}$, where $x_1' \!\in \alpha'(x_0', s_1)$,
$x_1 \!\in \alpha_{det}(x_{0det}, s_1)$ and $k'\!=0, 1, \cdots k$.
Hence there is $s_2 \in \Sigma_{uc}^{*}$ such that $x_2' \!\in
\alpha'(x_1', s_2)$ and $x_2 \!\in \alpha_{det}(x_1, s_2)$ with
$(x_2', x_2) \!\in Z_{0}$, which implies $s_1s_2 \!\in L(G)
\!\backslash L(F_{syn}(G)||R)$. Moreover,
$L(F_{syn}(G)||R)\!=L(F_{syn}(G)) \!\cap L(R)$ and $L_2
\!\subseteq L(F_{syn}(G)) \!\cap L(R)$. It follows that $s_1s_2
\!\notin L_2$. If $s_2 \!= \epsilon$, then $s_1 \!\notin L_2$,
which implies $s \!\notin L_2$. If $s_2 \!\neq \epsilon$, then
$s_1s_2(1)\cdots s_2(|s_2|-1) \!\notin L_2$ because $L_2$ is
language controllable w.r.t. $L(G)$ and $\Sigma_{uc}$, $s_2(|s_2|)
\!\in \Sigma_{uc}$ and $s_1s_2 \!\in L(G)\!\setminus L_2$. It in
turn follows that $s_1s_2(1)\!\cdots s_2(|s_2|-2) \!\notin L_2$,
$s_1s_2(1)\!\cdots s_2(|s_2|-3) \!\notin L_2$, $\cdots$, $s_1
\!\notin L_2$. Hence $s \!\notin L_2$. So there is a
contradiction, which implies the assumption is not correct. Then
$L_2 \!\subseteq L(R'')$ for any $L_2 \!\in C_2$. As a result,
$L(R'')\!=supC_2$. It remains to show that $L_{m}(R'')\!=supC_3$.
By the definition of $R''$ and the fact that $L_{m}(F_{syn}(G))
\!\subseteq L_{m}(G)$, we have $L_{m}(R'')\!=L(R'')\!\cap
L_{m}(F_{syn}(G))\! \cap L_{m}(R)\!=supC_2 \!\cap L_{m}$
$(F_{syn}(G)) \!\cap L_{m}(R)\!=supC_3$. It follows that $R''$ is
a deterministic automaton such that $L(R'')\!=supC_2$ and
$L_{m}(R'')$ $=\!supC_3$. By Theorem \ref{supeq}, we have $R''
\!\in supC_1$.
\end{proof}
\begin{Remark}
Algorithm \ref{alg2} can be terminated because the state set $X''$
is finite. Because the state numbers of $F_{syn}(G)$ and $det(G)$
are both $O(2^{|X|})$. Therefore, the complexity of Algorithm
\ref{alg2} is $O(2^{2|X|}|Q||\Sigma|)$.
\end{Remark}
Furthermore, the supremal synchronously simulation-based
controllable sub-specification can be calculated by formulas
without applying the recursive algorithm.
\begin{Theorem}\label{calsupsub}
Given a plant $G$ and a deterministic specification $R$, if
$M=L(R)\cap L(F_{syn}(G))-[(L(G)-L(R) \cap
L(F_{syn}(G)))/\Sigma_{uc}^{*}]\Sigma^{*}\neq\emptyset$, then
$G_{(M, M')}$ is a supremal synchronously simulation-based
controllable sub-specification with respect to $G$ and
$\Sigma_{uc}$, where $M'=M \cap L_{m}(R) \cap L_{m}(F_{syn}(G))$.
\end{Theorem}
\begin{proof}
According to Theorem 1 and Theorem 2 in
\citep{brandt1990formulas}, we obtain $supC_2=L(R) \cap
L(F_{syn}(G))-[(L(G)-L(R) \cap
L(F_{syn}(G)))/\Sigma_{uc}^{*}]\Sigma^{*}=M$. It follows that
$M'=supC_3$. From Theorem \ref{supeq}, $G_{(M, M')}$ is a supremal
synchronously simulation-based controllable sub-specification
w.r.t. $G$ and $\Sigma_{uc}$.
\end{proof}
Now we revisit Example \ref{checksync}.
\begin{figure}[!htb]
\begin{center}
\includegraphics*[scale=.5]{chk2.eps}
\caption{ $F_{syn}(G)$ (Left) and $det(G)$ (Right)} \label{chk2}
\end{center}
\end{figure}
\begin{Example}\label{sub2}
Example \ref{checksync} indicates that $R$ is not synchronously
simulation-based controllable with respect to $G$ and
$\Sigma_{uc}$. Thus, we would like to calculate the supremal
synchronously simulation-based controllable sub-specification with
respect to $G$ and $\Sigma_{uc}$ by the proposed methods.
(1) Recursive Method: From Algorithm \ref{alg2}, we establish
$F_{syn}(G)$ and $det(G)$, shown in Fig. \ref{chk2}. Then
$G''\!=(X'', \Sigma, x_{0}'', \alpha'',
X_{m}'')\!=(F_{syn}(G)||R)_{uc}||det(G)$ is achieved in (Fig.
\ref{chksup} (Left)). We obtain $Z_0\!=\!\{(D_{d}, x_{10}')\}$,
$Z_1\!=Z_0 \!\cup \!\{ (\{x_7, x_8\}, q_7, x_7'), (\{x_4\},q_4,
x_4')\}$ \\ and $Z_2\!= \!Z_1 \! \cup \! \{(\{x_2\}, q_2,
x_2')\}\!=\!Z_3$. Therefore, the supremal synchronously
simulation-based controllable sub-specification
$F_{G''}(X''\!-\!Z_{2})$ is obtained in Fig. \ref{chksup}.
\begin{figure}[!htb]
\begin{center}
\includegraphics*[scale=.5]{chksup1.eps}
\caption{ $(F_{syn}(G)|| R)_{uc}||det(G)$ (Left) and
$F_{G''}(X''-Z_2)$ (Right)} \label{chksup}
\end{center}
\end{figure}
(2) Formula-based Method: First we construct $F_{syn}(G)$, which
can be seen in Fig. \ref{chk2} (Left). Hence $L(R) \cap
L(F_{syn}(G)) =\overline{(d(fm+eg)n+cfgn+fgn)^{*}ab}$. Thus,
$M=L(R) \cap L(F_{syn}(G))-[(L(G)-L(R) \cap
L(F_{syn}(G)))/\Sigma_{uc}^{*}]\Sigma^{*}$=$\overline{(d(fm+eg)n+cfgn
}$\\$\overline{+fgn)^{*}ab}$-$(d(fm+eg)n+cfgn+fgn)^{*}ab\Sigma^{*}$
-$(d(fm+eg)n+cfgn+fgn)^{*}a\Sigma^{*}$-$(d(fm+eg)n+cfgn+fgn)^{*}f\Sigma^{*}$
=$\overline{(d(fm+eg)n+cfgn)^{*}}\neq\emptyset$ and $M'=M \cap
L_{m}(R) \cap
L_{m}(F_{syn}(G))$=$(d(fm+eg)n+cfgn)^{*}(d(fm+eg)+cfg)$. The
supremal synchronously simulation-based controllable
sub-specification $G_{(M, M')}\!=F_{G''}(X''-Z_2)$ is achieved in
Fig. \ref{chksup} (Right).
\end{Example}
\section{Conclusion}
In this paper, we investigated the bisimilarity enforcing
supervisory control of nondeterministic plants for deterministic
specifications. A necessary and sufficient condition for the
existence of a bisimilarity enforcing supervisor was deduced from
synchronous simulation-based controllability of the specification,
which can be verified by a polynomial algorithm. For those
specifications fulling the existence condition, a bisimilarity
enforcing supervisor has been constructed. Contrarily, when the
existence condition does not hold, a recursive method and a
formula-based method have been developed to calculate the maximal
permissive sub-specifications.
\bibliographystyle{model5-names}
|
{
"timestamp": "2012-03-09T02:02:15",
"yymm": "1203",
"arxiv_id": "1203.1745",
"language": "en",
"url": "https://arxiv.org/abs/1203.1745"
}
|
\section{Introduction}
physiabsorption of small molecules in multi-porous materials such as
zeolites and metal organic framework (MOF) materials has experienced a
surge of interest due to its potential for hydrogen-storage and
gas-separation applications.\citep{Zeolitic, Morris, Sholl, Rosi,
Rowsell, YunLiu, HughesMOF5, Caskey, HoffmannBreath, Valenzano,
HWuPCL10, bpdc, Nour_H2inter, Kong_rvH, Kong_ted, HWuMetal, WZhouMetal,
XiangMe, SunMetal, SumidaMgMOF74, DietzelNimof74, ChavanNiMOF74,
FitzZnmof74, Mgcell, FitzMOF5, synth1, WeiZ, synth3, synth4, synth5} MOF
structures have been widely investigated in order to find faster
absorption and higher storage densities, as well as proper binding
energies.\citep{Rosi, Rowsell, YunLiu, HughesMOF5, HoffmannBreath,
Caskey, Valenzano, HWuPCL10, bpdc, Nour_H2inter, Kong_rvH, Kong_ted,
HWuMetal, WZhouMetal, XiangMe, SunMetal, SumidaMgMOF74, DietzelNimof74,
ChavanNiMOF74, FitzZnmof74, Mgcell, FitzMOF5, synth1, WeiZ, synth3,
synth4, synth5} In order to design new MOFs with improved properties, it
is of critical importance to understand the nature of the interaction
between the absorbed molecule and the MOF host. Such understanding can
either be gained through theory, using first-principles
simulations,\citep{HoffmannBreath, Valenzano, HWuPCL10, bpdc,
Nour_H2inter, Kong_rvH, Kong_ted, HWuMetal, WZhouMetal, XiangMe,
SunMetal} or through experimental probes, such as infrared (IR)
absorption and Raman scattering.\citep{HoffmannBreath, Valenzano,
HWuPCL10, bpdc, Nour_H2inter, DietzelNimof74, ChavanNiMOF74,
FitzZnmof74,FitzMOF5, synth1}
Much progress has been made in improving the properties of MOFs. For
example, it has been shown that using unsaturated metal centers, such as
MOF74 with open Mg, Mn, Zn, Ni, Cu, or HKUST-1 with Cu, results in
higher absorption density for hydrogen and faster absorption at small
partial CO$_2$ pressures, the latter of which is highly desirable for
CO$_2$ capturing applications. \citep{Rowsell, YunLiu, HWuPCL10, Caskey,
Nour_H2inter, HWuMetal, WZhouMetal, XiangMe, SunMetal, SumidaMgMOF74,
DietzelNimof74, ChavanNiMOF74, FitzZnmof74} It has further been shown
that iso-structural MOFs with different open metal centers can have very
different absorption rates at low pressure.\citep{Caskey} In particular,
the electronically similar metal centers Mg and Zn result in a much
faster CO$_2$ uptake rate in Mg-MOF74 than in Zn-MOF74 at a pressure
smaller than 0.1~atm. \citep{Caskey} However, when previous research
yields contradictory results, it becomes difficult to gain further
insight; e.g.\ while the frequency shift of the asymmetric stretch mode
of absorbed CO$_2$ in Mg-MOF74 has been reported to be blue shifted in
one work using IR spectra and B3LYP-D$^*$ calculation,\citep{Valenzano}
it was reported as a red shift in another work using density functional
theory (DFT) with local density approximation (LDA)
simulations.\citep{HWuPCL10} Furthermore, a clear correlation between the
frequency shifts of the absorbed molecules and other absorption
properties such as the binding energy or the adsorption site is still
missing, which makes it difficult to directly correlate the observed
results with the physical nature of the absorption process.
Van der Waals density functional theory\citep{vdW-DF, potential_PRB,
vdW-DF2} (vdW-DF2) can be used as a very effective tool to understand the
molecule/MOF interactions. Unlike previous simulations
using LDA or GGA \citep{HWuPCL10} where long-range van der Waals
interactions are not included consistently, or B3LYP-D$^*$ \citep{Valenzano} where
the empirical parameters are used to incorporate the long-range
dispersion terms, in our vdW-DF2 method, the exchange-correlation
functional includes the---for these systems so important---long-range
van der Waals interactions between the MOF structure and the
physiadsorbed CO$_2$ molecules in a seamless\citep{vdW-DF} and fully
self consistent way. \citep{potential_PRB} vdW-DF2 and its predecessor
vdW-DF have been successfully applied to many van der Waals
systems,\citep{Langreth_rev} ranging from simple dimers \citep{dimers} and
physiadsorbed molecules \citep{physisorption} to DNA and drug
design.\citep{drug} In particular, it has been demonstrated that vdW-DF2
can correctly capture the interaction and determine the adsorption
sites, binding energy, and vibrational frequencies for small molecules
absorbed in different MOFs.\citep{Kong_rvH,bpdc,Kong_ted,Nour_H2inter}
To resolve the contradicting results in literature, and to achieve a
better understanding of the physics that determines the frequency shift
of the absorbed molecule, in this work we combine both theoretical
first-principles electronic-structure simulation using vdW-DF2 and
experimental IR and Raman spectroscopy to study the CO$_2$ absorption in
the iso-structural MOFs Mg-MOF74 and Zn-MOF74. In the first step, we use
our experimental probes to determine the absorption behavior and the
vibrational frequencies of CO$_2$ absorbed in these MOFs. Then, in the
next step, we use vdW-DF2 to calculate the corresponding shifts and
determine the mechanisms that cause them. Unlike in experiments,
first-principles simulations allow us to artificially freeze different
degrees of freedom of the system, enabling us to understand the
importance of different physical contributions to the CO$_2$ frequency
shifts. By analyzing the CO$_2$ asymmetric stretch frequency under
different geometries, we identify three different factors determining
the frequency shift of physiadsorbed CO$_2$, namely, the length of the
molecule, the asymmetric distortion, and the metal center.
\section{Method}
\subsection{Metal organic framework synthesis}
Mg-MOF74,\citep{synth3} Zn-MOF74,\citep{synth4} Co-MOF74,\citep{synth5} and
Ni-MOF74\citep{Caskey} were synthesized according to procedures described
in the literature;\citep{Nour_H2inter, synth1}
\begin{description} \item[Mg-MOF74] 2,5-Dihydroxyterephthalic acid
(H$_2$DHB DC) (99 mg, 0.5 mmol) and Mg(NO$_3$)$_2$$\cdot$6H$_2$O
(257~mg, 1.0~mmol) were dissolved in the mixture of THF (7~mL), 1M NaOH
solution (2~mL), and water (3~mL) with stirring. The mixture was then
sealed in a Teflon-lined autoclave and heated at 110$^\circ$C for 3
days. The product was collected by filtration as a light-yellow
substance. Yield: 115~mg, 83\%.
\item[Zn-MOF74] The preparation of Zn-MOF74 was similar to that of
Mg-MOF74 except that Zn(NO$_3$)$_2$$\cdot$6H$_2$O (298~mg, 1.0~mmol) was
used instead of Mg(NO$_3$)$_2$$\cdot$ 6H$_2$O. Yield: 160~mg, 87\%.
\item[Co-MOF74] H$_2$DHBDC (150~mg, 0.75~mmol) and
[Co(NO$_3$)$_2$]$\cdot$6H$_2$O (186~mg, 0.75~mmol) were dissolved in
15~mL of THF-H$_2$O solution (50:50, v:v) with stirring. The mixture was
transferred to a Teflon-lined autoclave, which was then sealed and
heated at 110$^\circ$C for 3 days. Brown-red rod-shape crystals were
isolated by filtration and dried under vacuum. Yield: 130~mg, 50\%.
\item[Ni-MOF74] A mixture of H$_2$DHBDC (60~mg, 0.3~mmol),
[Ni(NO$_3$)$_2$]$\cdot$6H$_2$O (174~mg, 0.6~mmol), DMF (9~mL), and
H$_2$O (1~mL) was transferred to a Teflon-lined autoclave and heated in
an oven at 100$^\circ$C for 3 days. Brown crystalline powder was
collected by filtration and dried under vacuum. Yield: 75~mg, 72\%.
\end{description}
\noindent All as-synthesized materials were exchanged with fresh
methanol four times in a duration of 4 days, followed by drying in an
vacuum oven at room temperature, and annealing at 480~K overnight under
high vacuum before spectroscopic measurements.
\subsection{IR and Raman spectroscopy}
IR absorption spectroscopy of CO$_2$ absorption in the MOFs was
performed in transmission at room temperature using a
liquid-N$_2$-cooled InSb detector. Approximately 12 mg of MOF powder was
pressed on a KBr support and mounted in a high-temperature high-pressure
cell (Specac product P/N 5850c) and heated to 200$^\circ$C in vacuum
(100 mTorr) overnight for complete desolvation. MOF74 samples were
activated by solvent exchange in methanol and drying in vacuum at room
temperature. Subtraction of the gas phase CO$_2$ contribution to the IR
spectra was performed as described in Ref.~\onlinecite{bpdc}.
Raman spectroscopy measurements were performed using a solid state 532
nm laser. The activated sample was loaded into a Linkam FTIR
cooling/heating stage, and the sample was heated to 120$^\circ$C in
vacuum (900~mTorr) for complete dehydration. A laser power of
0.113--1.23~mWatt was used to avoid sample burning from the laser.
\subsection{First-principles calculations}
For our first-principles calculations we used DFT as implemented in
$ABINIT$,\citep{ABINIT1,ABINIT2} utilizing vdW-DF2 to describe exchange and
correlation effects.\citep{vdW-DF2} Troulier-Martin type norm-conserving
pseudopotentials and a plane-wave basis are used,
\citep{Troulier} where the Zn 3$d$ semicore electrons are also considered
as valence electrons. To ensure a full convergence of the structure and
energy, a kinetic energy cutoff of 45 Hartree is used for the plane-wave
basis. For structural relaxation, we start from the experimental atomic
positions and relax them using vdW-DF2 until the force on each atom is
less than $0.05$~eV/\AA. The unit cell parameters are fixed to the
experimental values, where for the hexagonal unit cell
$a=25.881$~\AA and $c=6.8789$~\AA\ for Mg-MOF74\citep{Mgcell} and
$a=25.887$~\AA and $c=6.816$~\AA\ for Zn-MOF74,\citep{YunLiu} respectively.
Due to the complex interaction between CO$_2$ and MOF74, many local
minimum energy sites exist, making it difficult to identify the low-energy adsorption site, which corresponds to the low pressure absorption.
Indeed, multiple adsorption sites and binding energies are determined
from our simulations. In this work, we performed calculations with
multiple initial positions and orientations of CO$_2$ relative to the
MOF74, and choose the site with the lowest total energy as the one for
our analysis. To guarantee that CO$_2$ is absorbed at the equivalent
lowest energy site in both Mg-MOF74 and Zn-MOF74, we start the atomic
position relaxation of CO$_2$ in Mg-MOF74 with the coordinates of the
relaxed CO$_2$ in Zn-MOF74, which is possible due to the isostructure
and symmetry of the two MOF systems. This ensures that we are comparing
the effect of the open metal site on CO$_2$ in the two different systems
on the same footing.
\begin{figure*}
\begin{center}
\includegraphics[width=11cm]{Fig1.PS}
\end{center}
\caption{\label{fig:MOF74}(Color online)Illustration figure of CO$_2$ absorbed in Zn-MOF74.
Calculated values of $d_1$, $d_2$, $d_3$, and $\theta$ for CO$_2$ in
Zn-MOF74 and Mg-MOF74 can be found in
Table~\ref{binding}.}
\end{figure*}
To calculate the frequency shifts we use a frozen-phonon approach in the
CO$_2$ molecules, where the MOF74 atoms are kept fixed, while each of
the atoms in the CO$_2$ is distorted in $\pm dx, \pm dy, \pm dz$
directions by small distortions $\Delta r=0.02$ \AA, to calculate the
force on each atom. The symmetrized force matrix $2F(\Delta r)=F(\Delta
r)-F(-\Delta r)$ is constructed for only the atoms within the CO$_2$
molecule, based on the approximation that the interaction between the
CO$_2$ molecule and the MOF is weak and thus the effect of the vibration
of the MOF74 atoms on the CO$_2$ frequencies are negligible. However,
notice that, while the MOF74 atoms are kept fixed, the van der Waals
forces experienced by the CO$_2$ due to the presence of the MOF74 are
included by our vdW-DF2 calculation. To evaluate the effects of the surrounding MOF lattice, tests have been performed to allow
vibrations of several MOF74 atoms in the vicinity of the absorbed CO$_2$
molecule; frozen phonon calculations with such an extended force matrix
show no observable effect on the CO$_2$ frequencies. In general, our
studies show that calculated frequencies are converged to within less
than 1~cm$^{-1}$.
\section{Results and discussion}
\subsection{Structure and binding energy}
\begin{table}
\caption{\label{binding}Calculated binding energy $\Delta E$ (kJ/mol),
angle $\theta$~(deg), and various distances $d_i$ and $l$ (\AA)
(where $l=d_2+d_3$) of the CO$_2$ molecule physiadsorbed in the MOF ;
see Fig.~\ref{fig:MOF74} for further details. The calculated free CO$_2$
parameters are $d_0=1.1630$ \AA\ and $l_0=2.3260$ \AA.}
\begin{tabular*}{\columnwidth}{@{\extracolsep{\fill}}lcccccr@{}}\hline\hline
MOF & $\Delta E$ & $\theta$ & $d_1$ & $d_2-d_0$ & $d_3-d_0$ & $l-l_0$\\\hline
Mg-MOF74 & 35.4 & 120.8 & 2.53 & +0.0062 & --0.0059 & +0.0003\\
Zn-MOF74 & 26.9 & 116.4 & 2.69 & +0.0048 & --0.0039 & +0.0009\\\hline\hline
\end{tabular*}
\end{table}
The structure of CO$_2$ absorbed in MOF74 is illustrated in
Fig.~\ref{fig:MOF74}. In this work, we consider only CO$_2$ absorption
under low pressure smaller than 7 Torr, where the open metal site in
MOF74 is far from fully occupied. In our first-principles simulations
one CO$_2$ per six metal sites absorption is considered, corresponding
to the low-loading situation observed experimentally. For the binding
energy of CO$_2$ absorbed in Mg-MOF74, we find 35.4~kJ/mol, while it
binds somewhat weaker in Zn-MOF74 with 26.9~kJ/mol. Comparing the
distance of the CO$_2$ molecules with the open metal site, we find a
metal-oxygen distance ($d_1$ in Fig.~\ref{fig:MOF74}) of 2.53~\AA\ for Mg
versus 2.69~\AA\ for Zn. In other words, CO$_2$ binds stronger and closer
to the metal center in Mg-MOF74 than it does in Zn-MOF74. At the same
time, similar metal-CO$_2$ angles ensure that the CO$_2$ is absorbed at
equivalent sites in both MOFs, with slightly different metal-CO$_2$
angles of 120.8$^\circ$ and 116.4$^\circ$ in Mg-MOF74 and Zn-MOF74,
respectively.
Further analyzing the structure of the absorbed CO$_2$ molecule, we see
that the C=O bond closer to the metal site ($d_2$ in
Fig.~\ref{fig:MOF74}) is elongated in both systems with a value of
1.1692~\AA\ in Mg-MOF74 and 1.1678~\AA\ in Zn-MOF74; in free CO$_2$ it
is 1.1630~\AA\ (denoted as $d_0$ in Table~\ref{binding}). On the other
hand, the C=O bond farther away from the metal center ($d_3$ in
Fig.~~\ref{fig:MOF74}) are both shortened, with a value 1.1571~\AA\ in
Mg-MOF74 and 1.1591~\AA\ in Zn-MOF74. A summary of distortions is shown
in Table~\ref{binding}. This distortion can be understood intuitively
via the interaction between the CO$_2$ and the metal center, where the
attraction from the metal center weakens (and thus elongates) the nearby
C=O bond, shortening the remaining C=O bond. By comparing with the free
CO$_2$ value of 1.1630~\AA, we see that the CO$_2$ asymmetric distortion
in Mg-MOF74 is stronger than in Zn-MOF74. Summing up the two C=O bonds
yields the overall length of the CO$_2$ molecule, which shows an
elongation of +0.0003~\AA\ and +0.0009~\AA\ for Mg-MOF74 and Zn-MOF74
absorption, as shown in Table~\ref{binding}. It has been reported previously that CO$_2$ molecules absorbed in the MOF structure might experience some nonlinear distortion, i.e., an O-C-O angle differing from 180$^o$\citep{HWuPCL10}. In this work, we also alow nonlinear distortion of the CO$_2$ molecule, and we find that the relaxed CO$_2$ molecule are only slightly bent after adsorbed within MOF74, featuring an O-C-O angle of $179.25^o$ in Zn-MOF74 and $178.97^o$ in Mg-MOF74.
As mentioned above, multiple local minimum adsorption sites are obtained
during the relaxation of CO$_2$ within the MOF74, leading to different
metal-CO$_2$ distances and binding energies. For instance, in Zn-MOF74
positions with similar location but Zn-CO$_2$ distances of 2.96~\AA\ and
3.09~\AA\ are found with binding energies of 25.6~kJ/mol and
24.8~kJ/mol, respectively.
In addition to these sites similar to the global minimum
metal site, we also find a secondary non-metal site for CO$_2$
absorption, which lies between two neighboring equivalent metal sites
along the direction of the MOF74 pore, with a distance of 3.99~\AA\ to
the nearest metal Zn site and a binding energy of 21.8~kJ/mol---clearly
much smaller than at the metal site. The distance between two equivalent
metal sites along the pore direction in MOF74 is the $c$ length of the
hexagonal unit cell, which is approximately 6.8~\AA, as mentioned
above. It is thus possible that at high CO$_2$ pressure, when all the
metal sites are occupied (1 CO$_2$ per 1 metal), more CO$_2$ can still
be absorbed within the MOF74 occupying the secondary site, with a
CO$_2$--CO$_2$ distance of approximately 3.4~\AA\ along the pore
direction, thus resulting in a higher (2 CO$_2$ per 1 metal) storage
density of CO$_2$ within MOF74.
\subsection{CO$_2$ frequencies from experiment}
\begin{figure}[t]
\centering
\includegraphics[width=0.8\columnwidth]{Fig2_a.PS}\\[4ex]
\includegraphics[width=0.8\columnwidth]{Fig2_b.PS}
\caption{\label{Asy}IR absorption spectra of CO$_2$ absorbed into
Zn-MOF74 (top) and in Mg-MOF74 (bottom) at changing CO$_2$ pressure
(1--6 Torr).}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=0.8\columnwidth]{Fig3.PS}
\caption{\label{Inte}Integrated areas of IR absorption peaks for the
asymmetric stretch modes for CO$_2$ absorbed in Mg-MOF74 and Zn-MOF74 as
a function of CO$_2$ pressure.}
\end{figure}
IR absorption spectra of CO$_2$ absorbed in Mg-MOF74 and Zn-MOF74 are
shown in Fig.~\ref{Asy}, depicting a main IR absorption band attributed
to the asymmetric stretch of CO$_2$ at 2338 and 2352~cm$^{-1}$
for Zn-MOF74 and Mg-MOF74. The integrated areas of the IR band of the
asymmetric CO$_2$ stretch mode can serve as an indication of the
relative amount of CO$_2$ absorbed in MOF74. Figure \ref{Inte}
summarizes the integrated area as a function of CO$_2$ pressure for both
Mg-MOF74 and Zn-MOF74, showing that under the same pressure, the amount
of absorbed CO$_2$ is more for the case of Mg-MOF74 than it is for
Zn-MOF74. These results are consistent with isotherm measurements of
CO$_2$ reported in Ref.~\onlinecite{Caskey}. Similar IR measurements
are also performed on Co-MOF74 and Ni-MOF74, showing shifts of the
asymmetric CO$_2$ stretch ($\sim$2340~cm$^{-1}$) similar to that in
Zn-MOF74.
In Fig.~\ref{Asy}, the shoulder peaks at 2325 and
2341~cm$^{-1}$ are attributed to the combination mode of the stretch
mode and the two non-degenerate bending modes (denoted as $\delta_1$ and
$\delta_2$ in Fig.~\ref{Asy}). Experimentally, it is difficult to
identify the bending modes in the low frequency range because of their
weak intensity as compared to the MOF vibrations. Therefore, we could
only resolve one of these non-degenerate bending modes at
$\sim$658~cm$^{-1}$. The similarity of the bending mode for the Mg and
Zn cases is consistent with the close calculated values for both systems
shown in Table~\ref{frequency}.
Raman spectroscopy measurements were preformed to find the experimental
value of the symmetric stretch mode of absorbed CO$_2$ for the Co-MOF74
and Ni-MOF74. The symmetric stretch was found to be approximately
--1380 and --1382~cm$^{-1}$ for Co-MOF74 and Ni-MOF74,
respectively. Because of the strong MOF Raman modes, the symmetric
stretch for Mg-MOF74 and Zn-MOF74 is not available. A summary of the IR
and Raman data for the frequencies of the absorbed CO$_2$ within MOF74
can be found in Table ~\ref{frequency}.
\subsection{Analyzing the frequency shift}
When we consider the frequency shifts from the free CO$_2$ situation,
comparing with 2349~cm$^{-1}$ for free CO$_2$, the asymmetric stretch
for CO$_2$ absorbed in MOF74 has been shifted by $+3$ and
--11~cm$^{-1}$ for Mg and Zn-MOF74, respectively. Our first-principles
frozen-phonon calculations are in good agreement with this result and we
find the CO$_2$ asymmetric stretch mode experiences a slight redshift of
approximately --0.5~cm$^{-1}$ in Mg-MOF74 and --8.1~cm$^{-1}$ in
Zn-MOF74. Furthermore, the bending modes calculated also experience
redshifts of about --10~cm$^{-1}$ in both Mg-MOF74 and Zn-MOF74, which
is in excellent agreement with the experimentally observed --9~cm$^{-1}$
redshifts in these systems. The calculated results for all the CO$_2$
modes are summarized in Table~\ref{frequency}.
We have shown above that CO$_2$ is binding stronger in Mg-MOF74, with a
closer metal-CO$_2$ distance and larger molecule distortions. It is
thus puzzling to observe that the $\nu_3$ frequency shift for CO$_2$ in
Mg-MOF74 is much smaller compared with that in Zn-MOF74. There have been
many experiments and theoretical simulations studying the frequency
shifts of small molecules absorbed in MOFs, such as H$_2$, CO$_2$, CO,
etc.\citep{Valenzano, HWuPCL10, bpdc, Nour_H2inter, Kong_rvH, Kong_ted,
SumidaMgMOF74, DietzelNimof74, synth1, ChavanNiMOF74, FitzZnmof74,
FitzMOF5} It is well known that the frequency shift of small molecule
vibrations has no obvious correlations with either the binding energy or
the adsorption site of the molecule within the MOF. In our example here
with iso-structural Mg-MOF74 and Zn-MOF74, this behavior is even more
obvious, as Mg and Zn have similar electronic structures: the metal
centers have valence electrons of 3$s$ and 4$s$, respectively, except that
Zn has the additional fully occupied 3$d$ orbital electrons. Thus the
question arises as to what the physical reason is that determines the
frequency shift of the small molecules absorbed in the MOF.
\begin{table}
\caption{\label{frequency}Vibrational frequencies (cm$^{-1}$) of CO$_2$
physiadsorbed in MOF74. Higher accuracy is kept for the calculated $\nu_3$ for further detailed analysis.}
\begin{tabular*}{\columnwidth}{@{\extracolsep{\fill}}llccr@{}}\hline\hline
& system & sym. $\nu_1$ & bend $\nu_2$ & asym. $\nu_3$ \\\hline
& free CO$_2^\dagger$ & 1388 & 667 & 2349 \\
& Mg-MOF74 & --- & 658 & 2352 \\
exp. & Zn-MOF74 & --- & 658 & 2338 \\
& Co-MOF74 & 1380 & --- & 2340 \\
& Ni-MOF74 & 1382 & --- & 2340 \\\hline
& free CO$_2$ & 1298 & 646, 639 & 2288.5 \\
calc. & Mg-MOF74 & 1294 & 636, 630 & 2288.0 \\
& Zn-MOF74 & 1296 & 637, 632 & 2280.4 \\\hline\hline
\end{tabular*}
\raggedright$^\dagger$Taken from Ref.~\onlinecite{NIST}.
\end{table}
Several reasons may contribute to the frequency shift. For instance, by
analyzing the geometries of the absorbed CO$_2$, we noticed that the
absorbed CO$_2$ molecules have been distorted from their free molecule
geometry, exhibiting an off-center shift of the carbon atom, as well as
an elongated overall length of the molecule, as summarized in
Table~\ref{binding}(The slight nonlinear distortion of the CO$_2$ molecule is ignored here in this analysis). The change in the length of the molecule and the
asymmetric distortion can both affect the vibrational frequencies.
Beyond the effect of the change in the molecule geometry, the nearby
open metal center might also play a direct role in the frequency shift,
by attracting or repelling the nearby oxygen atom in the CO$_2$
molecules during the vibration. To understand these different
contributions, in the following we analyze the influences of each of
these factors on the frequency of the CO$_2$ vibrations separately
and summarize their contributions in Table~\ref{shift}.
\subsubsection{Change in molecule length}
We first analyze the effect of the changing molecule length. To do this,
we perform frozen-phonon calculations on the free CO$_2$ molecules fixed
to the lengths of the CO$_2$ absorbed in Mg-MOF (shown in the last
column of Table~\ref{binding}), while keeping the carbon atom at the
center of the molecule. With this geometry, the CO$_2$ asymmetric
stretch is shifted by --1.6~cm$^{-1}$. A similar calculation by setting
the CO$_2$ molecule length to the value of that absorbed in Zn-MOF74
yields a redshift in $\nu_3$ of --3.7~cm$^{-1}$. Note that overall the
CO$_2$ in Mg-MOF74 is elongated by 0.0003~\AA, while that in Zn-MOF74 is
elongated by 0.0009~\AA. The CO$_2$ with the longer length (in
Zn-MOF74) has a larger redshift of approximately --2.1~cm$^{-1}$ than
that in Mg-MOF74. That is, a longer molecule has more redshift, as
suggested by common sense.
\subsubsection{Asymmetric distortion of the molecule}
Next, we take the asymmetric effect into consideration by placing the
CO$_2$ at exactly the same geometries as they are in the two MOFs, while
removing the surrounding MOFs. The frequency shift thus comes from the
change in both the length of the CO$_2$ molecule and the asymmetric
shift of the carbon atom. Our frozen-phonon calculations give an overall
frequency shift of \mbox{--0.5} and --3.0~cm$^{-1}$ for the
$\nu_3$ mode of CO$_2$ from Mg-MOF74 and Zn-MOF74, respectively.
Subtracting the previous results of only considering the length of the
molecule, the asymmetric distortion of the carbon atom causes a slight
blue shift in the asymmetric stretch with a value of 1.1 and
0.7~cm$^{-1}$ in Mg-MOF74 and Zn-MOF74.
To confirm these results, we also perform calculations using another
setup, where we keep the optimized CO$_2$ within the MOF and shift the
carbon atom to the center of the CO$_2$ molecule. In this way, we
include the effect of the MOF environment and the length effect. The
only difference between this new setup and the optimized CO$_2$ in MOFs
is that the carbon atom off-center asymmetric effects are eliminated.
Our simulations show that the CO$_2$ in such a geometry has a frequency
shift of \mbox{--1.7} and --8.7~cm$^{-1}$ in Mg-MOF74 and
Zn-MOF74. Comparing these results with the frequency shifts for relaxed
CO$_2$ in the same MOFs, we find similar differences of approximately
1.2 and 0.6~cm$^{-1}$ for the case of Mg-MOF74 and Zn-MOF74,
which are consistent with our previous results. It is thus clear that
the carbon off-center distortion indeed causes slight blue shifts in the
$\nu_3$ frequency. This result can also be understood in the following
way: by shifting the carbon atom off-center, the CO$_2$ atom is
distorted asymmetrically in a similar pattern as the asymmetric stretch
mode, which thus favors the asymmetric stretch. As shown in
Table~\ref{binding}, the asymmetric distortion in Mg-MOF74 is stronger
than that in Zn-MOF74, which is consistent with a larger blueshift
effect in the CO$_2$ $\nu_3$ frequency.
\subsubsection{Effect of the metal center}
By comparing with the results for optimized CO$_2$ in the MOFs, the
previous results of the effects of the CO$_2$ molecule geometry change
on the $\nu_3$ frequency shifts yield the influence of the metal center
on the $\nu_3$ frequencies. A direct comparison gives a frequency shift
of --5.1~cm$^{-1}$ for the Zn-MOF74 and 0~cm$^{-1}$ for Mg-MOF74. This
result is quite a surprise, claiming that the open Zn center has a
strong redshift influence on the $\nu_3$ frequency, while the Mg center
has no effect at all.
To confirm this result, we place the undistorted CO$_2$ molecule at the
same position and angle of the adsorption site, with the position of the
oxygen atom near the metal center fixed, while the positions of the
carbon and the other oxygen atom shifted slightly along the line to
achieve the free CO$_2$ geometry. The frequency shift of the CO$_2$
asymmetric stretch thus results mostly from the direct effect of the
metal center. With this geometry, our calculation shows that, while the
asymmetric stretch of CO$_2$ in Zn-MOF74 is shifted by approximately
--5.0~cm$^{-1}$, the one in Mg-MOF74 has a negligible shift of about
--0.6~cm$^{-1}$, confirming our previous results. In other words, the
fact that the oxygen atom in CO$_2$ is being close to the open metal
center has a significantly different effect depending on the metal
atoms. While the Zn atom affects the frequency strongly, the Mg atom has
almost no effect at all. This result is quite striking, since the metal
centers Mg and Zn have a very similar valence electronic structure with
3$s$ and 4$s$ electrons as the outermost valence states. The result thus
shows that fully occupied semicore 3$d$ electrons in Zn have an important
effect in the interaction with the absorbed CO$_2$ molecules. In fact,
in Co-MOF74 and Ni-MOF74 where 3$d$ electrons are also present in the
metal center, experimentally we observed that the asymmetric stretch of
CO$_2$ has been similarly red-shifted by --10~cm$^{-1}$ as that in
Zn-MOF74, as summarized in Table~\ref{frequency} , indicating that the
3$d$ orbitals are indeed playing a similar role for the CO$_2$ frequencies
in these MOF74 systems.
The contributions of these three effects to the $\nu_3$ frequency shifts
are summarized in Table~\ref{shift}.
\begin{table}
\caption{\label{shift}Frequency shifts of $\nu_3$ (cm$^{-1}$) for
different geometries. See main text for details about these geometry.}
\begin{tabular*}{\columnwidth}{@{\extracolsep{\fill}}lcr@{}}\hline\hline
effect & Mg-MOF74 & Zn-MOF74 \\\hline
length & --1.6 & --3.7 \\
length+asym & --0.5 & --3.0 \\
length+metal & --1.7 & --8.7 \\
metal & --0.6 & --5.0 \\\hline
overall & --0.5 & --8.1 \\
\hline\hline
\end{tabular*}
\end{table}
\subsection{Microscopic insight and implication of bonding}
\begin{figure}[t]
\centering
\includegraphics[width=0.8\columnwidth]{Fig4_a.PS}\\[4ex]
\includegraphics[width=0.8\columnwidth]{Fig4_b.PS}
\caption{\label{Den}(Color online)Charge density differences $\Delta\rho=\rho_{MOF74+CO_2}-\rho_{MOF74}-\rho_{CO_2}$ from before and after CO$_2$ adsorbed into Zn-MOF74 (top) and in Mg-MOF74 (bottom).}
\end{figure}
In order to gain more insight understanding for the differences in the interaction between CO$_2$ and the two different MOF74s, we investigate the charge density differences before and after CO$_2$ adsorbed inside the MOF74. To do this, we calculate the charge densities of the CO$_2$ and MOF74 separately, by removing CO$_2$ from the MOF74, while keeping the atomic positions. The charge density difference is defined by
\begin{equation*}
\Delta\rho=\rho_{MOF74+CO_2}-\rho_{MOF74}-\rho_{CO_2}.
\end{equation*}
The charge density difference $\Delta\rho$ thus illustrates the effect of the interaction by placing the CO$_2$ inside the MOF74. The obtained charge density differences are plotted in Fig. \ref{Den}.
Not surprisingly, by introducing CO$_2$ inside the MOF74, the electrons in CO$_2$ are attracted toward the metal side, featuring electron deficiency in the far end (light blue color) and electron gaining near the metal site (yellow color). This can be understood via the attraction from the positively charged metal centers, similar for both Mg and Zn centers. However, one can also observe differences between the Zn-MOF74 and Mg-MOF74 systems, where the electron transfers more in the later than the former. As we look at the metal center, we can observe that near Zn atom, the electrons shift slightly away from the CO$_2$ adsorption position. This charge density change near the Zn centers are missing from those Mg center, clearly showing that it is an effect of the $d$-orbitals of the Zn atoms. It is thus clear that the presence of the $d$-orbitals prevents the charge density transfer within the adsorbed CO$_2$ molecules, leading to smaller charge transfer as compared with those near Mg center. As a result, the bonding energy between CO$_2$ and Zn-MOF74 is smaller than that of CO$_2$ in Mg-MOF74. On the other hand, this clear effect of the Zn $d$-orbital effectively affect the IR frequency shift of the adsorbed CO$_2$ molecule, leading to the differences between the two systems as we discussed above.
\subsection{Discussion}
From the previous analysis, it is now clear that the $\nu_3$ frequency
shifts of the CO$_2$ absorbed in MOF74 can be understood by the three
distinct contributions, namely, the change of the molecule length, the
off-center asymmetric distortion, and the direct effect of the open
metal site. The interaction between CO$_2$ and the MOF causes
distortions in the molecule geometry, which affect the vibration modes.
The absorbed CO$_2$ molecules experience a stronger asymmetric
distortion in Mg-MOF74, as shown in Table~\ref{binding}, which is
consistent with the larger bonding energy calculated, as well as the
larger integrated area for the asymmetric stretch IR peak. However, such
an asymmetric distortion only has a small effect on the $\nu_3$
frequency shifts. On the other hand, although CO$_2$ is less
asymmetrically distorted in Zn-MOF74, it experiences a larger elongation
in the overall length, which affects the $\nu_3$ frequency shift more
strongly. In addition, the nearby open metal site can play a quite
different role in affecting the CO$_2$ vibrations, where in this work,
the Zn affects strongly while Mg has barely any effect on the $\nu_3$
frequency shift. The differences between these two MOF74 can be intuitively understood by the presence of the $d$-orbitals within the Zn atoms and the missing of those within Mg centers, as illustrated in the charge density difference plots.
The results of this work provide insight to the factors that determine
the frequency shifts of the absorbed CO$_2$ in MOF, helping to
understand the puzzling frequency shifts observed experimentally. More
importantly, the analysis method of this work can serve as a new way to
understand the more widely examined molecule-MOF interactions and
frequency shifts. However, one must keep in mind that frequency shifts
obtained through such fixed geometries and environments reflect the
influence of different factors on the force matrix and can only give an
estimation of the influence of certain factors. In reality, originating
in the molecule-MOF interaction, all three factors are closely connected
intrinsically and it is impossible to exactly separate these different
effects.
\section{Summary}
In this work, we analyzed the physics determining the asymmetric
frequency shift of the CO$_2$ molecules physiadsorbed in MOFs. Our
specific findings are summarized as follows: (i)first-principles vdW-DF2 simulations determine that the CO$_2$'s have a closer distance to the Mg center and a larger binding energy within Mg-MOF74 comparing with those in Zn-MOF74. (ii) Contrary to our intuition, and despite the isostructure and the similarity of the open metals Mg and Zn, the
asymmetric stretch frequency of physiadsorbed CO$_2$ has been shifted stronger in Zn-MOF74 (--11~cm$^{-1}$ by IR and --8.1~cm$^{-1}$ by theory) than that in Mg-MOF74 (\mbox{$+3$~cm$^{-1}$} by IR and --0.5~cm$^{-1}$ by theory) .(iii) By comparing the response in two isostructure MOFs,
namely Zn-MOF74 and Mg-MOF74, we identified the three most important
factors contributing to the frequency shifts: the elongated CO$_2$
molecule, the off-center asymmetric distortion of the carbon atoms, and
the effect of the metal center. (iv) The asymmetric stretch frequency
is very sensitive to the overall length of the CO$_2$ molecule. Absorbed
in the MOF, the CO$_2$ molecules are elongated, which leads to a
redshift in the frequency. This elongation effect and resulting
redshift are more significant for CO$_2$ absorbed in Zn-MOF74 compared
with those in Mg-MOF74. (v) The slight off-center asymmetric
distortion, on the other hand, favors the asymmetric stretch and causes
a slight blueshift in the frequency. (vi) Aside from changing the
geometries of the CO$_2$ molecule (i.e.\ elongating the molecule,
causing off-center asymmetric distortion of the carbon atom) and
depending on the species of the open metal site, the direct interaction
of the oxygen atom with the metal center can have very different effects
on the frequency of the asymmetric stretch, where the Zn center leads to
a redshift of about --5~cm$^{-1}$ and the Mg center has a negligible
effect on the frequency. (vii) The observed different effects of the Zn-MOF74 and Mg-MOF74 can be understood by the presence of $d$-orbital electrons in the Zn-MOF74.
\section*{Acknowledgments}
We would like to thank David Langreth, the father of vdW-DF, for his inspirational research. We thank
Professors \ K.\ Rabe and D.\ Vanderbilt for very helpful discussions
throughout the whole project. This work was supported in full by the
Department of Energy Grant, Office of Basic Energy Sciences, Materials
Sciences and Engineering Division, Grant No. DE-FG02-08ER46491.
|
{
"timestamp": "2012-03-09T02:04:37",
"yymm": "1203",
"arxiv_id": "1203.1899",
"language": "en",
"url": "https://arxiv.org/abs/1203.1899"
}
|
\section[npgpgc]{Non perturbative gluon propagator and gluon condensates}
In QCD, polarization tensor is not
transverse in general, \(K^\mu \Pi_{\mu \nu}\left(K\right) \neq 0\).
For an $O(3)$ invariant gauge fixing condition, the most
general tensorial structure of the in-medium gluon self-energy can be
written as (we omit trivial color factor)~\cite{Heinz:1986kz}
\begin{equation}
\label{self_energy_general}
\pi_{\mu\nu} \left(\omega, k\right) = \pi_l\left(\omega, k\right)
\mathcal{P}^l_{\mu\nu} + \pi_t\left(\omega, k\right) \mathcal{P}^t_{\mu\nu} +
\pi_m \left(\omega, k\right) \mathcal{M}_{\mu\nu} +
\tilde{\pi} \left(\omega, k\right) l_{\mu\nu}\,.
\end{equation}
Here, $\omega$ and $k$ are the Lorentz invariant single particle energy and momentum respectively,
\begin{eqnarray}
\omega &=& u \cdot K\,, \nonumber \\
k &=& \left[\left(u \cdot K\right)^2 - K^2\right]^\frac{1}{2}\,,
\end{eqnarray}
and $u^\mu$ is the four velocity of the heat bath. In the rest frame of the medium $u^\mu = \delta^\mu_0$.
The projection operators are defined as~\cite{Weldon:1982aq,Heinz:1986kz},
\begin{subequations}
\begin{align}
\mathcal{P}^l_{\mu\nu} &= -\frac{1}{k^2 K^2}\left(k^2 u_\mu +
\omega\widetilde{K_\mu}\right)
\left(k^2 u_\nu + \omega\widetilde{K_\nu}\right) = \frac{K^2}{\widetilde{K}^2}
\bar{u}^\mu \bar{u}^\nu \,, \\
\mathcal{P}^t_{\mu\nu} &= \eta_{\mu\nu} - u_\mu u_\nu
- \frac{\widetilde{K_\mu} \widetilde{K_\nu}}{\widetilde{K}^2}\,,\\
\mathcal{M}_{\mu\nu} &= - \frac{1}{\sqrt{-2 \widetilde{K}^2}} \left(
\bar{u}_\mu K_\nu + \bar{u}_\nu K_\mu
\right)\,,\\
l_{\mu\nu} &= \frac{K_\mu K_\nu}{K^2}\,.
\end{align}
\end{subequations}
where \ensuremath{\widetilde{K_\mu} = K_\mu - \omega u_\mu} and $\bar{u}^\mu
= u^\mu - \frac{\omega}{K^2}K^\mu$. The projectors $\mathcal{P}_l^{\mu \nu}$
and $\mathcal{P}_t^{\mu \nu}$ are transverse with respect to $K^\mu$ and
$\mathcal{M}^{\mu \nu}$ satisfies a weaker condition
$K_\mu \mathcal{M}^{\mu \nu} K_\nu = 0$. The scalar structure functions in the self
energy are extracted through appropriate projections,
\begin{subequations}
\begin{align}
\pi_l &= \mathcal{P}_l^{\mu\nu} \pi_{\mu\nu}\,, \\
\pi_t &= \frac{1}{2} \mathcal{P}_t^{\mu\nu} \pi_{\mu\nu}\,, \\
\pi_m &= - \mathcal{M}^{\mu\nu} \pi_{\mu\nu}\,, \\
\widetilde{\pi} &= l^{\mu\nu} \pi_{\mu\nu}\,.
\end{align}
\label{pi_structures}
\end{subequations}
Now from (\ref{self_energy_general}), the most general form of the gluon
propagator $\mathcal{D}_{\mu\nu} = \mathcal{D}_{0,\mu\nu}\left(1 +
\pi_{\mu\nu} \mathcal{D}_{0,\mu\nu}\right)^{-1}$ can be written as,
\begin{eqnarray}
\mathcal{D}_{\mu\nu}
&=& - \frac{\mathcal{P}_{\mu\nu}^t}{K^2 - \pi_t} - \frac{2}{2\left(K^2
- \pi_l\right)\left(\xi^{-1}K^2 - \widetilde{\pi}\right) +
\pi_m^2} \nonumber \\
& \times & \left[\left(\xi^{-1} K^2 - \widetilde{\pi}\right)
\mathcal{P}^l_{\mu\nu} + \pi_m \mathcal{M}_{\mu\nu} + \left(K^2 - \pi_l\right)
l_{\mu\nu}\right]\,.
\label{non_tr_prop}
\end{eqnarray}
In covariant gauges, the Slavnov Taylor identity reads
$K^\mu \mathcal{D}_{\mu\nu} K^\nu = K^\mu \mathcal{D}_{0,\mu\nu} K^\nu =
-\xi$, we get
\begin{equation}
\label{sti_rel}
\frac{2\left(K^2 -\pi_l\right) K^2}{2\left(K^2 -\pi_l\right)\left(\xi^{-1} K^2 -
\tilde{\pi}\right) + \pi_m^2} = \xi\,.
\end{equation}
For $\xi \neq 0$. Eq.~(\ref{sti_rel}) can be written as,
\begin{equation}
\label{sti2}
\left(K^2 - \pi_l\right) \tilde{\pi} = \frac{\pi_m^2}{2}
\end{equation}
This leaves three independent components in (\ref{non_tr_prop}). So the
most general form of the `non perturbative' gluon propagator should be,
\begin{eqnarray}
D^{ab}_{\mu\nu} &\stackrel{\rm def}{=}& \mathcal{D}^{ab,{\rm }}_{\mu\nu} -
\mathcal{D}^{ab,{\rm pert}}_{\mu\nu}\nonumber\\
&=& P^l_{\mu\nu}
D_l\left(\omega,k\right) + P^t_{\mu\nu}
D_t\left(\omega,k\right) + \mathcal{M}_{\mu\nu} D_m\left(\omega,k\right)
\,, \label{non-pt-gluon-prop}
\end{eqnarray}
in an obvious notation. Note that, $D_m$ is absent in covariant gauges if
1) $\xi = 0$ (Landau gauge) or 2) the self energy is transverse $\pi_m =
\tilde{\pi} = 0$.
\begin{figure}[t]
\begin{minipage}[t]{\textwidth}
\centering{
\includegraphics[width=.50\textwidth,keepaspectratio]{defn_np_gp.pdf}
}
\end{minipage}
\caption{\label{non_pt_gluon_prop} Graphical representation of Eq.
\ref{non-pt-gluon-prop}. Full and nonperturbative propagators are denoted by circular and oval
blobs respectively.}
\end{figure}
Condensates of various dimensions can be related to the moments of
the non-perturbative gluon propagator.
In the rest frame of the medium (\ensuremath{u^\mu = \delta^\mu_0}),
the dimension two condensates are given by,
\begin{subequations}
\label{A2_condensate}
\begin{align}
\left\langle A_0^2 \right\rangle_T & =
- T\left(N_c^2 -1\right)\int\,\frac{d^3k}{\left(2\pi\right)^3}
D_l\left(0, k\right)\,, \\
\left\langle A_i^2 \right\rangle_T & = 2 T\ \left(N_c^2 -1\right)
\int\,\frac{d^3k}{\left(2\pi\right)^3} D_t\left(0, k\right)\,,
\end{align}
\end{subequations}
where $N_c$ is the number of color. We have restricted here to the lowest \
Matsubara mode ($k_0 = 0$) in the spirit of planewave method~\cite{Schaefer:1998wd,Chakraborty:2011uw}.
\section{Quark self energy}
\begin{figure}[!htbp]
\begin{minipage}[t]{\textwidth}
\centering{
\includegraphics[width=.25\textwidth,keepaspectratio]{fermion_self.pdf}
}
\end{minipage}
\caption{\label{quark_self_gluon_condensate} Gluon condensate contribution
to quark self energy.}
\end{figure}
The $D=4$ gluon condensate contribution to the quark self energy has been
studied in~\cite{Schaefer:1998wd}. Here we shall evaluate the $D=2$ condensate
contribution.
The general expression for in-medium fermion self energy in the chiral limit
is given by,
\begin{equation}
\Sigma \left(P\right) = - a\left(\omega,p\right) \slashed{P} - b\left(\omega,p\right)
\slashed{u}\,,
\end{equation}
where the scalar functions are given by,
\begin{subequations}
\begin{align}
a \left(\omega,p\right) &= \frac{1}{4p^2}\left[\mathrm{Tr}\left(\slashed{P}\Sigma\right)
- \omega \mathrm{Tr}\left(\slashed{u}\Sigma\right)\right]\,,\\
b \left(\omega,p\right) &= \frac{1}{4p^2}\left[P^2\mathrm{Tr}\left(\slashed{u}\Sigma\right)
- \omega \mathrm{Tr}\left(\slashed{P}\Sigma\right)\right]\,.
\end{align}
\end{subequations}
Using the plane wave method~\cite{Reinders:1984sr,Schaefer:1998wd} and
nonperturbative propagator from Eq.~(\ref{non-pt-gluon-prop}), we get from
Fig.~\ref{quark_self_gluon_condensate}
\begin{subequations}
\begin{align}
a\left(p_0,p\right) &= -\frac{2\pi\alpha_s}{N_c P^2} \left[\frac{1}{3}
\left\langle A_i^2 \right\rangle_T - \left\langle A_0^2 \right\rangle_T
\right]\,, \\
b\left(p_0,p\right) &= -\frac{2\pi\alpha_s p_0}{N_c P^2} \left[
2\left\langle A_0^2 \right\rangle_T +\frac{2}{3} \left\langle A_i^2 \right\rangle_T
\right]\,.
\end{align}
\end{subequations}
Note that, $a$ and $b$ contain nonperturbative contribution from
$D=4$ gluon condensate and other higher dimensional condensates
and perturbative contribution given by HTL corrections as,
\begin{eqnarray}
a &=& a^{\left\langle A^2 \right\rangle} + a^{\left\langle G^2 \right\rangle} + \cdots +
a^{\rm HTL} \,,\nonumber\\
b &=& b^{\left\langle A^2 \right\rangle} + b^{\left\langle G^2 \right\rangle} + \cdots +
b^{\rm HTL }\,.
\end{eqnarray}
Then, the chiral quark propagator \ensuremath{S^{-1}(P) = \slashed{P} - \Sigma}
follows as
\begin{eqnarray}
S\left(p_0,p\right) = \frac{\gamma_0 -\gamma\cdot\hat{\vec{p}}}{2D_+\left(p_0,p\right)} + \frac{\gamma_0 + \gamma\cdot\hat{\vec{p}}}{2D_-\left(p_0,p\right)}\,,
\end{eqnarray}
where,
\begin{equation}
D_{\pm}\left(p_0,p\right) = \left(-p_0 \pm p \right)\left(1 + a\right) - b\,.
\end{equation}
\section{Gluon self Energy}
\begin{figure}[!h]
\begin{minipage}[t]{\textwidth}
\hfill
\begin{minipage}[t]{.45\textwidth}
\begin{flushright}
\includegraphics[width=.5\columnwidth]{4gluon}
\end{flushright}
\end{minipage}
\hfill
\begin{minipage}[t]{.45\textwidth}
\begin{flushleft}
\includegraphics[width=.5\columnwidth]{3gluon}
\end{flushleft}
\end{minipage}
\end{minipage}
\caption{\label{3g_and_4g} Gluon self energy with non-perturbative gluon
propagator. }
\end{figure}
The gluon self energy with $D=2$ gluon condensate follows from
Fig.~\ref{3g_and_4g}. Various scalar structure functions in the self energy
as given in~(\ref{pi_structures}) are obtained as,
\begin{eqnarray}
\pi_l^{\left\langle A^2 \right\rangle} &=&
f_i \frac{\alpha_s}{\pi}
\left\langle A_i^2 \right \rangle_T -
f_0 \frac{\alpha_s}{\pi}\left\langle A_0^2 \right \rangle_T\,, \\
\pi_t^{\left\langle A^2 \right\rangle} &=&
g_i \frac{\alpha_s}{\pi}
\left\langle A_i^2 \right \rangle_T -
g_0 \frac{\alpha_s}{\pi} \left\langle A_0^2 \right\rangle_T\,, \\
\pi_m^{\left\langle A^2 \right\rangle} &=&
\sqrt{2} h_i \frac{\alpha_s}{\pi}\left\langle A_i^2 \right \rangle_T -
\sqrt{2} h_0 \frac{\alpha_s}{\pi}\left\langle A_0^2 \right
\rangle_T\,,\\
\widetilde{\pi}^{\left\langle A^2 \right\rangle} &=&
w_i \frac{\alpha_s}{\pi}\left\langle A_i^2 \right \rangle_T -
w_0 \frac{\alpha_s}{\pi}\left\langle A_0^2 \right
\rangle_T\,.
\end{eqnarray}
with
\begin{subequations}
\begin{align}
f_0 \left(p_0,p\right) & = \frac{4 \pi^2 N_c}{\left(N_c^2 -1\right)} \left\{
\left(-\frac{\vec{p}^2}{P^2} + \frac{4p_0^2}{P^2} + \frac{p_0^2\vec{p}^2}{P^4}
- \frac{4p_0^4}{P^4} - \frac{3p_0^4}{\vec{p}^2 P^2} + \frac{3p_0^6}{\vec{p}^2
P^4}\right) \right.\nonumber \\
& \left. + \chi \left( \frac{\vec{p}^2}{P^2} - \frac{2 p_0^2}{P^2} -
\frac{2 p_0^2 \vec{p}^2}{P^4} + \frac{4p_0^4}{P^4} + \frac{p_0^4}{\vec{p}^2
P^2} + \frac{p_0^4 \vec{p}^2}{P^6} + \frac{2p_0^6}{P^6}
\right. \right. \nonumber \\
& \left. \left. - \frac{2p_0^6}{\vec{p}^2 P^4} + \frac{p_0^8}{\vec{p}^2 P^6}
\right) \right\}\,, \\
f_i \left(p_0,p\right) & = \frac{2 \pi^2 N_c}{\left(N_c^2 -1\right)} \left\{
\left(-\frac{2\vec{p}^2}{P^2} + \frac{2p_0^2}{P^2} - \frac{4 \vec{p}^4}{3 P^4}
+ \frac{4 p_0^2 \vec{p}^2}{P^4} - \frac{8 p_0^4}{3 P^4} \right) + \chi \left(
\frac{p_0^2 \vec{p}^2}{P^4} + \frac{p_0^2 \vec{p}^4}{3 P^6}\right. \right. \nonumber \\
& \left. \left. - \frac{2 p_0^4}{P^4} -
\frac{5 p_0^4 \vec{p}^2}{3 P^6} + \frac{7 p_0^6}{3 P^6} + \frac{p_0^6}{\vec{p}^2
P^4} - \frac{p_0^8} {\vec{p}^2 P^6}
\right) \right\}\,,
\end{align}
\end{subequations}
\begin{subequations}
\begin{align}
g_0 \left(p_0,p\right) & = \frac{4 \pi^2 N_c}{\left(N_c^2 -1\right)} \left\{
\left(-1 + \frac{11 p_0^2}{2 P^2}\right) + \chi \left(-\frac{p_0^2}{P^2} +
\frac{p_0^4}{2 P^4} - \frac{p_0^4}{2 \vec{p}^2 P^2} -
\frac{p_0^6}{2 \vec{p}^2 P^4}
\right)\right\}\,\\
g_i \left(p_0,p\right) & = \frac{2 \pi^2 N_c}{\left(N_c^2 -1\right)} \left\{
\left(\frac{5}{2} + \frac{\vec{p}^2}{P^2} - \frac{11 p_0^2}{3 P^2}\right) +
\chi\left(\frac{\vec{p}^2}{6 P^2} +
\frac{p_0^2 \vec{p}^2}{6 P^4} + \frac{2p_0^4}{3 P^4} -
\frac{p_0^4}{2 \vec{p}^2 P^4} + \frac{p_0^6}{2 \vec{p}^2 P^4}
\right)\right\}\,
\end{align}
\end{subequations}
\begin{subequations}
\begin{align}
h_0 \left(p_0,p\right) & = \frac{4 \pi^2 N_c}{\left(N_c^2 -1\right)} \left\{
-\frac{p_0}{\left|\vec{p}\right|} +\frac{p_0^3}{\left|\vec{p}P^2\right|}
\right\}\,,\\
h_i \left(p_0,p\right) & = \frac{2 \pi^2 N_c}{\left(N_c^2 -1\right)} \left\{
\frac{p_0}{\left|\vec{p}\right|} + \frac{p_0 \left|\vec{p}\right|}{3P^2} -
\frac{p_0^3}{\left|\vec{p}P^2\right|}\right\}\,,
\end{align}
\end{subequations}
\begin{subequations}
\begin{align}
w_0 \left(p_0,p\right) & = 0\,,\\
w_i \left(p_0,p\right) & = 0.
\end{align}
\end{subequations}
Here $\chi = 1 - \xi$ and $\xi$ is gauge parameter.
At zero temperature, $k^\mu \pi^{\langle A^2 \rangle}_{\mu\nu} = 0 $. We find
that the transversality is weaker at finite temperature
$k^\mu \pi^{\langle A^2 \rangle}_{\mu\nu}k^\nu = 0$.
The $D=4$ condensate contribution to the gluon self energy will be
presented elsewhere~\cite{cmt:2011b}.
\section{Ghost self energy}
\begin{figure}[!htbp]
\begin{minipage}[t]{\textwidth}
\centering{
\includegraphics[width=.25\textwidth,keepaspectratio]{ghost_self_gluon.pdf}
}
\end{minipage}
\caption{\label{ghost_self_gluon_condensate}Gluon condensate contribution to ghost self energy}
\end{figure}
Similarly, $D=2$ gluon condensate contribution to ghost self energy in
covariant gauge can be evaluated from Fig.~\ref{ghost_self_gluon_condensate} as,
\begin{equation}
\Pi \left(p_0, p\right) = \frac{4 \pi \alpha_s N_c}{N_c^2 - 1} \left[
\frac{p_0^2}{P^2} \left\langle A_0^2 \right\rangle_T - \frac{1}{2} \left(
\frac{4}{3} - \frac{p_0^2}{P^2}\right) \left\langle A_i^2 \right\rangle_T\right]\,.
\end{equation}
There is no HTL correction for ghost, but ghost can receive non-perturbative
mass corrections in a background characterized by non-vanishing
condensates. One can similarly calculate Wilson coefficients of
$D=4$ gluon condensates to ghost self energy.
To fix
these coefficients uniquely, one has to go beyond one loop and calculate
Wilson coefficients of $D=4$ ghost-anti ghost
$\left(\bar{\eta} \Box \eta\right)$, and mixed ghost-gluon
condensate $\left(f^{abc} \partial_\mu \bar{\eta}^a A^{\mu,b} \eta^c\right)$.
This is similar to gluon self energy calculation at one
loop~\cite{Chakraborty:2011uw} and details will be presented
elsewhere~\cite{cmt:2011b}.
\section{Summary}
We have calculated $D=2$ gluon condensate contribution to quark, gluon and
ghost self
energies at finite temperature. With the values of condensates
taken as input from lattice QCD, one can quantitatively predict
the analytical structure of these propagators near the transition
temperature. As correlated applications, this will be a good starting point
to estimate nonperturbative dilepton rate, transport properties of both
heavy and light quarks etc. in the deconfined system of quark-gluon matter.
|
{
"timestamp": "2012-04-17T02:00:51",
"yymm": "1203",
"arxiv_id": "1203.2068",
"language": "en",
"url": "https://arxiv.org/abs/1203.2068"
}
|
\section{Introduction.}
The Lithium Purple Bronze, (Li$_{0.9}$Mo$_{6}$O$_{17}$) is
an unusual material which draws attention of research for nearly
three decades \cite{McC,Schlen}. It is a compound
with a rather complicated layered structure \cite{onoda} and
highly 1-dimensional (1D) electronic interactions
\cite{green,whang,popo}. Indeed band calculations with 3D
interactions show an electronic structure with 1D character
\cite{whang,popo}, and with large band dispersion in the direction
perpendicular to the layers. The band structure shows that two
bands are very close together near the crossing of the Fermi energy, $E_F$.
Because of its one dimensional electronic properties, the purple bronze is a
good playground to study the physics of one-dimensional interacting fermionic
systems, which is described by the Tomonaga-Luttinger liquid (TLL) universality class
\cite{QC}. In fact, several experimental results \cite{wang1,wang,hager,xue,gweon}
are consistent with the observation of such one-dimensional physics.
However some deviations from the simple TLL scaling are also
present. Recent results from angular resolved photoemission
spectroscopy (ARPES), done with very high resolution by Wang {\it
et al} \cite{wang1,wang}, indicate some deviations from TLL
predictions. The exponents for $\omega$ and $T$ dependence are
different from predictions from TLL theory, and yet another
exponent is deduced from scanning tunnelling measurements (STM)
\cite{hager}.
It is thus important to examine the possible sources from such
deviations. One possible origin, as with any quasi-one dimensional
system can come from the inter-chain coupling, which drives the
system away from the one dimensional fixed point. We determined in
another paper, starting from the limit of high energy, a
microscopic model incorporating the effects of interactions and
hopping between the chains \cite{cjg}. Such model applicable for
energies above $30$ meV gives results compatible with a Luttinger
liquid description.
In the present paper we focuss on the low energy limit, and
examine using Density Functional Theory (DFT) if additional
ingredients should be incorporated in this previous description
based on a \emph{rigid} band structure for the material. In
particular we examine whether the thermal motion of the atoms in
the solid can lead to a significant enough source of disorder that
could blur the 1D description. Such effects, in three
dimension, are indeed important for the properties of certain classes of compounds
with sharp structures of the density-of-states (DOS) near $E_F$,
such as in the B20 compound FeSi \cite{fesi}, and also for the appearance of the bands
in other materials \cite{wilk,giust,marini}.
We thus reexamine, in this light, in the present paper the band
structure of the purple bronze. In order to ascertain the effects
of the perpendicular hopping we compute carefully the dispersion
in the transverse direction. In addition we examine the effects of
thermal fluctuations and spin fluctuations.
The plan of the paper is as follows. In Sec.~\ref{sec:band} we
describe the method with the results for ordered structure. The
effect of deviations from the ideal atomic structure, due e.g.
from thermal fluctuations, are given in Sec.~\ref{sec:disorder}.
Section Sec.\ref{sec:stat-disor} is dedicated to the effects of
static, substitutional disorder. Results for spin fluctuations and
a discussion of their effects are given in Sec.~\ref{sec:spin}. In
Sec.~\ref{sec:arpes} we present models of how smearing and partial
gaps can modify photoemission intensities.
\section{Band structure.} \label{sec:band}
In this section we apply density functional band theory (DFT) in
order to see to what extent it can explain the unusual
photoemission data. In doing so it is important to note that
effects of thermal disorder and spin-fluctuations may be important
and have to be included in the density functional approach. The
electronic structure of Li$_2$Mo$_{12}$O$_{34}$ (two formula units
of stoichiometric purple bronze) has been calculated using the
Linear Muffin-Tin Method (LMTO, \cite{oka,arb}) in the local
density approximation \cite{lda}(LDA), with special attention to
effects of structural disorder. The lattice dimension and atomic
positions of the structure have been taken from Onoda {\it et al}
\cite{onoda}. The lattice constant in the conducting
$y$-direction, $b_0$, is 5.52 \AA and more than two times larger
along the least conducting $x$-direction $a_0=12.76$ \AA. Thus the
structure consist of well separated slabs. In order to adapt the
LMTO basis for an open structure as purple bronze we inserted 56
empty spheres in the most open parts of the structure. This makes
totally 104 sites within the unit cell. The basis consists of
s-,p- and d-waves for Mo, and s- and p-states for Li, O and empty
spheres, with one $\ell$ higher for the 3-center terms.
Corrections for the overlapping atomic spheres are included. All
atomic
sites are assumed to be fully occupied (except for the case with a
vacant Li atom, see section Sec.\ref{sec:stat-disor}), and they
are all considered as inequivalent in the calculations.
Self-consistency is made using 125 k-points, with more points for
selected paths for the band plots.
The band structure for the undistorted structure is shown in
Figs.~\ref{bndplt}-\ref{path5}. The total DOS
at $E_F$, $N(E_F)$, is not very large, 2.1 states per
cell and eV. The states at $E_F$ are mainly of Mo-d character
coming from sites far from the region containing Li, where locally
$N(E_F)$ amounts to about 0.2 st./eV/Mo. This is less than 1/3 of
$N(E_F)$ in bcc Mo.
Two bands, number 139 and 140 (counted from the lowest of the
valence bands), are the only ones crossing $E_F$. It was already
pointed out in Ref.\cite{whang} that these two bands originate
from zig-zag chains (oriented along y-axis) which are grouped in
pairs (along z-axis). This is the way each structural slab is
built. The two bands have similar DOS at $E_F$ to within $~$10
percent accuracy. The Fermi velocity, $v_F^y$, along the
conducting $y$-direction is normal as for a good metal, about
5$\cdot 10^5$ m/s, but the ratio between the Fermi velocity in y-
and z-directions, about 50, is compatible with the reported
anisotropic 1D-like resistivity \cite{green}. The velocity along
$\vec{x}$ is even smaller \cite{xyz}. The bands and
the DOS agree reasonably with the bands calculated by Whangbo and
Canadell \cite{whang} and Popovi\'{c} and Satpathy \cite{popo}
using different DFT methods. In Figs.~\ref{pathPK}-\ref{path5} we
show the details of the bands near $E_F$, where the band
separation and the electronic interactions (or $t$-integrals in a
tight-binding language) can be extracted. The band dispersion
along the conducting the $\Gamma-Y$ (or $P-K$) direction agree
well with the measured results obtained by ARPES \cite{wang} showing a
flattening of the two dispersive bands at about 0.4-0.5 eV below
$E_F$. Other bands are found about 0.25 eV below $E_F$. The
calculated structures A,B,C and D are identified on the bands in
the $P-K$ direction in Fig.~\ref{pathPK} at 0.6 (0.6), 0.4 (0.3),
0.6 (0.5) and 0.5 (0.4) eV below $E_F$, respectively, where the
values within parentheses are taken from the photoemission data of
Ref.~\onlinecite{wang}. Thus, there is an upward shift of the
order 0.1 eV in photoemission compared to the calculated bands.
Such trends are typical, for instance in ARPES on the cuprates,
and can to some extent be attributed to electron-hole interaction
in the excitation process \cite{bost}. The overall agreement
between different band results and photoemission is reassuring for
this complicated structure, while we now should focus on finer
details of the bands near $E_F$.
\begin{figure}
\includegraphics[height=6.0cm,width=8.0cm]{path25jan.ps}
\caption{Band structure of of purple bronze
Li$_2$Mo$_{12}$O$_{34}$ computed within DFT-LDA approximation for
the rigid structure with parameters taken from Ref.\cite{onoda}.
Bands are shown along symmetry lines in a 1eV window around the
Fermi energy $E_F$.} \label{bndplt}
\end{figure}
\begin{figure}
\includegraphics[height=6.0cm,width=8.0cm]{pathpk.eps}
\caption{Band dispersion along $PK$ (parallel to $\Gamma-Y$
halfway inside the zone) showing that Fermi momentum $k_F$ is very
close to half of the $P-K$ distance (at the vertical line). The
two bands crossing $E-F$ are very close for momenta just below
$k_F$ (for details see Fig.\ref{path5}) but they are separated by
$\sim$ 100 meV for $|k| \leq \frac{1}{2}|k_F|$.
chosen because the separation between the two bands (C and D) is
the largest for this value of $k_z$.
Notation as in
Ref.\cite{wang}; C and D corresponds to bands 139 and 140
respectively.} \label{pathPK}
\end{figure}
\begin{figure}
\includegraphics[height=6.0cm,width=8.0cm]{path5.ps}
\caption{Band dispersion of the two bands crossing $E_F$ shown
along the second conducting $z$-direction. Because of lattice
symmetry only a half of a dispersion curve ($k$-values in units
of 2$\pi$/$b_0$) is shown. Three different cuts of relevant portions of
the Brillouin zone are given.} \label{path5}
\end{figure}
The 1-D character of the band structure is revealed from the
comparison between Figs.~\ref{bndplt}-\ref{path5}. While the band
width for the bands crossing $E_F$ is of the order 2 eV in the
$y$-direction, it is not larger than 0.02 eV along $z$ and $x$.
These low values and the shape of the $z$- and $x$-dispersions are
of importance for a Luttinger description of bands close to $E_F$
\cite{cjg}. It should be noted that the in-plane band dispersion
is very small, and that the 2 bands have similar but not identical
shapes along the $z$-direction, see Fig.~\ref{path5}. The
dispersion is quite unusual with minimum in the middle of the
Brillouin zone instead at $q_c =0$. This suggests that there exist
not only one but several equally important, competing
hybridization paths between different chains. A brief analysis of
the structure\cite{xyz} indicates that several paths are possible.
First, for the closest pair of zig-zag chains there is a direct hopping between
Mo(1) and Mo(4) sites, and an indirect one going through Mo(2) and
Mo(5) atoms.
Secondly, the hopping between these pairs (of chains) always has to go
through Mo(2) atoms, while the next nearest neighbor hopping goes through
M(2) and M(5) atoms.
The direct inter-chain hopping (between Mo(1) and Mo(4) sites) is
weakened because it goes through rather weak $\delta$-bond, while
hoppings through M(2) and M(5) sites are enhanced because their
octahedra are more distorted. The cuts along different $k_b$ are
nearly identical, which shows that this competition does not
depend on the momentum of the propagating wavepacket. The
amplitude of the inter-chain hopping, as computed from band
structure shows that the one dimensionality of the compound is
certain for energies above $\sim 20$ meV, scale of the bare
inter-chain coupling. Below this energy the question is open, and
will be strongly dependent on how the transverse hopping can be
affected by strong correlations \cite{cjg}.
Another check was also made by calculating the bands for the
structural $a$-,$b$-, and $c$-parameters corresponding to low T.
It has been reported that the thermal expansion is quite unusual,
where almost no expansions are found along $b$ and $c$ between 0 K
up to room temperature \cite{santo,luz}. A calculation for the ordered
structure with all structural $a$-parameters downscaled by 0.3
percent, to correspond to the structure at 0 K, was made. The
effects on the bands are very small and not important for the
properties discussed in this work.
The average band separation decreases, but the change is
not significant compared to the original band separation of the
of order 30 meV.
\section{Thermal disorder and zero-point motion.} \label{sec:disorder}
The previous results were obtained by assuming an ideal periodic
structure. Band structure calculations rarely take into
account thermal distortions of the lattice positions. However,
structural disorder due to thermal vibrations is important at high
$T$, and properties for materials with particular fine structures
in the DOS near $E_F$ may even be affected at low $T$
\cite{fesi,ped}. Here for purple bronze, effects of thermal
fluctuations might be pertinent on the degree of dimensionality,
the band overlap between the two bands at $E_F$.
Phonons are excited thermally following the
Bose-Einstein occupation of the phonon density-of-states (DOS),
$F(\omega)$.
The averaged atomic displacement amplitude, $\sigma$, can be calculated
as function of $T$ \cite{zim,grim}. The result is approximately
that $\sigma_Z^2 \rightarrow 3\hbar\omega_D/2K$ at low $T$ due to zero point motion (ZPM)
and $\sigma_T^2 \rightarrow 3 k_BT/K$ at high $T$
(``thermal excitations''), where $\omega_D$ is a weighted average of $F(\omega)$. The force constant, $K=M_A\omega^2$,
where $M_A$ is an atomic mass (here the mass of Mo is used because of its
dominant role in the DOS), can be calculated as
$K = d^2E/du^2$ ($E$ is the total energy), or it can be taken from experiment.
We use the measurements
of the phonon DOS of the related blue bronze K$_{0.3}$MoO$_3$ \cite{requ} to estimate $K$ and the
average displacements of Mo atoms, as will be explained later.
The individual displacements $u$ follow a Gaussian distribution function.
\begin{equation}
g(u) = (\frac{1}{2\pi\sigma^2})^{3/2} exp(-u^2/2\sigma^2)
\label{Neffeq}
\end{equation}
where $\sigma$, the standard deviation, will be a parameter in the
different sets of calculations. In order to get an estimate to the
effect of such atomic displacements on the band structure, each
atomic site in the unit cell is assigned a random displacement
along $x,y$ and $z$ following the Gaussian distribution function.
Band calculations are made for a total of nine different
disordered configurations. The effects will be a shift of the band
position, to which both ZPM and thermal fluctuations contribute
and a broadening of the bands. Our
calculation, in which we will displace atoms from their natural
position, will also give information on the nature of the band,
and their sensitivity in touching a certain type of atoms.
For these investigations we have performed two types of
simulations of thermal effects: first including all atoms (treated
on equal footing) and second only for atoms around the zig-zag
chain where the DOS at $E_F$ is high. In the latter case, the
distortion, $\sigma_W$, is averaged over those sites only, see
later. These calculations confirm, as expected from the static
band structure, that mostly atoms around the zig-zag chains
contributes to observed effects. No general correlation between
the displacements of nearest neighbors is taken into account, but
extreme values of $u$ are limited in order to avoid that two
atomic spheres make a ``head-on'' collision, which of course would
not occur in the real material. Further refinements of the
disorder could involve different disorder for different atomic
mass, and anisotropic disorder. Purple bronze is a layered
material, 1D-like, and it is probable that vibrational amplitudes
are different perpendicular to the planes compared to within the
layers. However, such information is missing and here we assume
equal isotropic disorder for all atom types. The present
calculation already gives an estimate of the typical effects of
such atomic displacements.
\begin{figure}
\includegraphics[height=6.0cm,width=8.0cm]{bronzfig3.eps}
\caption{The difference of average shifts of the two bands between ordered and disordered structures
as function of the site weighted disorder parameter $\sigma_W$. The zero-point motion corresponds
approximately to $\sigma_w$ lower than 0.7
means that the two bands move closer for that disorder. Band broadening in the case for $\sigma_W \approx$ 0.7,
when there is no significant average shift, is only because of internal wiggling of the bands, see fig. \ref{fig4}.
}
\label{fig3}
\end{figure}
\begin{figure}
\includegraphics[height=6.0cm,width=8.0cm]{bronzfig4.eps}
\caption{Internal peak-to-peak energy difference between undistorted and distorted
bands (black * band 139, red + band 140). The green circles show the maximal difference
between the two previous values. The site weighted disorder parameter $\sigma_W$
is defined in the text.
}
\label{fig4}
\end{figure}
The broadening parameter $\sigma$ depends on $T$ and the
properties of the material. From the experimental data in blue
bronze \cite{requ} we estimate the $T$-dependence of $\sigma$ for
purple bronze. From this we find that $\sigma_{Z}/b_0$ is of the
order 0.7 percent for Mo, and thermal vibrations become larger
than $\sigma_Z$ from about 120 K. For oxygen sites $\sigma_Z/b_0$
is of the order 1 percent. However, as will be discussed later the
band structure is more sensitive to disorder on the Mo sites.
Two measures of the band disorder are shown in
Figs.~\ref{fig3}-\ref{fig4} as function of the weighted
displacements, $\sigma_W$, calculated as the average of
distortions for the 22 sites (far from Li) with the highest
$N(E_F)$. The broadening for each of the two bands is calculated
from the changes in band energies ($\epsilon(k)$) at 25 k-points,
which are found within $\sim$20 meV from $E_F$ in the undistorted
case. The difference between average energy shifts of the two
bands are shown in Fig.~\ref{fig3} as function of $\sigma_W$. This
can be thought as a good measure of the thermally induced effects.
There are more often positive energy differences, indicating that
in average the two bands tend to separate because of the atoms'
random displacements. The net effect is rather weak, we predict
only 10meV difference between room and helium temperature, but it
could be observable. What is more the thermal expansion of the
crystal in b-direction in anomalously weak, so it should not
affect the above result. This outcome is quite unusual, but the
origin of it becomes clear when one analyze the structure of a
crystal. It has been suggested \cite{whang} that $t_{\perp}$
hopping along c-axis is particularly weak in purple bronze because
of certain cancellation in hopping integrals (the $\delta$-bonds
mentioned in Sec.\ref{sec:band}), when the ions reside in a high
symmetry points. Our finding puts that statement on firm ground:
we clearly notice that when ions are slightly shifted the
perpendicular hopping can benefit noticeably.
We can gain even more information when studying the internal
behavior of the two bands separately. Measures of the "wiggling"
of each band because of disorder are displayed in Fig.~\ref{fig4}.
Peak-to-peak differences are given by
$$\Delta E_{max}^j = |max
[\Delta \epsilon^j(k_n)]-min[\Delta \epsilon^j(k_n)]|$$
$j$ = 139 or 140, where $\Delta \epsilon^j(k_n)$ is the deviation
in energy (with respect to undistorted band) at a certain point
$k_n$ for band $j$. The search for extremes goes through the
set of k-points $k_n$ ($n= [1,25]$). These differences of
extremes are shown for the two bands. For the largest disorder,
band 139 has changed some 100 meV at one k-point relative to the
change of the same band at some other k-point, however this quite
large value has to be taken with caution. In general the bands'
wiggling is of the same order as the average difference between
the two bands (shown on Fig. \ref{fig3}). We suspect that the
origin of wiggling is the same as the origin of increased band
separation, namely the activation of certain overlap integrals.
However, the fact that the wiggling is equally strong as the band splitting implies, at least
within naive tight-binding interpretation, that either the
inter-ladder hopping is equally strong or that the next nearest
neighbor hopping is of the same order as the standard $t_{\perp}$.
This second implication goes along the same lines than the ones
discussed in the context of Fig.~\ref{path5}.
The lower band appears to be more sensitive than the upper band in this respect. In Fig.~\ref{fig4} is
also shown the maximum difference,
$$\Delta\varepsilon = max|\Delta \epsilon^{139}(k_n)-\Delta \epsilon^{140}(k_n)|$$
It is defined as relative bands deviations at a certain point
$k_n$. In particular case when the two bands would change in the
same way because of thermal disorder (\emph{e.g.} shift
homogenously), then this last value would be equal to the absolute
value of what is shown in Fig.~\ref{fig3}. This is not the case,
which means that the two bands will change quite independently of
each other. In other words, most of the k-points are concerned by
the energy changes, and the wiggling of the two bands is
different. The interpretation of this subtle difference, not
visible within standard DFT band structure result, is quite
difficult. It implies that bands 139 and 140 do not have identical
site dependence. In other words one should not naively assume that
they originate from two zig-zag chains which are very weakly
hybridized as it is commonly done in the literature. The
microscopic complexity of effective interactions, at least at the
lowest energy scales of order $\sim 10meV$, needs to be rich.
In our calculation we have considered displacements of all sites,
but the bands at $E_F$ are mostly sensitive to disorder of Mo
sites with a high DOS. This is reflected by calculations in which
\emph{only} the four Mo with the highest DOS are displaced 0.7
percent of $b_0$ in a few selected directions. When these sites
are displaced perpendicular to the diagonal
($\vec{x},\vec{z}$)-direction the average band shift (equivalent
to the shifts displayed in Fig.~\ref{fig3} for general disorder)
is about 10 meV, while if the displacement is in the directions
parallel to the diagonal the shift is about 50 meV. The effect of
movements in the $\vec{y}$-directions is typically one order of
magnitude smaller. This indicates that the phonon mode whose
displacements are in the plane perpendicular to zig-zag planes
with the movements in the direction of the closest Mo neighbors,
is particularly strongly coupled to electronic liquid.
These values of energy changes can be compared with the
distance between the bands near $E_F$ in the undistorted case, shown
above in Fig.~\ref{path5} for the relevant paths in k-space. The
undistorted bands are separated by 30-40 meV, and the two bands
are confined within 10-15 meV along $k_x$ and $k_z$. The
additional wiggling for distorted cases is of the same order, 20-40
meV, as seen in Fig.~\ref{fig4}. However, this effect
coexists with a small increase of the band separation. The band separation,
will generally increase when thermal fluctuations are at play.
This is an indication of increased electronic delocalization
within the layers towards a more 3D band structure, but this also
indicates that any disorder will strongly affect carriers
propagation along c-axis. As the temperature increases the thermal
fluctuation effects will increase which means that, if this
picture is valid, the experimentally observed band broadening
should noticeably increase with temperature. This becomes
specially important when thermal activation of phonons becomes dominant at
about 120 K. In addition, the relatively rapid dispersion along
$k_y$ makes the band overlap important in k-space. For instance, a
band separation of 25 meV corresponds to a shift of $k_y$ of only
0.01 of the $\Gamma - Y$ distance.
\section{Static disorder}\label{sec:stat-disor}
In addition to the thermal effects, on which we concentrated in
the previous section, atomic distortions can come from a variety
of other sources. In particular imperfections of the crystal (from
non-stoichiometry, vacancies, site exchange etc.) should also
contribute to the effect. One can also worry about the fact that
one out of ten Li atoms are missing in the real material. The unit
cell considered here contains two Li, and the influence on the
electronic bands from the replacement of one of these with an
empty sphere (without taking structural distortion into account)
is moderately large. The Fermi level goes down because one
electron is removed, but if one neglects the chemical potential
shift he finds an increase of the average band separation between
band 139 and 140 by about 40 meV. This is comparable with largest
calculated effects of thermal disorder, see Fig.~\ref{fig3}.
However, a more realistic estimate is obtained in virtual crystal
calculations where one of the Li is replaced by a virtual atom
with nuclear and electron charges of 2.8 (instead of 3.0 for the
other Li). This set-up has the correct electron count
corresponding to 90 percent occupation of Li. Now the average
shifts of the two bands is much smaller, and the wiggling of the
bands are not even comparable ($\sim$ less than half) to what is
found from ZPM. Therefore it can be expected that effects from
thermal disorder will overcome those from structural disorder in
high quality samples.
The smallness of this effect has important implications if one
looks from the 1D Luttinger liquid perspective. The random Li
vacancies are placed relatively far from zig-zag chains where 1D
liquid resides. This implies that interaction will have Coulomb
character, the small momentum exchange events shall dominate. Thus
substitutional disorder will have primarily forward scattering
character with an amplitude $\approx 15meV$ as determined above.
This situation can be modelled as a Luttinger liquid with forward
disorder. In this case the spectral function is can be given
\cite{cjg}. This implies that substitutional disorder cannot be
invoked to explain phenomena taking place at energy scales larger
than 15 meV. For these larger energies the standard Luttinger
liquid behavior is expected.
\section{Spin fluctuations.} \label{sec:spin}
In Figs.~\ref{bndplt} and \ref{pathPK} it is seen that the two
free-electron like bands cross $E_F$ very close to half of the
$\Gamma-Y$- or $P-K$-distance along the conducting
$\vec{y}$-direction of the structure. A doubling of the real space
periodicity in this direction would open a gap in the DOS near
$E_F$ and lead to a gain in total energy
\cite{htc,Peierls,beni,pytte}, suggesting that this material might
have intrinsic spin density waves type instabilities. In strongly
correlated materials solving these question is of course
complicated, specially in a low dimensional material, since the
proper spin exchanges and quantum fluctuations have to be taken
into account. Some of these issues can be addressed at the level
of the microscopic model derived in Ref.~\onlinecite{cjg}. Here we
look at the possibility of such a spin instability, at the band
structure level and low temperatures, where the two- and three-
dimensional aspects of the system can a priori play a more
important role, and thus the effect of interactions can be reduced
\cite{QC}
A complete verification of how a cell doubling with phonons or
magnetic waves affects the band near $E_F$ would require more
complex band calculations, and will be a demanding undertaking.
Instead we propose at this stage to extract information from
calculations for the 104-site cell, and to apply a free electron
model of the band dispersion in the $y$-direction to see if at all
magnetic fluctuations might be of interest for a band gap. We
noted that displacements of some Mo with the highest local
$N(E_F)$ contribute much to the band distortion. But the size of
the potential shifts at these sites are limited by rather
conservative force constants. Therefore, even if a selected phonon
can contribute to a gap near $E_F$, it might be more effective to
open gaps through spin waves since the potential shifts in this
case can diverge near a magnetic instability. In order to estimate
the strength of the exchange enhancement on Mo we extend our
calculations for the ordered structure to be spin polarized with
an anti-ferromagnetic spin arrangement of the moments on the Mo
with the highest $N(E_F)$. This is made by application of positive
of negative magnetic fields within the atomic spheres of two
groups of four Mo in the cell with the highest DOS.
The propensity for fluctuations of anti-ferromagnetic moments
within the cell is surprisingly large according to the
calculations. The local exchange enhancement on Mo, corresponding
to the Stoner factor for ferromagnetism (i.e. the ratio between
exchange splitting and applied magnetic field), is close to 4.5 in
calculations at low field and temperature. The total energy $E(m)$
is fitted to a harmonic expansion of the moment amplitude $m$,
$E_m = K_m m^2$, where $K_m$ is the ``force constant''. In analogy
with phonon displacement amplitudes one can estimate $m^2 =
k_BT/K_m$ as a measure of the amplitude of moment fluctuations
\cite{htc2}. While phonon distortions $u$ always increase
with $T$, in a Fermi liquid framework one expects that magnetic
moments $m$ will be quenched at a certain $T$ \cite{htc}. This is
because of a self-supporting process where $m$ is a function of
the exchange splitting $\xi$. The latter depends on $m$ and the
local spin density, and the mixing of states above and below $E_F$
given by the Fermi-Dirac function will reduce $m$ at high $T$, so
$m$ and $\xi$ can drop quite suddenly. Calculations at an
electronic temperature of 200K make $K_m \approx 2 eV/\mu_B^2$.
>From this one can estimate that $m$ will be of the order 0.1
$\mu_B$ per Mo at room temperature corresponds to $\xi \approx 0.2
eV$.
Another complication is that the thermal disorder of the lattice
are mixing states across $E_F$ too, and this is another
diminishing factor for spin polarization. Nevertheless, it
suggests that spin fluctuations can play a role at low or
intermediate temperatures. It is interesting to note that the
value of $\xi$ is of the same order of magnitude than the
superexchange parameter calculated from the strong correlations
perspective \cite{Nishi,cjg}. If one compares this value with the
previously determined strength of disorder and effects of thermal
fluctuations, one would realize that spin fluctuations play a much
more prominent role in the low energy physics. Such fluctuations
can potentially lead to pseudogap features in the band dispersion,
thus understanding better their properties is an interesting
challenge, clearly going way beyond the scope of this paper. On
the experimental side, there was only one report of such large gap
($\Delta >10meV$) in the low energy spectrum \cite{xue} and in the
light of more recent experiments \cite{gweon} done with the same
method this finding is highly controversial. An extremely small
gap has been observed in transport experiments \cite{Hussey}, and
probably in STM \cite{hager}, however the spin sector probed by
static susceptibility \cite{green,choi} and muon spectroscopy
\cite{Mandrus} certainly is not gapped. Further theoretical studies are
necessary to understand this situation, where high magnetic
propensity does not lead to a gap for spin excitations.
\section{Comparison with ARPES} \label{sec:arpes}
We show in this section some of the consequences of the band
structure calculated above for the ARPES data. Note that the above
calculation do not take into account the effects of strong
correlations. Thus the deviations from the picture presented below
should thus be direct measure and consequence of such effects.
This will of course depend on the range of temperature and/or
energy.
In order to simulate ARPES intensities for the free-electron bands near $E_F$ we ignore
matrix elements and energy relaxations due to electron-hole interaction. We consider
one-particle excitations directly from the band occupied according to the Fermi-Dirac
distribution, and with pyramidal broadening functions for energy (experimental and intrinsic broadening
due to disorder) and momentum, $k$. The experimental broadenings have the same FWHM values
as in the work of Wang {\it et al}, and we chose to show cuts in momentum of the
same step size as in their work \cite{wang}. The free electron band is fitted to the LMTO
results so that $E_F$ is 0.6 eV at $k_F$. The splitting into
two bands of $\sim$30 meV,
is assumed constant everywhere, which is approximately true for the real bands near $E_F$.
A linear background is added to the intensities in order to make a more
realistic display of photoemission with secondary excitations. Five cuts in momentum below $k_F$,
in equal steps as in Ref.~\onlinecite{wang},
descend to about -0.2 eV. The five cuts appear almost equally spaced in energy, since the
the dispersion is almost linear within the narrow energy interval.
First, as shown in Fig.~\ref{inten1}
the band splitting of about 30 meV should be visible in the ARPES data if only the experimental
broadening functions were at work. Secondly, the broken lines in Fig.~\ref{inten2} show
wider distributions, because of additional broadening coming from the thermal disorder
of the lattice. These distributions are not as wide as in Ref.~\cite{wang}. This strongly
suggests that other physical effects (effects of interactions, static disorder due to non-stoichiometry, spin fluctuations)
are at play.
As an extreme case we can consider a cell doubling (e.g. due to
spin ordering) which would create a gap near $k_y = k_F$. The
full lines in Fig.~\ref{inten2} show what would happen to the
spectra if $\xi$ in the free-electron model goes up to about 70
meV. In this case it is seen that the band retracts from $E_F$,
since the peaks for $k$ closest to $k_F$ are lower in energy than
the broken lines. Far below $E_F$ there is not much difference
between full and broken lines. In the end the presence of a gap
leads to a more non-uniform energy distribution of the five k-cuts
of the intensity. Comparison of the experimental features, with
the one obtained by such a band structure analysis could thus be
useful to investigate the low energy properties of purple bronze,
and in particular the existence of a gap or pseudogap in the
dispersion relation. The analysis in Ref.~\cite{wang} revealed a
$T^{0.6}$ scaling of the intensity near $E_F$ over a wide
$T$-interval. Ascertaining whether this behavior at low energy
comes from one dimensional fluctuations or some pseudogap regime
is an important question.
\begin{figure}
\includegraphics[height=6.0cm,width=8.0cm]{intensity1.eps}
\caption{Free electron intensities for 5 $k$-values going from $k_F$ in steps of 3.6 percent
of $\Gamma-Y$ added to an arbitrarily chosen background.
The energy and momentum resolutions $\Delta E$ and $\Delta k$ are similar to the experimental values in
ref. \cite{wang}. It is seen that a band splitting of more than $\sim$30 meV should be seen in
photoemission.
}
\label{inten1}
\end{figure}
\begin{figure}
\includegraphics[height=6.0cm,width=8.0cm]{intensity2.eps}
\caption{Calculated intensities for a band splitting of 30 meV as in fig. \ref{inten1}, but
where an band broadening from ZPM of $\sim$25 meV has been included (broken lines). One intermediate
cut with $k$ at 1.8 percent from $k_F$ is added in this figure. The band broadening
makes the band splitting undetected. Full lines: Including spin fluctuations as described in the text. Note
the energy lowering close to $E_F$.
}
\label{inten2}
\end{figure}
\section{Conclusion.}
In this paper we have reexamined the band structure of purple
bronze in the ordered lattice and found that the main features of
the bands agree well with previous calculations and ARPES results.
In particular, only two bands cross the Fermi level. In our study
we focus on these two bands. The perpendicular interactions along
$z$ and $x$ (``perpendicular hoppings'') are weak and appear to
lay out the conditions of TLL-like behavior of the bands near
$E_F$, for energy down to at least $20$ meV.
We have investigated whether additional effects, giving small
energy distortions at these energy scales could cause an overlap
between these two bands at low $T$. The energy separation between
the two essentially one dimensional bands increases below $E_F$,
as can be seen in Fig.~\ref{bndplt} along the $\Gamma - Y$
direction. For instance, the two considered bands are separated
$\sim 0.1$ eV at 0.4 eV below $E_F$, similarly to what is seen in
ARPES at this energy \cite{wang}. When bands are closer to $E_F$
then they are also closer together, however as shown in
Sec.\ref{sec:arpes} within single particle picture (LDA-DFT) they
should be distinguishable at low $T$ provided that intrinsic
disorder is not too large and if the matrix elements for
transitions from both bands are comparable \cite{matr}. As $T$
increases beyond $\sim$120 K or more, the band broadening
increases because of additional thermal disorder so that the two
bands seem to merge. We find that the thermal effects can
potentially play a role in the band separations however one has to
keep in mind the temperature dependence of the effect: the bands
becomes less and less distinguishable as the temperature
increases. There is obviously a complicated cooperation between
increase in splitting and wiggling but in any case the following
effect can be used to verify experimentally this picture: as the
temperature increases the ARPES lines should become noticeably
thicker.
Additional calculations for Li deficient purple bronze show no
important modifications of the bands crossing $E_F$, at least as
long as it is not associated with static disorder. Likewise the
band structure for a unit cell with modified $x/y$- and
$x/z$-ratios, appropriate for the structure at room temperature,
is not very different from what is shown in Fig.~\ref{bndplt}.
Recent photoemission result of Wang {\it et al} \cite{wang}, made
at low $T$, only detects one band at $E_F$. The interpretation of
these result is still an exciting issue given the fact that
electron-electron interactions can also suppress tunnelling and
reinforce the one dimensionality of the material \cite{cjg}. What
we showed in this paper is that at a one body level, the effect of
distortion of the electronic structure, also go in the direction
of a smearing of the difference between the two bands and
thickening observed ARPES spectra. Disentangling the two effects
is thus an important question, and can potentially be done by
comparing the predicted band separations, at the level of band
structure, with the actual ARPES data.
We have also explored the possibility of anti-ferromagnetic spin
fluctuations. Here, our investigations are not complete, but our
first results show surprisingly large anti-ferromagnetic exchange
enhancements on Mo within the basic unit cell. This and other
facts motivate further studies of fluctuations within larger
cells, in order to see if they can lead to gap or pseudogap
features near $E_F$.
It is interesting to note that the existence of thermal
fluctuations and imperfections of the atomic structure, makes --
at the band structure level -- the system more three dimensional.
This effect will be in competitions with the renormalization of
the inter-chain hopping coming from the electron-electron
correlations. The competition between these two effects leads to
an intermediate and low temperature physics which is still
mysterious in this compound.
Aknowlegments: We would like to thank Jim Allen and Enric Canadell for shearing their knowledge
about purple bronze with us. We also aknowlege an additional insight provided by Sachi Satpathy, Tanusri Saha-
Dasgupta and Maurits W. Haverkort. This work was supported by the Swiss NSF under
MaNEP and Division II.
|
{
"timestamp": "2012-03-09T02:03:20",
"yymm": "1203",
"arxiv_id": "1203.1827",
"language": "en",
"url": "https://arxiv.org/abs/1203.1827"
}
|
\section{Introduction}
Three-dimensional (3D) topological insulators (TIs) are characterized by
a novel topological order \cite{Moore, Fu-Kane, Roy, QHZ} which dictates
the appearance of spin-filtered massless Dirac fermions on the
surface.\cite{RMP_TI_10, Moore_Nature10, Qi_RMP11} To experimentally
address the peculiar physics of 3D TIs, it is desirable to access the
Dirac point of the surface state (SS).\cite{RMP_TI_10, Moore_Nature10,
Qi_RMP11} This is relatively easy with the surface-sensitive
spectroscopies such as the angle-resolved photoemission
\cite{Hsieh_Nature08, NishidePRB10, Xia_Nphys09, SatoPRL10,
Hiroshima_TBE_PRL10, ChenPRL10, Xue-NatCom, Arakane_NComm12} and the
scanning tunneling microscope,\cite{HanaguriPRB10, PCheng_PRL10} but it
is more challenging for the bulk-sensitive transport experiments because
the chemical potential is always pinned to the bulk bands (including the
impurity band) in real materials. To tune the chemical potential to a
desirable position for transport experiments, two approaches have been
employed: one is the tuning of the chemical compositions upon
synthesizing crystals,\cite{HorPRB09, RenPRB10, Taskin_BSTS_PRL11,
Ren_BSTS_PRB11, Ren_Cd_PRB11, Jia_PRB11} and the other is the gating to
control the surface carriers.\cite{ChenJ_PRL10, Steinberg_NL10,
CheckelskyPRL11, Kong_NNano11, Yuan_NL11} Among the latter approach, the
electric-double-layer gating (EDLG) method is a promising new technique
\cite{Yuan_NL11} to allow application of a large electric
field.\cite{Dhoot}
In the EDLG configuration, either cations or anions in a liquid
electrolyte are accumulated near the surface of a sample by application
of an electric field, and they form an electric double layer which
generates a very strong electric field locally on the surface. This leads
to the induced surface carrier density of as high as $\sim$10$^{15}$
cm$^{-2}$.\cite{Dhoot} Such a large tunability of the surface carrier
density is the merit of the EDLG method. In the context of TIs, Yuan
{\it et al.} showed \cite{Yuan_NL11} that the EDLG method allows an
ambipolar doping control in ultrathin films of Bi$_2$Te$_3$; however,
the surface state was obviously gapped in the ultrathin films used in
Ref. \onlinecite{Yuan_NL11}, and consequently the transport properties
were likely to be dominated by the bulk state. In the present paper, we
show that the EDLG method can achieve ambipolar transport even in bulk
single crystals, if one uses the highly bulk-insulating
Bi$_{2-x}$Sb$_{x}$Te$_{3-y}$Se$_{y}$ (BSTS)
system.\cite{Taskin_BSTS_PRL11, Ren_BSTS_PRB11} In our experiment, the
control of the chemical potential was possible on the {\it whole surface
of a bulk 3D sample}, opening new experimental opportunities for TIs. In
addition, our data suggest that the chemical potential is moving not
only at the surface but also in the bulk, possibly due to a nearly
reversible electrochemical reaction to cause apparent bulk doping during
the EDLG process.
\section{Sample Preparations and measurements}
\begin{figure*}
\includegraphics*[width=11cm,clip]{EDL_Fig1.eps}
\caption{(color online)
(a) Schematic picture of the experimental setup. Actual ionic liquid has
no color. (b)-(e) Changes in parameters with time in the procedure of
applying the gate voltage $V_G$ with the target value of $-3.5$ V on a
BSTS sample. First, $V_G$ is gradually changed from 0 to $-3.5$ V while
keeping the temperature at 220 K, and we wait for at least 10 min for the
electric double layer to develop; the system is then cooled down, and the
gate current $I_G$ vanishes completely after the ionic liquid
solidifies. (f)-(h) Changes in $T$, $I_G$, and the integrated charge in the
relaxation process from $V_G$ = $-3.5$ V, in which $V_G$ is set to 0 V and
the system is warmed to $\sim$240 K. (i) $V_G$ vs $I_G$ curve at 220 K
with the voltage sweep rate of $\sim$$-0.5$ V/min.}
\end{figure*}
A series of BSTS single crystals were grown by a modified Bridgman
method.\cite{Taskin_BSTS_PRL11, Ren_BSTS_PRB11} Six gold wires were
attached to each sample by a spot welding technique, and magnetotransport
measurements were performed by a conventional ac six-probe method by
sweeping the magnetic field between $\pm$9 T. An electrically insulating
cup made of Stycast 1266 was used as a sample container, in which the
sample with gold wires was submerged into an ionic liquid (IL)
electrolyte, as schematically shown in Fig. 1(a). We used a specially
purified ionic liquid [EtMeIm][BF$_4$] as the electrolyte.\cite{IL} As
the gate electrode, an additional piece of gold wire was dipped into the
IL without touching the sample. In this paper, we adopt the convention
of defining the gate voltage by taking the sample as the reference
point, as was done in most of the previous works. In this convention,
$n$-type carriers are supposed to be doped to the surface when a
positive gate voltage is applied.
The gate voltage was applied in the following procedure: first, the
temperature $T$ of the sample was stabilized at 220 K; then, the gate
voltage $V_G$ was swept slowly, and the temperature was kept at 220 K
for at least 10 min after the voltage reached the set value;
finally, the sample was cooled down slowly. When changing the gate
voltage, we cycled the above procedure. Figures 1(b)-1(e) show an
example of the history of $T$, $V_G$, gate current $I_G$, and the
accumulated charge, during the procedure for the target $V_G$ of $-3.5$
V. With the IL used in the present experiment, gate current was never
detected at temperatures below 200 K, even when the gate voltage was
changed. This is because when the IL solidifies, ions are not mobile at
all.
The accumulated ions can be released when the gate voltage is set to
zero and the system is warmed up to 220 K. Such a relaxation process
from $V_G$ = $-3.5$ V is shown in Figs. 1(f)-1(h). One can see that the
amount of released charge [$\sim$32 $\mu$C in Fig. 1(h)] is comparable
to that of the accumulated charge [$\sim$34 $\mu$C in Fig. 1(e)]. The
small difference between the released and accumulated charge is probably
an indication of an irreversible electrochemical reaction during the
gating processes at a high gate voltage. In the above example, the total
surface area of the sample was 13.4 mm$^2$, so that the $\sim$32 $\mu$C
of charge accumulated on the surface corresponds to the accumulated ion
density of $\sim$1.6$\times 10^{15}$ cm$^{-2}$ and the capacitance of
the unit area 80 $\mu$F cm$^{-2}$, which is comparable to that
previously reported for EDLG.\cite{YuanZnO} Figure 1(i) shows $V_G$ vs
$I_G$ curve at 220 K, which indicates that there is a threshold $|V_G|$
of $\sim$2 V below which little current flows, suggesting that the ions
are not mobile below this threshold voltage; similar behavior was
previously reported for a different IL.\cite{Ueno} The origin of this
behavior is currently not clear, but it may well be a characteristic of the
ionic liquid used here. Also, it was difficult to obtain reproducible
results of the transport properties for $|V_G|$ between 2 and 3 V,
probably because the formation of the electric double layer is unstable
in this gate-voltage region. Therefore, we closely measured the
transport properties only in the range of $|V_G|\ge 3$ V and $V_G$ = 0
V.
\section{EDLG experiment on $\mathbf{Bi_{1.5}Sb_{0.5}Te_{1.7}Se_{1.3}}$ single crystals}
\begin{figure}[t]
\includegraphics[width=8.5cm,clip]{EDL_Fig2.eps}
\caption{(color online)
(a),(b) Temperature dependences of the resistivity of a
Bi$_{1.5}$Sb$_{0.5}$Te$_{1.7}$Se$_{1.3}$ single crystal
for $V_G=-3.5$, 0, and +3.5 V.
(c) Magnetic-filed dependences of $\rho_{yx}$ at 1.8 K.
In these three figures (a)-(c), the dotted, solid, and broken lines correspond to
$V_G$ of $-3.5$, 0, and +3.5 V, respectively.
(d) Fitting of the two-band conduction model to the $\rho_{yx}(H)$ data
at $V_G$ = 0 V, which yields $\rho_{\rm bulk}$ = 3.2 $\Omega \,{\rm
cm}$, $n_{\rm bulk}$ = 9.1 $\times$ 10$^{16}$ cm$^{-3}$, $\mu_{\rm
bulk}$ = 29 cm$^2$/V$\,$s, $\rho_{\rm SS}^{\rm 2D}$ = 8.1 ${\rm
k}\Omega$, $n_{\rm SS}$ = 1.6 $\times$ 10$^{12}$ ${\rm cm}^{-2}$, and
$\mu_{\rm SS}$ = 4.7 $\times$ 10$^{3}$ cm$^2$/V$\,$s.
The gray thick line represents the experimental result and the solid line
is the fitted result.}
\label{rhochiplot}
\end{figure}
First, we show the results of the EDLG experiments on
Bi$_{1.5}$Sb$_{0.5}$Te$_{1.7}$Se$_{1.3}$, which is currently one of the
most bulk-insulating TI materials. \cite{Taskin_BSTS_PRL11,
Ren_BSTS_PRB11} Figure 2(a) shows the temperature dependences of the
resistivity $\rho$ with the gate voltage ($V_G$) of $-3.5$, 0, and +3.5
V. Even though this is a bulk single crystal, at low temperatures $\rho$
presents a clear change upon application of $V_G$; namely, below
$\sim$50 K, $\rho$ changes with changing $V_G$, and it tends to decrease
(increase) for positive (negative) $V_G$. Figure 2(b) magnifies this
change for $T \le$ 20 K. Figure 2(c) shows the magnetic-field
dependences of the Hall resistivity, $\rho_{yx}(H)$. The sign of the
charge carriers remains negative in this sample, although a significant
non-linearity suggests the coexistence of surface and bulk conduction
channels. Note that the surface conduction in TIs can involve both the
topological surface states and topologically-trivial two-dimensional
electron-gas (2DEG) states that appear as a result of band bending.
\cite{Bianchi} In the present experiment, when the gate
voltage is applied, the 2DEG states are most likely contributing to the
conduction alongside of the topological surface states. Unfortunately,
we have not been able to elucidate the contributions of the topological
and non-topological surface states because no Shubnikov-de Haas (SdH)
oscillation has been observed in our gated samples. (To successfully
separate the contributions of the two, detailed information obtained
from SdH oscillations is necessary.\cite{Taskin_BSTS_PRL11,Taskin_MBE})
We therefore make no claim of the composition of the surface carriers in
the present paper.
In Fig. 2(c), one can see a clear tendency that both the absolute value
of $\rho_{yx}$ and the slope of $\rho_{yx}(H)$ decrease upon increasing
$V_G$ from $-3.5$ to +3.5 V; this means that the apparent electron
concentration increases with increasing $V_G$. Since $n$-type carriers
are expected to be doped when a positive $V_G$ is applied, the present
observation can be understood as a natural consequence of the EDLG. The
nonlinear $H$-dependence observed in $\rho_{yx}$ is useful for gaining
insights into the respective roles of bulk and surface transport
channels, because its analysis based on a simple two-band model
\cite{RenPRB10} gives a crude idea about the relevant transport
parameters. For example, the $\rho_{yx}(H)$ data at $V_G$ = 0 V give the
following estimate based on the fitting shown in Fig. 2(d):
For the bulk channel, the bulk resistivity
$\rho_{\rm bulk} \simeq$ 3 $\Omega \,{\rm cm}$, the bulk carrier density
$n_{\rm bulk} \simeq$ 9 $\times$ 10$^{16}$ cm$^{-3}$, and the bulk
mobility $\mu_{\rm bulk} \simeq$ 30 cm$^2$/V$\,$s; for the surface
channel, the sheet resistance $\rho_{\rm SS}^{\rm 2D} \simeq$ 8 ${\rm
k}\Omega$, the surface carrier density $n_{\rm SS} \simeq$ 2 $\times$
10$^{12}$ ${\rm cm}^{-2}$, and the surface mobility $\mu_{\rm SS}
\simeq$ 5 $\times$ 10$^{3}$ cm$^2$/V$\,$s. In this fitting, the
constraint imposed by the presence of sharp kinks at $\sim\pm$1 T helps
reduce the ambiguity in the fitting parameters, and in fact, the three
parameters, $n_{\rm bulk}$, $\mu_{\rm bulk}$, and $n_{\rm SS}$, are in
reasonable agreement with our previous transport studies of BSTS
involving SdH oscillations.\cite{Taskin_BSTS_PRL11, Ren_BSTS_PRB11} The
large value of $\mu_{\rm SS}$ would imply that the SdH oscillations be
observed, but we did not observe any SdH oscillations in this sample;
this is possibly because the surface chemical potential (and hence
$n_{\rm SS}$) is not very uniform throughout the sample and the SdH
oscillations with various different frequencies add up to smear visible
oscillations.
The above result of the two-band analysis suggests that the surface
contribution in the total conductance was only $\sim$1\%, which is
reasonable because the measured sample had a considerable thickness of
332 $\mu$m. Nevertheless, if one looks at the EDLG effect in resistivity
[Figs. 2(a) and 2(b)], one notices a very puzzling fact: for the
negative $V_G$ of $-3.5$ V, the number of surface electrons are expected
to be reduced and indeed, the slope of $\rho_{yx}$ gets larger; however,
the resistivity increase is as much as 25\%. Since the surface
contribution in the total conductance is only $\sim$1\%, even when the
surface conduction is completely suppressed by EDLG, one can expect an
increase in $\rho$ of $\sim$1\% at most, as long as the bulk channel is
not affected by EDLG. Therefore, the observed large increase in $\rho$
strongly suggests that the bulk channel must also be affected by EDLG.
Indeed, the $\rho_{yx}(H)$ data shown in Fig. 2(c) presents a clear
change in the slope at high fields for different $V_G$, which implies
that not only $n_{\rm SS}$ but also $n_{\rm bulk}$ is changing. To
corroborate this inference, the two-band analyses of the $\rho_{yx}(H)$
data at finite $V_G$ suggests that the bulk carriers decreases
(increases) by 5\% (10\%) for $V_G$ of $-3.5$ V ($+3.5$ V).
It is very surprising that the EDLG affects the density of bulk carriers
by a noticeable amount in a sample as thick as 332 $\mu$m, but our
transport data can hardly be understood if one does not accept this
possibility. Given that the electric field generated by $\sim$1 $\times$
10$^{15}$ cm$^{-2}$ of ions on the surface is shielded in less than 100
nm,\cite{penetration} the only possibility is that some bulk doping into
the BSTS sample is taking place during the EDLG process. In this
respect, the slow time scale of the change in the measured current
during the EDLG process seems to support the idea that some
electrochemical reaction is taking place.
In passing, we note that we have not successfully measured SdH
oscillations in any of the gated samples. This is likely to be due to an
inhomogeneous distribution of the local electric field (which is
conceivable because our samples have a lot of macroscopic terraces on
the surface) or some chemical degradation of the surface caused by the
ionic-liquid gating. Hence a more definitive analysis of the transport
data beyond the simple two-band analysis is currently unavailable.
Nevertheless, the bulk doping due to the EDLG seems to be an inevitable
conclusion of our result.
\section{Ambipolar transport in $\mathbf{BiSbTeSe_{2}}$ single crystals}
It was recently found \cite{Arakane_NComm12} that the position of the
chemical potential with respect to the Dirac point of the SS is tunable
within the bulk band gap in BSTS when one follows a particular series of
$x$ and $y$ that were identified in Ref. \onlinecite{Ren_BSTS_PRB11}. In
particular, the chemical potential was found to be close to both the
Dirac point of the SS and the middle of the bulk band gap in BiSbTeSe$_2$.
Therefore, for our EDLG experiment, to maximize the possibility of
achieving ambipolar transport, we mainly measured the BiSbTeSe$_2$
system. (The gating data shown in Fig. 1 were taken during the
experiment on the BiSbTeSe$_2$ sample reported below.)
\begin{figure}[t]
\includegraphics[width=8.5cm,clip]{EDL_Fig3.eps}
\caption{(color online)
(a) Temperature dependence of $\rho$ in the BiSbTeSe$_{2}$ single
crystal used for the EDLG experiment at $V_G$ = 0 V in a semi-log plot;
inset shows the $V_G$ dependence of $\rho$ at 1.8 K, where the arrows indicate
the order of experiments.
(b),(c) Low-temperature part of $\rho(T)$ for various $V_G$.}
\end{figure}
\begin{table}[]
\centering
\begin{tabular}{c c c c}
\hline\hline
Experiment & $V_G$ & $\rho$ ($\Omega$ cm) & $R_H^{\rm LF}$ (cm$^3$/C) \\
order & (V) & at 1.8 K & at 1.8 K\\
[0.5ex]
\hline
1 & 0 & 13.8 & $-3940$ \\
2 & $-3.125$ & 15.7 & $-2290$ \\
3 & $-3.5$ & 14.9 & 699 \\
4 & $-3.3$ & 15.2 & 165 \\
5 & 0 & 12.4 & $-1440$ \\
6 & $+3.0$ & 10.9 & $-767$ \\
7 & $-3.3$ & 14.2 & 289 \\
[1ex]
\hline
\end{tabular}
\caption{Experimental order of applied $V_G$ and resultant transport properties
at low temperatures.
$R_H^{\rm LF}$ is the low field limit of the Hall coefficient.}
\end{table}
Figure 3(a) shows the temperature dependence of the resistivity $\rho$
of a BiSbTeSe$_2$ single crystal before applying the gate voltage. One
can see that $\rho$ exceeds 10 $\Omega$cm at low temperature, testifying
to a high quality of this sample. As was the case with
Bi$_{1.5}$Sb$_{0.5}$Te$_{1.7}$Se$_{1.3}$, at low temperatures we
observed clear change in $\rho$ upon application of $V_G$ [Figs. 3(b) and 3(c)]
even though this is a bulk single crystal. However, it also turned out
that the transport properties do not completely recover after cycling
$V_G$. To illustrate the situation, we show in Table I the measured
transport properties at 1.8 K for various values of $V_G$ in the order
of the measurements. As one can see in this table, the transport
properties for $V_G$ = 0 were measured twice, before and after applying
$V_G$ = $-3.5$ V; also, there are two different data for $V_G$ = $-3.3$
V, which were taken before and after $V_G$ = $+3.0$ V was applied. In
both cases, the resistivity decreased after a high voltage was applied.
Such a decrease in resistivity after application of a high voltage was
also observed in other samples, so it appears to be an unavoidable
effect in BSTS crystals; this is probably due to some irreversible
electrochemical reaction taking place in the bulk, which gradually
spoils the bulk-insulating property. However, as one can
see in the inset of Fig. 3(a), a large part of the change in resistivity
in response to $V_G$ is reversible. This reversible part of the change
is consistent with the picture that the number of $n$-type carriers
increases with positive $V_G$ and decreases with negative $V_G$ due to
the EDLG effect.
An interesting feature in our resistivity data is that the resistivity
value presents a maximum around $-3.2$ V [see inset of Fig. 3(a) and also
Fig. 3(b)]; namely, the resistivity is {\it smaller} at $V_G$ of $-3.5$
V compared to that at $-3.3$ V, despite the overall trend that negative
voltage increases $\rho$. In fact, when the system is in the $n$-type
regime, a more negative $V_G$ value would lead to a smaller number of
$n$-type carriers, and one would expect the resistivity to increase; the
opposite behavior observed for $V_G < -3.3$ V suggests that the system
is changing from $n$-type to $p$-type. To confirm this possibility, one
must look at the Hall data.
\begin{figure}[t]
\includegraphics[width=8.5cm,clip]{EDL_Fig4.eps}
\caption{(color online)
(a) Magnetic-field dependences of $\rho_{yx}$ at 1.8 K for various $V_G$.
(b) $V_G$ dependence of $R_H$ at 1.8 K.
(c) Temperature dependences of $R_H$ for various $V_G$.
Arrows indicate the experimental order.}
\end{figure}
Figure 4(a) shows how the behavior of $\rho_{yx}(H)$
changes with $V_G$. Initially, at $V_G$ = 0
V (first) and at $-3.125$ V, the charge carriers are clearly $n$-type,
but the carriers become $p$-type at large negative $V_G$ values of
$-3.5$ and $-3.3$ V. This is a signature of an ambipolar transport in a
bulk crystal, and confirms the idea that the sign change of charge
carriers takes place between $V_G$ = $-3.125$ and $-3.3$ V. As noted
above, the apparent maximum in the resistivity near $V_G \simeq -3.2$ V
is consistent with this interpretation. The $n$-type doping is recovered when
$V_G$ was set to 0 V again, and the slope of $\rho_{yx}(H)$ was found to
decrease with increasing $V_G$ up to $+3.0$ V, suggesting an increase in
the $n$-type carriers, as expected.
It should be noted that we observed the sign change in this sample again
after setting $V_G$ to $+3.0$ V and then bringing it back to $-3.3$ V,
as shown by a broken line in Fig. 4(a). Thus, the sign change of the
carriers is obviously reproducible. By defining the Hall coefficient
$R_H$ as the slope of $\rho_{yx}(H)$ at low field, we summarize the
gate-voltage dependences of $R_H$ in Fig. 4(b). One can see that the
sign change in $R_H$ is reproducibly observed, although the exact
value of $R_H$ at a given $V_G$ shows a history dependence.
The temperature dependences of $R_H$ for various $V_G$ measured in the
successive five experiments [Fig. 4(c)] indicate that the ambipolar
transport is observed only below $\sim$30 K where thermal activations of
bulk carriers are negligible. Since the experiment on
Bi$_{1.5}$Sb$_{0.5}$Te$_{1.7}$Se$_{1.3}$ discussed in the previous
section indicated that the carrier densities in both the bulk and
surface transport channels are changing with EDLG, it is important to
elucidate whether the sign change of the carriers observed in
BiSbTeSe$_2$ is occurring in the bulk or on the surface, or both. To
infer the origin of the sign change, we have analyzed the $\rho_{yx}(H)$
data for various values of $V_G$ with the simple two-band model.
\begin{figure}[]
\includegraphics[width=6.5cm,clip]{EDL_Fig5.eps}
\caption{(color online)
Results of the two-band model fitting to the $\rho_{yx}(H)$ data of
BiSbTeSe$_2$ at various $V_G$. Thick lines are the data at 1.8 K and
solid lines are the fitting results.}
\end{figure}
The results of the two-band fitting are summarized in Fig. 5 and Table
II. For example, the fitting to the data at $V_G$ = 0 V shown
in Fig. 5 gives the following crude estimate for the
transport parameters: For the bulk channel, $\rho_{\rm bulk} \simeq$ 13
$\Omega$cm, $n_{\rm bulk} \simeq$ 1 $\times$ $10^{16}$
cm$^{-3}$, and $\mu_{\rm bulk} \simeq$ 40 cm$^2$/V$\,$s; for the surface
channel, $\rho_{\rm SS}^{\rm 2D}$ = $\rho_{\rm SS}/d \simeq$ 30
k$\Omega$, $n_{\rm SS} \simeq$ 2 $\times$ $10^{11}$ cm$^{-2}$, and
$\mu_{\rm SS} \simeq$ 1 $\times$ $10^{3}$ cm$^2$/V$\,$s. Here, $d$ (=
181 $\mu$m) is the thickness of the sample and the sign of charge
carriers is negative for both the bulk and surface channels. Given that
the chemical potential is very close to the Dirac point in
BiSbTeSe$_{2}$, it seems that the estimate of the surface carrier
density indicated in this analysis is reasonable, in spite of the
weakness of the non-linearity in $\rho_{yx}(H)$. With the above
parameters, the contribution of the surface channel to the total
conductance is calculated to be $\sim$2\%.
\begin{table}[]
\centering
\begin{tabular}{c c c c c}
\hline\hline
$V_G$ & Bulk carrier & $\mu_{\rm bulk}$ & Surface carrier & $\mu_{\rm SS}$ \\
(V) & density (cm$^{-3}$) & (cm$^2$/Vs) & density (cm$^{-2}$) & (cm$^2$/Vs) \\
[0.5ex]
\hline
$+3.0$ & $-4.7$ $\times$ 10$^{16}$ & 12 & $-1.5$ $\times$ 10$^{11}$ & 1.2 $\times$10$^{3}$ \\
0 & $-1.3$ $\times$ 10$^{16}$ & 38 & $-2.2$ $\times$ 10$^{11}$ & 1.0 $\times$10$^{3}$ \\
$-3.125$ & $-1.8$ $\times$ 10$^{16}$ & 21 & $-2.0$ $\times$ 10$^{11}$ & 2.0 $\times$10$^{3}$ \\
$-3.3$ & 5.0 $\times$ 10$^{16}$ & 8 & 1.6 $\times$ 10$^{11}$ & 4.6 $\times$10$^{2}$ \\
$-3.5$ & 9.8 $\times$ 10$^{15}$ & 42 & 1.8 $\times$ 10$^{11}$ & 8.4 $\times$10$^{2}$ \\
[1ex]
\hline
\end{tabular}
\caption{Parameters of the two-band model fitting of the $\rho_{yx}(H)$ data
at various $V_G$; the fitted curves are shown in Fig. 5.}
\end{table}
Looking at the results of the two-band analyses of the data at finite
$V_G$, one can see that the Hall data strongly suggests that both the
bulk and surface carriers change sign simultaneously at $V_G \le -3.3$
V. Indeed, we found that it is impossible to fit the $\rho_{yx}$ data
for $V_G \le -3.3$ V by assuming that only one of the two channels
changes sign. Therefore, it appears that in the present EDLG experiment
the chemical potential is swung from the $n$-type regime to the $p$-type
regime not only on the surface but also in the bulk. Most likely, what
is happening in the bulk at negative $V_G$ is compensation due to
electrochemical $p$-type doping, which eventually overwhelms the
preexisting $n$-type bulk carriers.
\begin{figure}[t]
\includegraphics[width=6.5cm,clip]{EDL_Fig6.eps}
\caption{(color online)
Magnetoresistance data at various stages of the EDLG experiment of
BiSbTeSe$_2$ at 1.7 K.}
\end{figure}
Finally, we show in Fig. 6 the magnetoresistance (MR) data for various
$V_G$ values. Obviously, the data present the weak antilocalization (WAL)
effect at low fields;\cite{WAL1,WAL2,Taskin_MBE} since the WAL effect is
a signature of two-dimensional transport and is not usually observed in
bulk TI crystals dominated three-dimensional transport, the MR data give
additional evidence for a sizable contribution of the surface transport
to the total conductance. At the same time, this WAL effect makes it
difficult to examine the consistency of the parameters obtained from the
two-band analysis of $\rho_{yx}(H)$ in the MR data. Also, the behavior
of MR qualitatively changed after the application of $V_G$ = $+3$ V, the
origin of which is not clear at the moment. Since the MR is too
complicated and not very reproducible, we did not try to make a detailed
analysis.
\section{Discussion}
It is important to mention that there are missing charges in our EDLG
experiment; namely, the total amount of charge induced by gating in the
sample is much smaller than the total amount of ions accumulated on the
surface. For example, in the case of BiSbTeSe$_2$, the total amount of
accumulated charges measured by the current is $\sim$34 $\mu$C, which
corresponds to the ion density on the surface of $\sim$1.6 $\times$
10$^{15}$ cm$^{-2}$. On the other hand, the change in the surface
carrier density in BiSbTeSe$_2$ was $\sim$4 $\times$ 10$^{11}$ cm$^{-2}$
and its bulk carrier density changed by less than 1 $\times$ 10$^{17}$
cm$^{-3}$, which amounts to the total charge of less than 20 $\mu$C.
Therefore, obviously the gating is not very efficiently performed. In
this regard, the doping control of the surface carriers in the present
experiment is similar to another EDLG experiment on Bi$_2$Te$_3$ thin
film,\cite{Yuan_NL11} where $\sim$7 $\times$ 10$^{11}$ cm$^{-2}$ of
surface carriers were doped with $V_G \simeq$ $-3$ V. For other
materials, the amount of surface carrier doping by EDLG is of the order
of 10$^{13}$--10$^{15}$ cm$^{-2}$,
\cite{Dhoot,YuanZnO,Ye_PNAS11,EndoAPL10} and thus the electric-field
effect on Bi-based topological insulators appears to be exceptionally
ineffective.\cite{note}
The mechanism to cause these missing charges is not clear at the moment,
but we speculate that some electrochemical redox reaction involving
adsorbed molecules on the surface causes a layer of immobile charges
that shields some fraction of the electric field created by the ions,
leading to a weakening of the electric field for inducing mobile
carriers in the sample.
Also, the bulk doping that accompanies the EDLG in our BSTS samples is
surprising. Remarkably, the data for BiSbTeSe$_2$ suggests that the bulk
doping process is nearly reversible and it takes place in the time scale
of the order of 10 min. The chemical mechanism of this bulk doping is
not clear at the moment, but the possible cause might be intercalation
of ions into the van-der-Waals gap in the BSTS crystal. Obviously, there
is a lot to understand about the electrochemistry accompanying the EDLG
on Bi-based tetradymite TI materials.
\section{conclusion}
In conclusion, the electric-double-layer gating (EDLG) using ionic
liquid was applied to bulk single crystals of BSTS to control the
chemical potential, and ambipolar transport was observed in a sample of
BiSbTeSe$_2$ as thick as 181 $\mu$m. The gating was successfully applied
to tune the chemical potential on the whole surface of a
three-dimensional sample, and surprisingly, it appears that the EDLG on
BSTS crystals is accompanied by a nearly reversible electrochemical
reaction that caused bulk carrier doping. It turned out that the EDLG is
exceptionally inefficient for the BSTS system, with the maximum change
in the surface carrier density of $\sim$4 $\times$ 10$^{11}$ cm$^{-2}$
despite the ion density on the surface of $\sim$1.6 $\times$ 10$^{15}$
cm$^{-2}$. The key to the successful ambipolar carrier control in
the present experiment was the use of BiSbTeSe$_2$ crystals in which
the chemical potential is located close to the middle of the bulk band
gap \cite{Arakane_NComm12} and the residual bulk carrier density was
only $\sim$1 $\times$ 10$^{16}$ cm$^{-3}$. In combination with a
technique to open a gap on the surface,\cite{Nomura,Sato} the present
experiment paves the way for topological magnetoelectric-effect
experiments \cite{QHZ} that require the chemical-potential control on
the whole surface of a bulk topological insulator, although the
mechanism of the bulk doping associated with EDLG needs to be understood
before this technique is comfortably applied.
\section{acknowledgment}
This work was supported by JSPS (NEXT Program), MEXT (Innovative Area
``Topological Quantum Phenomena" KAKENHI 22103004 and KAKENHI 20371297),
and AFOSR (AOARD 104103 and 124038).
|
{
"timestamp": "2012-08-23T02:03:19",
"yymm": "1203",
"arxiv_id": "1203.2047",
"language": "en",
"url": "https://arxiv.org/abs/1203.2047"
}
|
\section{Introduction}
AU~Mic is a nearby \citep[$9.91\pm 0.10$~pc,][]{van07} M1Ve flare star
\citep{tor06} with the young age of $12^{+8}_{-4}$~Myr \citep{zuc01}, a key
epoch in the formation of planetary systems. The star is surrounded by a nearly
edge-on circumstellar disk discovered in coronographic images of scattered
starlight that extends to a radius of at least 210~AU \citep{kal04}. Since the
small grains responsible for scattering should be removed rapidly by stellar
pressure forces, the disk is thought to consist of the collisional debris of
unseen planetesimals experiencing ongoing impacts \citep[for recent reviews of
the debris disk phenomenon, see][]{wya08,kri10}. Because of its proximity, the
AU~Mic debris disk has become one of the best studied examples at optical and
infrared wavelengths, including detailed imagery of both intensity and
polarization from the {\em Hubble Space Telescope} \citep{kri05,gra07}.
Many observational properties of the AU~Mic debris disk bear striking
similarities to the archetype debris disk surrounding $\beta$~Pic, also viewed
nearly edge-on and located in the same young moving group. In particular, the
midplane optical surface brightness profiles of these disks are remarkably
alike, with shallow inner slopes ($\sim 1/r^{1-2}$) that steepen substantially
($\sim 1/r^{4-5}$) in the outer regions, near 35~AU and 100~AU for AU~Mic and
$\beta$~Pic, respectively \citep{kri05,fit07,hea00,gol06}. These similarities
inspired the development of a unified framework for debris disks based on a
localized belt of planetesimals, or ``birth ring'', that produces dust in a
collisional cascade \citep{str06,aug06}. The smallest grains are blown out from
the belt by radiation and winds from the star, larger grains are launched into
eccentric orbits with the same periastron as the belt, and the largest grains,
which are minimally affected by non-gravitational forces, do not travel far
from the belt before being ground down. The grains are therefore segregated
according to their size, which gives rise to the characteristic scattered light
profile. For these dynamics to prevail, the disks must be substantially free of
gas \citep{the05}, a property confirmed by measurements at ultraviolet
wavelengths \citep{rob05,fra07}.
A ``birth ring'', if present, would remain hidden in optical and infrared
images dominated by small grains that populate an extended halo
\citep[e.g.,][]{su05}. By contrast, observations at (sub)millimeter wavelengths
highlight thermal emission from the largest grains and hence the location of
the dust-producing planetesimals \citep{wya06}. To date, the best case for a
``birth ring'' comes from millimeter observations of $\beta$~Pic \citep{wil11}.
While the optical disk of $\beta$~Pic extends more than $1000$~AU from the
star, the millimeter imaging reveals a much more compact belt of emission at
$\sim95$~AU radius. This millimeter emission belt coincides with the change in
the slope of the optical surface brightness, which in the models marks the
outer extent of the colliding planetesimals.
Previous (sub)millimeter-wave observations of AU Mic did not have sufficient
angular resolution to reveal much about the emission morphology. A detection at
850~$\mu$m using JCMT/SCUBA ($14''$ beam) indicated a reservoir of cold dust
with mass $\sim0.01$~M$_{\oplus}$, but did not resolve any structure
\citep{liu04b}. Subsequent observations at 350~$\mu$m using CSO/SHARC~II
($10''$ beam) marginally resolved an orientation compatible with the scattered
light, but were otherwise limited \citep{che05}. In this {\em Letter}, we
present imaging observations of AU Mic at 1.3~millimeters using the
Submillimeter Array (SMA)\footnote{The Submillimeter Array is a joint project
between the Smithsonian Astrophysical Observatory and the Academica Sinica
Institute of Astronomy and Astrophysics and is funded by the Smithsonian
Institution and the Academica Sinica.} that obtain $3''$ resolution and provide
evidence for a planetesimal belt.
\section{Observations} \label{sec:obs}
We observed AU~Mic with the SMA \citep{ho04} on Mauna Kea, Hawaii at
1.3~millimeters wavelength using the compact and extended configurations of the
array. Table~\ref{tab:obs} provides basic information about these observations,
including the observing dates, baseline lengths, and atmospheric opacities. The
weather conditions were good for all of these observations and best for the two
compact configuration tracks when only 6 of the 8 array antennas were
available. The phase center was $\alpha = 20^h45^m09\fs53$, $\delta =
-31\degr20\arcmin27\farcs2$ (J2000), offset by $\sim5\farcs3$ from the location
of the star. The $\sim54''$ (FWHM) field of view is set by the primary beam
size of the 6~meter diameter array antennas. The total bandwidth available was
8~GHz derived from two sidebands spanning $\pm4$~to~$8$~GHz from the LO
frequency. Time dependent complex gains were calibrated using observations of
two quasars, J2101-295 (3.9 degrees away) and J1924-292 (17.4 degrees away),
interleaved with observations of AU Mic in a 15~minute cycle. The passband
shape for each track was calibrated using available bright sources, mainly
J1924-292. The absolute flux scale was set with an accuracy of $\sim10\%$ using
observations of Callisto or Ganymede in each track. All of the calibration
steps were performed using the IDL based MIR software, and imaging and
deconvolution were done with standard routines in the MIRIAD package. We made a
series of images with a wide range of visibility weighting schemes to explore
compromises between higher angular resolution and better surface brightness
sensitivity.
\begin{deluxetable}{lcccc}
\tablecaption{
Submillimeter Array Observations of AU Mic
\label{tab:obs}
}
\tablewidth{0pt}
\tablehead{
\colhead{Observation} & \colhead{2011 Jul 31} & \colhead{2011 Sep 26} &
\colhead{2011 Sep 27} & \colhead{2011 Oct 25}
}
\startdata
Array Configuration & Extended & Compact & Compact & Compact \\
Number of Antennas & 8 & 6 & 6 & 7 \\
Baseline Lengths (m) & 10--189 & 8--68 & 8--68 & 8--68 \\
LO frequency (GHz) & 235.6 & 225.4 & 225.4 & 235.6 \\
225 GHz atm. opacity\tablenotemark{a} & 0.10 & 0.10--0.06 & 0.06 & 0.08 \\
\enddata
\tablenotetext{a}{Measured at the nearby Caltech Submillimeter Observatory.}
\end{deluxetable}
\section{Results and Analysis} \label{sec:result}
\subsection{1.3 Millimeter Emission} \label{sec:continuum}
Figure~\ref{fig:image} shows a contour image of the 1.3~millimeter emission
overlaid on a {\it Hubble Space Telescope}/ACS coronographic image of optical
scattered light (F606W filter) from \citet{gra07}. The synthesized beam size
for this 1.3~millimeter image is $3\farcs3 \times 2\farcs6$ ($33\times26$~AU),
position angle $-12{\degr}$, obtained with natural weighting and an elliptical
Gaussian taper $1\farcs5 \times 1\farcs0$ (FWHM) oriented east-west to make the
beam shape more circular. The rms noise in this image is 0.40~mJy~beam$^{-1}$,
and the individual peaks have a signal-to-noise $\gtrsim7$. The star symbol
marks the stellar position corrected for proper motion, offset by $(3\farcs20,
-4\farcs25)$ from the phase center. The image shows a resolved band of 1.3
millimeter emission that extends approximately symmetrically from the stellar
position to the southeast and northwest. By fitting a line to the positions of
the two peaks in the image, we estimate the position angle of the emission
structure to be $130\pm 2\degr$, in excellent agreement with the disk
orientation inferred from scattered light observations \citep{liu04a}. From its
position angle and double peaked morphology, we identify the emission structure
as a limb-brightened dust belt. Note that thermal emission from the stellar
photosphere contributes only $0.06$~mJy at this wavelength, and even the
strongest synchrotron radio flares from stellar activity should remain well
below the noise \citep{kun87}. The total flux density obtained by integrating
over a box that surrounds the emission is $8.5\pm2$~mJy, where the uncertainty
is estimated conservatively by evaluating boxes of the same size in
emission-free regions of the image. No other significant features are detected
in the field of view.
An extrapolation of the measurements at shorter wavelengths indicates that
missing flux due to the spatial filtering by the SMA is not a significant issue
for these observations. The spectral index between 350~$\mu$m
\citep{che05} and 850~$\mu$m
\citep{liu04b} is $1.7^{+0.3}_{-0.5}$, which predicts
$7.1^{+2.0}_{-1.2}$~mJy at 1.3~millimeters, in good agreement with the SMA
measurement. This shallow index is consistent, within the uncertainties, with
values near 2.0 determined for a sample of debris disks by \citet{gas11},
notably for stars later than F-type where dust temperatures are so low that the
Rayleigh-Jeans approximation in this part of the spectrum is invalid. We
conclude that the compact structure detected by the SMA accounts for all of the
1.3~millimeter emission in the AU~Mic system.
\subsection{A Simple Disk Model} \label{sec:models}
To characterize the millimeter emission structure, we use a simple parametric
disk model, following closely the method used by \citet{wil11} for analyzing
similar observations of the $\beta$~Pic disk. We assume the emission arises
from a geometrically thin, axisymmetric belt with a radial surface brightness
profile given by $I\propto f(r)r^{-q}$. This functional form is intended to
capture the essence of low optical depth emission from a disk with a spatially
invariant dust emissivity, surface density profile $f(r)$, and temperature
profile that falls off with radius as a power-law. We fix the power-law index
$q=0.5$ to approximate radiative heating from the central star, and we consider
two shapes for $f(r)$: (1) an annulus with power law slope, $f(r) \propto
r^{-p}$ for $R \pm \Delta R/2$, $p \in {0,1}$, and (2) a Gaussian, $f(r)
\propto \exp{[-((r-R)/ \sqrt{2} \Delta R)^2]}$. In each case, the model is
described by three parameters: a center radius $R$, width $\Delta R$, and total
flux density $F = \int I d\Omega$. We fix the disk inclination at $89\fdg5$ and
position angle at $130\degr$, as determined from scattered light data
\citep{kri05}, and place the origin within $0\farcs1$ of the nominal stellar
position (within the SMA astrometric accuracy). Note that the vertical
thickness of the millimeter grain population is expected to be $<0\farcs25$
\citep{the09}, negligible compared to the resolution of the data. The utility
of this simple modeling approach is to provide basic constraints while avoiding
the multitudinous assumptions about grain properties, dynamics, and radiative
transfer required in more sophisticated treatments.
To estimate the model parameters and their uncertainties, we calculate
three-dimensional grids spanning appropriate ranges of the parameter values.
For the Gaussian model, the grid covers $20 < R < 47$~AU and $1 < \Delta R <
40$~AU in 1~AU steps, and $5 < F < 11$~mJy in 0.2~mJy steps. For each position
in a model grid, we compute a set of synthetic SMA visibilities and compare
directly to the data with a $\chi^2$ value (the sum of real and imaginary
components over all spatial frequencies) that uses natural weights modified by
the Gaussian taper used for the image in Figure~\ref{fig:image}. The rightmost
panel of Figure~\ref{fig:model} shows the $\chi^2$ surface in the $(R,\Delta
R)$ plane for the Gaussian model after marginalizing over $F$. The best-fit
model is marked by a cross, and the contours delineate (marginalized) $1
\sigma$ intervals in $\Delta \chi^2$. The best-fit center radius is $R =
36^{+7}_{-16}$~AU, with corresponding width $\Delta R = 10^{+13}_{-8}$~AU (or
FWHM = $2\sqrt{2\ln2} \Delta R = 23^{+31}_{-19}$~AU) and flux density $F = 8.0
\pm 1.2$~mJy. The three lefthand panels of Figure~\ref{fig:model} show the
original 1.3~millimeter image from Figure~\ref{fig:image}, the Gaussian model
image obtained using the best-fit parameter values and the same visibility
weighting scheme, and the imaged residuals (where data$-$model subtraction is
conducted on the visibilities). This comparison shows clearly that the
best-fit model reproduces the main features of the image and leaves no
systematic residuals. The power-law models for $f(r)$ give similar best-fit
parameter values: for $p=0$, we find $R=35\pm6$~AU and $\Delta R =
38^{+26}_{-18}$~AU (where the upper limit is set by the computational grid
boundary). For $p=1$, the best-fit locus is shifted to roughly $10\%$ larger
values of $R$, a manifestation of the degeneracy between the radial gradient
and emission extent. All of these best-fit models reproduce the observed SMA
visibilities equally well (i.e., have effectively the same minimum $\chi^2$
values).
The modest signal-to-noise of the observations preclude placing tighter
constraints on the model parameters. Nonetheless, it is reassuring that
consistent results are obtained from the various assumptions about the emission
morphology. Each gives a similar center radius, as well as an effectively
cleared central cavity with size compatible with that inferred from scattered
light \citep[$\sim12$~AU][]{kri05} and the lack of mid-infrared emission
\citep[$\sim17$~AU][]{liu04b}. The best-fit Gaussian model reproduces better
the contrast of the maxima to the extended emission of the disk in the image
plane, but there is no clear preference for any of the functional forms. Of
course, none of these {\em ad hoc} models are perfect, and more sensitive data
will be needed to constrain further the details of the emission distribution.
\section{Discussion} \label{sec:discussion}
We have spatially resolved the thermal emission at 1.3 millimeters from the
AU~Mic debris disk, thereby revealing the distribution of its millimeter grain
population. These large grains are expected to have dynamics similar to the
unseen dust-producing planetesimals, and the emission at this long wavelength
provides a direct link to their spatial distribution.
The overall structure of the AU~Mic millimeter emission is reproduced well by a
belt surrounding the star centered near a radius of $35$~AU. The inferred
location of this emission belt coincides with the region over which the
scattered light brightness profile of the disk steepens markedly, the feature
that provided impetus for developing the ``birth ring'' model. Like AU~Mic's
more massive sister star $\beta$~Pic \citep{wil11}, the resolved
multi-wavelength data are consistent with a scenario where destructive
collisions of planetesimals within the belt create grains with a spectrum of
sizes, and the effects of size-dependent dust dynamics generate the millimeter
emission and spatially extended scattered light nebula. For $\beta$~Pic, the
intense stellar radiation pressure can account for the large halo of small
grains in the outer disk. But for AU~Mic, which is about two orders of
magnitude less luminous, the much weaker stellar radiation field is apparently
augmented by a stellar wind to expel small grains \citep{str06,aug06}.
Though the fractional radial extent of the AU~Mic millimeter emission is not
strongly constrained, the best-fit models suggest that it could be broad,
$\Delta R/R \sim 1$. For $\beta$~Pic, the millimeter emission is characterized
by $\Delta R/R < 0.5$, with the upper end favored for the width of the
underlying dust-producing planetesimal belt from detailed models of the knee of
the scattered light profile \citep{hah10}. Such broad belts are not unusual
features of debris disks, in particular among the few resolved at millimeter
wavelengths \citep{hug11}. An inner cavity devoid of dust is plausibly
maintained by the combination of grinding collisions in the belt and clearing
by stellar radiation forces, though a tenable alternative is that the inner
regions are swept clear by a planet. Planets seem to be the most viable
explanation for the millimeter cavities common to the gas-rich transition disks
around younger, pre-main-sequence stars \citep{and11}. While a planet has been
imaged to orbit within the cavity in the disk around $\beta$~Pic \citep{lag10},
direct evidence for planets in the AU Mic system remains elusive, both from
high contrast imaging observations \citep{mas05,met05} and from transit
searches \citep{heb07}. The outer boundary of the emission belts may reflect
the outer extent of successful planetesimal formation, or it may simply
mark the edge of the region where the planetesimal stirring mechanism has
been successful at initiating dust-producing collisions, as in the models of
\citet{ken04}.
These new millimeter observations of AU~Mic show the global structure of large
grains in its debris disk, but they do not have sufficient resolution and
sensitivity to reveal any significant departures from axisymmetry or
substructure that might result from gravitational interactions with unseen
planets. The higher resolution images of scattered light clearly show several
radial and vertical inhomogeneities at subarcsecond scales \citep[features A-E,
see][]{liu04a,met05,fit07}. If these substructures have counterparts at
millimeter wavelengths, then future observations with the Atacama Large
Millimeter/Submillimeter Array (ALMA) may be able to detect them, pinpoint the
location of parent bodies, and constrain their origins.
\acknowledgments{
We thank James Graham for providing the {\it Hubble Space Telescope} image in
Figure 1. A. M. H. is supported by a fellowship from the Miller Institute for
Basic Research in Science.
}
{\it Facility:} \facility{Submillimeter Array}
|
{
"timestamp": "2012-03-09T02:04:35",
"yymm": "1203",
"arxiv_id": "1203.1896",
"language": "en",
"url": "https://arxiv.org/abs/1203.1896"
}
|
\section{Introduction}
The Large Hadron Collider (LHC) at CERN has to date delivered an integrated luminosity of up to 5~${\rm fb}^{-1}$
of proton-proton collisions at a center-of-mass beam energy of $\mathbf{\sqrt{s} = 7}$~TeV,
with a further 15 ${\rm fb}^{-1}$ anticipated during 2012, in tandem with an upgrade to $\mathbf{\sqrt{s} = 8}$~TeV.
The CMS and ATLAS Collaborations have analyzed the first 1--2 ${\rm fb}^{-1}$ of data in their
search for signals of supersymmetry (SUSY), thus far revealing no significant excesses beyond the
Standard Model (SM) expectations. The absence of any definitive signal has imposed severe constraints
onto the viable parameter spaces of the baseline supersymmetric models, leading pessimists to question whether
there is even a SUSY framework extant in nature to discover at all. Nevertheless, a few select search
methodologies have exhibited curious indications of strain between the maximum reasonable expectation for the
background and the number of observations in the
data~\cite{PAS-SUS-11-003,ATLAS-CONF-2011-130,Aad:2011ib,Aad:2011qa,PAS-SUS-11-019}. This
begs the question of whether such tensions are mere statistical fluctuations, the inevitable
``look elsewhere'' styled distribution tails that become probable {\it somewhere} within an exhaustive search,
or perhaps rather early warning indicators of something much more significant yet to come. Given any reasonable likelihood
that we {\it may} indeed be witnessing the commencement, in its earliest nascent phase, of a legitimate SUSY signal
effervescing to the surface, then it becomes imperative to closely examine these particular search strategies in unison
to uncover clues as to whether they might originate from a common underlying physics. Such a multi-axis study could imply
a key connection not apparent from the individual searches in isolation, allowing for the directed {\it correlation}
across independent selection spaces to statistically distinguish a signal from the noise. The demonstration
of a specifically detailed supersymmetric model that can expose tendencies for high SUSY visibility in the
appropriate searches, without damaging event overproduction in those cases where the observations are in stricter
accord with the expected backgrounds, would suggestively hint that the currently observed stresses might
represent an authentic physical correlation after all.
In a previous work entitled {\it Profumo di SUSY}~\cite{Li:2011av}, we investigated a pair of early single
inverse femtobarn LHC reports from CMS~\cite{PAS-SUS-11-003} and ATLAS~\cite{Aad:2011qa}, where data
observations for events with ultra-high jet multiplicities, minimally seven to nine, could be readily
extracted. In the time frame preceding release of the CMS and ATLAS reports~\cite{PAS-SUS-11-003,Aad:2011qa},
we had been strongly advocating for close scrutiny of precisely these sort of
events~\cite{Li:2011hr,Maxin:2011hy,Li:2011gh,Li:2011rp}, following a realization that the unique
spectrum of our favored SUSY construction, and in particular its rather light stop squark $\widetilde{t}_1$
and gluino $\widetilde{g}$, would lead to a strong and distinctive signal in these particular channels.
Consequently, we were gratified that an initial inquiry into such events could be undertaken at the LHC,
and buoyed by the observation of scant, yet nonetheless tantalizing, excesses in events with $\ge$7--9
jets. We thus undertook a carefully detailed study in Ref.~\cite{Li:2011av}. During the intermission between those
early CMS and ATLAS reports~\cite{PAS-SUS-11-003,Aad:2011qa} and the present time frame, numerous parallel
high luminosity studies from CMS and ATLAS have appeared to compliment those discriminated by jet count. In
the present work, we thus broaden our horizon to encompass the wider panorama of search strategies squarely directed
at the detection of pair-produced gluinos and light stops, including light stops transpiring from gluino mediated
decays. This is a calculated augmentation, given that our model is auspiciously predisposed to the production of gluinos
and light stops in the present running phase of the LHC. Moreover, it facilitates precisely the sort of
cross-correlation between independent experimental signatures previously described. This will be quantified in
the present work by application of a $\chi^2$ statistical test in seven degrees of freedom. If the prior fleeting
scent~\cite{Li:2011av} of supersymmetry in the multijet data was corporeal, then that same aroma should linger
also in certain of the wider stop and gluino searches; if it was a wishfully conjured mirage, then even the memory
must fade in the cleansing light of new data.
\section{The No-Scale $\mathbf{\cal{F}}$-$\mathbf{SU(5)}$ Model}
The context of our study on the correlation of light squark and gluino SUSY searches is a model named No-Scale
$\cal{F}$-$SU(5)$~\cite{Li:2010ws, Li:2010mi,Li:2010uu,Li:2011dw, Li:2011hr, Maxin:2011hy, Li:2011xu,Li:2011in,Li:2011gh,Li:2011rp,Li:2011fu,Li:2011xg,Li:2011ex,Li:2011av,Li:2011ab}.
No-Scale $\cal{F}$-$SU(5)$ is defined by the convergence of the ${\cal F}$-lipped $SU(5)$~\cite{Barr:1981qv,Derendinger:1983aj,Antoniadis:1987dx}
grand unified theory (GUT), two pairs of hypothetical TeV scale vector-like supersymmetric multiplets (dubbed {\it flippons}) of mass $M_{\rm V}$
with origins in local ${\cal F}$-theory~\cite{Jiang:2006hf,Jiang:2009zza,Jiang:2009za,Li:2010dp,Li:2010rz} model building, and the dynamically
established boundary conditions of No-Scale Supergravity~\cite{Cremmer:1983bf,Ellis:1983sf, Ellis:1983ei, Ellis:1984bm, Lahanas:1986uc}.
This construction inherits all of the most beneficial phenomenology of the flipped $SU(5) \times U(1)_{\rm X}$~\cite{Nanopoulos:2002qk,Barr:1981qv,Derendinger:1983aj,Antoniadis:1987dx}
gauge group structure, as well as all of the valuable theoretical motivation of No-Scale Supergravity~\cite{Cremmer:1983bf,Ellis:1983sf, Ellis:1983ei, Ellis:1984bm, Lahanas:1986uc}.
A substantially more detailed theoretical treatment of the model under analysis is available in the cited references,
including a rather thorough summary in the Appendix of Ref.~\cite{Maxin:2011hy}.
Since mass degenerate superpartners for the known SM fields are not observed,
SUSY must itself be broken around the TeV scale. In the Constrained Minimal
Supersymmetric Standard Model (CMSSM) and minimal supergravities
(mSUGRA)~\cite{Chamseddine:1982jx}, this occurs first in a hidden sector, and the secondary
propagation by gravitational interactions into the observable sector is parameterized by universal
SUSY-breaking ``soft terms'' which include the gaugino mass $M_{1/2}$, scalar mass $M_0$ and the
trilinear coupling $A$. The ratio of the low energy Higgs vacuum expectation values (VEVs) $\tan \beta$,
and the sign of the SUSY-preserving Higgs bilinear mass term $\mu$ are also undetermined, while the
magnitude of the $\mu$ term and its bilinear soft term $B_{\mu}$ are determined by the $Z$-boson mass $M_Z$
and $\tan \beta$ after electroweak symmetry breaking (EWSB). In the simplest No-Scale scenario,
$M_0$=A=$B_{\mu}$=0 at the unification boundary, while the complete collection of low energy SUSY
breaking soft-terms evolve down with a single non-zero parameter $M_{1/2}$. Consequently, the particle
spectrum will be proportional to $M_{1/2}$ at leading order, rendering the bulk ``internal'' physical
properties invariant under an overall rescaling. The matching condition between the low-energy value of
$B_\mu$ that is demanded by EWSB and the high-energy $B_\mu = 0$ boundary is notoriously difficult to
reconcile under the renormalization group equation (RGE) running. The present solution relies on
modifications to the $\beta$-function coefficients that are generated by radiative loops containing
the vector-like {\it flippon} multiplets. By coupling to the Higgs boson, the {\it flippons} will moreover
have an impact on the Higgs boson mass $m_h$~\cite{Moroi:1992zk,Babu:2008ge,Huo:2011zt}, resulting in a 3-4~GeV
upward shift in $m_h$, handily generating a Higgs mass of 124-126~GeV~\cite{Li:2011ab} that is in fine
accord with the recent ATLAS and CMS reports~\cite{Collaboration:2012tx,Collaboration:2012si}.
Pertinent to the present work, we have previously demonstrated the range of the $\cal{F}$-$SU(5)$ model
space that is adherent to a set of firm ``bare-minimal'' phenomenological constraints~\cite{Li:2011xu},
including consistency with the world average top-quark mass $m_{\rm t}$~\cite{:1900yx},
the dynamically established boundary conditions of No-Scale supergravity, radiative electroweak
symmetry breaking, the centrally observed WMAP7 CDM relic density~\cite{Komatsu:2010fb}, and precision LEP
constraints on the lightest CP-even Higgs boson $m_{h}$~\cite{Barate:2003sz,Yao:2006px} and other light
SUSY chargino and neutralino mass content. We have moreover established a highly constrained
subspace, dubbed the {\it Golden Strip}~\cite{Li:2010mi,Li:2011xu,Li:2011xg}, that is noteworthy for
its capacity to additionally conform to the phenomenological limits on rare processes that are established
by measurement of the muon anomalous magnetic moment $(g_{\mu}-2)/2$ and the branching ratios of the
flavor-changing neutral current decays $b \to s\gamma$ and $B_S^0 \rightarrow \mu^+\mu^-$. A similarly
favorable {\it Silver Strip} slightly relaxes the constraints imposed by $(g_{\mu}-2)$.
\section{$\cal{F}$-$SU(5)$ Stops and Gluinos}
Production of the light stop $\widetilde{t}_1$ and gluino $\widetilde{g}$ at the early LHC in the first
5~${\rm fb^{-1}}$ of integrated luminosity and naturally accounting for a 125~GeV Higgs boson tend to be mutually exclusive
goals for the traditional MSSM constructions. In particular, the mechanism for elevation of the Higgs mass will
typically correspond to squark and gluino masses which are far too heavy to have yet crept above the SM
background for the initial $\sqrt{s} = 7$~TeV operating phase of the LHC. For instance, it has been
suggested that in the CMSSM and mSUGRA, the only viable remnant of solution space that has thus far
survived the rapidly encroaching LHC constraints while delivering a 125~GeV Higgs boson mass requires a super-heavy scalar
mass $m_0$ of 10--20 TeV~\cite{Baer:2012uy}, which is well beyond reach of the LHC operating energy, both currently and
into the future. The additional contributions from the vector-like {\it flippons} are the key
to differentiating $\cal{F}$-$SU(5)$ from the CMSSM and mSUGRA. The {\it flippon} loops allow a 125~GeV Higgs boson in
conjunction with a light TeV-scale SUSY spectrum. In contrast, the CMSSM and mSUGRA require very large values
of the trilinear A-term, which pushes $m_0$ to very large values, such that the extremely massive stop and sbottom
squark masses lead to severe electroweak fine-tuning~\cite{Baer:2012uy}. No-Scale $\cal{F}$-$SU(5)$ takes
advantage of the same strongness of the Higgs to top quark coupling that provides the primary lifting of the
SUSY Higgs mass to generate a hierarchically light partner stop in the SUSY mass-splitting. However, this
rather generic mechanism is not in itself enough. The model further leverages the same vector-like
multiplets which provide the secondary Higgs mass perturbation to flatten the RGE running of universal
color-charged gaugino mass $M_3$, blocking the standard logarithmic enhancement of the gluino mass at
low energies, and producing the distinctive mass ordering
$M({\widetilde{t_1}}) < M({\widetilde{g}}) < M({\widetilde{q}})$ of a light stop and gluino, both
substantially lighter than all other squarks. The stability of this distinctive mass hierarchy is
manifest across the entire model space, a hierarchy that is not
precisely replicated in any MSSM constructions of which we are aware.
Indeed, it is specifically because the light stop $\widetilde{t}_1$ and gluino $\widetilde{g}$ are less
massive than the heavier bottom squarks $\widetilde{b}_1$ and $\widetilde{b}_2$ and the first and second generation
left and right squarks $\widetilde{q}_R$ and $\widetilde{q}_L$ that we are afforded a uniquely
distinctive test signature for $\cal{F}$-$SU(5)$ at the LHC. This spectrum generates a characteristic event
topology starting from the pair production of heavy squarks $\widetilde{q}$ and/or gluinos
$\widetilde{g}$ in the initial hard scattering process, with each squark likely to yield a quark-gluino
pair $\widetilde{q} \rightarrow q \widetilde{g}$ in the cascade decay. The gluino will tend
to decay via QCD to a typical 2-jet final state as $\widetilde{g} \rightarrow q \overline{q} \widetilde{\chi}_1^0$,
though at an atypically low 60\% branching ratio. The weakly interacting
lightest neutralino $\widetilde{\chi}_1^0$ escapes the detector unseen, leaving only an imprint of
missing energy. This leaves allowance for the production of light stops through gluino decays $\widetilde{g} \rightarrow \widetilde{t}_1 \overline{t}$
at a relatively high rate of 40\%, where the light stops decay as $\widetilde{t}_1 \rightarrow t \widetilde{\chi}_1^0$
at 58\% and as $\widetilde{t}_1 \rightarrow b \widetilde{\chi}_1^{\pm}$ at 32\%. We note that the
intermediate light stop may tend to be off shell, particularly for the lighter $\cal{F}$-$SU(5)$ spectra,
below a gaugino mass $M_{1/2}$ of about 700~GeV.
The repercussions of these final states are two-fold. Firstly, it is expected that each gluino will produce
{\it at least} four hard jets 40\% of the time. Processes such as this may then consistently exhibit a net
product of eight or more hard jets emergent from a single squark-squark, squark-gluino, or gluino-gluino
pair production event. When the further process of jet fragmentation is allowed after the primary hard
scattering events and the sequential cascade decay chain, this will ultimately result in a
spectacular signal of ultra-high multiplicity final state events. Events with very high multiplicity
jets have received little study in legacy experiments such as LEP and those at the Tevatron, though
fortunately such analyses are now beginning to receive more than just sporadic attention at LHC.
Recognizing the prospect of a conveniently encoded SUSY signal within such multijet events, we
optimistically anticipate a near-term expansion search horizon in the high multiplicity regime.
A second impact of the $\cal{F}$-$SU(5)$ final states could be discovery of the light stop and gluino production itself.
Considering the high production rate of gluino mediated light stops at 40\%, those SUSY searches
currently focused more intently on the $\widetilde{g} \rightarrow \widetilde{t}_1 t$,
$\widetilde{t}_1 \rightarrow b \widetilde{\chi}_1^{\pm}$, and $\widetilde{t}_1 \rightarrow t \widetilde{\chi}_1^0$
channels may also expect to reap tangible benefits within this construction. In contrast
to the relatively unexplored region of ultra-high jet multiplicities, searches more directly focused on
gluino and light stop production are gaining maturity. These searches
typically concentrate on final product states of b-jets (heavy flavor tagging) and leptons, along with smaller multiplicities
of jets. Thus, we would not be surprised at all if an initial conclusive signal discovery emanated from
these search methodologies. In fact, a dual signal emergent in gluino and stop production search
strategies and in the ultra-high jet multiplicity events will be highly suggestive of $\cal{F}$-$SU(5)$ origins.
A further consequence of the accessible mass of the $\cal{F}$-$SU(5)$ light stop in the present operational phase of the
LHC is the more pronounced direct production cross-section of the light stops from the hard scattering
event. An inspection of the direct production cross-sections for squarks, gluinos, and light stops,
along with the branching ratios, yields an expectation that about 15-20\% of light stop production at the LHC
in an $\cal{F}$-$SU(5)$ framework would be pair-production directly from the hard scattering collision. This is in
contrast to other MSSM based constructions, where it is not uncommon for less than 1\% of all light stops to be produced directly.
\section{The LHC SUSY Search\label{sct:search}}
We now focus on seven ongoing LHC SUSY search strategies, of which five are substantially orthogonal in construction,
that are sensitive to the $\cal{F}$-$SU(5)$ final states comprised of stops and gluinos decaying into some quantity of jets.
Each one of these seven event selection methodologies exhibits at least slight positive strain against the expectation
for the SM background, a correlation that we shall demonstrate may not be coincidental. First, we offer a concise summary
of each search strategy, then present the $\cal{F}$-$SU(5)$ contribution to each in Section~\ref{sct:correlations}, followed by a multi-axis
$\chi^2$ best fit against the full contingent of selection strategies in Section~\ref{sct:chi2}.
\subsection{CMS Purely Hadronic Large Jet Multiplicities}
This search is detailed in Ref.~\cite{PAS-SUS-11-003} and based upon a data sample of 1.1~${\rm fb^{-1}}$.
All hadronic events with high $p_T$ are discriminated by jet count, allowing for smooth
extrapolation of events with very high jet multiplicities. The primary cuts are $H_T \ge$375~GeV,
$E_T^{Miss} \ge$100~GeV, and $p_T >$50~GeV. We use the data sample with no $\alpha_T$ cut, which we have
argued is actively biased against events with high multiplicities of jets~\cite{Maxin:2011hy}. We apply a further
cut on jet count, retaining only those events with greater than or equal to nine jets. This search strategy
is very favorable for exposing an $\cal{F}$-$SU(5)$ signal emanating from the sequential cascade decays of gluinos,
squarks, and light stops to many jets. We have studied this search methodology in some detail in Ref.~\cite{Li:2011av},
and we now reprise that analysis with the intent of revealing any potentially hidden correlations with a much
more broad sampling of contemporary LHC SUSY searches.
\subsection{ATLAS Large Jet Multiplicities}
The fine points for this search can be found in Ref.~\cite{Aad:2011qa}, in a study based upon
1.34~${\rm fb^{-1}}$ of data. Here again, all events are segregated by jet count, permitting a straightforward extraction
of events of all high multiplicity jet counts. Additionally, four key combinations of the jet count and transverse momentum $p_T$
per jet thresholds are isolated in the tables for detailed study. We choose to keep only those events with at least 7 jets with
$p_T >$80~GeV for the case of $E_T^{Miss}/\sqrt{H_T} >$3.5. As with the preceding SUSY search, this scenario will also
be sensitive to the large $\cal{F}$-$SU(5)$ multijet final states. Likewise, we invested much detail in the
analysis of this search strategy in Ref.~\cite{Li:2011av}, which we again carry over in the interest of
exposing correlations within a more comprehensive range of possible channels for a SUSY discovery.
The present incarnation of this search does differ with our prior report in one regard: we have opted
in this work to employ the cone jet finding algorithm provided with {\tt PGS4}~\cite{PGS4} rather
than the $k_t$ jet alternative.
\subsection{ATLAS B-jets plus Lepton}
The first undertaking of this ATLAS search strategy is defined in Ref.~\cite{ATLAS-CONF-2011-130}, employing a
data sample of 1.03 ${\rm fb^{-1}}$. The requirement here is at least four jets, a minimum of one b-jet,
precisely one lepton, $p_T >$50~GeV for all jets, $E_T^{Miss} >$80~GeV, and $M_{eff} >$600~GeV. In our
analysis here, we choose to harness the data-driven background findings. This strategy will be sensitive
to large cross-sections of $\widetilde{g}\widetilde{g}$ production with large branching ratios for
$\widetilde{g} \rightarrow \widetilde{t}_1 t$, as is expected in $\cal{F}$-$SU(5)$. Therefore, this strategy is
very sensitive to gluino-mediated light stop production in $\cal{F}$-$SU(5)$, which is currently in a very favorable production
phase at the LHC. However, we must note that the simplified model interpretation in Ref.~\cite{ATLAS-CONF-2011-130}
assumes a 100\% branching ratio for $\widetilde{t}_1 \rightarrow b \widetilde{\chi}_1^{\pm}$,
whereas the $\cal{F}$-$SU(5)$ branching ratio is only 32\%. Thus, we caution that the $\cal{F}$-$SU(5)$ spectra may be
misrepresented in the generic imposition of model limits.
\subsection{ATLAS B-jets plus Lepton SR1-D}
The ATLAS B-jets plus Lepton search in the previous subsection has been updated to an extent in
Ref.~\cite{ATLAS-CONF-2012-003} for 2.05 ${\rm fb^{-1}}$. While still implementing the same
pre-selection cuts as Ref.~\cite{ATLAS-CONF-2011-130}, the cuts on the leading jet $p_T$ and $M_{eff}$
have been altered, in turn significantly affecting the surviving background sample, and possibly any
embedded signal as well. Therefore, we consider the shift in the final results to be consequential enough
to warrant an independent analysis of Refs.~\cite{ATLAS-CONF-2011-130}
and~\cite{ATLAS-CONF-2012-003}. In Ref.~\cite{ATLAS-CONF-2012-003}, the two strategies of
interest here are the SR1-D and SR1-E. In the case of SR1-D, the search parameters have been updated such
that the $p_T$ for the leading jet has been raised to $p_T>$60~GeV and the cut on effective mass has been
increased to $M_{eff} >$ 700~GeV.
\subsection{ATLAS B-jets plus Lepton SR1-E}
The additional SR1-E scenario of Ref.~\cite{ATLAS-CONF-2011-130}, over and above the further cuts
implemented in SR1-D, has elevated the missing energy component to $E_T^{miss} >$ 200~GeV. In our analysis
to follow in Section~\ref{sct:correlations}, we shall clearly discriminate between the three ATLAS B-jets plus Lepton cases
due to the substantial impact that the retuned cuts have on the data sample. Each scenario will therefore
illustrate a unique state of the SUSY discovery program.
\subsection{ATLAS Purely Hadronic Events}
This study is based upon the search methodology in Ref.~\cite{Aad:2011ib}, using a data sample of 1.04~${\rm fb^{-1}}$.
We apply the ``High Mass'' cuts, consisting of at least four jets, $p_T >$80~GeV ($p_T >$130~GeV
for the leading jet), no lepton, $E_T^{Miss} >$130~GeV, and $M_{eff} >$1100~GeV. The intent of the ``High
Mass'' strategy is to extend a maximal reach into the SUSY mass spectrum. Sensitivity will be high for
models with large cross-sections of pair-produced $\widetilde{g}\widetilde{g}$,
$\widetilde{g}\widetilde{q}$, and $\widetilde{q}\widetilde{q}$, where $\widetilde{q} \rightarrow q \widetilde{\chi}_1^0$
and $\widetilde{g} \rightarrow q \overline{q} \widetilde{\chi}_1^0$. As indicated
earlier, the gluinos in $\cal{F}$-$SU(5)$ are lighter than all squarks except
the light stop, hence the $\widetilde{q} \rightarrow q \widetilde{g}$ channel will prevail more than 90\% of the
time for the $\widetilde{q}_R$ and two-thirds of the time for the $\widetilde{q}_L$. Thus, the
$\widetilde{q} \rightarrow q \widetilde{\chi}_1^0$ path is comparatively suppressed in $\cal{F}$-$SU(5)$. However,
with the rate of gluino to jets at 60\% for $\widetilde{g} \rightarrow q \overline{q} \widetilde{\chi}_1^0$, this
search should remain sensitive to the $\cal{F}$-$SU(5)$ gluino production. Since leptons are explicitly excluded, the QCD
background here is expected to be troublesome. The implementation of the ``High Mass'' cuts certainly
alleviate the QCD predicament to some degree, although the signal may also be diminished in scope as well.
\subsection{CMS Jet-Z Balance}
We employ the search of Ref.~\cite{PAS-SUS-11-019} here, which is based upon a data sample of 2.1~${\rm fb^{-1}}$.
The JZB method concentrates on states containing a Z-boson, jets and missing energy. The
advantage here is that the contribution from $Z \rightarrow l^+l^-$ is clean, and the Z+jets contribution
can be predicted. This is sensitive to SUSY $\widetilde{g}\widetilde{g}$ production, with the gluino
decay to a neutralino via $\widetilde{g} \rightarrow q \overline{q} \widetilde{\chi}_2^0$, followed
by $\widetilde{\chi}_2^0 \rightarrow Z \widetilde{\chi}_1^0$. However, the branching ratio of
$\widetilde{g} \rightarrow q \overline{q} \widetilde{\chi}_2^0$ is only 18\% in $\cal{F}$-$SU(5)$, while the
$\widetilde{\chi}_2^0 \rightarrow Z \widetilde{\chi}_1^0$ is a mere 0.34\%. Thus, with only a 0.06\%
probability of a $\widetilde{g} \rightarrow q \overline{q} Z \widetilde{\chi}_1^0$ transition,
expectations are that this channel will experience high suppression in $\cal{F}$-$SU(5)$, and hence provide no
observable SUSY signals within the 2.1 ${\rm fb^{-1}}$ data sample.
\section{Simulation and Error Analysis\label{sct:simulation}}
The explicit event selection scenarios from each of the seven CMS and ATLAS search strategies A--G
discussed in the previous section are applied to a representative sampling of the viable
$\cal{F}$-$SU(5)$ parameter space satisfying the bare-minimal constraints of Ref.~\cite{Li:2011xu}. The
resulting event counts are extrapolated to the full phenomenologically viable model space for each of the
seven cases, as depicted in Fig.~\ref{fig:7plex}. To achieve this result, we employ a detailed Monte Carlo
collider-detector simulation of all 2-body SUSY processes based on the standard
{\tt MadGraph}~\cite{Stelzer:1994ta,MGME} suite, including the {\tt MadEvent}~\cite{Alwall:2007st},
{\tt PYTHIA}~\cite{Sjostrand:2006za} and {\tt PGS4}~\cite{PGS4} chain. We employ the ATLAS and CMS
detector specification cards provided with {\tt PGS4}, and specify the cone jet clustering algorithm in all cases,
with an angular scale parameter $\Delta R$ of 0.5 for CMS, and 0.4 for ATLAS. The results are
filtered according to a careful replication of the individual SUSY search selection cuts,
using a script {\tt CutLHCO} of our own design~\cite{cutlhco}. SUSY particle mass calculations were performed using
{\tt MicrOMEGAs 2.1}~\cite{Belanger:2008sj}, employing a proprietary modification of
the {\tt SuSpect 2.34}~\cite{Djouadi:2002ze} codebase to run the {\it flippon}-enhanced RGEs.
\begin{figure*}[htp]
\centering
\includegraphics[width=0.70\textwidth]{7plex.eps}
\caption{Event counts for $\cal{F}$-$SU(5)$ are plotted as a function of the gaugino mass $M_{1/2}$. The span of $M_{1/2}$
in each plot space consists of the minimum $M_{1/2}$ = 385~GeV and maximum $M_{1/2}$ = 900~GeV allowed
by application of the bare-minimal phenomenological constraints of Ref.~\cite{Li:2011xu}. The thickness of
each curve is the consequence of a superposition of statistical uncertainty and the flexible range on the
{\it flippon} mass $M_V$, where variance of $M_V$ has a minor effect on the event counts. The median line transversing
the thickness is the best nominal fit to the $\cal{F}$-$SU(5)$ event count data. The rectangular shaded regions identify
the maximum and minimum number of events allowed to maintain consistency with the CMS and ATLAS reported
SM background and data observations. Therefore, the intersection of the $\cal{F}$-$SU(5)$ curves with the rectangular
range of uncertainty isolates the estimated upper and lower boundaries on $M_{1/2}$ in $\cal{F}$-$SU(5)$ that preserve
uniformity with the CMS and ATLAS results for each individual search.}
\label{fig:7plex}
\end{figure*}
An estimated uncertainty on the fitting between the simulated $\cal{F}$-$SU(5)$ event counts and the mass scale $M_{1/2}$ is
computed, representing a 99\% confidence level. This narrow region of uncertainty on
the $\cal{F}$-$SU(5)$ simulations, represented by the band width in Fig.~\ref{fig:7plex}, characterizes a combination of statistical
uncertainty and variations in the vector-like {\it flippon} mass due to a yet unknown resolution of this
$M_V$ parameter. Although the {\it flippon} mass can in
principle be limited to a more constrained range in order to facilitate a 124-126~GeV Higgs boson mass, we have
depicted in Fig.~\ref{fig:7plex} a range of uncertainty that would be inclusive of all {\it flippon}
masses, for the sake of completeness. For reference, the nominal best $M_{1/2}$ fit to each individual study is
further included. We overlay the upper and lower boundaries of uncertainty on the CMS and ATLAS derived background
and data observations, displayed as rectangular shaded regions in Fig.~\ref{fig:7plex}. This allows us
to clearly demonstrate those regions of the $\cal{F}$-$SU(5)$ model space that comply with each individual search
methodology, noting estimated lower bounds on $M_{1/2}$, and in one particular case, also an upper bound on $M_{1/2}$.
We have attempted to normalize the treatment of error propagation across the
various studies under consideration. In all seven cases, we are provided an
uncertainty on the SM background estimate by the collaboration. In one of
these cases~\cite{PAS-SUS-11-003}, the uncertainty is carefully extracted bin-by-bin
from the graphical presentation and summed in quadrature, while the remaining
six reports provide direct numerical values. Whenever statistical and systematic
errors are provided separately, we likewise combine these in quadrature. When
given a choice between data-driven and Monte Carlo analyses, we favor the background
estimate and associated (reduced) error provided by the data-driven methodology.
If differential upper and lower bounds are provided, we adopt the larger of the
error statistics. The central background estimates and associated 1-$\sigma$
deviations are listed in the second column of Table~\ref{tab:signals} in Section~\ref{sct:correlations}
for each study.
In general, the collaboration studies have not provided an explicit uncertainty on
the observed data counts, but one may confidently expect Poisson statistics to apply
here, and an error scaling that goes like the square root of the recorded events.
Specifically, we adopt a factor of $\sqrt{(1+{\rm Data})}$ across the board, which compares
satisfactorily with a graphical extraction of the event frequency bounds published in
Ref.~\cite{PAS-SUS-11-003}. The relevant observations are summarized in the third column
of Table~\ref{tab:signals}. Since the nominal target for SUSY event contributions is
the observed excess of the recorded data over the SM background estimate, this statistical
uncertainty on the event count must be combined in quadrature with the previously
described uncertainty of the background estimate itself. Reassuringly, a doubling
of this statistic to the 2-$\sigma$ level generates a very favorable match to the
95\% confidence outer bounds on SUSY counts over observed excesses that are reported in
Refs.\cite{Aad:2011qa,ATLAS-CONF-2012-003}.
Finally, any reported excess must be compared against our Monte Carlo simulation of
the $\cal{F}$-$SU(5)$ model space. The statistical errors on our procedure are very small, being
minimized by substantial oversampling of the integrated luminosity. Moreover, the
application of a combined full-model space fit of the expected event count against the
gaugino mass $M_{1/2}$ further smoothes out statistical variations. There remains some
uncertainty due to variation of the vector-like {\it flippon} mass scale $M_{\rm V}$,
but this higher order effect is nicely accounted for by the demonstrated width of the curves
in Fig.~\ref{fig:7plex}, and is thus not further considered outside that context. On the other hand, the
systematic errors on our procedure may be rather large, and are quite difficult to reliably
estimate in a systematic manner. We have opted to employ a factor of
$\sqrt{(1+{\rm Observed~Excess})}$, where the observed excess that the experiments report
over the expected backgrounds are tallied in the fifth column of Table~\ref{tab:signals}.
Although this quantity is actually more naturally suited to describe the sources of statistical
error than the systematic, it may still be a reasonable estimate for the systematic error, in
the absence of other concrete options. Since the comparison of our simulation against the
experimental results constitutes an implicit second level of subtraction, the corresponding error
must again be combined in quadrature with the net experimental error on the data excess.
It is this final statistic that is employed to tally the minimum and maximum permissible event
count variation at the 1-$\sigma$ level in columns 4 and 6 of Table~\ref{tab:signals}. It is
also used as the denominator of our subsequent multi-axis $\chi^2$ best fit against the
seven LHC SUSY search strategies that have been outlined.
\section{The $\cal{F}$-$SU(5)$ Correlations\label{sct:correlations}}
Notable in Fig.~\ref{fig:7plex} is the consistency which the $\cal{F}$-$SU(5)$ model enjoys with all seven search
schemes, with each generally sharing a similar locally favored region of the parameter space. Excluding the highly
suppressed CMS JZB search, the smallest lower bound is given by the ATLAS bjet and lepton search, at a little less
than $M_{1/2} = 440$~GeV, with the ATLAS and CMS multijet searches also
posting sub-500~GeV gaugino masses. In close proximity, the ATLAS bjet and lepton SR1-D search does
however set a lower bound on $M_{1/2}$ just slightly above 500~GeV. As forecasted, the minuscule
production of $q \overline{q} Z \widetilde{\chi}_1^0$ does in fact mightily subdue the number of $\cal{F}$-$SU(5)$
observations in the CMS JZB cutting technique, such that no lower boundary can be ascribed to $M_{1/2}$
using the JZB tactic. Thus, it is interesting that five of the seven searches fix the lower bound on
$M_{1/2}$ at about 500~GeV or less. The residual two probes, namely the ATLAS purely hadronic ``High Mass'' cuts and
ATLAS bjet and lepton SR1-E, call for a lower limit on $M_{1/2}$ in the neighborhood of 550~GeV.
Consequently, we can conclude that the $\cal{F}$-$SU(5)$ model space just above about $M_{1/2} = 550$~GeV is alive and
well after application of all CMS and ATLAS 1-2 ${\rm fb^{-1}}$ constraints, with the model space above
$M_{1/2} = 500$~GeV perfectly tolerated in five of the seven searches. Linking these $M_{1/2}$ model
parameter values to experimentally vital scales, $M_{1/2} = 550$~GeV corresponds to a light stop
$\widetilde{t}_1$ mass of about 600~GeV and a gluino $\widetilde{g}$ of about 750~GeV; $M_{1/2} = 500$~GeV
correlates to a light stop of about 540~GeV and a gluino of around 690~GeV.
Six of the seven schemes require no upper bound on $M_{1/2}$ as a result of there existing no inordinate
number of excess events. Nonetheless, the ATLAS bjet and lepton study does exhibit an excess even at the
maximum 1-$\sigma$ Standard Model limit, when applying the data-driven background statistics. The experimental
collaborations at LHC are striving for data-driven backgrounds in their SUSY searches, hence we believe
the choice of data-driven over Monte-Carlo generated to be justified. If this small residue is indeed substantive,
then we are compelled to enforce an upper boundary on $M_{1/2}$ at about 710~GeV, which
corresponds to a 785~GeV light stop mass and 970~GeV gluino mass. The interesting material
result is the constitution of a narrow strip between 565 $\lesssim M_{1/2} \lesssim$ 710~GeV corresponding
to an overlapping ``Discovery Region'' that is favorable for a potential finding of SUSY at ATLAS and CMS.
In the upper panel of Fig.~\ref{fig:discovery}, we visually demonstrate this overlap by resketching those segments of
the curves from Fig.~\ref{fig:7plex} that maintain compatibility at 1-$\sigma$ with experimental uncertainties on each search. We
remark that the upper limit imposed here is somewhat provisional, pending a more substantial
accumulation of collision data. In particular, the collaborations themselves maintain that the SM alone remains
essentially compatible with the data, to an acceptable statistical significance. Nevertheless, the current single standard
deviation limits taken at face value suggest an upper boundary for substantiation or exclusion on the $\cal{F}$-$SU(5)$ model space at
about $M_{1/2} \simeq$~710~GeV ($\widetilde{t}_1 \simeq$~785~GeV and $\widetilde{g} \simeq$~970~GeV).
\begin{figure*}[htp]
\centering
\includegraphics[width=0.80\textwidth]{Discovery_Chi2.eps}
\caption{In the upper panel, we superimpose event counts for the seven search methodologies studied in this work,
labeling the 1-$\sigma$ overlap between these strategies as the ``Discovery Region''.
The lower panel depicts a multi-axis $\chi^2$ fit to the same set of seven search strategies, where the
cumulative distribution function percentage demarcated on the right-hand axis dips to a minimum of 5.5\%
for the best overall fit at $M_{1/2} = 610$~GeV. The range of the parameter space that provides a
better fit than the median at 1-$\sigma$ significance is in broad agreement with the previously noted
Discovery Region, and both notably intersect with the phenomenologically favored {\it Golden Strip} and
{\it Silver Strip} regions. The light stop and gluino masses (in~GeV) corresponding to each $M_{1/2}$ value
have been inserted onto the lower horizontal axes. Uncertainties of a few GeV exist in the mapping of
the axis labels for both cases, due to higher order fluctuations arising from variation in the top quark
mass $m_t$ and {\it flippon} mass $M_V$.}
\label{fig:discovery}
\end{figure*}
The spatial synchronicity displayed by the Discovery Region in Fig.~\ref{fig:discovery} with the
phenomenologically derived {\it Golden Strip} at $555 \le M_{1/2} \le 580$~GeV
and {\it Silver Strip} at $580 \le M_{1/2} \le$~658~GeV is rather striking,
and embodies the recurring weight of strong correlation between ostensibly independent experimental data
points that we have become increasingly accustomed to observing throughout our extended study
of the No-Scale $\cal{F}$-$SU(5)$ model. It is an essential prerequisite that any high-energy framework of nature
discovered by precise measurements at LHC must correctly simultaneously account for the WMAP-7 measured
relic density, the top quark mass and other precision electroweak parameters, and the rare-process constraints.
This is a test that No-Scale $\cal{F}$-$SU(5)$ seems well poised to pass.
The SM background, data observations, and uncertainty statistics are detailed in Table~\ref{tab:signals}.
The ``Total SM'' and ``Data'' tabulations are those reported by CMS and ATLAS. The
``${\rm Signal_{Min}}$'' and ``${\rm Signal_{Max}}$'' entries describe the 1-$\sigma$ confines on the range of
excess SUSY events, as shown in Fig.~\ref{fig:7plex}. We further display in this table nine markers of the expected
event count in terms of $M_{1/2}$ to numerically illustrate the relevant progression through the region of interest.
Those entries highlighted in green depict consistency with the allowed range of uncertainty. For the ranges
displayed in Table~\ref{tab:signals}, those points in the vicinity of $M_{1/2} =$~600--650~GeV are best capable of
simultaneously satisfying all seven LHC search strategies. Note that the width of the Discovery Region in
Fig.~\ref{fig:discovery} is somewhat wider than this boundary, due to the thickness of the fitting to event counts.
Five benchmark spectra are listed in Table~\ref{tab:masses} by their model parameters, along with those supersymmetric
particle masses directly relevant to our discussion. The presented benchmarks are chosen specifically to highlight
the nominal best fits to various SUSY search strategies from Section~\ref{sct:search}.
Specifically note the Higgs boson masses near 125~GeV for each, in tandem with the light stop and gluino.
\begin{table*}[htpb]
\caption{A comparison of the $\cal{F}$-$SU(5)$ event counts for each of the seven search strategies discussed in
Section~\ref{sct:search}. The ``Total SM'' and ``Data'' columns display the reported SM background and
data observations by the CMS and ATLAS Collaborations, with the ``Observed Excess'' column being the
difference between the nominal values of the SM and data. The ``${\rm Signal_{Min}}$'' and ``${\rm Signal_{Max}}$''
values are the minimum and maximum allowed event counts in order to be consistent with the background and observations,
where the error analysis determination is elaborated in Section~\ref{sct:simulation}.
The ``$M_{1/2}$'' columns (in~GeV) contain the $\cal{F}$-$SU(5)$ event counts for each of the $M_{1/2}$ given, for a mass
range chosen to represent those regions of the parameter space consistent with the LHC searches under study. The
highlighted event counts are those that conform to the stated range within the ``${\rm Signal_{Min}}$''
and ``${\rm Signal_{Max}}$'' upper and lower boundaries.}
{\centering
\footnotesize
\begin{tabular}{|c||c|c||c|c|c||c|c|c|c|c|c|c|c|c|c|}\cline{7-15}
\multicolumn{6}{c|}{} & \multicolumn{9}{c|}{\rm $M_{1/2}$}\\ \cline{1-15}
$\rm Search$&$\rm~~ Total~SM ~~$&$\rm Data$&$\rm Signal_{Min}$&$\rm Observed~ Excess$&$\rm Signal_{Max}$&$~~475~~$&$~~500~~$&$~~525~~$&$~~550~~$&$~~575~~$&$~~600~~$&$~~625~~$&$~~650~~$&$~~675~~$ \\ \hline \hline
$\rm A$&$3.4 \pm 0.7$&$8$&$0.7$&$4.6$&$8.4$&${\color{green4} 6.9}$&${\color{green4} 5.7}$&${\color{green4} 4.7}$&${\color{green4} 3.8}$&${\color{green4} 3.1}$&${\color{green4} 2.5}$&${\color{green4} 2.0}$&${\color{green4} 1.5}$&${\color{green4} 1.2}$ \\ \hline
$\rm B$&$1.3 \pm 0.9$&$3$&$0.0$&$1.7$&$4.4$&$4.8$&${\color{green4} 4.0} $&${\color{green4} 3.3}$&${\color{green4} 2.7} $&$ {\color{green4} 2.2}$&${\color{green4} 1.8}$&${\color{green4} 1.5}$&${\color{green4} 1.2}$&${\color{green4} 0.9}$ \\ \hline
$\rm C$&$54.9 \pm13.6$&$74$&$2.4$&$19.1$&$35.8$&${\color{green4} 22.7} $&${\color{green4} 17.1} $&${\color{green4} 12.8} $&${\color{green4} 9.5} $&$ {\color{green4} 7.0}$&${\color{green4} 5.0}$&${\color{green4} 3.6}$&${\color{green4} 2.4}$&$1.6$ \\ \hline
$\rm D$&$77.0 \pm 18.4$&$81$&$0.0$&$4.0$&$24.7$&$40.9 $&$31.3 $&${\color{green4} 23.8} $&${\color{green4} 18.0} $&$ {\color{green4} 13.5}$&${\color{green4} 10.0}$&${\color{green4} 7.2}$&${\color{green4} 5.1}$&${\color{green4} 3.4}$ \\ \hline
$\rm E$&$14.4 \pm 5.4$&$17$&$0.0$&$2.6$&$9.7$&$26.3 $&$ 21.0 $&$ 16.6 $&$ 13.1 $&$ 10.2 $&${\color{green4} 7.9}$&$ {\color{green4} 6.0}$&${\color{green4} 4.5}$&${\color{green4} 3.2}$ \\ \hline
$\rm F$&$13.1 \pm 3.1$&$18$&$0.0$&$4.9$&$10.8$&$26.6$&$20.9 $&$16.3$&$12.7 $&$ {\color{green4}9.8}$&${\color{green4}7.5}$&${\color{green4}5.7}$&${\color{green4} 4.3}$&${\color{green4} 3.1}$ \\ \hline
$\rm G$&$7.0 \pm 2.6$&$11$&$0.0$&$ 4.0$&$8.9$&${\color{green4} 1.2} $&${\color{green4} 0.9} $&${\color{green4} 0.7} $&$ {\color{green4} 0.6}$&${\color{green4} 0.4} $&${\color{green4} 0.3} $&$ {\color{green4} 0.2}$&${\color{green4} 0.2}$&${\color{green4} 0.1}$ \\ \hline
\end{tabular}}
\label{tab:signals}
\end{table*}
\begin{table*}[htpb]
\caption{Higgs boson and sparticle masses (in~GeV) are given for five benchmark gaugino masses $M_{1/2}$,
representative of a best fit for each search methodology examined in this work. The light stop
$\widetilde{t}_1$ and gluino $\widetilde{g}$ columns have been highlighted to reflect their discovery
mass ranges, all of which should be accessible at the $\sqrt{s} = 8$~TeV LHC in 2012, whereas the lighter
points may have already been substantively probed by the $\sqrt{s} = 7$~TeV LHC during the 2011 run. The $M_{1/2} = 518$~GeV point
is the representative benchmark of Ref.~\cite{Li:2011ab}. The $M_{1/2}$ = 610~GeV point is also
indicative of the precise minimum of the multi-axis $\chi^2$ fit described in Section~\ref{sct:chi2}.
Significantly each of the five benchmarks cataloged here can moreover handily generate a 124-126~GeV Higgs mass.}
\centering
\footnotesize
\begin{tabular}{|c|c|c|c|c||c|c|c|c|c|c|c|c|c|c|c|c|c|}\hline
$\rm Search$&$\rm M_{1/2}$&$\rm M_V$&$\rm m_t$&$\rm tan\beta$&$\rm \widetilde{\chi}_1^0$&$\rm Higgs$&$\rm \widetilde{\chi}_2^0$&$\rm \widetilde{\chi}_1^{\pm}$&$\rm \widetilde{t}_1$&$\rm \widetilde{g}$&$\rm \widetilde{b}_1$&$\rm \widetilde{t}_2$&$\rm \widetilde{b}_2$&$\rm \widetilde{u}_R$&$\rm \widetilde{d}_R$&$\rm \widetilde{u}_L$&$\rm \widetilde{d}_L$ \\ \hline \hline
$\rm A$&$ 518$&$ 1640$&$ 174.4$&$ 20.65$&$99 $&$ \textbf{125.4}$&$216 $&$216 $&${\color {blue} 558} $&${\color {blue} 704} $&$934 $&$982 $&$1046 $&$1053 $&$1094 $&$1144 $&$1147 $ \\ \hline
$\rm B$&$ 610$&$ 2500 $&$ 174.3 $&$21.44 $&$ 121 $&$ \textbf{124.4}$&$260 $&$260 $&${\color {blue} 669 } $&${\color {blue} 826 }$&$1076 $&$ 1117 $&$1194 $&$ 1207$&$ 1252 $&$ 1312$&$ 1314 $ \\ \hline
$\rm C$&$ ~485~$&$~1475~ $&$~174.3~ $&$ ~20.40~$&$~92~ $&$~ \textbf{125.5}~ $&$~200~ $&$ ~200~$&$ ~{\color {blue} 518}~$&$~{\color {blue} 661}~ $&$~881 ~$&$~932~ $&$ ~989~$&$ ~994~$&$~1034~ $&$~1080 ~$&$~1083~ $ \\ \hline
$\rm D,E$&$ 675$&$2950 $&$ 174.4 $&$ 21.87 $&$136 $&$\textbf{124.6} $&$291 $&$291 $&${\color {blue} 746 } $&${\color {blue} 910 } $&$1179 $&$1215 $&$ 1301 $&$1318 $&$ 1367 $&$ 1433 $&$ 1435 $ \\ \hline
$\rm F$&$ 638$&$ 2505$&$174.4 $&$ 21.63$&$ 127$&$ \textbf{124.9}$&$ 273$&$ 273$&$ {\color {blue} 703}$&${\color {blue} 861} $&$1123 $&$1161 $&$1243 $&$1258 $&$1305 $&$1367 $&$1369 $ \\ \hline
\end{tabular}
\label{tab:masses}
\end{table*}
\section{A Multi-Axis $\chi^2$ Fit\label{sct:chi2}}
We have implemented a $\chi^2$ test in order to establish the optimal correspondence
between the No-Scale $\cal{F}$-$SU(5)$ model space and the ongoing LHC SUSY search, and also to
gauge the overall statistical significance of the resulting best fit. To facilitate
this task, it was necessary to first establish a continuous functional relationship
between the gaugino mass $M_{1/2}$ and the expected event count for each SUSY search
strategy under consideration. It should be noted that this process is greatly simplified,
and the result thereby lent much greater parsimony, by the fact that the spectrum is
generated at leading order by only the single mass parameter. To proceed, we generously
sampled the $\cal{F}$-$SU(5)$ model space at nineteen representative benchmark combinations of
$(M_{1/2},M_{\rm V},m_{\rm t}~{\rm and}~\tan\beta)$, generating a detailed Monte Carlo
collider-detector simulation, including the careful application of relevant selection cuts,
as described in Section~\ref{sct:simulation}.
For each search strategy, a satisfactory continuous empirical fit was obtained
by linear regression at quadratic order to the log-log distribution of event counts
vs. $M_{1/2}$.
The $\chi^2$ test statistic, which is expected to asymptotically approach the
formal $\chi^2_N$ distribution, is defined as
\begin{equation}
\chi^2 (M_{1/2}) = \sum_{i=1}^N \left\{ \frac{{\left({\rm Events}(M_{1/2})_i - {\rm Excess}_i \right)}^2}{\sigma_i^2} \right\} \,
\end{equation}
where ${\rm Events}(M_{1/2})_i$ is the continuous fit to the number of SUSY events expected
from $\cal{F}$-$SU(5)$ at the given mass scale, ${\rm Excess}_i$ is the observed excess from Table~\ref{tab:signals}
and $\sigma_i^2$ is the square of the single standard deviation error described in Section~\ref{sct:correlations},
all under the $i^{th}$ set of selection cuts, with $N = 7$.
The resulting function is plotted in the lower panel of Fig.~\ref{fig:discovery}, demonstrating a distinct
minimum in the vicinity of $M_{1/2} = 610$~GeV, corresponding to light stop and gluino masses of approximately
665~GeV and 830~GeV. The significance of the fit is established by comparison
with the formal $\chi^2_N$ probability distribution with $N$ degrees of freedom, which establishes
the likelihood of a given value for the $\chi^2$ test statistic under application of the ``null hypothesis'',
{\it i.e.}~where all signal deviations from the observed excess are attributed to uncorrelated Gaussian
fluctuations about the $\sigma_i$.
The figure of merit is the cumulative distribution function (CDF) of $\chi^2_N$,
which indicates the fraction of randomized trials under action of the null hypothesis that should be expected to
produce a lower value (better fit) of the $\chi^2$ test statistic than some given threshold value. The median
value of the CDF for $N=7$ is $6.35$, and the double-sided $\pm 1,2 \sigma$ CDF values, corresponding to the traditional
Gaussian percentage thresholds of 2.28\%, 15.87\%, 84.13\% and 97.73\% (centrally encapsulating 68\% and 95\%),
fall at $\chi^2 = (1.64, 3.44, 10.57~{\rm and}~16.27)$, respectively. The best fit value of $M_{1/2}$ produces
a rather small $\chi^2$ of 2.24, corresponding to a CDF of 5.5\%, immediately on the cusp of the range generally
considered to represent a statistically significant deviation from the null hypothesis. The fit produced in the
SM limit, the asymptote of the soft high-mass boundary in the lower panel of Fig.~\ref{fig:discovery}, is also
reasonably satisfactory, with $\chi^2 = 4.62$, and a CDF of 29.4\%. Models disfavored for overproduction at
the 2- and 1-$\sigma$ limits have mass scales $M_{1/2}$ of 501~GeV and 518~GeV, respectively, while the median fit
occurs at 538~GeV. Models favored by a negative deviation of one standard deviation or more in the CDF exist within the
range from $M_{1/2} = 564$~GeV to $M_{1/2} = 709$~GeV, with the best fit, again, around $M_{1/2} = 610$~GeV.
\section{The Once and Future LHC}
We close our analysis with a brief glance in postscript toward the future $\sqrt{s}$ = 8 TeV beam energy and 15 ${\rm fb^{-1}}$
of data expected in 2012, as well as the already collected 5 ${\rm fb^{-1}}$ at $\sqrt{s} = 7$~TeV that remains to be reported.
If the results presented in our study in fact represent persistent correlations in the data, and not merely statistical fluctuations,
then evidence of their verity should continue to ripen. The ``Observed Excess'' in Table~\ref{tab:signals} exemplifies the slender
corridor with which we are attempting to extricate a signal of supersymmetry's presence in the data. All excesses but one
are less then five events, making precise extrapolations tenuous. Moreover, the subtraction of large numbers (the net event count
and the expected SM background) to yield a small differential implies rather large proportional uncertainties, pushing the statistical
machinery to extremes. This is a key reason that the SM asymptote maintains a reasonably favorably $\chi^2$ value in
Fig.~\ref{fig:discovery}.
Although (or because) we remain in a fledgling phase of the LHC data collection mission, and the status of plausible signal candidates
is still tentative, this is also a period of incredibly rapid and dynamic development at the high energy and high intensity frontiers.
Firstly, the fact that the beam quality is still being tuned means that the time integrated luminosity continues to grow much more
rapidly than a linear trend. Secondly, since the doubling interval for data collection is still somewhat low, newly accumulated statistics
make extremely strong fractional contributions to the combined knowledge over reasonable time scales. A growth of the excess to a
minimum of ten events will assist in sharpening the analysis to a degree, and we suspect that a full analysis of the 2012 statistics
should put us well on the way toward a conclusive resolution to the matter. Thirdly, the shortly anticipated upgrade to a $\sqrt{s} = 8$~TeV
beam will substantially enhance the expected SUSY event cross-sections; our Monte Carlo simulations attribute an improved
time efficiency in the collection of productive data on the order of two to this upgrade.
Table~\ref{tab:610} reports the extrapolated signal significance at $5~{\rm fb}^{-1}$ for $\sqrt{s} = 7$~TeV,
computed as the ratio $S/\sqrt{B+1}$ of signal events $S$ to the square root of one plus the expected background $B$,
for each of the seven considered SUSY searches A--G, assuming viability of the central $\chi^2$ fit at $M_{1/2} = 610$~GeV.
The ``discovery index'' (DI) calculates the required scaling on luminosity, reported in inverse femtobarns, that would be required
to establish a nominal signal significance of five. Also shown is the signal significance at $15~{\rm fb}^{-1}$ for $\sqrt{s} = 8$~TeV, summed with the statistics for $5~{\rm fb}^{-1}$ for $\sqrt{s} = 7$~TeV, for a total of $20~{\rm fb}^{-1}$. We see that four of the seven searches exceed the gold standard of 5.0 for signal significance for the total collected luminosity of $20~{\rm fb}^{-1}$ expected by the close of 2012.
\begin{table*}[htb]
\caption{The extrapolated signal significance $S/\sqrt{B+1}$ at $5~{\rm fb}^{-1}$ for $\sqrt{s} = 7$~TeV is presented for
each of the seven considered SUSY searches A--G, for $M_{1/2} = 610$~GeV. The ``discovery index'' (DI)
calculates the required scaling on luminosity, reported in inverse femtobarns, that would be required to
establish a nominal signal significance of five. Also shown is the signal significance at $15~{\rm fb}^{-1}$ for $\sqrt{s} = 8$~TeV, summed with the statistics for $5~{\rm fb}^{-1}$ for $\sqrt{s} = 7$~TeV, for a total of $20~{\rm fb}^{-1}$.}
{\centering
\footnotesize
\begin{tabular}{|c||c|c|c|c||c|c||c|}\cline{2-8}
\multicolumn{1}{c|}{} & \multicolumn{3}{c|}{$5~{\rm fb^{-1}}~@~7~{\rm TeV}$} & \multicolumn{1}{c||}{$7~{\rm TeV}$} & \multicolumn{2}{c||}{$15~{\rm fb^{-1}}~@~8~{\rm TeV}$} & \multicolumn{1}{c|}{$5~{\rm fb^{-1}}~@~7~{\rm TeV}~+~15~{\rm fb^{-1}}~@~8~{\rm TeV}$}\\ \cline{1-8}
$\rm ~~Search~~$&$~~{\cal F}-SU(5)~~$&$\rm ~~SM~~$&$\rm ~~S/ \sqrt{B+1}~~$&$\rm ~~DI~({\rm fb^{-1}})~~$&$~~{\cal F}-SU(5)~~$&$\rm ~~SM~~$&$\rm ~~S/ \sqrt{B+1}~~$ \\ \hline \hline
$\rm A$&$ 8.2$&$ 15.5$&$ 2.0$&$ 29~ $&$57$&$107$&$5.8$ \\ \hline
$\rm B$&$ 4.5$&$ 4.9 $&$ 1.9 $&$31~ $&$31$&$34$&$5.7$ \\ \hline
$\rm C$&$ 15.5$&$267 $&$0.9 $&$ 139~ $&$107$&$1842$&$2.7$ \\ \hline
$\rm D$&$ 16.3$&$188 $&$ 1.2 $&$ 88~ $&$113$&$1297$&$3.3$ \\ \hline
$\rm E$&$ 12.9$&$ 35$&$2.2 $&$ 26~ $&$89$&$242$&$6.1$ \\ \hline
$\rm F$&$ 24.0$&$ 63$&$3.0 $&$ 13.8~ $&$166$&$435$&$8.5$ \\ \hline
$\rm G$&$ 0.17$&$ 16.0$&$0.1 $&$ 69,204~ $&$1.2$&$110$&$0.1$ \\ \hline
\end{tabular}}
\label{tab:610}
\end{table*}
\section{Conclusions}
No conclusive indication of supersymmetry (SUSY) has been observed at the early LHC as the accumulation of
data advances toward 5~${\rm fb^{-1}}$, yet could enticing clues be germinating in the massive collection
of observations? This is the challenging question that presently attracts our interest, and which we may currently
attempt to address only by leaning upon the limited 1--2~${\rm fb^{-1}}$ of integrated luminosity amassed thus far.
Although we cannot argue for incontrovertible evidence of SUSY peeping beyond the Standard Model veil, we can
safely suggest that No-Scale $\cal{F}$-$SU(5)$ is a better global fit to the data than the SM alone, and moreover that
its predictions appear to be meaningfully correlated with observed low-statistics excesses across a wide variety
of specialized search strategies, while gracefully avoiding devastating overproduction where events are not observed.
This is a strong statement in an era when the portion of phenomenologically viable MSSM and mSUGRA constructions
is diminishing rapidly, choked out by inconsistency with the Higgs measurements and advancing squark and gluino exclusion limits.
The No-Scale $\cal{F}$-$SU(5)$ model, by virtue of its distinctive supersymmetric mass hierarchy of
$M({\widetilde{t_1}}) < M({\widetilde{g}}) < M({\widetilde{q}})$, possesses the signature event
fingerprint of a very high multiplicity of hadronic jets. Moreover, the light stop and gluino masses,
possibly within reach of data collected during the $\sqrt{s} = 7$~TeV LHC run, renders
presently maturing searches aimed at stops and gluinos quite pertinent to the testing of $\cal{F}$-$SU(5)$ as well.
Building upon a prior analysis of two existing searches conducted by CMS and ATLAS, we studied
five additional CMS and ATLAS search strategies, variously employing cuts on jets,
b-jets and leptons, designed to reveal light stop, gluino, and squark production. A number of
interesting conclusions were established.
Notably, we found that the entire region of the model
space for $M_{1/2} \gtrsim$ 440~GeV ($m_{\widetilde{t}_1} \gtrsim$ 460~GeV and
$m_{\widetilde{g}} \gtrsim$ 600~GeV) thrives in at least one of the seven search methodologies.
When all seven searches are combined, a lower bound of $M_{1/2} \gtrsim$ 565~GeV ($m_{\widetilde{t}_1} \gtrsim$ 615~GeV and
$m_{\widetilde{g}} \gtrsim$ 770~GeV) can be tentatively set.
Also intriguing is the potential for one experiment, the ATLAS search requiring at least
one b-jet and exactly one lepton, to demand an {\it upper} bound of $M_{1/2} \lesssim$ 710~GeV
($m_{\widetilde{t}_1} \lesssim$ 785~GeV and $m_{\widetilde{g}} \lesssim$ 970~GeV). However, it should be
mentioned that more recent ATLAS results (also included in the present study) on similar, though not identical,
selection cuts do alleviate the need to mandate an upper bound on $M_{1/2}$. The values of $M_{1/2}$ which are compatible
with all search strategies under present consideration in the 1-$\sigma$ overlap exist within the
range of 565 $\lesssim M_{1/2} \lesssim$ 710~GeV, which we refer to as the Discovery Region.
In order to test the statistical significance of any correlations across the simulated $\cal{F}$-$SU(5)$ collider
response in these seven search strategies, we implemented a multi-axis $\chi^2$ fitting procedure. The best
overall match was obtained in the vicinity of $M_{1/2} = 610$~GeV (corresponding to light stop and gluino
masses of approximately 665~GeV and 830~GeV), where the $\chi^2_N$ cumulative distribution
function in seven parameters was reduced to 5.5\%. The range of masses having a better fit than the median
at 1-$\sigma$ significance is 564--709~GeV, which is in excellent agreement with the simpler overlap statistic
just reported.
Both mechanisms produce a conspicuous overlap with the highly phenomenologically favorable
{\it Golden Strip} and {\it Silver Strip}, which add good agreement with rare process constraints on flavor
changing neutral currents and the anomalous magnetic moment of the muon to the broader capacity of the model
for respecting the WMAP7 relic density, the world average top-quark mass, radiative electroweak symmetry breaking,
precision LEP Higgs and SUSY constraints, and the dynamically established boundary conditions of No-Scale supergravity.
No less propitious is the ability to handily generate a 125~GeV Higgs boson
mass through additional loop process contributions from the vector-like {\it flippon} multiplets, producing
a fine accord with the statistical excesses recently reported by CMS and ATLAS in the mass range of 124-126~GeV.
The damage exacted onto the supersymmetric model landscape by the swiftly progressing LHC constraints
has been severe. The tension between the growing likelihood of a 125~GeV Higgs boson mass, developing CMS
and ATLAS exclusion zones, and a supersymmetric spectrum light enough to be within reach of the current
operational phase of the LHC has greatly altered the conventional wisdom as to how a discovery
of supersymmetry would manifest at the LHC. While the validation prospects for almost all prospective
models has greatly withered, the outlook for $\cal{F}$-$SU(5)$ appears to have in fact brightened; in stark contrast,
it is perfectly capable of simultaneously striking each of these three targets. A rare feat nowadays, indeed.
\begin{acknowledgments}
This research was supported in part
by the DOE grant DE-FG03-95-Er-40917 (TL and DVN),
by the Natural Science Foundation of China
under grant numbers 10821504 and 11075194 (TL),
by the Mitchell-Heep Chair in High Energy Physics (JAM),
and by the Sam Houston State University
2011 Enhancement Research Grant program (JWW).
We also thank Sam Houston State University
for providing high performance computing resources.
\end{acknowledgments}
|
{
"timestamp": "2012-03-09T02:04:56",
"yymm": "1203",
"arxiv_id": "1203.1918",
"language": "en",
"url": "https://arxiv.org/abs/1203.1918"
}
|
\section{Introduction}
We have entered the so-called era of accurate cosmology \citep{Peebles:2002iq}.
The aim is to understand the composition and expansion history of the universe at the percent level. In particular, one of the goals is to establish if the observed acceleration of the universe \citep{Perlmutter:1998np, Riess:1998cb} is driven by the cosmological constant or by a fluid with negative pressure, dark energy.
At the most basic level the question is if the observed equation of state $w$ is compatible with $-1$ or not, the value corresponding to the cosmological constant.
It is therefore crucial to study all possible systematic effects on $w$ \citep{Amendola:2010ub,Sinclair:2010sb, Marra:2010pg, deLavallaz:2011tj, Romano:2011mx}.
As our observations are confined to the light cone, there is an intrinsic degeneracy between temporal evolution and spatial variation around us.
In particular, inhomogeneities around us are degenerate with the properties of dark energy, most importantly its equation of state.
A clear example of how intertwined are attempts to detect any evolution of dark energy to large-scale structures is given by the so-called ``void models''.
An observer inside a spherical underdensity expanding faster than the background sees indeed apparent acceleration, thus removing the need for dark energy \cite[see e.g.][and references therein]{Marra:2011ct}.
Void models strongly violate the Copernican principle and have been ruled out -- at least in their simplest incarnation -- as they predict a too strong kinematic Sunyaev-Zel'Dovich effect \citep{GarciaBellido:2008gd,Zhang:2010fa,Moss:2011ze,Zibin:2011ma}.
While on one hand this strengthens the case for dark energy as the likely explanation for the acceleration of the universe, on the other hand it illustrates how large-scale structure can alter the determination of cosmological parameters.
Therefore, it is necessary to adequately model large-scale structures if one has to achieve the grand goal of accurately determining the composition of the universe.
In \citet{Valkenburg:2011ty} it was shown by means of mock data that a local inhomogeneity, of proportions similar to a structure on the surface of last scattering that could cause the CMB Cold Spot, can have strong effects on our perception of the equation of state of dark energy. Here we extend that analysis to real data using the model of \citet{Marra:2011zp}.
More precisely, we consider a $w$CDM model endowed with a local almost-linear inhomogeneity surrounding the observer and test it against supernova observations, CMB anisotropies and local measurements of the Hubble parameter. By $w$CDM we mean a universe containing dark matter and a dark-energy fluid with equation of state $w$.
In this way we can show how the observed large-scale structure of the universe could impact the reconstruction of the dark-energy parameters.
Following \citet{Valkenburg:2011ty}, we consider an inhomogeneity inspired by the observed Cold Spot in the CMB, which has a radius of roughly $5^{\circ}$ and a temperature deviation of roughly $\mathcal{O}(50\sim200)$ $\mu$K.\footnote{See e.g.~\cite{Cruz:2006sv,Zhang:2009qg,Bennett:2010jb}.}
The idea is that the Cold Spot is a primary CMB anisotropy due to an object on the surface of last scattering, and not a secondary effect caused by an object along the line of sight \citep{Tomita:2005nu,Inoue:2006rd,Inoue:2006fn,Masina:2008zv}.
Such an inhomogeneity has a radius of roughly 1 Gpc and a density contrast today of roughly $-0.1$ \citep{Valkenburg:2011ty}.
It is therefore too shallow to bias the $w$CDM model to the point of removing the need for dark energy, as in the void scenario.
Nonetheless, as argued above, such a structure may bias the value of the dark-energy parameters to a level that may be important if one wants to determine whether dark energy is a cosmological constant or not.
Moreover, as we briefly argue in the body of this paper, structures of radius $\sim$1 Gpc and density contrast today $\sim$0.01 -- which still give an interesting effect -- are not at more than three times the dispersion of the density perturbations arising from a close to scale-invariant primordial spectrum.
Therefore, the setup considered in this paper is not in conflict with standard cosmology and, in particular, the Copernican principle~\citep{Valkenburg:2012td}.
We model the inhomogeneity with a particular case of the spherically symmetric solution presented in \citet{Marra:2011zp}, which features a pressureless matter component and a dark-energy fluid with constant equation of state and negligible sound speed $c_s$.
The possibility of a dark-energy fluid with negligible sound speed has been investigated in the literature under various assumptions. This generally requires a non canonical scalar field like $k$-essence and kinetic gravity braiding, as opposed to standard quintessence models with canonical scalar fields which always have $c_{s}=1$ \citep[see e.g.][and references therein]{Creminelli:2008wc, Bertacca:2008uf, Lim:2010yk,Bertacca:2010ct,Deffayet:2010qz,Li:2011sd}.
Here we choose this particular model because it significantly simplifies the dynamical equations and the numerical analysis as there are no pressure gradients that can generate peculiar velocities from an initially comoving motion.
We use this model phenomenologically so as to minimally extend the parameter space of the $w$CDM model by adding only two extra parameters: the radius of the inhomogeneity and its overall contrast. All the other initial conditions follow indeed rigidly.
The paper is organized as follows.
In Section \ref{model} we go briefly through the formalism of the model and its initial conditions, and in Section \ref{analysis} we explain how the cosmological data analysis has been performed.
We show in Section \ref{results} that the effect of local inhomogeneity on the dark-energy parameters can be important and that the inhomogeneity is not unlikely to occur. We conclude in Section \ref{conclusions}.
\section{The model} \label{model}
We consider the case of an observer located at the center of an inhomogeneous sphere embedded in a flat $w$CDM universe.
The inhomogeneities are given by a pressureless matter component and by a dark-energy fluid with constant equation of state $w_{\rm out}$ and negligible sound speed (the subscript ``out'' refers to values at $r>r_{b}$ where $r_{b}$ is the comoving radius of the inhomogeneity). To be more precise, the sound horizon is much smaller than the inhomogeneity scale considered so that we can set $c_s=0$ throughout the paper.
In terms of the radius dependent non-adiabatic equation of state this means that we consider:
\begin{equation}
w(r,t) = w_{\rm out} \, {\rho_{X, {\rm out}} (t) \over \rho_{X}(r,t)} \,,
\end{equation}
that is, the pressure is homogenous:
\begin{equation}
p_{X} (r,t)=p_{X, {\rm out}} (t) = w_{\rm out} \, \rho_{X, {\rm out}} (t) \,,
\end{equation}
where $r$ is the coordinate radius, $t$ is cosmic time, $\rho$ denotes energy density, $p$ denotes pressure and the label $X$ refers to the dark-energy fluid.
Since pressure gradients are absent, matter and dark energy evolve along geodesics.
Moreover, we set initial conditions such that dust and dark energy are initially comoving. Therefore, the absence of pressure gradients implies that peculiar velocities between the two fluids will never develop and that the matter and dark-energy reference frames always coincide.
Next, we discuss the equations governing the dynamics of the model and the relevant initial conditions.
\subsection{Dynamical equations} \label{equa}
We adopt the exact spherically symmetric inhomogeneous solution with $n$ perfect fluids presented in \citet{Marra:2011zp}, which we limit to the case discussed above.
We will now briefly introduce the relevant equations, and we refer to \citet{Marra:2011zp} for the general equations and more details.
Using the reference frame of the dust and dark-energy components, the metric describing our model is:
\begin{equation} \label{metric}
\textrm{d}s^2=-\textrm{d}t^2+\frac{Y'(r,t)^2}{1-k(r) r^2}\textrm{d}r^2+Y(r,t)^2\textrm{d}\Omega^{2} \,,
\end{equation}
where $Y(r,t)$ is the scale function, $k(r)$ is the curvature function, $\textrm{d}\Omega^{2}= \textrm{d} \theta^{2} + \sin^{2}\theta \, \textrm{d} \phi^{2}$ and we have set $c=1$.
A prime denotes partial derivation with respect to the coordinate radius $r$, whereas a dot denotes partial derivation with respect to the coordinate time $t$.
The curvature function is time independent because of the adopted reference frame and sound speed~\citep{Marra:2011zp}.
The metric (\ref{metric}) is written in the same form as the Lema\^itre-Tolman-Bondi (LTB) metric. However, with the inclusion of a dark-energy fluid, the dynamics is no longer that of the LTB metric.
The metric of Eq.~(\ref{metric}) reduces to the Friedman-Lema\^itre-Robertson-Walker (FLRW) metric if $k(r) =$const and $Y(r,t)=r \, a(t) $, where $a(t)$ is the scale factor.
We label the dust component with $M$ and the dark-energy component with $X$.
The conservation equation for the dust source can be solved directly and gives:
\begin{equation} \label{dustevo}
{\rho_{M}(r, t) \over \rho_{M}(r, \bar t)} = {Y^{2}(r, \bar t) Y'(r, \bar t) \over Y^{2}(r, t) Y'(r, t) } \,,
\end{equation}
where $\bar t$ is the initial time at which we give the initial conditions.
As explained before, there are no peculiar velocities between the two fluids and the remaining dynamical equations reduce to:
\begin{align}
\dot Y^2(r,t) =& \frac{2 G F(r,t)}{Y(r,t)} -k(r) r^2 , \label{set1} \\
\dot Y'(r,t) =& \frac{G F'(r,t)}{Y(r,t) \dot Y(r,t)} -\frac{G F(r,t) Y'(r,t)}{Y^2(r,t) \dot Y(r,t)} - { [k(r)r^2]' \over 2 \dot Y(r,t)} , \label{set2} \\
\dot F(r,t) =&- 4 \pi Y^{2}(r,t) \dot Y(r,t) \; p_{X, {\rm out}} (t) , \label{set3}\\
\dot \rho_{X}(r,t) =& - \Big [\rho_{X}(r,t)+p_{X, {\rm out}} (t) \Big ] \Big [H_R(r,t) + 2 H_A(r,t) \Big] , \label{set4}
\end{align}
where the radial and angular expansion rates are $H_R = \dot{Y}' /Y'$ and $H_A=\dot{Y} /Y$, and $F$ is the total effective gravitating mass which also satisfies the following consistency equation:
\begin{equation}
F'(r,t) = 4 \pi Y^{2}(r,t) Y'(r,t) \Big [ \rho_M (r,t) + \rho_{X}(r,t) \Big ] \,.
\end{equation}
Eq.~(\ref{set2}) is the $r$-derivative of Eq.~(\ref{set1}) and allows us to solve directly for the unknown functions $Y$, $Y'$, $F$ and $\rho_X$ without having to take numerical derivatives.
If $w=w_{\rm out}=-1$, this solution becomes the usual $\Lambda$LTB model which has been studied recently in, e.g.,~\citet{Enqvist:2006cg, Sinclair:2010sb, Marra:2010pg, Valkenburg:2011tm, Romano:2011mx}.
Finally, in this particular case the light-cone equations have the same form as in the LTB model:
\begin{equation}
\frac{dt}{dz} = -\frac{Y'}{(1+z)\dot{Y}'} \,, \qquad
\frac{dr}{dz} =\frac{\sqrt{1-k(r) r^2}}{(1+z)\dot{Y}'} \,.
\end{equation}
\subsection{Initial and boundary conditions} \label{inico}
\begin{figure}
\begin{center}
\includegraphics[width= \columnwidth]{profile}
\caption{Shape of the auxiliary function $W_3\left(x,0\right)$ that is used in Eq.~(\ref{profi}) to model the curvature profile.}
\label{fig:profile}
\end{center}
\end{figure}
We fix the flat $w$CDM background model by setting $h$, $\Omega_{X}$ and $w_{\rm out}$, where $\Omega_{X}$ is the present-day background dark-energy density parameter and $h$ is the present-day dimensionless Hubble rate defined by $H_0=100 \, h$ km s$^{-1}$ Mpc$^{-1}$.
We fix the gauge for the radial coordinate in Eq.~\eqref{metric} such that \mbox{$Y(r, \bar t)= r \, a(\bar t)$} at some initial time $\bar t$.
We choose to parametrize the curvature by means of
\begin{align} \label{profi}
k(r) =& k_c \; W_3\left(\frac{r}{r_b},0\right) \,,
\end{align}
where $r_b$ is the comoving radius of the spherical inhomogeneity and:
\begin{align} \label{eq:theprofile}
W_3\left(x,0\right)=\left\{\begin{array}{ll}
\frac{1}{4\pi^2} + 1 - 2 x^2 - {\cos \left( 4\pi x \right) \over 4\pi^2} & \mbox{for } 0 \le x < \frac{1}{2}\\
\frac{-1}{4\pi^2} + 2 \left (1 -x \right)^2 + {\cos \left( 4\pi x \right) \over 4\pi^2} & \mbox{for } \frac{1}{2} \le x < 1,\\
0& \mbox{for } x\geq 1
\end{array}\right.
\end{align}
is the third order of the function $W_n(x,\alpha)$, which has been defined in~\citet{Valkenburg:2011tm} and interpolates from $1$ to $0$ in the interval $\alpha < x < 1$ while remaining $C^n$ everywhere.
Hence $k(r)$ is $C^3$ everywhere, such that the metric is $C^2$ and the Riemann curvature is $C^0$.
Although the function $W_3\left(x,0\right)$ looks rather complicated, it has actually a very simple shape as one can see in Fig.~\ref{fig:profile}.
The curvature profile of Eq.~(\ref{profi}) is exactly zero for $r \ge r_b$ and so the metric is correctly matched to the exterior spatially-flat $w$CDM model, which means that the central over- or under-density is automatically compensated by a surrounding under- or over-dense shell.
The constant $k_c$ gives the curvature at the center of the inhomogeneity and will determine its density contrast.
A spherical inhomogeneity depends crucially on only two {\it physical} parameters: the radius and the overall density contrast.
Therefore, the precise shape of the density profile should not be essential and our analysis should be representative also of other possible curvature or density profiles.
Next, we have to give initial conditions at $t=\bar t$ for $F(r, \bar t)=\bar F_{M}(r)+\bar F_{X}(r)$.
We choose $\bar t$ such that matter is dominant over dark energy, $\bar F_{X}(r) \ll \bar F_{M}(r)$, and so the model becomes the standard dust LTB model.
We can then link the curvature function $k(r)$ to the initial condition for $\bar F_{M}(r)$ by demanding that the universe has the same age $\bar t$ for any $r$. That is, we demand a homogeneous Big Bang, implying the absence of decaying modes in the matter density \citep{Zibin:2008vj}.
In particular we can use the following analytic result of \citet{VanAcoleyen:2008cy} valid for a linear matter density contrast:
\begin{equation}
k(r) \simeq {5\over 3} a^2(\bar t) H_{\rm out}(\bar t) ^2 \; \bar \delta_{F_M}(r) \,,
\end{equation}
which clearly relates the curvature at the center $k_c$ to the matter contrast at the center $\bar \delta_{M} =\bar \delta_{F_M}(0)$.
The latter term is the contrast in the gravitating mass and is defined as:
\begin{equation}
\bar \delta_{F_M}(r) = {\bar F_{M}(r) \over \bar F_{M, {\rm out}}(r)} -1 \,,
\end{equation}
where $\bar F_{M, {\rm out}}= {4 \pi \over 3} a^{3}(\bar t) r^{3} \rho_{M, {\rm out}} (\bar t)$ is the corresponding background gravitating mass and the gauge $Y(r, \bar t)= r \, a(\bar t)$ has been used.
The initial matter density is then:
\begin{equation}
\rho_{M}(r,\bar t)= {\bar F_{M}'(r) \over 4 \pi a^3(\bar t) r^2 } \,.
\end{equation}
In order to have initial conditions without decaying modes in the dark-energy component, we have to set its initial profile according to the following relation valid during matter domination and $c_{s}^{2}\ll1$ \citep{Ballesteros:2010ks}:
\begin{equation}
{\bar \delta_{F_X}(r) \over \bar \delta_{F_M}(r) }= {\bar \delta_{X} \over \bar \delta_{M}} = {1+w_{\rm out} \over 1-3 w_{\rm out}} \,,
\end{equation}
where, analogous to the matter contrast, we have,
\begin{equation}
\bar \delta_{F_X}(r) = {\bar F_{X}(r) \over \bar F_{X, {\rm out}}(r)} -1 \,,
\end{equation}
$\bar F_{X, {\rm out}}= {4 \pi \over 3} a^{3}(\bar t) r^{3} \rho_{X, {\rm out}} (\bar t)$ and $\bar \delta_{X} =\bar \delta_{F_X}(0)$.
The initial dark-energy density is then,
\begin{equation}
\rho_{X}(r,\bar t)= {\bar F_{X}'(r) \over 4 \pi a^3(\bar t) r^2 } \,.
\end{equation}
Hence, all the initial conditions relative to the inhomogeneous patch are indeed specified by a given curvature profile~$k(r)$.
\section{Cosmological data analysis} \label{analysis}
In this Section we explain how we compare the predictions of this model with supernovae, Hubble rate and cosmic microwave background observations.
We decided not to include baryon acoustic oscillations in the analysis as perturbation theory in an inhomogeneous background has not been thoroughly understood yet\footnote{See, however, \citet{Nishikawa:2012we} for a recent development.} \citep{Zibin:2008vj,Clarkson:2009sc,Alonso:2010zv}.
\subsection{Hubble rate}\label{subsec:h0}
The Hubble rate is obtained by measuring cosmological standard candles mostly within a redshift range with median value $z_h \sim 0.05$.
We compare the observed value to the theoretical quantity,
\begin{equation} \label{hloco}
H_{\text{loc}}= {1 \over z_{\rm max} - z_{\rm min}} \int_{z_{\rm min}}^{z_{\rm max}} H_A(r(z),t(z)) \, dz \,.
\end{equation}
The values $z_{\rm max}$ and $z_{\rm min}$ depend on the redshift volume that is probed by a given experiment.
The reason we compare an averaged expansion rate to the data is primarily because the observed expansions rate in fact is an averaged quantity, so this should be a fair comparison. Moreover, when the redshift $z_b$ of the boundary of the inhomogeneity is close in value to~$z_h$, the averaged $H_{\text{loc}}$ may differ significantly from $H_A(r(z_h),t(z_h))$, which falsely would lead to a bad fit.
The approach of Eq.~(\ref{hloco}) is to some extent arbitrary and one should instead reanalyze the raw data without assuming a FLRW fiducial model as it is usually done \citep[see e.g. the discussion in][]{Zumalacarregui:2012pq}. However, as we will see in Section \ref{scmb} our results depend weakly on the Hubble parameter constraint and so this caveat should not sizably affect our findings.
We mainly consider the
determination of the Hubble rate from \citealt{Riess:2009pu} (R09). However, in order to study a possible sensitive dependence on this datum, we will also consider the results from \citealt{Freedman:2000cf} (F01) and \citealt{Sandage:2006cv} (S06). The three measurements are:
\begin{align}
H_{\rm{F01}}&=72\phantom{.0} \pm 8\phantom{.0} \; \frac{\rm{km/s}}{\rm{Mpc}}, &0.005<z<0.1, \label{F01} \\
H_{\rm{S06}}&=62.3 \pm 6.3 \; \frac{\rm{km/s}}{\rm{Mpc}}, &0.01<z<0.07, \label{S06} \\
H_{\rm{R09}}&=74.2 \pm 3.6 \; \frac{\rm{km/s}}{\rm{Mpc}}, &0.023<z<0.1. \label{R09}
\end{align}
\subsection{Supernova observations}
We use the Union2 SN Compilation \citep{Amanullah:2010vv}, which consists of 557 type Ia supernovae in the redshift range $z=0.015-1.4$.
As we are considering an almost-linear inhomogeneity surrounding the observer, we are not departing strongly from the standard model.
Therefore it should be a good approximation to use the magnitude-redshift and correlation tables provided by \citet{Amanullah:2010vv}.
\subsection{Cosmic microwave background} \label{scmb}
The metric of Eq.~(\ref{metric}) is matched to the background FLRW metric at a redshift at which radiation is still negligible.
In this way the last scattering surface, which is responsible for most of the CMB anisotropies, is outside the inhomogeneous patch and a standard analysis of the primordial CMB power spectrum is possible.
One has to replace the inhomogeneous model with an {\it effective} FLRW metric which accounts for the different angular diameter distance to the surface of last scattering of the CMB as compared to the homogeneous background model.
This is done by placing an FLRW observer ($\delta_0\equiv0$) in the same coordinate system at $r=0$ but at a different time than $t_0$, such that this observer's angular diameter distance to the surface of last scattering, which lies at some constant time $t_{\rm LS}$, agrees with the actual LTB observer's angular diameter distance to the surface of last scattering. The physics at last scattering itself is unaffected, since the {\em effective} FLRW observer is placed in the same FLRW universe in which the LTB patch is embedded, albeit at a different time. The CMB spectrum is then calculated using {\sc camb} \citep{Lewis:1999bs}.
See \citet{Biswas:2010xm, Moss:2010jx,Marra:2010pg} for more explicit details about how the effective model is obtained.
Note that there are other contributions to the CMB coming from secondary effects, which in the inhomogeneity may differ from those in the effective FLRW metric, and are due to the photons traveling through inhomogeneities inside the void, such as the late-time ISW effect and weak lensing. For the same reason as for which we ignore the baryon acoustic oscillations, we ignore these secondary effects, since they are subdominant and studying them would require knowledge of the growth of perturbations in an inhomogeneous background.
We fit the theoretical predictions of our model to the WMAP 7-year data release \citep{Komatsu:2010fb}.
\subsection{Parameter estimation}
We perform a Markov-Chain Monte-Carlo likelihood analysis using {\sc CosmoMC} \citep{Lewis:2002ah}. We calculate all distance measures using an improved version of {\sc VoidDistancesII}{\footnote{\url{http://web.physik.rwth-aachen.de/download/valkenburg/}}} \citep{Biswas:2010xm}, which now acts as a wrapper around {\sc camb} \citep{Lewis:1999bs}, necessitating no changes to {\sc camb}'s source code and minimal changes to {\sc CosmoMC}'s source code. We combine this module with the $\Lambda$LTB module {\sc ColLambda}\footnote{\url{http://web.physik.rwth-aachen.de/download/valkenburg/ColLambda/}} \citep{Valkenburg:2011tm} for calculating all metric functions, which we extended to include the numerical solutions to the scenario discussed here, with $w_{\rm out}\neq -1$ and $c_s = 0$.
With this setup, for every selected vector of parameter values, we calculate the theoretical predictions for supernova distances, the local Hubble rate
and the CMB power spectrum.
For reference, next to the inhomogeneous model we analyze its homogeneous background model, $w$CDM, which is described by the same model but has $z_b\equiv0$ and $\delta_0\equiv0$.
\newlength{\mywidth}
\setlength{\mywidth}{.9 \columnwidth}
\begin{table}
\begin{center}
\begin{tabular}{c}
\begin{tabular*}{\mywidth}{c}
\hline
Flat priors\\
\hline \hline
\begin{tabular}{rcl}
$0.4\quad <$ &$h$ & $< \quad 1$\\
$0.005 \quad <$&$\Omega_{\rm b}h^2$ & $< \quad 0.1$ \\
$0.001\quad <$&$\Omega_{\rm dm}h^2$ & $< \quad 0.99$ \\
$-2\quad <$&$w_{\rm out}$ & $< \quad -0.4$ \\
$0.01 \quad <$ & $\tau$ & $ < \quad 0.8$ \\
$2.7 \quad <$ & $\log 10^{10} A_S$ & $ < \quad 4$ \\
$0.5 \quad <$ & $n_S$ & $ < \quad 1.5$ \\
$-0.2 \quad <$ & $\alpha_S$ & $ < \quad 0.2$ \\
$-0.2 \quad <$&$ \delta_0 $ & $< \quad 0.2$ \\
$100$~Mpc $\quad <$&$d(r_b) $ & $< \quad3$~Gpc
\end{tabular}\\\hline\\ \hline
\hline
Additional constraints\\
\hline \hline
\begin{tabular}{rl}
$\Omega_{X}$&$ >0$\\
$\Omega_{k}$&$ =0$
\end{tabular}\\
\hline
\end{tabular*}
\end{tabular}
\end{center}
\caption{Priors imposed on the parameters in the numerical analysis. The size of the LTB patch $d(r_b)$ is defined in Eq.~\eqref{eq:defdr}. The additional constraint $\Omega_X>0$ with $\Omega_k=0$, in fact implies a non-flat prior on both $\Omega_{\rm dm} h^2$ and $h$, as explained in Appendix~\ref{prioco}.}\label{tab:priors}
\end{table}
We take flat priors on the parameters listed in Table~\ref{tab:priors}. Most of these are the usual cosmological parameters: the background present-day Hubble rate $h$, the background baryon density $\Omega_{\rm b}h^2$, the background dark-matter density $\Omega_{\rm dm}h^2$, the equation of state of dark energy $w_{\rm out}$, the optical depth to re-ionization $\tau$, the amplitude of primordial scalar perturbations $A_S$, the tilt of the spectrum of primordial scalar perturbations $n_S$ and its running $\alpha_S$. We set the spatial curvature outside the LTB patch, $\Omega_k$, as well as the amplitude of primordial tensor perturbations to zero.
As discussed in Section~\ref{inico}, two additional parameters describe the LTB patch: the curvature at the center $k_c$ and the comoving radius $r_b$.
As shown in Table~\ref{tab:priors}, we will use as actual parameters the present-day total density contrast and the present-day proper size of the radius, respectively.
The latter is given by
\begin{align}
d(r_{b}) &\equiv \int_0^{r_{b}} \frac{Y'(r,t_0)}{\sqrt{1-k(r) r^2}} \,\textrm{d}r \,, \label{eq:defdr}
\end{align}
and we define the former as
\begin{align} \label{totcont}
\delta_0 \equiv \frac{\rho_{\rm in} - \rho_{\rm out}}{ \max(\rho_{\rm in},\rho_{\rm out})} \,,
\end{align}
where $\rho_{\rm in}= \rho_M (0,t_0) + \rho_X (0,t_0)$ and $\rho_{\rm out}=\rho_{M, {\rm out}} (t_0)+\rho_{X, {\rm out}} (t_0)$.
We calculate the contrast using the total density as in principle the dark-energy component can be as inhomogeneous as the matter component, and $\delta_0$ should be a relevant physical quantity to be used.
We chose a somewhat unusual definition by using $\max(\rho_{\rm in},\rho_{\rm out})$ in the denominator. With this definition, this quantity fundamentally satisfies $-1<\delta_0<1$, such that the parameter-space volume in over- and under-densities is equally distributed.
With the more usual definition we would have had $-1<\frac{\rho_{\rm in} - \rho_{\rm out}}{ \rho_{\rm out}}<\infty$, which in the Bayesian parameter estimation
induces a strong prior favouring large over-densities, possibly excluding under-densities from the analysis. In practice, however, the preferred values for this parameter are small, such that the difference between both definitions is almost negligible.
We discuss the priors in more detail in Appendix~\ref{prioco}. In the next section, where we discuss the results of the MCMC parameter estimation, we explore the full parameter space, never fixing the background parameters to some central value. Therefore we can always marginalize over all parameters, and do not bias the result in any way.
That is, any possible degeneracy, expected or unexpected, between the cosmological parameters and the LTB parameters will show up and will not influence the results without being noticed.
\section{Results} \label{results}
The four left panes in Figure~\ref{fig:2Dtango} show the parameters on which the main focus in this paper lies.
This figure shows the two-dimensional marginalized posterior probability distributions of the background dark-energy density $\Omega_X$, the dark-energy equation of state $w_{\rm out}$, the boundary of the inhomogeneity in redshift space $z_b$, and the total density contrast in the inhomogeneity $\delta_0$ (see Eq.~(\ref{totcont})).
Alternatively, in the four right panes of Figure~\ref{fig:2Dtango} we use as proxies for $z_b$ and $\delta_0$ the apparent size that the inhomogeneity would subtend if located at the last scattering surface and its corresponding temperature anisotropy, respectively. Here we define the temperature perturbation $\Delta T/T$ as the relative difference in CMB temperature at the center of the inhomogeneity and outside, in the homogeneous (average) background. This is not necessarily representative for the average temperature in the spot.
The most interesting result is the clear degeneracy between $\delta_0$ and $w_{\rm out}$ in the lower right pane in the left of Figure~\ref{fig:2Dtango}, and
the degeneracy between $\delta_0$ and $\Omega_X$ in the upper right pane. This graph explicates the necessity of properly modeling the inhomogeneity of the local universe, before any conclusion can be drawn on the properties of dark energy, in particular about the fundamental value of its equation of state: if the local density is ignored, the value of $w_{\rm out}$ can be misestimated by possibly 50\%. Note also that the redshift up to which the inhomogeneity extends, $z_b$, is hardly of any influence on the central value of $w_{\rm out}$ or $\Omega_X$.
The bias on the parameters is indeed of opposite sign for opposite $\delta_0$. Therefore, as we marginalize over $\delta_0$ in the combined posterior of $w_{\rm out}$ and $z_b$ (lower left pane in Fig.~\ref{fig:2Dtango}), the total effect is compensated (we will come back to this point with Figure~\ref{fig:1Ddiffdelta}). Note, however, that the scatter does increase with the size of the inhomogeneity.
The right of Figure~\ref{fig:2Dtango} shows the dimensions that the inhomogeneity would have on the observed CMB temperature map, if it were centered on the observer's surface of last scattering. In this situation there are hence two identical inhomogeneities in the universe: one surrounding the observer, one centered on the observer's surface of last scattering.
This figure shows that even spots that do not violate the observables of Section \ref{analysis}, do induce a strong bias on $w_{\rm out}$, following the findings of \citet{Valkenburg:2011ty}.
As before we see that the apparent size of the inhomogeneity has almost no effect on the value of $w_{\rm out}$ or~$\Omega_X$.
\begin{figure*}
\begin{center}
\includegraphics[width= .48\textwidth]{riessdeltavswcdm_2D}
\qquad
\includegraphics[width= .48\textwidth]{riessdeltavswcdm_tempdeg_2D}
\caption{Two-dimensional marginalized posterior probability distributions for the most interesting parameters characterizing the $w$CDM model endowed with a local inhomogeneity considered in this paper. The parameters are constrained by the Union2 SN Compilation \citep{Amanullah:2010vv}, the WMAP 7-year CMB power spectrum \citep{Komatsu:2010fb}
and the recent determination of the Hubble rate by \citet{Riess:2009pu}. Inner tangerine-tango coloured contours are 68\% confidence level (c.l.) contours, while the outer dark-red coloured contours are 95\% c.l. The left and right figures show the same information, however the redshift of the boundary of the inhomogeneity, $z_b$, and the total density contrast, $\delta_0$, on the left, are on the right traded in for the angular diameter that the inhomogeneity would have if it were located at the observer's last scattering surface and the temperature fluctuation that it would induce, respectively.}
\label{fig:2Dtango}
\end{center}
\end{figure*}
\begin{figure*}
\begin{center}
\includegraphics[width= .7\textwidth]{riessdeltavswcdm}
\caption{One-dimensional marginalized posterior probabilities for $\Omega_X$ and $w_{\rm out}$, constrained by CMB, SNe, and $H_{\text{loc}}$.
The standard $w$CDM constraints on these parameters are shown in red dashed lines and are the same in the different panes. The blue vertical line serves as a guide for the eye, always going through the maximum of the $w$CDM value.
The constraints on the $w$CDM model endowed with a local inhomogeneity considered in this paper are given under different priors on $\delta_0$: $-0.2<\delta_0<0.2$ (top), $0<\delta_0<0.2$ (second row from top), $-0.2<\delta_0<0$ (third row from top), and for two constraining priors on $\delta_0$ (bottom): $\delta_0=0.1$ (solid black line) and $\delta_0=-0.15$ (dashed-dotted black line). A prior on $\delta_0$ that averages around zero, widens but does not shift the posterior distributions. If we know instead the sign of $\delta_0$, constraints on $w_{\rm out}$ and $\Omega_X$ shift by as much as $5\%\sim10\%$. In the bottom row, for $\delta_0=0.1$ the 95\% c.l.~upper bound on $w_{\rm out}$ is -1.03. For $\delta_0=0.15$ the 95\% c.l.~lower bound on $w_{\rm out}$ is -0.98. Both priors hence rule out the cosmological constant at 95\% c.l., given current observations. }
\label{fig:1Ddiffdelta}
\end{center}
\end{figure*}
In Figure~\ref{fig:1Ddiffdelta} we show the one-dimensional marginalized posterior probabilities for $\Omega_X$ and $w_{\rm out}$, under different priors on $\delta_0$, imposed by means of importance sampling on the MCMC chains that explore the full range of $\delta_0$. The result is displayed in black, and for comparison the constraints on these parameters for the homogeneous $w$CDM model ($z_b\equiv 0$, $\delta_0\equiv0$) are displayed in dashed red lines. The top row shows $\Omega_X$ and $w_{\rm out}$ marginalized over all values of $\delta_0$, both positive and negative. Since positive and negative values of $\delta_0$ have opposing effects on $w_{\rm out}$ and $\Omega_X$, marginalizing over $\delta_0$ mostly widens the tails of the distributions.
This is already an important result showing how inhomogeneities contribute to the error budget in the cosmological parameters. This uncertainty from large-scale structures is expected to become more important when future data will tighten the confidence regions of the parameters of interest.
If we impose, however, the prior that $\delta_0>0$ (second row in Figure~\ref{fig:1Ddiffdelta}) or $\delta_0<0$ (third row from top), we find even stronger results: we see indeed that both the tails and the central values of $\Omega_X$ and $w_{\rm out}$ shift. This shift is significant if one wants to progress towards not just precision but also accurate cosmology. If we push the magnitude even further, pretending we know that the local density must be either $\delta_0=0.1$ or $\delta_0=-0.15$, as in the bottom row in Figure~\ref{fig:1Ddiffdelta}, then we find that $w=-1$ is excluded at 95\% confidence level (c.l.) in both cases: $w<-1.03$ and $w>-0.98$, respectively. However, these particular models are only included at 99.7\% c.l.
In order to better show the degeneracy of $\Omega_X$ and $w_{\rm out}$ with $\delta_0$ we plot again in Figure~\ref{fig:degendirs} the corresponding two-dimensional marginalized posterior probabilities with 68\%, 95\%, 99.7 \% and 99.99\% confidence level contours. Also plotted for comparison are the 95\% c.l.~one-dimensional constraints on $\Omega_X$ and $w_{\rm out}$ for the standard $w$CDM model.
This plot is meant to justify the values $\delta_0=0.1$ and $\delta_0=-0.15$ used in the bottom row of Figure~\ref{fig:1Ddiffdelta}.
It shows indeed that large values of $\left| \delta_0 \right|$ that can significantly bias $\Omega_X$ and $w_{\rm out}$ are still within the 99.7\% c.l.
\begin{figure*}
\begin{center}
\includegraphics[width= .47\textwidth]{d0lplotlambda}
\qquad
\includegraphics[width= .47\textwidth]{d0lplotw}
\caption{Two-dimensional marginalized posterior probabilities for $\Omega_X$ and $w_{\rm out}$ with $\delta_0$, constrained by CMB, SNe, and $H_{\text{loc}}$. The color shaded regions correspond from the innermost region to the outmost region to 68\% c.l., 95\% c.l., 99.7 \% c.l.~and 99.99\% c.l., respectively. The blue horizontal band corresponds to the 95\% c.l.~one-dimensional constraints on $\Omega_X$ and $w_{\rm out}$ for the standard $w$CDM model.
This plot shows that if future data will constrain $\left| \delta_0 \right|$ to be large, then the inclusion of such data will shift the best fit region towards values of $w$ that are far from $-1$.}
\label{fig:degendirs}
\end{center}
\end{figure*}
\subsection{Sensitivity to local Hubble-rate constraints}
In Figure~\ref{fig:H0dependence} we compare the effect of different Hubble-rate observations on the resulting posterior probabilities of $\Omega_X$ and $w_{\rm out}$, when we fit the $w$CDM model endowed with a local inhomogeneity to $H_{\text{loc}}$, SNe and CMB. As explained in Section~\ref{subsec:h0}, we compare the three different values from \citet{Riess:2009pu} (solid black), \citet{Freedman:2000cf} (dashed red) and \citet{Sandage:2006cv} (dashed dotted blue).
We see that the constraints on $\Omega_X$ do depend on the chosen measurement for $H_{\text{loc}}$, while the resulting constraints on $w_{\rm out}$ hardly depend on $H_{\text{loc}}$. This should be expected as SNe observations (constraining both $\Omega_X$ and $w_{\rm out}$) are insensitive to $H_{\text{loc}}$ while CMB observations (constraining $\Omega_X$ but weakly $w_{\rm out}$) are instead sensitive to $H_{\text{loc}}$.
These results imply that our conclusions with regard to $w_{\rm out}$ are robust against different observational determinations of~$H_{\text{loc}}$, and the constraints hence mostly follow from the SNe and~CMB.
\begin{figure}
\begin{center}
\includegraphics[width= \columnwidth]{vdii_CHS_AllLTB_Riess}
\caption{One-dimensional marginalized posterior probabilities of $\Omega_X$ and $w_{\rm out}$ for the inhomogeneous model, given CMB, SN and $H_{\text{loc}}$ observations, comparing different constraints on $H_{\text{loc}}$: \citet{Riess:2009pu} (solid black), \citet{Freedman:2000cf} (dashed red) and \citet{Sandage:2006cv} (dashed dotted blue).
The conclusions about the effect of the inhomogeneity on $w_{\rm out}$ are robust against different observational determinations of $H_{\text{loc}}$.}
\label{fig:H0dependence}
\end{center}
\end{figure}
\subsection{kSZ effect} \label{kSZe}
In Figure~\ref{fig:dipoles} we show the CMB dipole that observers at different radii would observe, for a given configuration; an over-density on the left, an under-density on the right, in both cases the LTB patch has a radius of 1 Gpc, but with different values for $w_{\rm out}$. Both cases fit the data roughly as well as the standard $w$CDM, while still giving an interesting bias on $w_{\rm out}$.
We obtained these figures by -- at each radius -- starting an integration of the geodesic equations in two directions (negative and positive $r$-direction), back to the surface of last scattering, which lies at constant time in the synchronous gauge of the LTB metric. The difference in redshift to the surface of last scattering in both directions is then translated into a $\Delta T _{\rm CMB}/ T _{\rm CMB}=(z_{+}-z_{-}) / (2+z_{ +}+z_{-}) $.
Only for small radii this is to a good approximation equal to the dipole in a spherical harmonics expansion of the CMB temperature map. On the vertical axis on the righthand side we list the corresponding peculiar velocity that an observed temperature difference corresponds to, if it were the effect of peculiar velocity alone. For both panes, left and right, the peculiar velocities do not exceed the magnitude of expected random peculiar velocities. Therefore the kinematic Sunyaev-Zel'dovich effect that such velocities induce on CMB photons~\citep{GarciaBellido:2008gd, Zhang:2010fa, Moss:2011ze} should be at present undetectable.
So as to strengthen this claim it is useful to look at the findings of \citet{Valkenburg:2012td} where constraints on the $\Lambda$LTB model from kSZ observations have been computed.
While in the present paper dark energy is not the cosmological constant, the above analysis should give nevertheless an estimate of the kSZ signal.
Therefore, this suggests that structures with a contrast of roughly $\sim$0.1 extending
for a radius of 1-2 Gpc are not excluded by present observations. A thorough
study of the kSZ effect in these models is left to future work.
\begin{figure*}
\includegraphics[width= .44\textwidth]{VoidDipole_vs_r_od}
\qquad \qquad \qquad
\includegraphics[width= .44\textwidth]{VoidDipole_vs_r_ud}
\caption{The CMB-dipole observed by observers at different radii $d(r)$ in an LTB patch with a radius of 1 Gpc, for an over-density (left) and an under-density (right). On the right vertical axis we list the corresponding peculiar velocity that is derived assuming that the observed CMB dipole is caused solely by the peculiar velocity of the observer. The magnitude of the velocities does not exceed the magnitude of expected random velocities
See Section \ref{kSZe} for more details.
}
\label{fig:dipoles}
\end{figure*}
\subsection{FLRW Observer's $w(z)$} \label{wofz}
\begin{figure*}
\includegraphics[width= .45\textwidth]{Void_wofz_od}
\qquad
\includegraphics[width= .45\textwidth]{Void_wofz_ud}
\caption{The function $w_{\rm obs}(z)$ as defined in \citet{Clarkson:2007bc}, which is the equation of state of dark energy that an observer thinks to see if the observer falsely assumes that the universe is described by the FLRW metric. The examples shown here are the same two cosmologies as in Fig~\ref{fig:dipoles}; an over-density (left) and an under-density (right). In both cases the observed $w_{\rm obs}(z)$ (solid red) matches closely with the fundamental $w_{\rm out}$ (dashed blue) once the radius is reached where the spherically symmetric metric matches to the surrounding FLRW metric. The inhomogeneity causes clear features in $w_{\rm obs}(z)$: the contracting core of the over-density increases the magnitude of $w_{\rm obs}(z)$, while the under-dense compensating shell has the opposite effect. For the under-dense center, the inverse holds. In this picture $w_{\rm obs}(z)$ is computed using the exact solutions, while taking second derivatives of observed distances would not necessarily reveal these features.}
\label{fig:wofz}
\end{figure*}
Following \citet{Clarkson:2007bc} one can, given a luminosity distance-redshift relation in a homogeneous universe, compute what the underlying $w(z)$ of the dark-energy fluid is. In the homogeneous universe (described by the FLRW metric), one can find indeed an exact relation between $w(z)$ and the first and second derivatives of the luminosity distance with respect to redshift and two more parameters, $\Omega_k$ and $\Omega_{\rm m}$. If an observer knows the latter two parameters from other observations, and deduces the first and second derivatives of the luminosity distance from SN observations, the observer can derive $w(z)$. In the scenario studied here, at background level $w$ is not a function of time or redshift. However, the inhomogeneity comes into play in the luminosity distance-redshift relation. Therefore, an observer that falsely assumes that the metric surrounding him/her is FLRW will in fact see a redshift dependence in $w$. We calculate the observed $w_{\rm obs}(z)$ (Eq. (3) in ~\citealt{Clarkson:2007bc}) for the two example models of Fig.~\ref{fig:dipoles}, and show the result in Fig.~\ref{fig:wofz}. Inside the inhomogeneity, $w_{\rm obs}(z)$ shows a very clear signature of the matter distribution: the contracting core and expanding compensating shell for the over-density show corresponding effects on $w_{\rm obs}(z)$. The inverse holds for the under-density.
Therefore, if one performs an analysis such as in \citet{Shafieloo:2009ti, Zhao:2012aw}, one may find a significant deviation from a constant $w$, while fundamentally $w$ is constant at the background level.
In particular, it is very interesting to note that the $w(z)$ reconstruction by \citealt{Zhao:2012aw} (see e.g. the pane (A2) of Fig.~1 in that reference), if interpreted within this framework, could indicate the presence of a large-scale underdensity around us, and not of a possibly time-dependent equation of state.
It is indeed worth noting the similarity of the result by \citet{Zhao:2012aw} with the observed $w_{\rm obs}(z)$ shown in Fig.~\ref{fig:Void_wofz_zhao}, which corresponds to a $\Lambda$CDM model endowed with a local underdensity of central contrast $\delta_{0}=-0.06$ and redshift boundary $z_{b}=0.4$.
\begin{figure}
\begin{center}
\includegraphics[width= \columnwidth]{Void_wofz_zhao}
\caption{As in Fig.~\ref{fig:wofz} but for a $\Lambda$CDM model endowed with a local underdensity of central contrast $\delta_{0}=-0.06$ and redshift boundary $z_{b}=0.4$ such that the observed $w_{\rm obs}(z)$ appears qualitatively similar to the $w(z)$ reconstruction by \citet{Zhao:2012aw} (see e.g. the pane (A2) of Fig.~1 in that reference).
If interpreted within this framework, the results of \citet{Zhao:2012aw} could indicate the presence of a large-scale underdensity around us, rather than a possibly time-dependent dark-energy equation of state.}
\label{fig:Void_wofz_zhao}
\end{center}
\end{figure}
\subsection{Other parameters} \label{showoff}
In Figure~\ref{fig:otherpars} we show posterior probabilities of $H_0$, which
is the expansion rate of the background universe (and does {\em not} correspond to the locally observed expansion rate), $\delta_0 \equiv \frac{\rho_{\rm in} - \rho_{\rm out}}{ \max(\rho_{\rm in},\rho_{\rm out})}$, $\delta_{M,0} \equiv \frac{\rho_{M,{\rm in}}}{\rho_{M,{\rm out}}}-1$, and $\delta_{X,0} \equiv \frac{\rho_{X,{\rm in}}}{\rho_{X,{\rm out}}}-1$. All other parameters that we allowed to vary, listed in Table~\ref{tab:priors}, show practically no deviation in their constraints in the presence and absence of the inhomogeneity. The constraint on $H_0$ is significantly weakened when one takes into account the possibility that we may live in a local inhomogeneity. This result is similar to the findings in \citet{Valkenburg:2012ds}, even if the local inhomogeneity considered is orders of magnitude different.
Moreover, the actual matter perturbation that is allowed by the data can be as large as $\left|\delta_{M, 0}\right|\simeq 0.5$, leaving the total density perturbation around $\left|\delta_0\right|\simeq0.1$, because the energy perturbation in the dark-energy fluid is generally small as $w_{\rm out}$ is never very far from -1.
\begin{figure*}
\begin{center}
\includegraphics[width= \textwidth]{riess_otherpars}
\caption{One-dimensional marginalized posterior probabilities of $H_0$, which describes the age of the universe and does {\em not} correspond to the locally observed expansion rate, $\delta_0 \equiv \frac{\rho_{\rm in} - \rho_{\rm out}}{ \max(\rho_{\rm in},\rho_{\rm out})}$, $\delta_{M,0} \equiv \frac{\rho_{M,{\rm in}}}{\rho_{M,{\rm out}}}-1$, and $\delta_{X,0} \equiv \frac{\rho_{X,{\rm in}}}{\rho_{X,{\rm out}}}-1$. The presence of the spherical structure weakens significantly the bounds on the background expansion rate (in red we show the constraint on $H_0$ in the homogeneous $w$CDM model). Secondly, the local energy perturbation consists of an almost negligible dark-energy perturbation and a significant dust perturbation, which is not apparent if one considers the total $\delta_0$ alone. }
\label{fig:otherpars}
\end{center}
\end{figure*}
\subsection{Probability under homogeneous initial conditions} \label{gaussianperv}
Given the fact that we observe several large spots in the CMB, and the possibility that these spots are the result of density perturbations on the surface of last scattering, we can argue that living in such a perturbation must have a non-zero probability.\footnote{We would like to point out that for the almost-linear models considered in this paper the observer does not need be very close to the center so as not to see a too large CMB dipole. For the cases of Fig.~\ref{fig:dipoles}, for example, the dipole is never larger than the observed value of $\sim 10^{-3}$. For larger contrasts there will be regions where observers would see a larger-than-measured dipole, but these regions will not occupy the majority of the inhomogeneity. Finally, we would like to stress that we have placed the observer at the center simply to simplify the numerical calculations.} If all perturbations arise from a smooth, close to scale invariant spectrum of primordial perturbations, it is not obvious that large cold and hot spots should exist, and there is an ongoing debate about this topic \citep{Cruz:2006sv,Zhang:2009qg,Ayaita:2009xm}.
Let us nonetheless quantify the probability of having the density perturbations that we took as examples for Figs.~\ref{fig:dipoles}~and~\ref{fig:wofz}. If these perturbations come from the same spectrum as the perturbations that we observe in the CMB, then the probability of their existence can be approximated by the variance of the gaussian density field, smoothed by a top hat filter with a radius that corresponds to the radius of the density perturbation under consideration \citep{Kolb:1990vq}. It must be noted that, because of the compensated shape of the density profile that we consider, taking the full radius of the spherical patch, $r_b$, as the radius of the top hat filter would give almost zero density perturbation. Therefore we choose the radius at which the density changes sign as the smoothing radius. This is roughly at $r_b/2$, but we use the numerically obtained exact value. Comparing at the time of decoupling when dark energy is negligible, and with the primordial spectrum of perturbations of the two models respectively (since the models were fit to the CMB, they carry spectral parameters), we find that the over-dense model considered in Figs.~\ref{fig:dipoles}~and~\ref{fig:wofz} is at three times the dispersion of the smoothed density field of its cosmology (CMB spectrum), and the under-dense model is at six times the dispersion of its cosmology. Notably the over-dense model is not at all unlikely to occur, while it does give a large effect on both $w_{\rm out}$ and $w_{\rm obs}$ as shown in Fig.~\ref{fig:wofz}.
\section{Conclusion} \label{conclusions}
We have analyzed present observations of the local expansion rate, distant supernovae and the cosmic microwave background within a flat $w$CDM model endowed with a local almost-linear inhomogeneity surrounding the observer. We have found a significant impact on the dark-energy parameters, in particular on the equation of state which is strongly degenerate with the inhomogeneity contrast. The implications of this degeneracy are twofold. On one hand we have shown that with prior knowledge on the inhomogeneity, to be obtained possibly with some future probe, it is already possible to rule out the case of the cosmological constant with current data.
On the other hand, even if future probes exclude the case of the cosmological constant in a homogeneous universe, this still may be due to a poor modeling of the large-scale structure of the universe.
The same conclusions apply to constraints on the time variation of the equation of state.
The analysis in the present paper is but a first step towards a more accurate reconstruction of the cosmological parameters.
We have indeed chosen, for technical reasons, a very specific dark-energy model and inhomogeneity profile. Before drawing definitive conclusions, a more comprehensive analysis should be performed.
Firstly, it would be particularly interesting (even though perhaps challenging) to consider a nonzero dark-energy sound speed. However, since we found that the dark-energy component is only very mildly inhomogeneous, we do not expect our results to be strongly dependent on the assumption of a negligible sound speed.
Secondly, it would be interesting to consider more inhomogeneous patches with more general density profiles, possibly with the observer at randomized positions \citep{Marra:2007pm, Marra:2007gc, Valkenburg:2009iw,Szybka:2010ky,Flanagan:2011tr}.
\section*{Acknowledgments}
It is a pleasure to thank Luca Amendola and Ignacy Sawicki for useful comments and discussions.
MP acknowledges financial support from the Magnus Ehrnrooth Foundation.
VM and WV acknowledge funding from DFG through the project TRR33 ``The Dark Universe''.
\bibliographystyle{mn2e_eprint}
|
{
"timestamp": "2013-03-21T01:02:17",
"yymm": "1203",
"arxiv_id": "1203.2180",
"language": "en",
"url": "https://arxiv.org/abs/1203.2180"
}
|
\section{Introduction}
An accurate determination of the meridional flow speed in both the solar
photosphere and the solar interior is crucial to the understanding of
solar dynamo and predicting solar cycle variations. For example,
\citet{dik06} suggested that the slow-down of the meridional circulation
during the solar-cycle maximum could change the duration of the following
minimum and delay the onset of the next solar cycle. \citet{hat10} found
that the meridional flow speed was substantially faster during the solar
minimum of Cycle 23 than during the previous minimum, and suggested that
this might explain the prolonged minimum of Cycle 23. Using flux-transport
dynamo model simulations, \citet{dik10} suggested that the prolonged
minimum of Cycle 23 might be due to that the poleward meridional flow extended
all the way to the pole in Cycle 23, unlike in Cycle 22 the flow switched to
equator-ward near the latitude of $60\degr$. All these works demonstrate
that an accurate measurement of the meridional flow speed is very important.
The photospheric
meridional flow speed can be determined by tracking certain
photospheric features, such as magnetic structures and supergranules
\citep[e.g.,][]{kom93, hat10, hat10b, giz04, sva06}, although it is
not quite clear how well the motions of these surface features represent
the photospheric plasma flows. The photospheric meridional flow speed can
also be inferred from Doppler-shift measurements \citep[e.g.,][]{hat96,
ulr10}, and recently \citet{ulr10} made extensive comparisons of
the meridional flow speeds obtained by different methods. The meridional
flows in the solar interior are primarily determined by helioseismology,
i.e., by measuring frequency shifts between poleward and equator-ward
traveling acoustic waves \citep[e.g.,][]{bra98, kri07, rot08}, and by use of
local helioseismology techniques, namely, ring-diagram analysis and
time-distance helioseismology \citep[e.g.,][]{gil97, cho01, hab02,
zha04, gon08}.
The subsurface meridional flow speeds obtained by the two different
local helioseismology techniques are in reasonable agreement for at least
the upper 20 Mm of the convection zone \citep[e.g.,][]{hin04}. However,
the agreement between the two analysis techniques cannot rule out
that both techniques may be affected by the same or similar systematic
effects. In this Letter, we report on a systematic center-to-limb
variation in helioseismic travel times measured by the time-distance
helioseismology technique, which was previously unnoticed but must
be taken into account in the inference of the subsurface meridional flows.
Other helioseismology techniques, such as the ring-diagram analysis, may
also be affected by a similar systematic effect (Rick Bogart, private communication).
We develop an empirical correction procedure by measuring the center-to-limb
variation along the equatorial area during the periods when the solar
rotation axis is perpendicular to the line-of-sight, i.e., when the solar
B-angle is close to $0\degr$. This correction scheme provides consistent
results for the acoustic travel times in the north-south directions measured
from different HMI observables and AIA chromospheric intensity variations.
We introduce our data analysis procedure and present results in \S2, and
discuss the results and their implications in \S3.
\section{Data Analysis and Results}
\label{sec2}
\subsection{Data Analysis Tools}
\label{sec21}
To facilitate analysis of the large amount of data from the
Helioseismic and Magnetic Imager \citep[HMI;][]{scherrer12, schou12}
onboard {\it Solar Dynamics Observatory} \citep[SDO;][]{pes12}, a
time-distance helioseismology data-analysis pipeline was developed
and implemented at the HMI-AIA Joint Science Operation Center
\citep{zha12}. Every 8 hours, the pipeline provides measurements of
acoustic travel times, and generates maps of subsurface flow and
wave-speed perturbations by inversion of the measured travel times,
covering nearly the full-disk Sun with an area of $120\degr\times120\degr$.
For the pipeline processing, we select 25 overlapping areas on the solar disk
and analyze the 8-hr sequences of solar oscillation data separately for each
region. Acoustic travel-time maps are obtained for 11 selected wave travel
distances, and are inverted to derive maps of subsurface velocity and
wave-speed perturbations from the surface to about 20~Mm in depth
\citep{zha12}. Then the
results for individual regions are merged into nearly-full-disk maps covering
$120\degr$ in both longitude and latitude, with a spatial sampling of
$0\fdg12$ pixel$^{-1}$ on a uniform longitude-latitude grid. The pipeline
gives acoustic travel time measurements from two different fitting techniques
\citep{cou12}, and inversion results based on ray-path and Born-approximation
sensitivity kernels. Both measurement uncertainties for different
distances and inversion error estimates for different inversion depths
are given in \citet{zha12}. In this Letter,
only the acoustic travel times obtained from Gabor-wavelet fitting
\citep{kos97} and inversion results based on the ray-path approximation
kernels are presented.
Although the pipeline is designed to analyze the HMI Dopplergrams,
it can nevertheless be used to analyze other HMI observables
that carry solar oscillation signals, e.g., continuum intensity, line-core
intensity, and line-depth. The full-disk data from the 1600~\AA\ and
1700~\AA\ channels of Atmospheric Imaging Assembly \citep[AIA;][]{lem12}
onboard {\it SDO} can also be used for helioseismology studies with
good accuracy \citep{how11}. In this study, we compare
results obtained from the four HMI observables and the AIA 1600~\AA\
data following the same analysis procedure. This comparison helps us to
identify the systematic center-to-limb variation and develop a correction
method.
\subsection{Center-to-Limb Variation in Measured Travel Times}
\label{sec22}
The north-south ($\tau_\mathrm{ns}$) and west-east ($\tau_\mathrm{we}$) acoustic travel-time differences
approximately represent the north-south and west-east flow components,
respectively, although a full inversion is required to determine more
precisely these flows. We first show the measured travel
time differences and then present the inversion results. We choose a 10-day
period of December 1 through December 10, 2010, when the solar B-angle
between the equator and the ecliptic is close to $0\degr$ to avoid
complications caused by leakage of the solar rotation signal into
the meridional flow measurements.
The upper panels of Figure~\ref{map} show the nearly-full-disk map of $\tau_\mathrm{ns}$,
averaged over the 10-day period, measured from the HMI Dopplergram and
continuum intensity data for an acoustic travel distance of $1\fdg08 - 1\fdg38$.
The general pattern of positive $\tau_\mathrm{ns}$\ in the southern hemisphere
and negative $\tau_\mathrm{ns}$\ in the northern hemisphere is usually thought
to be caused by the interior poleward meridional flows. However, the
apparent differences between the magnitude of $\tau_\mathrm{ns}$\ obtained from the Doppler
data and that obtained from the continuum intensity data indicate
that there are additional systematic variations. The lower panels
show the averaged $\tau_\mathrm{we}$\ maps measured from the same observables
after the latitude-dependent travel times caused by the differential
rotation, obtained by averaging the measurements of all longitudes, are
subtracted.
One would expect $\tau_\mathrm{we}$\ be relatively flat along same latitudes because
the solar rotation does not vary significantly with longitude, but both
$\tau_\mathrm{we}$\ maps show systematic travel-time variations along the same latitudes,
positive in the eastern hemisphere and negative in the western hemisphere.
The longitudinal variation
is quite significant for the measurements from the intensity data
(Figure~\ref{map}d) and small but not negligible for the measurements
from the Dopplergram data (Figure~\ref{map}c).
To more quantitatively illustrate this systematic variation, we average
$\tau_\mathrm{ns}$\ in a $20\degr$-wide band along the central meridian as a function of
latitude, and display the averaged curves in the top panels of
Figure~\ref{curves} for three selected measurement distances and
for the four HMI observables: Doppler velocity, continuum intensity,
line-core intensity, and line-depth. In the middle-row panels, we show
the corresponding $\tau_\mathrm{we}$\ curves obtained by averaging over a $20\degr$-wide
band along the equator as a function of longitude (hereafter, longitude
is relative to the central meridian). If there were no systematic
center-to-limb variation in the measured acoustic travel times,
$\tau_\mathrm{ns}$\ obtained from the different observables would agree with each
other, and $\tau_\mathrm{we}$\ would remain flat for all observables. But clearly,
the measurements are not as expected. Among all these measurements, the
$\tau_\mathrm{ns}$\ curves
show not only different magnitude of travel-time shifts but sometimes
also opposite
variation trends. For $\tau_\mathrm{we}$, it is found that the measurements from the
Dopplergrams have the smallest systematic variations and are similar
to the line-core intensity measurements at larger distances. However,
the travel times measured from the continuum intensity and line-depth data show
not just substantial travel-time shifts for all measurement distances but
also opposite center-to-limb variations. It is quite clear that the travel-time
variations along the equator are not caused by solar flow but represent
a systematic center-to-limb variation. If we treat the longitude-dependent
$\tau_\mathrm{we}$\ variations along the equatorial area as the systematic center-to-limb
variations, and subtract these variations from the latitude-dependent $\tau_\mathrm{ns}$\
measured using the corresponding observables,
we get the residual curves shown in the bottom-row panels of
Figure~\ref{curves}. It is remarkable that for all the measurement
distances, the results from all four HMI observables are in reasonable
agreement. This suggests that the residual travel times correspond
to the subsurface meridional flow signals.
It was demonstrated that the {\it SDO}/AIA 1600~\AA\ data are also suitable
to perform helioseismology analysis \citep{how11}. Figure~\ref{AIA}
shows a comparison of the results obtained from the HMI Dopplergrams and
the AIA 1600~\AA\ data for one selected measurement distance, $3\fdg12 -
3\fdg84$. Similarly,
$\tau_\mathrm{ns}$\ measured from the AIA 1600~\AA\ data also differ from that measured
from the HMI Dopplergram. The $\tau_\mathrm{we}$\ measured from the AIA data also show
systematic variations, though its longitudinal trend is opposite to that
measured from HMI continuum intensity and the variation magnitude is
much smaller. The residual, obtained after subtracting the
longitude-dependent $\tau_\mathrm{we}$\ from the latitude-dependent $\tau_\mathrm{ns}$\ for AIA
1600~\AA, is quite similar to that obtained from the HMI Dopplergrams
and other HMI observables. This result is particularly remarkable because
HMI and AIA are different instruments observing in different spectral
lines: AIA in 1600~\AA\ and HMI in \ion{Fe}{1}~6173~\AA, which are formed
at different heights in the solar atmosphere.
\subsection{Effect on Meridional Flow Inversions}
\label{sec23}
The systematic center-to-limb travel-time variations would have
an apparent effect on the inference of subsurface meridional flows
obtained by inversion. Here, we investigate how the inferred
meridional flow speed changes after removal of this systematic effect.
We employ the time-distance helioseismology pipeline code developed for
inversion of the acoustic travel times \citep{zha12}.
The inversion results are shown in Figure~\ref{invert}. The middle row
of Figure~\ref{invert} presents inversion results obtained from travel-time
measurements shown in Figure~\ref{curves}, but with an extension of
$15\degr$ closer to both poles and both limbs to better illustrate that
the removal of the systematic effect also helps to improve inversions
in high-latitude areas. The top and bottom rows of Figure~\ref{invert}
show results obtained from June 1 -- 10, 2010 and June 1 -- 10, 2011,
when the solar B-angle were also close to 0$\degr$. It can be seen
from left columns of Figure~\ref{invert} that above $\sim 55\degr$
latitude, the inferred meridional flow drops in speed and becomes
equator-ward for the depth of 0 -- 1 Mm, but remains pole-ward deeper
than 3 Mm. This is a suspicious behavior for the meridional flows.
We then antisymmetrize the east-west direction velocity caused by
the center-to-limb variations (middle column of Figure~\ref{invert}),
and subtract it from the inferred meridional velocity. The residual
subsurface meridional flows show much more consistent behaviors at different
depths in high-latitude areas (right column of Figure~\ref{invert}).
From Figure~\ref{invert}, one can also find that the center-to-limb
variation measured from equatorial area also slightly change with
time, and why there is such a change is not understood and worth further
studies.
The removal of the center-to-limb variation in the measured travel times
also decreases the inferred flow speed by nearly 10~m~s$^{-1}$.
This indicates that the subsurface meridional flows derived
from the previous time-distance studies \citep[e.g.,][]{zha04} might
have overestimated the flow speed by a large fraction.
Figure~\ref{all_merid} shows that after removal of the systematic effect,
the meridional flow speed is closer in magnitude to the results obtained
by the magnetic feature tracking \citep{hat10} and surface Doppler
measurements \citep{ulr10}. Note that the results from the magnetic
feature tracking and from the MWO Doppler observations shown in
Figure~\ref{all_merid} are both averaged over a 6-month period with
our analyzed 10-day period in the middle. The difference of analysis
period may result in some differences seen in the figure. The meridional
flows obtained from this study are displayed in Figure~\ref{all_merid}
after a $2\degr$ spatial averaging to remove the strong fluctuations
caused by supergranular flows.
\section{Discussion}
\label{sec3}
The analysis of acoustic travel times obtained from different observables
of the HMI and AIA instruments on SDO by the time-distance helioseismology
technique has revealed a systematic center-to-limb variation.
The systematic variation
is different for different observables, and range from $\sim$2~sec for
the HMI Dopplergram to $\sim$10~sec for the continuum intensity measurements.
For an accurate determination of subsurface meridional flows, and also for
more accurate inference of full-disk subsurface flow fields,
this systematic effect should be removed. We have developed an empirical
correction procedure by removing the systematic variation measured along
the equatorial area during the period when solar B-angle is close to $0\degr$.
This correction reconciles the latitude-dependent $\tau_\mathrm{ns}$\ measured from
different observables and reduces the inferred meridional flow speed by
about 10~m~s$^{-1}$.
It is not quite clear what causes this systematic center-to-limb
effect. However, this effect is unlikely caused by instrumental or
data calibration, because it is observed in the data from two different
instruments, HMI and AIA, and it exists in different observables of
HMI. We also rule out the following factors as causes of this effect:
finite speed of light, contribution of horizontal wave component,
and foreshortening effect. Due to the finite speed of light, acoustic
wave signals observed at high latitude and observed near the equator
are not simultaneous, but this effect is negligible given
the small measurement distances used in this study. The horizontal wave
component \citep{nig07} is used to explain the center-to-limb variation
in mean travel times observed by \citet{duv03}, but this effect is
not significant in travel-time differences for waves traveling in
opposite directions and in small distances as used in this study.
To check foreshortening effects, we mimicked the high-latitude data
by reducing the spatial resolution of the data observed at lower latitudes,
and our measurements did not show systematic changes in the measured
travel times similar to the center-to-limb variations reported here.
It is more likely that this effect has a physical origin related to
properties of solar acoustic waves in the solar atmosphere or response
of the spectral lines to the solar oscillations. It is well known that
for a given spectral line the Sun is observed at the same optical depth
but not at the same geometrical height. The observed height gradually
increases with distance from the disk center. Thus it is possible that
the acoustic waves traveling in opposite directions and different
height will give different measured travel times due to the subsurface
location of the wave source and the waves evanescent behavior above
the photosphere \citep{nag09}. However, this cannot explain why different
observables give different magnitude and trend of the center-to-limb
variations. Another interesting fact is that the largest systematic
effect exists in the measurements using HMI continuum that forms at the lowest
height in the atmosphere. As one moves up to the height where Doppler
velocity is measured, the systematic effect is reduced. As the measurements
move further up to where AIA~1600~\AA\ is formed, the effect reverses sign.
This may give us some indication of that this center-to-limb variation
is related to the line formation height. It is also possible that
this systematic effect is related to differences in the acoustic power
distributions, line-asymmetry of solar modes \citep{duv93}, and the correlated
noise effect \citep{nig98}. The cause of this systematic effect is worth
further studies, and may be resolved by numerical modeling of solar
oscillations including the spectral line-formation simulation and
spherical geometry of the Sun. It is worth attention that when inferring
meridional flow velocity from surface Doppler measurements, a systematic
center-to-limb velocity profile also needs to be removed \citep{ulr88,
ulr10}. A similar limb-shift effect very likely exists in the HMI
Dopplergrams that are used in our time-distance measurements, and
it is not clear whether this effect accounts for some of the systematic
variations in our measured acoustic travel times. This is worth further
studies.
Although the cause of this systematic center-to-limb variation is
not well understood, it is demonstrated that the empirical correction
procedure can improve the inferred subsurface meridional flows.
Figures~\ref{curves} -- \ref{invert} demonstrate
that subtracting the longitude-dependent $\tau_\mathrm{we}$\ from the latitude-dependent
$\tau_\mathrm{ns}$, measured from same observables, is an effective way to remove
this systematic effect. This correction procedure helps to reconcile
the $\tau_\mathrm{ns}$\ measured from different observables, and also helps to remove
the inconsistent behaviors of meridional flows in high-latitude
areas. As an effect, the newly obtained subsurface
meridional flows at shallow depths are approximately 10~m~s$^{-1}$
slower than the speed previously derived following
a similar analysis procedure.
This systematic center-to-limb variation in measured acoustic travel times
has an important implication in the long-searched deep equator-ward
meridional flows, and this is currently under investigation.
\acknowledgments
SDO is a NASA mission, and HMI project is supported by NASA contract
NAS5-02139. We thank Drs.~Roger Ulrich and David Hathaway for providing
us their analysis results used to make Figure~\ref{all_merid}. We also
thank an anonymous referee whose suggestions help to improve the quality
of this paper.
|
{
"timestamp": "2012-03-09T02:04:42",
"yymm": "1203",
"arxiv_id": "1203.1904",
"language": "en",
"url": "https://arxiv.org/abs/1203.1904"
}
|
\section{Introduction}\label{sec1}
General relativity (GR) is established in the framework of the Levi-Civita connection,
therefore there is only curvature rather than torsion in the spacetime. On the other hand,
one can also introduce other connections, such as the Weitzenb\"{o}ck connection, into the same
spacetime where only torsion is reserved.
Thus, there is no such a thing as curvature or torsion of
spacetime, but only curvature or torsion of connection. Basing on
the Weitzenb\"{o}ck connection, Einstein~\cite{Einstein} introduced
firstly the Teleparallel Gravity (TG) in his endeavor to unify
gravity and electromagnetism with the introduction of a tetrad
field. TG can, as is well known, show up as a theory completely
equivalent to GR since the difference between their actions (the
actions of TG and GR are the torsion scalar $T$ and Ricci scalar
$R$, respectively) is just a derivative term~\cite{FNGtnb, FNGtn1, FNGtn2, FNGtn3, FNGtn4, FNGtne}.
Recently, a modification of TG, called $f(T)$
theory~\cite{Bengochea2009, Ferraro2007, Linder, Zheng2011,
Ferraro2008, Ferraro2011, pwhy2011, Wu2011,Bamba2011,Wu2010a,Ben2011ab,Wu2010b, Zhang2011bb, FTbe, FT1,
FT2, FT3, FT4, FT5, FT6, FT7,FT8,FT9, FT10, FT11, FT12, FT13, FT14, FT15, FT16,FT17, FT18, FT19, FT20,
FT21, FT22, FT23,FT24,FT25, FT26, FT27, FT28, FT29, FT30, FT31, FT32, FT33, FT34, FT35, FT36, FT37,
FT38, FTed, Ferraro2011a, Li2011aa,Li2011bb,Li2011b, LiM2011, Miao2011}, has spurred an increasing deal of
attention, as it can explain the present accelerated cosmic
expansion discovered from observations (the Type Ia
supernova~\cite{R98, P99}, the cosmic microwave background
radiation~\cite{Spa, Spb}, and the large scale structure ~\cite{T2004, E2005},
etc.) without the need of dark energy. $f(T)$ theory is obtained by
generalizing the action $T$ of TG to an arbitrary function $f$ of
$T$, which is very analogous to $f(R)$ theory~(see
\cite{FeNoj08,FeSot10,Felice2010,FeNoj11,FebCli} for recent review) where the action $R$ of GR is
generalized to be $f(R)$. An advantage of $f(T)$ theory is that its
field equation is only second order, while in $f(R)$ gravity it is
forth order.
It has been found that $f(T)$ theory can give an inflation without
an inflaton~\cite{Ferraro2007, Ferraro2008}, avoid the big bang
singularity problem in the standard cosmological
model~\cite{Ferraro2011}, realize the crossing of phantom divide
line for the effective equation of state~\cite{Wu2011, Bamba2011},
and yield an usual early cosmic evolution~\cite{Wu2010b, Zhang2011bb}. But, at
the same, this theory lacks the local Lorentz
invariance~\cite{Li2011aa, Li2011bb}, and this results in the appearance of
extra degrees of freedom~\cite{LiM2011}, the broken down of the
first law of black hole thermodynamic~\cite{Miao2011}, and the
problem in cosmic large scale structure~\cite{Li2011b}.
In this paper, we plan to study the causality issue of $f(T)$ theory by examining the possibility of existence of the closed
timelike curves in
the G\"{o}del spacetime~\cite{Godel}. The G\"{o}del metric is the first cosmological solution with rotating matter to the
Einstein equation in GR. Since the G\"{o}del solution is very convenient for studying whether the closed timelike curves exist, it has been
used widely to test the causality issue.
For example, G\"{o}del found that the closed timelike solution cannot be excluded in GR, assuming a cosmological constant or a perfect fluid with its pressure equal to the energy density. G\"{o}del's work has been generalized to include other matter sources, such as, the vector field~\cite{Sombe, SomRe79, SomRay80}, scalar field~\cite{Hiscockbe, HisCha, HisPan}, spinor field~\cite{Villalbabe, VilPim,VilKre,VilLea,VilHered, Reboucas1983} and tachyon field~\cite{Reboucas}. In addition, the G\"{o}del-type universes~\cite{mjrteibe,mjrtei1,mjrteied, Reboucas1983} have also been studied in the framework of other theories
of gravitation, such as TG~\cite{TGGod}, $f (R)$ gravity~\cite{RebClif05e, Reb09b1, Reb10ed} and string-inspired gravitational theory~\cite{stri,barrow1998}.
Here, assuming that the matter source is the perfect fluid or a scalar field, we aim to find out the condition for non-violation of causality in $f(T)$ gravity. The paper is organized as follows. We give, in Sec. II, a brief review of $f(T)$ theory and the vierbein of a general cylindrical symmetry metric in Sec.III. The G\"{o}del-type universe in $f(T)$ theory is discussed in Sec. IV. With an assumption of different matter sources, we investigate the issue of causality in Sec. V. Finally, we present our conclusions in Sec. VI.
\section{ $f(T)$ gravity}\label{ftgravi}
In this section, we give a brief view of $f(T)$ gravity. We use the Greek alphabet ($\mu $, $\nu $,
$\cdots$= 0, 1, 2, 3) to denote tensor indices, that is, indices related to
spacetime, and middle part of the Latin alphabet ($i$, $j$, $\cdots$= 0, 1, 2, 3) to
denote tangent space (local Lorentzian) indices. TG, instead of using the metric tensor, uses tetrad, $e_{\mu }^{i}$ or $e_{i}^{\mu }$ (frame or coframe), as the dynamical object. The
relation between frame and coframe is
\begin{equation}\label{tetradralation}
e_{i}^{\mu }e_{\mu }^{j}=\delta _{i}^{j}\;,\qquad e_{i}^{\mu }e_{\nu
}^{i}=\delta _{\nu }^{\mu },\end{equation}
and the relation between tetrad and metric tensor is
\begin{equation}\label{vierbien}
g_{\mu \nu }=e_{\mu }^{i}e_{\nu }^{j}\eta _{ij}\;,\qquad \eta _{ij}=e_{i}^{\mu }e_{j}^{\nu }g_{\mu
\nu }\;,
\end{equation}
where $\eta _{ij}=diag(1,-1,-1,-1)$ is the Minkowski metric.
Different from GR, the Weitzenb\"{o}ck connection is used in TG
\begin{equation}\label{conection}
{\Gamma }_{\mu \nu }^{\lambda }=e_{i}^{\lambda }\partial _{\nu }e_{\mu
}^{i}=-e_{\mu }^{i}\partial _{\nu }e_{i}^{\lambda }\;.
\end{equation}
As a result, the convariant derivative, denoted by $D_{\mu }$, satisfies:
\begin{equation}\label{drive}
D_{\mu }e_{\nu }^{i}=\partial _{\mu }e_{\nu }^{i}-\Gamma _{\nu \mu
}^{\lambda }e_{\lambda }^{i}=0\;.
\end{equation}
To describe the difference between Weitzenb\"{o}ck and Levi-Civita connections, a contorsion tensor
$K_{\;\;\mu \nu }^{\rho }$ needs to be introduced:
\begin{equation}\label{K}
K_{\;\;\mu \nu }^{\rho }\equiv \Gamma _{\mu \nu }^{\rho }-\overset{\circ }{\Gamma }{}_{\mu \nu }^{\rho
}=\frac{1}{2}(T_{\mu }{}^{\rho }{}_{\nu }+T_{\nu
}{}^{\rho }{}_{\mu }-T_{\;\;\mu \nu }^{\rho })\;.
\end{equation}
Here $T_{\;\;\mu \nu }^{\rho }$ is the torsion tensor
\begin{equation}\label{T}
T_{\;\;\mu \nu }^{\rho }={\Gamma }_{\nu \mu }^{\rho }-{\Gamma }_{\mu \nu
}^{\rho }=e_{i}^{\rho }(\partial _{\mu }e_{\nu }^{i}-\partial _{\nu }e_{\mu
}^{i})\;,
\end{equation}
and $\overset{\circ }{\Gamma }{}_{\mu \nu }^{\rho }$ denotes the
Levi-Civita connection
\begin{equation}
\overset{\circ }{\Gamma }{}_{\mu \nu }^{\rho }=\frac{1}{2}g^{\rho \sigma
}(\partial _{\mu }g_{\sigma \nu }+\partial _{\nu }g_{\sigma \mu }-\partial
_{\sigma }g_{\mu \nu }).
\end{equation}
By defining the super-potential $S_{\sigma }^{\;\;\mu \nu }$
\begin{equation}\label{S}
S_{\sigma }^{\;\;\mu \nu }\equiv K_{\;\;\;\;\sigma }^{\mu \nu
}+\delta _{\sigma }^{\mu }T_{\;\;\;\;\;\alpha }^{\alpha \nu }-\delta _{\sigma
}^{\nu }T_{\;\;\;\;\;\alpha }^{\alpha \mu }\;,
\end{equation}
we obtain the torsion scalar $T$
\begin{equation}\label{scalarT}
T\equiv \frac{1}{2}S_{\sigma }^{\;\;\mu \nu }T_{\;\;\mu \nu }^{\sigma }=
\frac{1}{4}T^{\alpha \mu \nu }T_{\alpha \mu \nu }+\frac{1}{2}T^{\alpha \mu
\nu }T_{\nu \mu \alpha }-T_{\alpha \mu }^{\;\;\;\;\alpha }T_{\;\;\;\;\;\nu }^{\nu \mu
}\;.
\end{equation}
In TG, the Lagrangian density is given by:
\begin{equation}
L_{T}=\frac{eT}{2\kappa ^{2}}\;,
\end{equation}
where, $e=\det (e_\mu^{i})=\sqrt{-g}\;, \kappa ^2{\equiv }8\pi G$. Generalizing $T$ to be an arbitrary function $f$ of $T$ in the above expression, we obtain the Lagrangian
density of $f(T)$ theory
\begin{equation}\label{f1}
L_{T}=\frac{ef(T)}{2\kappa ^2}\;.
\end{equation}
Adding a matter Lagrangian density $L_M$ to Eq.~(\ref{f1}), and
varying the action with respect to the vierbein, one finds the
following field equation of $f(T)$ theory:
\begin{eqnarray}\label{motion1}
&&[e^{-1}\partial_\mu (ee^\rho_i S^{\;\;\nu\mu}_\rho)-e^\lambda_i S^{\rho\mu\nu} T_{\rho\mu\lambda}]f_T(T)
+e^\rho_i S^{\;\;\nu\mu}_\rho \partial_\mu (T)f_{TT}(T)\\ \nonumber
&&+\frac{1}{2}e^\nu_i f(T)=\kappa^2 e^\rho_i \overset{em}{T}{}^{\;\;\nu}_{\rho}.
\end{eqnarray}
Here $f_T= df(T)/dT$, $f_{TT}= d^2f(T)/dT^2$, and $\overset{em}{T}{}^\nu_{\rho}$ is the matter energy-momentum tensor. In a coordinate system, this field equation can be rewritten as
\begin{eqnarray}\label{motion2}
A_{\mu\nu}f_T(T)+S^{\;\;\;\;\;\sigma}_{\nu\mu}(\nabla_\sigma T)f_{TT}(T)+\frac{1}{2}g_{\mu\nu} f(T)=\kappa^2\overset{em}{T}{}_{\mu\nu}\;,
\end{eqnarray}
where
\begin{eqnarray}
&&A_{\mu \nu }=g_{\sigma\mu}e^i_\nu[e^{-1}\partial_\xi(ee^\rho_iS^{\;\;\sigma\xi}_\rho)-e^\lambda_iS^{\rho\xi\sigma} T_{\rho\xi\lambda}]\\ \nonumber
&&\qquad=G_{\mu \nu }-\frac{1}{2}g_{\mu \nu }T=-\nabla^\sigma S_{\nu\sigma\mu }
-S_{\;\;\;\;\mu }^{\rho\lambda }K_{\lambda \rho \nu }\;,
\end{eqnarray}
$G_{\mu\nu}$ is the Einstein tensor, and $\nabla_{\sigma}$ is the covariant derivative associated with the Levi-Civita connection.
The trace of Eq.~(\ref{motion1}) or (\ref{motion2}), which can be used to
simplify and constrain the field equation, can be expressed as
\begin{eqnarray}\label{trace}
-[2e^{-1}\partial_\sigma (eT^{\;\;\rho\sigma}_\rho)+T]f_T(T)+S^{\;\;\rho\sigma}_\rho (\partial_\sigma T)f_{TT}(T)+2f(T)=\kappa^2\overset{em}{T}\;,
\end{eqnarray}
where, $\overset{em}{T}=\overset{em}{T}{}^{\mu}_{\;\;\mu}=g^{\mu \nu }\overset{em}{T}{}_{\mu \nu }$ is the trace
of the energy-momentum tensor.
Clearly, in the case of TG, $f(T)=T$, and Eq.~(\ref{trace}) reduces to
\begin{equation}\label{trTEGR}
T-2e^{-1}\partial _{\sigma }(eT_{\rho }^{\;\;\rho \sigma })=\kappa^2\overset{em}{T}\;,
\end{equation}
which shows an equivalence between GR and TG since \begin{equation}\label{T+R}
-R=T-2e^{-1}\partial _{\sigma }(eT_{\rho }^{\;\;\rho \sigma })\;.
\end{equation}
\section{ vierbein for cylindrical symmetry metric }
Since the G\"{o}del-type metric is usually expressed
in cylindrical coordinates $[(r,\phi,z)]$, we consider a general cylindrical symmetry metric
\begin{eqnarray}\label{ds21}
ds^2=dt^2+2H(r)dtd\phi-dr^2-G(r)d\phi^2-dz^2\;,
\end{eqnarray}
where $H$ and $G$ are the arbitrary functions of $r$. This metric
can be re-expressed in the following form
\begin{eqnarray}
ds^2=[dt+H(r)d\phi]^2-D^2(r)d\phi^2-dr^2-dz^2\;,
\end{eqnarray}
where
\begin{eqnarray}
D(r)=\sqrt{G(r)+H^2(r)}\;.
\end{eqnarray}
Since the local Lorentz invariance is violated in $f(T)$ theory and the
vierbein have six degrees of freedom more than the metric, one
should be careful in choosing a physically reasonable tetrad in
terms of Eq.(\ref{vierbien}). Here, we choose the tetrad anstaz of
the cylindrical symmetry metric to be:
\begin{eqnarray}\label{tet}
e^i_\mu\equiv \left(\begin{array}{cccc}
1&0&H&0\\0&1&0&0\\0&0&D&0\\0&0&0&1
\end{array}\right) \;,\;\;\;
e^\mu_i\equiv\left(\begin{array}{cccc}
1&0&-\frac{H}{D}&0\\0&1&0&0\\0&0&\frac{1}{D}&0\\0&0&0&1\\
\end{array}\right)\;.
\end{eqnarray}
Using Eqs.~(\ref{conection}--\ref{scalarT}), one can find that the
Weitzenb\"{o}ck invariant $T$ is
\begin{eqnarray}\label{T}
T=\frac{1}{2}\left(\frac{H'}{D}\right)^2\;,
\end{eqnarray}
where a prime presents a derivative with respect to $r$.
Substituting the vierbein given in Eq.~(\ref{tet}) into Eq.~(\ref{motion2}), we obtain the following non-zero components of the $f(T)$ field equation:
\\$\nu=0, i=0$
\begin{eqnarray}\label{ft00}
\bigg(T-\frac{D''}{D}+\frac{HT'}{2H'}\bigg)f_T(T)+\bigg(\frac{HT}{H'}-\frac{D'}{D}\bigg)T'f_{TT}(T) +\frac{1}{2}f(T)=\kappa^2\overset{em}{T}{}^0_{\;\;0}
\end{eqnarray}
$\nu=0, i=2$
\begin{eqnarray}\label{ft02}
\bigg(HT+\frac{T'D^2}{2H'}\bigg)f_T(T)+\frac{T'H'}{2}f_{TT}(T)-\frac{H}{2}f(T)=\kappa^2\bigg(\overset{em}{T}{}^0_{\;\;2}-
H\overset{em}{T}{}^0_{\;\;0}\bigg)\;,
\end{eqnarray}
$\nu=1, i=1$
\begin{eqnarray}\label{ft11}
-T f_T(T)+\frac{1}{2}f(T)=\kappa^2\overset{em}{T}{}^1_{\;\;1}\;,
\end{eqnarray}
$\nu=2, i=0$
\begin{eqnarray}\label{ft20}
T'\bigg[\frac{1}{2H'}f_T(T)+\sqrt{\frac{T}{2}}f_{TT}(T)\bigg]=\kappa^2\overset{em}{T}{}^2_{\;\;0}\;,
\end{eqnarray}
$\nu=2, i=2$
\begin{eqnarray}\label{ft22}
-Tf_T(T)+\frac{1}{2}f(T)=\kappa^2\bigg(\overset{em}{T}{}^2_{\;\;2}-H\overset{em}{T}{}^2_{\;\;0}\bigg)\;,
\end{eqnarray}
$\nu=3, i=3$
\begin{eqnarray}\label{ft33}
-\frac{D''}{D}f_T(T)-\frac{T'D'}{D}T'f_{TT}(T)+\frac{1}{2}f(T)=\kappa^2\overset{em}{T}{}^3_{\;\;3}\;.
\end{eqnarray}
Apparently, the non-symmetric components of the modified Einstein
equation are consistent with the tetrad anstaz given in
Eq.~(\ref{tet}). In the above equations, all other components of
$\overset{em}{T}{}^\mu_{\;\;\nu}$ must be zero, which means that,
$\overset{em}{T}{}_{\mu\nu}$, has the cylindrical symmetry as
expected. In a G\"{o}del-type spacetime, the energy-momentum tensor
in a local basis, $\overset{em}{T}_{ab}$ given in~(\ref{Tab}), has a
general form: $\overset{em}{T}{}_{ab}=diag(\rho, p_1,p_2,p_3)$.
Using $\overset{em}{T}{}_{\mu\nu}=e^a_\mu e^b_\nu
\overset{em}{T}{}_{ab} $, we have
\begin{eqnarray}
\overset{em}{T}{}_{00}=\rho,\;\;\overset{em}{T}{}_{11}=p_1,\;\;\;\overset{em}{T}{}_{22}=H^2\rho+D^2p_2,\;\;\;\overset{em}{T}{}_{33}=p_3,\;\;\;\overset{em}{T}{}_{02}=\overset{em}{T}{}_{20}=H\rho\;.
\end{eqnarray}
One can then find easily
\begin{eqnarray}\label{costraint}
\overset{em}{T}{}^2_{\;\;0}=0,\;\;\;\;\overset{em}{T}{}^0_{\;\;2}=H\bigg(\overset{em}{T}{}^0_{\;\;0}-\overset{em}{T}{}^2_{\;\;2}\bigg)\;\;.
\end{eqnarray}
Thus, Eq.~(\ref{ft02}) seems to give an extra constraint on $f(T)$
gravity. This equation is satisfied automatically in a
G\"{o}del-type spacetime, since $T$, as shown in Eq.~(\ref{TG}), is
a constant in a G\"{o}del-type universe. Furthermore, it is easy
to see that, in a G\"{o}del-type spacetime, Eq.~(\ref{ft02}) gives
the same expression as Eq.~(\ref{ft22}). Four independent field
equations are obtained, which is consistent with the anstaz of
tetrad. In addition, one can check that the field equations (23-28)
for the vierbein given in (\ref{tet}) can also be obtained from an
action constructed by replacing the specific form of $T$ (\ref{T})
with the general action of $f(T)$ theory. Therefore, the dynamical
equations are consistent, which means that the tetrad anstaz given
in Eq.~(\ref{tet}) is a good guess for the G\"{o}del-type
spacetime.
\section{G\"{o}del-type universe in $f(T)$ theory}\label{ftgodel}
To show the possibility of existence of the closed timelike curves and
the causality feature in $f(T)$ gravity, we consider the G\"{o}del-type metric, which has the form of Eq.~(\ref{ds21}) with $H$ and $G$ being:
\begin{eqnarray}
H(r)=\frac{4\omega}{m^2}\sinh^2\bigg(\frac{mr}{2}\bigg)\;,
\end{eqnarray}
\begin{eqnarray}
G(r)=\frac{4}{m^2}\sinh^4\bigg(\frac{mr}{2}\bigg)\bigg[\coth^2\bigg(\frac{mr}{2}\bigg)-\frac{4\omega^2}{m^2}\bigg]\;,
\end{eqnarray}
where $\omega$ and $m$ ($-\infty< m^2<+\infty, 0<\omega^2$) are two constant parameters
used to classify different G\"{o}del-type geometries.
Thus, we have
\begin{eqnarray}
D(r)=\frac{1}{m}\sinh(mr)\;.
\end{eqnarray}
Substituting the expressions of $H$ and $D$ into Eq.~(\ref{T}), one can obtain easily
\begin{eqnarray}\label{TG}
T=2\omega^2\;,
\end{eqnarray}
which is a positive constant.
If $G(r)<0$, Eq.~(\ref{ds21}) shows that one type of closed
timelike curve, called noncausal G\"{o}del circle \cite{Godel},
exists in the case of $t, z, r=const$. This means a violation of
causality. For a particular case of $0<m^2<4\omega^2$, the causality
violation region, i.e., $G(r)<0$ region, exists if
\begin{eqnarray}
\tanh^2\frac{mr}{2}<\frac{m^2}{4\omega^2}\;.
\end{eqnarray}\label{critcalr}
Thus, one can define a critical radius $r_c$~\cite{Godel, RebClif05e, Reb09b1, Reb10ed}
\begin{eqnarray}\label{crirad}
\tanh^2\frac{mr_c}{2}=\frac{m^2}{4\omega^2}\;,
\end{eqnarray}
beyond which, $G(r)<0$ and causality is violated. When $m=0$, the critical radius is $r_c=1/\omega$.
When $m^2=4\omega^2$, $r_c=+\infty$,
which means that a breakdown of causality is avoided.
Thus, the codomain range of $r_c$ is
$r_c \in(1/\omega, +\infty)$. Therefore, the condition for non-violation of causality is $m^2\geq 4\omega^2$ or $r<r_c$.
For the case in which $m^2=-\mu^2<0$, both
$H(r)=\frac{4\omega}{\mu^2}\sin^2(\frac{\mu r}{2})$ and
$G(r)=\frac{4}{\mu^2}\sin^4(\frac{\mu r}{2})[\cot^2(\frac{\mu r}{2})-\frac{4\omega^2}{\mu^2}]$
are periodic functions. Thus, an infinite circulation of causal and noncausal ranges
appears~\cite{Reb09b1, Reb10ed}.
It is easy to see that, if one further defines a set of bases $\{\theta^a\}$:
\begin{eqnarray}
\theta^0=dt+H(r)d\phi,\qquad \theta^1=dr,
\end{eqnarray}
\begin{eqnarray}
\theta^2=D(r)d\phi,\qquad \theta^3=dz,
\end{eqnarray}
the Go\"{o}del-type line element can be simplified to be:
\begin{eqnarray}
ds^2=\eta_{ab}\theta^a\theta^b\;,
\end{eqnarray}
where $\eta_{ab}=diag(1,-1,-1,-1)$ is the Minkowski metric.
By choosing $\{\theta^a\}$ as basis,
the $f(T)$ field equation~(\ref{motion2}) becomes:
\begin{eqnarray}\label{ds23}
A_{ab}f_T(T)+\frac{1}{2}\eta_{ab}f(T)=\kappa^2\overset{em}{T}_{ab}\;.
\end{eqnarray}
Here, both $f(T)$ and $f_T(T)$ are evaluated at $T=2\omega^2$. The second term of Eq.~(\ref{motion2}) is discarded in obtaining the above equation since the torsion scalar $T$ is a constant.
We find that the nonzero components of $A_{ab}$ are
\begin{eqnarray}\label{A}
A_{00}=2\omega^2-m^2, \quad A_{11}=A_{22}=2\omega^2, \quad A_{33}=m^2\;.
\end{eqnarray}
Thus, we obtain a very simple form of the field equation in $f(T)$ gravity, which will help us discuss the causality issue.
\section{Causality Problem in $f(T)$ theory}\label{ftcondition}
One can see, from Eq.~(\ref{ds23}), that, in order to discuss the causality problem, the matter source is a very important component. As was obtained in \cite{RebClif05e, Reb09b1, Reb10ed}, different matter sources may lead to different results. In this paper, we assume that the matter source consists of two different components: a perfect fluid and a scalar field. Thus, the energy-momentum tensor $\overset{em}{
T}{}_{ab}$ has the form
\begin{eqnarray}\label{Tab}
\overset{em}{T}_{ab}=\overset{m}{T}_{ab}+\overset{s}{T}_{ab}\;,
\end{eqnarray}
where, $\overset{m}{T}_{ab}$ and $\overset{s}{T}_{ab}$ correspond to the energy-momentum tensors of the
perfect-fluid and the scalar field, respectively. In basis $\{\theta^a\}$, $\overset{m}{T}_{ab}$ and $\overset{s}{T}_{ab}$ can be expressed as
\begin{eqnarray}
\overset{m}{T}_{ab}=(\rho+p)u_a u_b-p\eta_{ab}\;,
\end{eqnarray}
\begin{eqnarray}
\overset{s}{T}_{ab}=D_a\Phi D_b\Phi-\frac{1}{2}\eta_{ab}D_c\Phi
D_d\Phi\eta^{cd}\;,
\end{eqnarray}
where $u_a=(1,0,0,0)$, $\rho$ and $p$ are the energy density and
pressure of the perfect fluid, respectively, and $p=\text{w}\rho$
with $\text{w}$ being the equation of state parameter. $\Phi$ is
the scalar field, and $D_a$ denotes the covariant derivative
relative to the local basis $\theta^a$.
The scalar field equation is
$\square \,\Phi = \eta^{ab}_{}\,\nabla_{a} \nabla_{b} \,\Phi\,=0$. It
is easy to prove that
$\Phi (z)= \varepsilon z + \text{const}$ with a constant amplitude $\varepsilon$ satisfies this field equation
~\cite{Reboucas1983}. Using the solution $\Phi (z)=\varepsilon z + \text{const}$, one can obtain the nonvanishing
components of $\overset{s}{T}_{ab}$
\begin{equation} \label{S-comp}
\overset{s}T_{00} = - \overset{s}T_{11} = - \overset{s}T_{22} = \overset{s}T_{33} = \frac{\varepsilon^2}{2}\,,
\end{equation}
Thus, the energy-momentum
tensor of matter source becomes
\begin{eqnarray}
\overset{em}{T}_{ab}=diag\bigg(\rho+\frac{\varepsilon^2}{2}\;, \text{w}\rho-\frac{\varepsilon^2}{2
}\;, \text{w}\rho-\frac{\varepsilon^2}{2}\;, \text{w}\rho+\frac{\varepsilon^2}{2}\bigg) \label{emt}\;.
\end{eqnarray}
Substituting Eqs.~(\ref{A}) and~(\ref{emt}) into the $f(T)$ field equation (Eq.~(\ref{ds23})), we find
\begin{eqnarray}\label{cfe1}
(2\omega^2-m^2)f_T(T)+\frac{1}{2}f(T)=\kappa^2(\rho+\frac{\varepsilon^2}{2})\;;
\end{eqnarray}
\begin{eqnarray}\label{cfe2}
2\omega^2f_T(T)-\frac{1}{2}f(T)=\kappa^2(\text{w}\rho-\frac{\varepsilon^2}{2})\;;
\end{eqnarray}
\begin{eqnarray}\label{cfe3}
m^2f_T(T)-\frac{1}{2}f(T)=\kappa^2(\text{w}\rho+\frac{\varepsilon^2}{2})\;.
\end{eqnarray}
Since the effective Newton gravity constant in $f(T)$ gravity becomes $G_{N,eff}=G_N/f_T(T)$~\cite{Zheng2011}, only the case $f_T(T)>0$ will be considered in the following in order to ensure a positive $G_{N,eff}$.
From Eqs.~(\ref{cfe1}) and (\ref{cfe2}), one can derive a relation between $m$ and $\omega$:
\begin{eqnarray}
m^2=2\omega^2\bigg[1+\frac{\varepsilon^2}{\rho(1+\text{w})+\varepsilon^2}\bigg]\;,
\end{eqnarray}
which implies that the critical radius of the G\"{o}del's circle, Eq.~(\ref{crirad}), satisfies
\begin{eqnarray} \label{r_c}
\tanh^2\left(\frac{mr_c}{2}\right)=1-\frac{\rho(1+\text{w})}{2[\rho(1+\text{w})+\varepsilon^2]}\;.
\end{eqnarray}
Obviously, different matter sources give rise to different critical
radii and therefore different causality structures, e.g. when
$\varepsilon\rightarrow0$, we have a finite $r_c$, while for
$\rho\rightarrow 0$, $r_c=\infty$. Therefore, a violation of
causality may occur for the case of a perfect fluid as the matter
source, whereas causality is preserved in the case of a scalar
field. In order to show the causality feature in more detail and the
conditions for obtaining the G\"{o}del-type solutions, we will
divide our discussion into two special cases:
$\varepsilon\rightarrow0$ and $\rho\rightarrow0$. In addition, a
concrete $f(T)$ model will be considered.
\subsection{ $\varepsilon^2\rightarrow0$}\label{ftlimit1}
$\varepsilon^2\rightarrow0$ corresponds to the case that the universe only contains a perfect fluid.
Since $f_T(T)>0$, Eqs.~(\ref{cfe1}), (\ref{cfe2}), and (\ref{cfe3})
reduce to:
\begin{eqnarray}\label{mfe3}
m^2=2\omega^2\;;
\end{eqnarray}
\begin{eqnarray}\label{mfe2}
Tf_T(T)=\kappa^2\rho(1+\text{w})\;;
\end{eqnarray}
\begin{eqnarray}\label{mfe1}
f(T)=2\kappa^2\rho\;.
\end{eqnarray}
From Eqs.~(\ref{mfe2}, \ref{mfe1}), it is easy to see that, in
the limit of general relativity without a cosmological constant
($f(T)=T$), $\text{w}=1$ is required to ensure the existence of the
G\"{o}del-type solutions~\cite{godelnote1978, RebClif05e, Reb09b1, Reb10ed}. This means that a violation of causality in general relativity is
only possible for the so-called stiff fluid ($\text{w}=1$) which is not a normal fluid in our Universe.
In $f(T)$ theory, $Tf_T(T)>0$ and $\rho>0$ lead to $\text{w}>-1$.
So, the perfect fluid must satisfy the weak energy condition
($\rho>0$ and $\rho(1+\text{w})>0$). Using the above results, the
equation of state can be expressed as a function of the torsion
scalar:
\begin{eqnarray}\label{wofft}
\text{w}=\frac{2Tf_T(T)}{f(T)}-1\;.
\end{eqnarray}
Different from general relativity that requires $\text{w}= 1$
for perfect-fluid G\"{o}del solutions, the equation of state
parameter of the fluid $\text{w}$ in $f(T)$ gravity can differ from
one and its value is determined by concrete $f(T)$ models. For
example, a special $f(T)=\lambda T^{\delta}$ gives
$\text{w}=2\delta-1$,
from which one can see that $\text{w}$ can be an arbitrary number
for an arbitrary $\delta$. So, even normal matter, such as
pressureless matter or radiation, can lead to a violation of
causality in certain $f(T)$ theories. This indicates that the
issue of causality violation seems more severe in $f(T)$ gravity
than in general relativity where only an exotic stiff fluid allows
the existence of G\"{o}del-type solutions. From Eqs.~(\ref{mfe3}),
(\ref{mfe2}) and (\ref{mfe1}), and using $T=2\omega^2$, we find that
the critical radius given in Eq.~(\ref{r_c}) becomes
\begin{eqnarray}
r_c=2\text{tanh}^{-1}\bigg(\frac{1}{\sqrt{2}}\bigg)\cdot\sqrt{\frac{f_T(T)}{(1+\text{w})\kappa^2\rho}}\;,
\end{eqnarray}
which is dependent both on the specifics of $f(T)$ theory and the properties of the perfect fluid.
Now, let us consider a concrete power law $f(T)$ model~\cite{Linder}
\begin{eqnarray}
f(T)=T-\alpha T_* \left(\frac{T}{T_*}\right) ^n\;,
\end{eqnarray}
where $\alpha$ and $n$ are model parameters, and $T_*$ is a special
value of the torsion scalar, which is introduced to make $\alpha$
dimensionless. $|n|\ll 1$ is required in order to obtain an usual
early cosmic evolution~\cite{Wu2010b}. The current cosmic
observations give that $\alpha=-0.79^{+0.35}_{-0.79}$ and
$n=0.04^{+0.22}_{-0.33}$ at the $68.3\%$ confidence
level~\cite{Wu2010a}. Thus, a negative $\alpha$ is favored by
observations. In term of Eq.~(\ref{wofft}), the equation of state
of the perfect fluid becomes
\begin{eqnarray}\label{wpow}\label{consofwpow1}
\text{w}=1-\frac{2\alpha(n-1)T^{1-n}_*}{T^{1-n}-\alpha T^{1-n}}\;.
\end{eqnarray}
The equation above can be re-expressed as
\begin{eqnarray}\label{consofwpow2}
\frac{\alpha(2n-1-\text{w})}{1-\text{w}}=\left(\frac{T}{T_*}\right)^{1-n}>0\;,
\end{eqnarray}
where a positive $T/T_*$ is considered. Recalling $\alpha<0$ and $\text{w}>-1$, from Eq.~(\ref{consofwpow2}) one can obtain the possible ranges of $\text{w}$ for the G\"{o}del-type universes
\begin{eqnarray}\label{wpowrange}
1>\text{w}>-1+2n \quad(1>n>0)\;,\qquad 1>\text{w}>-1 \quad(n<0)\;.
\end{eqnarray}
For this power law model, the critical radius has the form
\begin{eqnarray}\label{powrc}
r_c=2\left[\frac{\alpha(2n-1-\text{w})}{1-\text{w}}\right]^{\frac{1}{2(n-1)}}\text{tanh}^{-1}(1/\sqrt{2})\;,
\end{eqnarray}
which is determined completely by the model parameters and the equation of state of the perfect fluid.
\subsection{ $\protect\rho\rightarrow0$}\label{ftlimit2}
This is the case of a scalar field as the matter source.
Eqs.~(\ref{cfe1}),~(\ref{cfe2}), and (\ref{cfe3}) now reduce to
\begin{eqnarray}\label{sfe0}
m^2=4\omega^2\;,
\end{eqnarray}
\begin{eqnarray}\label{sfe1}
Tf_T(T)=\kappa^2\varepsilon^2\;,
\end{eqnarray}
\begin{eqnarray}\label{sfe2}
f(T)=3\kappa^2\varepsilon^2\;.
\end{eqnarray}
Note that (\ref{sfe1}) and (\ref{sfe2})
combined together admit a relation between $T$ and $f(T)$:
\begin{eqnarray} \label{sig}
3Tf_T(T)-f(T)=0\;,
\end{eqnarray}
which constrains the class of solutions with no violation
of causality.
For the power law model, the causal G\"{o}del-type solution gives that the torsion scalar should satisfy
\begin{eqnarray}\label{ssfpow}
T=2\omega^2=\left[-\frac{(1-3n)\alpha}{2}\right]^{\frac{1}{1-n}}T_*\;.
\end{eqnarray}
Thus, $n <1/3$ is required if the numerator of $\frac{1}{1-n}$ is not even since the observations show $\alpha<0$.
\section{Conclusions}\label{ftconclusion}
$f(T)$ theory, a new modified gravity, provides an alternative way
to explain the present accelerated cosmic acceleration with no need
of dark energy. Some problems, including large scale structure,
local Lorentz invariance, and so on, of this modified gravity have
been discussed. In this paper, we study the issue of causality in
$f(T)$ theory by examining the possibility of existence of the
closed timelike curves in the G\"{o}del metric. Assuming that the
matter source is a scalar field or a perfect fluid, we examine the
existence of the G\"{o}del-type solutions. For the scalar field
case, we find that $f(T)$ gravity allows a particular G\"{o}del-type
solution with $r_c\rightarrow\infty$, where $r_c$ is the critical
radius beyond which the causality is broken down. Thus, the
violation of causality can be forbidden.
In the case of a perfect fluid as the matter source, we find
that the fluid must have an equation of state parameter greater than minus one
and this parameter should satisfy Eq.~(\ref{wofft}) for the
G\"{o}del-type solutions to exist. For certain $f(T)$ models, the
perfect fluid that allows the G\"{o}del-type solutions can even be
normal matter, such as pressureless matter or radiation. Since the
critical radius $r_c$ of perfect fluid G\"{o}del-type solutions
which depends on both matter and gravity is finite, the issue of
causality violation seems more severe in $f(T)$ gravity than in
general relativity where only an exotic stiff fluid allows the
existence of G\"{o}del-type solutions.
\begin{acknowledgments}
PXW would like to thank Prof.
Qingguo Huang for helpful discussions. This work was supported by
the National Natural Science Foundation of China under Grants Nos.
10935013, 11175093 and 11075083, Zhejiang Provincial Natural Science
Foundation of China under Grants Nos. Z6100077 and R6110518, the
FANEDD under Grant No. 200922, the National Basic Research Program
of China under Grant No. 2010CB832803, the NCET under Grant No.
09-0144, and K.C. Wong Magna Fund in Ningbo University.
\end{acknowledgments}
|
{
"timestamp": "2012-10-15T02:02:15",
"yymm": "1203",
"arxiv_id": "1203.2016",
"language": "en",
"url": "https://arxiv.org/abs/1203.2016"
}
|
\section{Introduction}
Optical lattice ultracold atoms continue to be of interest for more and more researches of different branches of physics \cite{Dalibard}. Big attention is given for the realization of different condensed matter models that provide a test system for achieving a deep understanding of fundamental physics and answering open questions in the subject \cite{Lewenstein}, beside their applications for quantum information processing \cite{Zeilinger}. In general, the main objective is to consider optical lattice ultracold atoms as artificial crystals with a wide range of controllable parameters.
Optical lattices form of counter propagating laser beams to get standing waves in which ground state ultracold atoms are loaded \cite{Bloch,Jaksch}. The atoms experience optical lattice potential with lattice constant of half wave length of the laser. Low dimensional lattices can be achieved with different geometric structures and symmetries \cite{Spielman}. In conventional solid crystals the lattice constant and the symmetry of the lattice is fixed through the different chemical bonds that responsible for the formation of the crystal. The advantage of optical lattices is due to the controllability of the lattice constant and symmetry through controlling the external laser field \cite{Dalibard}.
Collective states of electronic excitations play a central rule in solid crystals and molecular clusters and they usually termed excitons \cite{Davydov,Agranovich}. They induced by electrostatic interactions among the lattice atoms or molecules, where an electronic excitation can be delocalized in the crystal through energy transfer. Collective states can dominate the electrical and optical properties of the material, and especially they strongly affect the excitation lifetimes and give rise to dark and superradiant states. In such material the lattice constant is few angstroms which is much smaller than the electronic transition wavelength, and hence one can use electrostatic interactions, e.g. resonance dipole-dipole interactions, and to neglect radiative corrections altogether.
Electronic excitations in optical lattice ultracold atoms are of big importance, e.g., for optical lattice clocks \cite{Katori}, and for optical lattice Rydberg atoms \cite{Arimondo}. In our previous work we introduced excitons for optical lattice ultracold atoms in one and two dimensional set-ups \cite{ZoubiA,ZoubiB}. We concentrated mainly in the Mott insulator phase with one and two atoms per lattice site. We treated both large and finite atomic chains \cite{ZoubiC,ZoubiD,ZoubiE}, and we calculated the damping rate of excitons into free space and their emission pattern \cite{ZoubiF,ZoubiG,ZoubiH}. In all of our previous researches we exploited electrostatic interactions for the formation of collective states, mainly resonance dipole-dipole interactions. But for typical optical lattices the lattice constant is few thousands of angstroms, which can be of the order of the electronic transition wavelength, and hence radiative corrections can be significant.
In the present paper we investigate a one dimensional finite chain of atoms where the lattice constant can take any value relative to the atomic transition wavelength. Finite atomic chains have been realized recently in a number of optical lattice experiments \cite{Vetsch,Weitenberg}. We emphasize the influence of radiative corrections on the formation of collective sates and their damping rates, where we exploit general collective states with emphasize on the most symmetric one. We derive the condition for the validity of applying electrostatic interactions, which we used in our previous work. Few studies treated the collective effect on the optical properties of finite atomic chain of several atoms \cite{Mewton}, but extensive study done for two atoms in the radiative regime \cite{Ficek}, and in which we compare our results. We extract how the damping rate depends on the chain size, namely on the number of atoms in the lattice. Furthermore, we calculate the emission pattern off a finite atomic chain.
The paper is organized as follows: in section 2 we present a finite one dimensional atomic chain and discuss the energy transfer parameter due to dipole-dipole interactions in the radiative regime. Then in section 3 we calculate the damping rates for different collective states and several chain sizes. The emission pattern for collective states is calculated in section 4. The summary appears in section 5.
\section{Finite One-Dimensional Atomic Chain}
We consider a finite one dimensional atomic lattice, where the number of atoms is $N$ with lattice constant $a$, as seen in figure (1). The atoms are considered to be two-level systems with electronic transition energy $E_A=\hbar\omega_A$. An electronic excitation can delocalize in the lattice by transferring among the atoms. The electronic excitation Hamiltonian is given by
\begin{equation}
H_{ex}=\sum_n\hbar\omega_A\ B_n^{\dagger}B_n+\sum_{nm}\ \hbar J_{nm}B_n^{\dagger}B_m,
\end{equation}
where $B_n^{\dagger}$ and $B_n$ are the creation and annihilation operators of an electronic excitation at atom $n$. For a single excitation the operators can be assumed to obey boson commutation relations.
\begin{figure}[h!]
\centerline{\epsfxsize=7cm \epsfbox{Plot1.eps}}
\caption{A finite lattice of $N$ atoms. The lattice constant is $a$, and the transition dipole $\mbox{\boldmath$\mu$}$ makes an angle $\varphi$ with the lattice direction.}
\end{figure}
The energy transfer among two atoms, $n$ and $m$, is a function of the interatomic distance and given by \cite{Craig}
\begin{eqnarray}\label{Exact}
J\left(q_AR_{nm}\right)&=&\frac{3}{4}\Gamma_A\left\{\left[\frac{\sin\left(q_AR_{nm}\right)}{\left(q_AR_{nm}\right)^2}+\frac{\cos\left(q_AR_{nm}\right)}{\left(q_AR_{nm}\right)^3}\right]\right. \nonumber \\
&\times&\left.\left(1-3\cos^2\varphi\right)\right. \nonumber \\
&-&\left.\frac{\cos\left(q_AR_{nm}\right)}{q_AR_{nm}}\left(1-\cos^2\varphi\right)\right\},
\end{eqnarray}
where the distance between the two atoms is $R_{nm}=|n-m|a$, and $\mu$ is the magnitude of the electronic excitation transition dipole, which makes an angle $\varphi$ with the lattice direction, see figure (1). $q_A$ is the atomic transition wave number given by $E_A=\hbar cq_A$. Here $\Gamma_A$ is the single excited atom damping rate
\begin{equation}
\Gamma_A=\frac{\omega^3_A\mu^2}{3\pi\epsilon_0\hbar c^3}.
\end{equation}
In the limit of $\lambda_A>a$, where $\lambda_A$ is the atomic transition wave length defined by $E_A=hc/\lambda_A$, we can consider only energy transfer among nearest neighbor atoms with $J\left(q_Aa\right)$ where we take $R_{nm}=a$.
In figure (2) we plot $J\left(q_Aa\right)/\Gamma_A$ as a function of $q_Aa$ for two different polarization directions. Note that for typical optical lattice we have $E_A=1\ eV$, with $\lambda_A\approx 12405\ \AA$, and $q_A\approx 4\times10^{-4}\ \AA^{-1}$. For $a=1000\ \AA$ we get $q_Aa\approx 0.5$, and $a/\lambda_A\approx 0.08$. For $\varphi=0^{\circ}$ we obtain $J(0.5)/\Gamma_A\approx -13.4$, and for $\varphi=90^{\circ}$ we get $J(0.5)/\Gamma_A\approx 5.4$. For large $q_Aa$ the coupling tend to zero with oscillations, and the atoms are almost independent.
\begin{figure}[h!]
\centerline{\epsfxsize=8cm \epsfbox{Plot2.eps}}
\caption{The scaled interaction $J\left(q_Aa\right)/\Gamma_A$ vs. $q_Aa$. The full line is for $\varphi=0^{\circ}$, and the dashed line for $\varphi=90^{\circ}$.}
\end{figure}
In the limit $\lambda_A\gg a$, or $q_Aa\ll1$, we can neglect the radiative terms (as we did in our previous works \cite{ZoubiA,ZoubiB,ZoubiC,ZoubiD,ZoubiE,ZoubiF,ZoubiG,ZoubiH}), to get the electrostatic resonance dipole-dipole interaction
\begin{equation}\label{Appr}
J\approx\frac{3}{4}\frac{\Gamma_A}{(q_Aa)^3}\left(1-3\cos^2\varphi\right).
\end{equation}
Using the previous numbers, $\varphi=0^{\circ}$ yields $J(0.5)/\Gamma_A\approx -12$, and $\varphi=90^{\circ}$ yields $J(0.5)/\Gamma_A\approx 6$, which are slightly different from the above exact results. For smaller $q_Aa$ we get much better agreement. In figure $(3)$ we plot equations (\ref{Exact}) and (\ref{Appr}) for $\varphi=0^{\circ}$, and in figure $(4)$ for $\varphi=90^{\circ}$. The results justify the use of electrostatic dipole-dipole interactions for optical lattice ultracold atoms when $q_Aa<1$.
\begin{figure}[h!]
\centerline{\epsfxsize=8cm \epsfbox{Plot3.eps}}
\caption{The scaled interaction $J\left(q_Aa\right)/\Gamma_A$ vs. $q_Aa$ for $\varphi=0^{\circ}$. The full line is for equation (\ref{Exact}), and the dashed line for equation (\ref{Appr}).}
\end{figure}
\begin{figure}[h!]
\centerline{\epsfxsize=8cm \epsfbox{Plot4.eps}}
\caption{The scaled interaction $J\left(q_Aa\right)/\Gamma_A$ vs. $q_Aa$ for $\varphi=90^{\circ}$. The full line is for equation (\ref{Exact}), and the dashed line for equation (\ref{Appr}).}
\end{figure}
\section{Collective Excitation Damping Rate}
We start in presenting the free space radiation field and its coupling to a finite atomic chain. The free space radiation field Hamiltonian is
\begin{equation}
H_{rad}=\sum_{{\bf q}\lambda}E_{ph}(q)\ a_{{\bf q}\lambda}^{\dagger}a_{{\bf q}\lambda},
\end{equation}
where $a_{{\bf q}\lambda}^{\dagger}$ and $a_{{\bf q}\lambda}$ are the creation and annihilation operators of a photon with wave vector ${\bf q}$ and polarization $\lambda$, respectively. The photon energy is $E_{ph}(q)=\hbar cq$. The electric field operator is
\begin{equation}
\hat{\bf E}({\bf r})=i\sum_{{\bf q}\lambda}\sqrt{\frac{\hbar cq}{2\epsilon_0 V}}\left\{a_{{\bf q}\lambda}\ {\bf e}_{{\bf q}\lambda}e^{i{\bf q}\cdot{\bf r}}-a_{{\bf q}\lambda}^{\dagger}\ {\bf e}_{{\bf q}\lambda}^{\ast}e^{-i{\bf q}\cdot{\bf r}}\right\},
\end{equation}
where ${\bf e}_{{\bf q}\lambda}$ is the photon polarization unit vector, and $V$ is the normalization volume.
The atomic transition dipole operator is
\begin{equation}
\hat{\mbox{\boldmath$\mu$}}=\mbox{\boldmath$\mu$}\sum_{n=1}^N\left(B_n+B_n^{\dagger}\right).
\end{equation}
The matter-field coupling is given formally by the electric dipole interaction $H_I=-\hat{\mbox{\boldmath$\mu$}}\cdot\hat{\bf E}$. In the rotating wave approximation and for linear polarization, we get
\begin{eqnarray}
H_I&=&-i\sum_{{\bf q}\lambda,n}\sqrt{\frac{\hbar cq}{2\epsilon_0 V}}\left(\mbox{\boldmath$\mu$}\cdot{\bf e}_{{\bf q}\lambda}\right) \nonumber \\
&\times&\left\{a_{{\bf q}\lambda}B_n^{\dagger}\ e^{iq_zna}-a_{{\bf q}\lambda}^{\dagger}B_n\ e^{-iq_zna}\right\}.
\end{eqnarray}
In the following we treat a single electronic excitation in the atomic chain. We start in treating the most symmetric collective state and then the general collective state.
\subsection{Symmetric Collective Excitation}
We consider a single excitation in the system with the symmetric collective state
\begin{equation}
|i\rangle_s=\frac{1}{\sqrt{N}}\sum_i|g_1,\cdots,e_i,\cdots,g_N\rangle.
\end{equation}
This state is an eigenstate of the Hamiltonian in the limit of $q_Aa>1$ with $J/\Gamma_A<1$, where the atoms are almost independent. The other limit of $q_Aa<1$ treated by us in other work \cite{ZoubiA,ZoubiB,ZoubiC,ZoubiD,ZoubiE,ZoubiF,ZoubiG,ZoubiH}.
We calculate the damping rate of such collective state through the emission of a photon into free space and the damping into the final ground state
\begin{equation}
|f\rangle=|g_1,\cdots,g_N\rangle.
\end{equation}
We apply the Fermi golden rule to calculate the collective symmetric state damping rate
\begin{equation}
\Gamma_s=\frac{2\pi}{\hbar}\sum_{{\bf q}\lambda}|\langle f|H_I|i\rangle|^2\delta(E_A-E_{ph}),
\end{equation}
which in the present case reads
\begin{equation}
\Gamma_s=\sum_{{\bf q}\lambda}\frac{\pi cq}{\epsilon_0 VN}\left(\mbox{\boldmath$\mu$}\cdot{\bf e}_{{\bf q}\lambda}\right)^2\left|\sum_{n=1}^Ne^{-iq_zna}\right|^2\delta(E_A-E_{ph}).
\end{equation}
The summation over the photon polarization yields
\begin{equation}
\sum_{\lambda}\left(\mbox{\boldmath$\mu$}\cdot{\bf e}_{{\bf q}\lambda}\right)^2=\mu^2-\frac{\left({\bf q}\cdot\mbox{\boldmath$\mu$}\right)^2}{q^2}.
\end{equation}
The summation over ${\bf q}$ can be converted into the integral
\begin{equation}
\sum_{\bf q}\rightarrow \frac{V}{(2\pi)^3}\int_{0}^{2\pi}d\phi\int_{0}^{\pi}d\theta\sin\theta\int_0^{\infty}q^2 dq.
\end{equation}
We use
\begin{equation}
{\bf q}=q(\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta),
\end{equation}
and the transition dipole is taken to be
\begin{equation}
\mbox{\boldmath$\mu$}=\mu(\sin\varphi,0,\cos\varphi).
\end{equation}
The integration over $\phi$, and the change of the variable $y=q_Aa\ cos\theta$, gives
\begin{eqnarray}\label{Gamma}
\Gamma_s&=&\frac{\mu^2q_A^2}{8\pi\epsilon_0 \hbar aN}\int_{-q_Aa}^{+q_Aa}dy\left|\sum_{n=1}^Ne^{-iny}\right|^2 \nonumber \\
&\times&\left[\left(1+\cos^2\varphi\right)-\frac{y^2}{(q_Aa)^2}\left(3\cos^2\varphi-1\right)\right].
\end{eqnarray}
Using the relation
\begin{equation}
\left|\sum_{n=1}^Ne^{-iny}\right|^2=N+\sum_{n<m=1}^N2\cos[(n-m)y],
\end{equation}
we reach, after the integration over $y$, the result
\begin{equation}
\Gamma_s=\Gamma_A\left\{1+\frac{2}{N}\sum_{n<m=1}^NF[q_Aa(n-m)]\right\},
\end{equation}
where
\begin{eqnarray}
F(x)&=&\frac{3}{2}\left\{\frac{\sin x}{x}\left(1-\cos^2\varphi\right)\right. \nonumber \\
&+&\left.\left[\frac{\cos x}{x^2}-\frac{\sin x}{x^3}\right]\left(1-3\cos^2\varphi\right)\right\}.
\end{eqnarray}
In figure (5) we plot the function $F(x)$, for two different polarization directions. Using the previous numbers, for $\varphi=0^{\circ}$ we get $F(0.5)=0.9752$, and for $\varphi=90^{\circ}$ we get $0.9507$, which justifies the use of $F=1$ for optical lattice ultracold atoms in our previous works \cite{ZoubiA,ZoubiB,ZoubiC,ZoubiD,ZoubiE,ZoubiF,ZoubiG,ZoubiH}.
\begin{figure}[h!]
\centerline{\epsfxsize=8cm \epsfbox{Plot5.eps}}
\caption{The function $F(x)$ vs. $x$. The full line is for $\varphi=0^{\circ}$, and the dashed line for $\varphi=90^{\circ}$.}
\end{figure}
For different number of atoms we get
\begin{eqnarray}
\Gamma_s(1)&=&\Gamma_A, \nonumber \\
\Gamma_s(2)&=&\Gamma_A\left\{1+F(q_Aa)\right\}, \nonumber \\
\Gamma_s(3)&=&\Gamma_A\left\{1+\frac{2}{3}\left[2F(q_Aa)+F(2q_Aa)\right]\right\}, \nonumber \\
&\cdots&
\end{eqnarray}
The symmetric damping rate can be written in the form
\begin{equation}
\Gamma_s(N)=\Gamma_A\left\{1+2\sum_{n=1}^{N-1}\frac{(N-n)}{N}F(q_Aan)\right\}.
\end{equation}
In figure $(6)$ we plot $\Gamma/\Gamma_A$ as a function of $q_Aa$ for $N=5$, and for the polarizations $\varphi=0^{\circ}$ and $\varphi=90^{\circ}$.
\begin{figure}[h!]
\centerline{\epsfxsize=8cm \epsfbox{Plot6.eps}}
\caption{The symmetric scaled damping $\Gamma/\Gamma_A$ vs. $q_Aa$, for $N=5$. The full line is for $\varphi=0^{\circ}$, and the dashed line for $\varphi=90^{\circ}$.}
\end{figure}
Lets consider $F(q_Aan)$ to represent a bond between two atoms that separated by a distance $(an)$, then in the above summation the function $F(q_Aan)$ is multiplied by the number of bonds of this length which is $(N-n)$. In the limit of $q_Aa\ll1$ we get $F\simeq 1$, and then $\Gamma_s(N)\approx N\Gamma_A$. In the limit of $q_Aa\gg1$ we get $F\simeq 0$, and then $\Gamma_s(N)\approx\Gamma_A$ with oscillations.
Now we emphasize the dependence of the symmetric state damping rate as a function of the number of atoms $N$. We plot the scaled damping rate $\Gamma_s/\Gamma_A$ as a function of $N$ for different values of $q_Aa$. In figures $(7-9)$ we plot for $q_Aa=0.001$, $q_Aa=0.1$, and $q_Aa=1$, in the two cases of $\varphi=0^{\circ}$ and $\varphi=90^{\circ}$. The damping rate of the symmetric state grows linearly with the number of atoms for small $N$, and approach a finite value for large $N$. For $q_Aa\geq1$ the damping rate approach the finite value faster than for $q_Aa\ll1$. In figure $(10)$ we plot $\Gamma_s/\Gamma_A$ as a function of $\phi$ for $N=100$ at $q_Aa=0.1$. Significant difference appears between the damping rates for $\varphi=0^{\circ}$ and $\varphi=90^{\circ}$.
\begin{figure}[h!]
\centerline{\epsfxsize=8cm \epsfbox{Plot7.eps}}
\caption{The symmetric scaled damping rate $\Gamma/\Gamma_A$ vs. $N$, for $q_Aa=0.001$. The full line is for $\varphi=0^{\circ}$, and the dashed line for $\varphi=90^{\circ}$.}
\end{figure}
\begin{figure}[h!]
\centerline{\epsfxsize=8cm \epsfbox{Plot8.eps}}
\caption{The symmetric scaled damping rate $\Gamma/\Gamma_A$ vs. $N$, for $q_Aa=0.1$. The full line is for $\varphi=0^{\circ}$, and the dashed line for $\varphi=90^{\circ}$.}
\end{figure}
\begin{figure}[h!]
\centerline{\epsfxsize=8cm \epsfbox{Plot9.eps}}
\caption{The symmetric scaled damping rate $\Gamma/\Gamma_A$ vs. $N$, for $q_Aa=1$. The full line is for $\varphi=0^{\circ}$, and the dashed line for $\varphi=90^{\circ}$.}
\end{figure}
\begin{figure}[h!]
\centerline{\epsfxsize=8cm \epsfbox{Plot10.eps}}
\caption{The symmetric scaled damping rate $\Gamma/\Gamma_A$ vs. $\varphi$, for $N=100$ at $q_Aa=0.1$.}
\end{figure}
\subsection{General Collective Excitation}
Here we consider the case of a single excitation but for a general collective state, which is given by
\begin{equation}
|i\rangle=\frac{1}{\sqrt{N}}\sum_iC_i|g_1,\cdots,e_i,\cdots,g_N\rangle,
\end{equation}
where $C_i=\pm 1$, and $\sum_iC_i^2=N$. Equation (\ref{Gamma}) reads
\begin{eqnarray}
\Gamma&=&\frac{\mu^2q_A^2}{8\pi\epsilon_0 \hbar aN}\int_{-q_Aa}^{+q_Aa}dy\left|\sum_{n=1}^NC_ne^{-iny}\right|^2 \nonumber \\
&\times&\left[\left(1+\cos^2\varphi\right)-\frac{y^2}{(q_Aa)^2}\left(3\cos^2\varphi-1\right)\right].
\end{eqnarray}
We use
\begin{equation}
\left|\sum_{n=1}^NC_ne^{-iny}\right|^2=N+\sum_{n\neq m=1}^NC_nC_me^{-i(n-m)y},
\end{equation}
where $C_nC_m=\pm 1$. After integration over $y$, we get
\begin{equation}
\Gamma(N)=\Gamma_A\left\{1+\frac{2}{N}\sum_{n<m=1}^NC_nC_m\ F[q_Aa(n-m)]\right\}.
\end{equation}
Here we present the results for two examples. For $N=2$ we have the symmetric state
\begin{equation}
|i\rangle_s=\frac{|e_1,g_2\rangle+|g_1,e_2\rangle}{\sqrt{2}},
\end{equation}
with the damping rate
\begin{equation}
\Gamma_s=\Gamma_A\left\{1+F(q_Aa)\right\},
\end{equation}
and the antisymmetric state
\begin{equation}
|i\rangle_a=\frac{|e_1,g_2\rangle-|g_1,e_2\rangle}{\sqrt{2}},
\end{equation}
with the damping rate
\begin{equation}
\Gamma_a=\Gamma_A\left\{1-F(q_Aa)\right\}.
\end{equation}
The results for $N=2$ agree with the known results \cite{Ficek}. In figure $(11)$ we plot $\Gamma/\Gamma_A$ for the symmetric state of $N=2$ as a function of $q_Aa$, for the polarizations $\varphi=0^{\circ}$ and $\varphi=90^{\circ}$. In figure $(12)$ the plot is for the antisymmetric state.
\begin{figure}[h!]
\centerline{\epsfxsize=8cm \epsfbox{Plot11.eps}}
\caption{The scaled damping $\Gamma/\Gamma_A$ vs. $q_Aa$, for $N=2$ with the symmetric state. The full line is for $\varphi=0^{\circ}$, and the dashed line for $\varphi=90^{\circ}$.}
\end{figure}
\begin{figure}[h!]
\centerline{\epsfxsize=8cm \epsfbox{Plot12.eps}}
\caption{The scaled damping $\Gamma/\Gamma_A$ vs. $q_Aa$, for $N=2$ with the antisymmetric state. The full line is for $\varphi=0^{\circ}$, and the dashed line for $\varphi=90^{\circ}$.}
\end{figure}
For $N=3$, for the symmetric state
\begin{equation}
|i\rangle_s=\frac{|e_1,g_2,g_3\rangle+|g_1,e_2,g_3\rangle+|g_1,g_2,e_3\rangle}{\sqrt{3}},
\end{equation}
we get
\begin{equation}
\Gamma_s=\Gamma_A\left\{1+\frac{2}{3}\left[2F(q_Aa)+F(2q_Aa)\right]\right\}.
\end{equation}
For the antisymmetric state
\begin{equation}
|i\rangle_a=\frac{|e_1,g_2,g_3\rangle-|g_1,e_2,g_3\rangle+|g_1,g_2,e_3\rangle}{\sqrt{3}},
\end{equation}
we get
\begin{equation}
\Gamma_a=\Gamma_A\left\{1-\frac{2}{3}\left[2F(q_Aa)-F(2q_Aa)\right]\right\}.
\end{equation}
In the limit of $q_Aa\ll1$ we have $F\simeq 1$, then for the symmetric state we get $\Gamma\approx 3\Gamma_A$, and for the antisymmetric one we get $\Gamma\approx\Gamma_A/3$. In the limit of $q_Aa\gg1$ we have $F\simeq 0$, then for the symmetric and antisymmetric states we get $\Gamma\approx\Gamma_A$.
In figures $(13)$ we plot $\Gamma/\Gamma_A$ for the symmetric state of $N=3$ as a function of $q_Aa$, for the polarizations $\varphi=0^{\circ}$ and $\varphi=90^{\circ}$. In figure $(14)$ the plot is for the antisymmetric state.
\begin{figure}[h!]
\centerline{\epsfxsize=8cm \epsfbox{Plot13.eps}}
\caption{The scaled damping $\Gamma/\Gamma_A$ vs. $q_Aa$, of the symmetric state for $N=3$. The full line is for $\varphi=0^{\circ}$, and the dashed line for $\varphi=90^{\circ}$.}
\end{figure}
\begin{figure}[h!]
\centerline{\epsfxsize=8cm \epsfbox{Plot14.eps}}
\caption{The scaled damping $\Gamma/\Gamma_A$ vs. $q_Aa$, of the antisymmetric state for $N=3$. The full line is for $\varphi=0^{\circ}$, and the dashed line for $\varphi=90^{\circ}$.}
\end{figure}
\section{Collective Excitation Emission Pattern}
Here we calculate the emission pattern of a collective state in a chain of $N$ atoms separated by a distance $a$. The transition dipole of each atom is $\mbox{\boldmath$\mu$}=\mu(\sin\varphi,0,\cos\varphi)$, at positions ${\bf R}_n=(0,0,R_n)$. For simplicity the observation point is taken to be at ${\bf r}=(x,0,0)$, as seen in figure $(15)$. We concentrate here in the limit of $q_Aa>1$ where the atoms can be treated independently. The other limit of $q_Aa<1$ investigated by us in previous work \cite{ZoubiH}. The positive electric field operator of the atom $(n)$, in the far zone field where $x\gg\lambda_A$, is given by \cite{Loudon}
\begin{equation}
\hat{\bf E}^{(+)}_n({\bf r},t)=\frac{\mu q_A^2}{4\pi\epsilon_0}\frac{\sin\phi_n}{|{\bf r}-{\bf R}_n|}B\left(t-\frac{|{\bf r}-{\bf R}_n|}{c}\right)\hat{\bf e}_n,
\end{equation}
where $\phi_n$ is the angle between $\mbox{\boldmath$\mu$}$ and ${\bf r}-{\bf R}_n$, and the unit vector $\hat{\bf e}_n$ is defined by
\begin{equation}
\hat{\bf e}_n=\hat{\bf n}_n\times\hat{\bf y},\ \hat{\bf n}_n=\frac{{\bf r}-{\bf R}_n}{|{\bf r}-{\bf R}_n|}.
\end{equation}
We have
\begin{equation}
{\bf r}-{\bf R}_n=(x,0,-R_n),\ |{\bf r}-{\bf R}_n|^2=x^2+R_n^2,
\end{equation}
and
\begin{equation}
\phi_n=\pi-\varphi-\alpha_n,\ \tan\alpha_n=x/R_n,
\end{equation}
with
\begin{equation}
\hat{\bf n}_n=\frac{(x,0,-R_n)}{\sqrt{x^2+R_n^2}},\ \hat{\bf e}_n=\frac{(R_n,0,x)}{\sqrt{x^2+R_n^2}}.
\end{equation}
\begin{figure}[h!]
\centerline{\epsfxsize=6cm \epsfbox{Plot15.eps}}
\caption{The observation point at ${\bf r}$ along the ${\bf x}$ axis, and the lattice is along the ${\bf z}$ axis. The angle between the transition dipole $\mbox{\boldmath$\mu$}$ and ${\bf r}-{\bf R}_n$ is $\phi_n$. The electric field direction off atom $(n)$ is $\hat{\bf e}_n$.}
\end{figure}
For the atomic transition operators we use the expectation values
\begin{eqnarray}
\langle B_i(t-t_i)\rangle&=&\langle B_i(0)\rangle e^{-i\omega_A(t-t_i)} e^{-\Gamma_A(t-t_i)/2}, \nonumber \\
\langle B_i^{\dagger}(t-t_i)B_i(t-t_i)\rangle&=&\langle B_i^{\dagger}(0)B_i(0)\rangle e^{-\Gamma_A(t-t_i)},
\end{eqnarray}
and
\begin{eqnarray}
\langle B_i^{\dagger}(t-t_i)B_j(t-t_j)\rangle&=&\langle B_i^{\dagger}(0)B_j(0)\rangle e^{-\Gamma_A\left[t-(t_i+t_j)/2\right]} \nonumber \\
&\times&e^{-i\omega_A(t_i-t_j)},
\end{eqnarray}
where $t_i=|{\bf r}-{\bf R}_i|/c$. In the limit $q_Aa>1$ the single excitation collective states decay with the single excited atom damping rate $\Gamma_A$.
The total electric field at the observation point is
\begin{equation}
\hat{\bf E}^{(+)}({\bf r},t)=\sum_i\hat{\bf E}_i^{(+)}({\bf r},t),
\end{equation}
and the intensity is
\begin{equation}
I({\bf r},t)=\frac{1}{2}\epsilon_0c\langle\hat{\bf E}^{(-)}({\bf r},t)\mbox{\boldmath$\cdot$}\hat{\bf E}^{(+)}({\bf r},t)\rangle.
\end{equation}
Explicitly we can write
\begin{equation}
I({\bf r},t)=\sum_iI_i({\bf r},t)+\sum_{i\neq j}G_{ij}({\bf r},t),
\end{equation}
where the $i$-th intensity is
\begin{equation}
I_i({\bf r},t)=\frac{1}{2}\epsilon_0c\langle\hat{\bf E}_i^{(-)}({\bf r},t)\hat{\bf E}_i^{(+)}({\bf r},t)\rangle,
\end{equation}
and the correlation function is
\begin{equation}
G_{ij}({\bf r},t)=\frac{1}{2}\epsilon_0c\langle\hat{\bf E}_i^{(-)}({\bf r},t)\mbox{\boldmath$\cdot$}\hat{\bf E}_j^{(+)}({\bf r},t)\rangle.
\end{equation}
We get
\begin{equation}
I_i({\bf r},t)=\frac{\mu^2 \omega_A^4}{32\pi^2\epsilon_0c^3}\frac{\sin^2\phi_i}{|{\bf r}-{\bf R}_i|^2}\langle B_i^{\dagger}(0)B_i(0)\rangle e^{-\Gamma_A(t-t_i)},
\end{equation}
and
\begin{eqnarray}
G_{ij}({\bf r},t)&=&\frac{\mu^2 \omega_A^4}{32\pi^2\epsilon_0c^3}e^{-\Gamma_A\left[t-(t_i+t_j)/2\right]}e^{-i\omega_A(t_i-t_j)} \nonumber \\
&\times& \langle B_i^{\dagger}(0)B_j(0)\rangle\frac{\sin\phi_i}{|{\bf r}-{\bf R}_i|}\frac{\sin\phi_j}{|{\bf r}-{\bf R}_j|}\left(\hat{\bf n}_i\mbox{\boldmath$\cdot$}\hat{\bf n}_j\right). \nonumber \\
\end{eqnarray}
\subsection{Two-Atoms Chain}
We present the results for the simple case of two atoms. One atom is located at the origin ${\bf R}_1=(0,0,0)$, and the second at ${\bf R}_2=(0,0,a)$. The observation point is at ${\bf r}=(x,0,0)$, where ${\bf r}-{\bf R}_1=(x,0,0)$, and ${\bf r}-{\bf R}_2=(x,0,-a)$, with $|{\bf r}-{\bf R}_1|=x$, and $|{\bf r}-{\bf R}_2|=\sqrt{x^2+a^2}$. We have the angles $\phi_1=\frac{\pi}{2}-\varphi$, and $\phi_2=\pi-\varphi-\alpha$, where $\tan\alpha=x/a$. We get the times $t_1=x/c$, and $t_2=\sqrt{x^2+a^2}/c$. Also we have the unit vectors $\hat{\bf e}_1=(0,0,1)$, and $\hat{\bf e}_2=\frac{(a,0,x)}{\sqrt{x^2+a^2}}$, then $\hat{\bf n}_1=(1,0,0)$, and $\hat{\bf n}_2=\frac{(x,0,-a)}{\sqrt{x^2+a^2}}$, hence $\left(\hat{\bf n}_1\mbox{\boldmath$\cdot$}\hat{\bf n}_2\right)=\frac{x}{\sqrt{x^2+a^2}}$. We obtain
\begin{eqnarray}
I_1({\bf r},t)&=&\frac{\mu^2 \omega_A^4}{32\pi^2\epsilon_0c^3}\ \frac{\sin^2\phi_1}{x^2}\ \langle B_1^{\dagger}(0)B_1(0)\rangle\ e^{-\Gamma_A\left(t-\frac{x}{c}\right)}, \nonumber \\
I_2({\bf r},t)&=&\frac{\mu^2 \omega_A^4}{32\pi^2\epsilon_0c^3}\ \frac{\sin^2\phi_2}{x^2+a^2}\ \langle B_2^{\dagger}(0)B_2(0)\rangle \nonumber \\
&\times&e^{-\Gamma_A\left(t-\frac{\sqrt{x^2+a^2}}{c}\right)},
\end{eqnarray}
and
\begin{eqnarray}
G_{12}({\bf r},t)&=&\frac{\mu^2 \omega_A^4}{32\pi^2\epsilon_0c^3}\ \frac{\sin\phi_1\sin\phi_2}{x^2+a^2}\ \langle B_1^{\dagger}(0)B_2(0)\rangle \nonumber \\
&\times&e^{-\Gamma_A\left[t-\left(\frac{x+\sqrt{x^2+a^2}}{2c}\right)\right]}\ e^{-i\omega_A\left(\frac{x-\sqrt{x^2+a^2}}{c}\right)}, \nonumber \\
G_{21}({\bf r},t)&=&\frac{\mu^2 \omega_A^4}{32\pi^2\epsilon_0c^3}\ \frac{\sin\phi_1\sin\phi_2}{x^2+a^2}\ \langle B_2^{\dagger}(0)B_1(0)\rangle \nonumber \\
&\times&e^{-\Gamma_A\left[t-\left(\frac{x+\sqrt{x^2+a^2}}{2c}\right)\right]}\ e^{i\omega_A\left(\frac{x-\sqrt{x^2+a^2}}{c}\right)}.
\end{eqnarray}
Now we consider the two initial states of symmetric and antisymmetric collective states.
For the symmetric collective state
\begin{equation}
|i\rangle=\frac{|e_1,g_2\rangle+|g_1,e_2\rangle}{\sqrt{2}},
\end{equation}
we have
\begin{eqnarray}
\langle B_1^{\dagger}(0)B_1(0)\rangle&=&\langle B_2^{\dagger}(0)B_2(0)\rangle \nonumber \\
=\langle B_1^{\dagger}(0)B_2(0)\rangle&=&\langle B_2^{\dagger}(0)B_1(0)\rangle=\frac{1}{2},
\end{eqnarray}
then we get
\begin{eqnarray}
I({\bf r},t)&=&\frac{I_0(x)}{4}\left\{\sin^2\phi_1\ e^{-\Gamma_A\left(t-\frac{x}{c}\right)}+\frac{x^2\sin^2\phi_2}{x^2+a^2}\right. \nonumber \\
&\times&\left.e^{-\Gamma_A\left(t-\frac{\sqrt{x^2+a^2}}{c}\right)}\right. \nonumber \\
&+&\left.\frac{x^2\sin\phi_1\sin\phi_2}{x^2+a^2}\ 2\cos\left[\omega_A\left(\frac{x-\sqrt{x^2+a^2}}{c}\right)\right]\right. \nonumber \\
&\times&\left.e^{-\Gamma_A\left[t-\left(\frac{x+\sqrt{x^2+a^2}}{2c}\right)\right]}\right\},
\end{eqnarray}
where we defined the intensity
\begin{equation}
I_0(x)=\frac{\mu^2 \omega_A^4}{16\pi^2\epsilon_0c^3x^2}.
\end{equation}
In figures $(16-18)$ we plot the relative intensity $I({\bf r},t)/I_0(x)$ as a function of $a$ for the angles $\varphi=0^{\circ}$, $\varphi=45^{\circ}$ and $\varphi=90^{\circ}$, at the observation point $x=10^{6}\ \AA$ at the moment $t=2x/c$. We use $E_A=1\ eV$, $\mu=1\ e\AA$ and $\Gamma_A=10^{8}\ Hz$.
For small $a$ the relative intensity is maximum for the polarization angle $\varphi=0^{\circ}$ and decreases for larger angles. It is half for $\varphi=45^{\circ}$, and becomes zero for $\varphi=90^{\circ}$. The maximum of the relative intensity moves into larger $a$ with increasing the angle $\varphi$. The relative intensity oscillates in changing $a$ and tend to a finite value for large $a$. Interesting case is for $\varphi=90^{\circ}$, where the intensity is zero for small $a$ and increases with increasing $a$ till it reach a maximum at $a=10^{6}\ \AA$ (for the given numbers), and decreases back towards a finite value for larger $a$.
In the limit of $x\gg a$, where $\sqrt{x^2+a^2}\sim x+\frac{a^2}{2x}$, and as $\phi\approx\frac{\pi}{2}-\varphi$, we can write
\begin{eqnarray}
I({\bf r},t)&\simeq&\frac{\mu^2 \omega_A^4}{64\pi^2\epsilon_0c^3x^2}\ e^{-\Gamma_A\left(t-\frac{x}{c}\right)}\cos^2\varphi \nonumber \\
&\times&\left\{1+e^{\Gamma_A\frac{{a^2}}{2cx}}+2\cos\left(\omega_A\frac{a^2}{2cx}\right)\ e^{\Gamma_A\frac{a^2}{4cx}}\right\}.
\end{eqnarray}
\begin{figure}[h!]
\centerline{\epsfxsize=8cm \epsfbox{Plot16.eps}}
\caption{The symmetric state scaled intensity $I({\bf r},t)/I_0(x)$ vs. $a$, for $\varphi=0^{\circ}$ at $x=10^{6}\ \AA$ and $t=2x/c$.}
\end{figure}
\begin{figure}[h!]
\centerline{\epsfxsize=8cm \epsfbox{Plot17.eps}}
\caption{The symmetric state scaled intensity $I({\bf r},t)/I_0(x)$ vs. $a$, for $\varphi=45^{\circ}$ at $x=10^{6}\ \AA$ and $t=2x/c$.}
\end{figure}
\begin{figure}[h!]
\centerline{\epsfxsize=8cm \epsfbox{Plot18.eps}}
\caption{The symmetric or antisymmetric state scaled intensity $I({\bf r},t)/I_0(x)$ vs. $a$, for $\varphi=90^{\circ}$ at $x=10^{6}\ \AA$ and $t=2x/c$.}
\end{figure}
For the antisymmetric collective state
\begin{equation}
|i\rangle=\frac{|e_1,g_2\rangle-|g_1,e_2\rangle}{\sqrt{2}},
\end{equation}
we have
\begin{eqnarray}
\langle B_1^{\dagger}(0)B_1(0)\rangle&=&\langle B_2^{\dagger}(0)B_2(0)\rangle=\frac{1}{2}, \nonumber \\
\langle B_1^{\dagger}(0)B_2(0)\rangle&=&\langle B_2^{\dagger}(0)B_1(0)\rangle=-\frac{1}{2},
\end{eqnarray}
then we can write
\begin{eqnarray}
I({\bf r},t)&=&\frac{I_0(x)}{4}\left\{\sin^2\phi_1\ e^{-\Gamma_A\left(t-\frac{x}{c}\right)}+\frac{x^2\sin^2\phi_2}{x^2+a^2}\right. \nonumber \\
&\times&\left.e^{-\Gamma_A\left(t-\frac{\sqrt{x^2+a^2}}{c}\right)}\right. \nonumber \\
&-&\left.\frac{x^2\sin\phi_1\sin\phi_2}{x^2+a^2}\ 2\cos\left[\omega_A\left(\frac{x-\sqrt{x^2+a^2}}{c}\right)\right]\right. \nonumber \\
&\times&\left.e^{-\Gamma_A\left[t-\left(\frac{x+\sqrt{x^2+a^2}}{2c}\right)\right]}\right\}.
\end{eqnarray}
In figures $(19-20)$ we plot the relative intensity $I({\bf r},t)/I_0(x)$ as a function of $a$ for the angles $\varphi=0^{\circ}$ and $\varphi=45^{\circ}$. The case of $\varphi=90^{\circ}$ is the same as in figure $(23)$. As before, the observation point is at $x=10^{6}\ \AA$ at the moment $t=2x/c$, with the other previous numbers. The results are similar to the symmetric ones except from the case of small $a$ where the relative intensity tends to zero as expected.
In the limit of $x\gg a$, as $\phi\approx\frac{\pi}{2}-\varphi$, we can write
\begin{eqnarray}
I({\bf r},t)&\simeq&\frac{\mu^2 \omega_A^4}{64\pi^2\epsilon_0c^3x^2}\ e^{-\Gamma_A\left(t-\frac{x}{c}\right)}\cos^2\varphi \nonumber \\
&\times&\left\{1+e^{\Gamma_A\frac{{a^2}}{2cx}}-2\cos\left(\omega_A\frac{a^2}{2cx}\right)\ e^{\Gamma_A\frac{a^2}{4cx}}\right\}.
\end{eqnarray}
\begin{figure}[h!]
\centerline{\epsfxsize=8cm \epsfbox{Plot19.eps}}
\caption{The antisymmetric state scaled intensity $I({\bf r},t)/I_0(x)$ vs. $a$, for $\varphi=0^{\circ}$ at $x=10^{6}\ \AA$ and $t=2x/c$.}
\end{figure}
\begin{figure}[h!]
\centerline{\epsfxsize=8cm \epsfbox{Plot20.eps}}
\caption{The antisymmetric state scaled intensity $I({\bf r},t)/I_0(x)$ vs. $a$, for $\varphi=45^{\circ}$ at $x=10^{6}\ \AA$ and $t=2x/c$.}
\end{figure}
\section{Summary}
In the present paper we investigated optical properties of a one dimensional atomic chain, in which the lattice constant can range from a few angstroms up to thousands of angstroms. Namely, the lattice constant can change from being smaller than the atomic transition wavelength up to much larger one. In the limit of lattice constant smaller than the atomic transition wavelength the electrostatic interactions are applicable, which found useful for most of the typical experiments on optical lattice ultracold atoms. In our previous work we limited the discussion to electrostatic interactions, where we considered only resonance dipole-dipole interactions, and which is justified in the present work. For small lattice constant the electrostatic interactions are responsible for the formation of excitons, where we did extensive study in this regime with emphasize on the exciton life times. For large lattice constant the inclusion of radiative corrections are necessary, which is the main issue in the present paper.
For large lattice constant the radiative corrections are included, and in this regime we found that the coupling parameter for the energy transfer among even the nearest neighbor atom sites is smaller than a single excited atom damping rate. Hence, the energy transfer is not favorable, and we treated the atoms as independently setting on the lattice sites. Then, we calculated the damping rates of different collective electronic excitations in including the radiative corrections by considering the effect of the existence of all the other atom sites, despite their large distances from the excited atom. Big attention we gave for the most symmetric state, where we emphasized the dependence of its damping rate on the number of atoms for different lattice constant. We found the symmetric damping rate to behave linearly at small atom numbers and saturate at large numbers. The damping rate of symmetric and antisymmetric collective states tend to that of a single excited atom with oscillations due to the radiative effect through the exchange of virtual photons. The differences between damping rates of collective states appear for small lattice constant, in which the symmetric states have superradiant damping rate, that is $N$ time the single excited atom rate. Here, part of the antisymmetric states become dark with zero damping rate, and other part is metastable with a fraction of the single excited atom damping rate. Moreover we calculated the emission pattern off a chain of atoms with a large lattice constant in which the atoms can be considered independently. The emission intensities off two atoms with symmetric and antisymmetric states are presented as a function of the interatomic distance.
The results of the present paper are illustrated in terms of optical lattice ultracold atoms, but they are general and can be adopted for any chain of optically active material. For example, chains of semiconductor quantum dots fit exactly in the regime of large lattice constant, where radiative corrections are unavoidable, and the life times of their collective states can be treated according to the present paper. Other system that exploits the regime of the present paper is a lattice of large organic molecules sitting on a matrix with a given large lattice constant, the collective damping rate and emission pattern are expected to behave according to our present results.
\
The author acknowledge very fruitful discussions with Helmut Ritsch. The work was supported by the Austrian Science Funds (FWF) via the project (P21101), and by the DARPA QuASAR program.
|
{
"timestamp": "2012-03-12T01:01:44",
"yymm": "1203",
"arxiv_id": "1203.2094",
"language": "en",
"url": "https://arxiv.org/abs/1203.2094"
}
|
\section{Introduction}
\label{s:int}
Ultra-high energy cosmic-ray (UHECR) particles are the highest energy
particles observed by humankind, well beyond what is accessible at the
Large Hadron Collider. Current experiments \cite{Auger,HiRes,TA} are
recording large amounts of high quality data. Recent observations do
not draw a simple nor consistent picture of the nature of UHECR
particles. There are hints of anisotropy in the arrival
directions~\cite{AugerAnisotropy}, favouring the presence of light primary
particles. Using the current high energy hadronic interaction models, typical air-shower observables do not generally
support the light component hypothesis: the Pierre Auger Collaboration observes the depth of the electromagnetic
shower maximum, muon production maximum, rise-time of the shower front,
etc.\ that are all pointing in the direction of a primary composition
dominated by heavy particles at ultra-high
energies~\cite{AugerER,Auger_ICRC}. On the other hand, the HiRes
Collaboration claims that their measurement of the depth of the shower maximum is compatible with
protons up to the highest energies~\cite{Abbasi:2009nf}, which is currently supported by
the first TA data~\cite{FirstTAdata,TAER}.
It is important to notice that whereas Auger publishes $\langle{X_{\rm max}}\rangle$ in the atmosphere by using a fiducial volume selection to minimize acceptance biases, HiRes and TA presently do not apply such corrections and present $\langle{X_{\rm max}}\rangle$ at the detector level.
Also the possible difference of the energy scale between Auger and HiRes/TA ($\approx$25\,\%)~\cite{Abraham:2010mj} contribute to the difficulties to explain the data~\cite{Engel:2011zz,OlintoTAUP2011}.
The current situation could point to exciting physics aspects that are about to emerge with higher accumulated event statistics, as for instance, systematic differences of the cosmic-ray flux on the northern and southern hemisphere. The Pierre Auger Observatory observes the southern, while HiRes/TA the norther sky, with very
different astrophysical objects in their direct field-of-view.
It is clear, that the understanding of the primary mass composition of
UHECR is one of the major steps towards the final solution of the UHECR
puzzle. Depending on a reliable measurement of the mass composition
very different scenarios of the nature of UHECR will finally emerge.
In this article we discuss the concept of \emph{observed} versus \emph{true} $X_{\rm max}$-distributions to finally demonstrate that the \emph{observed} average of depth of shower maxima, $\langleX_{\rm max}\rangle$, can result in a non-linear relation with the average logarithm of the mass number, $\langle \ln A \rangle$, of the UHECR. These effects are very specific for a
given experimental setup and have to be accounted for in order to
allow a comparison or interpretation of these data.
\section{Extensive Air-Showers and Fluorescence Telescope}
Fluorescence telescopes measure the ultraviolet light, which is proportional to the energy deposited by the passage of the charged particles of the air-shower cascade through the atmosphere. This makes possible to reconstruct the longitudinal profile of the electromagnetic shower~\cite{Unger}.
The atmospheric depth at which the energy deposit is maximal is the shower maximum, $X_{\rm max}$, and is related to the nature of the primary particle.
At the same primary energy, primary nuclei with mass $A$ produce air-showers with shower maxima at different atmospheric depths. This can be approximated as~\cite{Matthews}
\begin{eqnarray}
\nonumber
X_{\rm max}\approx \lambda_{\rm int} + \ln2\, X_0 \ln \frac{E}{A},
\end{eqnarray}
where $\lambda_{\rm int}$ is the cross-section of cosmic-ray primaries with air and $X_0\sim\unit[37]{g/cm^2}$ is the electromagnetic radiation length in air.
The structure of this equation, $X_{\rm max}=a + b \ln E/A$, holds also for more general considerations~\cite{AlvarezMuniz:2002ne}. It has been typically used to compute the average logarithmic mass from the data of the average depth of the shower maximu
\begin{eqnarray}
\nonumber
\langle\ln A\rangle = \frac{1}{b} \left[ a + b\ln E - \langleX_{\rm max}\rangle \right] = \frac{1}{b} \left[ \langleX_{\rm max}\rangle_{p}-\langleX_{\rm max}\rangle \right] .
\label{eq:lnAXmax}
\end{eqnarray}
Thus, there is a linear dependence of the measured average shower maximum
from the average mass of the cosmic-ray primary particles.
Instead of calculating $\langle\ln A\rangle$ one can also compute the component fractions in a model with two primary cosmic-ray species. This is typically done for the proton fraction $f$ under the assumption of a simple proton/iron mixture.
For this case the relation of $\langleX_{\rm max}\rangle$ is
\begin{equation}
\langleX_{\rm max}\rangle=f \langleX_{\rm max}\rangle_p+(1-f)\langleX_{\rm max}\rangle_{\rm Fe} ,
\label{eq:linear}
\end{equation}
which is linear in $f$.
Thus, the distinction between an admixture of different primaries or a pure primary of an intermediate mass, are indistinguishable on the base of a $\langleX_{\rm max}\rangle$ value alone. The resulting equivalence is
\begin{eqnarray}
\nonumber
\langle \ln A \rangle= \frac{1}{b} \left[\langleX_{\rm max}\rangle_p-\langleX_{\rm max}\rangle_{\rm Fe}\right] (1-f).
\end{eqnarray}
\section{The effect of the telescope acceptance}
The objective of this paper is not to reproduce any particular detector configuration, but to demonstrate that the relation between $\langle X_{\rm max} \rangle$ and $\langle \ln A \rangle$ depends critically on the detector acceptance, which varies with the primary energy. Thus, the telescope acceptance has an impact on the interpretation of $\langleX_{\rm max}\rangle$ data, and in general any other momentum of the $X_{\rm max}$-distribution~(see also~\cite{unger2}).
Fluorescence telescopes have a limited viewing angle defined by the optics. Showers falling outside the field-of-view cannot be detected. But even for observed showers the shower maximum can be outside the field-of-view of the telescope and can then not be reconstructed reliably.
\begin{figure}[!]
\centering
\includegraphics[width=.45\textwidth]{fov.eps}~\hfill
\includegraphics[width=.5\textwidth]{distance.eps}\\
\caption{Left panel: Geometry of our simulation study. Right panel: Resulting efficiency, $N_{\rm obs}/N_{\rm gen}$, depending on the distance of air-shower cores from the telescope. \label{fig:fov}}
\end{figure}
The angular field-of-view defines a volume in the atmosphere where
showers can effectively be fully reconstructed. One of the most important
criteria of this is that the shower maximum is located within the field-of-view.
This volume is depicted in Fig.~\ref{fig:fov}~(left). When showers fall very close to the telescope, the transverse area of the field-of-view is small, meaning that many shallow and/or deep showers are not fully reconstructed. This effect is naturally most severe in the energy range close to the lower detection threshold for air-showers where the distance to the events is limited by the small amount of generated fluorescence light. For air-showers detected at larger distance to the telescope the geometrical field-of-view cone is much larger. However, fluorescence light can travel only a limited distance in the atmosphere before being absorbed. At some point, light emitted at the shower axis is attenuated too much and cannot be observed any more. This is responsible for the characteristic rapid drop of the efficiency as shown in Fig.~\ref{fig:fov}~(right) beyond 20\,km.
The Rayleigh absorption rate is proportional to the density of the atmosphere,
and thus depends on changing air pressure as well as on the height in the atmosphere
at which a particular shower is developing. In this work we do not discuss
aerosol related absorption. It is much smaller than Rayleigh absorption and
to include it here does not help the argument, but just adds additional complexity.
By means of a toy Monte Carlo we reproduce all parameters relevant for the modelling of air-shower observation with fluorescence telescopes at an arbitrary primary energy. Our simulations focus on the geometric and atmospheric effects and simplifies the situation as much as possible. We do not attempt to reproduce a specific detector setup, but only want to show the inevitable impact on the interpretation of the observed data.
The geometry of our simulation is shown in Fig.~\ref{fig:fov}. The
vertical depth of the telescope is at $\unit[\approx880]{g/cm^2}$,
which corresponds to h=1.5\,km, as explained below, and the
opening angle of the field-of-view is 20$^\circ$. We consider air-showers of fixed primary
energy, with random arrival directions, sampled from ${\rm d}N/{\rm
d}\cos\theta=$const up to 60$^\circ$ zenith angle, as well as distances $l$
from the telescope, sampled from ${\rm d}N/{\rm
d}l\propto{l}$. The requirement of a particular shower to be
observed is that at least $N_{\rm det}=100$ photons reach the
telescope from the location of the shower maximum on the shower
axis. The number of photons is computed as
\begin{eqnarray}
N_{\rm det} = N_{\rm ph} A_{\rm dia} / r^2 \exp\left(-\frac{t}{\lambda_{\rm abs}}\right),
\label{eq:ndet}
\end{eqnarray}
where $N_{\rm ph}$ is the number of fluorescence photons emitted at
the shower maximum, $A_{\rm dia}=10\,$m$^2$ is the aperture of the
telescope, $r$ is the geometric distance, $t=\int\rho(z){\rm
d}\vec{r}$ the integrated depth distance from the telescope to the
location of $X_{\rm max}$, and $\lambda_{\rm abs}=\unit[1000]{g/cm^2}$
is the photon absorption length. For air showers of primary energy
$E_0$ and a fluorescence yield of $Y_{\rm fluo}=5/$MeV the number of photons
emitted at $X_{\rm max}$ is
\begin{equation}
N_{\rm ph}=\left(\frac{{\rm d}E}{{\rm d}X}\right)_{\rm max}\; Y_{\rm fluo} \; \rho \; {\rm c} \; \Delta t\,,
\label{eq:nph}
\end{equation}
where $\Delta t=100\,$ns is the telescope sampling time and $({\rm
d}E/{\rm d}X)_{\rm max}$ is the energy deposit at the shower
maximum, which is $10^{-11.77 + \lg(E_0/{\rm eV})}\,$GeV\,cm$^2$/g in
very good approximation\footnote{This is obtained with the SIBYLL~\cite{Ahn:2009wx}
interaction model.}. For air showers with $E_0=10^{18.5}\,$eV this
yields $N_{\rm ph}\approx2\cdot10^{10}$, which is the default value
throughout this paper if not stated otherwise. The atmospheric
density profile $\rho(z)$ is exponential with a scale height of
\unit[8]{km} and a pressure at the height above sea level,
$h=\unit[1.5]{km}$, of the telescope of $\unit[\approx 880]{g/cm^2}$.
\begin{figure}[bth!]
\begin{center}
\includegraphics[width=.8\textwidth]{efficiency.eps}
\caption{Efficiency $\epsilon(X_{\rm max})$ of the $X_{\rm max}$ reconstruction as a function of
$X_{\rm max}$ for a particular detector, as
calculated by our toy Monte Carlo.\label{fig:eff}}
\end{center}
\end{figure}
When we generate events with a flat distribution of $X_{\rm max}$, we
find that air-showers with different $X_{\rm max}$ are observed with
different efficiency $\epsilon(X_{\rm max})=N_{\rm obs}/N_{\rm
gen}$. In Fig.~\ref{fig:eff} we show this efficiency as a function
of $X_{\rm max}$. The peak of this distribution is related to the
\emph{average atmospheric slant depth} in the volume observed by the
telescope. Every telescope setup has limits both for very small as
well as very large values of $X_{\rm max}$ (see
e.g.\ \cite{flyseye}). While large integrated atmospheric depths can
in principle always be achieved by very inclined geometries, there is
a strict bound on the minimal observed depth even for vertical events,
which is related to the maximum possible observation distance and the
upper elevation boundary of the field-of-view. This effect becomes
more relevant for lower energy air-showers, since here the maximum
observation distance is smaller.
\begin{figure}[tb!]
\begin{center}
\includegraphics[width=.7\textwidth]{cprimaries.eps}
\caption{The two input distributions corresponding to light and
heavy primary cosmic-ray particles in our study. The
distributions are Exponential convoluted with a Gaussian. The
used parameters of Eq.~(\ref{eq:modelxmax}) are
$\mu=\unit[815]{g/cm^2}$, $\sigma=\unit[50]{g/cm^2}$ and
$\tau=\unit[45]{g/cm^2}$ for the light component as well as
$\mu=\unit[1005]{g/cm^2}$, $\sigma=\unit[30]{g/cm^2}$ and
$\tau=\unit[5]{g/cm^2}$ for the heavy component.
\label{fig:input}}
\end{center}
\end{figure}
Given the average efficiency $\epsilon(X_{\rm max})$ of a telescope
setup, the measured distribution of $X_{\rm max}$ is related to the parent distribution via
\begin{eqnarray}
\nonumber \left(\frac{{\rm d}N}{{\rm d}X_{\rm max}}\right)_{\rm
measured} = \epsilon(X_{\rm max}) \left(\frac{{\rm d}N}{{\rm
d}X_{\rm max}}\right)_{\rm true}.
\end{eqnarray}
Small changes in the telescope setup can yield very different
$\epsilon(X_{\rm max})$. If also energy, $X_{\rm max}$ resolution,
and possible $X_{\rm max}$ reconstruction biases effects were included
the relation between the measured and parent distribution would become
more complex~\cite{resolution, vitor}, but this does not help the
clarity of our argument.
In the following we demonstrate how the telescope acceptance,
$\epsilon(X_{\rm max})$, affects in particular also \emph{simple}
analyses as for example related to $\langle X_{\rm max}\rangle$. For
this purpose we generate input $X_{\rm max}$ distributions derived
from an Exponential convoluted with a Gaussian function
\begin{eqnarray}
\frac{{\rm d}N}{{\rm d}X_{\rm max}} = \frac{N_{\rm evt}}{2\tau} \;\;\,{\rm e}^{\sigma^2/(2\tau^2)}\;\; {\rm e}^{(\mu-X_{\rm max})/\tau} \;\; {\rm erfc}\left(\frac{\mu-X_{\rm max}-\sigma^2/\tau}{\sqrt{2}\sigma}\right).
\label{eq:modelxmax}
\end{eqnarray}
In total this function has three shape parameters: the mean and width
of the Gaussian, $\mu$ and $\sigma$, and the exponential slope,
$\tau$. We generate two distributions (c.f.\ Fig.~\ref{fig:input}),
the first one corresponding to \emph{light} cosmic-ray primaries has
an average of $\unit[750]{g/cm^2}$ and an RMS of $\unit[66.5]{g/cm^2}$,
the second one corresponds to \emph{heavy} cosmic-ray primaries has an
average of $\unit[650]{g/cm^2}$ and an RMS of $\unit[30]{g/cm^2}$.
These values are chosen since
they correspond roughly to typical values for proton or iron induced
air-showers respectively. The average efficiencies
\begin{eqnarray}
\nonumber
\epsilon_{A}=\frac{\int\limits_0^{\infty} \left(\frac{{\rm d}N}{{\rm d}X_{\rm max}}\right)_{\rm measured} {\rm d}X_{\rm max}}{\int\limits_0^{\infty} \left(\frac{{\rm d}N}{{\rm d}X_{\rm max}}\right)_{\rm true} {\rm d}X_{\rm max}} = \int\limits_0^\infty \epsilon(X_{\rm max}) {\rm d}X_{\rm max}
\end{eqnarray}
for the observation of $X_{\rm max}$ with our telescope setup for
these two distributions are $\epsilon_{\rm light}=33.1\,\%$ and
$\epsilon_{\rm heavy}=16.7\,\%$. The average $X_{\rm max}$ observed by
the telescope setup are $\langleX_{\rm max}\rangle^\prime_{\rm
light}=\unit[770]{g/cm^2}$ for the light and
$\langleX_{\rm max}\rangle^\prime_{\rm heavy}=\unit[660]{g/cm^2}$ for the
heavy component, which is $\unit[10-20]{g/cm^2}$ biased with respect
to the input distributions.
\begin{figure}[bt!]
\begin{center}
\includegraphics[width=.9\linewidth]{fiducial_energy.eps}
\caption{Dependence of the bias $\Delta\langle X_{\rm max}\rangle$
from the energy and thus the distance of air-showers from the
telescope. The upper axis indicates the resulting average
distance of showers from the telescope depending on the
primary energy.\label{fig:energy}}
\end{center}
\end{figure}
\begin{figure}[bt!]
\begin{center}
\includegraphics[width=.9\linewidth]{composition.eps}
\caption{The average $X_{\rm max}$ as a function of the light cosmic-ray
fraction, $f$. The \emph{observed} $\langle{X_{\rm max}}\rangle$ values for pure
light and pure heavy compositions are shifted with respect to
the \emph{true} $\langle{X_{\rm max}}\rangle$. The relation between $\langle{X_{\rm max}}\rangle$ and
$f$, and therefore with $\langle \ln A \rangle$ by
Eq. \ref{eq:lnAXmax}, is linear for the true $X_{\rm max}$
distributions, whereas it is non-linear for the case of the
observed $\langle{X_{\rm max}}\rangle$. The curve labeled ``Observation A''
indicates the effect on a pure composition of intermediate mass.}
\label{f:fXmax}
\end{center}
\end{figure}
The bias $\Delta\langle X_{\rm max}\rangle$, defined as the difference
between the observed and true $\langle X_{\rm max}\rangle$ value,
changes with distance to the telescope and thus also with the primary
energy of the cosmic-ray particles according to Eqs.~(\ref{eq:ndet}) and (\ref{eq:nph}). In Fig.~\ref{fig:energy} this is
shown for the cases of pure light, pure heavy and an equal mixture of
light and heavy primaries. For this study we consider a change of
$\langle X_{\rm max}\rangle$ with energy of ${\rm d}\langle X_{\rm
max}\rangle/{\rm d}\lg{E}=\unit[60]{g/cm^2}$ with respect to the
default values used otherwise in this paper. It is interesting to
note that the resulting bias depends on the underlying mass
composition. In general, for smaller energies the showers are located
closer to the telescope and the effect becomes stronger. For example,
the low energy fluorescence telescope extensions HEAT~\cite{heat} at
the Pierre Auger Observatory and TALE~\cite{tale} at the Telescope
Array are important to limit these biases by providing a much wider
field-of-view for showers at close distances. Furthermore, the bias
can be positive as well as negative, depending on the overlap of the
true $X_{\rm max}$ distribution with the telescope efficiency
function, c.f.\ Fig.~\ref{fig:eff}. However, the impact will
qualitatively be always as shown in Fig.~\ref{fig:energy}: at short
distances the field-of-view cuts shallow showers and thus $\langle
X_{\rm max}\rangle$ is overestimated, at larger distances this effects
becomes smaller and eventually might even reverse leading to an
underestimation of the true $\langle X_{\rm max}\rangle$ value.
It is important to realize that the primary effect is due to
a different distance of showers, and that any parameter that affects
the typical observation distance will have a similar impact. Such
parameters are typically related to the detector setup, for example
the diaphragm opening $A_{\rm dia}$, the sampling time $\Delta t$, but
also the atmospheric density profile and optical absorption
characteristics.
\begin{figure}[bt!]
\begin{center}
\includegraphics[width=.9\linewidth]{scan.eps}
\caption{Impact of different values of $r=\epsilon_{\rm light}/\epsilon_{\rm
heavy}$ on the non-linearity.}
\label{fig:scan}
\end{center}
\end{figure}
If we consider a mixture of light and heavy particles at fixed primary
energy with a given fraction $f$ of light and $1-f$ of heavy primary
particles, the average observed shower maximum becomes
\begin{equation}
\langleX_{\rm max}\rangle^\prime=\frac{f
r \langleX_{\rm max}\rangle^\prime_{\rm light} +(1-f) \langleX_{\rm max}\rangle^\prime_{\rm heavy} }{fr + 1-f}
\label{eq:nonlinear}
\end{equation}
where $r=\epsilon_{\rm light}/\epsilon_{\rm heavy}$. This
relation is non-linear in $f$ and thus not in $\ln A$. In
Figure~\ref{f:fXmax} we show the results of this study. The true as
well as the observed $\langle{X_{\rm max}}\rangle$ are shown, together with the linear
interpolation from Eq.~(\ref{eq:linear}) and the non-linear version
Eq.~(\ref{eq:nonlinear}). The latter describes perfectly well the
results of our Monte Carlo study.
We also study how a different detector setup affects our result. In general, by
changing parameters of the telescope detector setup, very different
values of $r$ can be obtained. In Fig.~\ref{fig:scan} we show
how the predictions of Eq.~(\ref{eq:nonlinear}) change for a wide range of different
values of $r$. It summarizes the effect described in this
paper, and the miss-interpretation it can induce on the inference of
the primary mass composition from $\langle X_{\rm max} \rangle$. For
instance, $r=3$ would imply that a real fraction 50\,\% proton
and 50\,\% iron, that is f=0.5 and $\langle\ln A\rangle=2$, would be
interpreted as 75\,\% proton and 25\,\% iron, $\langle\ln A\rangle=1$, if
just the linear relation Eq.~(\ref{eq:linear}) is assumed.
\section{Summary}
The data collected by a fluorescence telescope are affected by
acceptance effects, which can have an important impact on the detailed
interpretation of the data. This is problematic for researchers
not from within a particular experimental collaboration who, for
example, have no access to a detailed Monte Carlo simulation
of the detector setup.
The average efficiency $\epsilon(X_{\rm max})$, which describes the
response to a flat distribution in $X_{\rm max}$, is one of the crucial
properties of a fluorescence telescope. However, in addition to the
acceptance effects also the detector resolution and $X_{\rm max}$
reconstruction biases have to be known for a
full data analysis.
We have demonstrated that the bias in the average shower maximum
introduced by the acceptance of fluorescence telescopes induces a non
linearity between $\langle{X_{\rm max}}\rangle$ and $\langle \ln A \rangle$.
\section{Acknowledgments}
We would like to thank A. Olinto for the initial discussions that
inspired this work during the TAUP2011 meeting at Munich. J.~Bellido, D.~Harari, M.~Bueno, M.~Unger,
R.~Concei\c{c}\~{a}o and M.~Pimenta for careful reading of this manuscript and their
comments. Finally we thank our colleagues from the Pierre Auger
Collaboration for helping us to get the necessary insight on the
fluorescence detection techniques. L.C.\ wants to thank fundings by
Funda\c{c}\~{a}o para a Ci\^{e}ncia e Tecnologia (CERN/FP11633/2010),
and fundings of MCTES through POPH-QREN Tipologia 4.2, Portugal, and
European Social Fund.
\bibliographystyle{elsarticle-num}
|
{
"timestamp": "2012-10-01T02:01:50",
"yymm": "1203",
"arxiv_id": "1203.1781",
"language": "en",
"url": "https://arxiv.org/abs/1203.1781"
}
|
\section*{Abstract}
From the viewpoint of networks, a ranking system for players or teams in sports is equivalent to a centrality measure for sports networks, whereby a directed link represents the result of a single game. Previously proposed network-based ranking systems are derived from static networks, i.e., aggregation of the results of games over time. However, the score of a player (or team) fluctuates over time. Defeating a renowned player in the peak performance is intuitively more rewarding than defeating the same player in other periods. To account for this factor, we propose a dynamic variant of such a network-based ranking system and apply it to professional men's tennis data. We derive a set of linear online update equations for the score of each player. The proposed ranking system predicts the outcome of the future games with a higher accuracy than the static counterparts.
\newpage
\section{Introduction}\label{sec:introduction}
Ranking of individual players or teams in sports, both professional
and amateur, is a tool for entertaining fans and developing sports
business. Depending on the type of sports, different ranking systems
are in use \cite{Stefani1997JAS}. A challenge in sports ranking is
that it is often impossible for all the pairs of players or teams (we
refer only to players in the following. However, the discussion also
applies to team sports) to fight against each other. This is the case
for most individual sports and some team sports in which a league
contains many teams, such as American college football
and soccer at an international level. Then, the
set of opponents depends on players such that ranking players by
simply counting the number of wins and losses is inappropriate.
In this situation, several ranking systems on the basis of
networks have been proposed. A player is regarded to be a node in
a network, and a directed link from the winning player
to the losing player (or the converse) represents the result of a single
game. Once the directed network of players is generated,
ranking the players is equivalent to defining a
centrality measure for the network.
A crux in constructing a network-based ranking system is to let
a player that beats a strong player gain a high score.
Examples of network-based ranking systems include
those derived from the Laplacian matrix of the network
\cite{Daniels1969Biom,Moon1970SiamRev,Borm2002AOR,Saavedra2010PhysicaA}, the PageRank
\cite{Radicchi2011PlosOne}, a random walk that is different from those
implied by the Laplacian or PageRank \cite{Callaghan2004NAMS},
a combination of node degree and global structure of networks
\cite{Herings2005SCW}, and the so-called win-lose score
\cite{Park2005JSM}.
Previous network-based ranking systems do not account for
fluctuations of rankings.
In fact, a player, even a history making strong player, referred to as
$X$, is often weak
in the beginning of the career. Player $X$ may also be
weak past the most brilliant period in the $X$'s career,
suggestive of the retirement in a near future.
For other players, it is more rewarding to beat $X$
when $X$ is in the peak performance than when $X$ is novice,
near the retirement, or in the slump.
It may be preferable to take into account the
dynamics of players' strengths for defining a ranking system.
In the present study, we extend the win-lose score, a network-based ranking system proposed by Park and Newman \cite{Park2005JSM} to the dynamical case.
Then, we apply the proposed ranking system
to the professional men's tennis data.
In broader contexts, the current study is related to at least two other
lineages of researches.
First, a dynamic network-based ranking implies that we exploit the
temporal information about the data, i.e., the times when games are played.
Therefore, such a ranking system is equivalent to
a dynamic centrality measure for temporal networks, in which
sequences of pairwise interaction events with time stamps are building units
of the network
\cite{HolmeSaramaki2012PhysRep}.
Although some centrality measures specialized in temporal
networks have been proposed \cite{Tang2010SNS,Pan2011PRE,Grindrod2011PRE},
they are not for ranking purposes.
In addition, they are constant valued centrality measures for dynamic (i.e., temporal) data of pairwise interaction.
In the context of temporal networks, we propose
a dynamically changing centrality measure
for temporal networks.
Second, statistical approaches to sports ranking have a much longer history
than network
approaches. Representative statistical ranking systems include the Elo
system \cite{Elo1978book} and the Bradley-Terry model (see
\cite{Bradley1976Biom} for a review). Variants of these models have
been used to construct dynamic ranking systems. Empirical Bayes
framework naturally fits this problem
\cite{Glickman1993thesis,Fahrmeir1994JASA,Glickman1999JRSSSC,Knorrheld2000Stat,Coulom2008LNCS,Herbrich2007NIPS}. Because the Bayesian estimators cannot be obtained analytically, or
even numerically owing to the computational cost, in these models,
techniques for obtaining Bayes estimators such as the Gaussian
assumption of the posterior distribution
\cite{Glickman1999JRSSSC,Herbrich2007NIPS}, approximate message
passing \cite{Herbrich2007NIPS}, and Kalman filter
\cite{Fahrmeir1994JASA,Glickman1999JRSSSC,Knorrheld2000Stat}, have
been employed. In a non-Bayesian statistical ranking system,
the pseudo likelihood, which is defined such that the contribution of the
past game results to the current pseudo likelihood decays
exponentially in time,
is numerically maximized \cite{Dixon1997ApplStat}.
In general, the parameter set of a statistical ranking system that accounts
for dynamics of players' strengths is composed of dynamically
changing strength parameters for all the players and perhaps other
auxiliary parameters.
Therefore, the number of parameters to be statistically estimated
may be large relative to the amount of data.
In other words, the instantaneous ranks of players have to be
estimated before the players play sufficiently many games with others under
fixed strengths.
Even under a Bayesian framework with which updating of the parameter values
is naturally implemented, it may be difficult to reliably estimate
dynamic ranks of players due to relative paucity of data.
In addition, in sports played by individuals, such as tennis,
it frequently occurs that new players begin and old and underperforming
players leave. This factor also increases the
number of parameters of a ranking system.
In contrast, ours and other
network-based ranking systems, both static and dynamic ones,
are not founded on statistical methods.
Network-based ranking systems
can be also simpler and more transparent than statistical counterparts.
\section*{Results}\label{sec:results}
\subsection*{Dynamic win-lose score}\label{sub:dwl}
We extend the
win-lose score \cite{Park2005JSM} (see Methods) to
account for the fact that the strengths of players fluctuate over time.
In the following, we refer to the win-lose score as the original
win-lose score and the extended one as the dynamic
win-lose score.
The original win-lose score overestimates
the real strength of a player $i$ when $i$ defeated an opponent $j$ that is
now strong and was weak at the time of the match between $i$ and $j$.
Because $j$ defeats many strong opponents afterward,
$i$ unjustly receives many indirect wins through $j$.
The same logic also applies to other network-based
static ranking systems \cite{Daniels1969Biom,Moon1970SiamRev,Borm2002AOR,Saavedra2010PhysicaA,Radicchi2011PlosOne,Callaghan2004NAMS,Herings2005SCW}.
To remedy this feature, we pose two assumptions.
First, we assume that
the increment of the win score of player $i$ through the $i$'s winning
against player $j$ depends on the $j$'s win score at that moment. It
does not explicitly depend on the $j$'s score in the past or future.
The same holds true for the lose score.
Second, we assume that each player's win and lose scores
decay exponentially in time. This assumption is also employed in a
Bayesian dynamic ranking system \cite{Dixon1997ApplStat}.
Let $A_{t_n}$ be the win-lose matrix for the game that occurs at time $t_n$
($1\le n\le n_{\max}$).
In the analysis of the tennis data carried out in the following,
the resolution of $t_n$ is equal to one day. Therefore, players'
scores change even within a single tournament.
If player $j$ wins against player $i$ at time $t_n$,
we set the $(i,j)$ element of the matrix $A_{t_n}$ to be 1. All the other elements of $A_{t_n}$
are set to 0.
We define the dynamic win score at time $t_n$ in vector form, denoted
by $\bm w_{t_n}$, as follows:
\begin{align}
W_{t_n} =& A_{t_n} + e^{-\beta (t_n - t_{n-1})} \sum _ {m_n \in \{ 0,1 \}} \alpha^{m_n} A_{t_{n-1}}A_{t_n}^{m_n} \notag\\
&+ e^{-\beta (t_n - t_{n-2})} \sum _ {m_{n-1},m_n \in \{ 0,1 \}} \alpha^{m_{n-1}+m_n} A_{t_{n-2}}A_{t_{n-1}}^{m_{n-1}}A_{t_n}^{m_n} \notag\\
&+ \cdots + e^{-\beta (t_n - t_1)} \sum _ {m_2, \ldots, m_n \in \{ 0,1 \}}
\alpha^{\sum_{i=2}^n m_i} A_{t_1}A_{t_2}^{m_2}\cdots A_{t_n}^{m_n}
\label{eq:def of W_{t_n}}
\end{align}
and
\begin{equation}
\bm w_{t_n} = W_{t_n}^{\top}\bm 1,
\label{eq:def of w_{t_n}}
\end{equation}
where $\alpha$ is the weight of the indirect win, which is the same as
the case of the original win-lose score (Methods),
and $\beta\ge 0$ represents the decay rate of the score.
The first term on the right-hand side of Eq.~\eqref{eq:def of
W_{t_n}} (i.e., $A_{t_n}$) represents the effect of the direct win at
time $t_n$. The second term consists of two contributions. For
$m_n=0$, the quantity inside the summation represents the direct win
at time $t_{n-1}$, which results in weight $e^{-\beta (t_n-t_{n-1})}$.
For $m_n=1$, the quantity represents the
indirect win. The ($i$, $j$) element of $A_{t_{n-1}}A_{t_n}$ is positive
if and only if
player $j$ wins against a player $k$ at time $t_n$ and $k$ wins
against $i$ at time $t_{n-1}$. Player $i$ gains score
$e^{-\beta (t_n-t_{n-1})} \alpha$ out of this situation.
For both cases $m_n=0$ and $m_n=1$,
the $j$th column of the second term accounts for the effect of the
$j$'s win at time $t_{n-1}$.
The third term covers four cases. For $m_{n-1}=m_n=0$,
the quantity inside the summation represents the direct win at $t_{n-2}$,
resulting in weight $e^{-\beta (t_n-t_{n-2})}$. For $m_{n-1}=0$ and $m_n=1$,
the quantity represents the indirect win based on the games at $t_{n-2}$ and
$t_n$, resulting in weight $e^{-\beta (t_n-t_{n-2})}\alpha$.
For $m_{n-1}=1$ and $m_n=0$, the quantity represents the indirect win
based on the games at $t_{n-2}$ and $t_{n-1}$, resulting in weight
$e^{-\beta (t_n-t_{n-2})}\alpha$. For $m_{n-1}=m_n=1$, the quantity
represents the indirect win based on the games at $t_{n-2}$,
$t_{n-1}$, and $t_n$, resulting in weight
$e^{-\beta (t_n-t_{n-2})}\alpha^2$.
In either of the four cases, the $j$th column of the third term accounts for the effect of the $j$'s win at time $t_{n-2}$.
To see the difference between the original and dynamic win scores,
consider the exemplary data with $N=3$ players shown in
\FIG\ref{fig:N=3 example}. The original
win-lose scores calculated from the aggregation of the data up
to time $t_n$ ($n=1, 2$, and 3), denoted by $w_{t_n}(i)$ for player $i$, are given by
\begin{equation}
\begin{cases}
w_{t_1}(1)=1,\\
w_{t_1}(2)=0,\\
w_{t_1}(3)=0,
\end{cases}
\begin{cases}
w_{t_2}(1)=1+\alpha,\\
w_{t_2}(2)=1,\\
w_{t_2}(3)=0,
\end{cases}
\begin{cases}
w_{t_3}(1)=1+\alpha+\alpha^2+\cdots,\\
w_{t_3}(2)=1+\alpha+\alpha^2+\cdots,\\
w_{t_3}(3)=1+\alpha+\alpha^2+\cdots.
\end{cases}
\end{equation}
The scores of the three players are the same at $t=t_3$ because the aggregated network
is symmetric (i.e., directed cycle) if we discard the information
about the time.
The dynamic win-lose scores for the same data are given by
\begin{equation}
\begin{cases}
w_{t_1}(1)=1,\\
w_{t_1}(2)=0,\\
w_{t_1}(3)=0,
\end{cases}
\begin{cases}
w_{t_2}(1)=e^{-\beta(t_2-t_1)},\\
w_{t_2}(2)=1,\\
w_{t_2}(3)=0,
\end{cases}
\begin{cases}
w_{t_3}(1)=e^{-\beta(t_3-t_1)},\\
w_{t_3}(2)=e^{-\beta(t_3-t_2)},\\
w_{t_3}(3)=1+\alpha e^{-\beta(t_3-t_1)}.
\end{cases}
\end{equation}
The score of player 1 at $t_2$ (i.e., $w_{t_2}(1)$)
differs from the original win-lose score in two aspects.
First, it is discounted by factor
$e^{-\beta(t_2-t_1)}$. Second, the value of $w_{t_2}(1)$ indicates that
player 1 does not gain an indirect win. This is
because it is after player 1 defeated player 2 that player 2
defeats player 3.
In contrast, player 3 gains an indirect win at $t=t_3$ because player 3
defeats player 1, which defeated player 2 before (i.e.,
at $t=t_1$).
It should be noted that the win scores of the three players are
different at $t=t_3$ although the aggregated network is symmetric.
Equation~\eqref{eq:def of W_{t_n}} leads to
\begin{align}
W_{t_n} =& A_{t_n} + e^{-\beta (t_n - t_{n-1})} \left[ A_{t_{n-1}}
+ e^{-\beta (t_{n-1} - t_{n-2})} \sum_{m_{n-1} \in \{ 0,1 \}}
\alpha^{m_{n-1}} A_{t_{n-2}}A_{t_{n-1}}^{m_{n-1}}\right.\notag\\
&+\cdots\notag\\
& \left. + e^{-\beta (t_{n-1} - t_1)} \sum_{m_2, \ldots, m_{n-1} \in \{ 0,1
\}} \alpha^{\sum _{i=2}^{n-1} m_i} A_{t_1}A_{t_2}^{m_2}\cdots A_{t_{n-1}}^{m_{n-1}} \right] \sum_{m_n \in \{ 0,1 \}} \alpha^{m_n} A_{t_n}^{m_n}\notag\\
=& A_{t_n} + e^{-\beta (t_n - t_{n-1})}W_{t_{n-1}}(I + \alpha A_{t_n}).
\label{eq:convenient W_{t_n}}
\end{align}
Therefore, by combining Eqs.~\eqref{eq:def of w_{t_n}} and \eqref{eq:convenient W_{t_n}}, we obtain
the update equation for the dynamic win score as follows:
\begin{equation}
\bm w_{t_n}=\begin{cases}
A_{t_1}^{\top}\bm 1 & (n=1),\\
A_{t_n}^{\top} \bm 1 + e^{-\beta (t_n - t_{n-1})} (I + \alpha
A_{t_n}^{\top})\bm w_{t_{n-1}} & (n>1).
\end{cases}
\label{eq:wtn}
\end{equation}
The dynamic lose score at time $t_n$ is denoted
in vector form by $\bm \ell_{t_n}$.
We obtain the update equation for $\bm \ell_{t_n}$
by replacing $A_{t_n}$ in \EQ\eqref{eq:wtn} by $A_{t_n}^{\top}$
as follows:
\begin{equation}
\bm \ell_{t_n}=\begin{cases}
A_{t_1}\bm 1 & (n=1), \\
A_{t_n} \bm 1 + e^{-\beta (t_n - t_{n-1})} (I + \alpha A_{t_n})\bm \ell_{t_{n-1}} & (n>1).\end{cases}
\label{eq:ltn}
\end{equation}
Finally, the dynamic win-lose score at time $t_n$, denoted by $\bm s_{t_n}$, is
given by
\begin{equation}
\bm s_{t_n} = \bm w_{t_n} -\bm \ell_{t_n}.
\end{equation}
It should be noted that we do not treat retired players in special ways. Players' scores exponentially decay after retirement.
\subsection*{Predictability}\label{sub:predictability}
We apply the dynamic win-lose score to results of professional men's tennis.
The nature of the data is described in Methods.
In this section, we predict the outcomes of future games
based on different ranking systems.
The frequency of violations, whereby
a lower ranked player wins against a higher ranked
player in a game, quantifies the degree of predictability
\cite{Martinich2002Inter,Bennaim2006JQAS}.
In other literature,
the retrodictive version of the frequency of violations is also used
for assessing the performance of ranking systems
\cite{Martinich2002Inter,Lundh2006JQAS,Park2010JSM,Coleman2005Interface}.
We compare the predictability of the
dynamic win-lose score, the original win-lose score
\cite{Park2005JSM}, and the prestige score (Methods).
The prestige score, proposed by Radicchi
and applied to
professional men's tennis data \cite{Radicchi2011PlosOne},
is a static ranking system and
is a version of the PageRank originally proposed for ranking webpages
\cite{Brin98}. We also implement a dynamic version of the prestige score (Methods) and compare its performance of prediction with that of the dynamic win-lose score.
We define the frequency of violations as follows.
We calculate the score of each player at $t_n$ ($1\le n\le n_{\max}-1$)
on the basis of the results up to $t_n$.
For the original win-lose score and prestige score, we
aggregate the directed links
from $t=t_1$ to $t=t_n$ to construct a static network and
calculate the players' scores.
If the result of each game at $t_{n+1}$ is
inconsistent with the calculated ranking,
we regard that a violation occurs.
If the two players involved in the game at $t_{n+1}$
have exactly the same score,
we regard that a tie occurs irrespective of the result of the game.
We define the prediction accuracy at the $N_{\rm gp}$th game
as the fraction of correct prediction when the results of the
games from $t=t_2$ through the $N_{\rm gp}$th game
are predicted. The prediction accuracy is given by
$\left(N_{\rm gp}^{\prime}-e-v\right)/\left(N_{\rm gp}^{\prime}-e\right)$,
where $N_{\rm gp}^{\prime} (<N_{\rm gp})$ is the number of predicted games,
$v$ is the number of violations, and $e$ is the number of ties.
For the prestige score and its dynamic variant,
we exclude the games in which either
player plays for the first time because the score is not defined for
the players that have never played. In this case, we increment $e$ by
one.
The original and dynamic win-lose scores can be negative
valued. Equations~\eqref{eq:def of w_{t_n}}
and \eqref{eq:w original win score} guarantee that the initial score is equal to zero for all the players for the dynamic and original win-lose scores, respectively.
Furthermore, any player has a zero win-lose score
when the player fights a game for the first time.
Even though we do not treat such a game
as tie unless both players involved in the game have zero scores, treating it as tie
little affects the following results.
The prediction accuracy for the dynamic win-lose score,
original win-lose score, prestige score, and dynamic prestige score
are shown in
\FIGS\ref{fig:performance}(a), \ref{fig:performance}(b),
\ref{fig:performance}(c),
and \ref{fig:performance}(d), respectively, for various parameter values.
Figure~\ref{fig:performance}(a) indicates that the prediction accuracy
for the dynamic win-lose score is the largest for
$\alpha=0.13$ except when the number of games (i.e., $N_{\rm gp}$) is
small. The accuracy is insensitive to $\alpha$ when $0.08\le \alpha\le
0.2$. In this range of $\alpha$, we confirmed by additional numerical simulations that
the results for $\beta=1/365$ and
those for $\beta=0$ are indistinguishable. Therefore, we
conclude that the performance of prediction has some robustness with
respect to $\alpha$ and $\beta$. We also confirmed that
the accuracy monotonically increases between $\alpha\approx 0.03$ and
$\alpha\approx 0.13$. However, for an unknown reason, the accuracy
with $\alpha\approx 0.03$ is smaller than that with $\alpha=0$ (results
not shown).
Figure~\ref{fig:performance}(b) indicates that the prediction accuracy
for the original win-lose score is larger for $\alpha=0$
than $\alpha=0.004835$. The latter
$\alpha$ value is very close to the upper
limit calculated from the largest eigenvalue of $A$ (see subsection ``Parameter values'' in Methods). We also found that the prediction accuracy monotonically
decreases with
$\alpha$. Nevertheless, except for small $N_{\rm gp}$,
the accuracy with
$\alpha=0$ is lower than that for the dynamic win-lose score
with $\alpha=0$ and $0.08\le \alpha\le 0.2$ (\FIG\ref{fig:performance}(a)).
Figure~\ref{fig:performance}(c) indicates that
the prediction by the prestige score is better for a smaller
value of $q$ (see Methods for the meaning of $q$).
We confirmed that this is the case for other values of
$q$ and that the results with $q\le 0.05$ little
differ from those with $q=0.05$.
Except for small $N_{\rm gp}$, the prediction
accuracy with $q=0.05$ is lower than that for the dynamic win-lose score with
$0.08\le \alpha\le 0.2$ (\FIG\ref{fig:performance}(a)).
Figure~\ref{fig:performance}(d) indicates that
the prediction by the dynamic variant of the prestige score is more accurate than that by the dynamic win-lose score, in particular for small $N_{\rm gp}$. Similar to the case of the original prestige score, the prediction accuracy decreases with $q$.
The findings obtained from \FIG\ref{fig:performance} are summarized as follows. When
$\alpha$ is between $\approx 0.08$ and $\approx 0.2$ and
$\beta$ is between 0 and $1/365$,
the dynamic win-lose score outperforms the original win-lose score and the prestige score in the prediction accuracy. For example,
at the end of the data,
the accuracy is equal to 0.659, 0.661, 0.661, and 0.659
for the dynamic win-lose score with
($\alpha$, $\beta$) $=$ (0.08, $1/365$),
(0.1, $1/365$), (0.13, $1/365$), and (0.2, $1/365$), respectively,
while it is equal to 0.623 for the original win-lose score with $\alpha=0$
and 0.631 for the prestige score with $q=0.05$.
However, the accuracy for
the dynamic variant of the prestige score with $q=0.05$ (i.e., 0.668) is slightly larger than the largest value obtained by the dynamic win-lose score.
We also compare the prediction accuracy for the dynamic win-lose score with that for the official Association of Tennis
Professionals (ATP) rankings. Because the calculation of the ATP rankings involves relatively minor games that do not belong to ATP World Tour tournaments, which we used for \FIG\ref{fig:performance}, we use a different data set for the present comparison (see ``Data'' in Methods). The prediction accuracy at the end of the data is equal to 0.637 for the ATP rankings and 0.588, 0.629, 0.646, 0.650, and 0.649 for the dynamic win-lose score with
($\alpha$, $\beta$) $=$ (0.08, $1/365$),
(0.1, $1/365$), (0.13, $1/365$), (0.17, $1/365$), and (0.2, $1/365$), respectively. The prediction accuracy for the dynamic win-lose score is larger than that for the ATP rankings in a wide range of $\alpha$ (i.e., $0.11\le \alpha\le 0.39$).
\subsection*{Robustness against parameter variation}\label{sub:sensitivity}
Figure~\ref{fig:performance}(a) indicates that
the prediction accuracy for the dynamic win-lose score is robust against some variations in the $\alpha$ and $\beta$ values.
In this section, we examine the robustness of the dynamic win-lose score
more extensively by examining the rank correlation between
the scores derived from different $\alpha$ and $\beta$ values.
The Kendall's tau is a standard method to quantify the rank
correlation \cite{Kendall1938Biom}. In our data,
the full ranking containing all the players, to which the Kendall's
tau applies, contains players that only appear in a few games.
In fact, most players are such players \cite{Radicchi2011PlosOne},
and their ranks are inherently
unstable. In addition, it is usually the list of top ranked players that are
of practical interests.
Therefore,
we use a generalized Kendall's tau for comparing top $k$ lists
of the full ranking \cite{Fagin2003SIAMDM}.
We denote the sets of the top $k$ players, i.e., $k$ players with the
largest scores, in the two full rankings by
$\bm R_1$ and $\bm R_2$.
In general, $\bm R_1$ and $\bm R_2$ can be different.
For an arbitrarily chosen pair of players
$r_1$, $r_2$ $\in \bm R_1 \cup \bm R_2$, $r_1\neq r_2$,
we set $\overline{K}_{r_1 , r_2}(\bm R_1,\bm R_2)=1$ if
(1) $r_1$ and $r_2$ appear in both top $k$ lists $R_1$ and $R_2$, and $r_1$ and $r_2$ are in the opposite order in the two top $k$ lists,
(2) $r_1$ has a higher rank than $r_2$ in one of the top $k$ lists, and
$r_2$, but not $r_1$, is contained in the other top $k$ list,
(3) $r_1$ exists only in one of the two top $k$ lists, and $r_2$
exists only in the other top $k$ list.
Otherwise, we set $\overline{K}_{r_1, r_2}(\bm R_1,\bm R_2)=0$.
$\overline{K}_{r_1, r_2}(\bm R_1,\bm R_2)$ is a penalty imposed on the inconsistency between the two top $k$ lists. We use the so-called
optimistic variant of the Kendall distance
$K_{\tau}^{(0)}(\bm R_1,\bm R_2)$ defined as follows \cite{Fagin2003SIAMDM}:
\begin{equation}
K_{\tau}^{(0)}(\bm R_1,\bm R_2) = \sum_{r_1,r_2 \in \bm R_1 \cup \bm R_2} \overline{K} _{r_1, r_2}(\bm R_1,\bm R_2).
\end{equation}
We normalize the distance between the two rankings as follows
\cite{Mccown2007JCDL}:
\begin{equation}
K = 1 - \frac{K_{\tau}^{(0)}(\bm R_1,\bm R_2)}{k^2}.
\end{equation}
A large value of $K$ indicates a higher correlation between the two
top $k$ lists.
It should be noted that $0\le K\le 1$. In particular, when there is no overlap
between the two top $k$ lists,
we obtain $K=0$.
For the dynamic win-lose scores at $t_{n_{\max}}$,
i.e., at the end of the entire period,
we calculate $K$ with $k=300$ for different pairs of $\alpha$ and $\beta$
values.
The results for $\beta=1/365$ and
different values of $\alpha$ are shown in
\FIG\ref{fig:robustness alpha}. The top $k$ lists are similar (i.e., $K \ge 0.85$) for any $\alpha$ larger than $\approx 0.06$.
This finding is consistent with the fact
that the prediction accuracy is high
and robust when $\alpha$ falls between $\approx 0.08$ and $\approx 0.2$
(\FIG\ref{fig:performance}(a)).
For fixed values of $\alpha$, the $K$ values between the ranking
with $\beta=1/365$ and that with various values of $\beta$ are shown
in \FIG\ref{fig:robustness alpha beta}. $K$ is almost
unity at least in the range $0\le\beta\le 2/365$. Therefore, removing
the assumption of the exponential
decay of score in time (i.e., $\beta=0$) little changes the top
300 list. This finding is consistent with the result
that the prediction accuracy is almost the same between $\beta=0$ and
$\beta=1/365$ if $0.1\le\alpha\le 0.2$ (see the previous subsection).
Nevertheless, this observation does not imply that we can ignore the
temporal aspect of the data. Keeping the order of the
games contributes to the performance of prediction, as suggested by
the comparison between the prediction results for
the dynamic (\FIG\ref{fig:performance}(a))
and original (\FIG\ref{fig:performance}(b)) win-lose scores.
\subsection*{Dynamics of scores for individual players}\label{sub:dynamics individual}
In contrast to the original win-lose score and prestige score, the
dynamic win-lose score can track dynamics of the strength of each
player. It should be noted that the summation of the scores over the
individuals, i.e., $\sum_{i=1}^N s_{t_n}(i)$, depends on time. In particular,
it grows almost exponentially for the parameter values with which the
prediction accuracy is high (i.e., $\alpha$ larger than $\approx
0.08$), as shown in \FIG\ref{fig:sum scores}. $\sum_{i=1}^N s_{t_n}(i)$
increases with the number of games, or equivalently, with time because
more recent players take more advantage of indirect wins than older
players.
The increase in $\sum_{i=1}^N s_{t_n}(i)$ is not owing to
the number of players or games observed per year;
in fact, the latter numbers do not increase in time \cite{Radicchi2011PlosOne}.
Therefore, for clarity, we normalize the win-lose score of each player by dividing it by the instantaneous $\sum_{i=1}^N s_{t_n}(i)$ value.
The time courses of the normalized win-lose scores for four
renowned players are shown in
\FIG\ref{fig:famous players}(a).
We set $\alpha=0.13$ and $\beta=1/365$, for
which the prediction is approximately the most accurate. The ATP rankings of the four players during the same period are shown in \FIG\ref{fig:famous players}(b) for comparison. The time courses of
the dynamic win-lose score and those of the ATP rankings are similar.
In particular, the times at which the strength of one player (e.g., Federer) begins to exceed another player (e.g., Agassi) are similar between
\FIGS\ref{fig:famous players}(a) and \ref{fig:famous players}(b).
Figure~\ref{fig:famous players} suggests
that the dynamic win-lose score appositely captures
rises and falls of these players.
\section*{Discussion}
We extended the win-lose score for static sports networks
\cite{Park2005JSM} to the case of dynamic networks. By assuming that
the score decays exponentially in time, we could derive closed
online update equations for the win and lose scores. The proposed dynamic win-lose
score realizes a higher prediction accuracy than the original win-lose
score and the prestige score.
It is straightforward to extend the dynamic win-lose score to incorporate
factors such as the importance of each tournament or game via modifications
of the game matrix $A_{t_n}$.
We also confirmed the robustness of the
ranking against variation in the two parameter values in the model.
Finally, the dynamic win-lose score is capable of
tracking dynamics of players' strengths.
It seems that network-based ranking systems are easier to understand and
implement, and more scalable than those based on statistical methods.
The dynamic win-lose score share these desirable features with
static network-based ranking systems.
The applicability of the idea behind the dynamic win-lose score is not
limited to the case of the win-lose score. In fact, we implemented a dynamic variant of the prestige score. It even yielded a larger prediction accuracy than the dynamic win-lose score did. This result implies that
the idea of network-based dynamic ranking systems may be a powerful approach
to assessing strengths of sports players and teams, which fluctuate over time. The dynamic win-lose score is better than our version of the dynamic prestige score in that only the former allows for a set of closed online update equations. Establishing similar update equations
for other network-based ranking systems
such as the prestige score and the Laplacian
centrality (see Introduction) is warranted for future work.
Prospective results obtained through
this line of researches may be also useful in
systematically deriving dynamic centrality measures for
temporal networks in general.
\section*{Methods}
\subsection*{Park \& Newman's win-lose score}\label{sub:Park}
The win-lose score by Park and Newman \cite{Park2005JSM} is a
network-based static ranking system defined as follows.
We assume $N$ players and denote by $A_{ij}$ ($1\le i, j\le N$)
the number of times that player $j$ wins against player $i$ during the entire period.
We let $\alpha$ ($0\le \alpha<1$) be a constant representing the
weight of indirect wins.
For example,
if player $i$ wins against $j$ and $j$ wins against $k$,
$i$ gains score 1 from the direct win against $j$ and score $\alpha$
from the indirect win against $k$. Therefore, the
$i$'s win score is equal to $1+\alpha$.
If $k$ wins against yet another player $\ell$,
the $i$'s win score
is altered to $1+\alpha+\alpha^2$.
The win scores of the players are given by
\begin{align}
W =& A + \alpha A^2 + \alpha^2 A^3 + \cdots\notag\\
=& A(I+\alpha A + \alpha^2 A^2 + \alpha^3 A^3 + \cdots)\notag\\
=& A(I - \alpha A)^{-1},\\
\bm w =& W^{\top} \bm 1 = (I-\alpha A^{\top})^{-1}A^{\top} \bm 1,
\label{eq:w original win score}
\end{align}
where $W$ is the $N\times N$ matrix whose $(i,j)$ element represents
the score that player $j$ obtains via
direct and indirect wins against player $i$,
$\bm w$ is the $N$ dimensional column vector whose $i$th
element represents the win score of player $i$, and
$\bm 1$ is the $N$ dimensional column vector defined by
\begin{equation}
\bm 1=(1\; 1\; \cdots \; 1)^{\top}.
\end{equation}
We similarly obtain
the lose scores of the $N$ players
in vector form by replacing $A$ with $A^{\top}$ as follows:
\begin{equation}
\bm \ell = (I-\alpha A)^{-1}A \bm 1.
\end{equation}
The total win-lose score is given in vector form by
\begin{equation}
\bm s = \bm w -\bm \ell.
\end{equation}
\subsection*{Prestige score}
The prestige score of player $i$, denoted by $P_i$, is defined by
\begin{equation}
P_i = (1-q)\sum_{j=1}^N P_j\frac{\tilde{w}_{ji}}{s_j^{\rm out}} + \frac{q}{N} + \frac{1-q}{N}\sum_{j=1}^N P_j\delta (s_j^{\rm out})\quad (1\le i\le N),
\label{eq:prestige score}
\end{equation}
where $q$ is a constant,
$\tilde{w}_{ji}$ is the number of times player $i$ defeats player $j$ during the entire
period (it should be noted that $\tilde{w}_{ji}$ has nothing to do with the win scores denoted by $\bm w$ in \EQS\eqref{eq:def of w_{t_n}} and \eqref{eq:w original win score}),
$s_j^{\rm out}\equiv\sum_{i^{\prime}=1}^N \tilde{w}_{ji^{\prime}}$ is equal to the number of losses for player $j$, $\delta(s_j^{\rm out})=1$ if $s_j^{\rm out}=0$,
and $\delta(s_j^{\rm out})=0$
if $s_j^{\rm out}\ge 1$. The normalization is given by
$\sum_{i=1}^N P_i = 1$.
We set $q=0.15$, as in
\cite{Radicchi2011PlosOne}, and also $q=0.05$ and $q=0.30$.
To define a dynamic variant of the prestige score,
we let $\tilde{w}_{ij}$ used in \EQ\eqref{eq:prestige score}
depend on time. We define $\tilde{w}_{ij}$
at time $t$ by
\begin{equation}
\tilde{w}_{ji}\equiv \sum_n A_{t_n}(j,i)e^{-\beta(t-t_n)},
\label{eq:tilde w(t)}
\end{equation}
where $A_{t_n}(j,i)$ is the $(j,i)$ element of the win-lose matrix $A_{t_n}$, and the summation over $n$ is taken over the games that occur before time $t$.
Substituting \EQ\eqref{eq:tilde w(t)} in
\EQ\eqref{eq:prestige score} yields the dynamic prestige score $P_i$
($1\le i\le N$) at time $t$. We set $\beta=1/365$, which is the same value as that used for the dynamic win-lose score.
\subsection*{Data}
We collected the data from the website of ATP \cite{ATPWorldTour}.
Except when we compared the prediction accuracy for the
dynamic win-lose score with that for the ATP rankings,
we used single games in
ATP World Tour tournaments recorded on this website.
The data set contains 137842 singles games from December 1972 to May
2010 and involves 5039 players that
participated in at least one game.
Because the source of our data set is the same as that of
Radicchi's data set \cite{Radicchi2011PlosOne} and the
period of the data is similar,
the number of games contained in our data and that in
Radicchi's are close to each other.
In the comparison between the dynamic win-lose score and the ATP rankings,
we used all the types of single games recorded on the website of ATP.
They include the games belonging to
ATP Challenger Tours and ITF Futures tournaments in addition to
ATP World Tour tournament games. We used this data set because it corresponds to the games on which the calculation of the ATP rankings is based. The ATP rankings are not available
on a regular basis in early years. Therefore, we used the data from July 23, 1984 to August 15, 2011. The data set contains 330796 games and involves 13077 players
that participated in at least one game.
\subsection*{Parameter values for the dynamic win-lose score}\label{sub:parameter choice}
A guiding principle for setting
the parameter values of a ranking system is to select the values that
maximize the performance of prediction
\cite{Dixon1997ApplStat,Knorrheld2000Stat}. Instead,
we set $\alpha$ and $\beta$ as follows.
In the original win-lose score, it is recommended that $\alpha$ is set
to the value smaller than and close to the inverse of the largest eigenvalue of
$A$ \cite{Park2005JSM}. If $\alpha$ exceeds this upper limit, the original win-lose score diverges. For our data, the upper limit according to this criterion is equal to $1/206.80=0.0048355$.
However, the dynamic win-lose score converges irrespective of the values of $\alpha$ and $\beta$ for the following reason. For expository purposes, let us assign different nodes to the same player at different times $t_n$ ($1\le n\le n_{\max}$). Then, \EQ\eqref{eq:def of W_{t_n}} implies that
any link in the network, which represents a game at time $t_n$,
is directed from the winner at $t_n$ to the loser at $t_{n}$ or earlier times.
Because there is no time-reversed link (i.e., from $t_n$ to $t_{n^{\prime}}$, where $t_n<t_{n^{\prime}}$) and any pair of players play at most once at any $t_n$, the network is acyclic.
The upper limit of $\alpha$
is infinite when the network is acyclic \cite{Park2005JSM}.
On the basis of this observation,
we examine the behavior of the dynamic win-lose score for
various values of $\alpha$.
In the official ATP ranking, the score
of a player is calculated from the player's performance in the last
52 weeks $\approx$ one year \cite{ATPWorldTour}. The results of the games in this time window
contribute to the current ranking of the player with the same weight if the other conditions are equal.
The dynamic win-lose score uses
the results of all the games in the past, and the contribution of the
game decays exponentially in time. By equating the contribution
of a single game in the two ranking systems, we assume $1\times 365 =
\int_{0}^{\infty } e^{-\beta t}dt$, which leads to $\beta = 1/365$.
In Results, we also investigated the robustness of the
ranking results against variations in
the $\alpha$ and $\beta$ values.
|
{
"timestamp": "2012-12-07T02:01:00",
"yymm": "1203",
"arxiv_id": "1203.2228",
"language": "en",
"url": "https://arxiv.org/abs/1203.2228"
}
|
\section{Introduction}
In formal language theory, quotient is a basic and very important operation and plays a fundamental role in the construction of minimal deterministic finite automata (DFA). Given a formal language $L$ over an alphabet $\Sigma$, the left quotient $u^{-1}L$ of $L$ by a word $u$ is defined as the language $\{v\in \Sigma^\ast | uv\in L\}$, where $\Sigma^\ast$ is the free monoid of words over $\Sigma$. The famous Myhill-Nerode Theorem then states that $L$ is a regular language if and only if the number of different left quotients of $L$ (also called the quotient complexity \cite{Brz09} of $L$) is finite. Moreover, a minimal DFA which recognizes $L$ can be constructed in a natural way by using left quotients as states. In particular, this means that the quotient complexity of $L$ is equal to the size of the minimal DFA which recognizes $L$.
The notion of left quotient of a formal language by a word can be extended to quotients by a formal language in two ways. Given two formal languages $L,X$, the left quotient of $L$ by $X$, denoted by $X^{-1}L$, is defined as the union of $u^{-1}L$ for all words $u$ in $X$. Another extension is less well-known, if not undefined at all. We define the \emph{left residual} of $L$ by $X$, denoted by $X\backslash L$, as the intersection of $u^{-1}L$ of all words in $X$. Similarly we have $LX^{-1}$, the right quotient of $L$ by $X$, and $L/X$, the right residual of $L$ by $X$. Regarding each left residual of $L$ as a state, there is a natural way to define an automaton, which is called the \emph{universal automaton} \cite{conway71,sakarovitch09} of $L$. The universal automaton of a formal language $L$ contains many interesting information (e.g. factoraization) of $L$ \cite{lombardy08} and plays a very important role in constructing the minimal nondeterministic finite automaton (NFA) of $L$ \cite{ADN92,polak05}.
Former power series are extensions of formal languages, which are used to describe the behaviour of weighted automata (i.e. finite automata with weights). Weighted automata were introduced in 1961 by Sch\"{u}tzenberger in his seminal paper \cite{schutzenberger61}. A formal power series is a mapping from $\Sigma^\ast$, the free monoid of words over $\Sigma$, into a semiring $S$. Depending on the choice of the semiring $S$, formal power series can be viewed as weighted, multivalued or quantified languages where each word is assigned a weight, a number, or some quantity. Weighted automata have been used to describe quantitative properties in areas such as probabilistic systems, digital image compression, natural language processing. We refer to \cite{droste09} for an detailed introduction of weighted automata and their applications.
Despite that a very large amount of work has been devoted to the study of formal power series and weighted automata (see e.g. \cite{kuich86, salomaa78, berstel11, droste09,esik10} for surveys), the important concept of quotient as well as universal automata has not been systematically investigated in this weighted context. The only exception seems to be \cite{berstel11}, where the quotient of formal power series (by word) was discussed in pages 10-11. When the semiring is complete, it is straightforward to extend the definition of the quotient of a formal power series $A$ from words to series: we only need to take the weighted sum of all left quotients of $A$ by word in $\Sigma$. Our attempt to characterize the residual of a formal power series $A$ by a formal power series as the weighted intersection of all left quotients of $A$ by word in $\Sigma$ is, however, unsuccessful. Several important and nice properties fail to hold anymore.
The aim of this paper is to introduce the quotient and residual operations in formal power series and study their application in the minimization of weighted automata. To overcome the above obstacle with residuals, we require the semiring to be a complete c-semiring (to be defined in Section~2), and then give a characterization of residuals in terms of quotients by word. Many nice properties and useful notions then follow in a natural way.
The remainder of this paper is organized as follows. Section 2 introduces basic notions and properties of semirings, formal power series, and weighted automata. Quotients of formal power series are introduced in Section 3, where we also show how to construct the minimal deterministic weighted automata effectively. In Section 4, we introduce the residuals and factorizations of formal power series. Using the left residuals, we define the universal weighted automaton $\mathcal{U}_A$ for arbitrary formal power series $A$ in Section~5, and justify its universality in Section~6.
An effective method for constructing the universal automaton is described in Section~7, which is followed by a comparison of the quotient and the residual operations. The last section concludes the paper.
\section {Preliminaries}
We recall in this section the notions of semirings, formal power series, weighted automata, and weighted contex-free grammar. Interested readers are referred to \cite{droste09,kuich86,salomaa78,esik10} for more information.
\subsection{Semirings}
A 5-tuple ${\cal S}=(S, \oplus, \otimes, 0,1)$ is called a \emph{ semiring} if $S$ is a set containing at least two different elements $0$ and $1$, and $\oplus $ and $\otimes$ are two binary operations on $S$ such that
\begin{itemize}
\item [(i)] $\oplus $ is associative and is commutative and has identity $0$;
\item [(ii)] $\otimes$ is associative and has identity $1$ and null element $0$ (i.e., $a\otimes 0=0\otimes a=0$ for all $a\in S$); and
\item [(iii)] $\otimes$ distributes over $\oplus $, i.e., for all $a,b,c\in S$, $a\otimes (b\oplus c)=(a\otimes b)\oplus (a \otimes c)$ and $(b\oplus c)\otimes a=(b\otimes a)\oplus (c\otimes a)$.
\end{itemize}
Intuitively, a semiring is a ring (with unity) without subtraction. All rings (with unity), as well as all fields, are
semirings, e.g., the integers $\mathbb{Z}$, rationals $\mathbb{Q}$, reals $\mathbb{R}$, complex numbers $\mathbb{C}$. Lattices provide another important type of semirings. Recall that a partially ordered set $(L, \leq)$ is a \emph{lattice} if for any two elements
$a, b\in L$, the least upper bound $a\vee b = \sup\{a, b\}$ and the greatest lower
bound $a\wedge b = \inf\{a, b\}$ exist in $(L,\leq)$. A lattice $(L, \leq)$ is \emph{distributive}, if
$a\wedge(b\vee c) = (a\wedge b)\vee (a\wedge c)$ for all $a, b, c\in L$; and \emph{bounded}, if $L$ contains a
smallest element, denoted 0, and a greatest element, denoted 1. Let $(L, \leq)$ be any bounded distributive lattice. Then $(L,\vee,\wedge, 0, 1)$ is a semiring. Because
a distributive lattice $L$ also satisfies the dual distributive law $a\vee(b\wedge c) = (a\vee b)\wedge (a\vee c)$ for
all $a, b, c\in L$, the structure $(L,\wedge,\vee, 1, 0)$ is also a semiring.
Other important examples of semirings include:
\begin{itemize}
\item [-] The Boolean semiring $\mathbb{B} =( \{0, 1\}, \vee, \wedge, 0, 1)$;
\item [-] The semiring of the natural numbers $(\mathbb{N}, +, \cdot, 0, 1)$ with the usual addition and multiplication;
\item [-] The tropic semiring $(\mathbb{N}\cup\{\infty\}, \min, +,\infty, 0)$ with min and $+$ extended to $\mathbb{N}\cup\{\infty\}$ in a natural way;
\item [-] The min-sum semiring of nonnegative reals $(\mathbb{R}^+\cup\{0, \infty\}, \min, +, \infty, 0)$;
\item [-] The semiring of (completely positive) super-operators on a Hilbert space $\cal H$ $({\cal SO}({\cal H}), +, \circ, 0_{\cal H}, {\cal I}_{\cal H})$.
\end{itemize}
We note in the last semiring the addition is not idempotent and the product is not commutative. This semiring is recently used in model checking quantum Markov chains \cite{Feng+12} and the study of finite automata with weights taken from this semiring just initiated.
\subsubsection{Complete Semiring}
Let $I$ be an index set and let $S$ be a semiring. An infinitary sum operation $\sum_I : S^I \rightarrow S$ is an operation that associates with every family $\{a_i | i\in I\}$ of elements of $S$ an element $\sum_{i\in I} a_i$ of $S$. A semiring $S$ is called \emph{complete} if it has an infinitary sum operation $\sum_I$
for each index set $I$ and the following conditions are satisfied:
\begin{itemize}
\item [(i)] $\sum_{i\in\emptyset}a_i= 0$, $\sum_{i\in\{j\}}a_i=a_j$, and $\sum_{i\in\{j,k\}}a_i=a_j\oplus a_k$ for $j\not=k$.
\item [(ii)] $\sum_{j\in J}\sum_{i\in I_j}a_i=\sum_{i\in I}a_i$ if $\bigcup_{j\in J}I_j=I$ and $I_j\cap I_k=\emptyset$ for $j\not=k$.
\item [(iii)] $\sum_{i\in I}(a\otimes a_i) = a \otimes\sum_{i\in I} a_i$ and $\sum_{i\in I}(a_i\otimes a) = \sum_{i\in I} a_i \otimes a$.
\end{itemize}
This means that a semiring $S$ is complete if it is possible to define infinite sums (i) that are extensions of the finite sums, (ii) that are associative and
commutative, and (iii) that satisfy the distributive laws.
\subsubsection{Complete c-Semiring}
A semiring $S$ is a \emph{c-semiring} if $\oplus $ is idempotent (i.e., $a\oplus a=a$ for all $a\in S$), $\otimes $ is commutative, and $1$ is the absorbing element of $\oplus $ (i.e., $a\oplus 1=1$ for any $a\in S$). In general, for a semiring $S$, we define a preorder $\leq_S$ over the set $S$ by
\begin{equation*}
\mbox{$a\leq _S b$ iff $a\oplus c=b$ for some $c\in S$.}
\end{equation*}
If $\oplus $ is idempotent, then $\leq_S$ is also a partial order. Suppose $S$ is a c-semiring. For any $a,b\in S$, we have $0\leq_S a\leq_S 1$ and $a\oplus b = a\vee b$ (the least upper bound of $a$ and $b$) in the poset $(S,\leq_S)$. If $S$ is clear from the context, then $S$ is omitted and we simply write $\leq$ for $\leq_S$ in the following.
A semiring $S$ is called a \emph{complete c-semiring} if $S$ is a complete semiring and a c-semring. In a complete c-semiring, the infinitary sum $\sum_{i\in I} a_i$ is exactly the least upper bound of $a_i$ ($i\in I$) in $S$ under the induced partial order $\leq_S$. In this case, $\sum_{i\in I} a_i$ is also written as $\bigvee_{i\in I}a_i$.
Complete c-semiring is a special kind of the notion of \emph{quantale} \cite{rothal}, which is a complete lattice $L$ equipped with a multiplication operator $\otimes$ such that $(L,\otimes)$ is a semigroup satisfying the following distributive laws:
\begin{equation}
\label{eq: quantale-dist}
\bigvee_{i\in I}(a\otimes a_i) = a \otimes\bigvee_{i\in I} a_i,\ \ \bigvee_{i\in I}(a_i\otimes a) = \bigvee_{i\in I} a_i \otimes a.
\end{equation}
Since the infinite distributive laws (Eq.~\ref{eq: quantale-dist}) holds, there are two \emph{adjunctions} or \emph{residuals}, denoted $a\backslash b$ (left residual) and $b/a$ (right residual), respectively, satisfying the following adjunction (residual) conditions,
\begin{equation}
\label{eq:quantale-adj}
x\leq a\backslash b\ \mbox{iff} \ a\otimes x\leq b,\ \mbox{and}\ x\leq b/a\ \mbox{iff}\ x\otimes a\leq b
\end{equation}
When the given quantale is commutative, i.e., the operation $\otimes$ is commutative, then the left residual is the same as the right residual. In this case, we call the left residual (and the right residual) the residual, denoted by $a\rightarrow b$. Then we have
\begin{equation}
\label{a-rw-b}
a\rightarrow b = \bigvee \{x | a\otimes x \leq b\}.
\end{equation}
We have the following proposition.
\begin{proposition}
Let $S$ be a complete c-semiring. Then $S$ is a commutative quantale with unit $1$ as the largest element of $S$.
\end{proposition}
\subsection{Formal Power Series}
Let $\Sigma$ be an alphabet and $S$ a semiring. Write $\Sigma^{\ast}$ for the set of all finite strings (or words) over $\Sigma$, and write $\varepsilon$ for the empty string. Then $\Sigma^{\ast}$ is the free monoid generated by $\Sigma$ under the operation of concatenation. We write $\Sigma^+$ for all finite non-empty strings over $\Sigma$.
A \emph{formal power series} $A$ is a mapping from $\Sigma^{\ast}$ into $S$. For simplicity, we also call a formal power series as a \emph{series}, or an \emph{$S$-subset} of $\Sigma^{\ast}$. The value of $A$ at a word $w\in\Sigma^{\ast}$ is denoted $(A,w)$ or $A(w)$ in this paper. We write $A$ as a formal sum
\begin{equation}\label{eq:formal-sum}
A=\sum_{w\in\Sigma^{\ast}}(A,w)w,
\end{equation}
where the values $(A,w)$ are referred as the \emph{coefficients} of $A$. The collection of all power series $A$ as defined above is denoted by $S\langle\!\langle\Sigma^{\ast}\rangle\!\rangle$. For a series $A$, if $A(\varepsilon)=0$, then $A$ is called \emph{proper}. For any series $A$, $A$ can be written as the sum of a proper series and and non-proper series, i.e.,
\begin{equation}
\label{eq:Aepsilon}
A=(A,\varepsilon)\varepsilon+\sum_{w\in \Sigma^+}(A,w)w.
\end{equation}
For s series $A$ on $\Sigma$, the \emph{support} of $A$ is defined as
\begin{equation}
\label{eq:supp}
supp(A)=\{w\in\Sigma^{\ast} | (A,w)\not=0\}.
\end{equation}
For two series $A$ and $B$ in $S\langle\!\langle\Sigma^{\ast}\rangle\!\rangle$, we define $A\leq B$ whenever $A(w)\leq_S B(w)$ for any $w\in\Sigma^{\ast}$.
\subsection{Weighted Automata}
Weighted automata are an extension of the classical finite automata.
\begin{definition}\label{def:l-vfa}
Let $S$ be a semiring. A \emph{weighted automaton} with weights in $S$ is a
5-tuple ${\cal A}=(Q,\Sigma,\delta,I,F)$, where $Q$ denotes a
set of states, $\Sigma$ is an input alphabet, $\delta$ is a mapping
from $Q\times \Sigma\times Q$ to $S$, and $I$ and $F$ are two mappings from $Q$ to $S$. We call ${\cal A}$ a \emph{finite} weighted automaton if both $Q$ and $\Sigma$ are finite sets; We call ${\cal A}$ a \emph{deterministic weighted automaton} (DWA for short) if $\delta$ is crisp and deterministic, i.e., $\delta$ is a mapping from $Q\times \Sigma$ into $Q$, and $I=q_0\in Q$.
The mapping $\delta$ is called the \emph{weighted (state) transition relation}. Intuitively, for
any $p,q\in Q$ and $\sigma\in \Sigma$, $\delta(p,\sigma,q)$ stands
for the weight that input $\sigma$ causes state $p$ to become $q$. $I$ and $F$ represent the (weighted) initial and, respectively, final state. For each $q\in Q$, $I(q)$ indicates the weight that $q$ is an initial state, $F(q)$ expresses the weight that $q$ is a final state.
\end{definition}
\begin{remark}
Our definition of deterministic weighted automaton is different from the one used in e.g. \cite{buchsbaum00}, where a weighted automaton ${\cal A}=(Q,\Sigma,\delta,I,F)$ is \emph{deterministic} or \emph{sequential} if there exists a unique state $q_0$ in $Q$ such that $I(q_0)\not=0$ and for all $q\in Q$ and all $\sigma\in\Sigma$, there is at most one $p\in Q$ such that $\delta(q,\sigma,p)\not=0$.
These two definitions are not identical in general.
In fact, deterministic weighted automaton called in this paper is just the \emph{simple} deterministic weighted automaton defined in \cite{buchsbaum00}, for the detail comparison, we refer to \cite{buchsbaum00,droste09}. A simple deterministic weighted automaton is obviously a sequential weighted automaton defined in \cite{buchsbaum00}. The following example shows that the converse does not hold in general. Let $S=(\mathbb{N}\cup\{\infty\}, \min, +,\infty, 0)$ be the tropical semiring. Suppose $\Sigma=\{a\}$ and $A$ is the formal power series over $\Sigma$ defined by $A(w)=|w|$, where $|w|$ denotes the length of the string $w$. Then $A$ can not be accepted by any simple deterministic weighted automaton (see Proposition \ref{pro:dffa} below). However, $A$ can be accepted by a sequential weighted automaton ${\cal A}=(\{q\},\Sigma,\delta,\{q\},\{q\})$, where $\delta(q,a,p)=1$ if $p=q$ and $\delta(q,a,p)=0$ otherwise.
If $S$ is \emph{locally finite}, however, then the two definitions are equivalent in the sense that they accept the same class of former power series, where a semiring is locally finite if every sub-semiring generated by a finite set is also finite \cite{droste10}.
\end{remark}
Two weighted automata can be compared by a morphism.
\begin{definition}\label{de:morphism}
A \emph{ homomorphism} (or \emph{ morphism}) between two weighted automata ${\cal A}=(Q,\Sigma,\delta,I,F)$ and ${\cal B}=(P,\Sigma,\eta,J,G)$ is a mapping $\varphi: Q\rightarrow P$, satisfying the following conditions:
\begin{equation}\label{eq:morphism}
\mbox{$I(p)\leq J(\varphi(p))$, $F(p)\leq G(\varphi(p))$ and $\delta(p,\sigma,q)\leq \eta(\varphi(p),\sigma,\varphi(q))$},
\end{equation}
for any $p,q\in Q$ and $\sigma\in\Sigma$.
A morphism $\varphi$ is \emph{ surjective} if $\varphi: Q\rightarrow P$ is onto and ${\cal B}=(P,\Sigma,\varphi(\delta),\varphi(I)$, $\varphi(F))$, where
\begin{eqnarray*}
\varphi(\delta)(p_1,\sigma,p_2) &=& \sum\{\delta(q_1,\sigma,q_2) | \varphi(p_1)=q_1, \varphi(p_2)=q_2\},\\
\varphi(I)(p) &=& \sum\{I(q)| \varphi(p)=q\},\\
\varphi(F)(p) &=& \sum\{F(q) | \varphi(p)=q\}.
\end{eqnarray*}
In this case, ${\cal B}$ is also called the \emph{morphic image} of ${\cal A}$.
If $\varphi:Q\rightarrow P$ is one-to-one, then we call ${\cal A}$ a \emph{sub-automaton} of ${\cal B}$.
A morphism $\varphi$ is called a \emph{strong homomorphism} if
\begin{eqnarray*}
J(p) &=& \sum\{I(q) | \varphi(q)=p\},\\
G(\varphi(q)) &=& F(q),\\
\eta(\varphi(q),\sigma,p) &=& \sum\{\delta(q,\sigma,r) | \varphi(r)=p\}.
\end{eqnarray*}
In case $\varphi$ is an onto strong homomorphism, we call ${\cal B}$ a \emph{quotient} of ${\cal A}$.
We say $\varphi$ is an \emph{isomorphism} if it is bijective and its inverse $\varphi^{-1}$ is also a morphism. In this case, we say ${\cal A}$ is isomorphic to ${\cal B}$.
\end{definition}
The behaviour of a weighted automaton is characterized by the formal power series it recognizes. To introduce this formal power series, we extend the weighted transition function $\delta: Q\times \Sigma \times Q \rightarrow S$ to a mapping $\delta^{\ast}: Q\times \Sigma^\ast \times Q\rightarrow S$ as follows:
\begin{itemize}
\item [(i)] For all $p\in Q$, set $\delta^{\ast}(q, \varepsilon, p)=1$ if $p=q$, and $\delta^{\ast}(q, \varepsilon, p)=0$ otherwise;
\item [(ii)] For all $\theta=\sigma_1\cdots \sigma_n\in\Sigma^{\ast}$, define
\begin{equation*}
\label{eq:ds}
\delta^{\ast}(q, \sigma_1\cdots \sigma_n, p)=\sum\{\delta(q,\sigma_1,q_1)\otimes\cdots\otimes\delta(q_{n-1},\sigma_n,p)| q_1,\cdots,q_{n-1}\in Q\}.
\end{equation*}
\end{itemize}
If ${\cal A}$ is deterministic, the extension $\delta^{\ast}$ of transition function $\delta$ is defined similar as in the classical case. It is easy to see that for any $\theta=\theta_1\theta_2\in \Sigma^{\ast}$ we have
\begin{equation}
\label{eq:theta1and2}
\delta^{\ast}(q, \theta_1\theta_2, p)=\sum_{r\in Q}[\delta^{\ast}(q, \theta_1, r)\otimes\delta^{\ast}(r, \theta_2, p)].
\end{equation}
\begin{definition}\label{dfn:recognized-fps}
For a weighted automaton ${\cal A}=(Q,\Sigma,\delta,I,F)$, the formal power series \emph{recognized} or \emph{accepted} by ${\cal A}$, written$|{\cal A}|: \Sigma^{\ast} \rightarrow S$, is defined as follows:
\begin{equation}
\label{eq:recognized-fps}
|{\cal A}|(\theta)=\sum\{I(p)\otimes\delta^{\ast}(p, \theta,q)\otimes F(q) | p, q\in Q\}. \hspace*{8mm} (\theta\in\Sigma^{\ast})
\end{equation}
If ${\cal A}$ is deterministic, then the formal power series recognized by $\mathcal{A}$ is defined as
\begin{equation}
\label{eq:recognized-fps-det}
|{\cal A}|(\theta)=F(\delta^{\ast}(q_0,\theta)). \hspace*{8mm} (\theta\in\Sigma^{\ast})
\end{equation}
We say a formal power series $A\in S\langle\!\langle\Sigma^{\ast}\rangle\!\rangle$ is a \emph{regular series} or an \emph{ $S$-regular language} on $\Sigma$ if it is recognized by a finite weighted automaton; and say $A$ is a \emph{DWA-regular series} or a \emph{DWA-regular language} on $\Sigma$ if it is recognized by a finite DWA.
\end{definition}
It was proved by Sch\"{u}tzenberger \cite{schutzenberger61} that regular series are precisely the rational formal power series for all semirings. So we also say a regular series as a rational series in this paper.
\subsection{Weighted Contex-Free Grammar}
We next recall the concept of weighted context-free grammar and weighted context-free series.
A \emph{weighted context-free grammar} is in essence a classical context-free grammar together with a function mapping rules of the grammar to weights in a certain semiring (\cite{droste09,esik10}). Let $S$ be a semiring. A weighted context-free grammar (WCFG) is defined as a tuple $G=(\Sigma,N,Z_0,S,P)$, where $\Sigma$ (the set of terminal symbols) and $N$ (the set of non-terminal symbols) are two finite sets that are disjoint, $Z_0$ (the start or initial symbol) is an element in $N$, $P$ is a mapping from $N \times (N \cup\Sigma)^{\ast}$ (the set of productions or rules) to $S$.
Similar to the classical case, we can define the induction of a weighted context-free grammar $G$. Suppose $Z\overset{r}{\rightarrow}\gamma$ is a weighted production, and $\alpha,\beta$ are elements in $(N\cup\Sigma)^{\ast}$. We say $\alpha\gamma\beta$ is a \emph{direct induction} of $\alpha Z\beta$ with weight $r$, denoted by $\alpha Z\beta\overset{r}{\Rightarrow}\alpha\gamma\beta$. For productions $\alpha_1,\cdots,\alpha_k$ in $(N\cup\Sigma)^{\ast}$, if $\alpha_1\overset{r_1}{\Rightarrow}\alpha_2, \cdots, \alpha_{k-1}\overset{r_{k-1}}{\Rightarrow}\alpha_k$, then we say $\alpha_k$ is an induction of $\alpha_1$ with weight $r=r_1\otimes\cdots\otimes r_{k-1}$, denoted by $\alpha_1\overset{r}{\Rightarrow}_{\ast}\alpha_k$.
The formal power series $|G|$ generated by $G$ is defined as $|G|(w)=\sum\{r | Z_0\overset{r}{\Rightarrow}_{\ast} w\}$ ($w\in\Sigma^{\ast}$). A series $A$ is called \emph{ context-free} if there is a weighted context-free grammar $G$ such that $A=|G|$.
\subsection{Operations of Formal Power Series}
We recall several well-know operations of formal power series (cf. \cite{li05}). For two formal power series $A,B\in S\langle\!\langle\Sigma^{\ast}\rangle\!\rangle$, a value $r\in S$, and $w\in \Sigma^\ast$, we define
\begin{eqnarray}
(A\oplus B)(w) &=& A(w)\oplus B(w),\\
(AB)(w) &=& \sum\{A(w_1)\otimes B(w_2) | w_1w_2=w\},\\
(rA)(w) &=& r\otimes A(w),\\
(Ar)(w) &=& A(w)\otimes r,\\
A^{\ast}(w) &=& \sum\{A(w_1)\otimes\cdots\otimes A(w_n)| n\geq 0, w_1\cdots w_n=w\},\\
\label{eq:A^R}
A^R(w) &=& A(w^R),
\end{eqnarray}
where $w^R=\sigma_n\cdots\sigma_2\sigma_1$ if $w=\sigma_1\sigma_2\cdots\sigma_n$.
We call $A\oplus B$ and $AB$ the \emph{sum} and, respectively, the \emph{concatenation} (or \emph{Cauchy product}) of $A$ and $B$, and call $rA$, $Ar$, $A^\ast$, and $A^R$ the \emph{left scalar product}, the \emph{right scalar product}, the \emph{Kleene closure}, and the \emph{reversal} of $A$, respectively.
Given a weighted automaton, ${\cal A}=(Q,\Sigma, \delta,I,F)$, three other operations can be defined for the formal power series recognized by $\mathcal{A}$. For any two states $p,q$ in $Q$, and any $\theta\in \Sigma^\ast$, we define
\begin{eqnarray}
Past_{{\cal A}}(q)(\theta) &=& \sum\{I(p)\otimes \delta^{\ast}(p,\theta,q)| p\in Q\}, \\
Fut_{{\cal A}}(q)(\theta) &=& \sum\{ \delta^{\ast}(q,\theta,p)\otimes F(p)| p\in Q\}, \\
Trans_{{\cal A}}(p,q)(\theta) &=& \delta^{\ast}(p,\theta,q).
\end{eqnarray}
The following result holds.
\begin{proposition}\label{pro:factor-automata}
Suppose ${\cal A}=(Q,\Sigma, \delta,I,F)$ is a weighted automaton. For any $q\in Q$, we have
\begin{equation}
Past_{{\cal A}}(q)Fut_{{\cal A}}(q)\leq |{\cal A}|.
\end{equation}
\end{proposition}
\begin{proof}
For any $\theta\in \Sigma^{\ast}$, we have
\begin{eqnarray*}
&& (Past_{{\cal A}}(q)Fut_{{\cal A}}(q))(\theta) \\
&=& \sum_{uv=\theta}Past_{{\cal A}}(q)(u)\otimes Fut_{{\cal A}}(q)(v)\\
&=& \sum_{uv=\theta}\sum_{q_0\in Q}I(q_0)\otimes \delta^{\ast}(q_0,u,q)\otimes \sum_{p\in Q}\delta^{\ast}(q,v,p)\otimes F(p)\\
&=& \sum_{uv=\theta}\sum_{q_0,p\in Q}I(q_0)\otimes\delta^{\ast}(q_0,u,q)\otimes\delta^{\ast}(q,v,p)\otimes F(p)\\
&\leq& \sum_{uv=\theta}\sum_{q_0,p,q\in Q}I(q_0)\otimes\delta^{\ast}(q_0,u,q)\otimes\delta^{\ast}(q,v,p)\otimes F(p)\\
&=& \sum_{uv=\theta}I(q_0)\otimes\delta^{\ast}(q_0,uv,p)\otimes F(p)\\
&=& |{\cal A}|(\theta).
\end{eqnarray*}
Hence, $Past_{{\cal A}}(q)Fut_{{\cal A}}(q)\leq |{\cal A}|$.
\end{proof}
\section{Quotients of Formal Power Series}
In this section, we first introduce quotients of formal power series and then, upon this operation, introduce for each formal power series $A$ a canonical weighted automaton that recognizes $A$. Properties of the quotient operation is also studied.
\subsection{Quotients and Minimal Weighted Automata}
Let $A: \Sigma^{\ast}\rightarrow S$ be a formal power series, $u\in\Sigma^{\ast}$ a word. The \emph{left quotient} of $A$ by $u$, written $u^{-1}A$, is the formal power series $u^{-1}A:\Sigma^{\ast}\rightarrow S$ defined as:
\begin{equation}
\label{eq: left-quotient}
u^{-1}A (v)=A(uv) \hspace{8mm} (v\in\Sigma^{\ast}).
\end{equation}
Dually, the \emph{right quotient} of $A$ by $u$, written $Au^{-1}$, is the formal power series $Au^{-1}: \Sigma^{\ast}\rightarrow S$ defined as:
\begin{equation}
\label{eq: left-quotient}
Au^{-1}(v)=A(vu) \hspace{8mm} (v\in\Sigma^{\ast}).
\end{equation}
for any $v\in\Sigma^{\ast}$.
The left quotient operation introduces an equivalent relation on $\Sigma^{\ast}$.
\begin{definition}\label{dfn:non-distinguishable-relation}
Suppose $A\in S\langle\!\langle\Sigma^{\ast}\rangle\!\rangle$. We say two words $u_1$ and $u_2$ are \emph{non-distinguishable} in $A$, written $u_1 \equiv_A u_2$, if $u_1^{-1} A=u_2^{-1}A$, i.e., if $A(u_1u)=A(u_2 u)$ for all $u\in\Sigma^{\ast}$.
\end{definition}
It is straightforward to show that $\equiv_A$ is an equivalent relation on $\Sigma^{\ast}$. For each word $u\in\Sigma^{\ast}$, we write $[u]_A$ for the equivalent class of $\equiv_A$ that contains $u$.
\begin{lemma}\label{le:mini DWA}
Suppose $A \in S\langle\!\langle\Sigma^{\ast}\rangle\!\rangle$. The mapping defined by $[u]\mapsto u^{-1}A$ $(u\in \Sigma^{\ast})$ is a bijection from the set of equivalent classes of $\equiv_A$ to the set of left quotients of $A$.
\end{lemma}
We define a deterministic weighted automaton ${\cal M}_A=(Q_A,\Sigma,\delta_A,I_A,F_A)$ as follows:
\begin{itemize}
\item $Q_A=\Sigma^{\ast}/\equiv_A$ is the quotient of $\Sigma^{\ast}$ modulo $\equiv_A$;
\item $\delta_A: Q_A\times \Sigma\rightarrow Q_A$ is defined as
\begin{equation}
\label{eq: delta-minimal-dwa}
\delta_A([\theta],\sigma)=[\theta\sigma], \hspace*{8mm} (\sigma\in\Sigma^{\ast})
\end{equation}
\item $I_A$ is the singleton state $[\varepsilon]$ in $Q_A$;
\item $F_A: Q_A\rightarrow S$ is defined by $F_A([\theta])=A(\theta)$ for $\theta\in \Sigma^{\ast}$.
\end{itemize}
\begin{proposition}\label{pro:minimal DWA}
Suppose $A\inS\langle\!\langle\Sigma^{\ast}\rangle\!\rangle$. Then ${\cal M}_A$ is the minimal DWA that recognizes $A$.
\end{proposition}
It is easy to see that $\delta_A$ is well-defined and, hence, ${\cal M}_A=(Q_A,\Sigma,\delta_A,I_A,F_A)$ is a DWA. It is straightforward to show that ${\cal M}_A$ recognizes $A$. In other words, there is a DWA ${\cal A}$ that accepts $A$ for any formal power series $A$. In general, ${\cal M}_A$ is not finite; but, when it is finite, ${\cal M}_A$ is the minimal DWA that recognizes $A$.
By Lemma \ref{le:mini DWA}, an equivalent minimal DWA
\begin{equation}
\label{eq:M-A'}
{\cal M}_A'=(Q_A',\Sigma,\delta_A',I_A',F_A')
\end{equation}
that recognizes $A$ can be constructed by using the left quotients of $A$ as follows
\begin{itemize}
\item $Q_A'=\{u^{-1}A | u\in\Sigma^{\ast}\}$, the set of all left quotients of $A$ by a word;
\item $\delta_A': Q_A'\times \Sigma\rightarrow Q_A'$ defined by
\begin{equation}
\label{eq:delta'}
\delta_A'(u^{-1}A,\sigma)=(u\sigma)^{-1}A
\end{equation}
\item $I_A'=A=\varepsilon^{-1}A$;
\item $F_A':Q_A'\rightarrow S$ is defined by $F_A'(u^{-1}A)=A(u)$ for $\theta\in \Sigma^{\ast}$.
\end{itemize}
The following proposition presents a characterization of formal power series that can be recognized by a finite DWA.
\begin{proposition}\label{pro:dffa}
Suppose $A\in S\langle\!\langle\Sigma^{\ast}\rangle\!\rangle$. Given $r\in S$, we write $Im(A)=\{A(\theta) | \theta\in\Sigma^{\ast}\}$, $A_r=\{ \theta\in\Sigma^{\ast} | A(\theta)\geq r\}$, and $A_{[r]}=\{ \theta\in\Sigma^{\ast} | A(\theta)=r\}$. Then the following statements are equivalent.
\begin{itemize}
\item [(i)] $A$ can be recognized by a finite DWA.
\item [(ii)] $Im(A)$ is finite and each $A_r$ is a regular language for $r\in Im(A)$.
\item [(iii)] $Im(A)$ is finite and each $A_{[r]}$ is a regular language for $r\in Im(A)$.
\item [(iv)] There exist $r_1,\cdots,r_k\in S\setminus\{0\}$ and $k$ regular languages $L_1,\cdots,L_k$ over $\Sigma$, which are pairwise disjoint, such that $A=\sum_{i=1}^k r_i L_i$.
\item [(v)] $\equiv_A$ has finite index, i.e., $Q_A$ is finite.
\end{itemize}
\end{proposition}
\begin{proof}
The proof is similar to that given for lattices in \cite{li05,li11}.
\end{proof}
But \emph{when can a regular series be recognized by a finite DWA}? This is closely related with the structure of semiring $S$. It can be shown that (cf.\cite{li05}), for any regular series $A$, $A$ can be recognized by a finite DWA iff the monoid $(S,\oplus,0)$ and $(S,\otimes,1)$ are both locally finite, where a monoid $(M,\times,1)$ is locally finite if every submonoid generated by a finite subset of $M$ is also finite. In particular, if $S$ is finite, regular series and DWA-regular series are the same.
\subsection{Properties of Quotients}
The left quotient of a series $A$ by a word $u$ can be regarded as a left action of $\Sigma^{\ast}$ on $S\langle\!\langle\Sigma^{\ast}\rangle\!\rangle$, i.e., a mapping $\Sigma^{\ast}\times S\langle\!\langle\Sigma^{\ast}\rangle\!\rangle\rightarrow S\langle\!\langle\Sigma^{\ast}\rangle\!\rangle$. The action satisfies the following conditions.
\begin{proposition}\label{pro:quotient-word}
Let $S$ be a semiring. Suppose $A\in S\langle\!\langle\Sigma^{\ast}\rangle\!\rangle$, $\varepsilon$ is the empty word in $\Sigma^{\ast}$, $u,v$ are words in $\Sigma^{\ast}$, and $k$ is a value in $S$. Then we have
\begin{itemize}
\item[(i)] $(uv)^{-1}A=v^{-1}(u^{-1}A)$, $A(uv)^{-1}=(Av^{-1})u^{-1}$, $\varepsilon^{-1}A=A=A\varepsilon^{-1}$;
\item[(ii)]$u^{-1}(A\oplus B)=u^{-1}A\oplus u^{-1}B$, $(A\oplus B)u^{-1}=Au^{-1}\oplus Bu^{-1}$;
\item [(iii)] $u^{-1}(kA)=k(u^{-1}A)$, $u^{-1}(Ak)=(u^{-1}A)k$, $(kA)u^{-1}=k(Au^{-1})$, $(Ak)u^{-1}$ $=(Au^{-1})k$.
\end{itemize}
\end{proposition}
\begin{proof}
Straightforward.
\end{proof}
When the semiring $S$ is complete, the left quotient operation can be extended from words to series in a natural way. Let $A, X$ be two formal power series in $S\langle\!\langle\Sigma^{\ast}\rangle\!\rangle$. We define the \emph{left quotient} of $A$ by $X$, denoted by $X^{-1}A$, as
\begin{equation}
\label{eq:X-1A}
X^{-1}A(v)=\sum_{u\in\Sigma^{\ast}}X(u) (u^{-1}A)(v)=\sum_{u\in\Sigma^{\ast}}X(u) A(uv). \hspace*{8mm} (v\in\Sigma^{\ast})
\end{equation}
Similarly, we can define the \emph{right quotient} of $A$ by $Y$ for any $Y\in S\langle\!\langle\Sigma^{\ast}\rangle\!\rangle$, denoted by $AY^{-1}$ as
\begin{equation}
\label{eq:AY-1}
AY^{-1}(v)=\sum_{u\in \Sigma^{\ast}}A(vu) Y(u). \hspace*{8mm} (v\in\Sigma^{\ast})
\end{equation}
It is easy to see that when the formal power series $X$ is a word $u$, then $X^{-1}A=u^{-1}A$ and $AX^{-1}=Au^{-1}$. We summarize some algebraic properties of the quotient operation.
\begin{proposition}\label{pro:quotient-fps}
Let $S$ be a complete semiring. Suppose $A,X_1,X_2 \in S\langle\!\langle\Sigma^{\ast}\rangle\!\rangle$. Then
\begin{itemize}
\item
[(i)] $(X_1\oplus X_2)^{-1}A=X_1^{-1}A\oplus X_2^{-1}A$, $A(Y_1\oplus Y_2)^{-1}=AY_1^{-1}\oplus AY_2^{-1}$;
\item
[(ii)] $X^{-1}(A_1\oplus A_2)=X^{-1}A_1\oplus X^{-1}A_2$, $(A_1\oplus A_2)Y^{-1}=A_1Y^{-1}\oplus A_2Y^{-1}$;
\item
[(iii)] $(X^{\ast})^{-1}A=A\oplus (X^{\ast})^{-1}(X^{-1}A)$, $A(Y^{\ast})^{-1}=A\oplus (AY^{-1})(Y^{\ast})^{-1}$;
\item[(iv)]$X^{-1}(AB)=(X^{-1}A)B\oplus (A^{-1}X)^{-1}B$, $(AB)Y^{-1}=A(BY^{-1})\oplus A(YB^{-1})^{-1}$;
\item [(v)] $(rX)^{-1}A=r(X^{-1}A)$, $X^{-1}(Ar)=(X^{-1}A)r$, $(rA)Y^{-1}=r(AY^{-1})$, $A(Yr)^{-1}=(AY^{-1})r$;
\item [(vi)] If $S$ is commutative, then $(X_1X_2)^{-1}A=X_2^{-1}(X_1^{-1}A)$, $A(Y_1Y_2)^{-1} = (AY_2^{-1})Y_1^{-1}$.
\end{itemize}
\end{proposition}
If the semiring $S$ is commutative, the following lemma shows that the right quotient is dual to the left quotient, where the reversal operation $X^R$ is defined (see Eq.~\ref{eq:A^R}) as $X^R(\sigma_1\sigma_2\cdots\sigma_n)=X(\sigma_n\cdots\sigma_2\sigma_1)$ for any $w=\sigma_1\sigma_2\cdots\sigma_n$.
\begin{lemma}
\label{lemma:right-quotient}
Suppose $S$ is a complete semiring. For $A,X, Y\inS\langle\!\langle\Sigma^{\ast}\rangle\!\rangle$, if $Y(u)A(vu)=A(vu)Y(u)$, $A(uv)X(u)=X(u)A(uv)$ for any $u,v\in\Sigma^{\ast}$, then
\begin{eqnarray}
AY^{-1} &=& ((Y^R)^{-1}A^R)^R\\
X^{-1}A &=& (A^R(X^{R})^{-1})^R.
\end{eqnarray}
In particular, if $A$ is a classical language, i.e., $Im(A)\subseteq \{0,1\}$ or if $S$ is commutative, the above equalities hold.
\end{lemma}
\begin{proof}
Straightforward.
\end{proof}
By the above result, if the semiring $S$ is commutative, then the right quotient is dual to the left quotient by the reversal operation. Properties of the right quotient can be dually obtained in this case.
\subsection{Closure Properties of Former Power Series under Quotient}
In this subsection, we study the language properties of the quotient of series. We first recall a preliminary result which will be used in the proof.
\begin{lemma} [cf. \cite{droste09}]\label{le:proper}
Suppose $Y$ is a proper and regular series in $S\langle\!\langle\Sigma^{\ast}\rangle\!\rangle$. Then there exists a weighted automaton ${\cal A}=(Q,\Sigma,\delta,q_0,\{q_f\})$ that recognizes $Y$ such that
$q_0\not= q_f$ and $\delta(q,\sigma,q_0)=0$ for any $q\in Q$.
\end{lemma}
\begin{proposition}\label{pro:quotient-closeness}
Let $S$ be a complete semiring. Suppose $A,X,Y$ are former power series in $S\langle\!\langle\Sigma^{\ast}\rangle\!\rangle$. Then
\begin{itemize}
\item [(i)] If $A$ is regular, then $X^{-1}A$ and $AY^{-1}$ are regular.
\item [(ii)] If $A$ is DWA-regular, then $X^{-1}A$ and $AY^{-1}$ are DWA-regular.
\item [(iii)] For a commutative semiring $S$, if $A$ is context-free, $X$ and $Y$ are regular, then $X^{-1}A$ and $AY^{-1}$ are context-free.
\item [(iv)] If $A$ is context-free, $X$ and $Y$ are DWA-regular, then $X^{-1}A$ and $AY^{-1}$ are context-free.
\end{itemize}
\end{proposition}
\begin{proof}
See Appendix A.
\end{proof}
Using the quotient of a series $A$ by a series, we have a related deterministic weighted automaton
\begin{equation}
\label{eq:B_A}
{\cal B}_A=(\{X^{-1}A | X\in S\langle\!\langle\Sigma^{\ast}\rangle\!\rangle\}, \Sigma, \delta,A,F),
\end{equation}
where $\delta(X^{-1}A,\sigma)=(X\sigma)^{-1}A$, and $F(X^{-1}A)=\sum_{u\in\Sigma^{\ast}}X(u)A(uv)$.
Because the accessible part of ${\cal B}_A$ is the minimal DWA ${\cal M}_A'$ (cf. (\ref{eq:M-A'})), hence $|{\cal B}_A|=A$.
Recall in the classical case, if $A$ is a regular language over $\Sigma$, then $\{X^{-1}A | X\subseteq \Sigma^{\ast}\}$ (the set of all left quotients of $A$ by languages) is a finite set. This does not hold in general. In fact, the finiteness of $Q=\{X^{-1}A | X\subseteq S\langle\!\langle\Sigma^{\ast}\rangle\!\rangle\}$ heavily depends on the finiteness of the semiring $S$.
\begin{proposition}\label{pro:quotient automaton}
Let $S$ be a semiring. Then $S$ is finite if and only if ${\cal B}_A$ is finite for any DWA-regular series $A$ in $S\langle\!\langle\Sigma^{\ast}\rangle\!\rangle$, where $\mathcal{B}_A$ is the weighted automaton that recognizes $A$ defined in (\ref{eq:B_A}).
\end{proposition}
\begin{proof}
Suppose ${\cal B}_A$ is finite for any DWA-regular series $A$. In particular, ${\cal B}_A$ is finite for $A=\Sigma^{\ast}$. Take $a\in \Sigma$, for any $r\in S$, let $X_r=ra$ (which is a series over $\Sigma$). By a simple calculation, the left quotient $X_r^{-1}A$ is $\sum_{w\in\Sigma^{\ast}}rw$, i.e., $X_r^{-1}A=\overline{r}$, where $\overline{r}(w)=r$ for any $w\in\Sigma^{\ast}$. Then, as a subset of $\{X^{-1}A | X\inS\langle\!\langle\Sigma^{\ast}\rangle\!\rangle\}$, the set $\{\overline{r} | r\in S\}$ is finite. Hence, $S$ is a finite set.
Conversely, suppose $S=\{r_1,\cdots,r_k\}$ is finite. For any $X\inS\langle\!\langle\Sigma^{\ast}\rangle\!\rangle$, let $X_i=\{w\in\Sigma^{\ast} | X(w)=r_i\}$. Then $X=\sum_{i=1}^k r_iX_i$. Since $A$ is regular, there exist finite regular languages $L_1,\cdots,L_k$ such that $A=\sum_{i=1}^k r_iL_i$. By Proposition \ref{pro:quotient-fps}, it follows that $X^{-1}A=\sum_{i=1}^k\sum_{j=1}^k(r_i\otimes r_j)X_i^{-1}L_j$. Since $L_j$ is regular, the set $\{X_i^{-1}L_j |X\subseteq \Sigma^{\ast}\}$ is finite. It follows that the set $\{X^{-1}A |X\in S\langle\!\langle\Sigma^{\ast}\rangle\!\rangle\}$ is also finite.
\end{proof}
\par
In the remainder of this paper, we will consider the residual operations on formal power series.
\section{Residuals and Factorizations}
In this section, we study the residuals and factorizations of formal power series. For $A,X\in S\langle\!\langle\Sigma^{\ast}\rangle\!\rangle$, the \emph{left residual} of $A$ by $X$, written as $X\backslash A$, is defined as the largest $Y\inS\langle\!\langle\Sigma^{\ast}\rangle\!\rangle$ such that $XY\leq A$. Similarly, we define the \emph{right residual} of $A$ by $X$, written as $A/X$, as the largest $Y\inS\langle\!\langle\Sigma^{\ast}\rangle\!\rangle$ such that $YX\leq A$. We note that $X\backslash A$ and $A/X$ need not exist for all $A,X$.
\subsection{Residuals of Formal Power Series}
The following proposition shows that residuals always exist when $S$ is a complete c-semiring. We recall that each c-semiring is commutative.
\begin{proposition}
Suppose $S$ is a complete c-semiring. For any $A,X,Y\inS\langle\!\langle\Sigma^{\ast}\rangle\!\rangle$, $X\backslash A$ and $A/Y$ do exist and we have
\begin{eqnarray}
X\backslash A &=& \sum\{Z | XZ\leq A\},\\
A/Y &=& \sum\{Z | ZY\leq A\}.
\end{eqnarray}
\end{proposition}
\begin{proof}
Because $S$ is a complete c-semiring, we know $\sum_{i\in I}Z_i$ exists for any subset $\{Z_i | i\in I\}$ of $S\langle\!\langle\Sigma^{\ast}\rangle\!\rangle$. Moreover, the concatenation operation satisfies the following conditions:
\begin{eqnarray}
X(\sum_{i\in I}Y_i) &=& \sum_{i\in I} XY_i,\\
(\sum_{i\in I}Y_i)X &=& \sum_{i\in I} Y_iX.
\end{eqnarray}
That is to say, $S\langle\!\langle\Sigma^{\ast}\rangle\!\rangle$ is a quantale under the operations $\sum$ and concatenation. It is then easy to see that $X (\sum\{Y | XY\leq A\}) \leq A$ and hence $\sum\{Y | XY\leq A\}$ is the largest $Z$ such that $XZ\leq A$, i.e., $A\backslash Y= \sum\{Y | XY\leq A\}$. Similarly, we have $A/Y=\sum\{X | XY\leq A\}$.
\end{proof}
The following proposition shows that left residual and right residual are dual to each other.
\begin{proposition}
\label{prop:dual-residuals}
Let $S$ be a complete c-semiring. Suppose $A,X,Y\inS\langle\!\langle\Sigma^{\ast}\rangle\!\rangle$. Then we have
$(X\backslash A)^R=A^R/X^R$ and $(A/Y)^R$ $=Y^R\backslash A^R$.
\end{proposition}
\begin{proof}
Straightforward.
\end{proof}
By the above result, in the following, we often consider only one (either left or right) residual. Dual results can be applied to the other residual.
Left residual has close connection with left quotient.
\begin{proposition}\label{pro:quotient-word}
Let $S$ be a complete c-semiring. Suppose $A$ is a formal power series in $S\langle\!\langle\Sigma^{\ast}\rangle\!\rangle$, $u$ is a word in $\Sigma^{\ast}$. Then $u^{-1}A$ is the largest language $Y$ such that $uY\leq A$, i.e., $u\backslash A = u^{-1}A$.
\end{proposition}
\begin{proof}
Note that $(u(u^{-1}A))(\theta)=u^{-1}A(v)=A(uv)$ if $\theta=uv$ and $0$ otherwise. It follows that $u(u^{-1}A)\leq A$. If $uY\leq A$, then $Y(v)=(uY)(uv)\leq A(uv)=(u^{-1}A)(v)$ for any $v\in\Sigma^{\ast}$. Hence, $Y\leq u^{-1}A$. Therefore $u^{-1}A$ is the largest $Y$ such that $uY\leq A$.
\end{proof}
In the classical case, given two languages $A,X$ over $\Sigma$, we always have
\begin{eqnarray}
X\backslash A &=& \bigcap \{u^{-1}A | u\in X\}\\
A/X &=& \bigcap\{Au^{-1} | u\in X\}.
\end{eqnarray}
In the weighted case, residuals are usually not the intersection of a set of quotients by word. But we can still represent residuals in terms of quotients by word. Before give the details, we summarize some basic algebraic properties of the residuals.
\begin{proposition}\label{pro:residual quotient}
Let $S$ be a complete c-semiring. For $A,X,Y\inS\langle\!\langle\Sigma^{\ast}\rangle\!\rangle$, we have
\begin{itemize}
\item
[(i)] $X(X\backslash A)\leq A$, $(A/Y)Y\leq A$.
\item [(ii)] The operations $cl,cr: S\langle\!\langle\Sigma^{\ast}\rangle\!\rangle\rightarrow S\langle\!\langle\Sigma^{\ast}\rangle\!\rangle$ defined by $cl(X)=A/(X\backslash A)$ and $cr(Y)=(A/Y)\backslash A$ are two closure operators, where $\tau: S\langle\!\langle\Sigma^{\ast}\rangle\!\rangle \rightarrow S\langle\!\langle\Sigma^{\ast}\rangle\!\rangle$ is a closure operator if $X\leq \tau(X)$, $\tau(X_1\vee X_2)=\tau(X_1)\vee \tau(X_2)$ and $\tau(\tau(X))=\tau(X)$.
\item [(iii)] $(XY)\backslash A=Y\backslash(X\backslash A)$, $A/(YX)=(A/X)/Y$.
\item [(iv)] $(X_1\oplus X_2)\backslash A=X_1\backslash A \wedge X_2\backslash A$, $A/(Y_1\oplus Y_2)=A/Y_1\wedge A/Y_2$.
\item [(v)] $X\backslash (A\wedge B)=(X\backslash A)\wedge (Y\backslash B)$.
\item [(vi)] $(A/(X\backslash A))\backslash A=X\backslash A$, $A/((A/Y)\backslash A)=A/Y$.
\end{itemize}
\end{proposition}
\begin{proof}
We give the proof of (vi), the others are straightforward. By (ii), $X\leq A/(X\backslash A)$, then it follows that $(A/(X\backslash A))\backslash A\leq X\backslash A$. On the other hand, since $A/(X\backslash A) (X\backslash A)\leq A$, it follows that $X\backslash A\leq (A/(X\backslash A))\backslash A$. Hence, $(A/(X\backslash A))\backslash A=X\backslash A$. Similarly, we have $A/((A/Y)\backslash A)=A/Y$.
\end{proof}
We next give the characterization of residuals in terms of quotients by word, where $\rightarrow$ is the residual operation in the quantale $S$ defined (cf. (\ref{a-rw-b})).
\begin{proposition}\label{pro:characterization of residual quotient}
Let $S$ be a complete c-semiring.
Suppose $X,Y,A\in S\langle\!\langle\Sigma^{\ast}\rangle\!\rangle$. Then, for any $v\in \Sigma^{\ast}$, we have
\begin{eqnarray}
\label{eq:X|A}
X\backslash A(v) &=& \bigwedge\{X(u)\rightarrow A(uv) | u\in \Sigma^{\ast}\}\\
\label{eq:A|Y}
A/Y(v) &=& \bigwedge\{Y(u)\rightarrow A(vu) | u\in \Sigma^{\ast}\}.
\end{eqnarray}
\end{proposition}
\begin{proof}
By Proposition~\ref{prop:dual-residuals}, we need only prove Eq.(\ref{eq:X|A}).
For $v\in\Sigma^{\ast}$, let $\overline{Y}(v)=\bigwedge\{X(u)\rightarrow A(uv) | u\in \Sigma^{\ast}\}$. Then $\overline{Y}$ is a series. We show $\overline{Y}=X\backslash A$, i.e., $\overline{Y}$ is the largest series $Y$ such that $XY\leq A$.
First,
\begin{eqnarray*}
X\overline{Y}(w) &=& \sum_{uv=w}X(u)\overline{Y}(v) \\
&\leq& \sum_{uv=w}X(u)(X(u)\rightarrow A(uv)) \\
&\leq& \sum_{uv=w}A(uv)=A(w).
\end{eqnarray*}
Second, if $XY\leq A$, then, for any $uv=w$, $X(u)Y(v)\leq A(uv)$. It follows that $Y(v)\leq X(u)\rightarrow A(uv)$ for any $u\in\Sigma^{\ast}$, thus, $Y(v)\leq \bigwedge\{X(u)\rightarrow A(uv) | u\in\Sigma^{\ast}\}=\overline{Y}(v)$, i.e., $Y\leq \overline{Y}$. This shows that $X\backslash A=\overline{Y}$.
\end{proof}
The following proposition shows that DWA-regular series are closed under residual operations.
\begin{proposition}\label{pro:implication-language}
Suppose $S$ is a complete c-semiring,
and $A,X,Y\in S\langle\!\langle\Sigma^{\ast}\rangle\!\rangle$. If $A$ is DWA-regular, then so are $X\backslash A$ and $A/Y$.
\end{proposition}
\begin{proof}
By Proposition~\ref{prop:dual-residuals}, we need only consider right residuals. Suppose that ${\cal A}=(Q,\Sigma,\delta,q_0,F)$ is a DWA accepting $A$. Then we have
$|{\cal A}|(\theta)=F(\delta^{\ast}(q_0,\theta))$ for any $\theta\in\Sigma^{\ast}$.
Define another weighted automaton, ${\cal A}^Y=(Q,\Sigma,\delta,q_0,F^Y)$, where
$$F^Y(q)=\bigwedge\{Y(u)\rightarrow F(\delta^{\ast}(q,u)) | u\in \Sigma^{\ast}\}.$$
Then
\begin{eqnarray*}
|{\cal A}^Y|(\theta) &=& F^Y(\delta^{\ast}(q_0,\theta))\\
&=& \bigwedge\{Y(u)\rightarrow F(\delta^{\ast}(\delta^{\ast}(q_0,\theta),u) | u \in \Sigma^{\ast}\}\\
&=& \bigwedge\{Y(u)\rightarrow F(\delta^{\ast}(q_0,\theta u) | u \in \Sigma^{\ast}\} \\
&=& \bigwedge\{Y(u)\rightarrow A(\theta u) | u \in \Sigma^{\ast}\} \\
&=& A/Y(\theta).
\end{eqnarray*}
Hence, $A/Y$ is DWA-regular.
\end{proof}
\subsection{Factorizations of Formal Power Series}
In formal language theory, factorization is an important notion that is closely related to quotients and residuals \cite{lombardy08}. This notion can be generalized to formal power series straightforwardly.
\begin{definition} \label{de:factor}
For any $X,Y\in S\langle\!\langle\Sigma^{\ast}\rangle\!\rangle$, if $XY\leq A$, then we call $(X,Y)$ a sub-factorization of $A$. Furthermore, if the sub-factorization $(X,Y)$ is maximal,
i.e., if $X\leq X^{\prime}$, $Y\leq Y^{\prime}$ and $X^{\prime}Y^{\prime}\leq A$, then $X=X^{\prime}$ and $Y=Y^{\prime}$. In this case we call $(X,Y)$ a factorization
of $A$, and write $R_A$ for the set of all factorizations of $A$.
\end{definition}
The relationship between residuals and factorizations of a series is as follows.
\begin{proposition}\label{pro:factor-language}
Let $S$ be a complete c-semiring. For $A,X,Y\in S\langle\!\langle\Sigma^{\ast}\rangle\!\rangle$, we have
\begin{itemize}
\item
[(i)] $(X,Y)\in R_A$ if and only if $X=A/Y$ and $Y=X\backslash A$.
\item [(ii)] If $WZ\leq A$, then there exists $(X,Y)\in R_A$ such that $W\leq X$ and $Z\leq Y$.
\end{itemize}
\end{proposition}
\begin{proof}
(i) Suppose $(X,Y)\in R_A$ we show $X=A/Y$ and $Y=X\backslash A$. For any $u,v\in\Sigma^{\ast}$, we note that $X(u)\otimes Y(v)\leq A(uv)$ if and only if $Y(v)\leq X(u)\rightarrow A(uv)$. Hence, $Y(v)\leq \bigwedge_{u\in \Sigma^{\ast}}X(u)\rightarrow A(uv)=X\backslash A(v)$. This shows that $Y\leq X\backslash A$. Conversely, by $X(X\backslash A)(\theta)=\bigvee_{uv=\theta}X(u)\otimes X\backslash A(v)\leq A(\theta)$ and the maximality of the factorization, we know $X\backslash A\leq Y$. Hence, $Y=X\backslash A$. Similarly, we can show $X=A/Y$.
On the other hand, suppose $X=A/Y$ and $Y=X \backslash A$. By definition, we know $Y$ is the largest $Z$ such that $XZ\leq A$, and $X$ is the largest $W$ such that $WY\leq A$. This shows that $(X,Y)$ is a factorization of $A$.
(ii) Let $X=A/Z$, $Y=(A/Z)\backslash A$. It is clear that $W\leq X$. By Prop.~\ref{pro:residual quotient} (ii) and (vi), we know $Z\leq (A/Z)\backslash A$ and $X=A/ [(A/Z)\backslash A] = A/Y$. Therefore $(X,Y)\in R_A$ and $W\leq X$ and $Z\leq Y$.
\end{proof}
By the above proposition, a factorization of a formal power series $A$ is just a pair $(X,Y)$ such that $X$ is the right residual of $A$ by $Y$ and $X$ is the left residual of $A$ by $X$. In this case, we also call $X$ the \emph{ left factor} of $A$ by $Y$ and $Y$ the \emph{ right factor} of $A$ by $X$, respectively. Since $\varepsilon A=A\varepsilon=A$, by Proposition~\ref{pro:factor-language} (ii) $A$ itself is both a left factor and a right factor. We denote the corresponding right factor and left factor by $X_s$ and
$Y_e$, respectively, where $X_s(\varepsilon)=Y_e(\varepsilon)=1$, and call $(X_s,A)$ and $(A,Y_e)$ the \emph{initial} and \emph{final} factorization, respectively.
We write $lR(A)$ ($rR(A)$) for the set of left (right) residuals of $A$, i.e.,
\begin{eqnarray*}
lR(A) &=& \{X\backslash A | X\in S\langle\!\langle\Sigma^{\ast}\rangle\!\rangle\},\\
rR(A) &=& \{A/Y | Y\in S\langle\!\langle\Sigma^{\ast}\rangle\!\rangle\}.
\end{eqnarray*}
Let $\varphi:lR(A)\rightarrow rR(A)$ be the mapping defined as $\varphi(X\backslash A)=A/(X\backslash A)$. By Proposition \ref{pro:residual quotient}, it is easy to see that $\varphi$ is a bijection.
Moreover, we have
$$R_A=\{(X,\varphi(X)) | X\in lR(A)\}=\{(\varphi^{-1}(Y),Y) | Y\in rR(A)\}.$$
\section{The Universal Weighted Automaton}
In this section, we use the residuals of a formal power series $A$ to construct a weighted automaton which recognizes $A$. To this end, we introduce the notion of the inclusion degree.
\begin{definition}\label{def:inclusion regree}
Suppose $S$ is a complete c-semiring. For two $S$-subsets $f,g:U \rightarrow S$, the \emph{inclusion degree} of $f$ into $g$, denoted by
$f\rightarrow_{incl} g\in S$, is defined as
\begin{equation}
f\rightarrow_{incl} g=\bigwedge\{f(u)\rightarrow g(u) | u\in U\}.
\end{equation}
\end{definition}
The following lemma summarizes several useful properties of the inclusion degree operator.
\begin{lemma}\label{le:inclusion}
Suppose $S$ is a complete c-semiring. For $X,X^\prime\in S\langle\!\langle\Sigma^{\ast}\rangle\!\rangle$, any $c\in S$, and any $w\in \Sigma^{\ast}$, we have
\begin{itemize}
\item
[(1)] $c\leq X\rightarrow_{incl} X^{\prime}$ iff $cX\leq X^{\prime}$.
\item [(2)] If $(X,Y),(X^{\prime}, Y^{\prime})\in R_A$, then $X\rightarrow_{incl} X^{\prime}=Y^{\prime}\rightarrow_{incl} Y$.
\item [(3)] $X\rightarrow_{incl} X^{\prime}=Xw\rightarrow_{incl} X^{\prime}w=wX\rightarrow_{incl} wX^{\prime}$.
\end{itemize}
\end{lemma}
\begin{definition}\label{def:universal automaton}
Suppose $S$ is a complete c-semiring. For $A\inS\langle\!\langle\Sigma^{\ast}\rangle\!\rangle$, the \emph{universal weighted automaton} of $A$, denoted by ${\cal U}_A$, is a weighted automaton
$(R_A,\Sigma, \eta_A$, $J_A, G_A)$, where
\begin{eqnarray}
J_A(X,Y) &=& X(\varepsilon),\\
G_A(X,Y) &=& Y(\varepsilon),\\
\eta_A((X,Y),\sigma,(X^{\prime}, Y^{\prime})) &=& X\sigma Y^{\prime}\rightarrow_{incl} A,
\end{eqnarray}
for any $(X,Y),(X^{\prime}, Y^{\prime})\in R_A$, $\sigma\in \Sigma$.
\end{definition}
\begin{proposition}\label{pro:ufa1} Suppose $S$ is a complete c-semiring. For $A\in S\langle\!\langle\Sigma^{\ast}\rangle\!\rangle$, and $(X,Y), (X',Y')\in R_A$ and $\sigma\in \Sigma$, we have
\begin{eqnarray}
X(\varepsilon) &=&Y\rightarrow_{incl} A,\\
Y(\varepsilon) &=& X\rightarrow_{incl} A,\\
X\sigma Y^{\prime}\rightarrow_{incl} A &=& X\sigma\rightarrow_{incl} X^{\prime}=\sigma Y^{\prime}\rightarrow_{incl} Y.
\end{eqnarray}
\end{proposition}
\begin{proof}
Since $(X,Y)\in R_A$,
\begin{eqnarray*}
X(\varepsilon)=A/Y(\varepsilon) &=& \bigwedge\{Y(v)\rightarrow A(\varepsilon v) | v\in \Sigma^{\ast}\}\\
&=& \bigwedge\{Y(v)\rightarrow A(v) | v\in \Sigma^{\ast}\}=Y\rightarrow_{incl} A.
\end{eqnarray*}
Similarly, we can prove the case for $G_A(X,Y)$.
As for $\eta_A$, it is sufficient to show that $c\leq
X\sigma Y^{\prime}\rightarrow_{incl} A$ iff $c\leq X\sigma\rightarrow_{incl} X^{\prime}$ for any $c\in S$. This is because, $c\leq X\sigma Y^{\prime}\rightarrow_{incl} A$ iff
$cX\sigma Y^{\prime}\leq A$, iff $(cX\sigma) Y^{\prime}\leq A$, iff
$cX\sigma\leq X'$, iff $c\leq X\sigma\rightarrow_{incl} X^{\prime}$. Hence,
$X\sigma Y^{\prime}\rightarrow_{incl} A=X\sigma\rightarrow_{incl} X^{\prime}$.
Similarly, we have $X\sigma Y^{\prime}\rightarrow_{incl} A=\sigma Y^{\prime}\rightarrow_{incl} Y$.
\end{proof}
The extension of $\eta$ has the following form.
\begin{proposition}\label{pro:extension}
Let $S$ be a complete c-semiring and suppose $A\inS\langle\!\langle\Sigma^{\ast}\rangle\!\rangle$.
For $(X,Y),(X^{\prime}$, $Y^{\prime})\in R_A$, and any $w\in \Sigma^{+}$, we have
\begin{equation}\label{eq:extension of eta}
\eta_A^{\ast}((X,Y),w,(X^{\prime}, Y^{\prime}))= XwY^{\prime}\rightarrow_{incl} A=Xw\rightarrow_{incl} X^{\prime}=wY^{\prime}\rightarrow_{incl} Y.
\end{equation}
\end{proposition}
\begin{proof} First, for any $c\in S$, $c\leq XwY^{\prime}\rightarrow_{incl} A$ iff $cXwY^{\prime}\leq A$, iff $(cXw)Y^{\prime}\leq A$, iff $cXw\leq X^{\prime}$, iff $c\leq Xw\rightarrow_{incl} X^{\prime}$.
Hence, $XwY'\rightarrow_{incl} A=Xw\rightarrow_{incl} X^{\prime}$.
Similarly, we have $XwY^{\prime}\rightarrow_{incl} A=wY^{\prime}\rightarrow_{incl} Y$.
We shall show $\eta_A^{\ast}((X,Y),w,(X^{\prime}, Y^{\prime}))=
XwY^{\prime}\rightarrow_{incl} A$ by induction on the length $|w|$ for $w\in \Sigma^{+}$.
If $|w|=1$, this is just the definition of $\eta_A$.
Given $w\in \Sigma^{+}$ and $\sigma\in\Sigma$, we show that (\ref{eq:extension of eta}) holds for $\sigma w$ if (\ref{eq:extension of eta}) holds for $w$.
\begin{eqnarray*}
&&\eta_A^{\ast}((X,Y),\sigma w,(X^{\prime}, Y^{\prime}))\\
&=& \bigvee_{(X'',Y'')\in R_A}\eta_A((X,Y),\sigma,(X'', Y''))\otimes \eta_A^{\ast}((X'',Y''),w,(X^{\prime}, Y^{\prime}))\\
&=& \bigvee_{(X'',Y'')\in R_A}(X\sigma Y''\rightarrow_{incl} A)\otimes (X''wY'\rightarrow_{incl} A)\\
&=& \bigvee_{(X'',Y'')\in R_A}(X\sigma\rightarrow_{incl} X'')\otimes (X''w\rightarrow_{incl} X')\\
&=& \bigvee_{(X'',Y'')\in R_A}(X\sigma w\rightarrow_{incl} X''w)\otimes (X''w\rightarrow_{incl} X')\\
&\leq& \bigwedge_{\theta\in\Sigma^{\ast}}(X\sigma w(\theta)\rightarrow X''w(\theta))\otimes(X''w(\theta)\rightarrow X'(\theta)) \\
&\leq& \bigwedge_{\theta\in\Sigma^{\ast}}(X\sigma w(\theta)\rightarrow X'(\theta))\\
&=& X\sigma w\rightarrow_{incl} X'.
\end{eqnarray*}
Conversely,
\begin{eqnarray*}
&& c\leq X\sigma w\rightarrow_{incl} X' = X\sigma w Y'\rightarrow_{incl} A \\
& \Longleftrightarrow& cX\sigma w Y'\leq A \\
& \Longleftrightarrow & (cX\sigma)(wY')\leq A \\
& \Longleftrightarrow & \mbox{there exists $(X'',Y'')\in R_A$ such that $cX\sigma\leq X''$, $wY'\leq Y''$} \\
& \Longleftrightarrow& \mbox{there exists $(X'',Y'')\in R_A$ such that $c\leq X\sigma\rightarrow_{incl} X''$, $1\leq wY'\rightarrow_{incl} Y''$},
\end{eqnarray*}
which implies
\begin{eqnarray*}
c &\leq& (X\sigma\rightarrow_{incl} X'')\otimes (wY'\rightarrow_{incl} Y'')\\
&=& (X\sigma Y''\rightarrow_{incl} A)\otimes (X''wY'\rightarrow_{incl} A) \\
&\leq& \eta_A^{\ast}((X,Y),\sigma w,(X^{\prime}, Y^{\prime})).
\end{eqnarray*}
Hence, $X\sigma w\rightarrow_{incl} X'\leq \eta_A^{\ast}((X,Y),\sigma w,(X^{\prime}, Y^{\prime}))$.
Therefore, the equality (\ref{eq:extension of eta}) holds.
\end{proof}
We give some properties of universal weighted automaton as follows.
\begin{proposition}\label{pro:future-past}
Let $S$ be a complete c-semiring and suppose $A\inS\langle\!\langle\Sigma^{\ast}\rangle\!\rangle$.
For $(X,Y)\in R_A$, we have $Past_{{\cal U}_A}(X,Y)=X$, $Fut_{{\cal U}_A}(X,Y)=Y$.
\end{proposition}
\begin{proof} For any $\theta\in\Sigma^{\ast}$,
\begin{eqnarray*}
Past_{{\cal U}_A}(X,Y)(\theta) &=& \bigvee_{(X',Y')\in R_A}J_A(X',Y')\otimes\eta^{\ast}((X',Y'),\theta,(X,Y))\\
&=& \bigvee_{(X',Y')\in R_A}X'(\varepsilon)\otimes (X'\theta\rightarrow_{incl} X)\\
&=& \bigvee_{(X',Y')\in R_A}X'(\varepsilon)\otimes \bigwedge_{u\in\Sigma^{\ast}}(X'\theta(u)\rightarrow X(u)) \\
&\leq& \bigvee_{(X',Y')\in R_A}X'(\varepsilon)\otimes (X'\theta(\theta)\rightarrow X(\theta))\\
&=& \bigvee_{(X',Y')\in R_A}X'(\varepsilon)\otimes (X'(\varepsilon)\rightarrow X(\theta)) \\
&\leq& X(\theta).
\end{eqnarray*}
On the other hand,
\begin{eqnarray*}
Past_{{\cal U}_A}(X,Y)(\theta) &=& \bigvee_{(X',Y')\in R_A}J_A(X',Y')\otimes\eta^{\ast} ((X',Y'),\theta,(X,Y)) \\
&\geq& J_A(X_s,A)\otimes \eta^{\ast}((X_s,A),\theta,(X,Y)) \\
&=& X_s(\varepsilon)\otimes (X_s\theta\rightarrow_{incl} X)\\
&=& X_s\theta\rightarrow_{incl} X=\bigwedge_{u\in \Sigma^{\ast}}X_s\theta(u)\rightarrow X(u)\\
&=& \bigwedge_{u\not=\theta}X_s\theta(u)\rightarrow X(u)\wedge (X_s\theta(\theta)\rightarrow X(\theta))\\
&=& \bigwedge_{u\not=\theta}(0\rightarrow X(u))\wedge (1\rightarrow X(\theta))\\
&=& X(\theta).
\end{eqnarray*}
Hence, $Past_{{\cal U}_A}(X,Y)=X$.
Similarly, we have $Fut_{{\cal U}_A}(X,Y)=Y$.
\end{proof}
\begin{theorem}\label{th:universal 1}
Let $S$ be a complete c-semiring and suppose $A\inS\langle\!\langle\Sigma^{\ast}\rangle\!\rangle$. Then we have $|{\cal U}_A|=A$.
\end{theorem}
\begin{proof}
For any $\theta\in\Sigma^{\ast}$,
\begin{eqnarray*}
|{\cal U}_A|(\theta) &=& \bigvee_{(X,Y)\in R_A}J_A(X,Y)\otimes Fut_{{\cal U}_A}(X,Y)(\theta) \\
&=& \bigvee_{(X,Y)\in R_A}X(\varepsilon)\otimes Y(\theta)\leq A(\theta).
\end{eqnarray*}
On the other hand,
\begin{eqnarray*}
|{\cal U}_A|(\theta) &=& \bigvee_{(X,Y)\in R_A}J_A(X,Y)\otimes Fut_{{\cal U}_A}(X,Y)(\theta)\\
& \geq & J_A(X_s,A)\otimes Fut_{{\cal U}_A}(X_s,A)(\theta) \\
&=& X_s(\varepsilon)\otimes A(\theta)=A(\theta).
\end{eqnarray*}
Therefore, $|{\cal U}_A|=A$.
\end{proof}
So far, we have defined two canonical weighted automata for each $A\inS\langle\!\langle\Sigma^{\ast}\rangle\!\rangle$, viz. the minimal DWA ${\cal M}_A$ and the universal weighted automaton ${\cal U}_A$. Both automata recognize $A$. Recall in the classical case, if $A$ is a regular language over $\Sigma$, then ${\cal M}_A$ and ${\cal U}_A$ are both finite automata. This does not hold in general for weighted automata. In the following we discuss when ${\cal U}_A$ is finite.
For $A\inS\langle\!\langle\Sigma^{\ast}\rangle\!\rangle$,
write
\begin{equation}\label{eq:S-A}
S_A=\{c\rightarrow a | c\in S, a\in Im(A)\},
\end{equation}
and let
\begin{equation}\label{eq:S-A^}
S_A^{\wedge}=\{\bigwedge X | X\subseteq S_A\}
\end{equation}
\noindent be the $\bigwedge$-sublattice of $S$ generated by $S_A$. Then it is well-known that $S_A^{\wedge}$ is finite iff $S_A$ is finite (cf. \cite{li93,davey02}).
The following proposition shows the relationship between the finiteness of ${\cal U}_A$ and the finiteness of ${\cal M}_A$.
\begin{proposition}\label{pro:finiteness}
Let $S$ be a complete c-semiring. For $A\inS\langle\!\langle\Sigma^{\ast}\rangle\!\rangle$, the following conditions are equivalent.
\begin{itemize}
\item
[(i)] ${\cal U}_A$ is finite, i.e., $R_A$ is finite.
\item
[(ii)] $A$ can be accepted by a finite DWA, and $S_A$ in Eq. (\ref{eq:S-A}) is finite.
\item
[(iii)] ${\cal M}_A$ is finite, i.e., $\equiv_A$ has finite index, and $S_A$ is finite.
\end{itemize}
In particular, if $S$ is finite c-semiring or a linear order lattice, the above conditions are equivalent, and the condition ``$S_A$ is finite'' can be omitted.
\end{proposition}
\begin{proof}
By Proposition \ref{pro:dffa}, $A$ is recognized by a finite DWA iff $\equiv_A$ has finite index. It remains to show that $A$ has finite factorizations iff $S_A$ is finite.
Assume that $A$ can be recognized by a finite DWA and $S_A$ is finite. This implies that $\equiv_A$ has finite index, i.e., $Q_A$ is a finite set. If $u_1\equiv_A u_2$, then $A(u_1v)=A(u_2v)$ for any $v\in\Sigma^{\ast}$.
It follows that
\begin{eqnarray*}
X(u_1)=A/Y(u_1) &=& \bigwedge\{Y(v)\rightarrow A(u_1v) | v\in\Sigma^{\ast}\} \\
&=& \bigwedge\{Y(v)\rightarrow A(u_2v) | v\in\Sigma^{\ast}\} \\
&=& A/Y(u_2)=X(u_2).
\end{eqnarray*}
Thus $X$ induces a unique mapping from $Q_A=\Sigma^{\ast}/\equiv_A$ into $S_A^{\wedge}$. Since $Q_A$ is finite and $S_A$ is finite, the latter implies that the set $S_A^{\wedge}$ is also finite. Then the set of the mappings from $Q_A$ into $S_A^{\wedge}$ is also finite. Therefore, $R_A$ is finite.
On the other hand, suppose $R_A$ is finite. We first show ${\cal M}_A$ is a finite DWA. Note that by Theorem \ref{th:universal 1}, $A$ is accepted by the finite universal weighted automaton ${\cal U}_A$. By Proposition \ref{pro:m-minimal} (the proof of which is independent to this proposition), ${\cal M}_A$ is a sub-automaton of ${\cal U}_A$. Therefore, $A$ can be accepted by a finite DWA.
To end the proof, we show $S_A$ is finite. We prove this by contradiction. Suppose $S_A$ is infinite. Since $Im(A)$ is finite, there is $a\in Im(A)$ such that the subset $S_1=\{c\rightarrow a | c\in S\}$ of $S_A$ is infinite. Assume that $A(u)=a$ for $u\in\Sigma^{\ast}$. For any $c\in S$, define a series $Y_c\inS\langle\!\langle\Sigma^{\ast}\rangle\!\rangle$ as, $Y_c(w)=c$ if $w=\varepsilon$ and $Y_c(w)=0$ otherwise. Consider the right residual $A/Y_c$ , by a simple calculation, we have $A/Y_c(u)=\bigwedge\{Y_c(v)\rightarrow A(uv) | v\in\Sigma^{\ast}\}=Y_c(\varepsilon)\rightarrow A(u)=c\rightarrow a$. It follows that the set $\{A/Y_c | c\in S\}$ is infinite, then $rR(A)$ is infinite, and thus $R_A$ is infinite. A contradiction. Therefore $S_A$ is finite.
\end{proof}
We give two examples to illustrate the construction of the universal weighted automaton of a formal power series.
\begin{example}\label{ex:finite}
{\rm Assume $\Sigma=\{a,b\}$, $S=(\mathbb{N}\cup\{\infty\},\max,\min,0,\infty)$, where $S$ is a linear order lattice under the natural order of the integer numbers. Consider the formal power series $A\inS\langle\!\langle\Sigma^{\ast}\rangle\!\rangle$ defined as follows,
\begin{displaymath}
{A(\theta)= \left\{ \begin{array}{ll}
$2$, & \textrm{if $\theta\in \Sigma^{\ast} ab \Sigma^{\ast}$}\\
$1$, & \textrm{otherwise.}\\
\end{array} \right.}
\end{displaymath}
$A$ can be recognized by the DWA ${\cal A}=(Q,\Sigma,\delta,q_0,F)$ presented in Figure 1, where $F=1\diagup q_0+1\diagup q_1+2\diagup q_2$.
\begin{figure}[ptb]
\begin{center}
\includegraphics[width=0.5\textwidth]{fig1}
\end{center}
\caption{The DWA recognizes the formal power series $A$}
\label{fig:1}
\end{figure}
The set of factorizations of
$A$ is $R_A=\{u_i|u_i=(X_i,Y_i),i=1,2,3,4.\}$, where
$X_1(\theta)=Y_2(\theta)=1$, $X_2(\theta)=Y_1(\theta)=2$, for all
$\theta \in \Sigma^*$;
\begin{displaymath}
{X_3(\theta)= \left\{ \begin{array}{ll}
$2$, & \textrm{if $\theta\in \Sigma^{\ast} a \Sigma^{\ast}$}\\
$1$, & \textrm{otherwise.}\\
\end{array} \right.},
\end{displaymath}
\begin{displaymath}
{X_4(\theta)=Y_3(\theta)= \left\{ \begin{array}{ll}
$2$, & \textrm{if $\theta\in \Sigma^{\ast} ab \Sigma^{\ast}$}\\
$1$, & \textrm{otherwise.}\\
\end{array} \right.},
\end{displaymath}
\begin{displaymath}
{Y_4(\theta)= \left\{ \begin{array}{ll}
$2$, & \textrm{if $\theta\in \Sigma^{\ast} b \Sigma^{\ast}$}\\
$1$, & \textrm{otherwise.}\\
\end{array} \right.}.
\end{displaymath}
\begin{figure}[ptb]
\begin{center}
\includegraphics[width=0.5\textwidth]{fig2.eps}
\end{center}
\caption{The universal weighted automaton ${\cal U}_A$ of the formal power series $A$}
\label{fig:2}
\end{figure}
By definition, the
universal weighted automaton of $A$ is ${\cal
U}_A=(R_A,\Sigma,\eta_A$, $J_A,G_A)$, where\footnote{We write $x\diagup u_i$ for the value $x$ of $u_{i}$ in the given $S$-subset.}
\begin{itemize}
\item [-] $J_A=1\diagup u_1+2\diagup
u_2+1\diagup u_3+1\diagup u_4$,
\item [-] $G_A=2\diagup u_1+1\diagup
u_2+1\diagup u_3+1\diagup u_4$,
\item [-] $\eta_A(u_i,x,u_j)$ is either 2 or 1, as shown in Figure 2.
\end{itemize}
}
\end{example}
\begin{example}\label{ex:infinite}
{\rm Assume $\Sigma=\{a\}$, $S=(\mathbb{N}\cup\{\infty\},\min,+,\infty,0)$ is the tropical semiring. Consider the formal power series $A\inS\langle\!\langle\Sigma^{\ast}\rangle\!\rangle$ defined as follows
\begin{displaymath}
{A(a^k)= \left\{ \begin{array}{ll}
$0$, & \textrm{if $k=0$}\\
k-1, & \textrm{if $k>0$.}\\
\end{array} \right.}
\end{displaymath}
By Proposition \ref{pro:dffa}, $A$ can not be recognized by any finite DWA. However, as a regular series, $A$ can be recognized by a finite weighted automaton ${\cal
B}=(\{q_0,q_1\},\Sigma,\eta, \{q_0\},\{q_1\})$, where $\eta(q_0,a,q_1)=0$, $\eta(q_1,a,q_1)=1$. The universal weighted automaton ${\cal U}_A$ has infinite states
$\{(X_i,Y_i)\}_{i=0}^{\infty}$, where
\begin{center}
$Y_0=A, X_0(a^k)=k$;
\end{center}
\noindent and for $i>0$,
\begin{displaymath}
{X_i(a^k)= \left\{ \begin{array}{ll}
\infty, & \textrm{if $k=0$}\\
max(k-i,0), & \textrm{if $k>0$,}\\
\end{array} \right.}
\end{displaymath}
\begin{center}
$Y_i(a^k)=k$, for any $k\geq 0$.
\end{center}
For any $i,j$, we have $\eta_A((X_i,Y_i),a,(X_j,Y_j))=0$ if $i\leq j+1$ and $\infty$ otherwise, $J_A=(X_0,Y_0)$ and $G_A=R_A\setminus J_A$. In this case, ${\cal U}_A$ is not a finite weighted automaton.}
\end{example}
\section {The Universality of the Universal Weighted Automaton}
In this section, we show the weighted automaton ${\cal U}_A$ defined in the previous section satisfies the following universal property.
\begin{definition}
\label{dfn:universal-property}
Suppose $A\inS\langle\!\langle\Sigma^{\ast}\rangle\!\rangle$ and $\cal U$ is a weighted automaton that recognizes $A$. We say $\cal U$ satisfies the \emph{universal property} if there exists a morphism from $\cal B$ to $\cal U$ for any weighted automaton $\cal B$ such that $|{\cal B}|\leq A$.
\end{definition}
To demonstrate the universality of ${\cal U}_A$, we introduce a canonical mapping for each weighted automaton $\cal B$ that recognizes a subset of $A$.
\begin{definition}
Let $S$ be a complete c-semiring. Suppose $A\inS\langle\!\langle\Sigma^{\ast}\rangle\!\rangle$ and ${\cal B}=(P,\Sigma,\eta,J,G)$ is a weighted automaton such that $|{\cal B}|\leq A$. We define $\varphi_{\cal B}:
P\rightarrow R_A$ by $\varphi_{\cal B}(p)=(X_p,Y_p)$ for any $p\in P$, where
\begin{equation}
\mbox{$Y_p=Past_{{\cal B}}(p)\backslash A$ and $X_p=A/Y_p$.}
\end{equation}
We call $\varphi_{\cal B}$ the canonical mapping from $\cal B$ to ${\cal U}_A$.
\end{definition}
The following lemma shows that $\varphi$ is a morphism.
\begin{lemma}\label{lemma:universality}
Let $S$ be a complete c-semiring. Suppose $A\inS\langle\!\langle\Sigma^{\ast}\rangle\!\rangle$. If ${\cal B}=(P,\Sigma,\eta,J,G)$ is a weighted automaton such that $|{\cal B}|\leq A$, then the canonical mapping $\varphi_{\cal B}$ is a morphism from ${\cal B}$ into ${\cal U}_A$.
\end{lemma}
\begin{proof}
If $\varphi_{\cal B}(p)=(X_p,Y_p)$, then it is obvious that $Past_{{\cal B}}(p)\leq X_p$ and $Fut_{{\cal B}}(p)\leq Y_p$. It follows that $J_A(\varphi_{\cal B}(p))=X_p(\varepsilon)\geq Past_{{\cal B}}(p)(\varepsilon)\geq J(p)$ and $G_A(\varphi_{\cal B}(p))=Y_p(\varepsilon)\geq Fut_{{\cal B}}(p)(\varepsilon)\geq G(p)$.
For the remainder part, notice that $\eta(p,\sigma,q)Past_{{\cal B}}(p)\sigma \leq Past_{{\cal B}}(q)$. Then
$$\eta(p,\sigma,q)Past_{{\cal B}}(p)\sigma Y_q\leq Past_{{\cal B}}(q)Y_q\leq A,$$
i.e., $Past_{{\cal B}}(p)(\eta(p,\sigma,q)\sigma Y_q)\leq A$. Hence, $\eta(p,\sigma,q)\sigma Y_q\leq Y_p$. Then it follows that $$\eta(p,\sigma,q)\leq \sigma Y_q\rightarrow_{incl} Y_p=\eta_A(\varphi(p),\sigma,\varphi(q)).$$
Therefore, $\varphi_{\cal B}$ is a morphism from ${\cal B}$ into ${\cal U}_A$.
\end{proof}
The universality of ${\cal U}_A$ follows immediately.
\begin{theorem}\label{th:universality}
Let $S$ be a complete c-semiring. For $A\inS\langle\!\langle\Sigma^{\ast}\rangle\!\rangle$, the universal weighted automaton ${\cal U}_A$ satisfies the universal property, i.e. there is a morphism to ${\cal U}_A$ from any weighted automaton ${\cal B}$ such that $|{\cal B}|\leq A$.
\end{theorem}
The following lemma establishes the connection between the formal power series accepted by two weighted automata connected by a morphism.
\begin{lemma}\label{lemma:morphism1}
Let $S$ be a complete c-semiring. Suppose ${\cal A}=(Q,\Sigma,\delta,I,F)$ and ${\cal B}=(P,\Sigma,\eta,J,G)$ are two weighted automata. If $\varphi$ is a morphism from ${\cal A}$ into ${\cal B}$, then we have
\begin{equation}
\mbox{$Past_{{\cal A}}(q)\leq Past_{{\cal B}}(\varphi(q))$, $Fut_{{\cal A}}(q)\leq Fut_{{\cal B}}(\varphi(q))$,}
\end{equation}
for any $q\in Q$, and thus, $|{\cal A}|\leq |{\cal B}|$. If $\varphi$ is a strong homomorphism, then $|{\cal A}|=|{\cal B}|$.
\end{lemma}
\begin{proof}
For any $\theta\in\Sigma^{\ast}$,
\begin{eqnarray*}
Past_{{\cal A}}(q)(\theta) &=& \bigvee_{i\in Q}I(i)\otimes \delta^{\ast}(i,\theta,q) \\
&\leq& \bigvee_{i\in Q}J(\varphi(i))\otimes \eta^{\ast}(\varphi(i),\theta,\varphi(q)) \\
&\leq& \bigvee_{j\in P}J(j)\otimes \eta^{\ast}(j,\theta,\varphi(q)) \\
&=& Past_{{\cal B}}(\varphi(q))(\theta). \\
Fut_{{\cal A}}(q)(\theta) &=& \bigvee_{t\in Q}\delta^{\ast}(q,\theta,t)\otimes F(t) \\
&\leq& \bigvee_{t\in Q}\eta^{\ast}(\varphi(q),\theta,\varphi(t))\otimes G(\varphi(t)) \\
&\leq& \bigvee_{p\in P}\eta^{\ast}(\varphi(q),\theta,p)\otimes G(p) \\
&=& Fut_{{\cal B}}(\varphi(q))(\theta)).
\end{eqnarray*}
Hence, $Past_{{\cal A}}(q)\leq Past_{{\cal B}}(\varphi(q))$ and $Fut_{{\cal A}}(q)\leq Fut_{{\cal B}}(\varphi(q))$. Then it follows that $|{\cal A}|\leq |{\cal B}|$.
If $\varphi$ is a strong homomorphism, then it can be easily verified that $\eta^{\ast}(\varphi(q),\theta,p)=\bigvee\{\delta^{\ast}(q,\theta,r)| \varphi(r)=p\}$ for any $\theta\in\Sigma^{\ast}$.
Then it follows that
\begin{eqnarray*}
|{\cal B}|(\theta) &=& \bigvee_{j,p\in P}J(j)\otimes \eta^{\ast}(j,\theta,p)\otimes G(p) \\
&=& \bigvee_{i\in Q,p\in P}I(i)\otimes \eta^{\ast}(\varphi(i),\theta,p)\otimes G(p) \\
&=& \bigvee_{i\in Q, \varphi(r)=p}I(i)\otimes\delta^{\ast}(i,\theta,r)\otimes G(\varphi(r)) \\
&=& \bigvee_{i,r\in Q}I(i)\otimes \delta^{\ast}(i,\theta,r)\otimes F(r)=|{\cal A}|(\theta).
\end{eqnarray*}
Hence, $|{\cal A}|=|{\cal B}|$.
\end{proof}
Suppose ${\cal A}=(Q,\Sigma,\delta,I,F)$ is a weighted automaton. We say two states $p$ and $q$ in $Q$ are \emph{mergible} in ${\cal A}$ if there exist a weighted automaton ${\cal B}=(P,\Sigma,\eta,J,G)$ that accepts the same language as ${\cal A}$ and a surjective morphism $\varphi: {\cal A}\rightarrow {\cal B}$ such that $\varphi(p)=\varphi(q)$.
\begin{proposition}\label{pro:mergible}
Let $S$ be a complete c-semiring. Suppose $A \in S\langle\!\langle\Sigma^{\ast}\rangle\!\rangle$. Then there is no mergible states in the universal weighted automaton ${\cal U}_A$.
\end{proposition}
\begin{proof}
Otherwise, there is a weighted automaton ${\cal C}$ and a surjective morphism $\varphi: {\cal U}_A\rightarrow {\cal C}$ such that $|{\cal U}_A|=| {\cal C}|=A$,
and $\varphi(X,Y)=\varphi(X',Y')=s$ for two distinct states $(X,Y)$ and $(X',Y')$ in ${\cal U}_A$. Then we have,
\begin{eqnarray*}
X=Past_{{\cal U}_A}(X,Y)\leq Past_{{\cal C}}(\varphi(X,Y))=Past_{{\cal C}}(s), \\
X'=Past_{{\cal U}_A}(X',Y')\leq Past_{{\cal C}}(\varphi(X',Y'))=Past_{{\cal C}}(s),
\end{eqnarray*}
and thus, $X\vee X'\leq Past_{{\cal C}}(s)$. Similarly, we have $Y\vee Y'\leq Fut_{{\cal C}}(s)$. Therefore,
$$(X\vee X')(Y\vee Y')\leq Past_{{\cal C}}(s)Fut_{{\cal C}}(s)\leq A.$$
This contradicts with the maximality of the factorization $(X,Y)$ and $(X',Y')$.
\end{proof}
In fact, ${\cal U}_A$ is the largest non-mergible weighted automaton.
\begin{corollary}\label{co:largest}
Let $S$ be a complete c-semiring. Suppose $A\inS\langle\!\langle\Sigma^{\ast}\rangle\!\rangle$. Then ${\cal U}_A$ is the largest weighted automaton among those that accept $A$ but have no mergible states.
\end{corollary}
\begin{proof}
A weighted automaton ${\cal B}$ accepting $A$ that has strictly more states than ${\cal U}_A$ is sent into ${\cal U}_A$ by a morphism which is necessarily non-injective.
\end{proof}
Moreover, ${\cal U}_A$ is the smallest `universal' weighted automaton.
\begin{proposition}\label{pro:least}
Let $S$ be a complete c-semiring. Suppose $A\inS\langle\!\langle\Sigma^{\ast}\rangle\!\rangle$. Then ${\cal U}_A$ is the smallest weighted automaton among those that accept $A$ and have the universal property.
\end{proposition}
\begin{proof}
Suppose that ${\cal C}$ has the universal property with respect to the $A$. As ${\cal U}_A$ accepts $A$, there should be a morphism from ${\cal U}_A$ into ${\cal C}$. As ${\cal U}_A$ has no mergible states, this morphism should be injective: ${\cal C}$ has at least as many states as ${\cal U}_A$.
\end{proof}
Applying Theorem \ref{th:universality} to weighted automata that accepts $A$, we obtain the following corollary.
\begin{corollary}\label{co:m-minimal}
Let $S$ be a complete c-semiring. Suppose $A\inS\langle\!\langle\Sigma^{\ast}\rangle\!\rangle$ and ${\cal B}$ is a weighted automaton that accepts $A$. If $\cal B$ has no mergible states, then ${\cal B}$ is a sub-automaton of ${\cal U}_A$.
\end{corollary}
\begin{proof}
By Theorem \ref{th:universality}, the canonical mapping $\varphi$ from ${\cal B}$ to ${\cal U}_A$ is a morphism. Because ${\cal B}$ has no mergible states, we know $\varphi$ must be one-to-one. Therefore, ${\cal B}$ is a sub-automaton of ${\cal U}_A$.
\end{proof}
It is not difficult to show that any minimal weighted (determinate or non-determinate) automaton that accepts $A$ has no mergible states. The above corollary then suggests a simple way for searching the minimal (non-determinate) weighted automaton that accepts $A$: It suffices to check the sub-automaton of the universal automaton ${\cal U}_A$ which accepts $A$ and has minimal states. The following proposition shows that ${\cal M}_A$, the minimal DWA that accepts $A$, is also a sub-automaton of ${\cal U}_A$.
\begin{proposition}\label{pro:m-minimal}
Let $S$ be a complete c-semiring. Suppose $A\in S\langle\!\langle\Sigma^{\ast}\rangle\!\rangle$. Then ${\cal M}_A$ is a sub-automaton of ${\cal U}_A$.
\end{proposition}
\begin{proof}
By Corollary~\ref{co:m-minimal}, we need only show that ${\cal M}_A$ has no mergible states.
Recall the minimal DWA that accepts $A$ is ${\cal M}_A=(Q,\Sigma,\delta,q_0,F)$, where
\begin{itemize}
\item [-] $Q=\{u^{-1}A | u\in\Sigma^{\ast}\}$,
\item [-] $\delta(u^{-1}A,\sigma)=(u\sigma)^{-1}A$,
\item [-] $\delta^{\ast}(u^{-1}A,v)=(uv)^{-1}A$,
\item [-] $q_0=(\varepsilon)^{-1}A=A$,
\item [-] $F:Q\rightarrow S$ is $F(u^{-1}A)=A(u)$.
\end{itemize}
Then for any state $u^{-1}A\in Q$, $\delta^{\ast}(A,u)=u^{-1}A$. We show that there are no mergible states in ${\cal A}$. Otherwise, there are two distinct states $u^{-1}A$, $v^{-1}A$ in $Q$, but there exists another weighted automaton ${\cal B}$ and a morphism $\varphi$ from ${\cal A}$ into ${\cal B}$ such that ${\cal B}$ is the morphic image of ${\cal A}$, $|{\cal A}|=|{\cal B}|$ and $\varphi(u^{-1}A)=\varphi(v^{-1}A)$.
Since $u^{-1}A\not=v^{-1}A$, there exists $w\in\Sigma^{\ast}$ such that $u^{-1}A(w)\not=v^{-1}A(w)$, i.e., $A(uw)\not=A(vw)$. Note that
\begin{eqnarray*}
A(uw)=|{\cal A}|(uw)=F(\delta^{\ast}(A,uw))=F(\delta^{\ast}(\delta^{\ast}(A,u),w))=F(\delta^{\ast}(u^{-1}A,w)),\\
A(vw)=|{\cal A}|(vw)=F(\delta^{\ast}(A,vw))=F(\delta^{\ast}(\delta^{\ast}(A,v),w))=F(\delta^{\ast}(v^{-1}A,w)).
\end{eqnarray*}
Then
\begin{eqnarray*}
|{\cal B}|(uw) &=& \varphi(F)(\delta^{\ast}(\varphi(A),uw)) \\
&=& \varphi(F)(\varphi(\delta^{\ast}(A,uw)))\\
&=&\varphi(F)(\varphi(\delta^{\ast}(u^{-1}A,w))) \\
&=& \varphi(F)(\delta^{\ast} (\varphi(u^{-1}A),w))) \\
&=& \varphi(F)(\delta^{\ast}(\varphi(v^{-1}A),w))) \\
&=& \varphi(F)(\delta^{\ast}(\varphi(A),vw))\\
&=& |{\cal B}|(vw).
\end{eqnarray*}
Since $|{\cal B}|=|{\cal A}|=A$, it follows that $A(uw)=A(vw)$, a contradiction occurs.
Therefore, ${\cal A}$ has no mergible states.
\end{proof}
\section{Construction of the Universal Weighted Automaton}
In general, it is not effective to construct all factorizations of $A$. Suppose ${\cal A}=(Q,\Sigma,\delta,q_0,F)$ is an arbitrary DWA accepting $A$. In this section, we give an effective method to construct ${\cal U}_A$ by using the DWA $\cal A$. Let $l_A$ be the $\bigvee$-sublattice generated by $S_A^{\wedge}$ as defined in Eq.(\ref{eq:S-A}), i.e.,
\begin{equation}\label{eq:l-A}
l_A=\{\bigvee X | X\subseteq S_A^{\wedge}\}.
\end{equation}
\noindent It is well known that $l_A$ is finite iff $S_A^{\wedge}$ is finite (cf.\cite{li93,davey02}) iff $S_A$ is finite. Write $Q_1=l_A^Q$. If ${\cal A}$ is finite and $S_A$ is finite, then $Q_1$ is also finite. We construct
a weighted automaton ${\cal A}_1=(Q_1,\Sigma,\eta,J,G)$ as follows:
\begin{eqnarray*}
J(f) &=&f(q_0),\\
G(f) &=& f\rightarrow_{incl} F,\\
\eta(f,\sigma,g) &=& f\sigma\rightarrow_{incl} g,
\end{eqnarray*}
where $f\sigma:Q\rightarrow l_A$ is defined by $f\sigma (q)=\bigvee\{f(p) | \delta(p,\sigma)=q\}$.
We next define a mapping $\varphi$ from ${\cal A}_1$ to ${\cal U}_A$. To this end, we first establish the correspondence between weighted states and factorizations of $A$.
\begin{proposition}
\label{prop: Xf&Yf}
Let $S$ be a complete c-semiring. Suppose $A\inS\langle\!\langle\Sigma^{\ast}\rangle\!\rangle$ and ${\cal A}=(Q,\Sigma,\delta,q_0,F)$ is an arbitrary DWA accepting $A$. Then $(X_f,Y_f)$ is a factorization of $A$ for any weighted state $f: Q\rightarrow l_A$, where
\begin{eqnarray}
\label{eq:Y_f}
Y_f &=& \bigwedge_{q\in Q} f(q)\rightarrow Fut_{{\cal A}}(q),\\
\label{eq:X_f}
X_f &=& A/Y_f,
\end{eqnarray}
and, for any $\theta\in\Sigma^{\ast}$,
\begin{eqnarray}
(f(q)\rightarrow Fut_{{\cal A}}(q))(\theta) &=& f(q)\rightarrow Fut_{{\cal A}}(q)(\theta).
\end{eqnarray}
Therefore, the mapping $\varphi$ defined by
$\varphi(f)=(X_f,Y_f)$
is a mapping from $Q_1=l_A^Q$ to $R_A$.
\end{proposition}
\begin{proof}
Without loss of generality, we assume that ${\cal A}$ is accessible. Since ${\cal A}$ is a DWA, for any $q\in Q$, there exists $u\in \Sigma^{\ast}$ such that $\delta^{\ast}(q_0,u)=q$. Moreover, if $\delta^{\ast}(q_0,v)=q$ for another $v\in \Sigma^{\ast}$, then $A(uw)=A(vw)$ for any $w\in \Sigma^{\ast}$. By this observation, we have, for any $v\in\Sigma^{\ast}$,
\begin{eqnarray*}
Y(v) &=& \bigwedge_{q\in Q}f(q)\rightarrow Fut_{{\cal A}}(q)(v)\\
& = & \bigwedge_{u\in\Sigma^{\ast},\delta^{\ast}(q_0,u)=q}f(q)\rightarrow A(uv) \\
& =& \bigwedge_{u\in\Sigma^{\ast}}f(\delta^{\ast}(q_0,u))\rightarrow A(uv).
\end{eqnarray*}
If we let $X'(u)=f(\delta^{\ast}(q_0,u))$ for any $u\in \Sigma^{\ast}$, then we obtain a series $X':\Sigma^{\ast}\rightarrow l_A$ such that $Y=X'\backslash A$.
\end{proof}
The mapping $\varphi: l_A^Q \rightarrow R_A$ is also onto.
\begin{proposition}\label{pro:facor-state1}
Let $S$ be a complete c-semiring. Suppose $A\inS\langle\!\langle\Sigma^{\ast}\rangle\!\rangle$ and ${\cal A}=(Q,\Sigma,\delta,q_0,F)$ is a DWA that accepts $A$. For any $(X,Y)\in R_A$, there is a weighted state $f:Q\rightarrow l_A$ such that $Y=Y_f$.
\end{proposition}
\begin{proof}
Define a weighted state $f:Q\rightarrow l_A$ as, for any $q\in Q$,
$$f(q)=\bigvee\{X(u) | \delta^{\ast}(q_0,u)=q\}.$$
\noindent By the proof of Proposition \ref{pro:finiteness}, $X$ induces a unique mapping from $Q_A=\Sigma^{\ast}/\equiv_A$ into $S_A^{\wedge}$, so $f$ is well-defined.
We show $Y=Y_f$ in the following.
For any $\theta\in\Sigma^{\ast}$, we have
\begin{eqnarray*}
Y_f(\theta) &=& \bigwedge_{q\in Q}f(q)\rightarrow Fut_{{\cal A}}(q)(\theta)\\
&=& \bigwedge_{q\in Q}(\bigvee\{X(u) | \delta^{\ast}(q_0,u)=q\})\rightarrow F(\delta^{\ast}(q,\theta)\\
&=& \bigwedge_{q\in Q, \delta^{\ast}(q_0,u)=q}X(u)\rightarrow F(\delta^{\ast}(q,\theta)) \\
&=& \bigwedge_{u\in\Sigma^{\ast}}X(u)\rightarrow A(u\theta) \\
&=& X\backslash A(\theta) \\
&=& Y(\theta).
\end{eqnarray*}
Hence, $Y=Y_f$.
\end{proof}
Furthermore, we have
\begin{proposition}\label{pro:strong homo}
Let $S$ be a complete c-semiring. Suppose $A\in S\langle\!\langle\Sigma^{\ast}\rangle\!\rangle$. The mapping $\varphi$ defined in Proposition~\ref{prop: Xf&Yf} is a strong homomorphism from weighted automaton ${\cal A}_1=(Q_1,\Sigma,\eta,J,G)$ \emph{onto} the universal weighted automaton ${\cal U}_A$, and thus $|{\cal A}_1|=|{\cal U}_A|=A$.
\end{proposition}
\begin{proof}
See Appendix B.
\end{proof}
Define an equivalence relation $\sim$ on $Q_1$ as follows:
\begin{equation}
\label{eq:sim-relation}
\mbox{$f\sim g$ iff $\varphi(f)=\varphi(g)$}
\end{equation}
It is clear that $f\sim g$ iff $Y_f=Y_g$. Using this equivalence relation, we obtain a quotient weighted automaton from ${\cal A}_1$, denoted by
${\cal A}^{\prime}$, which is isomorphic to ${\cal U}_A$.
\begin{corollary}\label{co:construction}
Let $A\inS\langle\!\langle\Sigma^{\ast}\rangle\!\rangle$ and ${\cal A}_1$ be as in Proposition~\ref{pro:strong homo}. Suppose ${\cal A}^{\prime}$ is the quotient weighted automaton of ${\cal A}_1$ modulo the equivalent relation $\sim$ on $Q_1$. Then ${\cal A}^\prime$ is isomorphic to ${\cal U}_A$.
\end{corollary}
Once the DWA ${\cal A}$ is finite and $S_A$ is finite (this condition can be guaranteed if $S$ is finite or $S$ is a linear-order lattice as declared in Proposition \ref{pro:finiteness}), the equivalence $\sim$ defined by Eq.(\ref{eq:sim-relation}) can be effectively constructed. This is because, $\delta^{\ast}(q,\theta)$ takes at most $|Q|=n$ states, i.e., the set $\{\delta^{\ast}(q,\theta) | \theta\in\Sigma^{\ast}\}$ as a subset of $Q$ has at most $n$ states, the corresponding $Fut_{{\cal A}}(q)=G_A(\delta^{\ast}(q,\theta))$ has at most $n$ values. Therefore, it is sufficient to check these states in Eq.(\ref{eq:sim-relation}). In turn, it is sufficient to check those $\theta\in\Sigma^{\ast}$ with $|\theta|<n$ in Eq.(\ref{eq:sim-relation}). Hence, the equivalence relation $\sim$ is decidable
and the weighted automaton ${\cal A}^{\prime}$ can be effectively constructed.
We next give one example.
\begin{example}\label{ex:finite-state}
{\rm Consider the formal power series $A$ in Example \ref{ex:finite}, which is recognized by the finite DWA as shown in Figure 1. Then the weighted automaton ${\cal A}'=(Q_1,\Sigma,\eta,J,G)$, where
$Q_1=\{f_1,f_2,f_3,f_4\}$ with $f_1(q)=1$ and $f_2(q)=2$ for any $q\in Q$,
\begin{displaymath}
{f_3(q)= \left\{ \begin{array}{ll}
$2$, & \textrm{$q = q_1,q_2$}\\
$1$, & \textrm{$q=q_0.$}\\
\end{array} \right.}
\end{displaymath}
\begin{displaymath}
{f_4(q)= \left\{ \begin{array}{ll}
$2$, & \textrm{$q = q_2$}\\
$1$, & \textrm{$q=q_0,q_1.$}\\
\end{array} \right.}
\end{displaymath}
and
\begin{eqnarray*}
J &=& 1\diagup f_1+2\diagup f_2+1\diagup f_3+1\diagup f_4,\\
G &=& 2\diagup f_1+1\diagup f_2+1\diagup f_3+1\diagup f_4;
\end{eqnarray*}
and
$\eta(f_i,x,f_j)$ is either 2 or 1 (see Figure 3). Clearly, ${\cal A}'$ is isomorphic to ${\cal U}_A$.
\begin{figure}[ptb]
\begin{center}
\includegraphics[width=0.5\textwidth]{fig3.eps}
\end{center}
\caption{The weighted automaton ${\cal A}'$}
\label{fig:2}
\end{figure} }
\end{example}
We next give a detailed examination of the equivalent relation $\sim$.
The correspondence $\varphi$ between the weighted states and the factorizations of $A$ stated in Proposition~\ref{prop: Xf&Yf} may be not one-to-one. There may have more than one weighted states correspond to a given factorization. However, the following proposition asserts that there exists a largest weighted states.
\begin{proposition}\label{pro:largest state}
Let $S$ be a complete c-semiring. Suppose $A\in S\langle\!\langle\Sigma^{\ast}\rangle\!\rangle$ and ${\cal A}=(Q,\Sigma,\delta,q_0,F)$ is a DWA that accepts $A$. If $Y=Y_{g_i}$ for all weighted states $g_i: Q\rightarrow l_A$ ($i\in I$), then $Y=Y_{\bigvee_{i\in I}g_i}$. Therefore, for each right residual $Y$, there exists a largest $h:Q\rightarrow l_A$ such that $Y=Y_h$.
\end{proposition}
By the above propositions, we know for each right residual $Y$ of $A$ there exists a (unique) largest weighted state $h:Q\rightarrow l_A$ such that $Y=Y_h$. Furthermore, suppose $(X,Y)\in R_A$. Then $h$ is defined by $h(q)=\bigvee\{X(u) | \delta^{\ast}(q_0,u)=q\}$ as implied in the proofs of Propositions \ref{prop: Xf&Yf} and \ref{pro:facor-state1}. In general, for a weighted state $f:Q\rightarrow l_A$, the structure of the largest weighted state $h$ such that $Y_f=Y_h$
is unclear. But we have the following estimation.
\begin{proposition}\label{pro:largest element1}
Let $S$ be a complete c-semiring. Suppose ${\cal A}=(Q,\Sigma,\delta$, $q_0,F)$ is a weighted automaton, and $f:Q\rightarrow l_A$ a weighted state. If $\{u_i\}_{i\in I}\subseteq\Sigma^{\ast}$ and $Y_f=Y_{\bigwedge_{i\in I}u_i\circ F}$,
then $f\leq \bigwedge_{i\in I}u_i\circ F$, where $u\circ F(q)=F(\delta^{\ast}(q,u))$.
\end{proposition}
\begin{proof}
Let $g=\bigwedge_{i\in I}u_i\circ F$. Then for each $j\in I$ we have
$$Y_g(u_j)=\bigwedge_{q\in Q}g(q)\rightarrow Fut_{{\cal A}}(q)(u_j)=\bigwedge_{q\in Q}(\bigwedge_{i\in I}u_i\circ F)(q)\rightarrow (u_j\circ F)(q)=1.$$
Thus $\bigwedge_{q\in Q}f(q)\rightarrow Fut_{{\cal A}}(q)(u_j)=1$. It follows that $f(q)\rightarrow Fut_{{\cal A}}(q)(u_j)=1$, i.e., $f(q)\leq Fut_{{\cal A}}(q)(u_j)=u_j\circ F(q)$ for any $q\in Q$.
Therefore, $f\leq u_j\circ F$ for any $j$, hence $f\leq \bigwedge_{i\in I}u_i\circ F$.
\end{proof}
Recall that if $S$ is the two elements Boolean algebra $\{0,1\}$, then $\{\bigwedge_{i\in I}u_i\circ F | u_i\in\Sigma^{\ast}\}$ forms the whole set of those largest elements (cf.\cite{lombardy08}). It is still unclear whether each largest weighted state can be represented as the intersection of $u_i\circ F$ for a set of $u_i$.
\section{A Comparison of Quotients and Residuals}
For a formal power series $A$, we have introduced the notions of (left and right) quotients and (left and right) residuals of $A$. As can be seen in the above discussion, these two kinds of operations have different behaviors when considering their algebraic and language properties. Especially, with the associated weighted automata, ${\cal B}_A$ and ${\cal U}_A$ have different structure, although they are equivalent as language recognizers. For example, there exists series $A$ such that ${\cal B}_A$ is infinite but ${\cal U}_A$ is finite.
In this section, we consider the order relation between the quotients and the residuals of a same series $A$. Because the duality between left and right quotients (residuals), we need only compare the left quotient $X^{-1}A$ and the left residual $X\backslash A$ of $A$ by a series $X$. Our results show that all the four possibilities are possible.
First, if $X$ is a word $u\in\Sigma^{\ast}$ or $\sum_{u\in\Sigma^{\ast}}X(u)=1$ and $A=0$, then it is obvious that $X^{-1}A=X\backslash A$.
Second, let $S=\{0,1\}$ be the two element Boolean algebra. Then for any languages $X,A\subseteq\Sigma^{\ast}$ we have $$X\backslash A=\bigcap_{u\in X} u^{-1}A\subseteq \bigcup_{u\in X}u^{-1}A=X^{-1}A.$$
Moreover, $X^{-1}A=X\backslash A$ iff $u\equiv_A v$ for any $u,v\in X$; but if there exists $u,v\in X$ such that $u\not\equiv_A v$, then the above inclusion is strict.
Third, let $S$ be the tropical semiring $(\mathbb{N}+\cup\{\infty\}$, $\min,+,\infty,0)$. For $A\inS\langle\!\langle\Sigma^{\ast}\rangle\!\rangle$, $u\in\Sigma^{\ast}$, and $k\in S$, we have $(ku)^{-1}A=k(u^{-1}A)\leq_S u^{-1}A$, while $(ku)\backslash A >_S u^{-1}A$ if $k\not=0$ and $k\not=\infty$. Therefore $(ku)\backslash A>_S(ku)^{-1}A$ if $k\not=0$ and $k\not=\infty$.
Fourth, let $S$ be the tropical semiring $(\mathbb{N}+\cup\{\infty\},\min,+,\infty,0)$. Take $\Sigma=\{a,b\}$. Consider the formal power series $A\inS\langle\!\langle\Sigma^{\ast}\rangle\!\rangle$ defined as follows
\begin{displaymath}
{A(\theta)= \left\{ \begin{array}{ll}
0, & \textrm{if $\theta=ba$ or $\theta=bb$}\\
10, & \textrm{if $\theta=aa$}\\
3, & \textrm{if $\theta=ab$}\\
\infty, & \textrm{otherwise.}\\
\end{array} \right.}
\end{displaymath}
Let $X=min((4+a),2+b)$. We show $X^{-1}A$ and $X\backslash A$ are incomparable. By
\begin{eqnarray*}
X^{-1}A(a) &=& min(4+A(aa),2+A(ba))=2, \\
X^{-1}A(b) &=& min(4+A(ab), 2+A(bb))=2, \\
X\backslash A(a) &=& max(4\rightarrow A(aa), 2\rightarrow A(ba))=6, \\
X\backslash A(b) &=& max(4\rightarrow A(ab), 2\rightarrow A(bb))=0,
\end{eqnarray*}
we know $X^{-1}A(a)>_S X\backslash A(a)$, but $X^{-1}A(b)<_S X\backslash A(b)$. Therefore $X^{-1}A$ and $X\backslash A$ are incomparable.
It is still unclear when (i.e. for what kind of $S$ and $A\inS\langle\!\langle\Sigma^{\ast}\rangle\!\rangle$) we will have an uniform order relation between $X^{-1}A$ and $X\backslash A$.
\section {Conclusions}
In this paper, we have defined the quotient and residual operations for formal power series. The algebraic and closure properties under these operations were discussed. Our results show that most nice properties are kept in formal power series. Moreover, we introduced two canonical weighted automata ${\cal M}_A$ and ${\cal U}_A$ for each formal power series $A$ using the quotients and, respectively, residuals of $A$. It was shown that ${\cal M}_A$ is the minimal DWA of $A$, and ${\cal U}_A$ is the universal weighted automaton of $A$ which contains as a sub-automaton any weighted automaton that accepts $A$ but has no mergible states. In particular, any minimal weighted (deterministic or non-deterministic) automaton of $A$ is a sub-automaton of ${\cal U}_A$. This suggests an efficient way to find (approximations of) the minimal weighted NFA of $A$. Last but not least, we also showed, under a rather weak restriction, that ${\cal U}_A$ is finite iff ${\cal M}_A$ is finite, and that ${\cal U}_A$ can be effectively constructed when we have a finite DWA that recognizes $A$.
There are still several open problems left unsolved. Suppose $X$ and $A$ are two formal power series. Is $X\backslash A$ regular (context-free) whenever $A$ is regular (context-free)? What is the precise relation between $X^{-1}A$ and $X\backslash A$? When characterizing residuals in terms of quotients by word, we require the underlying semiring to be a complete c-semiring. It is easy to see that this requirement is not necessary. Another problem then is, to what extent can we develop the related theory in a weaker semiring structure, e.g. in quantale?
Another interesting question is to develop an abstraction scheme for formal power series and weighted automata based on semiring homomorphism. Just like in soft constraint satisfaction \cite{BCR02,LiY08}, we expect that semiring homomorphisms can play an important role in approximately computing the minimal NFA and the universal weighted automaton of a formal power series.
|
{
"timestamp": "2012-03-13T01:00:56",
"yymm": "1203",
"arxiv_id": "1203.2236",
"language": "en",
"url": "https://arxiv.org/abs/1203.2236"
}
|
\section{Methods}
ARPES experiments were carried out at the SGM-3 beamline of the synchrotron radiation source ASTRID \cite{Hoffmann:2004}. Graphene was prepared on Ir(111) using a well
established procedure based on C$_2$H$_4$ dissociation \cite{Coraux:2009}. The quality of the graphene layer was controlled by low-energy electron diffraction and its spectral function was measured by ARPES. At low temperature, the photoemission linewidth of the features was found to be similar to published values \cite{Kralj:2011,Pletikosic:2009}. The temperature measurements were performed with a K-type thermocouple and an infrared pyrometer. The temperature-dependent data were taken such that the sample was heated by a filament mounted behind it. The filament current was pulsed and the data were acquired during the off-part of the heating cycle. The total energy and $k$ resolution during data acquisition were 18~meV and 0.01~\AA$^{-1}$, respectively. The MDC linewidth was determined as the average over an energy range between 250~meV and 550~meV below the Dirac point. An energy interval was chosen in order to improve the experimental uncertainties. The interval limits were chosen such that the lower limit is always more than a typical phonon energy ($\approx 200$~meV) away from $E_F$ and neither limit is too close to the Dirac point or the crossing points between the main Dirac cone and the replica bands, as this is known to lead to errors in the linewidth determination \cite{Nechaev:2009}.
The {\it ab initio} calculations were performed with the VASP
code~\cite{Kresse:1996}, the projector-augmented-wave method~\cite{Blochl:1994,Kresse:1999}, the
Perdew-Burke-Ernzerhof exchange-correlation energy~\cite{Perdew:1996}, and an
efficient extrapolation for the charge density~\cite{Alfe:1999}. Single
particle orbitals were expanded in plane waves with a cutoff of 400
eV. We used the NPT ensemble (constant particles number $N$, pressure $P$,
and temperature $T$), as recently implemented in
VASP~\cite{Hernandez:2001,Hernandez:2010}. For the present slab
calculations, we only applied the constant pressure algorithm to the
two lattice vectors parallel to the surface, leaving the third
unchanged during the simulation. Adsorption of graphene has been
modelled by overlaying a $10\times 10$ graphene sheet (200 C atoms)
over a $9 \times 9$ Ir(111) supercell~\cite{Pozzo:2011} and using a slab
of 3 layers where the two topmost layers were allowed to move while
the bottom layer was kept fixed. Molecular dynamics simulations were
performed with the $\Gamma $ point only at T = 300 K and T= 1000
K. Density of states were calculated on representative simulation
snapshots, using a $16 \times 16 \times 1$ grid of {\bf k}-points (128
points). The projected density of states on the carbon atoms were obtained by projecting the Bloch orbitals onto spherical harmonics with $l=$1, inside spheres of radius 0.86~\AA~ centered on the C atoms. The PDOS obtained in this way is representative of the density of states due to the $p$ orbitals of the carbon atoms.
The DOS of suspended graphene at 0 K was calculated only for the $p$ states, as the $s$ state contribution around the Fermi energy is very small. The DOS was rescaled such that it could be fitted to the analytical linear density of states prer unit cell of isolated graphene near the Dirac point. The same scaling factor was applied to all the calculated density of states data. The position of the Dirac point in the calculations for supported graphene was determined by fitting the analytical linear DOS with an offset in binding energy corresponding to the new position of the Dirac point.
\section{Acknowledgements}
This work was supported by The Danish Council for Independent Research / Technology and Production Sciences and the Lundbeck foundation. A.B. acknowledges the Universit\`a
degli Studi di Trieste for the \textit{Finanziamento per Ricercatori di Ateneo}-FRA2009. The \textit{ab-initio} calculations were performed on the HECToR national service in the U.K.
|
{
"timestamp": "2012-03-13T01:00:11",
"yymm": "1203",
"arxiv_id": "1203.2187",
"language": "en",
"url": "https://arxiv.org/abs/1203.2187"
}
|
\section{Introduction}\label{section: Introduction}
Lorentzian spacetimes with more than four dimensions are of current
interest in mathematical physics. It is consequently useful to have
higher dimensional
generalizations of the classification schemes which
have been successfully employed in four dimensions. In particular, the
introduction of the alignment theory
\cite{Coleyetal04,Milsonetal05,Coley08}, based on the concept of {\em boost
weight} (abbreviated {\em b.w.} throughout this
paper), has made it possible to algebraically classify any tensor in a
Lorentzian spacetime of arbitrary dimensions by its {\em (null) alignment
type}, including the classification of the Weyl tensor and the Ricci
tensor. To complement this, a higher dimensional generalization of the
Newman-Penrose formalism has been presented, which consists of the
Bianchi \cite{Pravdaetal04} and Ricci identities \cite{OrtPraPra07} and
of the commutator relations \cite{Coleyetal04vsi} written out for a null
frame. More recently, the corresponding GHP formalism has also been
developed \cite{Durkeeetal10}.
However, other mathematical tools for the study of higher dimensional
Lorentzian spacetimes can also be developed including, e.g., the classification of
tensors utilizing {\em bivectors}. From this viewpoint, the algebraic
(Segre type) classification of the Weyl tensor, considered as a linear
operator on bivector space, turns out to be equivalent to the algebraic
classification by alignment type in the special case of four dimensions
(i.e., the Petrov classification \cite{Stephanibook}); however, these two classification schemes are non-equivalent in higher dimensions. In particular, the alignment
classification is rather course, and developing the algebraic
classification of the Weyl bivector operator may lead to a more refined
scheme.
For this purpose the bivector formalism in higher dimensional
Lorentzian spacetimes was developed in \cite{ColHer09}. The Weyl
bivector operator was defined in a manner consistent with its b.w.
decomposition. Components of fixed b.w. were then
characterized in terms of basic constituents which transform under
irreducible representations of the spins.
This leads to another refinement of the alignment classification,
based on geometric relations between the highest b.w. constituents.
The types arising will be referred to as \emph{spin types}.
In this paper we study the general scheme of \cite{ColHer09} (and thus the two classification refinements mentioned above, and their interplay)
in the case of
five-dimensional (5D) Lorentzian spacetimes. These are of
particular interest for a number of reasons. First, they provide
the simplest arena in which properties of gravity qualitatively
differ from the well-known 4D case. Certain important new solutions
such as black rings, which are intrinsically higher-dimensional,
seem to admit a closed exact form only in five dimensions (see,
e.g., \cite{EmpRea08} and references therein). An alternative
spinor classification of the Weyl tensor has also been developed in
5D, and its connection with the b.w. approach has been
discussed~\cite{Desmet02,GarMar09,God10}. Since the two
classifications are not equivalent, the refinements we propose may
also be useful in the spinor classification. Finally, as a peculiar
feature of five dimensions (see also \cite{ColHer09}), the highest
b.w. constituents are represented by square matrices, vectors and a
single scalar. In this way both refinements of the 5D Weyl tensor
classification can be carried out in a fully explicit manner and its
main properties can be easily displayed.
The structure of the paper is as follows. In section \ref{section:
preliminaries} we review the algebraic properties of the 5D Weyl
tensor (i.e., the constituents, the null alignment types and the
Weyl bivector operator). The definitions and the main ideas
regarding the spin type and the Weyl operator refinements are
presented in section \ref{section: refinements}. In sections
\ref{section: type N}--\ref{section: type II} we elaborate on both
types of refinement and their intersections separately for each of
the primary alignment types {\bf {N}}, {\bf {III}} and {\bf {II}},
where special attention is given to the type {\bf {D}} subcase of
{\bf {II}}. In section \ref{section: type IG} we discuss the split
of the Weyl operator in electric and magnetic parts, which is most
useful for types {\bf {I}} and {\bf {G}}. We conclude in
section~\ref{section: Conclusion} with a discussion and we make some
brief remarks regarding future work. Finally, there are three
appendices. Appendix A summarizes some useful basic facts about the
Jordan normal form of square matrices. Appendix B provides further
details of the Weyl operator classification for type {\bf {III}}.
The intersection of the spin type and eigenvalue structure
classifications in the type {\bf {II}} case is exemplified in
appendix C.
\section{Preliminaries}\label{section: preliminaries}
In this section we recapitulate the necessary definitions and results
from earlier work, meanwhile introducing the notation and
conventions to be used.
\subsection{Boost weight and Weyl tensor constituents}\label{subsec: bw and constituents}
Consider a point $p$ of a 5D spacetime $(M,\gg)$ with Lorentzian metric signature $3$,
and assume that the Weyl tensor at $p$ is non-zero.
By definition this tensor inherits the basic Riemann tensor symmetries and is moreover traceless:
\be
C_{abcd}=C_{cdab},\quad C_{abcd}=-C_{bacd},\quad C_{a[bcd]}=0,\quad C^a{}_{bad}=0 , \label{symm1}
\ee
where round (square) brackets denote complete (anti-)symmetrization as usual.
Let $(\bl,\bn,{\bf m}_i,\,i=3\,..\,5)$ be a null frame of $T_p M$,
consisting of two null vectors $\bl$ and $\bn$, normalized by $l^an_a=1$,
and three spacelike orthonormal vectors $\mm_i$, orthogonal to the null vectors
($\mm_i{}^a\mm_j{}_a=\delta_{ij},\,\mm_i{}^a\bl_a=\mm_i{}^a\bn_a=0$, with $\delta_{ij}$
the Kronecker-delta). We take 0 and 1 to be the frame indices corresponding to $\bl$ and $\bn$,
respectively (e.g., $T_a l^a=T_0$), whereas $i,\,j,\,k,\ldots$ denote
spacelike indices, running from 3 to 5.~\footnote{We omit the
index 2. This is in accordance with \cite{ColHer09}, but in contrast
with \cite{Coleyetal04,Pravdaetal04} and \cite{GarMar09}, where (01,234), (12,345) are
used, respectively.} For a joint notation of the null frame indices we use capital Roman letters $A_i$ in the sequel.
Under a boost in the $(\bl,\bn)$-plane the frame vectors transform according to
\begin{eqnarray}
\bl\mapsto\l\bl,\quad \bn\mapsto\l^{-1}\bn,\quad {\bf m}_i\mapsto {\bf m}_i,\qquad \l\in\mathbb{R}\setminus \{0\},\label{boost}
\end{eqnarray}
such that the components of a rank $p$ tensor $T_{a_1\ldots a_p}$ change as follows:
\begin{equation}\label{boost tensor transform }
T_{A_1\ldots A_p}\mapsto \l^{b_{A_1\ldots A_p}}T_{A_1\ldots A_p},\qquad b_{A_1\ldots A_p}\equiv \sum_{i=1}^p (\delta_{A_i0}-\delta_{A_i1}),
\end{equation}
where $\delta_{AB}$ is the Kronecker delta symbol.
Thus the integer $b_{A_1\ldots A_p}$ is the difference between the numbers of 0- and 1-indices, and is called the
{\em boost weight} (b.w.) of the frame component $T_{A_1\ldots A_p}$ (or, rather, of the $p$-tuple $(A_1,\ldots,A_p)$).
For the Weyl tensor, the conditions (\ref{symm1}) imply that all
components of b.w.\ $\pm 4$ or $\pm 3$ are zero, as well as
algebraic relations between the Weyl components of fixed b.w.
($-2\leq \mbox{b.w.}\leq 2$)~\cite{Coleyetal04,Pravdaetal04}:
\begin{eqnarray}
&&\textrm{b.w.}\ 2:\,\,C_{0}{}^i{}_{0i}=0;\qquad \qquad \textrm{b.w.}\ -2:\,\,C_{1}{}^i{}_{1i}=0;\label{bw 2 rel}\\
&&\textrm{b.w.}\ 1:\,\,C_{010i}=C_{0}{}^j{}_{ij};\qquad \qquad \textrm{b.w.}\ -1:\,\,C_{101i}=C_{1}{}^j{}_{ij};\label{bw 1 rel}\\
&&\textrm{b.w.}\ 0:\,\,2C_{0(ij)1}=C_{i}{}^k{}_{jk},\qquad 2C_{0[ij]1}=-C_{01ij},\qquad 2C_{0101}=-C^{ij}{}_{ij}=2C_{0}{}^i{}_{1i}.\label{bw 0 rel}
\end{eqnarray}
Consider now the spin group, which is isomorphic to $O(3)$ and acts on the null frame according to
\begin{eqnarray}
\bl\mapsto\bl,\quad \bn\mapsto\bn,\quad {\bf m}_j\mapsto {\bf m}_iG^i{}_j,\qquad
G^i{}_jG_k{}^j=\delta^i_k.\label{spin}
\end{eqnarray}
The independent Weyl tensor components of a fixed b.w.\ $q$ define objects which transform under irreducible representations of the spin group.
These objects were presented in \cite{ColHer09}, for general spacetime dimension $n+2$, and are here referred to as the b.w.\ $q$ {\em (Weyl) constituents}.
~\footnote{We will use the notation of the Weyl tensor components and constituents used in \cite{ColHer09}. It may be useful to compare it with that employed in other works. For
negative b.w. components Ref.~\cite{Pravdaetal04} defined
$2\Psi_{ij}=C_{1i1j}$, $2\Psi_{jki}=C_{1ijk}$, and
$\Psi_{i}=-C_{011i}$, while for zero b.w. components, Ref.~\cite{PraPraOrt07} introduced $\Phi_{ij}=C_{0i1j}$,
with $\Phi_{ij}^S$ and $\Phi_{ij}^A$ for its symmetric, respectively, antisymmetric part
and $\Phi$ for its trace. These quantities have then appeared in several subsequent papers. A different set of symbols for the full set of Weyl components has been defined in \cite{Durkeeetal10}.
}
In particular, the components $C_{ijkl}$ (b.w.\ 0) and $C_{1ijk}$ (b.w.\ $-1$) are decomposed as follows ($n\geq 3$):
\begin{eqnarray}
&&C^{ij}{}_{kl}\equiv \Hb^{[ij]}{}_{[kl]}=\bar{C}^{ij}{}_{kl}+\frac{4}{n-2}\delta^{[i}_{[k}\Sb^{j]}{}_{l]}+\frac{2}{n(n-1)}\Rb\delta^{[i}_{[k}\delta^{j]}_{l]},\label{Cijkl}\\
&&C_{1ijk}\equiv \Lc_{i[jk]}=2\delta_{i[j}\check{v}_{k]}+\check{T}_{ijk},\qquad \check{T}^i{}_{ik}=0=\check{T}_{i(jk)}. \label{C1ijk}
\end{eqnarray}
Here $\Hb_{ijkl}$ symbolizes a $n$-dimensional Riemann-like tensor (i.e., a tensor exhibiting all the properties (\ref{symm1}), except for the last one), while $\bar{C}_{ijkl}$, $\Rb\equiv \Hb^{ij}{}_{ij}$ and $\Sb_{ij}\equiv \Hb^k{}_{ikj}-\tfrac 1n \Rb\delta_{ij}$ stand for the associated Weyl tensor, Ricci scalar and tracefree Ricci tensor, respectively. For the case of five dimensions ($n=3$) to be treated here, we have that the b.w.~0 Weyl constituent $\bar{C}_{ijkl}$ vanishes identically,
\begin{equation}\label{Cijkl=0}
\bar{C}_{ijkl}= 0,
\end{equation}
while the b.w.~-1 constituent $\check{T}$ is equivalent to a traceless symmetric matrix $\nc$:
\begin{equation}
\nc_{ij}\equiv \tfrac 12 \e^{kl}{}_{(i}\check{T}_{j)kl}\quad\Leftrightarrow\quad \check{T}^i{}_{jk}=\e_{jkl}\check{n}^{il},\qquad \nc_{ij}=\nc_{(ij)},\quad \nc^i{}_i=0,\label{redef Tcheck}
\end{equation}
where $\e_{ijk}$ denotes the sign of the permutation $(ijk)$ of (345). Analogously for $C_{0ijk}$, giving rise to b.w.~1 constituents $\hat v$ and $\nh$. Regarding the b.w.~0 constituent $A$, defined for general dimensions by $A_{ij}\equiv C_{01ij}$, we will use one more simplification specific to $n=3$ (not made explicit in \cite{ColHer09}): as $A_{ij}$ is antisymmetric in $ij$, we will use its dual vector $\wb$ as the equivalent Weyl constituent:
\begin{equation}\label{redef A}
\wb_i=\tfrac 12 \e_{ijk}A^{jk}\quad\Leftrightarrow\quad A_{ij}= \e_{ijk}\wb^k.
\end{equation}
The symbols of the Weyl constituents and their relation with the Weyl components are summarized in table \ref{dim5}, where the relations (\ref{bw 2 rel})-(\ref{bw 0 rel}) have been implicitly included. The two-index constituents $\Hh$, $\nh$, $\Sb$, $\nc$ and $\Hc$ are traceless and symmetric $3\times 3$ matrices;
those with one index define $3\times 1$ column vectors $\vh$, $\wb$ and $\vc$.
Together with $\Rb$ they add up to the $25+9+1=35$ independent components of the 5D Weyl tensor.
\begin{table}[ht]
\begin{tabular}{|r|l|l|}
\hline
b.w. & Constituents & \;\;\;\;\;\;\;\;Weyl tensor components \\
\hline
$+ 2$ & $\hat{H}_{ij}$ & \;\;$ C_{0i0j}\equiv \hat{H}_{ij}$ \\
\hline
$+1$ & $\hat{n}_{ij}$, $\hat{v}_i$ &
\begin{tabular}{l}$C_{0ijk}\equiv \Lh_{i[jk]}=2\delta_{i[j}\hat{v}_{k]}+\nh_i{}^l\varepsilon_{ljk}$\\
$C_{010i}\equiv \Kh_{i}=-2\hat{v}_i $\end{tabular} \\
\hline
$0$ & $\Sb_{ij}$, $\wb_i$, $\Rb$ & \begin{tabular}{l}
$C^{ij}{}_{kl}\equiv \Hb^{[ij]}{}_{[kl]}=4\delta^{[i}_{[k}\Sb^{j]}{}_{l]}+\frac{1}{3}\Rb\delta^{[i}_{[k}\delta^{j]}_{l]}$ \\
$C_{1i0j}\equiv M_{ij}=-\tfrac
12\Sb_{ij}-\tfrac 16 \Rb\delta_{ij}-\tfrac 12\e_{i
jk}\wb^k$\\ $C_{01ij}\equiv A_{[ij]}=\e_{ijk}\wb^k$\\
$C_{0101}\equiv \Phi=-\tfrac 12 \Rb$
\end{tabular}\\
\hline
$-1$ & $\check{v}_i$, $\check{n}_{ij}$ &
\begin{tabular}{l}$C_{1ijk}\equiv \Lc_{i[jk]}=2\delta_{i[j}\check{v}_{k]}+\check{n}_i{}^{l}\varepsilon_{ljk}$\\
$C_{101i}\equiv \Kc_i=-2\check{v}_i$\end{tabular} \\
\hline
$-2$ & $\check{H}_{ij}$ & \;\;$ C_{1i1j}\equiv\check{H}_{ij}$ \\
\hline
\end{tabular}
\caption{5D Weyl tensor components and constituents.}
\label{dim5}
\end{table}
\subsection{Null alignment type}\label{subsec: alignment type}
Given the null frame $(\bl,\bn,{\bf m}_i)$ of $T_p M$, the {\em boost order} of a rank $p$ tensor
$T_{a_1\ldots a_p}$ with respect to the frame is defined to be the maximal b.w.\ of its non-vanishing components
in the frame decomposition~\cite{Milsonetal05}. This integer is invariant under the subgroup $\mbox{Fix}([\bl])$ of Lorentz transformations fixing the null direction $[\bl]$.~\footnote{The subgroup of $\mbox{Fix}([\bl])$, consisting of special Lorentz transformations, is also known as Sim$(n)$ in $n+2$ dimensions.}
It follows that the boost order is a function of $[\bl]$ only, denoted by $b_T([\bl])$. For the Weyl tensor $C_{abcd}$ and for {\em generic} $\bl$ we have $b_C([\bl])=2$. If a null direction $[\bl]$ exists for which $b_C([\bl])\leq 1$, it is called a {\em Weyl aligned null direction} (WAND) of alignment order $1-b_C([\bl])$. A WAND is called single if its alignment order is 0, and multiple (double, triple, quadruple) if the alignment order is greater than zero (1, 2, 3). The integer
\begin{equation}\label{def zeta}
\zeta\equiv \min_{\bl}\, b_C([\bl])
\end{equation}
is a pointwise invariant of $(M,\gg)$, defining the {\em (Weyl) primary}
or {\em principal alignment type} $2-\zeta$ at $p$; if $\zeta=2,1,0,1$ or
$-2$ this type is still denoted by {\bf {G}}, {\bf {I}}, {\bf {II}}, {\bf {III}} or {\bf {N}},
respectively~\cite{Milsonetal05,Coleyetal04}.~\footnote{This generalizes
the Petrov types {\bf {I}}, {\bf {II}}, {\bf {D}}, {\bf {III}} and {\bf {N}} from four to
higher dimensions.
Petrov types {\bf {II}} and {\bf {D}} together correspond to primary alignment type {\bf {II}}.
Type {\bf {G}} does not occur in four dimensions, but is the generic situation in
higher dimensions~\cite{Milsonetal05}. Type {\bf {O}} corresponds to the trivial
case $C_{abcd}=0$, which we have excluded here.} For types {\bf {N}} and {\bf {III}} the
quadruple, respectively triple, WAND $[\bl]$ is in fact the unique multiple
WAND~\cite{Coleyetal04}.~\footnote{If there was another multiple WAND
$[\bl^*]$ we would get the contradiction $C_{abcd}=0$ by considering
components with respect to a null frame $(\bl,\bl^*,\mm_i)$.} If there is a unique double WAND
in the type {\bf {II}} case we will denote this by {\bf {II}}$_{0}$; if there are more
of them we denote this by {\bf {D}} $\equiv$ {\bf {II}}$_{ii}$, in accordance with
the secondary alignment type notation introduced in
\cite{Coleyetal04,Milsonetal05}.
In the present paper
we will focus on the {\em algebraically special types}
{\bf {N}}, {\bf {III}} and {\bf {II}} (or
{\bf {II}}$_0$ and {\bf {D}} separately) in the Weyl algebraic classification scheme (and
where the context is clear, we will refer
to these algebraically special types simply by
type {\bf {II}} or one of its specializations). Similarly, for type {\bf {I}} we will write {\bf {I}}$_0$ if there is a unique
single WAND, and {\bf {I}}$_i$ if there more than one. In general, a spacetime admits
no WANDs, and we
denote the general case by type {\bf {G}}.
\subsection{The Weyl bivector operator}\label{subsec: Weyl operator}
Let $\wedge^2T_pM$ be the 10-dimensional real vector space of
contravariant bivectors (antisymmetric two-tensors
$F^{ab}=F^{[ab]}$) at $p$. By the first couple of equations in
(\ref{symm1}), the map
\begin{equation}\label{Cop}
{\sf C}:\quad F^{ab}\mapsto \tfrac 12 C^{ab}{}_{cd}F^{cd}=\tfrac 12
F^{cd} C_{cd}{}^{ab}
\end{equation}
determines a linear operator (or endomorphism) on $\wedge^2T_pM$,
which we shall refer to as the {\em Weyl operator}.
A null frame of $T_p M$
\begin{equation}\label{5D null frame}
(\bl,\bn,\mm_i) ,
\end{equation}
where $i=3,4,5$, induces a null basis of $\wedge^2T_pM$:
\begin{equation}\label{calB bivector basis}
(\U_3,\U_4,\U_5,\W,\W_{[45]},\W_{[53]},\W_{[34]},\V_3,\V_4,\V_5),
\end{equation}
consisting of the simple bivectors (in abstract index notation):
\begin{equation}\label{calB def}
\U_i^{ab}=\bn^{[a}\mm_i{}^{b]},\quad \W^{ab}=\bl^{[a}\bn^{b]},\quad \W_{[jk]}^{ab}=\mm_j{}^{[a}\mm_k{}^{b]},\quad \V_i^{ab}=\bl^{[a}\mm_i{}^{b]}.
\end{equation}
The only non-zero scalar products among these (as induced by the spacetime metric) are given by
\[
2\U_i^{ab}\V_{j\,ab}=\delta_{ij},\quad 2\W^{ab}\W_{ab}=-1,\quad 2\W_{[jk]}^{ab}\W_{[lm]\,ab}=\delta_{jl}\delta_{km}-\delta_{jm}\delta_{kl}.
\]
With the notation of table \ref{dim5}, the matrix representation of ${\sf C}$ with respect to (\ref{calB bivector basis})-(\ref{calB def})
can be written in a (3+4+3) block form~\cite{ColHer09}:
\begin{eqnarray}
\label{WeylOperator}
&&{\cal C}\equiv\begin{bmatrix}
M^t & \Ch_K & \Hh \\
\Cc_K{}^t & \Omega & \Ch_{-K}{}^t \\
\Hc & \Cc_{-K}& M
\end{bmatrix},
\end{eqnarray}
where
\begin{eqnarray}
\Ch_{\pm K}\equiv [\pm\Kh\;\Lh],\quad\Omega\equiv\begin{bmatrix}-\Phi & -A^t \\
A & \bar{H} \end{bmatrix},\quad \Cc_{\pm K}\equiv [\pm\Kc\;\Lc].
\label{Omega_etc.}
\end{eqnarray}
Notice that, with respect to the given basis
(\ref{calB def}),
the subspaces
\begin{eqnarray*}
{\cal U}\equiv
\langle \U_3,\U_4,\U_5\rangle,\qquad {\cal W}\equiv
\langle\W,\W_{[45]},\W_{[53]},\W_{[34]}\rangle,\qquad {\cal V}\equiv \langle\V_3,\V_4,\V_5\rangle
\end{eqnarray*}
precisely contain the bivectors of b.w.\ $-1$, 0 and $+1$, respectively.
If we denote the projection operator onto ${\cal X}$ by $p_{\cal X}$ and the restriction to ${\cal X}$ by $\cdot|_{{\cal X}}$, then the block entries of $C$ are the respective matrix representations of the maps
\begin{eqnarray}
&&{\sf M}^t\equiv p_{\cal U}\circ {\sf C}|_{\cal U},\quad {\sf \Ch_{K}}\equiv p_{\cal U}\circ {\sf C}|_{\cal W} ,\quad {\sf \Hh}\equiv p_{\cal U}\circ {\sf C}|_{\cal V}, \label{p_U} \\
&&{\sf \Cc_{K}}{}^t\equiv p_{\cal W}\circ {\sf C}|_{\cal U},\quad {\sf \Omega}\equiv p_{\cal W}\circ {\sf C}|_{\cal W},\quad {\sf \Ch_{-K}}{}^t\equiv p_{\cal W}\circ {\sf C}|_{\cal V},\\
&&{\sf \Hc}\equiv p_{\cal V}\circ {\sf C}|_{\cal U},\quad {\sf \Cc_{-K}}\equiv p_{\cal V}\circ {\sf C}|_{\cal W},\quad {\sf M}\equiv p_{\cal V}\circ {\sf C}|_{\cal V}. \label{p_V}
\end{eqnarray}
\section{Refinements of the null alignment classification}\label{section: refinements}
\subsection{Spin type refinement}\label{subsec: spin type}
\subsubsection{Weyl spin type of a null direction}
The boost order of the Weyl tensor with respect to a null frame
$(\bl,\bn,\mm_i)$ is defined to be the maximal b.w.\ of its
non-vanishing components and is a function of $[\bl]$ only, denoted
by $b_C([\bl])$ (cf.\ section \ref{subsec: alignment type}). The
last statement follows by considering the induced action of an
element $g\in$ Fix$([\bl])$ on the leading terms. A more detailed
consideration leads to other invariants for this action, and thus to
other functions of (that is, properties associated with) $[\bl]$. To
this end, we (uniquely) decompose $g$ into a product
$S[G]\,A[\l]\,N[z]$, where $S[G]$ is a spin (\ref{spin}), $A[\l]$ a
boost (\ref{boost}), and $N(z)$ a null rotation about $[\bl]$,
acting on the null frame according to ($|z^2|\equiv z^iz_i$)
\begin{eqnarray}
\bl\mapsto\bl,\quad \bn\mapsto\bn+z^j{\bf m}_j-\frac{1}{2}|z|^2\bl,
\quad {\bf m}_i\mapsto{\bf m}_i-z_i\bl.\label{null rot l}
\end{eqnarray}
In conjunction with the vanishing of all b.w. $>b_C([\bl])$ terms,
(\ref{null rot l}) readily implies that $N(z)$ leaves the leading terms
(or, equivalently, the b.w.\ $b_C([\bl])$ constituents) invariant~\cite{Milsonetal05}.
Moreover, $A[\l]$ has the effect of simply multiplying the $b_C([\bl])$ constituent
components by a common factor $\l^{b_C([\bl])}$. It follows that spin-invariant quantities
defined by the b.w. $b_C([\bl])$ constituents are properties associated with $[\bl]$. More specifically for 5D Weyl tensors, the spin-irreducible constituents of b.w.\ $b_C([\bl])$ consist of a traceless, symmetric matrix $X$ (for all values of $b_C([\bl])$), a column vector $x$ (for $-1\leq b_C([\bl])\leq 1$) and a scalar $\Rb$ (for $b_C([\bl])=0$ only). This leads to the notion of {\em spin type of $[\bl]$}, as follows.\\
\underline{The matrix $X$} is either zero or (symmetric and thus) diagonalizable by applying spins. By tracelessness this leads to four possible {\em primary spin types of $[\bl]$}, symbolized in a Segre-like notation by
\begin{equation}
\{(000)\}, \qquad \{(11)1\}, \qquad\{110\}, \qquad
\{111\}.\label{Segre types}
\end{equation}
Here a zero (one) indicates a zero (non-zero) eigenvalue, and round brackets indicate equal eigenvalues. Hence, $\{(000)\}$ corresponds to the trivial case $X=0$. For primary spin type $\{(11)1\}$ there is one non-degenerate eigendirection and an eigenplane orthogonal to it. The last two types are the non-degenerate primary spin types characterized by three different eigenvalues, each with one corresponding eigendirection; if the distinction between zero and non-zero eigenvalues is irrelevant
the joint notation $\{111/0\}$ will be utilized.
If we denote the eigenvalues (without multiplicity) by $X_i$, $i=3\,..\,5$, we get $X=\diag[X_3,X_4]\equiv\diag(X_3,X_4,-X_3-X_4)$ after diagonalization.
\begin{comment}
For any cyclic permutation $(ijk)$ of $(345)$, define
\begin{equation}\label{xi_i def}
\xi_i\equiv X_j-X_k,
\end{equation}
such that $\xi_3+\xi_4+\xi_5=0$ (identically).
\end{comment}
For the non-degenerate primary types $\{111/0\}$ we write $X=\diff[X_3,X_4]$ to indicate that $X_i\neq X_j$ for $i\neq j$. We may interchange the $\mm_i$'s such that $X=\diff[X_3,-X_3]$ ($X_5=0$) in the case of spin type $\{110\}$, and $X=X_3\k_3$ in the case of spin type $\{(11)1\}$, where
\begin{eqnarray}
\k_3\equiv\diag\left(1,-\tfrac 12,-\tfrac 12\right)&\leftrightarrow&X_4=X_5=-\tfrac 12 X_3
\label{kappa3}
\end{eqnarray}
The possible normal forms for $X$ are summarized in table \ref{Table: X normal form}. Notice that for $b_C([\bl])=\pm 2$ the case $\{(000)\}$ should be excluded by definition of boost order (cf. Table~\ref{dim5}).\\
\begin{table}
\begin{tabular}{c|cccc}
Primary spin type& $\{(000)\}$ &$\{(11)1\}$&$\{110\}$&$\{111\}$ \\
\hline
Normal form of $X$&0&$X_3\kappa_3$& $\diff[X_3,-X_3]$&$\diff[X_3,X_4]$
\end{tabular}
\caption{Primary spin types of $[\bl]$ and normal forms for $X$ ($X_3X_4\neq 0$).}\label{Table: X normal form}
\end{table}
Next, if $-1\leq b_C([\bl])\leq 1$ we consider \underline{the vector $x$} and first suppose that $X\neq 0$.
If $x$ is non-zero, its position relative to the $X$-eigendirections is a Fix$([\bl])$-invariant, and one distinguishes three
qualitatively different cases. Together with the possibility $x=0$ we get four {\em secondary spin types of $[\bl]$}, which we symbolize by
\begin{itemize}
\item $0$: $x=0$;
\item $\para$: $x$ is parallel to a non-degenerate eigendirection;
\item $\bot$: $x$ is orthogonal to a non-degenerate eigendirection, and not coinciding with the other two such directions in the case of primary types $\{111/0\}$;
\item $\g$: $x$ is in `general' position; i.e., none of the above hold.
\end{itemize}
These symbols are added in subscript to the primary spin type to form the {\em (total) spin type of $[\bl]$}. For spin type $\{110\}_\para$ it may be important to indicate whether $x$ is parallel to the $0$-eigendirection or not, and we will symbolize this by $\{110\}_{\para0}$ and $\{110\}_{\para1}$, respectively. Likewise, for spin type $\{110\}_\p$ we will write $\{110\}_{\p0}$ if $x$ is orthogonal to the $0$-eigendirection, and $\{110\}_{\p1}$ if it is not.
In the case $X=0$ the spin type can be $\{(000)\}_\para$ ($x\neq 0$) for $-1\leq b_C([\bl])\leq 1$, and $\{(000)\}_0[R\neq 0]$ ($x=0$) for $b_C[\bl]=0$ (for this case we will add $\Rb=0$ or $\Rb\neq 0$ between square brackets after the spin type symbol whenever this distinction is important).\\
Regarding normal forms, it is possible and advantageous to order the $\mm_i$'s such that $x_4=0$ (except for spin type $\{111\}_\g$) and $x_3\neq 0$
whenever $x\neq 0$, instead of taking an arrangement where the $X$-normal forms of table \ref{Table: X normal form} are guaranteed. In fact, this has only implications for spin types $\{111\}_{\para 0}$, $\{110\}_{\bot 0}$ and $\{(11)1\}_\bot$, where the normal forms $X=\diff[0,X_4]$, $X=\diff[X_3,0]$ and, for example, $X=X_5\kappa_5$ should be taken instead, where
\begin{eqnarray}
\k_5\equiv\diag\left(-\tfrac 12,-\tfrac 12,1\right)&\leftrightarrow&X_3=X_4=-\tfrac 12 X_5
\label{kappa5}
\end{eqnarray}
Additionally, we will take $X=X_5\kappa_5$ for type $\{(11)1\}_\g$ as well.
The resulting normal forms for the highest b.w.\ constituents are summarized in table \ref{Table: Canforms bw zeta}. For $b_C([\bl])=\pm 1$ the case $\{(000)\}_0$ should be excluded by definition of boost order.
\begin{table}[ht]
\begin{tabular}{|c||l|l|c|}
\hline
Sec.\ type & $\begin{matrix}\{111/0\}\\X=\diff[X_3,X_4]\end{matrix}$ & \hspace{2cm} {$\{(11)1\}$} & $\begin{matrix}\{(000)\}\\X=0\end{matrix}$ \\
\hline
\hline
$0$ & \hspace{.5cm}$x=0$ & $ X=X_3\k_3$,\;\;\; $x=0$ & $x=0$ \\
\hline
$\para$
& $x=(x_3,0,0)$ & $ X=X_3\k_3$,\;\;\; $x=(x_3,0,0)$ & $x=(x_3,0,0)$\\% \hfill \quad [$45$]\\
\hline
$\bot$ & $x=(x_3,0
,x_5)$ & $X=X_5\k_5$,\;\;\; $x=(x_3,0,0)$ & -\\
\hline
$\g$ & $x=(x_3,x_4
,x_5)$ & $X=X_5\k_5$,\;\;\; $x=(x_3,0,x_5)$& -\\
\hline
\end{tabular}
\caption{Spin types of $[\bl]$ and normal forms for $X$ and $x$ ($-1\leq b_C([\bl])\leq 1$). The scalars $x_i$ are non-zero. For spin types $\{111\}_{\para 0}$ and $\{110\}_{\bot 0}$ we have $X_3=0$, resp.\ $X_4=0$, but in all other cases the scalars $X_i$ are non-zero as well.}
\label{Table: Canforms bw zeta}
\end{table}
\subsubsection{Weyl spin type at a point}
Since on the one hand Weyl-preferred null
directions may exist, and since there are only a finite number of possible spin types on the other,
it is natural to define the notion of Weyl spin type at a spacetime point $p$. It is possible to
distinguish four cases:\\
(1) If the Weyl tensor is of alignment type {\bf {N}}, {\bf {III}}, {\bf {II}}$_0$ or {\bf {I}}$_0$, then the WAND of maximal alignment order $1-\zeta$ (cf.\ definition (\ref{def zeta})) is unique.\\
(2) If the alignment type is {\bf {D}}, consider two distinct double WANDs $[\bl]\neq [\bn]$. With respect to null frames $(\bl,\bn,\mm_i)$ and $(\bn,\bl,\mm_i)$, the b.w.\ $\zeta=0$ constituents (which are then the only non-zero Weyl components) relate like $(\Sb,\wb,\Rb)$ and $(\Sb,-\wb,\Rb)$, and it immediately follows that the spin types of $[\bn]$ and $[\bl]$ are the same.\\
(3) If the alignment type is {\bf {I}}$_i$, one considers the single WANDs and, for instance, the total ordering
\begin{equation}
(X_1,x_1) < (X_2,x_2) \quad\Leftrightarrow\quad X_1 < X_2\quad\textrm{or}\quad X_1=X_2,\,x_1<x_2 \label{spin type ordering}
\end{equation}
on the set of possible spin types, where
\begin{eqnarray}
\{(000)\} < \{110\} < \{(11)1\} < \{111\},\qquad 0 \,< \,\,\para\,\, <\, \bot \,<\, \g.\label{Xx ordering}
\end{eqnarray}
(4) If the alignment type is {\bf {G}}, one considers all null directions and, for example, the ordering
\begin{eqnarray}
\{110\} < \{(11)1\} < \{111\} \label{X ordering}
\end{eqnarray}
on the set of possible spin types.
\begin{defn} The {\em Weyl spin type at a spacetime point $p$} is defined to be the spin
type of any maximally aligned null direction in the case of the algebraically special alignment
type {\bf {II}} (and its specializations) and alignment type {\bf {I}}$_0$, and the minimal spin type
of the single WANDs [of all null directions] with respect to the ordering (\ref{Xx ordering})
[(\ref{X ordering})] in the case of alignment type {\bf {I}}$_i$ [{\bf {G}}].
\end{defn}
\begin{rem}
(1) The notion of spin type, introduced here for a 5D Weyl tensor, may be readily
transferred (in principle) to any dimension and any tensor. Regarding the different
alignment types of the Weyl tensor in general dimensions, some spin type subcases have already been pointed out
in earlier work~\cite{Coleyetal04,Ortaggio09,ColHer09}, which will be commented on later.\\
(2) The spin type can be used as a classification tool. In
particular, we may try to determine all spacetimes with given
Ricci-Segre and Weyl alignment-spin types. In this respect, all 5D
Einstein spacetimes ($R_{ab}=\frac{R}{5} g_{ab}$) of alignment type
{\bf {D}} and spin type $\{(11)1\}_\para$, $\{(11)1\}_0$,
$\{(000)\}_\para$ or $\{(000)\}_0[\Rb\neq 0]$ have been invariantly
classified and partially integrated in \cite{ParradoWyll11}; the
collection of these spin types corresponds to the situation where
the Weyl tensor is isotropic in some spacelike plane, in addition
to the boost isotropy in any plane spanned by double WANDs.
\end{rem}
\subsection{Weyl operator refinement}\label{subsec: Weyl operator geometry}
The Weyl operator $\sf C$ on $\wedge^2T_pM$
is characterized by the list of elementary divisors
\[(x-\l_i)^{m_{ij}},\qquad i=1,\ldots,r,\quad j=1,\ldots,\nu_i,\]
where the $\l_i$ are the distinct, possibly complex, eigenvalues of $\sf C$ and $\nu_i\equiv\dim(E_{\l_i})$ is the dimension of the $\l_i$-eigenspace, which equals the number of Jordan blocks corresponding to $\l_i$ in the Jordan normal form of $\sf C$. The {\em Segre type} of $\sf C$ is the list of the orders $m_{ij}$ where, for fixed $i$, round brackets are used to enclose them in the case where $\nu_i>1$. For instance, $[(3211)12]$ would indicate that there are 3 distinct eigenvalues, the first one corresponding to four Jordan blocks of dimension 3, 2, 1 and 1, while the other two correspond to one Jordan block each, of dimensions 1 and 2, respectively. The integer $\sum_{1\leq j\leq\nu_i} m_{ij}$ (equalling 7, 1 and 2 for the respective eigenvalues in the example) is the dimension of the {\em generalized eigenspace} $M^{\l_i}$ corresponding to $\l_i$, which is a $\sf C$-invariant subspace of $\wedge^2T_pM$. A basis of the latter is built by concatenating $\nu_i$ {\em Jordan normal sequences} (JNSs)
of the form
\begin{equation}\label{F_j}
\F_j[m_{ij}]\equiv \left(\F_j,{\sf C}_\l(\F_j),\ldots,{\sf C}_\l^{m_{ij}-1}(\F_j)\right),\qquad {\sf C}_\l\equiv {\sf C}-\l\,\,\id_{\wedge^2T_pM}.
\end{equation}
We shall use the notation $W_k^{'\lambda_i}$ for the span of those bivectors $F_j$ for which the length $m_{ij}$ of the corresponding JNS equals $k$, and $p_k(\lambda_i)$ for the invariant $\dim(W_k^{'\lambda_i})$. The concatenation of the $M^{\l_i}$-bases yields a Jordan normal basis (JNB) realizing the Jordan normal form of ${\sf C}$. We refer to appendix A or standard text books on linear algebra (such as \cite{Lang}) for a further discussion, dealing with vector space endomorphisms in general.
In four dimensions (4D), the Segre type classification of the Weyl
bivector operator is fully equivalent to the alignment type
classification, and both reduce to the 6 distinct Petrov types. In
higher dimensions, however, a particular alignment type can allow
for different Segre types. In the present 5D analysis we shall focus
on the algebraically special alignment type {\bf {II}} (and its
specializations), and regard classification of a certain property of
the Weyl operator as a refinement thereof.
We will treat alignment types {\bf {N}} and {\bf {III}} in full
detail: we shall deduce the possible Segre types, and write down
the kernel $\Ker(\sf C)$, image $\Im(\sf C)$ and a JNB of $\sf C$.
As a type {\bf {N}} ({\bf {III}}) Weyl operator is nilpotent of
index 2 (3), it suffices to study the possibilities of $\rank(\sf
C)$ (and $\rank({\sf C}^2)$), according to formula (\ref{pk lA});
here and henceforth, we denote $\rank(Z)$ as the rank of a
matrix/linear operator $Z$. We also present the compatibility of
the Segre and spin type classifications.
Regarding type {\bf {II}} (covering {\bf {II}}$_0$ and {\bf {D}}), we shall emphasize the classification based on $\rank(M)$ and $\rank(\Omega)$,
and on the potential nilpotence of $\sf C$. We shall also determine the possible spin types in the case where $M$ is nilpotent and $\Om$ has a quadruple eigenvalue.
The determination of {\em all} possible Segre types for a given spin type would
involve a straightforward but tedious investigation.
Rather, for illustration, we shall present the complete eigenvalue degeneracies for
spin types $\{\cdot\}_0$, $\{\cdot\}_\para$ and $\{(11)1\}_\bot$ in
appendix \ref{subsec: type II eigenvalues}.
\section{Type {\bf {N}}}\label{section: type N}
In general dimensions, a type {\bf {N}} Weyl tensor is characterized by having a quadruple WAND $[\bl]$.
With respect to any null frame $(\bl,\bn,\mm_i)$ all components of b.w. greater than $\zeta=-2$
vanish; i.e., only the Weyl constituent $\Hc$ is non-zero:
\begin{equation}\label{type N constituents}
\Hh=0,\quad \nh=\vh=0,\quad \Sb=\wb=\Rb=0,\quad \nc=\vc=0,\quad \Hc\neq 0.
\end{equation}
In fact, the argument given in footnote $^{5}$ shows that $[\bl]$ is the only WAND.
\subsection{Spin types}\label{subsec: type N spin types}
The allowed spin types of the WAND, and thus of the spacetime, are $\{(11)1\}$, $\{110\}$ and $\{111\}$ (these were summed up in section 4.5 of \cite{ColHer09}).
We shall use the normal forms for $X=\Hc$ of table \ref{Table: X normal form}. By an additional boost-normalization, we could naturally take $\Hc=\diff[1,-1]$ in the case of spin type $\{110\}$, and $\Hc_3=\pm \kappa_3$ in the $\{(11)1\}$ case (where $\pm$ is the sign of $X_3$).
\subsection{Weyl operator}\label{subsec: type N Weyl operator}
With (\ref{type N constituents}) and the diagonal normal form of $\Hc$ the Weyl operator takes the form:
\beq\label{WeylOperatorN} {\cal
C}=\begin{bmatrix}
0 & 0 & 0 \\
0 & 0 & 0 \\
\Hc & 0 & 0
\end{bmatrix},\qquad \Hc=\diag(\Hc_3,\Hc_4,\Hc_5).
\eeq
Obviously ${\cal C}^2=0$; i.e., $\sf C$ is nilpotent of index 2 such that its only eigenvalue is 0,
and ${\cal V}\oplus{\cal W}\preceq \Ker({\sf C})$.
Also, ${\sf C}(\U_i)=\Hc_i\V_i$ (no sum over $i$), and by $\Hc_3\Hc_4\neq 0$ it
follows that $2\leq \rank({\sf C})=\rank(\Hc)\leq 3$,
where $\rank({\sf C})=2\Leftrightarrow \Hc_5=0$. From formula (\ref{pk lA}), with
$\l_A=0,\,s(0)=2,\,{\sf N}_A=\sf C$, we get
\[p_2(0)=\rank({\sf C}),\quad p_1(0)=10-2\rank(\sf C).\]
This leads to two possibilities (cf.\ section \ref{subsec: Weyl operator geometry} for the notation):
\begin{enumerate}
\item Segre type $[(2221111)]\,$ $\Leftrightarrow\,\rank({\sf C})=3\,\Leftrightarrow\, \Hc_5\neq 0$, with
\begin{eqnarray*}
&&\Im({\sf C})={\cal V}=\langle\V^3,\V^4,\V^5\rangle,\quad \Ker({\sf C})={\cal V}\oplus{\cal W},\\
&& JNB= (\U_3[2],\U_4[2],\U_5[2],[\W],[\W_{[45]}],[\W_{[53]}],[\W_{[34]}]),
\end{eqnarray*}
corresponding
to $W_2'^{(0)}={\cal U}$ and $W_1'^{(0)}={\cal W}$;
\item Segre type $[(22111111)]\,$ $\Leftrightarrow\,\rank({\sf C})=2\,\Leftrightarrow\,\Hc_5=0$, with
\begin{eqnarray*}
&& \Im({\sf C})=\langle\V^3,\V^4\rangle,\quad \Ker({\sf C})=\langle\U^5\rangle\oplus{\cal V}\oplus{\cal W},\\%\quad \X=[\U^5],[\V^5],
&& JNB= (\U_3[2],\U_4[2],[\U_5],[\V_5],[\W],[\W_{[45]}],[\W_{[53]}],[\W_{[34]}]),
\end{eqnarray*}
corresponding to $W_2'^{(0)}=\langle\U^3,\,\U^4\rangle$ and $W_1'^{(0)}=\langle\U^5\rangle\oplus\langle\V^5\rangle\oplus{\cal W}$.
\end{enumerate}
Here $\U_{i}[2]=[\U^{i},\Hc_i \V_{i}]$ for $\Hc_i\neq 0$ (cf.\ (\ref{F_j})).
\subsection{Intersection of the two refinements}\label{subsec: type N intersection}
The intersection of the spin type and Segre type classifications is trivial
and summarized in table \ref{Table: type N Segre-spin}: case 1 above covers
spin types $\{111\}$ and $\{(11)1\}$, whereas case 2 corresponds precisely to spin type $\{110\}$.
\begin{table}[t]
\begin{tabular}{c|ccc}
& $\{110\}$ & $\{(11)1\}$&$\{111\}$\\
\hline
$[(2221111)]$&-&x&x\\
$[(22111111)]$&x&-&-\\
\end{tabular}
\caption{Type {\bf {N}} Weyl tensors: spin types (columns) for given Segre types (rows). The symbols - and x indicate that the corresponding Segre type is not or is allowed, respectively.}
\label{Table: type N Segre-spin}
\end{table}
\subsection{Comparison with 4D}\label{subsec: type N 4D}
Consider a 4D spacetime which is of
(Weyl-Petrov) type {\bf {N}} at a point. With respect to
any null frame $(\bl,\bn,\mm_3,\mm_4)$ with $[\bl]$ the quadruple,
unique WAND, we formally have (\ref{type N constituents}), so that with respect to a bivector frame
\begin{equation}\label{type N 4D Weyl frame}
(\U_3,\U_4,\W,\W_{[34]},\V_3,\V_4),
\end{equation}
defined as in (\ref{calB def}), the Weyl operator ${\sf C}$ takes the (2+2+2)-block form formally equal to (\ref{WeylOperatorN}). Hence $\sf C$ is nilpotent of index 2. However, $\{11\}$ is the only possible spin type here, where we may take $\Hc=\diag(1,-1)$ after boost-normalization. Also, $\rank({\sf C})=2$ and the only possible Segre type of $\sf C$ is $[(2211)]$. On writing
\begin{eqnarray*}
{\cal V}\equiv \langle\V_3,\V_4\rangle,\quad {\cal W}\equiv
\langle\W,\W_{[34]}\rangle,\quad {\cal U}\equiv \langle
\U_3,\U_4\rangle
\end{eqnarray*}
we have
\begin{eqnarray*}
&&\mbox{Im}({\sf C})={\cal V},\quad \Ker({\sf C})={\cal V}\oplus{\cal W},\\
&& JNB=([\U_3,\V_3],[\U_4,-\V_4],[\W],[\W_{[34]}]).
\end{eqnarray*}
We note that certain four-dimensional results on the Weyl operator (including the rank properties) were given in \cite{Schell61,GolKer61}.
\begin{comment}
The Weyl tensor can be written as
\begin{eqnarray}
&&C^{abcd}=4(\V_3^{ab}\V_3^{cd}-\V_4^{ab}\V_4^{cd})=2\Re(\V^{ab} \V^{cd}),\nonumber\\
&&\V^{ab}\equiv \sqrt{2}\left(\V_3^{ab}-i\,
\V_4^{ab}\right)=2\bl^{[a}\mm^{b]},\qquad\mm^a\equiv
\frac{1}{\sqrt{2}}\left(\mm_3{}^a-i\,\mm_4{}^a\right).\;\;\;\label{Vab}
\end{eqnarray}
This is the well-known Petrov type {\bf {N}} normal form: with respect to the complex null tetrad $(\bl,\bn,\mm,\mmc)$ the Newman-Penrose Weyl coefficients $\Psi_i$, $i=0\,..\,3$, vanish since $[\bl]$ is the quadruple WAND, while (\ref{typeN normal Hc}) apparently corresponds to the normalization $\Psi_4=1$, cf.\ chapter 3 of \cite{Stephanibook}.
\end{comment}
\subsection{Discussion}\label{subsec: type N discussion}
From the above we conclude that {\em for alignment type {\bf {N}} Weyl tensors,
the spin type classification refines the Weyl operator Segre type
classification; the latter coincides with the classification based on the
rank of ${\sf C}$; i.e., on the number of non-zero eigenvalues of the
matrix $\Hc$.} Although only the $5D$ case has been treated here, it is
clear that these statements still hold in $n+2$ spacetime dimensions:
for $\rank({\sf C})=m\geq 2$ we will have precisely $m$ two-dimensional
and $n-2(m-1)$ one-dimensional Jordan blocks, reflected in the Segre
type; this Segre type is independent of the degeneracies of non-zero
$\Hc$-eigenvalues. Notice that $m\geq 2$ is due to the tracelessness of
$\Hc$; the case $m=2$ (Segre type $[(2211\cdots 1)]$)
corresponds precisely to the unique spin type $\{1100\cdots 0\}$, while for fixed
$m>2$ $\Hc$-eigenvalue degeneracies become possible and the corresponding
Segre type covers several spin types.
In 4D the matrix $\Hc$ is two-dimensional and $m=2$ is the only possibility, corresponding to spin type $\{11\}$. Most remarkably, {\em for type {\bf {N}}
non-Kundt Einstein spacetimes ($R_{ab}=\frac{R}{D} g_{ab}$) in {\em any} dimensions $D$, the case $m=2$ is the unique possibility} as well, which is due to the compatibility with the Bianchi identities~\cite{Pravdaetal04,Durkeeetal10}. Some explicit examples have been constructed in \cite{OrtPraPra10}.
However, such a constraint does not apply to Kundt Einstein spacetimes. As an illustration, let us recall the homogeneous plane-wave spacetimes,
\[ \d s^2=2\d u(\d v+a_{ij}x^ix^j\d u)+\delta_{ij}\d x^i\d x^j,\]
where $a_{ij}$ is a constant matrix. If the matrix $a_{ij}$ is traceless then this is Ricci-flat, otherwise some pure radiation will be present. In both cases this metric is of Weyl type {\bf {N}} and the eigenvalue type of $a_{ij}$ is directly related to the spin type of the Weyl tensor.
\section{Type {\bf {III}}}\label{section: type III}
In general dimensions, a type {\bf {III}} Weyl tensor is characterized by having a triple WAND $[\bl]$.
With respect to any null frame $(\bl,\bn,\mm_i)$ all components of b.w. greater than $\zeta=-1$ vanish, whereas those of b.w.~-1 are not all zero; in terms of constituents this is
\begin{equation}\label{type III constituents}
\Hh=0,\quad \nh=\vh=0,\quad \Sb=\wb=\Rb=0,\quad (\nc,\vc)\neq (0,0).
\end{equation}
The argument given in footnote $^{5}$ shows that any
other WAND must be single; i.e., $[\bl]$ is the unique triple WAND and there can be no double WANDs.
The existence of a single WAND corresponds to $\Hc=0$, and is symbolized within the full alignment type notation by {\bf {III}}$_i$~\cite{Coleyetal04}.
\subsection{Spin types}\label{subsec: type III spin types}
The allowed spin types of the unique triple WAND, and thus of the spacetime, are the combinations in table \ref{Table: X normal form}, with the exception of $\{(000)\}_0$ (which would yield type {\bf {N}} or {\bf {O}}).
The secondary spin type `0' (i.e, $\vc=0$) was denoted as type {\bf {III}}(a) in \cite{Coleyetal04,Ortaggio09} and {\bf {III}}(A) in \cite{ColHer09}. The primary spin types were also mentioned in \cite{ColHer09}, where $\{(000)\}$ ($\nc=0$) was denoted by {\bf {III}}(B).
\subsection{Weyl operator}\label{subsec: type III Weyl operator}
From (\ref{type III constituents}), and taking the diagonal normal form for $\nc$, the Weyl operator takes the form:
\begin{eqnarray}\label{WeylOperator III}
{\cal C}\equiv\begin{bmatrix}
0 & 0 & 0 \\
\Cc_K{}^t & 0 & 0 \\
\Hc & \Cc_{-K} & 0
\end{bmatrix},\quad
\Cc_{\pm K}\equiv [\pm \Kc\,\,\Lc]=\begin{bmatrix}\mp 2\vc_3&\nc_3&-\vc_5&\vc_4\\
\mp 2\vc_4&\vc_5&\nc_4&-\vc_3\\\mp 2\vc_5&-\vc_4&\vc_3&\nc_5\end{bmatrix}.
\end{eqnarray}
Obviously we have
${\cal C}^3=0$ and thus 0 is the only eigenvalue, just as for
type~{\bf {N}}. However, contrary to the type~{\bf N} case, a
type {\bf III} Weyl operator satisfies ${\cal C}^2\neq 0$. A proof
hereof, in general dimensions, was given in \cite{Coleyetal04vsi},
Lemma 12; however, let us present a shortcut, specifically for five
dimensions.
\begin{prop} A 5D type {\bf {III}} Weyl operator ${\sf C}$ is nilpotent of index 3; i.e., ${\sf C}^3=0\neq{\sf C}^2$. \label{prop_IIInilp}
\end{prop}
{\bf Proof.} We have
\begin{eqnarray}\label{WeylOperator IIIsquare}
{\cal C}^2\equiv\begin{bmatrix}
0 & 0 & 0 \\
0 & 0 & 0 \\
\Cc_{-K}.\Cc_K{}^t & 0 & 0
\end{bmatrix},
\end{eqnarray}
where a dot denotes matrix multiplication.
Suppose that ${\cal C}^2=0\Leftrightarrow \Cc_{-K}.\Cc_{K}{}^t=0$.
By $\Cc_{\pm K}=[\pm\Kc\,\Lc]$, this is equivalent to
\begin{equation}\label{rank C2 0}
\Lc.\Lc^t=\Kc.\Kc^t.
\end{equation}
This would imply $\rank(\Lc)=\rank(\Lc.\Lc^t)=1$ and hence
$\Lc=ax^t$, with $a,x\in\mathbb{R}^{3\times 1}$. Compatibility with
(\ref{rank C2 0}) then requires $\Lc=\Kc e^t$, with $e$ a unit
vector in $\mathbb{R}^{3\times 1}$.
But then $\Cc_{K}=\Kc[1\,\, e^t]$ such that $2\leq \rank(\Cc_{K})=1$: this is a contradiction.\vrk\\
The difference between the indices of nilpotence serves as an
easily testable criterion for distinguishing the alignment
types~{\bf III} and {\bf N}.
Next, from (\ref{WeylOperator III}) it is clear that the order 2
minors~\footnote{An order $p$ minor
$M\left(\begin{smallmatrix}i_1&i_2&\cdots&i_p\\j_1&j_2&\cdots&j_p\end{smallmatrix}\right)$
of a matrix $M\in\mathbb{R}^{m\times n}$, $1\leq i_1<\ldots<i_p\leq
m$, $1\leq j_1<\ldots<j_p\leq n$, is the determinant of the $p\times
p$ submatrix $A$ of $M$ with $A_{kl}=M_{i_kj_l}$, $1\leq k,l\leq
p$.}
$\Cc_K$$\left(\begin{smallmatrix}i_1&i_2\\j_1&j_2\end{smallmatrix}\right)$
of $\Cc_{\pm K}$ cannot all be zero, as this would lead to
$\vc=\nc=0$ and thus type~{\bf N}. Hence,
$\rank(\Cc_{K})=\rank(\Cck)=\rank(\Cckt)$ equals 2 or 3. Moreover,
from the structure of (\ref{WeylOperator III}) we immediately see
that ${\cal V}\precneqq\Ker({\sf C})$ and
\begin{eqnarray*}
4\leq 2\,\rank(\Cc_K)\leq \rank({\sf C})\leq 3+\rank(\Cc_K)\leq 6.
\end{eqnarray*}
Defining the order 3 minors
\begin{eqnarray}
&&d \equiv \det(\Lc)=\nc_3\nc_4\nc_5+\nc_3\vc_3^2+\nc_4\vc_4^2+\nc_5\vc_5^2,\label{ddef}\\
&&d_i \equiv
2[\vc_j\vc_k(\nc_j-\nc_k)-\vc_i(\vc_3^2+\vc_4^2+\vc_5^2+\nc_j\nc_k)],\label{didef}
\end{eqnarray}
we have
\begin{equation}\label{rankC=6}
\rank({\sf C})=6\quad
\Leftrightarrow\quad\rank(C_{K})=3\quad\Leftrightarrow\quad
D^6\equiv d^2+d_3^2+d_4^2+d_5^2\neq 0.
\end{equation}
Conforming to the canonical forms of table \ref{Table: Canforms bw
zeta}, we take $\vc_3\neq 0$ whenever $\vc\neq 0$, while in the case
$\vc=0$ we may permute the $\mm_i$'s such that $\nc_3\nc_4\neq 0$.
Then, the first and second columns of $\Cc_K{}^t$ are always
independent, as well as columns $j$ and $k$ of $\Cc_{-K}$, where
\begin{eqnarray}
&\vc=0\,\,(\nc_3\nc_4\neq 0):& j=2,\,k=3;\label{jk v=0}\\
&\vc\neq 0\,\,(\vc_3\neq 0):& j=1,\,k=4\,\,\mbox{or}\,\,3.\label{jk
v<>0}
\end{eqnarray}
It is then easily seen that only one order 5 minor is needed to
distinguish between $\rank({\sf C})=4$ and $\rank({\sf C})=5$:
\begin{equation}\label{D45def}
\rank({\sf C})=4\quad \Leftrightarrow\quad D^{45}\equiv {\cal
C}\left(\begin{smallmatrix}j+3&k+3&8&9&10\\1&2&3&j+3&k+3\end{smallmatrix}\right)=0.
\end{equation}
Finally, based on formula (\ref{pk lA}) with $\l_A=0,\,s(0)=3,\,{\sf
N}_A=\sf C$, the numbers $p_i(0)$ of Jordan blocks of dimension $i$
are determined by $\rank(\sf C)$ and $\rank({\sf C}^2)$:
\begin{equation}\label{pk III}
p_1(0)=10-2\rank({\sf C})+\rank({\sf C}^2),\qquad p_2(0)=\rank({\sf
C})-2\rank({\sf C}^2),\qquad p_3(0)=\rank({\sf C}^2).
\end{equation}
For $\rank({\sf C})=6$ we get $2\leq \rank({\sf C}^2)\leq 3$ from
$p_1(0)\geq 0$, where
\begin{eqnarray}\label{CK=3 C2=2}
&&\rank({\sf C}^2)=2\quad\Leftrightarrow\quad \Dth\equiv
\det\left(\Cc_{-K}.C_{K}{}^t\right)\equiv d^2-d_3^2-d_4^2-d_5^2=0.
\end{eqnarray}
For $\rank({\sf C})<6$, proposition \ref{prop_IIInilp} and
$p_2(0)\geq 0$ yield $1\leq \rank({\sf C}^2)\leq 2$, where
\begin{eqnarray*}
\rank({\sf C}^2)=1\quad\Leftrightarrow\quad\Dtw \equiv
(\nc_3^2+\vc_4^2+\vc_5^2-4\vc_3^2)(\nc_4^2+\vc^2-5\vc_4^2)-(\vc_5(\nc_3-\nc_4)-5\vc_3\vc_4)^2=0.
\end{eqnarray*}
This leads to Table~\ref{Table: type III summary}, which
summarizes the possible Segre types that arise in the classification
based on the Weyl operator geometry in the type {\bf III} case. For
further details (like $\Ker({\sf C})$, $\Im({\sf C})$ and the
determination of JNBs in the different cases) we refer to appendix
\ref{subsubsection: type III rank C} and \ref{subsubsection: type
III rank C2}.
\begin{table}[t]
\begin{tabular}{|l||c|l|c|}
\hline
& $\rank(\sf C^2)=3$ & $\rank(\sf C^2)=2$ & $\rank(\sf C^2)=1$ \\
\hline
\hline
$\rank({\sf C})=6$ ($D^6\neq 0$) & $[(3331)]$ ($D^{6<}\neq 0$) & $[(3322)]$ $\;\;$ ($D^{6<}=0$) & - \\
\hline
$\rank({\sf C})=5$ ($D^6=0\neq D^{45}$) & - & $[(33211)]$ \hfill ($D^{45<}\neq 0$) & $[(32221)]$ \hfill ($D^{45<}=0$) \\
\hline
$\rank({\sf C})=4$ ($D^6=0=D^{45}$) & - & $[(331111)]$ \hfill ($D^{45<}\neq 0$) & $[(322111)]$ \hfill ($D^{45<}=0$) \\
\hline
\end{tabular}
\caption{Type {\bf {III}} Weyl bivector operators: summary of the (six) possible Segre types and corresponding conditions on relevant determinants (defined in the text).}
\label{Table: type III summary}
\end{table}
\begin{rem}\label{remark type III_i and rank} If the full alignment type
is {\bf {III}}$_i$ ($\Hc=0$ in (\ref{WeylOperator III})) we clearly
have $\rank({\cal C})=2\rank(\Cc_K)=6$ or 4, i.e., $D^{45}$ vanishes
in this case (cf. (\ref{quant v=0}) and (\ref{gener_v_D45})).
\end{rem}
\subsection{Comparison with 4D}\label{subsec: type III 4D}
Consider a 4D, alignment (Weyl-Petrov) type {\bf {III}} Weyl tensor, and let $[\bl]$ be the unique triple WAND. Referring to the decomposition (\ref{C1ijk}) we have $\check{T}_{ijk}=0$ ($\Leftrightarrow \nc=0$) \cite{ColHer09}, such that the only possible spin type is $\{00\}_\para$. By applying a spin and a boost we may set $\vc\equiv(\vc_3,\vc_4)=(0,1)$, while $\Hc$ can be transformed to zero by a null rotation (\ref{null rot l}) about $[\bl]$, such that the full alignment type is {\bf {III}}$_i$~\cite{Stephanibook,ColHer09}. In this gauge
the Weyl operator
${\sf C}$ takes the (2+2+2)-block form
\begin{equation}
{\cal C}=\begin{bmatrix}
0 & 0 & 0 \\
\Cc_K{}^t & 0 & 0 \\
0 & \Cc_{-K} & 0
\end{bmatrix},\quad
\Cc_K{}^t=\begin{bmatrix}
0&1\\
1&0
\end{bmatrix},\quad \Cc_{-K}=\begin{bmatrix}
0&-1\\
1&0
\end{bmatrix}.
\end{equation}
Obviously ${\sf C}^3=0\neq {\sf C}^2$, $\rank({\sf C})=2\rank({\sf C}^2)=4$, and the only possible Segre type is $[(33)]$ and
\begin{equation*}
\mbox{Im}({\sf C})={\cal W}\oplus{\cal V},\quad \Ker({\sf C})={\cal V},\quad JNB=([\U_3,\W_{[34]},-\V_3],[\U_4,\W,\V_4]).
\end{equation*}
Cf. also \cite{Schell61,GolKer61}.
\begin{comment}
The Weyl tensor can be written as
\begin{eqnarray*}
&&C^{abcd}=8(\W_{[34]}^{\{ab}\V_3^{cd\}}+\W^{\{ab}\V_4^{cd\}})=2\sqrt{2}\Re(-i\V^{\{ab} \W_*^{cd\}}),
\end{eqnarray*}
with $\V^{ab}$ as in (\ref{Vab}) and
\begin{eqnarray*}
&&T^{\{abcd\}}\equiv \frac{1}{2}(T^{[ab][cd]}+T^{[cd][ab]}),\\
&&\W_*^{ab}\equiv 2\left(i
\W_{[34]}^{ab}-\W^{ab}\right)=2\left(\mm^{[a}\mmc^{b]}-\bl^{[a}\bn^{b]}\right).
\end{eqnarray*}
This is the well-known type {\bf {III}} normal form: with respect to the complex null tetrad $(\bl,\bn,\mm,\mmc)$ we have $\Psi_0=\Psi_1=\Psi_2=0$ ($[\bl]$ triple WAND) and $\Psi_4=0$ ($[\bn]$ single WAND), while the normalization $(\vc_3,\vc_4)=(0,1)$ corresponds to $\Psi_3=-i/\sqrt{2}$~\cite{Stephanibook}. {\bf Cf. again also \cite{Schell61,GolKer61}}.
\end{comment}
\subsection{Discussion}\label{subsec: type III discussion}
It is clear that the Weyl operator geometry approach distinguishes
between type {\bf {III}} and type {\bf {N}} (for this purpose it
suffices to consider $\rank({\sf C})$, equivalent to specifying the
number of eigenvectors, or simply the index of nilpotence of ${\sf
C}$). However, for type {\bf {III}} there exist different spin
types that are indistinguishable from the Segre type viewpoint (and
vice versa). We refer to appendix \ref{subsec: type III
intersection} for a full discussion. Table \ref{Table: type III
spin-Segre} summarizes the relation between the spin type and Segre
type refinement schemes for 5D alignment type {\bf {III}} Weyl
tensors. Just as for type {\bf {N}}, a certain Segre type covers
several spin types, and different spin types may allow for exactly
the same list of Segre types. However, except for spin types
$\{(11)1\}_0$, $\{111\}_0$ and $\{(000)\}_\para$, such a list does
not contain a unique element any more (as was the case for type {\bf
{N}}).
Even before studying more general Weyl types (and/or going to higher
dimensions), it is thus already clear from the above 5D type {\bf
{III}} analysis that the two schemes are essentially independent and
represent substantial refinements. This is in contrast to the 4D
case, where type {\bf {III}} exactly corresponds to a Segre type
$[(33)]$ Weyl operator and allows for a single spin type
($\{00\}_\para$); the same happens for the 4D types {\bf {II}} and
{\bf {I}} (see below).
Finally, we mention that some of the discussed features readily
generalize to $n+2$-dimensional type {\bf {III}} Weyl tensors. In particular,
the index of nilpotence for ${\sf C}$ is still 3~\cite{Coleyetal04vsi}
such that formula (\ref{pk III}) still holds, with 10 replaced by
$(n+2)(n+1)/2$. Also, $\rank({\sf C})\leq 2n,\,\rank({\sf C}^2)\leq n$ a
priori, but the lower bounds would need a detailed analysis. Constraints on type {\bf III} Ricci-flat spacetimes were derived in \cite{Pravdaetal04} using the Bianchi identities. Some examples of type {\bf III} Ricci-flat/Einstein spacetimes were constructed in \cite{OrtPraPra10}, where the ``asymptotic'' behavior of the Weyl tensor was also discussed.
\section{Type {\bf {II}}}\label{section: type II}
In general dimensions, a (primary) type {\bf {II}} Weyl tensor is characterized by having a
double WAND $[\bl]$. With respect to any null frame $(\bl,\bn,\mm_i)$ all components
of b.w. greater than $\zeta=0$ vanish, whereas those of b.w.~0 are not all zero; in terms of constituents
this is:
\begin{equation}\label{type II constituents}
\Hh=0,\quad \nh=\vh=0,\quad (\Sb,\wb,\Rb)\neq (0,0,0).
\end{equation}
If the (full) type is D, and if $\bn$ is a second double WAND, then the components of b.w.
less than $0$ also vanish:
\begin{equation}\label{type D constituents}
\nc=\vc=0,\quad \Hc=0.
\end{equation}
\subsection{Spin types}\label{subsec: type II spin types}
The allowed spin types of any double WAND, and thus of the spacetime
(cf.\ section \ref{subsec: spin type}), are the combinations in
table \ref{Table: X normal form}, with the exception of
$\{(000)\}_0[\Rb=0]$. In \cite{Coleyetal04,Ortaggio09,ColHer09} the
secondary spin type `0' (i.e, $\wb=0$) was denoted by subtype {\bf
{II}}(d), the primary spin type $\{(000)\}$ ($\Sb=0$) by {\bf
{II}}(b), and the case $\Rb=0$ by {\bf {II}}(a). In general
dimensions the subtype {\bf {II}}(c) is defined by equation
(\ref{Cijkl=0}); however, this is identically satisfied in five
dimensions.
\subsection{Weyl operator}\label{subsec: type II Weyl operator}
With (\ref{type II constituents}) the block representation of the Weyl operator reduces to
\begin{eqnarray}\label{WeylOperator II}
{\cal C}\equiv\begin{bmatrix}
M^t & 0 & 0 \\
\Cc_K{}^t & \Omega & 0 \\
\Hc & \Cc_{-K} & M
\end{bmatrix}.
\end{eqnarray}
The various submatrices are defined in table \ref{dim5} and (\ref{Omega_etc.}), and
we work with the diagonal normal form for $\Sb$, where
\[\Rb_i\equiv\Sb_{i}+\tfrac 13 \Rb,\]
with $\Sb_i$ being the diagonal entries (i.e., eigenvalues) of $\Sb$.
Then the diagonal block entries $M$ and $\Omega$ take the form
\begin{eqnarray}
M=\begin{bmatrix}-\tfrac{\Rb_3}{2}&-\tfrac{\wb_5}{2}&\tfrac{\wb_4}{2}\\
\tfrac{\wb_5}{2}&-\tfrac{\Rb_4}{2}&-\tfrac{\wb_3}{2}\\-\tfrac{\wb_4}{2}&\tfrac{\wb_3}{2}&-\tfrac{\Rb_5}{2}\end{bmatrix},\quad
\Omega=\begin{bmatrix}\tfrac{\Rb}{2}&-\wb_3&-\wb_4&-\wb_5\\
\wb_3&\tfrac{\Rb}{2}-\Rb_3&0&0\\\wb_4&0&\tfrac{\Rb}{2}-\Rb_4&0\\
\wb_5&0&0&\tfrac{\Rb}{2}-\Rb_5.
\end{bmatrix}.
\end{eqnarray}
We have
\begin{equation}\label{M=0 iff Om=0}
M=0\quad\Leftrightarrow\quad\Omega=0\quad\Leftrightarrow \quad\Rb_i=\wb_i=0,
\end{equation}
which is {\em not allowed for type {\bf II}}. Also notice that $M$, being the sum of an antisymmetric and a diagonalized symmetric part, may represent any $3\times 3$ matrix and thus may take any Jordan normal form, while this is not the case for $\Omega$.
We shall start by showing that a 5D, primary alignment type {\bf
{II}} Weyl operator ${\sf C}$ can be nilpotent, in contrast with the
4D case (cf.\ section \ref{subsec: type II 4D}). We will then
study its classification from different perspectives: Segre type of
$M$ or $\Om$, $\rank(M)$ and $\rank(\Om)$, and spin type. More
precisely, we will exemplify the Segre type classifications by
treating particular cases and determining all possibilities for the
other properties. We then give the conditions for when the ranks
$M$ and $\Om$ attain a given value, and comment on the relation with
$\rank({\sf C})$. We also describe the kernels and images of $M$
and $\Om$, and pay special attention to the type {\bf{ D}} subcase.
Finally, in appendix \ref{subsec: type II eigenvalues} we present
the possible eigenvalue degeneracies for spin types $\{\cdot\}_0$,
$\{\cdot\}_\para$ and $\{(11)1\}_\bot$ and comment on the more
general spin types.
We shall use an extended Segre type notation to indicate the Jordan
block dimensions (elementary divisors) of $M$, $\Om$ and ${\sf C}$
(cf.\ section \ref{subsec: Weyl operator geometry}). If there is a
single eigenvalue zero we indicate this by writing 0 instead of 1.
For a multiple zero eigenvalue, or if we want to emphasize a
specific eigenvalue, we write this value followed by the
corresponding Jordan block dimensions between round brackets. For a
pair of single complex eigenvalues we use $Z\Zc$ or $a\pm i b(Z\Zc)$
(and, e.g., $(Z21)(\Zc21)$ or $a+ib(Z21),a-ib(\Zc 21)$ if both
correspond to one Jordan block of dimension 2 and one of dimension
1). $W\Wc$ indicates a different pair of complex eigenvalues (for
instance, ${\sf C}[\Rb(2211),\pm 3 i\omega(Z\Zc),10]$ indicates that
${\sf C}$ has one eigenvalue $\Rb$ with Jordan block dimensions 2,
2, 1, 1, one pair of single complex eigenvalues $\pm 3 i\omega$, one
single zero and one single non-zero eigenvalue).
\subsubsection{Nilpotence of ${\sf C}$}
\label{subsubsec: type II C nilpotent}
The lower-triangular block structure of a type {\bf {II}} Weyl operator
(\ref{WeylOperator II}) is preserved by taking powers ${\cal C}^k$, which
have $((M^k)^t,\Omega^k,M^k)$ on the diagonal. Hence, if
$M^{k_0}=\Omega^{k_0}=0$ then ${\sf C}^{k_0}$ is (at most) a type {\bf {III}}
operator, and thus ${\sf C}^{3k_0}=0$. Therefore, {\em ${\sf C}$ is nilpotent
if and only if both $\sf M$ and $\sf\Omega$ are nilpotent}.
The characteristic polynomials of $M$ and $\Omega$ are:
\beq
&&k_M(x)=x^3-\s^1_M x^2+\s^2_M x-\s^3_M,\label{kM}\\
&&k_\Omega(x)=x^4-\s^1_\Omega x^3+\s^2_\Omega x^2-\s^3_\Omega x+\s^4_\Omega,\label{kOm}
\eeq
where
\beq
&-2\,\s^1_M=&-2\,\mbox{Tr}\,M=\s^1_\Omega=\mbox{Tr}\,\Omega=\Rb=\Rb_3+\Rb_4+\Rb_5,\label{s1M}\\
&4\s^2_M=&\s^2_\Omega=\Rb_4\Rb_5+\Rb_5\Rb_3+\Rb_3\Rb_4+\wb_3^2+\wb_4^2+\wb_5^2,\label{s2M}\\
&-8\s^3_M=&-8\det(M)=\Rb_3\Rb_4\Rb_5+\Rb_3\wb_3^2+\Rb_4\wb_4^2+\Rb_5\wb_5^2,\label{s3M}\\
&\s^3_\Omega=&-8\s^3_M+\tfrac 14 (\Rb-2\Rb_3)(\Rb-2\Rb_4)(\Rb-2\Rb_5),\label{s3Om}\\
&4\s^4_\Omega=&4\det(\Om)=(\Rb-2\Rb_4)(\Rb-2\Rb_5)\wb_3^2+(\Rb-2\Rb_5)(\Rb-2\Rb_3)\wb_4^2+(\Rb-2\Rb_3)(\Rb-2\Rb_4)\wb_5^2\nonumber\\
&&+\tfrac 14 \Rb(\Rb-2\Rb_3)(\Rb-2\Rb_4)(\Rb-2\Rb_5).\label{s4Om}
\eeq
For further purpose, notice that these coefficients are linear polynomials in
$\wb_3^2$, $\wb_4^2$ and $\wb_5^2$. The matrices $M$ and $\Omega$ are nilpotent
{\em if and only if} (abbreviated {\em iff} henceforward)
all their eigenvalues are zero, which is equivalent to the vanishing of all $\s^i_M$ and
$\s^i_\Omega$. From $\s^1_M=\s^3_\Omega+8\s^3_M=0$ it follows that $\Rb_4=0=\Rb_3+\Rb_5$
(by suitable axis permutation), and then from $\s^2_M=\s^3_M=\s^4_\Omega=0$ we obtain
\beq
\Rb_4=0, \qquad \Rb_3=-\Rb_5\neq 0, \qquad 2\wb_3^2=2\wb_5^2=\Rb_5^2 , \qquad \wb_4=0.
\label{C nilpotent}
\eeq
Hence, {\em ${\sf C}$ is nilpotent iff (\ref{C nilpotent}) is
satisfied.} Notice that the spin type is then $\{110\}_{\bot0}[\Rb=0]$. We easily find that $M^3=0\neq M^2$ and $\Omega^3=0\neq
\Omega^2$ in this case, such that the Segre types of $M$ and $\Omega$ are
$M[0(3)]$ and $\Omega[0(31)]$. Thus, the value of $k_0$ above equals 3, and
the operator ${\sf C}$ will be generically nilpotent of index $9$, but
lower indices $\geq 3$ can occur. In particular, we clearly have that {\em if a
5D, alignment type D Weyl bivector operator is nilpotent, its index of
nilpotence is 3, the Segre type being ${\sf C}[(3331)]$}. Thus, in this
case and from this viewpoint, it is undistinguishable from a generic type
{\bf {III}} Weyl operator. Let us discuss an easily
testable criterion distinguishing between these two situations.
From lemma 8 in \cite{Coleyetal04vsi} we know that any type {\bf
{D}} (and, in fact, any primary type {\bf II}) Weyl tensor (in
arbitrary dimensions) must have a non-zero polynomial invariant,
whereas all such invariants necessarily vanish in the type {\bf
{III}} (or {\bf N}) case.~\footnote{Indeed, this is true for any
tensor: the alignment theorem and the VSI corollary in
\cite{Her11-align} state that a tensor has only vanishing polynomial
invariants if and only if it is of type {\bf III}, or simpler.} The
polynomial invariants of the Weyl tensor $C_{abcd}$ are (by
definition) traces of {\em curvature operators} built from
$C_{abcd}$~\cite{HerCol10-op}. An example of a first order operator
is the aforementioned operator ${\sf C}$ acting on bivector space.
An example of a second order operator is $C^{abcd}C_{defg}$, acting
on the space of contravariant 3-tensors. Remarkably, there is
another natural first order operator associated to the Weyl tensor,
which does not seem to have been considered in the literature before
(not even in the 4D case). This is the following operator, acting on
the space $S^2 (T_p M)$ of {\em symmetric} two-tensors~\footnote{It
can be shown that the same operator acting on bivector space is
equivalent to ${\sf C}$.}:
\begin{equation}\label{Cs}
{\sf C}^s: Z^{ab}=Z^{(ab)}\mapsto C^{a}{}_{c}{}^b{}_d Z^{cd} .
\end{equation}
Now, both ${\sf C}$ and ${\sf C}^s$ are nilpotent in the type {\bf
III} (or {\bf N}) case, and it is easy to verify, in an {\em ad hoc}
manner in the present 5D context, that in the proper type {\bf II}
case where ${\sf C}$ is nilpotent (i.e., when (\ref{C nilpotent})
holds), then ${\sf C}^s$ {\em cannot} be nilpotent at the same time.
However, let us first explicitly prove the following more general result,
slightly strengthening lemma 8 in \cite{Coleyetal04vsi} and valid in
{\em any} dimension:
\begin{prop} Suppose that a rank 4 tensor
$T_{abcd}$ has symmetries $T_{abcd}=-T_{bacd}=T_{cdab}$, is of
primary type {\bf II} or more special and has nilpotent associated
operators ${\sf T}$ and ${\sf T}^s$. Then $T_{abcd}$ is necessarily
of type {\bf III} or {\bf N}, i.e., a frame exists in which all
components of b.w.\ 0, 1 and 2 vanish.
\end{prop}
{\em Note.} For a tensor $T_{abcd}$ satisfying the mentioned
symmetries we can indeed define the operators ${\sf T}$ and ${\sf
T}^s$ on $\Lambda^2 T_pM$ and $S^2(T_pM)$ by formally replacing
$C_{abcd}$ by $T_{abcd}$ in (\ref{Cop}) and (\ref{Cs}),
respectively. A null frame (\ref{5D null frame}) of $T_pM$, where
the indices $i$ now run from 3 to the dimension $(n+2)$ of the spacetime, induce a basis (\ref{calB def}) of
$\Lambda^2 T_pM$ wrt which ${\sf T}$ takes a $3\times 3$ block
matrix representation ${\cal T}$ as in (\ref{WeylOperator}), where
we will use the same symbols as there but with a tilde decoration.
On the other hand, (\ref{5D null frame}) induces the basis
\begin{eqnarray}
\label{S2 frame -21} &&{\bf P}^{ab}={\mbold n^{a}}{\mbold n}^{b},
\qquad \quad{\bf Q}_i^{ab}=\sqrt{2}\,{\mbold n^{(a}}{\bf m}_i^{b)},\\
\label{S2 frame 0}&&{\bf O}^{ab}=\sqrt{2}\,{\mbold n^{(a}}{\mbold
\ell}^{b)}, \quad {\bf O}^{ab}_{ij}=\sqrt{2}\,{\bf m}_i^{(a}{\bf
m}_j^{b)}\;(i<j), \quad {\bf
O}^{ab}_{ii}={\bf m}_i^{a}{\bf m}_i^{b}\;(\text{no sum over $i$}),\\
\label{S2 frame 12}&& {\bf R}_i^{ab}=\sqrt{2}\,{\mbold
\ell^{(a}}{\bf m}_i^{b)}, \qquad \quad {\bf S}^{ab}={\mbold
\ell^{a}}{\mbold \ell}^{b}
\end{eqnarray}
of $S^2(T_p M)$. Notice that the boost orders of the tensors
${\bf P}$, ${\bf Q}_i$, $({\bf O},{\bf O}_{ij})$, ${\bf R}_i$ and
${\bf S}$ along $\bl$ are -2,-1,0,1 and 2, respectively;
accordingly, wrt such a frame the operator ${\sf T}^s$ takes a
$5\times 5$ block matrix representation ${\cal T}^s$.
\begin{proof}
Since $T_{abcd}$ is of primary type {\bf {II}} or more special, a
null frame exists in which all its components of strictly positive
b.w.\ (1 or 2) vanish, such that the block matrices ${\cal T}$ and
${\cal T}^s$ are lower triangular.
Hence, the nilpotence of ${\sf T}$ and ${\sf T}^s$
is equivalent to the nilpotence of all block entries on the
diagonals of ${\cal T}$ and ${\cal T}^s$. Consider first the middle
(b.w.\ 0) block of ${\cal T}^s$, corresponding to (\ref{S2 frame
0}). It is easy to check that it is symmetric,~\footnote{The
$\sqrt{2}$ normalization factors in (\ref{S2 frame 0}) have been
introduced for this reason. Essentially this normalization implies
that the dual basis vectors of the (\ref{S2 frame 0}) frame vectors
are the metric dual ones.} and thus should vanish (since a symmetric
matrix is diagonalizable). In particular we get ${\bf
O}^{ab}T_{acbd}{\bf O}^{cd}=-T_{0101}=0$, and ${\bf
O}^{ab}T_{acbd}{\bf O}_{ij}^{cd}=\tilde M_{ij}+\tilde M_{ji}=0$ (for
all $i$ and $j$) implying that the matrix with entries $\tilde
M_{ij}:=T_{1i0j}$ is antisymmetric. Next, consider the first block
on the diagonal of ${\cal T}$ and the second block on the diagonal
of ${\cal T}^s$: these have entries $2({\bf V}_i)_{ab}{\sf T}({\bf
U}_j)^{ab}=-\tilde M_{ij}$ and $({\bf R}_i)_{ab}{\sf T}^s({\bf
Q}_j)^{ab}=T_{01ij}-\tilde M_{ij}$. Thus they are antisymmetric,
whence diagonalizable and 0. This gives $T_{01ij}=T_{1i0j}=0$.
Finally, the middle block $\tilde\Omega$ of ${\cal T}$ is now
symmetric and thus should also vanish; this gives $T_{ijkl}=0$.
Hence, all b.w.\ 0 components of $T_{abcd}$ also vanish in the
considered frame, and thus the tensor is of type {\bf III} or {\bf
N}.
\begin{comment}
\begin{eqnarray}
&&{\sf E}_1=0, \quad
{\sf E}_2=\begin{bmatrix}
\frac 12(-A-M^t)
\end{bmatrix}, \quad
{\sf E}_3=\begin{bmatrix}
0 & \frac 12(M+M^t) \\
\frac 12(M+M^t) & E
\end{bmatrix},\quad \text{where} \quad E=[ C^{(i\phantom{k}j)}_{\phantom{(i}k\phantom{j)}l}]\nonumber \\
&&
{\sf E}_4=\begin{bmatrix}
\frac 12(A-M)
\end{bmatrix}, \quad {\sf E}_5=0.
\end{eqnarray}
Since the eigenvalues of this operator is equal to the union of
eigenvalues of the block-diagonal pieces, we notice that this operator is
nilpotent only if $M=0$; since this is not allowed for type II, we conclude that ${\sf C}^s$ can never be nilpotent if Weyl is proper type {\bf II} and has therefore some non-zero eigenvalues. This can most easily be seen by first
considering the block ${\sf E}_3$ which is symmetric. A symmetric matrix
is nilpotent if and only if it is zero; i.e., $ \frac 12(M+M^t)=0$.
Inserting this into ${\sf E}_2$ we now conclude that, in addition, $A$ has to be zero
for this to be nilpotent.
\end{comment}
\end{proof}
In particular we thus conclude that \emph{if a proper type {\bf {II}}
Weyl tensor has a nilpotent Weyl operator ${\sf C}$, then the
operator ${\sf C}^s$ has non-zero eigenvalues, and consequently
non-zero polynomial invariants (contrary to any type {\bf III}
tensor).}
\subsubsection{Case of a nilpotent $\sf M$}
\label{subsubsec: type II}
In order to illustrate the classification of a type {\bf {II}} operator ${\sf C}$
from the viewpoint of the extended Segre type of $M$, we shall determine
the possible spin types and the extended Segre types of $\Omega$, in the
case of a nilpotent $M$; i.e., when the extended Segre type of $M$ is
either $M[0(3)]$ ($M^3=0\neq M^2\leftrightarrow\rank(M)=2$) or $M[0(21)]$
($M^2=0\neq M\leftrightarrow\rank(M)=1$).
The matrix $M\neq 0$ is nilpotent iff all of the eigenvalues are zero. This happens iff
\begin{equation}\label{M nilpot general}
\s^1_M=0,\qquad\s^2_M=0,\qquad\s^3_M=0.
\end{equation}
The first equation gives $\Rb=0$,
so that $\Rb_i=\Sb_i$, and the second equation together with (\ref{s2M}) implies that spin type $\{(000)\}_\para$ (all $\Rb_i=0$) is not allowed.
Therefore, when $\sf M$ is nilpotent at least two $\Rb_i$ must be non-zero, and at most
two of them can coincide. Hence, without loss of generality, we may assume that $\Rb_3\neq \Rb_5$.
We can then solve $\s^2_M=\s^3_M=0$ for $\wb_3^2$ and $\wb_5^2$, yielding the {\em necessary and sufficient conditions for $\sf M$ being nilpotent}:
\beq
\Rb_4=-\Rb_3-\Rb_5, \qquad \wb_3^2=\frac{\Rb_5^3-\wb_4^2(\Rb_3+2\Rb_5)}{\Rb_5-\Rb_3} , \qquad \wb_5^2=\frac{\Rb_3^3-\wb_4^2(2\Rb_3+\Rb_5)}{\Rb_3-\Rb_5}
\label{Mnilp} .
\eeq
To determine which spin types are allowed we use the normal
forms of table \ref{Table: Canforms bw zeta} with $(X,x)=(\Sb,\wb)$ and proceed by increasing
the number of zero $\wb$-components.
The generic spin type is $\{111/0\}_\g$, corresponding to $\wb_3\wb_4\wb_5\neq 0$.
If $\wb_4=0\neq \wb_3\wb_5$ we generically have spin type $\{111\}_\bot$; the special case of spin type $\{(11)1\}_g$ corresponds to the subcase
\begin{equation}\label{(11)1g}
\Rb_3=\Rb_4=-\tfrac 12 \Rb_5,\qquad \wb_4=0,\qquad 3\wb_3^2=24\wb_5^2=2\Rb_5^2
\end{equation}
of (\ref{Mnilp}), while $\{110\}_\bot$ leads to $\{110\}_{\bot0}$ and gives back (\ref{C nilpotent}).
If $\wb_4=\wb_5=0\neq \wb_3$, we find from (\ref{Mnilp}):
\beq
\Rb_3=0, \qquad \Rb_4=-\Rb_5\neq 0, \qquad \wb_4=\wb_5=0 , \qquad \wb_3^2=\Rb_5^2.
\label{M2=0}
\eeq
It can be readily verified that this subcase corresponds {\em precisely} to the case
where the nilpotence index of $M$ is 2 (Segre type $M[0(21)]$), whereas in all other
cases this index is 3 (Segre type $M[0(3)]$). Notice that (\ref{M2=0}) is a subcase
of spin type $\{110\}_{\para0}[\Rb=0]$. Finally, $\wb=0$ leads to a diagonal, and hence
non-nilpotent, $M$.
To determine the possible Segre types of $\Omega$, we first determine the possible multiplicities and nature (real or complex) of the eigenvalues. This is most easily done by means of the discriminant sequence of (\ref{kOm})~\cite{YangXia,ColHer11}. Given (\ref{Mnilp}) the coefficients (\ref{s1M})-(\ref{s4Om}) reduce to
\begin{eqnarray}
&&\s^\Omega_1=\s^\Omega_2=0,\label{s12 Om}\\
&&\s^\Omega_3=-2\Rb_3\Rb_4\Rb_5,\label{s3Om M nilp}\\
&&\s^\Omega_4=(\Rb_3-\Rb_4)(\Rb_5-\Rb_4)\wb_4^2-(\Rb_3^2+\Rb_5^2)\Rb_4^2,\label{s4Om M nilp}
\end{eqnarray}
where $\Rb_4=-\Rb_3-\Rb_5$, and the discriminant sequence list becomes
\begin{equation}
[4, 0, -36(\s^\Omega_3)^2, D_4],\qquad D_4 = 256(\s^\Omega_4)^3-27(\s^\Omega_3)^4.\label{Dseq4}
\end{equation}
Here $D_4$ is the classical discriminant of the degree 4 polynomial $k_\Omega(x)$ (given (\ref{s12 Om})) which vanishes iff the latter has multiple roots.
If $\s^\Omega_3=\s^\Omega_4=0$ (i.e., $0$ is a multiple root) we are in the case (\ref{C nilpotent}) of a nilpotent ${\sf C}$.
In all other cases it follows that the so called revised sign list (see \cite{YangXia,ColHer11})
must have at least one sign switch, such that $\Omega$ has at least one pair of complex eigenvalues.
If $D_4>0$ there are two (differing) complex pairs (Segre type $\Om[Z\Zc W\Wc]$), while for
$D_4\leq 0$ there are two additional real eigenvalues, which coincide iff $D_4=0$. In this
last case one can further show that this real (non-zero) eigenvalue must correspond to a
two-dimensional Jordan block (i.e., the Segre type of $\Omega$ must be $[Z\Zc 2]$); if $D_4<0$
we either have $[Z\Zc 11]\,(\s^\Omega_4\neq 0)$ or $[Z\Zc 10]\,(\s^\Omega_4= 0\neq \s^\Omega_3)$.
In the latter case we have $\rank(\Omega)=3$, while $\rank(\Omega)=2$ in the nilpotent case
(\ref{C nilpotent}); in all other cases we have $\rank(\Omega)=4$.
Regarding the compatibility of the spin types and the Segre types of
$\Omega$, we first observe from (\ref{Dseq4}) that $D_4<0$ is implied by
$\s^\Omega_4<0$. Primary spin type $\{110\}$ is equivalent to
$\s^\Omega_3=0$ (see (\ref{s3Om M nilp})), and we may take
$\Rb_4=\Rb_3+\Rb_5=0$ by permuting the axes; moreover, if $\wb_4\neq 0$ we
then see from (\ref{s4Om M nilp}) that $\s^\Omega_4<0$. The case
$\wb_4=0\neq\Rb_4$ immediately yields $\s^\Omega_4<0$ as well. It
follows that $D_4\geq 0$ is only consistent with the most generic spin
type $\{111\}_\g$ and the spin type $\{110\}_{\bot0}$ ($\wb_4=0=\Rb_4$)
corresponding to nilpotent ${\sf C}$; in all other case we have $D_4<0$
and thus Segre type $\Omega[Z\Zc 11]$.
A summary of the allowed spin types and Segre types of $M$ and $\Om$, and their compatibility, in the case of a nilpotent $M$ is given in table \ref{Table: type II M nilp}.
\begin{table}[t]
\begin{tabular}{|c|c|c|c|c|}
\hline
Spin types& Segre type of $M$ &$\rank(M)$&$\Om$ Segre type&$\rank(\Omega)$\\
\hline
\hline
$\{110\}_{\para 0}$ & $[0(21)]$ &1& $[Z\Zc 11]$&4 \\
\hline
$\{110\}_{\bot 0}$ & $[0(3)]$ &2& $[0(31)]$&2 \\
\hline
$\{(11)1\}_\g,\,
\{111\}_\bot,\,\{110\}_\g$ & $[0(3)]$&2 & $[Z\Zc 11]$ &4\\
\hline
$\{111\}_g$& $[0(3)]$&2 & $[Z\Zc 11],\,[Z\Zc W\Wc],\,[Z\Zc 2]$ &4\\
&&&$[Z\Zc 10]$&3\\
\hline
\end{tabular}
\caption{Type {\bf {II}} Weyl tensors: allowed spin types, and
corresponding Segre types and ranks of $M$ and $\Om$, in the case of
a {\em nilpotent matrix $M$}. Spin types $\{110\}_{\bot 0}$,
$\{(11)1\}_\g$ and $\{110\}_{\para 0}$ correspond to the special
cases (\ref{C nilpotent}), (\ref{(11)1g}) and (\ref{M2=0}) of the
characterizing equations (\ref{Mnilp}), respectively. {In all cases
$\Rb=0$.}} \label{Table: type II M nilp}
\end{table}
\subsubsection{Case of a quadruple $\sf \Omega$-eigenvalue}
\label{subsubsec: type II Omega nilpotent}
Given the analysis of the previous paragraph, we now illustrate the
classification of a type {\bf {II}} operator ${\sf C}$ from the viewpoint of the
extended Segre type of $\Omega$, by working out the case where $\Omega$
has a quadruple eigenvalue $x_0$. This comprises the subcase $x_0=0$ of a
nilpotent $\Omega$. Strikingly, we will see that the spin type
classification forms a pure refinement of the $\Om$-Segre type
classification (in the sense that a certain spin type is compatible with
at most one $\Omega$-Segre type) and that the extended Segre type of $M$
is almost always $M[Z\Zc 1]$.
\begin{comment}
The characteristic polynomial $k_\Om(x)$ of the matrix $\Omega$ has a quadruple root, namely $x_0=\tfrac 14 \Rb$,
if and only if
\beq
\s_1^\Om=4x_0,\qquad \s_2^\Om=6x_0^2,\qquad \s_3^\Om=4x_0^3,\qquad \s_4^\Om=x_0^4.\label{s234 spec}
\eeq
\end{comment}
The characteristic polynomial $k_\Om(x)$ of the matrix $\Omega$ has a quadruple root, namely
\begin{equation}
x_0=\tfrac{1}{4}\s^\Om_1=\tfrac 14 \Rb,
\end{equation}
if and only if
\beq
\s_2^\Om=\tfrac 38 \Rb^2,\qquad \s_3^\Om=\tfrac {1}{16} \Rb^3,\qquad \s_4^\Om=\tfrac {1}{256}\Rb^4.\label{s234 spec}
\eeq
By (\ref{s1M}), (\ref{s2M}) and (\ref{s3Om}) it follows that the $\s^M_i$ can be written in terms of the $\Rb_i$ only:
\beq\label{sMs}
\s^M_1=-\tfrac 12 \Rb,\qquad \s^M_2=\tfrac{3}{32}\Rb^2,\qquad \s^M_3=\tfrac {1}{32}(F-\tfrac{1}{4}\Rb^3),
\eeq
where
\beq
F\equiv (\Rb-2\Rb_3)(\Rb-2\Rb_4)(\Rb-2\Rb_5),\qquad \Rb=\Rb_3+\Rb_4+\Rb_5.\label{Fdef}
\eeq
Therefore, the cases where $k_M(x)$ has a root 0 or $x_0$ respectively correspond to
\begin{equation}\label{F conds}
F=\tfrac 14\Rb^3,\qquad F=\tfrac 52\Rb^3,
\end{equation}
and they occur simultaneously iff ${\sf C}$ is nilpotent. Also, the discriminant sequence of a general degree 3 polynomial (\ref{kM}), with arbitrary $\s^i_M$, is
\begin{equation}
[3,F_3,D_3],\qquad F_3\equiv 2(\s^M_1)^2-6\s^M_2,\qquad 27 D_3\equiv -[27\s^M_3-\s_M^1(9\s^M_2-2(\s_M^1)^2)]^2+4F_3^3.\label{Dseq3 gen}
\end{equation}
In the considered situation of a type {\bf {II}} Weyl operator with a quadruple $\Om$-eigenvalue, (\ref{s1M})-(\ref{s2M}) and the first equation of (\ref{s234 spec}) imply $F_3=-\tfrac{1}{16}\Rb^2\leq 0$, such that $D_3\leq 0$ from (\ref{Dseq3 gen}), where $F_3<0\Rightarrow D_3<0$. It follows that {\em either ${\sf C}$ is nilpotent or $k_M(x)$ has a pair of complex roots.} The nilpotent case is characterized here by $\Rb=\Rb_3\Rb_4\Rb_5=0$. Regarding the latter case we will indicate the respective subcases in (\ref{F conds}) by $M[Z\Zc0]$ and $M[Z\Zc1_c]$, and otherwise write $M[Z\Zc1]$.
Just as in the case of a nilpotent $M$, we can show that $\wb=0$ and spin type $\{(000)\}_\para$ are not allowed. The assumptions of $\wb_3\neq 0=\wb_4=\wb_5$ {\em or} primary spin type $\{(11)1\}$ both lead to the case (up to permutation of the axes):
\beq\label{Om(211)}
\wb_4=\wb_5=0,\qquad \tfrac 12\Rb_3=\Rb_4=\Rb_5=\wb_3\qquad (x_0=\wb_3),
\eeq
which is a subcase of spin type $\{(11)1\}_{\para}[\Rb\neq 0]$. It can be verified that this is the only case where $\Om$ has Jordan blocks of dimension at most 2, and that we have extended Segre types $\Om[\wb_3(211)]$ and
$M[-\tfrac{1\pm i}{2}(Z\Zc),-\wb_3(1)]$
and thus ${\sf C}[(ZZ)(\Zc\Zc)(11)(1111)]$. In all other cases,
given (\ref{s2M}), (\ref{s3Om}) and (\ref{s4Om}), (\ref{s234 spec}) can be solved for the $\wb_i^2$, yielding:
\begin{equation}\label{Om(4)}
\wb_3^2=\frac{\left(\Rb_3-\tfrac 14\Rb\right)^4}{(\Rb_3-\Rb_4)(\Rb_3-\Rb_5)} , \quad \wb_4^2=\frac{\left(\Rb_4-\tfrac 14\Rb\right)^4}{(\Rb_4-\Rb_5)(\Rb_4-\Rb_3)} , \quad \wb_5^2=\frac{\left(\Rb_5-\tfrac 14\Rb\right)^4}{(\Rb_5-\Rb_3)(\Rb_5-\Rb_4)}.
\end{equation}
By substituting this into $\left(\Om-\tfrac 14 \Rb\right)^3$ we find that Segre
type $\Om[(31)])$ corresponds precisely to spin type $\{111/0\}_\bot$ (i.e., the situation
where exactly one $\wb$-component vanishes); in agreement with table \ref{Table: Canforms bw zeta} this is for
\begin{equation}\label{wb4=0 cond}
\wb_4=0\neq \wb_3\wb_5\quad\Leftrightarrow\quad 4\Rb_4-\Rb=0\neq (4\Rb_3-\Rb)(4\Rb_5-\Rb).
\end{equation}
The case $\Rb=\Rb_4=0$ is equivalent to spin type $\{110\}_{\bot 0}$ and
gives the case (\ref{C nilpotent}) of nilpotent ${\sf C}$; for $\Rb\neq
0$ one necessarily has spin type $\{111\}_\bot$, (i.e., $\{110\}_{\bot
1}$ is not allowed) and extended Segre type $M[Z\Zc 1]$ (i.e.,
(\ref{wb4=0 cond}) is incompatible with (\ref{Fdef})-(\ref{F conds})).
Finally, the Segre type $\Om[4]$ allows for the spin types
$\{110\}_\g[\Rb\neq 0]$, $\{111\}_\g[\Rb=0]$ and $\{111\}_\g$, where the
second one gives a {\em nilpotent $\Om$} and only the last one is compatible with
either of the equations in (\ref{F conds}).
A summary of the allowed spin types and Segre types of $\Om$ and $M$, and their compatibility, in the case of a quadruple $\Om$-eigenvalue is given in table \ref{Table: type II Om quadr}.
\begin{table}[t]
\begin{tabular}{|c|c|c|c|c|}
\hline
Spin types& $\Om$ Segre type &$\rank(\Om)$&$M$ Segre type&$\rank(M)$\\
\hline
\hline
$\{(11)1\}_{\para}[\Rb\neq 0]$ & $[(211)]$ &4& $[Z\Zc 1]$&3 \\
\hline
$\{110\}_{\bot 0}[\Rb=0]$ & $[0(31)]$ &2& $[0(3)]$&2 \\
\hline
$\{111\}_\bot[\Rb\neq 0]$ & $[(31)]$&4 & $[Z\Zc 1]$ &3\\
\hline
$\{111\}_\g[\Rb=0]$ & $[0(4)]$&3 & $[Z\Zc 1]$ &3\\
\hline
$\{110\}_\g[\Rb\neq 0]$ & $[4]$&4 & $[Z\Zc 1]$ &3\\
\hline
$\{111\}_\g[\Rb\neq 0]$ & $[4]$&4 & $[Z\Zc 1]$, $[Z\Zc 1_c]$ &3\\
&&&$[Z\Zc 0]$&2\\
\hline
\end{tabular}
\caption{Type {\bf {II}} Weyl tensors: allowed spin types, and corresponding Segre types and ranks of $\Om$ and $M$, in the case where {\em $\Om$ has a quadruple eigenvalue}. The second row corresponds the the case of a nilpotent~${\sf C}$.}
\label{Table: type II Om quadr}
\end{table}
\subsubsection{Classification based on $\rank(M)$ and $\rank(\Om)$}\label{subsubsec: type II rank M}
The classification by the ranks of $M$ and $\Om$, and their intersection, gives a rather course subclassification of an alignment type {\bf {II}} Weyl tensor, but in combination with the determination of the (kernel and) image of ${\sf C}$ may be useful for particular purposes.
For example, the image has a particular implication for the holonomy group of the spacetime under consideration. The infinitesimal generators of the holonomy group are spanned by:
\[ R_{abcd}X^cY^d, \quad R_{abcd;e_1}X^cY^dZ^{e_1},\quad ...\]
for all vectors $X, ~Y,~Z,...\in T_pM$, through the isomorphsim $\bar\iota: \wedge^2T^*_pM\mapsto {\mf o}(1,n-1)$, where ${\mf o}(1,n-1)$ is the Lie algebra of the Lorentz group. The map $\bar\iota$ is explicitly given by raising an index: $X_{ab}\mapsto X^a_{~b}$. Similarly, by lowering an index we have an isomorphism ${\iota}: \wedge^2T_pM\mapsto {\mf o}(1,n-1)$. In particular, this implies that the image of the Riemann bivector operator generates a vector subspace of $\mf{h}\mf{o}\mf{l}$ (the Lie algebra of the holonomy group); i.e.,
\[{\iota}(\mathrm{Im}({\sf R}))\subset \mf{h}\mf{o}\mf{l}.\]
In the case of a Ricci-flat spacetime, this implies that
${\iota}(\mathrm{Im}({\sf C}))\subset \mf{h}\mf{o}\mf{l}$, and thus we can
obtain a minimal dimension for the holonomy algebra by considering the rank of ${\sf C}$.
Indeed, if the ${\iota}(\mathrm{Im}({\sf C}))$ does not close as a Lie algebra,
we can consider the algebra it generates (which must also be in $\mf{h}\mf{o}\mf{l}$). For example, since the dimension of the Lorentz group is 10, we immediately get that the vacuum cases where $\rho({\sf C})=10$ generate the whole Lorentz group (=holonomy group). Indeed, since there are no 9 or 8-dimensional proper subgroups of the Lorentz group, we also have that $\rho({\sf C})\geq 8$ must have the full Lorentz group as its holonomy group. Thus to conclude, the image $\Im({\sf C})$ generates a subalgebra of the infinitesimal holonomy algebra.
The allowed spin types in each case can be computed.
The generic case, of course, corresponds to having $\rank(M)=3$ and
$\rank(\Omega)=4$. We will now work out all possible special cases where
lower rank combinations are allowed. From (\ref{M=0 iff Om=0}) we have
$\rank(M)\neq 0 \neq\rank(\Omega)$ and cases of zero-rank matrices can
thus be ruled out from the start. Moreover, by considering the order~2
minors of $\Om$ it is easily shown that $\rank(\Omega)=1$ is not
allowed either. As a final preliminary remark we mention that $\rank(\Om)=2$
(all order 3 $\Om$-minors being zero) leads to the two cases (\ref{2_2b})
and (\ref{2_2a}) below, for which $\rank(M)=2$.
\begin{comment}
Let us first consider the classification based on $\rank(\Om)$ on its own. Firstly, $\rank(\Omega)=1$ is not allowed, which easily follows from considering the order 2 minors of $\Om$. Also, the order 3 minors yield that $\rank(\Omega)=2$ occurs in two qualitatively different cases:
\begin{eqnarray}
&& \Rb_3=\Rb_4=\Rb_5=0;\label{Rbi vanish}\\
&&\Rb_4=\Rb_3+\Rb_5,\quad \wb_4=0,\quad (\Rb_3+\Rb_5)\Rb_3\Rb_5+\Rb_3\wb_3^2+\Rb_5\wb_5^2=0.\label{second case}
\end{eqnarray}
The first case is spin type $\{(000)\}_\para[\Rb=0]$. For both cases we have $\det(M)=0$, such that
\begin{equation}\label{rank Om 2 and M}
\rank(\Omega)=2\Rightarrow \rank(M)<3.
\end{equation}
These are subcases of $\rank(\Om)<4$, which is expressed by $\det(\Om)=0$, i.e., by the vanishing of (\ref{s4Om}).
Suppose that $\rank(\Omega)<3$, i.e., all order 3 minors of $\Omega$ vanish. From $\Omega\left(\begin{smallmatrix}2&3&4\\2&3&4\end{smallmatrix}\right)=0$ it follows that e.g. $R_3=R_4+R_5$, and then
\[\Omega\left(\begin{smallmatrix}1&2&3\\1&2&3\end{smallmatrix}\right)=\wb_3^2\Rb_5=0,\quad \Omega\left(\begin{smallmatrix}1&2&4\\1&2&4\end{smallmatrix}\right)=\wb^2\Rb_4=0.\]
This leads to two cases:
\begin{enumerate}
\item $\Rb_3=\Rb_4=\Rb_5=0$, i.e.\ $\Sb=\Rb=0$.
Clearly, $\det(M)=0$ and $\rank(\Omega)=2$ (since $(\wb_3,\wb_4,\wb_5)\neq (0,0,0)$).
\item $R_3=R_4+R_5$ and $\wb_3=0$. Then the second row and column of $\Omega$ vanish, and we have
\begin{equation}
\rank(\Omega)<3\quad \Leftrightarrow\quad \Omega\left(\begin{smallmatrix}1&3&4\\1&3&4\end{smallmatrix}\right)=-8\det(M)=0.
\end{equation}
By considering the order 2 minors it now easily follows that $\rank(\Omega)=1$ is not allowed.
\end{enumerate}
\end{comment}
We now work out the classification based on $\rank(M)$, and its intersection with that of $\rank(\Om)$.
\paragraph{\underline{$\rank(M)=3$}} This is the case of a generic matrix $M$, corresponding to
\begin{equation}\label{detM}
-8\det(M)=\Rb_3\Rb_4\Rb_5+\Rb_3\wb_3^2+\Rb_4\wb_4^2+\Rb_5\wb_5^2\neq 0.
\end{equation}
If $\Omega$ is also generic then $\rank(\Omega)=4$. The case $\rank(\Omega)<4$ necessarily
gives $\rank(\Omega)=3$ by the above remark. This happens if $\det(\Om)=0$
(i.e., if (\ref{s4Om}) vanishes while (\ref{detM}) is valid), and occurs for either
\beqn
& & w_3^2=\frac{\Rb_4+\Rb_5-\Rb_3}{4(\Rb_4-\Rb_5-\Rb_3)(\Rb_4-\Rb_5+\Rb_3)}{\cal P} , \nonumber \label{3_3a} \\
& & {\cal P}= \Rb_3[\Rb_3^2+\Rb_3(\Rb_4+\Rb_5)-(\Rb_4-\Rb_5)^2+4(\wb_3^2+\wb_4^2)]-\Rb_5[\Rb_5^2-\Rb_4\Rb_5-\Rb_4^2+4(\wb_4^2-\wb_5^2)] \nonumber \\
& & \qquad\qquad\qquad\qquad {}-\Rb_4[\Rb_4^2-4(\wb_4^2-\wb_5^2)] ,
\eeqn
or
\beq
\Rb_3=0=\Rb_4-\Rb_5 .
\label{3_3b}
\eeq
In the latter case (\ref{detM}) reduces to $\Rb_5(\wb_4^2+\wb_5^2)\neq 0$.
It is understood that other cases can be obtained from these two by simple axis-permutations (hereafter we will not mention such possibilities any further).
The case of nilpotent $\Om$, corresponding to (\ref{Om(4)}) with $\Rb=0\neq \Rb_3\Rb_4\Rb_5$,
is a subcase of (\ref{3_3a}) (cf.\ also table \ref{Table: type II Om quadr}).
\paragraph{\underline{$\rank(M)=1$}} This is the case where all order 2 minors of $M$ vanish, which happens iff
\beq
\Rb_3=0, \qquad \wb_4=\wb_5=0, \qquad \wb_3^2=-\Rb_4\Rb_5,\quad (\Rb_4,\Rb_5)\neq (0,0).
\label{rankM=1}
\eeq
We have $\s^M_3=\s^M_2=0$, and generically the zero-$M$ eigenvalue is double
($\s^M_1\sim\Rb\neq 0$).
It becomes triple (i.e., $\sf M$ is nilpotent of index 2) for $\Rb=\Rb_4+\Rb_5=0$ (cf.~(\ref{M2=0})).
Notice that $\Rb_4\neq\Rb_5$, and since $16\det(\Omega)=-(\Rb_4-\Rb_5)^4$ by (\ref{s4Om}) we have
$\rank(\Omega)=4$.
\paragraph{\underline{$\rank(M)=2$}} By setting $\s^M_3=\det(M)=0$ to zero we find all cases
with $\rank(M)<3$, where we just have to exclude (\ref{rankM=1}) and its permutations
to get $\rank(M)=2$ (this is implicitly understood in the following). Generically,
$M$ has a single zero eigenvalue. This comprises the case where all $R_i$'s coincide;
by (\ref{detM}) it then follows that all of the $R_i$'s are zero, which is the case (\ref{2_2b}) below.
Otherwise, it is always possible to solve $\s^M_3=0$ for one $\wb_i^2$,
and we may assume $\Rb_3\neq \Rb_5$. The zero eigenvalue becomes at least double
iff $\s^M_3=\s^M_2=0$, which is the case iff
\beq\label{wb35}
\wb_3^2=\frac{-\Rb_5^2(\Rb_3+\Rb_4)+\wb_4^2(\Rb_4-\Rb_5)}{\Rb_5-\Rb_3} ,
\qquad \wb_5^2=\frac{-\Rb_3^2(\Rb_4+\Rb_5)+\wb_4^2(\Rb_4-\Rb_3)}{\Rb_3-\Rb_5}
\eeq
(note that different subcases have already been discussed in the tables).
There is a triple zero eigenvalue iff, in addition, $\s^M_1=0$, which is the case (\ref{Mnilp}) of a nilpotent $\sf M$ of index 3; the last three rows of table \ref{Table: type II M nilp} produce examples fitting in the subsequent discussion of $\rank(\Om)$.
Generically, we have $\rank(\Omega)=4$. For $\rank(\Omega)<4$ we have two possibilities.
If (\ref{3_3b}) holds then $\det(\Om)=0$ automatically,
and $\det(M)=0\Leftrightarrow \Rb_5(\wb_4^2+\wb_5^2)=0$, which leads to
(\ref{2_2b}) and a subcase of (\ref{2_2a}). If (\ref{3_3b}) or its permutations do not hold,
we necessarily have (e.g., $(\Rb_4-\Rb_5)(\Rb_3-\Rb_4-\Rb_5)\neq 0$) and the conditions $\det(M)=0=\det(\Omega)$ are solved simultaneously by
\beqn
& & \wb_4^2=\frac{(\Rb_3-\Rb_4+\Rb_5)^2}{\Rb_4-\Rb_5}\left[\frac{\Rb_5-\Rb_3}{(\Rb_3-\Rb_4-\Rb_5)^2}\wb_3^2+\frac{1}{4}\Rb_5\right] , \nonumber \label{2_3} \\
& & \wb_5^2=\frac{(\Rb_3+\Rb_4-\Rb_5)^2}{\Rb_4-\Rb_5}\left[\frac{\Rb_3-\Rb_4}{(\Rb_3-\Rb_4-\Rb_5)^2}\wb_3^2-\frac{1}{4}\Rb_4\right] .
\eeqn
This generically corresponds to $\rank(\Omega)=3$.
The case $\rank(\Omega)=2$ arises for two different choices of the parameters.
\begin{enumerate}
\item The first possibility is
\beq
\Rb_i=0 , \quad i=3,4,5 \label{2_2b} .
\eeq
This is precisely spin type $\{(000)\}_\para[\Rb=0]$. In this case the extended Segre types are $M[\pm i\omega(Z\Zc),0]$, $\Om[\pm i\omega(Z\Zc),0(11)]$ and thus ${\sf C}[i\omega(Z111),-i\omega(\Zc111),0(1111)]$.
\item The second possibility is
\beq
\Rb_5=\Rb_4-\Rb_3 , \qquad \wb_4=0, \qquad \wb_5^2=\frac{\Rb_3(\Rb_3\Rb_4-\Rb_4^2-\wb_3^2)}{\Rb_4-\Rb_3} , \label{2_2a}
\eeq
which can be understood as a subcase of~(\ref{2_3}). The further subcase
hereof
\beq
\Rb_4=0, \qquad \wb_3^2=\tfrac{1}{2}\Rb_3^2
,
\label{IInilpotent}
\eeq
is precisely the case (\ref{C nilpotent}) of a nilpotent ${\sf C}$ (cf.\ also table \ref{Table: type II M nilp}).
\end{enumerate}
A summary for the possible relative ranks of $M$ and $\Omega$ is given in table \ref{Table: type II M/Omega}.
\begin{table}[t]
\begin{tabular}{|l||c|c|c|c|}
\hline
& $\rank(\Omega)=4$ & $\rank(\Omega)=3$ & $\rank(\Omega)=2$ & $\rank(\Omega)=1$ \\
\hline
\hline
$\rank(M)=3$ $\Leftrightarrow$ (\ref{detM}) & generic case & (\ref{3_3a}) or (\ref{3_3b}) & - & - \\
\hline
$\rank(M)=2$ & (\ref{w3_M2}) & (\ref{2_3}) & (\ref{2_2b}) or (\ref{2_2a}) & - \\
\hline
$\rank(M)=1$ & (\ref{rankM=1}) & - & - & - \\
\hline
\end{tabular}
\caption{Weyl operator geometry for type {\bf {II}} spacetimes: possible relative ranks of $M$ and $\Omega$ and equation numbers of the corresponding conditions (up to permutations of the axes).}
\label{Table: type II M/Omega}
\end{table}
\subsubsection{\Ker({\sf M}) and $\Ker({\sf\Omega})$}\label{subsubsec: type II kernels M/Omega}
From (\ref{WeylOperator II}) it is easy to see that $\Ker({\sf M})\preceq\Ker({\sf C})$.
Of course, in the generic case ${\sf M}$ and ${\sf\Omega}$ have full rank (i.e., $\rank({\sf M})=3$ and $\rank({\sf\Omega})=4$), so that $\Ker({\sf M})=\{0\}$ and $\Ker({\sf \Omega})=\{0\}$. The spin type is generically $\{111\}_g[\Rb\neq0]$, but many special subcases are possible.
In particular, we have the type $\{(000)\}_\para[\Rb\neq0]$ if $\Rb_3=\Rb_4=\Rb_5$,
or $\{(000)\}_0[\Rb\neq0]$ if, additionally, $\wb=0$. For these two simple types the
conditions $\rank({\sf M})=3$ and $\rank({\sf\Omega})=4$ are, in fact, necessary.
The more special cases of table~\ref{Table: type II M/Omega} are now discussed.
\paragraph{\underline{$\rank(M)=3$, $\rank({\Omega})=3$}}
This case is defined by either (\ref{3_3a}) or (\ref{3_3b}). Clearly $\Ker({\sf M})=\{0\}$ here.
\begin{enumerate}
\item When (\ref{3_3a}) is satisfied we find
that
\beqn
\Ker({\sf\Omega})=\langle\W^+\rangle , \quad \W^+ = & & [(\Rb_4-\Rb_5)^2-\Rb_3^2]\W+2(\Rb_4-\Rb_5+\Rb_3)\wb_4\W_{[53]} \nonumber \\
& & {}+2(\Rb_5+\Rb_3-\Rb_4)\wb_5\W_{[34]}+\sqrt{\frac{{\cal P}[\Rb_3^2-(\Rb_4-\Rb_5)^2]}{\Rb_3-\Rb_4-\Rb_5}}\W_{[45]} .
\label{33}
\eeqn
The spin type is generically $\{111\}_g[\Rb\neq0]$, but many special subcases are possible.
\item If, instead, (\ref{3_3b}) holds we get
\beq
\Ker({\sf\Omega})=\langle\W^*\rangle , \qquad \W^* = -\wb_5\W_{[53]}+\wb_4\W_{[34]} .
\eeq
Here the spin type is $\{(11)1\}_g[\Rb\neq0]$; it specializes to $\{(11)1\}_\p[\Rb\neq0]$ if $\wb_3=0$.
\end{enumerate}
\paragraph{\underline{$\rank(M)=2$, $\rank(\Omega)=4$}} It is easy to see that if
$\Rb_3=\Rb_4=\Rb_5=0$ then $\rank(\Omega)=2$; therefore, here we can assume that
the $\Rb_i$ are not all identically zero. Let us take, for definiteness, $\Rb_3\neq0$. Then, from~(\ref{detM}) the condition $\det(M)=0$ gives
\beq
\wb_3^2=-\frac{\Rb_3\Rb_4\Rb_5+\Rb_4\wb_4^2+\Rb_5\wb_5^2}{\Rb_3} .
\label{w3_M2}
\eeq
It is now straightforward to see that
\beq
\Ker({\sf M})=\langle\V^+\rangle , \qquad \V^+=\frac{\Rb_4\wb_4^2+\Rb_5\wb_5^2}{\Rb_3}\V_3-(\Rb_5\wb_5+\wb_3\wb_4)\V_4+(\Rb_4\wb_4-\wb_3\wb_5)\V_5 ,
\label{24}
\eeq
where we substitute $\wb_3$ from~(\ref{w3_M2}). Again, the spin type is generically $\{111\}_g[\Rb\neq0]$, but many special subcases are possible.
\paragraph{\underline{$\rank(M)=2$, $\rank(\Omega)=3$}}
Using~(\ref{2_3}) we can easily compute the generators of the one-dimensional
spaces $\Ker({\sf M})$ and $\Ker({\sf \Omega})$; the expressions are rather
long and non-illuminating, and are therefore omitted.
The spin type is in general $\{111\}_g[\Rb\neq0]$. This specializes
to $\{(11)1\}_g[\Rb\neq0]$ (but in a non-canonical frame) if $\Rb_4=\Rb_3$, or
to $\{111\}_g[\Rb=0]$ if $\Rb_5=-\Rb_3-\Rb_4$. When $\wb_3=0$ we get the type $\{111\}_\p[\Rb\neq0]$, which becomes $\{111\}_\p[\Rb=0]$ if, additionally, $\Rb_5=-\Rb_3-\Rb_4$. Further, the type is $\{110\}_g[\Rb\neq0]$ if $\Rb_5=2\Rb_3-\Rb_4$; this becomes $\{110\}_{\p0}[\Rb\neq0]$ with the further condition $\wb_3=0$, or $\{110\}_{\p1}[\Rb\neq0]$ for $\wb_4=0$.
\paragraph{\underline{$\rank(M)=2$, $\rank(\Omega)=2$}}
There are two different possibilities corresponding to this situation.
\begin{enumerate}
\item When~(\ref{2_2b}) holds we get
\beqn
& & \Ker({\sf M})=\langle\V^*\rangle , \qquad \V^*=\wb_3\V_3+\wb_4\V_4+\wb_5\V_5 , \label{22b} \\
& & \Ker({\sf \Omega})=\langle\W^{*1},\W^{*2}\rangle , \qquad
\W^{*1}=-\wb_5\W_{[45]}+\wb_3\W_{[34]} , \quad \W^{*2}=-\wb_4\W_{[45]}+\wb_3\W_{[53]} . \nonumber
\eeqn
This corresponds univocally to spin type $\{(000)\}_\para[\Rb=0]$.
\item Assuming, instead, that~(\ref{2_2a}) is satisfied we obtain
\beqn
& & \Ker({\sf M})=\langle\V^\times\rangle , \qquad \V^\times=\wb_5(\Rb_3-\Rb_4)\V_3-\Rb_3(\Rb_3-\Rb_4)\V_4+\wb_3\Rb_3\V_5 , \label{22a} \\
& & \Ker({\sf \Omega})=\langle\W_{[53]},\W^\times\rangle , \qquad
\W^\times=\wb_5(\Rb_3-\Rb_4)\W+\wb_3\wb_5\W_{[45]}+(\Rb_3\Rb_4-\Rb_4^2-\wb_3^2)\W_{[34]} , \nonumber
\eeqn
where it is understood that $\wb_5$ is as in~(\ref{2_2a}).
We also recall the special subcase~(\ref{IInilpotent}) where both $\sf M$ and $\sf\Omega$
are nilpotent.
The spin type is in general $\{111\}_\p[\Rb\neq0]$. It becomes
$\{(11)1\}_\para[\Rb\neq0]$ when $\Rb_3=0$ (and $\{(11)1\}_0[\Rb\neq0]$ if,
additionally, $\wb_3=0$); this is the situation, for instance, for the Kerr black string (where $\wb_3=0$ in the equatorial plane). In addition, we may have the type $\{110\}_{\p0}[\Rb=0]$ for $\Rb_4=0$ (and $\{110\}_{0}[\Rb=0]$ if also $\wb_3=0$), which includes the case when $\sf C$ is nilpotent; or $\{111\}_{\para}[\Rb\neq0]$ if, instead, $\wb_3=0$.
\end{enumerate}
\paragraph{\underline{$\rank(M)=1$}} The conditions for this to occur were already given
in~(\ref{rankM=1}) (and correspond to a specialization of ~(\ref{w3_M2}) above). Now we have (without loss of generality we can assume $\Rb_4\neq0$)
\beq
\Ker({\sf M})=\langle\V_3,\V^0\rangle , \qquad \V^0=-\frac{\wb_3}{\Rb_4}\V_4+\V_5 ,\label{14}\\
\Im({\sf M})=\langle-\Rb_4\V_4+\wb_3\V_5\rangle ,
\eeq
while $\Ker({\sf \Omega})=\{0\}$. If $\Rb\neq 0$ then ${\sf M}$ also admits a non-zero eigenvalue $-\Rb/2=(\wb_3^2-\Rb_4^2)/(2\Rb_4)$, with eigendirection $\Im({\sf M})$.
If $\Rb=0$ then ${\sf M}$ is nilpotent and $\Im({\sf M})$ coincides with the subspace $\langle\V^0\rangle$ of $\Ker({\sf M})$.
Here the spin type is in general $\{111\}_\para[\Rb\neq0]$. It
specializes to $\{(11)1\}_0[\Rb\neq0]$ for $\Rb_5=0$, and to
$\{110\}_{\para0}[\Rb=0]$ for $\Rb_5=-\Rb_4$ (in which case $\sf M$ is
nilpotent). The Segre types of $M$, $\Om$ and ${\sf C}$ are easy to
determine (some cases are presented below).
\subsubsection{$\rank({\sf C})$ in the various subcases}\label{subsubsec: type II rank C}
We have
\beq
6\le 2\rank(M)+\rank(\Omega)\le\rank({\sf C})\le\mbox{min}\{\rank(M)+7,\rank(\Omega)+6\}\le 10 .
\eeq
All inequalities are a priori clear from (\ref{WeylOperator II}), except
for $6\le 2\rank(M)+\rank(\Omega)$ which follows from the detailed discussion above.
Considering the various previous subcases we thus obtain the possibilities summarized
in Table~\ref{Table:type_II_rank_C}. In general, the precise value of $\rank({\sf C})$ also
depends on the negative b.w. components.
\begin{table}[t]
\begin{tabular}{|l||c|c|c|}
\hline
& $\rank(\Omega)=4$ & $\rank(\Omega)=3$ & $\rank(\Omega)=2$ \\
\hline
\hline
$\rank(M)=3$ & $\rank({\sf C})=10$ & $\rank({\sf C})=9$ & - \\
\hline
$\rank(M)=2$ & $8\le\rank({\sf C})\le 9$ & $7\le\rank({\sf C})\le 9$ & $6\le\rank({\sf C})\le 8$ \\
\hline
$\rank(M)=1$ & $6\le\rank({\sf C})\le 8$ & - & - \\
\hline
\end{tabular}
\caption{Possible values of $\rank({\sf C})$ for all permitted combinations of values of $\rank(M)$ and $\rank(\Omega)$ in type {\bf {II}} spacetimes.}
\label{Table:type_II_rank_C}
\end{table}
\subsection{Type {\bf {D}} spacetimes}\label{subsec: type D}
Type {\bf {II}} spacetimes specialize to type {\bf {D}} when $\Cc_K{}^t=\Hc=\Cc_{-K}=0$ in~(\ref{WeylOperator II}).
In this case, ${\cal U}$, ${\cal W}$ and ${\cal V}$ are all invariant subspaces
(recall also~(\ref{p_U})--(\ref{p_V})) and the equality $\rank({\sf C})=2\rank(M)+\rank(\Omega)$
holds. Therefore, in each case $\rank({\sf C})$ sticks to the lower value given
in Table~\ref{Table:type_II_rank_C} and we can explicitly present the Kernel
and Image of ${\sf C}$, as follows:
\paragraph{\underline{$\rank(M)=3$, $\rank(\Omega)=4$}} Here $\rank({\sf C})=10$, so that
\beq
\Ker({\sf C})=\{0\}, \qquad \Im({\sf C})={\cal U}\oplus{\cal W}\oplus{\cal V} .
\eeq
Both the Schwarzschild-Tangherlini and Myers-Perry solutions belong to this class.
\paragraph{\underline{$\rank(M)=3$, $\rank(\Omega)=3$}} Here $\rank({\sf C})=9$, and we have
\beq
\Ker({\sf C})=\langle\W^+\rangle , \qquad \Im({\sf C})={\cal U}\oplus\Im({\sf\Omega}|_{{\cal W}\setminus\langle\W^+\rangle})\oplus{\cal V} .
\eeq
with $\W^+$ defined as in~(\ref{33}).
\paragraph{\underline{$\rank(M)=2$, $\rank(\Omega)=4$}} Here $\rank({\sf C})=8$ and
\beq
\Ker({\sf C})=\langle\V^+\rangle\oplus\langle\U^+\rangle , \qquad \Im({\sf C})=\Im({\sf M}|_{{\cal V}\setminus\langle\V^+\rangle})\oplus{\cal W}
\oplus\Im({\sf M}^t|_{{\cal U}\setminus\langle\U^+\rangle}) .
\eeq
with $\V^+$ as in~(\ref{24}) and $\U^+$ defined analogously (i.e., $\Ker({\sf M}^t)=\langle\U^+\rangle$).
\paragraph{\underline{$\rank(M)=2$, $\rank(\Omega)=3$}} Now $\rank({\sf C})=7$ and
\beq
& & \Ker({\sf C})=\Ker({\sf M})\oplus\Ker({\sf M}^t)\oplus\Ker({\sf\Omega}) , \nonumber \\
& & \Im({\sf C})=\Im({\sf M}|_{{\cal V}\setminus\Ker({\sf M})})\oplus\Im({\sf\Omega}|_{{\cal W}\setminus\Ker({\sf\Omega})})
\oplus\Im({\sf M}^t|_{{\cal U}\setminus\Ker({\sf M}^t)}) ,
\eeq
with long expressions for $\Ker({\sf M})$, $\Ker({\sf M}^t)$ and $\Ker({\sf\Omega})$, that we omit.
\paragraph{\underline{$\rank(M)=2$, $\rank(\Omega)=2$}} Now $\rank({\sf C})=6$ and there are two possibilities, so that either
\beq
& & \Ker({\sf C})=\langle\V^*\rangle\oplus\langle\U^*\rangle\oplus\langle\W^{*1},\W^{*2}\rangle , \nonumber \\
& & \Im({\sf C})=\Im({\sf M}|_{{\cal V}\setminus\langle\V^*\rangle})\oplus\Im({\sf\Omega}|_{{\cal W}\setminus\langle\W^{*1},\W^{*2}\rangle})\oplus\Im({\sf M}^t|_{{\cal U}\setminus\langle\U^*\rangle}) ,
\eeq
with the definitions of~(\ref{22b}) (and $\Ker({\sf M}^t)=\langle\U^*\rangle$), or
\beq
& & \Ker({\sf C})=\langle\V^\times\rangle\oplus\langle\U^\times\rangle\oplus\langle\W_{[53]},\W^\times\rangle , \nonumber \\
& & \Im({\sf C})=\Im({\sf M}|_{{\cal V}\setminus\langle\V^\times\rangle})\oplus\Im({\sf\Omega}|_{{\cal W}\setminus\langle\W_{[53]},\W^\times\rangle})
\oplus\Im({\sf M}^t|_{{\cal U}\setminus\langle\U^\times\rangle}) ,
\eeq
with the definitions of~(\ref{22a}) (and $\Ker({\sf M}^t)=\langle\U^\times\rangle$). For instance, the Kerr black string belongs to the latter class (see the text after (\ref{22a})).
\paragraph{\underline{$\rank(M)=1$, $\rank(\Omega)=4$}} Also in this case $\rank({\sf C})=6$, but now
\beq
\Ker({\sf C})=\langle\V_5,\V^0\rangle\oplus\langle\U_5,\U^0\rangle , \qquad \Im({\sf C})=\Im({\sf M}|_{{\cal V}\setminus\langle\V_5,\V^0\rangle})\oplus{\cal W}\oplus\Im({\sf M}^t|_{{\cal U}\setminus\langle\U_5,\U^0\rangle}) ,
\eeq with $\V^0$ as in~(\ref{14}) and $\U^0$ defined analogously
(i.e., $\Ker({\sf M}^t)=\langle\U_5,\U^0\rangle$). Cf.~(\ref{14})
for the explicit form of $\Im({\sf M})$.
\subsection{Comparison with 4D}\label{subsec: type II 4D}
For $n=2$ we have $\Sb=0$ in addition to (\ref{Cijkl=0}). Thus the only possible spin types for a 4D
primary alignment type {\bf {II}} (i.e., Petrov type {\bf {II}} or {\bf {D}}) Weyl bivector operator
are $\{(00)\}_\para[R\neq 0]$, $\{(00)\}_\para[R=0]$ and $\{(00)\}_0[R\neq 0]$.
The first one is the generic case. In the type {\bf {D}} subcase, in the second and third cases the Weyl tensor is dubbed purely magnetic and purely electric, respectively (see also section \ref{sec:PEPM}).~\footnote{Observe that $\Rb$ and $A_{34}$ correspond, in the Newman-Penrose notation, to the real and imaginary part of $\Psi_2$, respectively.}
The $2\times 2$ matrices $M$ and $\Omega$ take the form~\cite{ColHer09}:
\begin{equation}
M=\begin{bmatrix}-\tfrac 14 \Rb&-\tfrac 12 A_{34}\\\tfrac 12 A_{34}&-\tfrac 14 \Rb\end{bmatrix},\qquad \Om=\begin{bmatrix}\tfrac 12 \Rb&- A_{34}\\ A_{34}&\tfrac 12 \Rb\end{bmatrix},
\end{equation}
with respective eigenvalues $-\tfrac 14 (\Rb\pm 2i A_{34})$ and $\tfrac 12 (\Rb\pm 2i A_{34})$ . Thus the eigenvalue degeneracies are
\beq
\mbox{Spin type}\,\{(00)\}_\para:&&M[Z\Zc],\quad\Om[W\Wc],\quad {\sf C}[(ZZ)(\Zc\Zc)(W\Wc)];\\
\mbox{Spin type}\,\{(00)\}_0:&&M[(11)],\quad\Om[(11)],\quad {\sf C}[(1111)(11)].
\eeq
For Petrov type {\bf {D}} this also gives the extended Segre types for ${\sf C}$; for Petrov type {\bf {II}} the extended Segre types are ${\sf C}[Z[2]\Zc[2],W\Wc]$ and ${\sf C}[(22)(11)]$, respectively. Notice that {\em neither $M$ nor $\Om$ can be nilpotent}, basically due to the fact that they are the sum of an antisymmetric matrix and a multiple of the identity (instead of a general symmetric matrix).
\subsection{Discussion}\label{subsec: type II discussion}
We have studied the (refined) algebraic classification of the
5D primary alignment type
{\bf {II}} Weyl operator ${\sf C}$
from the perspectives of Segre type (of $M$ or $\Om$, and also of $\rank(M)$ and
$\rank(\Om)$) and spin type. In particular, we have
described the Segre type classifications by treating
particular cases, and then determined
other properties such as the values of the
ranks of $M$ and $\Om$, the
relation with $\rank({\sf C})$, and the kernels
and images of $M$ and $\Om$, within each particular case.
The nilpotence of ${\sf C}$ is of particular interest.
We found that the type {\bf {II}} Weyl
operator ${\sf C}$ is nilpotent if and only
if both $\sf M$ and $\sf\Omega$ are nilpotent (and
the Segre types of $M$ and $\Omega$ are $M[0(3)]$ and
$\Omega[0(31)]$, respectively),
and the operator ${\sf C}$ is generically nilpotent of
index $9$ (but lower indices can occur).
In order to illustrate the classification of a type {\bf {II}}
operator ${\sf C}$ we determined the possible spin types and
the extended Segre types of $\Omega$ in the case of a
nilpotent $M$;
a summary of the allowed spin types and Segre types of $M$
and $\Om$, and their compatibility, was given in table \ref{Table: type II M nilp}.
The classification of a type {\bf {II}} operator ${\sf C}$
from the viewpoint of the extended Segre type of $\Omega$ was also illustrated
in the case where $\Omega$ has a quadruple
eigenvalue. Remarkably, we found that the spin
type classification forms a pure refinement of the
$\Om$-Segre type classification
and that the extended Segre type of $M$ is almost always
$M[Z\Zc 1]$.
A summary of the allowed spin types and Segre types of $\Om$
and $M$, and their compatibility, in the case of a quadruple
$\Om$-eigenvalue was given in table \ref{Table: type II Om
quadr}.
We then focussed attention on the special
type {\bf {D}} subcase of
type {\bf {II}} spacetimes. In
this case,
the Kernel and Image of ${\sf C}$ were explicitly presented,
and the possible values of $\rank({\sf C})$ for all
permitted combinations of values of $\rank(M)$ and
$\rank(\Omega)$ in type {\bf {II}} spacetimes were given (see table~\ref{Table:type_II_rank_C}).
In appendix \ref{subsec: type II eigenvalues} we discuss the
classification of a type {\bf {II}} Weyl operator based on its spin
type. For spin types $\{\cdot\}_0$, $\{\cdot\}_\para$ and
$\{(11)1\}_\bot$ we present the degeneracies in the eigenvalue
spectra of ${\sf M}$, ${\sf \Omega}$ and ${\sf C}$ and briefly
comment on the more general spin types. As was the case for type
{\bf {III}}, a certain spin type gives rise to several possible
eigenvalue degeneracies, opposed to the situation in 4D.
Finally, let us point out that several other properties of type {\bf
{II}}/{\bf D} Einstein spacetimes in higher (in particular, five)
dimensions have been studied in \cite{PraPraOrt07,Durkee09}.
\section{Types {\bf {I}} and {\bf {G}}}
\label{section: type IG}
In general dimensions, a type {\bf {I}} Weyl tensor is characterized by having
a single WAND $[\bl]$ and no multiple WANDs. With respect to any null frame $(\bl,\bn,\mm_i)$
all components of b.w. greater than ~$\zeta=1$ vanish, whereas those of b.w.~1 are not all zero:
\begin{equation}\label{type I constituents}
\Hh=0,\quad (\nh,\vh)\neq (0,0).
\end{equation}
If this is the unique single WAND the type is {\bf {I}}$_0$,
otherwise the type is {\bf {I}}$_{i}$. If there are no WANDs at all,
the Weyl type is {\bf {G}}.
In 4D type {\bf {G}} does not occur (i.e., WANDs always exist), and if the type is {\bf {I}} then there
are exactly four WANDs (such that the type is automatically {\bf {I}}$_i$; this is the so called algebraically general case
in 4D). In higher than four dimensions, however, type {\bf {G}}
is generic~\cite{Milsonetal05}
and type {\bf {I}} is algebraically special with respect to type {\bf {G}}.
However, in many
applications the presence of a {\em multiple} WAND is important, and thus the distinction
between types {\bf {I}}/{\bf {G}} and types {\bf {II}}/{\bf {III}}/{\bf {N}} also appears to be significant.
We further note that type {\bf {I}}(A) \cite{ColHer09}
(i.e., type {\bf {I}} (a) of \cite{Coleyetal04}) corresponds to the collection of the
spin types I$\{\cdot\}_0$ (i.e., $\vh=0$), and type {\bf {I}}(B) to I$\{(000)\}_\para$
($\nh=0$). See, e.g., \cite{PraPra05,GodRea09} for examples of type {\bf {I}}/{\bf {G}} (vacuum) spacetimes in five-dimensions.
\subsection{The ``electric'' and ``magnetic'' parts of the Weyl operator}\label{sec:EMparts}
For the type {\bf {I}}/{\bf {G}} case there is another split which may be useful \cite{HOW}.
The split can be done for any of the types, but it is probably most useful for the
type {\bf {I}}/{\bf {G}} cases (and also possibly for type {\bf {D}}). The split utilises the existence of a \emph{Cartan involution} of the general linear group.
Consider the full Lorentz group $G=O(1,4)$. Let $K\cong O(4)$ be a maximal compact
subgroup of $O(1,4)$. Then there exists a unique Cartan involution $\theta$ of $G$ with
the following properties \cite{RS90}: (i) $\theta$ is invariant under the adjoint
action of $K$: $Ad_{K}(\theta)=\theta$. (ii) $O(1,4)$ is $\theta$-stable. (iii)
$\theta$ is the following automorphism of the Lie algebra $\mf{g}\mf{l}(n,\mb{R})$:
$X\mapsto -X^*$, where $^*$ denotes the adjoint (which is equal to the transpose
here since the coefficients
are real).
If $\theta_1$ and
$\theta_2$ are two such Cartan involutions of $G$, associated with maximal compact subgroups $K_1$ and $K_2$, then there exists a
$g\in G$ such that $\theta_1=\mathrm{Int}(g)\theta_2\mathrm{Int}(g^{-1})$,
where $\mathrm{Int}(g)$ is the inner automorphism by $g$.
By a slight abuse of notation, we will denote any representation of
$\theta$ simply by $\theta$.
First, let us consider the case when $\theta:T_pM\mapsto T_pM$.
The above-mentioned requirements enable us to choose a unit time-like vector
${\bf u}$ that is $K$-invariant, and let us then choose the $\theta$ which has ${\bf u}$
as an eigenvector. Therefore, in the orthonormal basis,
$\{{\bf u},{\bf m}_{i=2,...,5}\}$, we have the matrix representation:
\[ \theta=(\theta^a_{~b})=\diag (-1,1,1,1,1).\]
We note that $\theta^2={\sf 1}$. This consequently picks out a
special time-like direction and any other $\theta_2$ is related to a
(different) time-like vector ${\bf u}_2$.
Through the tensor structure of bivector space, we can let $\theta$
act on bivector space; explicitly, $\theta: F^{ab}\mapsto
\theta^a_{~c}\theta^b_{~d}F^{cd}$. By choosing the $\theta$ adapted
to the time-like vector ${\bf u}=({\mbold\ell}+{\mbold
n})/\sqrt{2}$, we get the matrix representation of $\theta$ acting
on bivector space in a $(3+4+3)$-block form relative to the basis
(\ref{calB bivector basis})-(\ref{calB def}):
\begin{eqnarray}
\theta=\begin{bmatrix}
0 & 0 & {\sf 1} \\
0 & \eta & 0 \\
{ {\sf 1} } & 0 & 0
\end{bmatrix}, \quad \eta=\diag (-1,1,1,1).
\end{eqnarray}
This will then act on the Weyl operator ${\sf C}$ through conjugation $\theta{\sf C}\theta$.
Since $\theta^2=1$, the eigenvalues of $\theta$ are $\pm 1$. We can
thus project the operator ${\sf C}$ along the eigenspaces of
$\theta$: \beq {\sf C}={\sf C}_++{\sf C}_-, \quad \text{where}\quad
{\sf C}_+=\frac 12({\sf C}+\theta{\sf C}\theta), \quad {\sf
C}_-=\frac 12({\sf C}-\theta{\sf C}\theta). \eeq Using the Weyl
operator (\ref{WeylOperator}) we can then compute the components of
the matrix representation of ${\sf C}_{\pm}$ relative to (\ref{calB
bivector basis})-(\ref{calB def}): \beq \mathcal{C}_+=
\begin{bmatrix}
\frac 12(M+M^t) & \frac 12(\Kh+\Kc) & \frac 12(\Lh+\Lc) & \frac 12(\Hh+\Hc) \\
\frac 12(\Kh^t+\Kc^t) &-\Phi& 0 &-\frac 12(\Kh^t+\Kc^t)\\
\frac 12(\Lh^t+\Lc^t) & 0 & \Hb & \frac 12(\Lh^t+\Lc^t) \\
\frac 12(\Hh+\Hc) & -\frac 12(\Kh+\Kc) & \frac 12(\Lh+\Lc) & \frac 12(M+M^t)
\end{bmatrix}\label{C+}
\\
\mathcal{C}_-=
\begin{bmatrix}
\frac 12(M-M^t) & \frac 12(\Kh-\Kc) & \frac 12(\Lh-\Lc) & \frac 12(\Hh-\Hc) \\
-\frac 12(\Kh^t-\Kc^t) &0 & -A^t &-\frac 12(\Kh^t-\Kc^t)\\
-\frac 12(\Lh^t-\Lc^t) & A & 0 & \frac 12(\Lh^t-\Lc^t) \\
-\frac 12(\Hh-\Hc) & \frac 12(\Kh-\Kc) & -\frac 12(\Lh-\Lc) & \frac 12(M^t-M)
\end{bmatrix}.\label{C-}
\eeq Therefore, we can see that the components ${\sf C}_{\pm}$ are
the symmetric and anti-symmetric parts of the Weyl operator with
respect to the Euclidean metric on bivector space. In 4D these
components are referred to as the electric and magnetic parts of the
Weyl tensor. In \cite{HOW} these were defined as the
higher-dimensional electric and magnetic part of the Weyl tensors;
thus, henceforth we will refer to the component ${\sf C}_+$ as the
\emph{electric part} of the Weyl operator (tensor), while ${\sf
C}_-$ will be referred to as the \emph{magnetic part}. Note that, as
in 4D, these parts depend on the choice of a time-like vector ${\bf
u}$ (and their representation with respect to a different time-like
vector ${\bf u}_2$ will change accordingly).
For type {\bf {I}}/{\bf {G}} Weyl tensors it is cumbersome to say anything general
about their eigenvalue structure; however, for purely electric or purely magnetic Weyl operators, we have the following \cite{HOW}:
\begin{thm} A purely electric (PE, ${\sf
C}_-=0$) Weyl operator has only real eigenvalues. A purely magnetic
(PM, ${\sf C}_+=0$) Weyl operator has at least 2 zero eigenvalues
while the remaining eigenvalues are purely imaginary.
\end{thm}
This can be seen more easily if we switch to an orthonormal frame
(see \cite{ColHer09}). Then using a $(4+6)$-block form:
\beq
\mathcal{C}_+=\begin{bmatrix}
S_1 & 0 \\ 0 & S_2
\end{bmatrix},
\quad \mathcal{C}_-=\begin{bmatrix}
0 & T \\ -T^t & 0
\end{bmatrix}, \quad S_1,~S_2 ~~\text{symmetric}, ~T \text{ a $4\times 6$ matrix}.
\eeq
In the purely electric case, the eigenvalues are the eigenvalues of $S_1$ and $S_2$, which are clearly real. In the purely magnetic case, we note that the matrix $T$ can be decomposed (using the singular value decomposition) as $T=g_1Dg_2$, where $g_1$ and $g_2$ are $SO(4)$ and $SO(6)$ matrices, respectively, and $D$ is a diagonal $4\times 6$ matrix $D=\diag(\lambda_1,\lambda_2,\lambda_3,\lambda_4)$. Consequently, a purely magnetic Weyl operator has eigenvalues $\{0,0,\pm i\lambda_1,\pm i \lambda_2,\pm i \lambda_3,\pm i \lambda_4\}$.
From the vanishing of the diagonal in (\ref{C-}) we see, in
particular, that the b.w.\ 0
part of a PE Weyl tensor has spin type $\{...\}_0$. The fact that PE
implies only real eigenvalues is illustrated, for example, by the
classification of the eigenvalue structure of the 5D type {\bf D}
Weyl operator in appendix \ref{subsec: type II eigenvalues}. On the
other hand, the converse is not true -- these are only
\emph{necessary} criteria, not sufficient; indeed, among
(\ref{App172})--(\ref{AppLast}) in Appendix \ref{app: spin
w3<>0=w4=w5}, one can find cases where the Weyl operator has only
real eigenvalues (i.e., only 1 and 0 occurring), which thus
constitute counterexamples to the converse (since $\wb\neq 0$ and
hence not PE).
For a PM Weyl tensor the b.w.\ 0 part has spin type
$\{(000)\}_\parallel[\Rb=0]$ (the diagonal of (\ref{C+}) vanishes)
which does indeed give purely imaginary eigenvalues, as illustrated by
(\ref{AppPM}) in Appendix \ref{app: spin w3<>0=w4=w5} in the type
{\bf D} case. Also for the PM case the converse is not true, as is
illustrated by (\ref{AppIm}) in Appendix \ref{app: spin w3<>0=w4=w5}
which has purely imaginary eigenvalues but is not PM.
\subsection{Exact purely electric/magnetic solutions}\label{sec:PEPM}
In 4D there have been many studies attempting to classify purely
electric or magnetic solutions with various sources. There is a
wealth of 4D purely electric Weyl spacetimes; e.g., all static
spacetimes (or, more generally, all those admitting a shear- and
vorticity-free timelike vector field), all spacetimes with
spherical, hyperbolic or planar symmetry, all Bianchi type I
spacetimes, and the Schwarzschild, C and G{\"o}del metrics.
However, only a very limited number of 4D purely magnetic Weyl
spacetimes are known to exist. It has even been conjectured that
purely magnetic vacuum or dust spacetimes do not exist (and this has
been proved under quite mild conditions, but not in general; see,
e.g.,
\cite{Hall73,Haddow95,VandenBergh03a,VandenBergh03b,Wylleman06}).
For an extensive review of known 4D purely magnetic Weyl solutions
see \cite{WyllVdB06}; see also \cite{Loz07,LozWyll11} for two more
recent contributions. All known purely magnetic metrics are
algebraically general (Petrov type {\bf I}), except for the locally
rotationally symmetric metrics given in \cite{LozCarm03,VdBWyll06},
which are of Petrov type {\bf D}.
There are also many examples of purely electric spacetimes in 5D,
including static spacetimes, spacetimes with an $\mb{R}^4$ spatial
translational invariance, spherically symmetric spacetimes, and
spacetimes with $SO(2)\times SO(2)$ isotropy or with spatial
isotropy $H\supset SO(2)\times SO(2)$ \cite{ColHer09}. Remarkably,
with our general definition of Weyl electric and magnetic parts and
of PE/PM spacetimes, many of the generic examples mentioned above
can be lifted to 5 or higher dimensions -- see \cite{HOW} for full
details. It would be useful to classify all such solutions. In
addition, the purely magnetic spacetimes are much harder to find.
The only purely magnetic spacetimes known so far in higher than 4
dimensions are conformally related to a two-parameter family of {\em
Riemann purely magnetic} spacetimes; these are all type {\bf I}$_i$,
see \cite{HOW}. A complete algebraic classification may be helpful
in the search for new exact 5D solutions.
\section{Conclusions and Discussion}\label{section: Conclusion}
In this paper we have presented a
refinement of the null alignment classification of the Weyl
tensor of a 5D spacetime based on
the notion of spin type of the
components of highest boost weight and
the Segre types of the Weyl operator acting on bivector space
(and we have examined the intersection between the two
(sub)classifications). We have presented a full treatment for types {\bf {N}} and
{\bf {III}}, and illustrated the classification from different
viewpoints (Segre type, rank, spin type) for types {\bf {II}} and {\bf {D}},
paying particular attention to possible nilpotence, since this is a completely new feature of higher dimensions. We also briefly discussed
alignment types {\bf {I}} and {\bf {G}}.
In future work we shall develop the algebraic classification further. In
particular, it is hoped that
canonical forms can be determined explicitly in each algebraic subcase. The analysis
may be used to study particular spacetimes of special interest in detail; for
example, stationary (static) spacetimes and
warped product spacetimes.
In particular, we could attempt to classify and analyse all vacuum Einstein type {\bf {III}}
spacetimes in 5D. We also note that the algebraic techniques may be of
use in other applications, since the
analysis is independent of any field equations.
This work is timely because of the
recent interest in the study of
general relativity (GR) in higher dimensions and, in
particular, in higher dimensional black holes \cite{EmpRea08}, motivated, in part,
by supergravity, string theory and the gauge-gravity
correspondence.
Indeed, even at the classical level gravity
in higher dimensions exhibits a much richer structure than in
4D. For example, there is no unique black hole
solution in higher dimensions. In fact, there now exist a
number of different asymptotically flat, higher-dimensional
vacuum black hole solutions \cite{EmpRea08},
including Myers-Perry black holes \cite{MyePer86}, black rings
\cite{EmpRea02prl,PomSen06}, and various solutions with multiple
horizons (e.g., \cite{ElvFig07,ElvRod08}).
Since the algebraic classification of spacetimes has
played such a crucial role in understanding exact solutions in 4D,
it is likely to play an important role also in higher
dimensions. However,
compared to 4D, the algebraic types defined by
the higher-dimensional alignment classification are rather
broad and it has proven more difficult to derive general
results. Therefore, it is important to develop more
refined algebraic classifications, and it is hoped that the work
in this paper will prove useful
in the search and analysis of exact higher dimensional (and, in particular, 5D)
black hole solutions.
It would also be useful to obtain a more constructive way
of accessing the invariant classification information in higher
dimensions. For example, in \cite{ColHer11} {\em {discriminants}}
were used to study the
necessary conditions for the Weyl
curvature operator (and hence the higher dimensional Weyl
tensor) to be of algebraic type {\bf II} or {\bf D} in terms of simple scalar
polynomial curvature invariants.
In particular, the Sorkin-Gross-Perry soliton, the supersymmetric black
ring, the doubly-spinning black ring, and a number of other higher
dimensional spacetimes were investigated using
discriminant techniques.~{\footnote {Indeed, it was found that
the Sorkin-Gross-Perry soliton spacetimes and the 5D supersymmetric
rotating black holes are
of type {\bf I} or {\bf G}, while the doubly spinning black ring can only
be of type {\bf II} or more special at
the horizon~\cite{Coleyetal11} (in fact, it is more generally known that at Killing horizons the Weyl type must be {\bf II} or more special, at least under some assumptions on the matter content \cite{LewPaw05,PraPraOrt07}).}}
\section*{Acknowledgments}
A. C. was supported, in part, by NSERC. M.O. has been
supported by research plan RVO: 67985840 and research grant GA\v CR
P203/10/0749. L.W.\ has been supported by an Yggdrasil mobility
grant No 211109 to Stavanger University, a BOF research grant of
Ghent University, and a FWO mobility grant No V4.356.10N to Utrecht
University, where parts of this work were performed. M.O.\ and L.W.\
are grateful to the Department of Mathematics and Statistics,
Dalhousie University, for hospitality during the initial stages of
this work.
|
{
"timestamp": "2012-09-25T02:01:58",
"yymm": "1203",
"arxiv_id": "1203.1846",
"language": "en",
"url": "https://arxiv.org/abs/1203.1846"
}
|
\section{Introduction}
Quantum impurity models
describe a finite set of interacting ``impurity'' orbitals coupled to a large number of non-interacting ``bath'' or ``lead'' states.
They were originally designed to describe the effect of magnetic impurities embedded in a non-magnetic host material\cite{Anderson61},
but have since found a wide variety of applications ranging from nanoscience,\cite{Hanson07} where they are used to describe quantum dots and molecular conductors, to surface science\cite{Brako81}, for the description of molecule adsorption on a substrate, to research in quantum field theories.\cite{Wilson75,Affleck08}
In recent years, they have gained an increasingly important role in condensed matter and materials science, where they appear as auxiliary models in the simulation of correlated lattice models within the so-called dynamical mean field theory (DMFT)\cite{Metzner89,Georges92,Georges96} approximation and its extensions.\cite{Kotliar06,Maier05}
A general quantum impurity model is described by the Hamiltonian
\begin{align}
H&=H_\text{loc}+H_\text{bath}+H_\text{hyb}, \\
\label{HQIM}
H_\text{loc} &= \sum_{pq} t_{pq}d^\dagger_p d_q +\sum_{pqrs}I_{pqrs}d^\dagger_p d^\dagger_q d_r d_s, \\
H_\text{bath} &=\sum_{k i} \varepsilon_{k i}c^\dagger_{k i}c_{k i},\\
H_\text{hyb} &=\sum_{k i p}{V}_{kip}c^\dagger_{ki}d_p +\text{h.c.}.
\end{align}
$H_\text{loc}$ describes the ``impurity'' itself, $H_\text{bath}$ a set of non-interacting ``bath'' or ``lead'' sites,
and the impurity-bath coupling or ``hybridization'' is contained in $H_\text{hyb}$. The operators $d^{(\dagger)}$ and $c^{(\dagger)}$ create and annihilate impurity and lead electrons, $t$ and $I$ describe the impurity hopping and interaction terms, $\varepsilon$ a bath dispersion and $V$ the impurity-bath hybridization strength.
The finite number of impurity interactions makes quantum impurity models numerically tractable.
The development of accurate and reliable numerical solvers for correlated quantum impurity models
is therefore one of the central challenges of computational many-body physics.
Many different approaches have been proposed.
Among those that can be made exact with sufficient computational effort, at least for some classes of models, are:
quantum Monte Carlo methods, such as the continuous-time quantum Monte Carlo (CT-QMC);\cite{GullRMP}
renormalization group (RG) methods, including numerical \cite{Bulla98,Bulla08b} and density matrix RG\cite{Garcia04,Nishimoto06}; and exact diagonalization (ED)\cite{Caffarel94}.
All these techniques have different strengths and weaknesses.
For example, CT-QMC is formulated in imaginary time and real-frequency data at high frequencies, obtained with analytic continuation, is notoriously unreliable, while
NRG has limited resolution in spectral quantities far from the Fermi surface and cannot be reliably extended beyond two impurity orbitals. ED
does not suffer from the above two difficulties, but introduces a finite size error associated with a discrete bath representation. For some special
Hamiltonians, such as those with density-density interactions and diagonal hybridizations,
CT-QMC has no sign problem, and thus affords a polynomial time solution of the impurity problem. However,
for general Hamiltonians, all the above techniques including CT-QMC exhibit an exponential scaling with the number of impurity orbitals and, in the case of ED, with the number of bath orbitals. Consequently, there is an urgent need to develop
controlled approximate solvers for general impurity models, where the exponential scaling is ameliorated or eliminated.
The dynamical mean-field theory and its cluster variants provide an ideal test bed for numerical quantum impurity solvers. DMFT is now established as a powerful theoretical framework for describing interacting quantum solids,
both in the context of single-site multi-orbital and single-orbital cluster model Hamiltonians, as well as with realistic interactions within the DFT+DMFT\cite{Kotliar06,Held06} framework.
In DMFT the bulk quantum problem is mapped onto a self-consistent quantum impurity model. Depending on the lattice model parameters, regions of weak, intermediate, and large correlation strengths can be accessed, and the wealth of previously computed data and the well-understood physics makes reliable comparisons possible.
Our present work presents controlled polynomial cost approximations to ED,
using the idea of configuration
interaction (CI) \cite{Helgaker00} that has long been studied in quantum chemistry. Recall that, at zero temperature,
the Green's function is
\begin{align}
i g_{ij}(\omega) &= \langle \Psi| a^\dag_i (\omega - H + E - i\eta)^{-1} a_j|\Psi \rangle \nonumber \\ &+
\langle \Psi | a_j (\omega + H-E +i\eta)^{-1} a^\dag_i |\Psi\rangle \label{eq:greensfn}
\end{align}
where $E$ and $\Psi$ are the ground-state energy and wavefunction of $H$. In ED, the true ground state wavefunction $\Psi$
is expanded in the complete space of Slater determinants, and the size of the complete space, which scales exponentially
as a function of the number of impurity and bath orbitals, is the primary limitation of the calculation.
Even using state-of-the-art ED (Lanczos) codes, no more than 16 electrons in 16 orbitals (32 spin-orbitals) can be treated. CI
approximates ED by solving for $\Psi$ within a {\it restricted} variational space of Slater determinants.
This variational space is constructed by including determinants based on their excitation level relative to a single,
or multiple, physically motivated reference determinants.
The various CI methods form a convergent hierarchy of approximations, where the variational space
is systematically increased, and thus their error, relative to the theoretical ED limit, can be controlled by monitoring the convergence of the hierarchy.
Furthermore, because CI methods exhibit a polynomial
scaling with respect to the number of impurity and bath orbitals, they have
the potential to treat much larger systems than ED.
Indeed, in quantum chemistry, CI calculations
with a thousand orbitals are routine.
The central question to answer in the context of correlated quantum impurity models is
whether or not CI approximations form a sufficiently rapidly convergent hierarchy for the
physical quantities of primary interest. If so, the ability to treat large numbers of orbitals and off-diagonal hybridizations, while retaining the strengths of ED,
can be expected to be of great utility in revealing the physics of complex quantum impurity models.
In Ref.~\onlinecite{Zgid11} we demonstrated that a very approximate CI solver
could reproduce exact diagonalization results in a simple quantum impurity problem arising from the DMFT approximation to the cubic hydrogen solid.
However, in that work our focus was not on the quality of the solver, but rather on chemical aspects of DMFT, such
as the use of realistic Hamiltonians which do not suffer from double-counting.
In the current work, we return to a systematic study of the CI approximations themselves.
Since our target is to assess the quality of our approximations, we concentrate here on well-studied
DMFT benchmark problems whose physics is understood, including
single-site DMFT of the 1D Hubbard model and 2$\times$2 cluster DMFT
of the 2D Hubbard model. The double-counting issue does not arise in these systems.
As we will demonstrate, in these systems the CI approximations
allow us to reproduce the ED calculations at a small fraction of the cost. Furthermore, because we can treat a larger number of orbitals than with ED, we will demonstrate that we can converge these models
with respect to their bath representation. In the cases of the 2$\times$2 cluster DMFT approximation to the Hubbard model this
has previously not been possible with ED (Lanczos).
The structure of this paper is as follows. In section \ref{sec:methsec} we first describe the theory behind CI approximations,
including a detailed description of the excitation space, single- and multi-reference CI approximations,
complete active spaces, and natural orbitals. In section \ref{sec:impsec}, we briefly describe some technical details of the implementation.
In section \ref{sec:results} we describe our application of CI solvers in model DMFT problems described above, using ED as a comparison
where possible, and we demonstrate further the ability of CI to converge systems with large numbers of bath orbitals. Finally, we describe
perspectives and conclusions in section \ref{sec:conclusions}.
\section{Configuration interaction approximations} \label{sec:methsec}
Configuration interaction (CI) wave functions $|\Psi_\text{CI}\rangle$ are a set of systematic approximations to the ED wave function $\Psi_\text{ED}$. Using CI wave functions, the ground state of the impurity model is determined in a truncated subset of the complete set of Slater determinants.
Once a ground state wave function is obtained, the impurity Green's function and self-energy, the central
quantities in DMFT, are evaluated through Eq.~(\ref{eq:greensfn}).
CI truncations rely on an {\it a priori} ranking of the importance of the determinants, in terms of excitation character
relative to a single starting determinant (single-reference CI), or multiple starting determinants (multi-reference CI).
This ranking is motivated by ordinary and degenerate perturbation theory, although CI approximations are not perturbative approximations {\it per se}.
Note that the accuracy of CI truncations of the determinant space depends on the choice of orbital basis, and this is also an important consideration in a CI calculation.
\begin{figure}[htb]
\begin{center}
\includegraphics[width=0.8\columnwidth]{srci.eps}
\caption{Schematic of determinants included in configuration interaction approximations. HF denotes
the Hartree-Fock determinant with a set of doubly occupied orbitals. S denotes a singles excitation, where
one particle (arrow, red online) is excited out of a doubly occupied orbital (leaving a hole, dot, blue online). D and T denote
doubly and triply excited configurations with respect to the Hartree Fock reference state. \label{fig:ci}}
\end{center}
\end{figure}
\begin{figure*}[htb]
\begin{center}
\begin{tabular}{cc}
(a) \includegraphics[width=0.8\columnwidth]{cas22.eps} & \hspace{2cm}
(b) \includegraphics[width=0.8\columnwidth]{cascisd.eps}
\end{tabular}
\caption{(a): determinants in a CAS(2,2) complete active space. The light (green online) lines are the 2 ``active'' orbitals; the rest
are denoted ``inactive''.
The four configurations correspond to the four ways of distributing 2 electrons across the 2 active orbitals. (b):
examples of four of the determinants contained in a CAS(2,2)CISD approximation. Excitations may be from doubly occupied, non-active orbitals (first determinant),
from active orbitals (second, fourth determinants), and from a mixture of active and inactive orbitals (third determinant). \label{fig:casci}}
\end{center}
\end{figure*}
We first motivate single-reference CI approximations using the Anderson model
in the limit of small $U$. Here, $\Psi$ is close to the
Hartree-Fock (HF) determinant $\Phi$. Consequently, we take $\Phi$ to be the (single) reference determinant in the CI.
We next change from the site basis ($d, d^\dag, c, c^\dag$) to the basis of
HF orbitals ($a, a^\dag$), which are a mixture of impurity and bath orbitals.
The determinants in the HF basis can be labeled in terms of excitation or particle-hole
character relative to $\Phi$. For example, a singly-excited determinant $\Phi_i^a=a^\dag_a a_i (\Phi)$
has one particle and one hole relative to the HF determinant; the doubly-excited determinant $\Phi_{ij}^{ab}=a^\dag_a a^\dag_b a_i a_j (\Phi)$
has two particles and holes, and so forth.
To construct a CI approximation, we truncate the complete determinant space
based on the maximum excitation character
of determinants in the expansion of $\Psi$. For example, in a CI singles and doubles (CISD) approximation,
we approximate $\Psi$ with an expansion with at most doubly excited determinants
(see Fig. \ref{fig:ci}),
\begin{align}
|\Psi\rangle \approx c_0 |\Phi\rangle + \sum_{i,a} c_i^a |\Phi_i^a\rangle + \sum_{ij,ab} c_{ij}^{ab} |\Phi_{ij}^{ab}\rangle.
\end{align}
More accurate CI approximations, with up to triple (CISDT), quadruple (CISDTQ) and higher excitations
can be formulated in a similar way, and these
systematically approach the full ED solution.
However, for small $U$, we can expect CISD to already be a good approximation, because it contains all classes of determinants
that couple with $\Phi$ through first-order perturbation theory in $U$. Similarly, CISDTQ contains
all classes of determinants that couple through second-order, and so on.
The above CI approximations (CISD, CISDT, etc.) are termed single-reference because
the truncation of the determinant space is based on excitations relative to a single reference determinant $\Phi$.
We expect this hierarchy of truncations to be rapidly convergent for small $U$, but for large $U$
multiple determinants can become degenerate with $\Phi$ on the scale of $U$ and contribute
with similar weights to $\Psi$. For example, in the single site Anderson model at large $U$, $\Psi$
is qualitatively given by a superposition of two determinants describing a ``Kondo'' singlet
coupling of electrons of opposite spin on the impurity orbital and in the bath.
In such cases, it is more reasonable to rank determinants with respect
to a {\it set} of near-degenerate determinants. This is the basis of
multi-reference CI approximations.
Denoting the near-degenerate determinants as $ \Phi(I)$ (where $I$ ranges
over the degenerate set), then for each $\Phi(I)$, we can
define singly, doubly, and higher excited determinants, $\Phi_i^a(I)=a^\dag_a a_i (\Phi(I))$, $\Phi_{ij}^{ab}(I)=a^\dag_a a^\dag_b a_i a_j (\Phi(I))$, and so on.
In the multi-reference CI singles doubles approximation, $\Psi$ is expanded in
\begin{align}
|\Psi\rangle &\approx \sum_I c_0(I) |\Phi(I)\rangle + \sum_{I} \sum_{i,a} c_i^a(I) |\Phi_i^a(I)\rangle \\ \nonumber &+ \sum_{I}\sum_{ij,ab} c_{ij}^{ab}(I)|\Phi_{ij}^{ab}(I)\rangle. \label{eq:casci}
\end{align}
Multi-reference approximations including triples and higher excitations can be defined analogously.
One drawback of multi-reference CI calculations
is that they are more difficult to set up and describe compactly, because of the need to
identify
the near-degenerate set of determinants $ \Phi(I)$.
A much simpler task is to specify only a set of near-degenerate {\it orbitals},
and to assume that all determinants obtained by considering
different occupancies of the near-degenerate orbitals (with
the remaining orbitals held empty or doubly occupied)
constitute a near-degenerate set of references.
This
is the basis of the complete active space (CAS) specification of the references $\Phi(I)$. A CAS$(n,m)$ set of references
is obtained by identifying $m$ near-degenerate orbitals, and constructing
the determinants consisting of all distributions of $n$ particles across the $m$ orbitals.
Once the CAS space is constructed, we can define a CASCI as above.
For example, the CASCISD approximation consists of expanding $\Psi$ using a space of singles and doubles excitations out of the CAS$(n,m)$ space as in Eq.~(\ref{eq:casci})
(see Fig.~\ref{fig:casci}), and higher analogues are similarly defined. In this work, we will exclusively use CAS
spaces when defining our multi-reference CI calculations.
\subsection{Improved orbitals}\label{subsec:orbital}
It is clear that the accuracy of a CI approximation
is dependent on the orbitals used to specify the determinants. For small $U$, the HF orbital basis
performs well. However, this is not always the best
choice when $U$ is large. The orbital basis which leads
to the most rapid convergence of a complete CI expansion, as measured by one-particle
density matrix norm, is called the {\it natural orbital} basis.
Natural orbitals are eigenfunctions of the one-particle density matrix
$D_{ij} = \langle \Psi | a^\dag_i a_j |\Psi\rangle$, where $\Psi$ is the exact or CI wavefunction; the corresponding eigenvalues are the natural occupancies.
The natural orbital basis is a commonly used basis for CI calculations.
Of course, the natural orbitals themselves are defined using $\Psi$ which
is not known until the CI calculation is performed. Consequently, a natural orbital based CI calculation
is usually carried out in two steps.\cite{ino} First, a CI calculation in the Hartree-Fock basis
is performed to obtain the density matrix. This is then diagonalized to obtain the natural orbitals, and the CI calculation
is repeated in this basis. In principle, this procedure can be iterated, although we have not done so in the calculations in this work.
These procedures are standard in modern quantum chemistry and are commonly used to treat finite molecular systems. Detailed descriptions can be found in Refs.~\onlinecite{Helgaker00,Sherrill99}.
\section{Implementation}\label{sec:impsec}
We have implemented the CI based approximations
described above, as well as ED, within the context of single-site
and cluster-based DMFT solvers.
Our code uses a modified version of the efficient
string-based CI algorithm in the \textsc{Dalton}
quantum chemistry package.\cite{Dalton05}
The solution of the CI and ED eigenvalue problems is carried out using iterative
Davidson diagonalization,\cite{Davidson75} while the determination of the Green's function in Eq.~(\ref{eq:greensfn})
is carried out using the Lanczos algorithm.\cite{Lanczos50} For the hybridization and bath fitting necessary in the DMFT context,
we have employed the procedures described in Ref.~\onlinecite{Zgid11}. The DMFT self-consistency was carried out until convergence in the self-energy was reached with a tolerance of less than $0.5\%$.
All calculations were performed at $T=0$, and all energies are in units of $t=1$. The $\beta$ used for fitting the dynamical mean field parameters were $20/t$ (single site DMFT) and $12.5/t$ (plaquette).
\section{Results}\label{sec:results}
We now assess the performance of CI approximations as quantum impurity solvers using established
benchmark problems. To recapitulate, the two central questions are:
how rapidly do the CI approximations converge to ED, for example, as a function of excitation level or orbital basis, and, do
CI approximations allow us to accurately treat a larger number of orbitals than ED?
We found the impurity models occurring in the dynamical mean field context to be more difficult to solve than simple impurity sites coupled to an analytically constructed density of states, and we therefore focus our presentation on impurity models obtained within this context. We first study an impurity model without self-consistency imposed. We then examine
two DMFT models. The first is the single site DMFT approximation to the 1D Hubbard model. Here, ED
calculations can be converged with respect to the number of bath orbitals, which allows us to
compare ED and CI approximations in the limit of a converged bath representation. Our second model is a 2$\times$2 plaquette (4-site) cellular dynamical mean field\cite{Kotliar01,Maier05} calculation
for the 2D Hubbard model. Such 4-site cluster models have been extensively studied with
ED~\cite{Lichtenstein00,Kotliar01,Civelli05,Kancharla08,Liebsch08,Liebsch09}
as well as with CT-QMC,\cite{Haule07,Gull08_plaquette,Park08plaquette}
and provide a standard calibration point. We begin by comparing ED and CI approximations using an
ED parametrization with 8 bath orbitals. Next, we demonstrate the ability of CI methods to treat large numbers of orbitals
by converging the 4-site cluster model with respect to the number of bath orbitals in the parametrization. Our largest
calculation involves 28 orbitals, significantly larger than can be treated with ED.
In the appendix we present
a three-orbital single site DMFT calculation of a model relevant for the physics of the $t_{2g}$ bands in transition metal oxides. This model uses
a Slater-Kanamori form of the impurity interaction.\cite{Mizokawa95,Imada98} We show that CI approximations can be used
with a general impurity Hamiltonian with non-density-density interactions and demonstrate convergence of the bath representation
with up to 24 orbitals.
\subsection{Anderson impurity model}
As a test case for a quantum impurity model we present in Tab.~\ref{tab:bathparm} the parametrization for typical hybridization strengths and energy level parameters as they arise in the DMFT context, for $U/t=4$ at half filling.
The Hamiltonian of this impurity model is
\begin{align}
H&=U\left(n_\uparrow n_\downarrow-\frac{n_\uparrow+n_\downarrow}{2}\right) +\sum_{i\sigma} \varepsilon_{i}c^\dagger_{i\sigma}c_{i\sigma}\\ \nonumber
&+\sum_{i\sigma}{V}_{i}c^\dagger_{i\sigma}d_{\sigma} +\text{h.c.}.
\end{align}
The impurity model has one impurity site and eleven bath sites.
In the particle-hole symmetric case, the choice of the active orbitals is motivated by the energetic degeneracy of the eigenvalues present already in the non-interacting Hamiltonian.
The active orbitals for the CAS calculation are the orbitals $6$ and $7$ of the natural orbitals displayed (as obtained in ED) in Tab.~\ref{tab:natorbocc}, which are singly occupied.
We first remark on the sizes of the CI determinant spaces and the corresponding run-times which are given in Table \ref{tab:spacetime}.
We see that all the CI approximations involve only a small fraction of the full ED determinantal space and take a much shorter
amount of time to run. All of these calculations are doable within minutes on a desktop PC.
\begin{table}
\begin{tabular}{|c|c|c|}
\hline
i&$\epsilon_i$&$V_i$\\\hline\hline
1 &0.558819356316&0.553263286885 \\
2 &-0.558819356316&0.553263286885 \\
3 &4.45759206721&0.541358378777\\
4 &-4.45759206721&0.541358378777\\
5 &-1.47891491526&0.488524003875\\
6 &1.47891491526&0.488524003875\\
7 &-0.185401954358&0.383193040171\\
8 &0.185401954358&0.383193040171\\
9 &0.0317683411165&0.23348635632\\
10 &-0.0317683411165&0.23348635632\\
11 &0.0&1.e-5\\\hline\hline
\end{tabular}
\caption{Bath parametrization for a typical impurity problem with $12$ sites ($1$ impurity site and $11$ bath sites) obtained from converging ED, for which the spectral function and impurity self energy are reproduced in Fig.~\ref{fig:QIMFig}.}
\label{tab:bathparm}
\end{table}
\begin{table}
\begin{tabular}{|l||c|c|c|c|c|c|}
\hline
orbital number&1-4&5&6&7&8&9-12\\\hline\hline
$U/t=4$&2.00&1.895 &1.020 &0.980 &0.105 &0.000\\
$U/t=6$&2.00 &1.738 &1.009 &0.991 &0.262 &0.000\\
$U/t=8$&2.00 &1.502 &1.001 &0.998 &0.498 &0.000\\
$U/t=20$&2.00&2.000 &1.000 &1.000 &0.000 &0.000\\
\hline
\end{tabular}
\caption{Orbital occupancies in the natural orbital basis, for the impurity model of Tab.~\ref{tab:bathparm}.
The choice of the active space is motivated by the
partially occupied natural orbitals.\label{tab:natorbocc}}
\end{table}
\begin{table}
\begin{tabular}{|c|c|c|c|}
\hline
\hline
method & space size & $t_{GS}/t_{ED_{GS}}$ & $t_{GF}/t_{ED_{GF}}$ \\
\hline
CISD & 1819 & 0.0026263 & 0.019265\\
CISDT & 18819 & 0.012848 & 0.057334\\
CAS(2,2)CISD & 6044 & 0.012242 & 0.035215\\
CAS(2,2)CISDT & 49644 & 0.12739 & 0.13240\\
\hline
ED & 853776 & 1&1\\
\hline
\hline
\end{tabular}
\caption{Size of determinant space for $12$ electrons and $12$ orbitals, and run-times
in the solution of $\Psi$, using various CI approximations and ED. $t_{GS}/t_{ED_{GS}}$ ($t_{GF}/t_{ED_{GF}}$): runtime of ground state (Green's function) calculation with respect to ED ground state (Green's function) calculation.}
\label{tab:spacetime}
\end{table}
\begin{figure}[htb]
\begin{tabular}{c}
\includegraphics[width=0.9\columnwidth]{pics/spectral_imp.eps}\\
\includegraphics[width=0.9\columnwidth]{pics/se.eps}
\end{tabular}
\caption{Spectral function $-\frac{1}{\pi}\text{Im} G(\omega)$ (upper panel) and the imaginary part of the self-energy $\text{Im}\Sigma(i\omega_n)$ (lower panel)
for an impurity model using the bath parametrization in Tab.~\ref{tab:bathparm}.
Solid lines (red online): ED. Light dashed line (green online): CISD. Dark dashed line (blue online): CISDT. Double dotted line (black online): CAS(2,2)CISD. Dotted line (magenta online): CAS(2,2)CISDT.
}
\label{fig:QIMFig}
\end{figure}
Fig.~\ref{fig:QIMFig} shows results for the spectral function (upper panel) and the imaginary part of the self energy (lower panel) for the methods of Tab.~\ref{tab:spacetime}. All methods recover both the high- and the low-energy part of the self-energy to high accuracy. Differences in the spectral function are visible for $\omega > 2$, where higher excitations that are not contained within the approximations become important. Note that
we intentionally use only a small imaginary broadening so as to preserve as much structure as possible and emphasize the difference between different approximations; This is why
the spectral functions do not appear smooth.
Fig.~\ref{fig:QIMFig} shows that for simple Anderson impurity models the truncated CI expansions are extremely robust, and even low excitation levels can recover the proper self-energy. In the dynamical mean field context, an additional complication arises: the self-consistency condition and the bath fitting procedure lead to an amplification of differences that make the final result more sensitive to differences in the impurity self energy.
\begin{figure*}[bth]
\begin{center}
\begin{tabular}{cc}
{ (a)} \includegraphics[angle=270,width=0.95\columnwidth]{pics/spec_U4t_new.eps} &
{ (b)} \includegraphics[angle=270,width=0.95\columnwidth]{pics/spec_U6t_new.eps} \\
{ (c)} \includegraphics[angle=270,width=0.95\columnwidth]{pics/spec_U8t_new.eps} &
{ (d)} \includegraphics[angle=270,width=0.95\columnwidth]{pics/spec_U20t_new.eps} \\
\end{tabular}
\caption{Single site DMFT approximation to the 1D Hubbard model at half-filling, using $11$ bath orbitals:
Spectral function (DOS) $A(\omega) = -\frac{1}{\pi}\text{Im} G(\omega)$.
Solid lines (red online): ED. Light dashed line (green online): CISD. Dark dashed line (blue online): CISDT. Double dotted line (black online): CAS(2,2)CISD. Dotted line (magenta online): CAS(2,2)CISDT.
(a) $U/t=4$, (b) $U/t=6$, (c) $U/t=8$, (d) $U/t=20$. Note that panel (a) is similar (but not completely identical) to Fig.~\ref{fig:QIMFig}, where all methods used the converged ED parameters of Tab.~\ref{tab:bathparm} for solving the impurity Hamiltonian.}
\label{fig:1D_dos_hf}
\end{center}
\end{figure*}
\subsection{Single site DMFT for the 1D Hubbard model}
\subsubsection{particle-hole symmetric case}
We carried out single site DMFT calculations for the 1D Hubbard model using
an 11 orbital bath parametrization (12 orbitals in total).
We used CISD, CISDT, CAS$(2,2)$CISD, CAS$(2,2)$CISDT approximations as well as
ED to obtain the spectral functions and impurity self-energies for
$U/t=4, 6, 8, 20$. All calculations are performed in the natural orbital basis, as described in Sec.~\ref{subsec:orbital}.
\begin{figure*}[htb]
\begin{center}
\begin{tabular}{cc}
{ (a)} \includegraphics[angle=270,width=0.95\columnwidth]{pics/se_Ut_4t.eps} &
{ (b)} \includegraphics[angle=270,width=0.95\columnwidth]{pics/se_Ut_6t.eps} \\
{ (c)} \includegraphics[angle=270,width=0.95\columnwidth]{pics/se_Ut_8t.eps} &
{ (d)} \includegraphics[angle=270,width=0.95\columnwidth]{pics/se_Ut_20t.eps} \\
\end{tabular}
\caption{single site DMFT approximation to the 1D Hubbard model at half-filling, using $11$ bath orbitals:
Imaginary part of self-energy $\text{Im} \Sigma(i\omega_n)$, with Matsubara frequencies $\omega_n = (2n+1)\pi/\beta$ for $\beta t=20$. Methods as in Fig.~\ref{fig:1D_dos_hf}.
(a) $U=4$, (b) $U=6$, (c) $U=8$, (d) $U=20$.
}
\label{fig:1s_se_error_hf}
\end{center}
\end{figure*}
The spectral functions at half-filling are shown in Fig.~\ref{fig:1D_dos_hf}. We observe good qualitative agreement of all CI methods with ED
for all values of $U/t$. If we include triple excitations (CISDT or
CAS(2,2)CISDT) the CI spectral
functions become indistinguishable to the eye from ED. In the case of CISDT, this is achieved using
only about $2\%$ of the complete determinant space of ED. Perhaps surprisingly, multi-reference CI approximations
are not necessary to obtain good agreement even for large $U/t$ where $\Psi$ contains large weights from
determinants other than the Hartree-Fock determinant.
This reflects the simplicity
of the 1D Hubbard model: the two main determinantal contributions to $\Psi$ at large $U$ differ
in the occupancies of only two electrons, which can be adequately described using doubles excitations.
It also reflects the non-perturbative nature of CI: so long as the determinants of interest are within the CI space,
they can assume arbitrarily large weights in $\Psi$, and
strongly interacting (large $U$) systems can be treated.
The corresponding self-energies of the various approximations are shown in Fig.~\ref{fig:1s_se_error_hf} (we plot only the imaginary part, $\text{Im}\Sigma(i \omega_n)$).
Again, good qualitative agreement between all the CI methods and ED is observed for all values of $U/t$. Indeed, for
$U/t=4,6,20$, even the simplest CI approximation (CISD) yields an essentially indistinguishable self-energy from ED. Only at $U/t=8$ (Fig.~\ref{fig:1s_se_error_hf}c)
do we see appreciable differences. Here we need to use CAS(2,2)CISDT to achieve less than 1\% error in the self-energy. Of course,
CAS(2,2)CISDT is also the most accurate approximation to ED as measured by the size of the excitation space.
As discussed in section \ref{subsec:orbital}, the accuracy of the CI expansions
can be improved by working in the natural orbital basis. Examining
the ED calculations at half-filling
we find that across the range of different $U/t$ only
$4$ natural orbitals have occupancies appreciably
different from 0 and 2. Consequently, we choose these 4 active orbitals for an active space
calculation in the natural orbital basis.
In Fig.~\ref{fig:natorb}, we show the spectral functions
at half-filling using the CAS(4,4) approximation, in the natural orbital basis of the ED calculation. Note that the
CAS(4,4) wavefunction involves \emph{ only 16 determinants} but the
spectral functions are still remarkably similar to the ED spectral functions. In fact, they are of similar quality to the CAS(2,2)CISD spectral functions (also in the natural orbital basis). This demonstrates the compactness of the natural orbital description.
\subsubsection{away from particle-hole symmetry}
We next consider the 1D Hubbard model away from half-filling. The corresponding imaginary parts of the self-energies,
for $U/t=6$ and dopings of $5\%-30\%$, are shown in Fig.~\ref{fig:1D_away_hf} for CISD, CISDT, and CAS(2,2)CISD. To better
illustrate the differences between the methods, here we plot the percentage error in the imaginary part of the self-energies, relative to ED. While
all the CI approximations yield qualitatively reasonable self-energies, we see that when we include triple excitations,
the errors become significantly less than $1\%$.
This is consistent with our expectation that away from half filling, the wave function of this model becomes more single-determinantal and therefore it is more advantageous to base the description on a single reference determinant (in this case the HF determinant) than to include multiple determinant reference wave functions as in CAS(2,2)CISD.
\begin{figure*}[htb]
\begin{center}
\begin{tabular}{cc}
{ (a)} \includegraphics[angle=270,width=0.95\columnwidth]{pics/spec_U4t_no.eps} &
{ (b)} \includegraphics[angle=270,width=0.95\columnwidth]{pics/spec_U6t_no.eps} \\
{ (c)} \includegraphics[angle=270,width=0.95\columnwidth]{pics/spec_U8t_no.eps} &
{ (d)} \includegraphics[angle=270,width=0.95\columnwidth]{pics/spec_U20t_no.eps} \\
\end{tabular}
\caption{Single site DMFT approximation to the 1D Hubbard model at half filling, using $11$ bath orbitals:
Spectral function (DOS) $A(\omega)=-\frac{1}{\pi}\text{Im} G(\omega)$
Comparison between CAS(4,4) in the natural orbital basis (black dots) and ED (solid line).
(a) $U/t=4$, (b) $U/t=6$, (c) $U/t=8$, (d) $U/t=20$.}
\label{fig:natorb}
\end{center}
\end{figure*}
\begin{figure*}[htb]
\begin{center}
\begin{tabular}{cc}
{ (a)} \includegraphics[angle=270,width=0.95\columnwidth]{pics/U6t_5.eps} &
{ (b)} \includegraphics[angle=270,width=0.95\columnwidth]{pics/U6t_10.eps} \\
{ (c)} \includegraphics[angle=270,width=0.95\columnwidth]{pics/U6t_15.eps} &
{ (d)} \includegraphics[angle=270,width=0.95\columnwidth]{pics/U6t_30.eps} \\
\end{tabular}
\caption{Single site DMFT approximation to the 1D Hubbard model away from half filling, using $11$ bath orbitals:
Percent error in the imaginary part of the self-energy $\text{Im}\Sigma(i\omega_n)$ (relative to ED) for CISD (solid lines, red online), CISDT (dashed lines, green online), and CAS(2,2)CISD (dotted lines).
(a) $5\%$ doping, (b) $10\%$ doping, (c) $15\%$ doping, (d) $30\%$ doping.
$\omega_n=(2n+1)\pi/\beta, \beta t=20$.
}
\label{fig:1D_away_hf}
\end{center}
\end{figure*}
\subsection{4-site cellular DMFT approximation to the 2D Hubbard model}
We now turn to cluster dynamical mean field theory, and in particular the
2$\times$2 cellular DMFT approximation\cite{Kotliar01,Maier05} of the 2D Hubbard model. We begin with a 12 orbital quantum impurity model using
8 bath orbitals, corresponding to 2 bath orbitals per impurity site, a model that
has been extensively studied in previous ED calculations.\cite{Civelli05,Kancharla08,Liebsch08,Liebsch09}
In common with these studies, we use the 4-fold
symmetry of the $2\times 2$ cluster and calculate the Green's function and self-energies
in the symmetry adapted basis of the cluster. In this basis, the Green's functions and self-energies become diagonal.
Labeling the sites of the 2$\times$2 cluster as $1\equiv(0,0)$, $2\equiv(1,0)$, $3\equiv(0,1)$, $4\equiv(1,1)$,
the symmetry orbitals $\Gamma$, $M$, $X$ (doubly degenerate) are given by
\begin{align}
\psi_\Gamma &= \frac{1}{2}(\phi_{1}+\phi_{2}+\phi_{3}+\phi_{4})\\
\phi_M &=\frac{1}{2}(\phi_{1}-\phi_{2}-\phi_{3}+\phi_{4})\\
\phi_X &=\frac{1}{2}(\phi_{1}+\phi_{2}-\phi_{3}-\phi_{4})\label{Xself}\\
\phi_{X'} &=\frac{1}{2}(\phi_{1}-\phi_{2}+\phi_{3}-\phi_{4})
\end{align}
The symmetry orbitals $\phi_X$ and $\phi_{X'}$ form a degenerate pair. For a detailed description of this model see e.g.~Refs.~\onlinecite{Liebsch08,Liebsch09}.
We carried out CISD, CISDT, CAS(2,2)CISD, CAS(2,2)CISDT, CAS(2,2)CISDTQ, and ED calculations of the spectral
functions and self-energies at half-filling. The CI calculations were carried out in the natural orbital basis of
a CAS(2,2)CISD calculation in the Hartree-Fock basis.
The local spectral functions $-\frac{1}{4\pi}\text{Tr} \text{Im} G(\omega)$ are shown in Fig.~\ref{fig:2D_dos}.
The imaginary parts of the $X$ self-energy, $\text{Im}\Sigma_X(\omega)$, corresponding to Eq.~\ref{Xself}, are shown in Fig.~\ref{fig:2D_se}. Similar to the 1D case, we find good agreement
between all the CI methods and ED for all studied values of $U/t$, although there are some visible differences
between CAS(2,2)CISD and ED. Once triple and higher excitations are included, however, the spectral functions
become indistinguishable to the eye. The same conclusion can be drawn from analyzing the self-energies.
While CAS(2,2)CISD is qualitatively similar to ED, the self-energy for $U/t=4$ is shifted from the ED self-energy,
with the errors largest at small frequencies. Once triples are included, the agreement becomes much better, and
with quadruples the self-energy is indistinguishable from that of ED. If we consider CAS(2,2)CISDT as yielding quantitative
agreement, then this is achieved using $49644$ determinants in the CI expansion, or only about $6\%$ of the ED determinantal space.
In this model, we find that the most difficult values of $U/t$ to achieve agreement between the CI methods and ED
are for $U/t=4$ and $U/t=5$. Here, the form of the self-energy is that of a correlated Fermi liquid with a large effective mass.
This behavior appears in the vicinity of the first-order cluster DMFT metal-insulator transition which, in CT-QMC simulations, is near $U/t=5.4$.\cite{Park08plaquette,Sordi10}
(Note that in this pseudogap region, ED calculations can actually
converge to two different correlated metallic solutions, depending
on the initial guess for the bath parametrization, a
feature which is repeated in the CI calculations. We have chosen to present the more insulating solution in Fig.~\ref{fig:2D_se}).
We now briefly turn to some calculations on this model which cannot be performed using ED. An essential
weakness of ED (and CI) solvers in the DMFT context is the need to parametrize the bath
using a finite number of bath orbitals. If the number of bath orbitals is too small, the
resolution of the spectral function and other quantities is very low, and furthermore, artifacts
can appear in the ED calculations due to a large fitting error at low frequencies.\cite{Koch08,Liebsch09,Senechal10} CI approximations, however, allow us to treat larger numbers
of orbitals, and thus potentially alleviate the bath parametrization problem by allowing us to
use a sufficient number of bath orbitals. We now demonstrate this for the $2\times 2$ cluster. In
Fig. \ref{fig:convergence} we plot the self-energies for $U/t=8$ from CAS(2,2)CISD calculations (using the CAS(2,2)CISD natural orbital basis)
and for $8, 16,$ and $24$ bath orbitals. For $12$ bath orbitals we used CAS(4,4)CISD rather than CAS(2,2)CISD for technical reasons due to the degeneracy of the reference wave function. The largest calculation with 24 bath orbitals (or a total of 28 orbitals in
the impurity model) is roughly twice the size of what can be treated with ED. Our studies confirm that convergence in this model
is achieved fairly rapidly, but that there are nonetheless quantitative differences
between the standard 8 bath orbital parametrization and larger bath representations, particularly
for small frequencies. The $16$ bath orbital and $24$ bath orbital parametrizations are indistinguishable, indicating that full convergence has been reached.
We have also carried out calculations for other values of $U/t$, where we observe similar convergence behavior.
The convergence of the bath parametrization appears slower for $U/t=5$ and $U/t=6$, which may once again be related to the proximity to a metal-insulator transition.
\begin{figure*}[htb]
\begin{center}
\begin{tabular}{cc}
{ (a)} \includegraphics[angle=270,width=0.95\columnwidth]{pics/spec_2d_U4t.eps} &
{ (b)} \includegraphics[angle=270,width=0.95\columnwidth]{pics/spec_2d_U5t.eps} \\
{ (c)} \includegraphics[angle=270,width=0.95\columnwidth]{pics/spec_2d_U6t.eps} &
{ (d)} \includegraphics[angle=270,width=0.95\columnwidth]{pics/spec_2d_U8t.eps} \\
\end{tabular}
\caption{Cellular dynamical mean field approximation to the $2D$ Hubbard model at half filling on a $2\times2$ cluster using $8$ bath orbitals:
Local spectral function (DOS) $A(\omega)=-\frac{1}{\pi}\text{Tr}\text{Im} G(\omega)$. For a description of the methods see text.
(a) $U/t=4$, (b) $U/t=5$, (c) $U/t=6$, (d) $U/t=8$.}
\label{fig:2D_dos}
\end{center}
\end{figure*}
\begin{figure*}[htb]
\begin{center}
\begin{tabular}{cc}
{ (a)} \includegraphics[angle=270,width=0.95\columnwidth]{pics/se_2d_Ut4_hf.eps} &
{ (b)} \includegraphics[angle=270,width=0.95\columnwidth]{pics/se_2d_Ut5_hf.eps} \\
{ (c)} \includegraphics[angle=270,width=0.95\columnwidth]{pics/se_2d_Ut6_hf.eps} &
{ (d)} \includegraphics[angle=270,width=0.95\columnwidth]{pics/se_2d_Ut8_hf.eps} \\
\end{tabular}
\caption{Cellular dynamical mean field approximation to the $2D$ Hubbard model at half filling on a $2\times2$ cluster using $8$ bath orbitals:
Imaginary part of the self-energy $\text{Im} \Sigma_{33}(i\omega_n)$. For a description of the methods see text.
(a) $U/t=4$, (b) $U/t=5$, (c) $U/t=6$, (d) $U/t=8$.
$\omega_n=(2n+1)\pi/\beta, \beta t=12.5$.}
\label{fig:2D_se}
\end{center}
\end{figure*}
\begin{figure}[htb]
\begin{center}
\includegraphics[angle=270,width=0.95\columnwidth]{pics/se_2d_Ut8_hf_all_bath_new}
\caption{Cellular dynamical mean field approximation to the $2D$ Hubbard model at half filling on a $2\times2$ cluster for a range of bath sites:
Imaginary part of the self-energy $\text{Im} \Sigma_{33}(i\omega_n)$
The CI method used was CAS(2,2)CISD in the natural orbital basis of CAS(2,2)CISD.}
\label{fig:convergence}
\end{center}
\end{figure}
\section{Conclusions}\label{sec:conclusions}
In the current work we have described how configuration interaction (CI) approximations
to exact diagonalization (ED) can be used as solvers for quantum impurity models,
such as those encountered in dynamical mean-field theory (DMFT). CI solvers
form a controlled hierarchy of polynomial cost approximations that
retain the main advantages of ED, such as the ability to treat
general interactions and obtain real spectral information. As we have demonstrated in this work,
the convergence of the CI hierarchy is sufficiently rapid that in many cases, they almost
exactly approximate the ED results, at a small fraction of the cost. This is true even in ``difficult'', ``strongly correlated'' regimes,
such as the pseudogap regime of the $2\times 2$ cluster DMFT of the Hubbard model. In addition, this great increase
in computational efficiency potentially allows us to treat considerably larger quantum impurity
models than have been considered in ED. In this work we used this ability to demonstrate
bath convergence for the $2\times 2$ cluster DMFT of the Hubbard model in a calculation with $28$
orbitals, for a case where previously only $12$ orbitals (Lanczos) were accessible.
Here we have focused on well studied DMFT problems in order to benchmark the CI approximations.
In future work, we plan to apply these CI approximations to study problems where existing solvers
have difficulties. Some of these include impurity models with a large number of orbitals and with
general interactions and off-diagonal hybridizations, for which CT-QMC methods encounter a severe sign problem. Another interesting direction to explore will be
to examine more sophisticated quantum chemistry approximations to ED. For example, for weak interactions,
coupled cluster approximations are known to be far superior to configuration interaction approximations
for a given computational cost. These and other directions are currently being pursued.
\acknowledgments{Dominika Zgid acknowledges helpful discussions with A.J.~Millis, D.R.~Reichman, A.I.~Lichtenstein, L.~de~Medici, and A.~Liebsch. Dominika Zgid and Garnet Kin-Lic Chan acknowledge support from the Department of Energy, Office of Science,
through Award DE-FG02-07ER46432. Emanuel Gull was partially supported by NSF-DMR-1006282.}
|
{
"timestamp": "2012-03-09T02:04:50",
"yymm": "1203",
"arxiv_id": "1203.1914",
"language": "en",
"url": "https://arxiv.org/abs/1203.1914"
}
|
\section[]{Introduction}
Over the past nearly 30 years the method of reverberation mapping (RM) (\citealt{Peterson88} and reference therein) has become one of the standard methods for studying the Active Galactic Nuclei (AGNs). It is based on the assumption that in a typical Seyfert galaxy the source of continuous radiation near a black hole, named as accretion disc (AD), is expected to be of order $10^{13}-10^{14}$~cm. Photoionization of the gas located at a distance of order $10^{16}$~cm produces broad emission lines. The relationship between the continuum and emission line fluxes can be represented by the equation \citep{BMcKee}
\[
L(t)=\int \Psi(\tau) C(t-\tau) dt,
\]
where $C(t)$ and $L(t)$ are the observed continuum and emission-line light curves,
and $\Psi(\tau)$ is the 1-d transfer function (TF). The TF determines the emission line
response to a $\delta$-function continuum pulse as seen by a distant observer. So, the emission lines "echo" or "reverberate" in response to the continuum changes with a delay $\tau$. The size
of the region where broad lines (BLR) are formed can be written as $R_{BLR}=c\tau$. The primary
task of the RM method is to use the observable $C(t)$ and $L(t)$ to solve the above integral
equation for the TF in order to obtain information about the geometry and physical conditions
in the BLR. Unfortunately, it is very difficult to find a unique and reliable solution to this
equation. However, it is possible to find a temporal shift (lag) between the continuum and
emission line light curves using the cross-correlation analysis.
Applying the virial assumption, the mass of the black hole can be determined when the BLR size
and the velocity dispersion of the BLR gas are known \citep{Peterson04}. The present tremendous
progress being made in black hole mass estimates can be attributed to the reverberation method.
On the other hand, different segments of a single emission line seem to be formed at
different effective distances from the ionizing source. In that case, the response in the flux
of emission line at line-of-sight velocity $V_r$ and time delay $\tau$ is caused by the 2-d
transfer function or "velocity delay map" \citep{Horne04}. The reverberation technique applied
to different parts of a single emission line allows one to make conclusions about the velocity field
of the BLR gas. To the present time, considerable progress was made in understanding the
direction of the BLR gas motion \citep{Gaskell09, Bentz08, Denney09, Bentz10}. For example,
some distinctive signatures for infalling gas motions in NGC~3516 and Arp~151 \citep{Bentz09b, Denney09} were revealed: the blue side of the line lagging the red side. NGC~5548 shows the virialized gas motions with the symmetric lags on both the red and blue sides of line. However, the BLR gas in NGC~3227 shows the signature of radial outflow: shorter lags for the blue-shifted gas and longer lags for the red-shifted gas \citep{Denney09}.
More than 40 AGNs have been studied by the RM up to now \citep{Peterson04, Bentz09b, Denney10}.
However, the Mrk~6 nucleus is absent in this list. Just a few studies of this galaxy have been
made by the reverberation method. We can mention the paper by \citet{DS03} based on the archive
spectra of Mrk~6 obtained from 1970--1991 using the image tube spectrograph at the 2.6-m
telescope of the Crimean Astrophysical Observatory (CrAO). There is also another paper by
\citet{Ser99} that includes the results of the 1992--1997 observations with the same spectrograph but with a CCD detector. In this paper a lag in the flux variations of hydrogen lines with respect to the adjacent continuum flux variations was reported for the first time and changes in the line profiles were studied.
Mrk~6 is a Seyfert 1.5 galaxy (Sy~1.5). This galaxy was one of the first galaxies in which the
strong variability of the \hb\, emission line profile was detected by \citet{KhW71}. Although
there have not been many optical observations of Mrk~6, there are some radio studies of Mrk~6
available \citep{Kuk96, Kharb06}, which revealed the complex structure of the radio emission. The
X-ray emission from the Mrk~6 nucleus \citep{Feldmeier99, Malizia03, Immler03, Schurch06}
exhibits a complex X-ray absorption and some authors assume that the BLR is a possible location
of this absorption complex \citep{Malizia03}.
In this paper we present the results of our optical spectroscopic observations of Mrk~6 during
the monitoring campaign from 1998 to 2008 performed after publishing our earlier papers with
observations made from 1970--1997. For completeness we apply our analysis to all the spectral
CCD observations that have been made since 1992 and we also use the results of our photometric
observations in the $V$ band. Throughout the entire work, we take $z=0.01865$ from our spectral estimates, the distance to Mrk~6 equal to $D=81$~Mpc, and $H_0=$70~km\,s$^{-1}~Mpc^{-1}$. The
observations and data reduction are described in Section~2. We present the cross-correlation
analysis in Section~\ref{fccf}, the line width measurements in Section~\ref{linewidth}; the estimates of the black hole mass, the mass--luminosity and lag--luminosity diagrams are presented in Sections~\ref{mass-bh} and \ref{slm}, the velocity-resolved reverberation lag analysis is performed in Section~\ref{vrrlag}. The results are summarized in Section~\ref{sum}.
\section{Observations}
\subsection{Spectral observations and data processing}
The \ha\ and \hb\ spectra of the Seyfert galaxy Mrk~6 were obtained from the CrAO 2.6-m Shajn
telescope. Prior to 2005 we used the Astro-550 CCD which had a size 580$\times520$~pixels and was
cooled by liquid nitrogen. The dispersion was 2.2~\AA\,pixel$^{-1}$, and the spectral resolution
was about 7--8~\AA. The working wavelength range was about 1200~\AA. The entrance slit width was
3\arcsec. For technical reasons it was not always possible to set the same position angle (PA) of the
entrance slit. Most of the observations were performed along PA$\sim125^{\rm o}$ or
PA$\sim90^{\rm o}$. The typical exposure time was about one hour. The "extraction window" was
equal to 11\arcsec.
In July 2005 the Astro-550 CCD was replaced with the SPEC-10 $1340\times100$~pixel CCD,
thermoelectrically cooled up to $-100$\degC. In this case the dispersion was
1.8~\AA\,pixel$^{-1}$. A 3\farcs0 slit with a 90\degr position angle was utilized for these
observations. The higher quantum efficiency (95\% max.) and the lower read noise of this CCD
allowed us to obtain higher quality spectra under shorter exposure times. The spectral wavelength
range for these data sets was about 2000~\AA\, near the \ha\, and \hb\ regions. However, the red
and blue edges of the CCD frame are unusable because of vignetting. Generally, all spectra in
both spectral regions were obtained with a single exposure during the whole night. Our final
data set for 1998--2008 consists of 135 spectra in the \hb\, region and 48 spectra in \ha.
The mean signal-to-noise ratio (S/N) is equal to 32 for the Astro-550 and 73 for the SPEC-10 in
the \hb\, region and is equal to 40 and 66 in the \ha\, region, respectively.
As a rule, each observation was preceded by four short-time ($\sim$10~s) exposures of the
standard star BS~3082, whose spectra were obtained at approximately the same zenith distance as
that of the galaxy. The spectral energy distribution for these spectra was taken from
\citet{Khar88}. The standard star spectra were used to remove telluric absorption features from
the spectra of Mrk~6, to provide the relative flux calibration, and to measure the seeing
parameter defined as the FWHM of the cross-dispersion profile on the CCD image. The description
of the primary data processing as well as some details of absolute calibration and measurements
of the spectra are given in \citet{Ser99}.
The flux calibration was carried out by assuming the narrow emission line fluxes to be constant.
We chose the narrow [OIII]$\lambda$5007 (for the \hb\, region) and [SII]$\lambda\lambda$6717,~6732 emission lines (for the \ha\, region) as the internal flux
standards. Their absolute fluxes were measured from the spectra obtained under photometric
conditions by using the spectra of the comparison star BS~3082. The mean fluxes in these lines
are given in \citet{Ser99}: F([OIII]$\lambda$5007)=$(6.90\pm0.11)\times10^{-13}$~erg~s$^{-1}$~cm$^{-2}$;
F[SII]~$\lambda\lambda(6717+6731)$=(1.61$\pm0.09)\times10^{-13}$~erg~s$^{-1}$~cm$^{-2}$, and
F[OI]~$\lambda$6300=(0.597$\pm0.035)\times10^{-13}$~erg s$^{-1}$~cm$^{-2}$. The [SII] lines
reside on the far wings of the broad \ha\, line. Thus, to measure their fluxes, we selected the
pseudo-continuum zones closely spaced around each line. The line fluxes were measured by
integrating the spectra over the specified wavelength intervals and above the continuum (or
local pseudo-continuum), which was fitted with a straight line in the selected zones. The mean
continuum flux per unit wavelength was determined in two windows: at 5162--5186~\AA\,
(designated as $F5170$) and 6985--7069~\AA\, (designated as $F7030$). The continuum zones and
integration limits are the same as in \citet{Ser99}. The line and continuum flux uncertainties contain errors related to the S/N of the source spectra, atmospheric dispersion, changes in the position angle of the slit, and seeing effects. Evaluation of these uncertainties is considered
in \citet{Ser99}. The mean spectrum of Mrk~6 produced by combining 23 quasi-simultaneous pairs of
spectra from the \ha\ and \hb\ regions is shown in Fig.\ref{M6-spe}. Figure \ref{M6mn-rms} shows the mean and rms spectra of Mrk~6 based on our observations in 1992--2008.
\begin{figure}
\includegraphics[width=84mm]{f01-spe.eps}
\caption{Mean Mrk~6 spectrum obtained by combining the quasi-simultaneous pairs of spectra
from the \ha\, and \hb\, spectral regions (SPEC-10 CCD only).}
\label{M6-spe}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=60mm]{f02mn-rms.eps}
\caption{Mean and rms spectra of Mrk~6 in the \hb\ region from our observations 1992--2008.}
\label{M6mn-rms}
\end{center}
\end{figure}
\subsection{Optical photometry}
\label{oph} In order to improve the time resolution of our data set in the F5170 continuum we
added the $V$-band photometry to our spectral observations. Photometric data came from two
sources: the $UBV$ observations were made at the Crimean Laboratory of the Sternberg
Astronomical Institute of Moscow University, and the $BVRI$ observations were obtained at the
CrAO. The $UBV$ observations were obtained in the standard Johnson photometric system and were carried out from 1986 to 2009 at the 60-cm Zeiss telescope with a photo-multiplier detector through the aperture A=27$\farcs5$. The mean uncertainty of Mrk~6 in the $V$-band is 0\fm022. These data were partially published by \citet{Dor2003}.
In 2001 we started regular observations of Mrk~6 using the CrAO 70-cm AZT-8 telescope and the
AP7p CCD. The CCD field covers 15\arcmin$\times$15\arcmin. Photometric fluxes were measured
within an aperture of 15\farcs0. The mean uncertainty of the $V$-band CCD observations is
0\fm009. Further details about the instrumentation, reductions, and measurements of the $BVRI$
photometric data can be found in \citet{dor05}. These data were partially published by
\citet{Ser05}.
\subsection{Light curves}
\label{comlc} Light curves in \ha\, and \hb, and the adjacent continuum are shown in Fig.~\ref{lc} for spectral observations corrected for seeing.
The continuum light curves obtained from the photometric $V$-band observations were scaled to
the flux density measured from the spectroscopic observations. To this end, we used the
observations made on the same nights almost simultaneously. We have 29 appropriate observational
nights at the 2.6-m and 70-cm telescopes (spectra plus CCD photometry) and 40 appropriate nights
at the 2.6-m and 60-cm telescopes (spectra plus the $UBV$ photoelectric photometry). The
correlation coefficient between the spectral continuum and the $V$-band CCD flux for the
appropriate nights is $r$=0.984 (n=29 points), and the correlation coefficient between the
spectral continuum and the $V$-band flux from the $UBV$ observations is $r$=0.993 (n=40 points).
Using the regression equations, we converted our $V$-band photometric fluxes to the spectral
continuum fluxes (F5170). For the nights where both spectral and photometric observations were
available, the continuum fluxes are calculated as the weighted average. The fluxes were not
corrected for the host starlight contamination and Galactic reddening.
\begin{figure}
\includegraphics[width=84mm]{F03lcspe.eps}
\caption{Light curves of Mrk~6 shown for the continuums and \ha, \hb, and \hg\ lines.
Narrow lines were not subtracted from the fluxes. Bottom panel shows the merged continuum light curve used for cross-correlation analysis. Units are 10$^{-13}$~erg~cm$^{-2}$~s$^{-1}$ and
10$^{-15}$~erg~cm$^{-2}$~s$^{-1}$~\AA$^{-1}$ for the lines and continua, respectively.
The vertical dash-dotted lines show the boundaries of the five time intervals considered in the present paper.}
\label{lc}
\end{figure}
{\bf Tables~\ref{hb-hgflx} and \ref{haflx} give the light curves from the spectral observations for 1998--2008 \citep[the data for 1991--1997 are available in][]{Ser99}.
The \hb\ and \hg\ line fluxes together with the spectral continuum fluxes $F_{5170}$
are shown in Table~\ref{hb-hgflx}. The \ha\ line fluxes and the spectral continuum fluxes $F_{7030}$
are shown in Table~\ref{haflx}. Table~\ref{cntflx} gives combined continuum fluxes from both the spectral observations and from $V$-band photometric measurements for 1991--2008. All the fluxes are seeing-corrected. These light curves have been used for the subsequent time-series analysis.}
\begin{table}
\caption{F5170 spectral continuum, the H$\beta$ and H$\gamma$ line fluxes.}
\label{hb-hgflx}
\begin{tabular}{@{}cccc}
\hline
JD-2,440,000 & $F5170$ & H$\beta$ & H$\gamma$ \\
\hline
10869.449 & $5.932\pm 0.057$ & $3.980\pm 0.079$ & -- \\
10875.449 & $6.279\pm 0.087$ & $3.987\pm 0.093$ & -- \\
10876.379 & $6.365\pm 0.085$ & $4.082\pm 0.112$ & -- \\
10905.395 & $6.451\pm 0.057$ & $4.420\pm 0.080$ & -- \\
10906.273 & $6.352\pm 0.088$ & $4.329\pm 0.098$ & -- \\
10924.432 & $5.684\pm 0.079$ & $4.511\pm 0.085$ & -- \\
10966.379 & $5.008\pm 0.058$ & $3.476\pm 0.065$ & -- \\
10982.395 & $5.570\pm 0.101$ & $3.792\pm 0.096$ & -- \\
10994.387 & $5.678\pm 0.062$ & $4.072\pm 0.085$ & -- \\
11014.516 & $5.302\pm 0.091$ & $3.971\pm 0.094$ & -- \\
11052.563 & $5.519\pm 0.081$ & $3.724\pm 0.100$ & -- \\
11074.574 & $5.665\pm 0.064$ & $3.939\pm 0.086$ & -- \\
11076.516 & $5.628\pm 0.080$ & $4.022\pm 0.078$ & -- \\
11085.570 & $6.117\pm 0.066$ & $4.102\pm 0.087$ & -- \\
11100.512 & $6.344\pm 0.076$ & $4.283\pm 0.079$ & -- \\
11141.406 & $5.155\pm 0.087$ & $4.316\pm 0.087$ & -- \\
11218.281 & $5.525\pm 0.117$ & $4.022\pm 0.133$ & -- \\
11278.395 & $6.743\pm 0.074$ & $4.795\pm 0.103$ & -- \\
11281.383 & $6.927\pm 0.087$ & $4.687\pm 0.099$ & -- \\
11290.402 & $5.878\pm 0.187$ & $4.668\pm 0.201$ & -- \\
11310.289 & $5.216\pm 0.067$ & $4.594\pm 0.090$ & -- \\
11319.313 & $4.817\pm 0.063$ & $4.276\pm 0.086$ & -- \\
11322.305 & $5.134\pm 0.148$ & $3.885\pm 0.113$ & -- \\
11349.324 & $4.884\pm 0.084$ & $3.615\pm 0.078$ & -- \\
11485.504 & $5.096\pm 0.072$ & $4.438\pm 0.089$ & -- \\
11497.434 & $5.160\pm 0.086$ & $4.141\pm 0.113$ & -- \\
11516.551 & $5.103\pm 0.073$ & $4.020\pm 0.078$ & -- \\
11557.418 & $4.465\pm 0.077$ & $3.188\pm 0.098$ & -- \\
11577.406 & $4.456\pm 0.066$ & $3.151\pm 0.087$ & -- \\
11587.293 & $4.514\pm 0.070$ & $3.119\pm 0.079$ & -- \\
11606.383 & $4.821\pm 0.112$ & $3.192\pm 0.127$ & -- \\
11608.285 & $4.540\pm 0.089$ & $3.187\pm 0.094$ & -- \\
11615.340 & $5.173\pm 0.114$ & $3.081\pm 0.108$ & -- \\
11636.270 & $5.252\pm 0.076$ & $3.561\pm 0.084$ & -- \\
11707.402 & $5.783\pm 0.095$ & $3.687\pm 0.092$ & -- \\
11720.336 & $5.875\pm 0.069$ & $3.671\pm 0.087$ & -- \\
11780.555 & $5.757\pm 0.102$ & $4.052\pm 0.123$ & -- \\
11782.555 & $5.868\pm 0.080$ & $3.989\pm 0.101$ & -- \\
11791.531 & $5.385\pm 0.093$ & $4.017\pm 0.126$ & -- \\
11810.531 & $5.463\pm 0.075$ & $3.751\pm 0.097$ & -- \\
11821.582 & $5.010\pm 0.092$ & $4.032\pm 0.094$ & -- \\
11823.496 & $5.049\pm 0.072$ & $4.068\pm 0.097$ & -- \\
11838.508 & $5.234\pm 0.183$ & $3.741\pm 0.181$ & -- \\
11840.539 & $5.027\pm 0.103$ & $3.574\pm 0.111$ & -- \\
11844.496 & $4.679\pm 0.072$ & $3.577\pm 0.098$ & -- \\
11847.473 & $4.688\pm 0.089$ & $3.433\pm 0.116$ & -- \\
11853.582 & $4.628\pm 0.067$ & $3.456\pm 0.089$ & -- \\
11869.496 & $5.184\pm 0.073$ & $3.363\pm 0.082$ & -- \\
11878.449 & $5.012\pm 0.080$ & $3.450\pm 0.102$ & -- \\
11901.316 & $5.436\pm 0.106$ & $3.354\pm 0.123$ & -- \\
11926.270 & $4.391\pm 0.084$ & $3.374\pm 0.080$ & -- \\
11959.227 & $4.689\pm 0.129$ & $3.360\pm 0.151$ & -- \\
11999.270 & $4.517\pm 0.112$ & $3.077\pm 0.103$ & -- \\
12030.422 & $5.191\pm 0.100$ & $3.149\pm 0.105$ & -- \\
12050.465 & $4.689\pm 0.087$ & $3.219\pm 0.081$ & -- \\
12104.316 & $4.394\pm 0.069$ & $2.805\pm 0.079$ & -- \\
12174.539 & $4.327\pm 0.069$ & $3.007\pm 0.087$ & -- \\
12175.531 & $4.153\pm 0.084$ & $3.094\pm 0.087$ & -- \\
12192.551 & $4.046\pm 0.095$ & $3.099\pm 0.104$ & -- \\
12201.598 & $3.845\pm 0.105$ & $2.994\pm 0.133$ & -- \\
12223.543 & $4.004\pm 0.085$ & $ 2.934\pm 0.107$ & -- \\
12323.375 & $3.899\pm 0.054$ & $ 2.424\pm 0.063$ & -- \\
12343.473 & $3.573\pm 0.107$ & $ 2.723\pm 0.118$ & -- \\
\hline
\end{tabular}
\end{table}
\begin{table}
\contcaption{F5170 spectral continuum, the H$\beta$ and H$\gamma$ line fluxes.}
\begin{tabular}{@{}cccc@{}}
\hline
JD-2,440,000& $F5170$ & H$\beta$ & H$\gamma$ \\
\hline
12381.262 & $4.039\pm 0.125$ & $ 2.356\pm 0.130$ & -- \\
12441.359 & $5.067\pm 0.078$ & $ 3.406\pm 0.082$ & -- \\
12613.543 & $5.260\pm 0.085$ & $ 3.500\pm 0.099$ & -- \\
12619.543 & $5.285\pm 0.063$ & $ 3.457\pm 0.075$ & -- \\
12675.410 & $5.032\pm 0.106$ & $ 2.854\pm 0.126$ & -- \\
12914.605 & $5.412\pm 0.152$ & $ 3.171\pm 0.156$ & -- \\
12915.531 & $5.223\pm 0.067$ & $ 3.445\pm 0.092$ & -- \\
13089.324 & $4.197\pm 0.062$ & $ 2.682\pm 0.066$ & -- \\
13097.301 & $4.849\pm 0.205$ & $ 2.603\pm 0.111$ & -- \\
13319.496 & $3.639\pm 0.065$ & $ 2.741\pm 0.055$ & -- \\
13356.555 & $3.175\pm 0.060$ & $ 2.219\pm 0.068$ & -- \\
13611.563 & $6.791\pm 0.081$ & $ 4.125\pm 0.075$ & $ 1.342\pm 0.057$ \\
13612.553 & $6.812\pm 0.062$ & $ 4.135\pm 0.058$ & $ 1.368\pm 0.046$ \\
13621.523 & $6.877\pm 0.082$ & $ 4.264\pm 0.079$ & $ 1.417\pm 0.064$ \\
13641.566 & $6.279\pm 0.096$ & $ 4.197\pm 0.078$ & $ 1.347\pm 0.069$ \\
13647.570 & $6.158\pm 0.095$ & $ 4.157\pm 0.075$ & $ 1.260\pm 0.062$ \\
13648.594 & $6.310\pm 0.084$ & $ 4.156\pm 0.077$ & $ 1.332\pm 0.069$ \\
13649.559 & $6.276\pm 0.081$ & $ 4.082\pm 0.073$ & $ 1.277\pm 0.063$ \\
13670.555 & $6.284\pm 0.083$ & $ 4.020\pm 0.068$ & $ 1.317\pm 0.060$ \\
13676.583 & $6.309\pm 0.079$ & $ 3.914\pm 0.065$ & $ 1.235\pm 0.055$ \\
13698.469 & $6.366\pm 0.103$ & $ 4.038\pm 0.081$ & $ 1.255\pm 0.067$ \\
13774.453 & $5.677\pm 0.189$ & $ 4.203\pm 0.114$ & $ 1.290\pm 0.104$ \\
13788.344 & $5.997\pm 0.091$ & $ 4.044\pm 0.067$ & $ 1.269\pm 0.058$ \\
13830.238 & $6.715\pm 0.113$ & $ 3.949\pm 0.086$ & $ 1.206\pm 0.070$ \\
13875.332 & $7.178\pm 0.086$ & $ 4.282\pm 0.086$ & $ 1.264\pm 0.071$ \\
13994.566 & $7.272\pm 0.080$ & $ 4.269\pm 0.077$ & $ 1.326\pm 0.062$ \\
14013.613 & $7.496\pm 0.089$ & $ 4.407\pm 0.087$ & $ 1.317\pm 0.066$ \\
14038.555 & $7.741\pm 0.089$ & $ 4.177\pm 0.086$ & $ 1.227\pm 0.073$ \\
14048.566 & $7.806\pm 0.136$ & $ 4.069\pm 0.100$ & $ 1.085\pm 0.082$ \\
14064.551 & $7.001\pm 0.085$ & $ 4.244\pm 0.081$ & $ 1.116\pm 0.057$ \\
14065.617 & $6.963\pm 0.079$ & $ 4.139\pm 0.075$ & $ 1.195\pm 0.063$ \\
14076.461 & $6.903\pm 0.156$ & $ 4.044\pm 0.082$ & $ 1.194\pm 0.076$ \\
14078.551 & $6.923\pm 0.101$ & $ 3.989\pm 0.090$ & $ 1.015\pm 0.072$ \\
14086.559 & $7.182\pm 0.097$ & $ 3.924\pm 0.080$ & $ 1.062\pm 0.065$ \\
14106.531 & $7.897\pm 0.243$ & $ 4.107\pm 0.186$ & -- \\
14111.473 & $7.469\pm 0.079$ & $ 4.316\pm 0.078$ & $ 1.241\pm 0.064$ \\
14142.594 & $7.459\pm 0.097$ & $ 4.418\pm 0.090$ & $ 1.120\pm 0.074$ \\
14156.270 & $7.366\pm 0.132$ & $ 4.429\pm 0.102$ & $ 1.220\pm 0.077$ \\
14331.543 & $8.416\pm 0.086$ & $ 5.376\pm 0.095$ & $ 1.647\pm 0.073$ \\
14369.516 & $7.978\pm 0.124$ & $ 4.843\pm 0.094$ & $ 1.556\pm 0.081$ \\
14370.539 & $7.390\pm 0.197$ & $ 4.736\pm 0.093$ & $ 1.335\pm 0.074$ \\
14377.582 & $7.570\pm 0.110$ & $ 4.714\pm 0.107$ & $ 1.410\pm 0.088$ \\
14386.457 & $7.490\pm 0.098$ & $ 4.184\pm 0.097$ & $ 1.294\pm 0.082$ \\
14391.551 & $6.816\pm 0.097$ & $ 4.384\pm 0.076$ & $ 1.327\pm 0.061$ \\
14392.539 & $7.115\pm 0.097$ & $ 4.410\pm 0.081$ & $ 1.331\pm 0.066$ \\
14393.504 & $7.178\pm 0.093$ & $ 4.517\pm 0.092$ & $ 1.396\pm 0.071$ \\
14423.559 & $6.261\pm 0.101$ & $ 3.715\pm 0.088$ & $ 1.149\pm 0.084$ \\
14438.438 & $5.934\pm 0.100$ & $ 3.612\pm 0.075$ & $ 1.043\pm 0.066$ \\
14480.414 & $7.435\pm 0.082$ & $ 4.110\pm 0.080$ & $ 1.166\pm 0.067$ \\
14481.398 & $7.342\pm 0.086$ & $ 4.108\pm 0.083$ & $ 1.049\pm 0.065$ \\
14482.434 & $7.498\pm 0.091$ & $ 4.016\pm 0.086$ & $ 1.179\pm 0.075$ \\
14496.367 & $7.517\pm 0.114$ & $ 4.244\pm 0.094$ & $ 1.209\pm 0.074$ \\
14497.418 & $7.561\pm 0.123$ & $ 3.955\pm 0.099$ & $ 1.155\pm 0.082$ \\
14499.328 & $7.615\pm 0.090$ & $ 4.125\pm 0.088$ & $ 1.168\pm 0.077$ \\
14508.418 & $7.071\pm 0.106$ & $ 4.272\pm 0.088$ & $ 1.258\pm 0.076$ \\
14537.340 & $6.605\pm 0.099$ & $ 3.742\pm 0.086$ & $ 1.233\pm 0.082$ \\
14554.324 & $5.749\pm 0.125$ & $ 3.736\pm 0.099$ & $ 0.942\pm 0.083$ \\
14574.309 & $6.132\pm 0.269$ & $ 3.303\pm 0.226$ & -- \\
14586.309 & $6.284\pm 0.086$ & $ 3.550\pm 0.074$ & $ 1.064\pm 0.069$ \\
14587.281 & $6.184\pm 0.092$ & $ 3.576\pm 0.080$ & $ 1.003\pm 0.067$ \\
14588.266 & $6.264\pm 0.098$ & $ 3.577\pm 0.082$ & $ 1.027\pm 0.068$ \\
14590.309 & $6.278\pm 0.098$ & $ 3.543\pm 0.076$ & $ 0.968\pm 0.060$ \\
\hline
\end{tabular}
\end{table}
\begin{table}
\contcaption{F5170 spectral continuum, the H$\beta$ and H$\gamma$ line fluxes.}
\begin{tabular}{@{}cccc}
\hline
JD-2,440,000& $F5170$ & H$\beta$ & H$\gamma$ \\
\hline
14591.273 & $6.440\pm 0.108$ & $ 3.556\pm 0.086$ & $ 1.020\pm 0.077$ \\
14617.344 & $7.076\pm 0.086$ & $ 3.632\pm 0.081$ & $ 0.955\pm 0.062$ \\
14620.301 & $7.053\pm 0.091$ & $ 3.923\pm 0.090$ & $ 1.053\pm 0.071$ \\
14622.426 & $7.141\pm 0.087$ & $ 3.682\pm 0.084$ & $ 1.048\pm 0.068$ \\
14687.547 & $6.498\pm 0.095$ & $ 4.099\pm 0.073$ & $ 1.091\pm 0.057$ \\
14779.629 & $7.584\pm 0.127$ & $ 4.711\pm 0.106$ & $ 1.328\pm 0.070$ \\
14804.566 & $8.214\pm 0.093$ & $ 4.650\pm 0.086$ & $ 1.305\pm 0.070$ \\
\hline
\end{tabular}
{\\\footnotesize
Units are $10^{-13}\,\ergs$ and $10^{-15}\,\ergsA$ for the
lines and continuum, respectively.
}
\end{table}
\begin{table}
\caption{F7030 spectral continuum and H$\alpha$ fluxes.}
\label{haflx}
\begin{tabular}{@{}ccc}
\hline
JD-2,440,000 & $F7030$ & H$\alpha$ \\
\hline
10869.359 & $ 6.858\pm 0.194$ & $31.959\pm 0.796$ \\
10876.453 & $ 7.063\pm 0.177$ & $32.400\pm 0.833$ \\
10905.332 & $ 7.403\pm 0.176$ & $33.322\pm 0.722$ \\
11053.559 & $ 6.227\pm 0.177$ & $30.750\pm 0.726$ \\
11075.582 & $ 6.408\pm 0.171$ & $30.532\pm 0.713$ \\
11220.219 & $ 6.271\pm 0.231$ & $29.815\pm 1.056$ \\
11279.324 & $ 7.851\pm 0.210$ & $37.009\pm 0.962$ \\
11321.301 & $ 5.697\pm 0.265$ & $33.947\pm 1.258$ \\
11517.488 & $ 5.327\pm 0.268$ & $31.821\pm 1.134$ \\
11587.355 & $ 5.174\pm 0.195$ & $27.152\pm 0.741$ \\
11638.250 & $ 6.202\pm 0.163$ & $29.285\pm 0.743$ \\
11707.344 & $ 6.400\pm 0.165$ & $30.001\pm 0.712$ \\
11721.324 & $ 6.689\pm 0.188$ & $32.688\pm 0.838$ \\
11823.543 & $ 6.115\pm 0.221$ & $32.067\pm 0.951$ \\
11847.531 & $ 5.611\pm 0.204$ & $30.121\pm 0.891$ \\
11878.512 & $ 6.134\pm 0.216$ & $30.642\pm 0.918$ \\
11902.461 & $ 6.325\pm 0.219$ & $30.126\pm 0.899$ \\
12000.254 & $ 4.590\pm 0.381$ & $28.123\pm 1.239$ \\
12031.402 & $ 6.159\pm 0.277$ & $27.928\pm 1.015$ \\
12105.324 & $ 5.290\pm 0.183$ & $28.077\pm 0.731$ \\
12176.520 & $ 4.917\pm 0.209$ & $27.194\pm 0.780$ \\
12193.508 & $ 4.927\pm 0.263$ & $28.147\pm 1.009$ \\
12225.438 & $ 3.968\pm 0.598$ & $28.053\pm 1.987$ \\
12323.457 & $ 4.401\pm 0.166$ & $22.116\pm 0.518$ \\
12343.551 & $ 4.813\pm 0.394$ & $25.054\pm 1.615$ \\
12613.594 & $ 5.715\pm 0.277$ & $26.821\pm 1.083$ \\
12675.477 & $ 5.779\pm 0.193$ & $26.451\pm 0.827$ \\
13090.277 & $ 4.505\pm 0.297$ & $24.783\pm 0.909$ \\
13356.465 & $ 3.857\pm 0.159$ & $20.999\pm 0.656$ \\
13610.559 & $ 8.018\pm 0.215$ & $31.853\pm 0.796$ \\
13621.555 & $ 8.487\pm 0.240$ & $32.350\pm 0.891$ \\
13642.594 & $ 7.377\pm 0.235$ & $31.616\pm 1.047$ \\
13648.551 & $ 7.688\pm 0.277$ & $30.374\pm 1.081$ \\
13648.563 & $ 7.474\pm 0.218$ & $30.213\pm 0.859$ \\
13789.316 & $ 7.176\pm 0.227$ & $32.545\pm 0.986$ \\
13831.234 & $ 7.418\pm 0.186$ & $30.618\pm 0.825$ \\
13994.586 & $ 8.940\pm 0.228$ & $34.663\pm 0.875$ \\
14066.457 & $ 8.472\pm 0.257$ & $34.872\pm 1.008$ \\
14370.563 & $ 8.707\pm 0.264$ & $37.536\pm 1.117$ \\
14392.563 & $ 8.377\pm 0.206$ & $36.161\pm 0.988$ \\
14393.527 & $ 8.125\pm 0.256$ & $35.540\pm 1.017$ \\
14481.387 & $ 8.255\pm 0.243$ & $34.970\pm 1.044$ \\
14508.445 & $ 8.117\pm 0.377$ & $36.963\pm 1.866$ \\
14617.324 & $ 7.455\pm 0.205$ & $31.689\pm 0.795$ \\
14620.316 & $ 8.052\pm 0.228$ & $33.558\pm 0.900$ \\
14622.367 & $ 7.785\pm 0.204$ & $32.227\pm 0.890$ \\
14804.543 & $10.004\pm 0.242$ & $38.158\pm 0.965$ \\
\hline
\end{tabular}
{\\\footnotesize
Units are $10^{-13}\,\ergs$ and $10^{-15}\,\ergsA$ for the
\ha\ line and continuum, respectively. \\
}
\end{table}
\begin{table}
\caption{Combined F5170 continuum fluxes from spectral and photometric observations.}
\label{cntflx}
\begin{tabular}{@{}lclc}
\hline
JD-2,440,000 & $F5170$ &Julian Date & $F5170$ \\
\hline
8630.5781 & $4.072\pm 0.107$ & 9783.2969 & $7.666\pm 0.098$ \\
8716.4258 & $5.003\pm 0.143$ & 9814.2656 & $7.775\pm 0.124$ \\
8717.3633 & $5.083\pm 0.129$ & 9838.3027 & $7.282\pm 0.356$ \\
8927.5117 & $4.125\pm 0.054$ & 9839.3398 & $6.827\pm 0.240$ \\
8983.4009 & $4.277\pm 0.050$ & 9867.3389 & $7.121\pm 0.303$ \\
9001.2852 & $4.483\pm 0.151$ & 9871.3545 & $6.683\pm 0.252$ \\
9031.2314 & $3.991\pm 0.150$ & 9872.3594 & $6.775\pm 0.080$ \\
9057.2441 & $4.225\pm 0.146$ & 9980.5746 & $6.749\pm 0.058$ \\
9059.3291 & $4.483\pm 0.171$ & 10008.5636 & $7.303\pm 0.052$ \\
9062.3330 & $4.331\pm 0.161$ & 10009.5860 & $7.380\pm 0.072$ \\
9070.2891 & $4.274\pm 0.068$ & 10010.5900 & $7.445\pm 0.083$ \\
9074.3203 & $4.378\pm 0.059$ & 10013.5390 & $7.236\pm 0.251$ \\
9088.2656 & $5.198\pm 0.076$ & 10015.5170 & $7.458\pm 0.255$ \\
9089.2578 & $5.019\pm 0.079$ & 10024.5360 & $7.791\pm 0.289$ \\
9100.2910 & $5.021\pm 0.186$ & 10036.5900 & $7.798\pm 0.119$ \\
9101.2930 & $5.116\pm 0.174$ & 10047.5526 & $8.101\pm 0.057$ \\
9141.3047 & $6.684\pm 0.119$ & 10064.5120 & $7.593\pm 0.072$ \\
9156.3203 & $7.048\pm 0.126$ & 10069.4550 & $6.782\pm 0.231$ \\
9250.5977 & $6.075\pm 0.170$ & 10092.2260 & $7.053\pm 0.241$ \\
9252.5757 & $6.082\pm 0.093$ & 10094.2990 & $7.399\pm 0.253$ \\
9255.4814 & $5.841\pm 0.210$ & 10096.3280 & $6.939\pm 0.234$ \\
9272.6016 & $5.799\pm 0.068$ & 10102.3720 & $7.282\pm 0.255$ \\
9273.4551 & $5.769\pm 0.211$ & 10133.2640 & $6.606\pm 0.229$ \\
9274.5796 & $5.568\pm 0.047$ & 10135.3040 & $7.540\pm 0.276$ \\
9275.4990 & $5.164\pm 0.184$ & 10139.4580 & $7.376\pm 0.252$ \\
9311.5742 & $5.665\pm 0.077$ & 10156.3134 & $7.602\pm 0.067$ \\
9313.4385 & $5.567\pm 0.061$ & 10159.3190 & $7.949\pm 0.275$ \\
9329.4219 & $5.841\pm 0.214$ & 10161.4450 & $7.790\pm 0.082$ \\
9331.4531 & $6.418\pm 0.068$ & 10201.5550 & $6.994\pm 0.165$ \\
9332.4521 & $6.027\pm 0.212$ & 10202.4060 & $7.008\pm 0.092$ \\
9341.3809 & $6.313\pm 0.219$ & 10212.2930 & $5.893\pm 0.229$ \\
9357.3135 & $6.540\pm 0.233$ & 10213.4450 & $6.383\pm 0.078$ \\
9359.3467 & $6.090\pm 0.211$ & 10218.3130 & $5.790\pm 0.225$ \\
9362.3525 & $6.017\pm 0.208$ & 10222.3060 & $6.101\pm 0.223$ \\
9364.3281 & $6.334\pm 0.067$ & 10225.3320 & $5.847\pm 0.077$ \\
9365.4180 & $6.448\pm 0.060$ & 10246.3710 & $5.474\pm 0.060$ \\
9395.3750 & $6.140\pm 0.068$ & 10258.3980 & $5.091\pm 0.110$ \\
9399.3984 & $6.155\pm 0.088$ & 10304.4220 & $6.479\pm 0.081$ \\
9450.3438 & $6.618\pm 0.062$ & 10361.5310 & $6.989\pm 0.068$ \\
9452.2461 & $6.651\pm 0.100$ & 10363.4840 & $7.053\pm 0.070$ \\
9454.2734 & $6.731\pm 0.077$ & 10364.5410 & $7.064\pm 0.274$ \\
9488.3398 & $7.461\pm 0.072$ & 10372.4804 & $7.427\pm 0.040$ \\
9520.3281 & $6.829\pm 0.076$ & 10392.5060 & $7.411\pm 0.271$ \\
9536.3867 & $6.575\pm 0.064$ & 10395.4687 & $7.013\pm 0.049$ \\
9548.5000 & $6.048\pm 0.099$ & 10396.4610 & $7.103\pm 0.063$ \\
9554.5234 & $5.900\pm 0.111$ & 10397.4610 & $7.274\pm 0.063$ \\
9555.5117 & $5.813\pm 0.124$ & 10399.4490 & $7.179\pm 0.078$ \\
9566.4570 & $5.304\pm 0.068$ & 10401.4600 & $7.018\pm 0.246$ \\
9578.3984 & $5.070\pm 0.058$ & 10403.4010 & $7.599\pm 0.267$ \\
9599.3672 & $4.653\pm 0.134$ & 10404.4920 & $7.635\pm 0.258$ \\
9622.2734 & $5.361\pm 0.094$ & 10406.5080 & $7.446\pm 0.240$ \\
9639.5547 & $5.878\pm 0.096$ & 10408.5900 & $7.202\pm 0.229$ \\
9653.5635 & $6.298\pm 0.074$ & 10430.3770 & $6.573\pm 0.224$ \\
9665.5331 & $5.784\pm 0.069$ & 10434.3200 & $6.386\pm 0.055$ \\
9685.3984 & $6.646\pm 0.147$ & 10435.3240 & $6.586\pm 0.072$ \\
9691.5078 & $6.915\pm 0.086$ & 10436.4100 & $6.371\pm 0.052$ \\
9713.4539 & $7.374\pm 0.052$ & 10461.5560 & $5.841\pm 0.195$ \\
9716.5488 & $7.064\pm 0.252$ & 10482.5040 & $6.621\pm 0.088$ \\
9723.4297 & $7.226\pm 0.081$ & 10483.4790 & $6.683\pm 0.220$ \\
9744.3984 & $8.089\pm 0.078$ & 10484.4480 & $6.628\pm 0.209$ \\
9753.3086 & $8.467\pm 0.074$ & 10487.3660 & $6.410\pm 0.213$ \\
9754.3789 & $8.249\pm 0.183$ & 10491.4380 & $6.573\pm 0.222$ \\
\hline
\end{tabular}
\end{table}
\begin{table}
\contcaption{Combined F5170 continuum fluxes from spectral and photometric observations.}
\begin{tabular}{@{}lclc}
\hline
JD-2,440,000 & $F5170$ &Julian Date & $F5170$ \\
\hline
9771.3086 & $7.988\pm 0.071$ & 10495.3670 & $6.383\pm 0.057$ \\
10509.3810 & $6.054\pm 0.062$ & 11085.5700 & $6.117\pm 0.066$ \\
10510.4262 & $5.990\pm 0.044$ & 11088.4720 & $5.330\pm 0.184$ \\
10511.3540 & $5.608\pm 0.196$ & 11100.5120 & $6.344\pm 0.076$ \\
10518.4320 & $5.408\pm 0.176$ & 11105.5480 & $6.584\pm 0.214$ \\
10519.4260 & $5.638\pm 0.190$ & 11110.5780 & $5.790\pm 0.195$ \\
10521.3830 & $5.709\pm 0.192$ & 11111.5760 & $5.924\pm 0.207$ \\
10522.3851 & $5.517\pm 0.067$ & 11141.4334 & $5.127\pm 0.081$ \\
10541.3549 & $5.524\pm 0.046$ & 11163.4850 & $5.729\pm 0.185$ \\
10543.2960 & $5.059\pm 0.188$ & 11164.3130 & $6.207\pm 0.220$ \\
10566.3030 & $6.464\pm 0.230$ & 11176.5790 & $5.478\pm 0.192$ \\
10569.3380 & $5.517\pm 0.196$ & 11192.3790 & $5.810\pm 0.201$ \\
10574.3634 & $5.080\pm 0.038$ & 11197.2630 & $5.934\pm 0.211$ \\
10575.3160 & $4.840\pm 0.054$ & 11199.3480 & $5.021\pm 0.174$ \\
10576.3520 & $4.868\pm 0.058$ & 11218.2810 & $5.525\pm 0.117$ \\
10580.3820 & $5.252\pm 0.210$ & 11261.4920 & $7.018\pm 0.250$ \\
10597.3852 & $5.460\pm 0.083$ & 11274.3120 & $6.951\pm 0.229$ \\
10601.3550 & $5.540\pm 0.064$ & 11278.3950 & $6.743\pm 0.074$ \\
10611.3440 & $6.296\pm 0.099$ & 11279.2870 & $6.650\pm 0.237$ \\
10628.3590 & $6.778\pm 0.081$ & 11281.3662 & $6.943\pm 0.081$ \\
10642.3320 & $6.824\pm 0.068$ & 11290.4020 & $5.878\pm 0.187$ \\
10654.3320 & $6.808\pm 0.153$ & 11306.3890 & $5.709\pm 0.203$ \\
10655.3280 & $6.764\pm 0.081$ & 11310.2890 & $5.216\pm 0.067$ \\
10687.5700 & $6.033\pm 0.116$ & 11319.3130 & $4.817\pm 0.063$ \\
10697.4208 & $5.833\pm 0.048$ & 11322.3050 & $5.134\pm 0.148$ \\
10699.5270 & $5.661\pm 0.057$ & 11346.3850 & $5.164\pm 0.203$ \\
10705.5590 & $5.587\pm 0.188$ & 11349.3240 & $4.884\pm 0.084$ \\
10714.5080 & $5.504\pm 0.063$ & 11400.5350 & $4.851\pm 0.247$ \\
10715.5200 & $5.526\pm 0.061$ & 11407.5290 & $5.021\pm 0.167$ \\
10728.5390 & $5.333\pm 0.052$ & 11409.5370 & $4.841\pm 0.191$ \\
10729.5080 & $5.399\pm 0.056$ & 11454.5560 & $5.658\pm 0.207$ \\
10747.5430 & $5.962\pm 0.076$ & 11467.4890 & $5.310\pm 0.181$ \\
10748.4650 & $5.950\pm 0.057$ & 11485.5040 & $5.096\pm 0.072$ \\
10755.5340 & $6.196\pm 0.211$ & 11488.6040 & $5.136\pm 0.193$ \\
10758.5249 & $5.924\pm 0.052$ & 11493.3450 & $5.438\pm 0.202$ \\
10759.4380 & $5.777\pm 0.076$ & 11497.4340 & $5.160\pm 0.086$ \\
10760.5280 & $5.924\pm 0.195$ & 11516.5510 & $5.103\pm 0.073$ \\
10761.5130 & $6.185\pm 0.206$ & 11522.5520 & $4.610\pm 0.162$ \\
10762.5670 & $6.196\pm 0.209$ & 11524.5720 & $5.184\pm 0.231$ \\
10777.4300 & $5.776\pm 0.067$ & 11525.4560 & $5.126\pm 0.190$ \\
10801.2660 & $5.955\pm 0.203$ & 11557.3624 & $4.581\pm 0.056$ \\
10817.3570 & $5.389\pm 0.177$ & 11577.4060 & $4.456\pm 0.066$ \\
10863.3710 & $6.562\pm 0.221$ & 11581.3890 & $4.547\pm 0.153$ \\
10866.3790 & $6.377\pm 0.234$ & 11586.4480 & $4.601\pm 0.157$ \\
10867.4230 & $6.291\pm 0.210$ & 11587.3295 & $4.542\pm 0.053$ \\
10868.4180 & $6.453\pm 0.212$ & 11588.5210 & $4.907\pm 0.163$ \\
10869.4520 & $5.996\pm 0.047$ & 11598.2560 & $4.721\pm 0.180$ \\
10873.4040 & $6.410\pm 0.208$ & 11603.2890 & $4.879\pm 0.192$ \\
10874.2920 & $6.324\pm 0.228$ & 11605.4510 & $5.145\pm 0.191$ \\
10875.4490 & $6.279\pm 0.209$ & 11606.3830 & $4.821\pm 0.112$ \\
10876.3961 & $6.207\pm 0.063$ & 11608.3236 & $4.493\pm 0.056$ \\
10905.3950 & $6.451\pm 0.057$ & 11612.3560 & $5.097\pm 0.181$ \\
10906.2730 & $6.352\pm 0.088$ & 11615.3400 & $5.173\pm 0.114$ \\
10924.4482 & $5.708\pm 0.074$ & 11628.3200 & $5.729\pm 0.209$ \\
10957.3660 & $5.136\pm 0.200$ & 11636.2700 & $5.252\pm 0.076$ \\
10966.3790 & $5.008\pm 0.058$ & 11661.2870 & $5.310\pm 0.194$ \\
10982.3950 & $5.570\pm 0.101$ & 11707.4020 & $5.783\pm 0.095$ \\
10994.3922 & $5.671\pm 0.053$ & 11720.3360 & $5.875\pm 0.069$ \\
11014.5160 & $5.302\pm 0.091$ & 11780.5550 & $5.757\pm 0.102$ \\
11044.3860 & $5.408\pm 0.220$ & 11782.5550 & $5.868\pm 0.080$ \\
11050.5250 & $5.498\pm 0.181$ & 11788.5360 & $5.638\pm 0.243$ \\
11052.5630 & $5.519\pm 0.081$ & 11791.5310 & $5.385\pm 0.093$ \\
\hline
\end{tabular}
\end{table}
\begin{table}
\contcaption{Combined F5170 continuum fluxes from spectral and photometric observations.}
\begin{tabular}{@{}lclc}
\hline
JD-2,440,000 & $F5170$ &Julian Date & $F5170$ \\
\hline
11074.5740 & $5.665\pm 0.064$ & 11810.5310 & $5.463\pm 0.075$ \\
11076.4995 & $5.650\pm 0.077$ & 11817.5320 & $5.986\pm 0.243$ \\
11818.5550 & $5.408\pm 0.185$ & 12349.3890 & $3.483\pm 0.133$ \\
11821.5820 & $5.010\pm 0.092$ & 12366.2620 & $3.890\pm 0.024$ \\
11823.4960 & $5.049\pm 0.072$ & 12367.2780 & $3.984\pm 0.026$ \\
11838.5080 & $5.234\pm 0.183$ & 12368.3450 & $4.077\pm 0.057$ \\
11840.5390 & $5.027\pm 0.103$ & 12369.2769 & $4.188\pm 0.022$ \\
11842.4730 & $5.388\pm 0.206$ & 12381.2620 & $4.039\pm 0.125$ \\
11843.5660 & $5.184\pm 0.439$ & 12385.3530 & $4.388\pm 0.033$ \\
11844.4960 & $4.679\pm 0.072$ & 12386.2840 & $4.481\pm 0.038$ \\
11847.5150 & $4.600\pm 0.062$ & 12387.2960 & $4.401\pm 0.034$ \\
11853.5681 & $4.767\pm 0.049$ & 12388.3290 & $4.504\pm 0.050$ \\
11866.4100 & $5.478\pm 0.233$ & 12399.3050 & $4.898\pm 0.169$ \\
11867.3210 & $5.155\pm 0.197$ & 12403.3180 & $5.097\pm 0.190$ \\
11868.3710 & $5.349\pm 0.190$ & 12404.2690 & $5.073\pm 0.029$ \\
11869.4960 & $5.184\pm 0.073$ & 12405.3055 & $5.015\pm 0.030$ \\
11878.4490 & $5.012\pm 0.080$ & 12406.3010 & $5.193\pm 0.227$ \\
11879.2630 & $5.222\pm 0.212$ & 12407.2730 & $4.971\pm 0.034$ \\
11882.4800 & $4.656\pm 0.170$ & 12408.2942 & $5.104\pm 0.025$ \\
11901.3160 & $5.436\pm 0.106$ & 12409.3480 & $5.078\pm 0.216$ \\
11902.2690 & $5.688\pm 0.205$ & 12410.2870 & $5.132\pm 0.028$ \\
11912.5120 & $5.116\pm 0.190$ & 12411.2860 & $5.119\pm 0.029$ \\
11926.2770 & $4.424\pm 0.056$ & 12417.2820 & $5.289\pm 0.052$ \\
11932.4920 & $4.795\pm 0.184$ & 12419.3020 & $5.272\pm 0.049$ \\
11959.2270 & $4.689\pm 0.129$ & 12421.2880 & $5.175\pm 0.058$ \\
11999.2700 & $4.517\pm 0.112$ & 12440.2860 & $5.165\pm 0.050$ \\
12030.4220 & $5.191\pm 0.100$ & 12441.3590 & $5.067\pm 0.078$ \\
12050.4650 & $4.689\pm 0.087$ & 12442.2850 & $5.123\pm 0.044$ \\
12104.3160 & $4.394\pm 0.069$ & 12459.2830 & $5.198\pm 0.064$ \\
12139.5420 & $4.138\pm 0.196$ & 12476.2920 & $5.265\pm 0.063$ \\
12144.5390 & $4.620\pm 0.200$ & 12484.3370 & $4.793\pm 0.064$ \\
12147.5200 & $4.060\pm 0.155$ & 12498.4950 & $4.379\pm 0.024$ \\
12166.5140 & $4.629\pm 0.167$ & 12530.5830 & $4.424\pm 0.031$ \\
12174.5390 & $4.327\pm 0.069$ & 12536.5580 & $4.640\pm 0.025$ \\
12175.5310 & $4.153\pm 0.084$ & 12539.5520 & $4.732\pm 0.040$ \\
12192.5510 & $4.046\pm 0.095$ & 12541.5710 & $4.653\pm 0.041$ \\
12199.5620 & $4.094\pm 0.151$ & 12557.5940 & $4.579\pm 0.027$ \\
12201.5449 & $3.888\pm 0.054$ & 12566.4570 & $4.422\pm 0.036$ \\
12210.5970 & $4.129\pm 0.149$ & 12569.5400 & $4.648\pm 0.043$ \\
12223.5055 & $4.090\pm 0.060$ & 12593.6361 & $4.954\pm 0.035$ \\
12225.3400 & $4.340\pm 0.161$ & 12595.3070 & $4.844\pm 0.035$ \\
12231.5060 & $4.060\pm 0.151$ & 12596.5600 & $4.890\pm 0.043$ \\
12263.4680 & $4.094\pm 0.024$ & 12597.5780 & $4.956\pm 0.044$ \\
12265.5230 & $4.182\pm 0.140$ & 12605.5350 & $5.416\pm 0.051$ \\
12280.5470 & $3.783\pm 0.020$ & 12608.5690 & $5.428\pm 0.034$ \\
12281.4510 & $3.725\pm 0.018$ & 12609.5610 & $5.467\pm 0.030$ \\
12283.4430 & $3.801\pm 0.020$ & 12610.4815 & $5.324\pm 0.028$ \\
12298.4070 & $3.623\pm 0.022$ & 12612.5340 & $5.550\pm 0.030$ \\
12301.3840 & $3.625\pm 0.028$ & 12613.4293 & $5.497\pm 0.030$ \\
12307.2820 & $3.427\pm 0.129$ & 12614.5590 & $5.548\pm 0.033$ \\
12308.3500 & $3.628\pm 0.019$ & 12618.3620 & $5.532\pm 0.026$ \\
12309.4384 & $3.599\pm 0.019$ & 12619.5547 & $5.452\pm 0.026$ \\
12310.4000 & $3.307\pm 0.128$ & 12620.4070 & $5.375\pm 0.035$ \\
12313.4260 & $3.307\pm 0.132$ & 12621.4710 & $5.477\pm 0.034$ \\
12314.3730 & $3.823\pm 0.030$ & 12625.4390 & $5.301\pm 0.036$ \\
12316.4110 & $3.906\pm 0.020$ & 12634.2880 & $4.730\pm 0.164$ \\
12321.4275 & $4.018\pm 0.023$ & 12635.4457 & $5.088\pm 0.155$ \\
12322.4080 & $4.144\pm 0.023$ & 12636.3986 & $5.035\pm 0.042$ \\
12323.3560 & $4.108\pm 0.021$ & 12665.3620 & $4.728\pm 0.026$ \\
12324.3990 & $4.152\pm 0.126$ & 12672.3350 & $4.974\pm 0.038$ \\
12336.2490 & $3.507\pm 0.110$ & 12674.4100 & $5.041\pm 0.031$ \\
12342.4700 & $3.402\pm 0.144$ & 12675.3846 & $5.002\pm 0.028$ \\
\hline
\end{tabular}
\end{table}
\begin{table}
\contcaption{Combined F5170 continuum fluxes from spectral and photometric observations.}
\begin{tabular}{@{}lclc}
\hline
JD-2,440,000 & $F5170$ &Julian Date & $F5170$ \\
\hline
12343.4263 & $3.559\pm 0.046$ & 12683.3540 & $4.635\pm 0.037$ \\
12346.3890 & $3.499\pm 0.132$ & 12684.3840 & $4.703\pm 0.034$ \\
12348.2790 & $3.315\pm 0.141$ & 12685.3890 & $4.632\pm 0.051$ \\
12689.3540 & $4.830\pm 0.041$ & 12998.4690 & $4.207\pm 0.024$ \\
12694.3330 & $4.772\pm 0.034$ & 13003.4349 & $4.187\pm 0.022$ \\
12696.3900 & $5.087\pm 0.195$ & 13006.4560 & $4.084\pm 0.056$ \\
12697.3620 & $4.836\pm 0.025$ & 13007.4330 & $4.046\pm 0.034$ \\
12698.3500 & $4.913\pm 0.027$ & 13015.4930 & $3.971\pm 0.057$ \\
12700.3786 & $4.836\pm 0.027$ & 13022.4180 & $4.391\pm 0.024$ \\
12701.3560 & $4.781\pm 0.027$ & 13023.3250 & $4.367\pm 0.143$ \\
12703.2680 & $4.711\pm 0.175$ & 13058.3160 & $4.776\pm 0.026$ \\
12710.2760 & $4.595\pm 0.028$ & 13071.4320 & $4.767\pm 0.072$ \\
12716.2540 & $4.167\pm 0.033$ & 13073.3260 & $4.521\pm 0.033$ \\
12722.3690 & $4.069\pm 0.145$ & 13077.3130 & $4.367\pm 0.136$ \\
12723.4030 & $3.889\pm 0.142$ & 13083.3410 & $4.438\pm 0.030$ \\
12724.3289 & $4.009\pm 0.024$ & 13084.3190 & $4.500\pm 0.029$ \\
12726.2940 & $3.771\pm 0.138$ & 13084.3190 & $4.500\pm 0.029$ \\
12727.3100 & $4.033\pm 0.028$ & 13085.3510 & $4.558\pm 0.028$ \\
12728.3020 & $4.123\pm 0.024$ & 13087.2780 & $4.425\pm 0.028$ \\
12729.3010 & $4.177\pm 0.024$ & 13089.3240 & $4.197\pm 0.062$ \\
12730.2840 & $4.232\pm 0.029$ & 13097.2904 & $4.606\pm 0.041$ \\
12739.2820 & $4.235\pm 0.028$ & 13098.2500 & $4.636\pm 0.038$ \\
12740.3020 & $4.185\pm 0.031$ & 13105.3080 & $4.556\pm 0.152$ \\
12742.3080 & $4.205\pm 0.035$ & 13111.3030 & $4.978\pm 0.027$ \\
12744.3070 & $4.221\pm 0.030$ & 13112.3115 & $4.799\pm 0.025$ \\
12745.2760 & $4.272\pm 0.037$ & 13113.3120 & $4.864\pm 0.028$ \\
12751.2799 & $4.242\pm 0.024$ & 13114.2551 & $5.012\pm 0.030$ \\
12752.2510 & $4.309\pm 0.024$ & 13115.2890 & $5.108\pm 0.029$ \\
12754.2770 & $4.383\pm 0.024$ & 13117.3010 & $4.814\pm 0.164$ \\
12756.2860 & $4.324\pm 0.024$ & 13130.2730 & $5.063\pm 0.044$ \\
12757.2770 & $4.296\pm 0.024$ & 13133.2930 & $5.075\pm 0.026$ \\
12759.2830 & $4.647\pm 0.165$ & 13135.2730 & $5.004\pm 0.029$ \\
12766.2680 & $4.454\pm 0.025$ & 13148.2890 & $4.770\pm 0.030$ \\
12767.2590 & $4.544\pm 0.029$ & 13149.2750 & $4.727\pm 0.045$ \\
12770.2590 & $4.320\pm 0.038$ & 13153.2910 & $4.802\pm 0.032$ \\
12771.2560 & $4.272\pm 0.051$ & 13154.3280 & $4.884\pm 0.052$ \\
12774.2750 & $4.356\pm 0.035$ & 13277.5570 & $5.045\pm 0.079$ \\
12775.2710 & $4.353\pm 0.043$ & 13291.5610 & $4.765\pm 0.029$ \\
12778.2980 & $4.492\pm 0.185$ & 13292.5770 & $4.817\pm 0.029$ \\
12790.3080 & $4.933\pm 0.028$ & 13296.6160 & $4.355\pm 0.027$ \\
12791.2820 & $4.859\pm 0.040$ & 13300.6100 & $4.245\pm 0.029$ \\
12841.5450 & $4.824\pm 0.104$ & 13302.6060 & $4.161\pm 0.029$ \\
12866.5290 & $5.115\pm 0.034$ & 13305.6230 & $4.073\pm 0.027$ \\
12883.5410 & $5.011\pm 0.171$ & 13307.6120 & $4.107\pm 0.053$ \\
12889.5360 & $5.408\pm 0.169$ & 13308.6250 & $4.040\pm 0.046$ \\
12890.5250 & $5.376\pm 0.032$ & 13309.6280 & $3.986\pm 0.039$ \\
12903.5440 & $5.300\pm 0.186$ & 13313.6020 & $4.011\pm 0.027$ \\
12906.5980 & $5.109\pm 0.031$ & 13314.5960 & $4.109\pm 0.027$ \\
12907.5800 & $4.926\pm 0.159$ & 13315.6030 & $4.046\pm 0.024$ \\
12912.5813 & $5.089\pm 0.030$ & 13317.5910 & $3.993\pm 0.027$ \\
12913.5490 & $5.132\pm 0.034$ & 13318.6095 & $3.912\pm 0.028$ \\
12914.5768 & $5.108\pm 0.028$ & 13319.4960 & $3.639\pm 0.065$ \\
12915.5268 & $5.260\pm 0.049$ & 13320.4660 & $3.581\pm 0.116$ \\
12945.6380 & $4.830\pm 0.047$ & 13323.5880 & $3.991\pm 0.133$ \\
12947.5950 & $4.664\pm 0.032$ & 13331.5011 & $3.818\pm 0.021$ \\
12965.5343 & $4.463\pm 0.022$ & 13355.4852 & $3.397\pm 0.019$ \\
12966.5492 & $4.309\pm 0.027$ & 13356.4777 & $3.397\pm 0.022$ \\
12967.5156 & $4.435\pm 0.026$ & 13357.4334 & $3.360\pm 0.018$ \\
12968.5385 & $4.372\pm 0.024$ & 13358.4690 & $3.454\pm 0.021$ \\
12973.4510 & $4.025\pm 0.132$ & 13365.5680 & $3.342\pm 0.037$ \\
12974.5110 & $3.982\pm 0.131$ & 13379.5508 & $3.316\pm 0.019$ \\
12983.6310 & $4.241\pm 0.045$ & 13383.4821 & $3.446\pm 0.019$ \\
\hline
\end{tabular}
\end{table}
\begin{table}
\contcaption{Combined F5170 continuum fluxes from spectral and photometric observations.}
\begin{tabular}{@{}lclc}
\hline
JD-2,440,000 & $F5170$ &Julian Date & $F5170$ \\
\hline
12984.5580 & $4.277\pm 0.034$ & 13384.4410 & $3.389\pm 0.020$ \\
12985.4810 & $4.329\pm 0.087$ & 13410.4190 & $3.370\pm 0.113$ \\
12988.5170 & $4.372\pm 0.029$ & 13411.5060 & $3.259\pm 0.106$ \\
12996.5188 & $4.248\pm 0.024$ & 13412.3370 & $2.904\pm 0.097$ \\
12997.4890 & $4.193\pm 0.025$ & 13419.3710 & $3.446\pm 0.047$ \\
13423.3570 & $3.454\pm 0.033$ & 13822.3430 & $6.565\pm 0.043$ \\
13424.4630 & $3.439\pm 0.068$ & 13823.3110 & $6.558\pm 0.044$ \\
13425.3360 & $3.493\pm 0.041$ & 13830.2380 & $6.715\pm 0.113$ \\
13434.3090 & $3.703\pm 0.022$ & 13837.3880 & $6.661\pm 0.051$ \\
13436.3070 & $3.791\pm 0.024$ & 13839.3400 & $6.759\pm 0.068$ \\
13437.2730 & $3.483\pm 0.114$ & 13844.3300 & $6.879\pm 0.040$ \\
13441.3348 & $3.908\pm 0.021$ & 13845.4410 & $6.959\pm 0.047$ \\
13445.2997 & $4.160\pm 0.024$ & 13849.2870 & $7.053\pm 0.251$ \\
13446.3470 & $4.388\pm 0.042$ & 13850.2917 & $7.117\pm 0.041$ \\
13449.3550 & $4.369\pm 0.034$ & 13854.2850 & $7.053\pm 0.220$ \\
13459.3238 & $4.875\pm 0.026$ & 13875.3320 & $7.178\pm 0.086$ \\
13460.2620 & $4.978\pm 0.028$ & 13880.3050 & $6.746\pm 0.036$ \\
13461.2693 & $5.067\pm 0.027$ & 13881.2850 & $6.795\pm 0.041$ \\
13462.3190 & $5.040\pm 0.032$ & 13953.5550 & $6.732\pm 0.106$ \\
13464.2240 & $5.285\pm 0.037$ & 13959.5530 & $6.737\pm 0.080$ \\
13465.3125 & $5.363\pm 0.031$ & 13967.5620 & $6.846\pm 0.056$ \\
13471.2810 & $5.882\pm 0.206$ & 13973.5680 & $6.731\pm 0.050$ \\
13476.3240 & $5.693\pm 0.037$ & 13986.5780 & $7.163\pm 0.090$ \\
13478.2290 & $5.844\pm 0.111$ & 13987.4830 & $7.138\pm 0.066$ \\
13487.2960 & $5.574\pm 0.029$ & 13989.5540 & $7.280\pm 0.091$ \\
13493.2990 & $5.485\pm 0.029$ & 13991.5330 & $7.168\pm 0.069$ \\
13495.2680 & $5.497\pm 0.034$ & 13994.5660 & $7.272\pm 0.080$ \\
13508.3130 & $5.755\pm 0.037$ & 13995.5930 & $7.363\pm 0.054$ \\
13509.2980 & $5.987\pm 0.045$ & 14010.5960 & $7.574\pm 0.066$ \\
13611.5630 & $6.791\pm 0.081$ & 14013.6130 & $7.494\pm 0.089$ \\
13612.5531 & $6.801\pm 0.083$ & 14022.5790 & $7.319\pm 0.058$ \\
13621.5230 & $6.877\pm 0.082$ & 14023.4520 & $7.557\pm 0.061$ \\
13641.5660 & $6.279\pm 0.096$ & 14038.5550 & $7.741\pm 0.089$ \\
13644.5290 & $5.884\pm 0.039$ & 14044.6030 & $8.011\pm 0.123$ \\
13645.6130 & $5.900\pm 0.076$ & 14048.5820 & $7.691\pm 0.097$ \\
13647.5700 & $6.158\pm 0.095$ & 14059.5450 & $7.104\pm 0.047$ \\
13648.6094 & $5.937\pm 0.048$ & 14060.6550 & $6.988\pm 0.147$ \\
13649.5913 & $6.174\pm 0.075$ & 14062.5450 & $7.171\pm 0.054$ \\
13650.6170 & $5.939\pm 0.054$ & 14064.5904 & $7.159\pm 0.049$ \\
13651.5770 & $5.914\pm 0.040$ & 14065.4730 & $7.052\pm 0.036$ \\
13653.6120 & $5.885\pm 0.031$ & 14067.5428 & $6.865\pm 0.045$ \\
13654.5950 & $5.972\pm 0.038$ & 14069.6400 & $6.920\pm 0.056$ \\
13670.5550 & $6.284\pm 0.083$ & 14076.4610 & $6.903\pm 0.156$ \\
13676.5860 & $6.309\pm 0.079$ & 14078.5510 & $6.923\pm 0.101$ \\
13680.5980 & $5.868\pm 0.033$ & 14086.5590 & $7.182\pm 0.097$ \\
13683.5360 & $5.688\pm 0.189$ & 14091.5530 & $7.180\pm 0.044$ \\
13698.4690 & $6.366\pm 0.103$ & 14106.5310 & $7.897\pm 0.243$ \\
13702.4780 & $5.872\pm 0.203$ & 14111.5780 & $7.450\pm 0.050$ \\
13708.5590 & $6.684\pm 0.036$ & 14116.4500 & $7.638\pm 0.059$ \\
13724.5730 & $7.009\pm 0.049$ & 14117.5520 & $7.552\pm 0.331$ \\
13728.3990 & $6.749\pm 0.217$ & 14118.4140 & $7.587\pm 0.055$ \\
13733.5160 & $6.918\pm 0.039$ & 14119.4440 & $7.704\pm 0.051$ \\
13738.4460 & $6.940\pm 0.046$ & 14121.5630 & $7.700\pm 0.047$ \\
13739.5320 & $6.813\pm 0.047$ & 14123.4870 & $7.720\pm 0.064$ \\
13744.5540 & $6.880\pm 0.047$ & 14142.5940 & $7.459\pm 0.097$ \\
13747.4950 & $6.735\pm 0.078$ & 14145.3596 & $7.674\pm 0.049$ \\
13749.4240 & $6.698\pm 0.070$ & 14146.4190 & $7.550\pm 0.044$ \\
13760.3710 & $6.082\pm 0.030$ & 14149.4380 & $7.735\pm 0.041$ \\
13761.4160 & $6.143\pm 0.042$ & 14150.4260 & $7.695\pm 0.038$ \\
13763.3990 & $5.901\pm 0.035$ & 14156.2700 & $7.366\pm 0.132$ \\
13774.4530 & $5.677\pm 0.189$ & 14167.3450 & $7.289\pm 0.050$ \\
13787.3870 & $6.090\pm 0.203$ & 14169.2930 & $7.252\pm 0.048$ \\
\hline
\end{tabular}
\end{table}
\begin{table}
\contcaption{Combined F5170 continuum fluxes from spectral and photometric observations.}
\begin{tabular}{@{}lclc}
\hline
JD-2,440,000 & $F5170$ &Julian Date & $F5170$ \\
\hline
13788.3440 & $5.997\pm 0.091$ & 14171.3070 & $7.250\pm 0.050$ \\
13790.2820 & $6.259\pm 0.199$ & 14174.4000 & $7.481\pm 0.243$ \\
13799.2870 & $6.168\pm 0.046$ & 14180.3510 & $7.516\pm 0.260$ \\
13807.3650 & $6.250\pm 0.057$ & 14181.2360 & $7.412\pm 0.042$ \\
13816.4150 & $6.349\pm 0.041$ & 14191.2710 & $7.835\pm 0.083$ \\
13820.3430 & $6.248\pm 0.208$ & 14200.3340 & $8.249\pm 0.058$ \\
14201.2850 & $7.755\pm 0.269$ & 14523.3480 & $6.185\pm 0.197$ \\
14204.2350 & $8.256\pm 0.058$ & 14530.3709 & $6.735\pm 0.064$ \\
14206.3000 & $8.379\pm 0.073$ & 14532.3500 & $6.735\pm 0.044$ \\
14213.2460 & $7.978\pm 0.049$ & 14534.3240 & $6.662\pm 0.036$ \\
14220.2400 & $8.159\pm 0.108$ & 14535.3485 & $6.570\pm 0.031$ \\
14234.2640 & $8.173\pm 0.060$ & 14536.3470 & $6.542\pm 0.038$ \\
14281.2840 & $8.751\pm 0.203$ & 14537.3175 & $6.452\pm 0.051$ \\
14283.2890 & $8.724\pm 0.117$ & 14538.3130 & $6.436\pm 0.062$ \\
14331.5430 & $8.416\pm 0.086$ & 14542.2750 & $6.073\pm 0.056$ \\
14337.5760 & $8.331\pm 0.118$ & 14554.3231 & $6.021\pm 0.030$ \\
14338.5040 & $8.265\pm 0.051$ & 14555.2970 & $5.994\pm 0.093$ \\
14369.5160 & $7.978\pm 0.124$ & 14565.4980 & $5.939\pm 0.037$ \\
14370.5390 & $7.390\pm 0.197$ & 14567.2910 & $5.906\pm 0.034$ \\
14371.4990 & $7.464\pm 0.073$ & 14568.2830 & $6.029\pm 0.043$ \\
14372.4860 & $7.718\pm 0.071$ & 14574.2652 & $6.262\pm 0.052$ \\
14376.5510 & $7.373\pm 0.061$ & 14582.4170 & $6.398\pm 0.035$ \\
14377.5820 & $7.570\pm 0.110$ & 14585.2780 & $6.093\pm 0.046$ \\
14386.4570 & $7.490\pm 0.098$ & 14586.3090 & $6.284\pm 0.086$ \\
14391.6032 & $7.187\pm 0.051$ & 14587.2927 & $6.252\pm 0.034$ \\
14392.5390 & $7.115\pm 0.097$ & 14588.2802 & $6.184\pm 0.054$ \\
14393.5960 & $7.037\pm 0.045$ & 14590.3090 & $6.278\pm 0.098$ \\
14423.5722 & $6.184\pm 0.037$ & 14591.2730 & $6.440\pm 0.108$ \\
14425.6040 & $6.198\pm 0.032$ & 14596.3030 & $6.331\pm 0.101$ \\
14426.5370 & $6.223\pm 0.047$ & 14600.2710 & $6.832\pm 0.055$ \\
14428.4230 & $6.298\pm 0.058$ & 14601.3560 & $6.695\pm 0.093$ \\
14434.4860 & $6.146\pm 0.044$ & 14602.3840 & $6.727\pm 0.057$ \\
14438.4380 & $5.934\pm 0.100$ & 14603.3040 & $6.667\pm 0.051$ \\
14439.6060 & $6.055\pm 0.034$ & 14604.3070 & $6.752\pm 0.056$ \\
14443.5395 & $6.118\pm 0.031$ & 14617.2919 & $6.978\pm 0.059$ \\
14444.5270 & $6.180\pm 0.033$ & 14618.2730 & $6.876\pm 0.087$ \\
14465.4540 & $7.011\pm 0.042$ & 14620.3010 & $7.053\pm 0.091$ \\
14467.5530 & $7.210\pm 0.043$ & 14622.4260 & $7.140\pm 0.087$ \\
14472.4660 & $7.411\pm 0.041$ & 14628.3120 & $6.876\pm 0.046$ \\
14475.4050 & $6.727\pm 0.227$ & 14632.3170 & $6.960\pm 0.075$ \\
14476.4683 & $7.358\pm 0.048$ & 14643.3460 & $7.117\pm 0.044$ \\
14477.4660 & $7.463\pm 0.058$ & 14647.3060 & $7.129\pm 0.041$ \\
14478.4300 & $7.452\pm 0.053$ & 14687.5470 & $6.498\pm 0.095$ \\
14479.4230 & $7.456\pm 0.053$ & 14713.5470 & $7.087\pm 0.267$ \\
14480.4251 & $7.368\pm 0.049$ & 14718.5400 & $7.288\pm 0.051$ \\
14481.4803 & $7.412\pm 0.054$ & 14720.5630 & $7.213\pm 0.227$ \\
14482.4074 & $7.451\pm 0.053$ & 14738.5060 & $7.030\pm 0.237$ \\
14483.4110 & $7.452\pm 0.069$ & 14740.4520 & $7.053\pm 0.241$ \\
14484.3690 & $7.183\pm 0.069$ & 14742.5900 & $7.451\pm 0.051$ \\
14488.4530 & $7.111\pm 0.120$ & 14769.5830 & $8.158\pm 0.265$ \\
14490.4630 & $7.314\pm 0.062$ & 14778.5960 & $8.054\pm 0.050$ \\
14496.4231 & $7.715\pm 0.044$ & 14779.6064 & $8.005\pm 0.041$ \\
14497.4230 & $7.654\pm 0.054$ & 14780.5790 & $8.052\pm 0.042$ \\
14498.4680 & $7.550\pm 0.051$ & 14781.5400 & $7.969\pm 0.048$ \\
14499.4361 & $7.641\pm 0.045$ & 14784.3780 & $8.013\pm 0.128$ \\
14500.4060 & $7.739\pm 0.050$ & 14786.4900 & $8.206\pm 0.084$ \\
14502.4057 & $7.653\pm 0.052$ & 14796.4990 & $8.174\pm 0.050$ \\
14503.4437 & $7.605\pm 0.045$ & 14799.6310 & $8.263\pm 0.061$ \\
14508.4223 & $7.310\pm 0.034$ & 14801.3500 & $8.467\pm 0.064$ \\
14512.5400 & $6.981\pm 0.131$ & 14802.5980 & $8.346\pm 0.052$ \\
14522.2720 & $6.496\pm 0.211$ & 14804.5660 & $8.214\pm 0.093$ \\
\hline
\end{tabular}
{\\\footnotesize
Continuum fluxes are in units $10^{-15}$\,ergs\,cm$^{-2}$\,s$^{-1}$\,\AA$^{-1}$
}
\end{table}
Bottom panel in Figure~\ref{lc} shows the combined light curve from different telescopes.
The combined light curve shows long time scale continuum variability in Mrk~6 as well as more rapid random changes. The flux maxima were observed in 1995--1996 and in 2007.
Statistical parameters of the light curves for the lines and continuum are listed in Table~\ref{stat}. Column~1 gives the spectral features; column~2 gives a number of data points; column~3 is the median time interval between the data points. The combined continuum light curve is sampled better than the \hb\ light curve, the \hb\ light curve is sampled better than \ha. The mean flux and standard deviation are given in columns~4 and 5, and column~6 lists the variance $F_{var}$ calculated as the ratio of the rms fluctuation, corrected for the effect of measurement errors, to the mean flux. $R_{max}$ in column~7 is the ratio between the maximum and minimum fluxes corrected for the measurement errors. Uncertainties in $F_{var}$ and $R_{max}$ were computed assuming that a light curve is a set of statistically dependent values, i.e., a random process. The $F_{var}$ values in Table~\ref{stat} are the lowest limits of actual $F_{var}$ because the observed fluxes were not corrected for starlight contamination.
\begin{table}
\caption{Light curve statistics.}
\label{stat}
{\scriptsize
\begin{tabular}{@{}ccccccc@{}}
\hline
Time & N & dt-med & Mean & std$^b$ & $F_{var}$& $R_{max}$ \\
series$^a$ & &(days)& Flux$^b$& & & \\
(1) &(2) & (3) &(4) & (5) & (6) & (7) \\
\hline
\multicolumn{7}{@{}c}{1992--2008, JD2448630--2454804}\\
F5170s & 235 &14 & 6.093 & 1.101& 0.18$\pm$0.03 & 2.65$\pm$0.06 \\
F5170scp & 742 & 3 & 5.780 & 1.277& 0.22$\pm$0.05 & 3.16$\pm$0.08 \\
\hb & 235 &14 & 4.065 & 0.658& 0.16$\pm$0.03 & 2.70$\pm$0.09 \\[3pt]
F7030s & 102 &29 & 6.897 & 1.261& 0.18$\pm$0.04 & 2.43$\pm$0.17 \\
\ha & 102 &29 & 32.06 & 3.77 & 0.11$\pm$0.02 & 1.83$\pm$0.09 \\[3pt]
\hline
\end{tabular}
} {\\\footnotesize $^a$ The letters in the first column indicate the origin of the continuum fluxes:
"s" -- from the spectral observations only and "scp" -- from combined spectral and photometric
observations ("c" -- CCD photometry, "p" -- photoelectric photometry).\\ $^b$ Mean fluxes and
standard deviations (std) are in units of
10$^{-13}$~erg~cm$^{-2}$~s$^{-1}$ and 10$^{-15}$~erg~cm$^{-2}$~s$^{-1}$~\AA$^{-1}$ for the
lines and the continuum, respectively.}
\end{table}
\section{Cross-correlation between the continuum and the integral Balmer line flux
variations}
\label{fccf}
As mentioned in the Introduction, estimating the light travel time delay between the continuum
and emission line flux variations is of special relevance for the determination of the BLR size,
which, in turn, can be used for black hole mass measurements \citep[see][]{WPM99, Peterson04}.
This time delay (or lag) is estimated through the cross-correlation function (CCF).
\citet{KorGask91} demonstrated that the CCF centroid gives the luminosity-weighted radius, in
contrast to the CCF peak, which is more influenced by gas at small radii, according to
\citet{Gaskell86}.
The time delays were computed using the interpolated cross-correlation function (ICCF)
\citep{Gaskell86, White94, Peterson01}. We computed both the lag related to the CCF peak ($\tau_{pk}$) and the CCF centroid ($\tau_{cn}$). The CCF centroid was adopted to be measured above the correlation level at $r\geq0.8 r_{max}$. The lag uncertainties were computed using the model-independent Monte Carlo flux randomization/random subset selection (FR/RSS) technique described by \citet{Peterson98}. The number of realizations was as large as 4000. The uncertainties were computed from the distribution function for $\tau_{pk}$ and $\tau_{cn}$ at the 68\% confidence, which corresponds to $\pm 1\sigma$ errors for the normal distribution.
The spectra for the 1992 season were obtained with the 2$\arcsec$ entrance slit and they were discarded from the CCF analysis in order to exclude the aperture effects. The results of the cross-correlation analysis for the 1993--2008 interval are presented in Table~\ref{ccf-ab}. The meaning of symbols "s" and "scp" in the first and second columns of Table~\ref{ccf-ab} is the same as in Table~\ref{stat}: "s" -- from spectral observations only and "scp" -- from combined spectral and photometric observations ("c" -- CCD, "p" -- photoelectric).
\begin{figure}
\includegraphics[width=84mm]{F04acf-ccf.eps}
\caption{CCFs of Mrk~6 between the \hb\ and continuum F5170 fluxes (left panel) as well
as between the \ha\ and continuum F7090 in 1993--2008 (thick lines). ACFs are shown by dashed
lines.}
\label{ccf}
\end{figure}
\begin{table}
\caption{Cross-correlation results for 1993--2008.}
\label{ccf-ab}
{\footnotesize
\begin{tabular}{@{}cccccc}
\hline
First set & Second set &$ r_{max}$&$\tau_{cn}$ & $\tau_{pk}$\\
\hline
\hb & F5170s & 0.84 & 21.5$^{+3.9}_{-2.1}$ & 21.0$^{+1.6}_{-4.9}$\\[5pt]
\hb & F5170scp & 0.83 & 21.9$^{+2.6}_{-2.7}$ & 21.0$^{+1.2}_{-4.0}$\\[5pt]
\ha & F7030s & 0.88 & 31.6$^{+10.9}_{-8.4}$ & 18.0$^{+8.3}_{-6.4}$\\[5pt]
\ha & F5170scp & 0.85 & 26.8$^{+10.6}_{-4.8}$ & 22.0$^{+6.9}_{-6.9}$\\[5pt]
\ha & \hb & 0.93 & 8.5$^{+8.6}_{-4.2}$ & 2.5$^{+0.4}_{-1.6}$\\[5pt]
F7030s & F5170s & 0.97 & 7.2$^{+3.7}_{-4.4}$ & 5.5$^{+2.5}_{-1.5}$\\[5pt]
F7030s & F5170scp & 0.96 & 8.4$^{+2.1}_{-3.6}$ & 3.2$^{+4.4}_{-1.8}$\\[5pt]
\hg & F5170sc & 0.53 & 33.1$^{+9.2}_{-11.6}$ & 27.1$^{+17.1}_{-5.9}$\\[5pt]
\hg &\hb & 0.82 & 8.9$^{+8.1}_{-9.1}$ & $-0.6^{+18.3}_{-2.3}$&\\[5pt]
\hline
\end{tabular}
}
{\\\footnotesize
Meaning of symbols "s"\ and "scp"\ the same as in Table\ref{stat}.}
\end{table}
Figure~\ref{ccf} shows the cross-correlation results for the \hb\, and \ha\, line fluxes with
the continuum as well as autocorrelation functions of the continuum for the 1993--2008 time
interval. Table~\ref{ccf-ab} gives the cross-correlation results. When the continuum light curve
is a combination of the spectral and the photometric data (designated as $F5170scp$ in Table~\ref{ccf-ab}) then the CCF computation has been carried out with the rebinning the $F5170scp$ light curve to the times of observations of the first time series. This is because the combined continuum light curve has much more data points than the continuum light curves which consist from the spectral data only (designated as $F5170s$ in Table~\ref{ccf-ab}). For the $F5170s$ light curve, we follow a standard method of the CCF computation with rebinnig both time series.
The variations of the \ha\ and \hb\ fluxes are tightly correlated as well as variations of the continuum fluxes near both lines (the correlation coefficients are equal to 0.93 and 0.97, respectively). The positive lag values for the `\ha--\hb' and `F7030--F5170' light curves
mean that the region of effective continuum emission at $\lambda$5170 is probably smaller than that at $\lambda$7030 and the region of effective \hb\ emission is probably smaller than that \ha. The probability that the delay for the `F7030--F5170' and `\ha--\hb' is less than zero equals to 0.023 and 0.036, respectively.
The time interval 1993--2008, that has been used for cross-correlation analysis, is very long.
We have divided it into five subinterval in order to check whether the lag values for individual sub-intervals are the same as for the entire 1993--2008 period, whether they are changed in time, and whether they are correlated with the fluxes and with line widths. In particular, the effective region of the broad-line emission can depend on the incident continuum flux, so the lag can depend on the continuum flux. Under a virialized motion of the line-emitting gas the expected relation between the lag $\tau$ and the line width $V$ is as follows: $V \propto \tau^{-1/2}$. To this end, we used only the \hb\ spectra because more reliable lag estimate for this line, and we carried out the cross-correlation analysis separately for each of the five time intervals listed in the Table \ref{time intervals}. The first and second intervals were taken the same as in the paper by \citet{Ser99}.
\begin{table}
\caption{Time intervals for time series analysis in the \hb\ region.}
\label{time intervals}
{\small
\begin{tabular}{@{}lccccc}
\hline
Time &JD2440000+ &\multicolumn{2}{c}{N}& \multicolumn{2}{c}{$\Delta t_{med}$} \\
Series & & \hb & cont$^a$ &\hb &cont \\
\hline
1& 09250--09872 & 38 & 51\,sp & 14.0 & 8.9\\
2& 09980--10777 & 54 & 90\,sp & 11.9 & 3.0\\
3& 10869--11516 & 27 & 59\,sp & 16.0 & 6.1\\
4& 11557--13356 & 47 &266\,scp & 19.1 & 2.0\\
5& 13611--14804 & 58 &242\,scp & 10.8 & 2.1\\
\hline
\end{tabular}
}
{\\\footnotesize $^a$ Meaning of symbols "sp"\ and "scp"\ is the same as in Table\ref{stat}.\\}
\end{table}
The cross-correlation results for the five time subintervals are given in Table~\ref{ccf-b}.
For the continuum light curve in Table~\ref{ccf-b} that consists of the spectral data only
(i.e., when the number of data points in both the \hb\ and continuum light curves is the same)
we have used the standard method of the CCF computation with the rebinning both time series.
When the photometric data are added to the continuum light curve, only the continuum light curve
was rebinned to the times of observation of the \hb\ light curve. As for the entire period, the lag uncertainties for the individual subintervals were computed by using the FR/RSS method as given by \citet{Peterson98}. The uncertainties for each subinterval were computed using 4000 FR/RSS realizations, and the probability distributions for both $\tau_{cn}$ and $\tau_{pk}$ were calculated. Each of these distributions was found to be very different from the normal distribution.
{\bf To obtain the unweighted mean lag and its uncertainties we
have sequentially (one after another) performed the convolution of the five individual distributions and then we have scaled the $\tau$-axis by dividing it by a number of subintervals (i.e., by five). After the convolution operation, the final probability distribution was found to be almost normal with the expectation and standard deviation as given in Table~\ref{ccf-b} for the mean lag.}
\begin{table}
\caption{Cross-correlation results for the \hb\ line for the five subintervals.}
\label{ccf-b}
{\footnotesize
\begin{tabular}{@{}lccccc}
\hline
Subset&\multicolumn{2}{c}{N points}&$\tau_{cn}$& $\tau_{pk}$ & $r_{max}$\\
Number& \hb & F5170 & & & \\
\hline
1 & 38 & 38 & 22.7$^{+7.4}_{-3.0}$ &20.5$^{+4.7}_{-2.8}$ & 0.939\\[5pt]
2 & 54 & 54 & 20.8$^{+3.0}_{-2.6}$ &18.1$^{+3.6}_{-1.0}$ & 0.942\\[5pt]
3 & 27 & 27 & 19.3$^{+4.0}_{-6.0}$ &18.2$^{+7.7}_{-6.1}$ & 0.761\\[5pt]
4 & 47 &199$^a$ & 26.2$^{+12.3}_{-6.8}$ &14.2$^{+14.6}_{-1.4}$ & 0.898\\[5pt]
5 &58 &230$^a$ & 20.2$^{+5.0}_{-3.9}$ &26.8$^{+0.3}_{-13.8}$ & 0.794\\[5pt]
\hline
\multicolumn{3}{l}{Mean value:}& 21.4$\pm$2.0& 19.3$\pm$1.9\\
\hline
1 & 38 & 51$^b$ & 21.2$^{+4.0}_{-3.2}$ &21.0$^{+4.0}_{-3.4}$ & 0.923\\[5pt]
2 & 54 & 90$^b$ & 20.7$^{+3.0}_{-2.4}$ &22.0$^{+1.3}_{-4.7}$ & 0.939\\[5pt]
3 & 27 & 59$^b$ & 20.5$^{+5.6}_{-7.0}$ &28.5$^{+1.4}_{-21.9}$ & 0.683\\[5pt]
4 & 47 &266$^c$ & 23.9$^{+17.0}_{-7.3}$ &9.5$^{+15.6}_{-1.2}$ & 0.881\\[5pt]
5 & 58 &242$^c$ & 20.4$^{+4.6}_{-4.1}$ &21.8$^{+5.1}_{-8.7}$ & 0.803\\[5pt]
\hline
\multicolumn{3}{l}{Mean value}& 21.1$\pm$1.9 & 20.5$\pm$2.2\\
\hline
\end{tabular}
}
{\\\footnotesize
$^a$ Continuum light curve was combined from the spectral data and from the CCD photometry.\\
$^b$ Continuum light curve was combined from the spectral data and from the photoelectric photometry.\\
$^c$ Continuum light curve was combined from the spectral data and from both the CCD and photoelectric photometry.\\
}
\end{table}
Since there are large gaps in our time series, we decided to investigate in more detail their effect on the lag uncertainties for Mrk~6. To make sure that the uncertainties in our lag estimates are realistic, we decided to verify the effect of the sampling of our time series to the lag determination and to compare results with those obtained by random subset selection (RSS) method. We have generated random time series with the same autocorrelation function as observed ACF. Stationary random process (or time series) with a given ACF can be generated from an array of {\em independent} random values $\xi_1, \xi_2, \xi_3, \ldots, \xi_n $. To do this, it is necessary to find a matrix $U_{ij}$, such that the multiplication of the matrix $U$ to the vector $\xi$ gives a vector of {\em dependent} random values $x_1, x_2, x_3, \ldots, x_n$, with a given correlation matrix $\rho_{ij} = ACF(t_j - t_i)$, where $t_1, t_2, t_3, \ldots, t_n$ are times of observations. The matrix $U$ is related to the correlation matrix $\rho$ as follows:
\begin {equation}
\rho = U\, U ^{\rm T},
\label{genrandproc}
\end {equation}
where $T$ denotes a transposition operation. We used our own algorithm to compute the matrix $U$ from ACF.
First we have generated 1000 realizations of the continuum light curve with a time resolution of 1 day over a period of 6452 days, which is longer than the real observed time interval (6175 days).
To simulate the \hb\ light curve from the continuum light curve we have experimented with the three kinds of transfer functions: (1) delta-function $\delta(\tau - 20)$, (2) $\Pi$-shaped function which is a constant for $0<\tau<40$, and (3) triangular function which is linearly decreased down to a zero value from $\tau = 0$ to $\tau=68.3$. Here the lag $\tau$ are in units of days. After the convolution with the simulated continuum light curves, all the transfer functions give the \hb\ light curve with a lag of about 20 days. Next the simulated continuum and line light curves were rebinned to real moments of observations and the cross-correlation functions were computed for each realization. The lag peak and centroid were measured. The largest uncertainties were obtained when we used the triangular transfer function. These uncertainties are only due to the sampling of the observation data. Table~\ref{simul} gives a comparison of both methods for uncertainty estimates (i.e., the random time series versus RSS method) for the triangular transfer function.
The columns designated as $\tau_{pk}$ and $\tau_{cn}$ are a lag with $\pm 1\sigma$ uncertainties computed from the random time series, while other two columns designated as RSS give $\pm 1\sigma$ uncertainties computed from RSS method for $\tau_{pk}$ and $\tau_{cn}$, respectively. The last column of the table is the mean CCF peak value obtained from the method of the random time series. As can be seen from this table, the RSS method gives comparable or larger (up to two times) uncertainties than the method of the random time series does. The random time series method seems to be more direct way to estimate lag uncertainties and since the lag uncertainties were computed in this paper by the RSS method, they seem to be realistic or slightly overestimated.
There is a contradiction between our lag estimate and the preliminary results on Mrk~6 published in a conference proceeding by Grier et al. (NASA ADS tag 2011nlsg.confE..52G). This new campaign had a nightly sampling rate and it spanned 125 nights beginning 2010 August 31 and ending on 2011 January 3. They claimed a lag of $8\pm 3$ days for the \hb\ line in Mrk 6. We have first checked the effect of the removing of linear trends from our light curves {\bf \citep[as recommended by some authors, e.g.,][]{Welsh}}. We have removed linear trends from the \hb\ and continuum light curves for each subinterval, even if there are no such trends exist. Then we have recalculated a mean lag value and it was to be $19$ days, i.e. two days less. Then we have generated random time series (continuum and \hb) by exactly the same way as described above, but for the sampling rate of Grier et al. in order to check whether the duration of the monitoring program is important for the lag measurements. We found that with the data sampling of Grier et al. our lag estimate must be less by one more day, and so a total difference between our and their results must be three days. The real difference is much larger than three days. An unexpected result of our simulation was very large lag uncertainties for the 125-days data sampling and for the triangular transfer function for the \hb\ line (see above). The lag uncertainties were found to be as large as $\pm 6$ days! However, for the $\delta(\tau-20)$ transfer function (i.e., when the line light curve is simply a shifted version of the continuum light curve) the lag uncertainties were found to be as small as $\sim 0.1$ days. So, for short-term campaigns and for transfer functions with a long tail (i.e., when the line light curve is not only shifted, but a strongly smoothed version of the continuum light curve), the uncertainties in lag estimates can be very large and they are connected to the extrapolation of the line and continuum fluxes when computing the CCF, not to the data sampling. We concluded that we can only explain the difference of three days between our and their lag measurements. Probably, the rest of the difference is due to real changes of lag or due to the measurement uncertainties and their underestimation.
It can be seen from Table~\ref{simul} that the simulated lag values are almost the same for all
subinterval. The expected lag value can be obtained by convolving the ACF with the transfer function and it was found to be: $\tau_{pk} = 20.7$\,days and $\tau_{cn} = 22.4$\,day in an excellent agreement with the simulation results. So, the large gaps in our time series do not shift the lag measurements.
\begin{table}
\caption{Comparison of the lag uncertainty estimates between the method of the random time series and the RSS method.}
\label{simul}
{\small
\begin{tabular}{@{}ccccccc}
\hline
Interval & $\tau_{pk}$& RSS & $\tau_{cn}$& RSS & $r_{sim}$\\
\hline
1& 20.8$^{+2.4}_{-2.1}$& 2.1& 22.2$^{+2.8}_{-2.8}$& 2.7& 0.968\\ [5pt]
2& 20.8$^{+1.5}_{-1.6}$& 1.6& 22.2$^{+2.8}_{-2.8}$& 2.1& 0.960\\[5pt]
3& 20.5$^{+2.8}_{-2.8}$& 6.0& 22.4$^{+4.6}_{-4.6}$& 4.3& 0.940\\[5pt]
4& 20.4$^{+2.6}_{-2.6}$& 3.4& 22.1$^{+4.4}_{-4.0}$& 8.5& 0.980\\[5pt]
5& 20.6$^{+2.4}_{-2.7}$& 6.9& 22.4$^{+3.7}_{-3.5}$& 3.3& 0.978\\[5pt]
\hline
\end{tabular}
}
\end{table}
\begin{table}
\caption{The broad \hb, and \ha\ line width measurements for 1993--2008.}
\label{width-lines}
{\small
\begin{tabular}{@{}lcc}
\hline
&FWHM & $\sigma_{line}$ \\
&\multicolumn{2}{c}{km\,s$^{-1}$} \\
\hline
\hb(mean-spectrum)& $6278 \pm 378 $&$ 2821 \pm 13 $\\
\hb(RMS-spectrum) & $5445 \pm 468 $&$ 2884 \pm 97 $\\
\hline \noalign{\smallskip}
\ha(mean-spectrum)& $5322 \pm 142 $&$ 2870 \pm 22 $\\
\ha(rms-spectrum) & $4443 \pm 338 $&$ 2780 \pm 35 $\\
\hline
\end{tabular}
}
\end{table}
\section{Line width measurements}
\label{linewidth}
It is well known that the emission-line profile evolution cannot be entirely attributed to the
reverberation effect and that the profile changes usually occur on a time scale that is much longer than the flux-variability time scale \citep[see][]{Wanders}. To decrease the effect of the long-term profile changes on the line width measurements and to get sufficient statistics, we measured the \hb\ and \ha\ line widths for the five subintervals presented in Table \ref{time intervals}. The line width is typically characterized by its full width at half maximum (FWHM) or by the second moment of the line profile, denoted as $\sigma_{line}$. To measure FWHM for the mean \ha\ and \hb\ line profiles, we removed narrow lines from the broad line profiles. This is not required for rms profiles. However, the spectra must be optimally aligned in wavelength and in spectral resolution in order to reduce the narrow line residuals in the rms profiles. It is difficult to measure FWHM because both the mean broad and rms \hb\ profiles are double-peaked, and so the scatter in the FWHM measurements is much larger than in $\sigma_{line}$ which is well defined for arbitrary line profiles. The uncertainties in the line width were obtained using bootstrap method described by \citet{Peterson04}. The \hb\ and \ha\ line width (both FWHM and $\sigma_{line}$) and their uncertainties are listed in Table~\ref{width-lines} for 1993--2008. Table~\ref{flx-ccf-width} gives the $\sigma_{line}$ computed separately for each considered subinterval and for the \hb\ line only.
We examined the relationship between the \hb\ lag, \hb\ width, and the continuum flux (see Fig.~\ref{lag-flx-wdt}). No significant correlations among above three parameters were found. In particular, the virial relationship between the lag and width does not contradict to our data, but the correlation coefficient between them does not differ significantly from the zero value.
More subintervals and less lag uncertainties are required.
\begin{figure}
\includegraphics[width=84mm]{F05lag-flx-wdt.eps}
\caption{The relation between the continuum flux and the lag ($\tau_{cn}$) for \hb\ in Mrk~6, as well as between the \hb\ width and the lag ($\tau_{cn}$).}
\label{lag-flx-wdt}
\end{figure}
\begin{table*}
\caption{Results for the five subintervals}
\label{flx-ccf-width}
{\small
\begin{tabular}{@{}lccccccc}
\hline
Time &\multicolumn{2}{c}{Mean Flux$^a$} &$\tau_{cn}$ &\multicolumn{2}{c}{$\sigma_{line}$km s$^{-1}$} &\multicolumn{2}{c}{M$_{BH}^{b}$in units 10$^8$M$_{\odot}$}\\
series &F(5170)& F(\hb)&days &(mean)&(rms)&(mean)&(rms)\\
1&2&3&4&5&6&7&8\\
\hline
1& 6.446 &3.632 &21.2$^{+4.0}_{-3.2} $ &2813$\pm$13 &2836$\pm$48 &1.77$^{+0.33}_{-0.27}$ &1.80$^{+0.34}_{-0.28}$\\[5PT]
2& 6.472 &3.976 &20.7$^{+3.0}_{-2.4} $ &2804$\pm$ 6 &2626$\pm$37 &1.72$^{+0.25}_{-0.20}$ &1.50$^{+0.22}_{-0.18}$\\[5PT]
3& 5.730 &3.587 &20.5$^{+5.6}_{-7.0} $ &2808$\pm$14 &2876$\pm$46 &1.70$^{+0.46}_{-0.58}$ &1.79$^{+0.49}_{-0.61}$\\[5PT]
4& 4.594 &2.677 &23.9$^{+17.0}_{-7.3}$ &2870$\pm$13 &3222$\pm$39 &2.07$^{+1.50}_{-0.63}$ &2.62$^{+1.80}_{-0.80}$\\[5PT]
5& 7.027 &3.523 &20.4$^{+4.6}_{-4.1} $ &2807$\pm$ 8 &2864$\pm$35 &1.69$^{+0.38}_{-0.34}$ &1.77$^{+0.40}_{-0.36}$\\[5PT]
\hline
\multicolumn{3}{l}{Average:}&$21.1\pm1.9$ & $2812\pm 10^c$ & $2882\pm100^c$ & $1.76\pm0.16$& $1.85\pm0.21$\\
\hline
\end{tabular}
}
{\\\footnotesize
\mbox{\hspace{2.5cm}}$^a$ Same flux units as in Table 1 for 5170 \AA\ continuum and \hb, respectively. \hfill \mbox{~}\\
\mbox{\hspace{2.5cm}}$^b$ Using Onken et al (2004) calibration, $f=5.5$. \hfill \mbox{~}\\
{\bf \mbox{\hspace{2.5cm}}$^c$ The line width and uncertainties were computed as weighted average and assuming \hfill \mbox{~}\\
\mbox{\hspace{2.5cm}}$^{~}$ different expectations of the line widths among individual periods of observations.\hfill \mbox{~}\\}}
\end{table*}
\section{Black hole mass of Mrk~6}
\label{mass-bh}
Determination of the black hole mass from reverberation mapping rests upon the assumption that
the gravity of the central super-massive black hole dominates over gas motions in the BLR. The black
hole mass is defined by the virial equation $$ M_{BH}=f \frac{c\tau(\Delta V)^2}{G},$$
where $\tau$ is the measured emission-line time delay, $c$ is the speed of light, $c\tau$ represents
the BLR size, and $\Delta V$ is the BLR velocity dispersion. The dimensionless parameter $f$ is
the scaling factor, which depends on the BLR structure, kinematics and inclination of BLR. \citet{Peterson04}
argued that $\tau_{cn}$ for the time delay $\tau$, and $\sigma_{line}$, measured from the \hb\ emission line in the rms spectrum for the emission line width $\Delta V$, provide the most robust estimates of the black hole mass with the reverberation technique. Later \citet{Collin2006} confirmed that in most cases for the black hole mass estimate the line dispersion $\sigma_{line}$ is more suitable than the FWHM, and $\sigma_{line}$ from the rms-spectrum is more suitable than the $\sigma_{line}$ from the mean spectrum.
We adopt an average value of $f = 5.5$ based on the assumption that AGNs follow the same $M_{BH}-\sigma_*$ relationship as quiescent galaxies \citep{Onken04}. This is consistent with \citet{Woo10} and allows easy comparison with previous results, but this is about a factor of
two larger than the value of $f$ computed by \citet{Graham11}. The value of $f$ can be more decreased due to the effect of radiation pressure, as was explored by \citet{Marconi08, Marconi09}. Marconi et al. suggested that neglecting the effect of radiation pressure can lead to underestimation of the true black hole mass, especially in objects close to their Eddington limit. Discussion between \citet{Marconi08, Marconi09} and \citet{Netzer09} shows that there are many unclear questions in this area. Naturally, a corrective term for radiation pressure will decrease the $f$-factor.
We calculated the black hole mass for Mrk~6 with the use of $\tau_{cn}=21.1\pm 1.9$ for the
time delay averaged over five time intervals and $\sigma_{line}=2882\pm100$ km\,s$^{-1}$ from the rms spectra for \hb. With $\tau_{cn}$ taken in days and $(\Delta V)$ in km\,s$^{-1}$, and taking into account the time dilation correction for the value of $\tau_{cn}$, the mass is equal to:
$$M_{BH}/M_{\odot}=0.1952\times f\times\frac{\tau_{cn}(obs)}{(1+z)}\times(\Delta V)^2.$$
The black hole mass calculated from the \hb\ line is $(1.85\pm0.21)\times 10^8$M$_{\odot}$. For the \ha\ line, the $\tau_{cn}=26.8\pm7.7$ days and $\sigma_{line}=2780\pm35$ km\,s$^{-1}$ and the black hole mass is equal to (2.2$\mathbf\pm0.6)\times10^8$M$_{\odot}$.
The black hole masses calculated for each of five periods of observations are listed in columns 7 and 8 of Table~\ref{flx-ccf-width} for the $\sigma_{line}$ from the mean and rms spectra. One can see that all the estimates of the black hole mass based on the \hb\ line are the same within the scatter.
\section{The BLR~Size--Luminosity and Mass--Luminosity Relationships}
\label{slm}
Many characteristics of the Mrk~6 galaxy are typical for active galaxies. In this connection, it
is of interest to see the localization of this galaxy on the BLR Radius--Luminosity and Mass--Luminosity diagrams. These diagrams determine a relationship between fundamental characteristics of AGNs. To this end, the Mrk~6 luminosity should be known. Up to the present, for many AGNs the luminosity in the rest-frame $\lambda_0=5100$ \AA\, has been corrected for host-galaxy starlight contribution within the apertures used in spectral observations \citep[see][]{Bentz09a}. These authors used high-resolution {\it Hubble Space Telescope} (HST) images to measure the starlight contribution. This contribution was found to be significant, especially for low-luminosity AGNs.
We tried to get at least a rough estimate of the host-galaxy contribution using the observations made by \citet{Neizvestny} at the Special Astrophysical Observatory (SAO) in October 1984 with different apertures from A=4$\farcs$3 to 55$\arcsec$. The surface brightness distribution in the host galaxy of the Mrk~6 nucleus calculated on the basis of these measurements is shown in Fig.~\ref{muV}.
\begin{figure}
\includegraphics[width=70mm]{F09-muv.eps}
\caption{Surface brightness distribution for Mrk~6 in the V-band on the basis of multi-aperture photoelectric photometry by \citet{Neizvestny}. The solid curve corresponds to the model $\mu (r)= a+br+cr^{1/4}$.}
\label{muV}
\end{figure}
The galaxy contribution in our 3\arcsec$\times$11\arcsec\ spectral window was found to be $V_{gal}$=15\fm6 or $F_{gal}$=2.08$\times10^{-15}$erg\,s$^{-1}$cm$^{-2}$\AA$^{-1}$. The mean flux observed in the continuum near $\lambda_0$=5100~\AA\, is $F_{(gal+nuc)}$=6.093$\times10^{-15}$ergs$^{-1}$cm$^{-2}$\AA$^{-1}$ (see Table~\ref{stat}) and, thus, the mean flux corrected for the galaxy contribution is equal to $F_{nuc}$=4.013$\times10^{-15}$ergs$^{-1}$cm$^{-2}$\AA$^{-1}$.
The variability amplitude $F_{var}$ increases from 18\% to 27\% after accounting for the galaxy contribution. The mean flux was also corrected for Galactic reddening according to the NASA/IPAC Extragalactic Database (NED) \citet{Schlegel98}. The luminosity was found to be $\lambda L_{\lambda}$(5100)=(2.51$\pm0.78)\times10^{43}$~erg\,s$^{-1}$ adopting the galaxy distance D=81\,Mpc and when the galaxy contribution is removed.
\begin{figure}
\includegraphics[width=84mm]{F10-sml.eps}
\caption{Top: \hb\ BLR~size vs. luminosity at 5100\AA\
according to \citet{Bentz09a} and \citet{Denney10}. The luminosity of all nuclei was corrected for host galaxy contribution. The solid line is the best fit to the relationship log(R$_{BLR})$=$-$21.3+0.519 log(L). Bottom: the Mass--Luminosity diagram. The black hole mass of the majority of AGNs was taken from \citet{Peterson04}, except for Mrk~290, Mrk~817, NGC~3227, NGC~3516, NGC~4051, and NGC~5548, for which new results by \citet{Denney10} were used. Solid lines show the Eddington limit $L_{Edd}$ and its 10\% and 1\% fractions. The position of Mrk~6 nucleus is indicated on both plots.}
\label{sml}
\end{figure}
The bolometric luminosity of the Mrk~6 nucleus was adopted to be $L_{bol}\simeq 9\lambda L_{\lambda}(5100\AA)$ according to \cite{Kaspi2000} and it is equal to $L_{bol}(nucl)$=2.26
$\times10^{44}$erg\,s$^{-1}$. This luminosity is far from the Eddington limit (L$_{Edd}$), which
is equal to $L_{Edd}$=2.16$\times10^{46}$ erg\,s$^{-1}$ for a black hole mass of
1.8$\times10^8$ M$_{\odot}$. In other words, the Eddington ratio for Mrk~6 is ${L_{bol}/L_{Edd}\simeq0.01}$. In this case and because there are no clear indications of gas outflow from the BLR, the radiation pressure has a negligible effect on the reverberation mass estimate.
The position of the Mrk~6 nucleus on the BLR~Size--Luminosity diagram is shown in Fig.~\ref{sml}.
The BLR size and the luminosity of other galaxies in Fig.~\ref{sml} are taken from \citet{Bentz09a} and \citet{Denney10}. The black hole masses in Fig.~\ref{sml} are taken from~\cite{Peterson04}, except for the galaxies Mrk~290, Mrk~817, NGC~3227, NGC~3516, NGC~4051, for which we used new data from \citet{Denney10}.
\section{Velocity-resolved reverberation lags}
\label{vrrlag}
\subsection{Entire time interval: 1993--2008}
\label{vel-res lag}
The question about whether the direction of gas motion can be determined from the response of the line profile to the continuum changes was firstly raised by \citet{Fabrika1980}. Generally speaking, the BLR gas velocity field can be random circular orbits, radial gas outflow or infall, or Keplerian motion. Examples demonstrating how the velocity resolved responses can be related to different types of BLR gas kinematics are given in \citet{Peterson01} and \citet{Bentz09b}. The random circular orbits generate a symmetric lag profile with the highest lag observed around zero velocity. The infall kinematics produces longer lags in the blue-shifted emission, and the outflow gas produces longer lags in the red-shifted emission.
Horne et al. (2004) formulated some important observational requirements for determining a
reliable velocity field of the BLR: (1) the time duration of observations should be at least
three times larger than the longest timescale of response, (2) the mean time between subsequent
observations should be at least two times less than the BLR light-crossing time, and (3) the
velocity sampling $\Delta V$ used for the velocity-delay maps should be no less than the
spectral resolution of the data. According to \citet{Horne04}, such conditions can allow one to distinguish clearly between alternative kinematic models of the BLR gas motion.
\begin{figure}
\includegraphics[width=84mm]{F06hab-lag.eps}
\caption{Two top panels show the \hb\ mean and rms profiles (left) and the \ha\
mean profile (right) divided into ten bins of equal flux separated by vertical dashed lines. The flux units for the mean and rms profiles are 10$^{-13}$~ergs~cm$^{-2}$~s$^{-1}$~\AA$^{-1}$. The third panels from the top show the corresponding velocity-resolved time lag response, where the delays are plotted at the flux centroid of each velocity bin. Here the vertical error bars are 1$\sigma$ uncertainties in the lag for each velocity bin denoted by the horizontal error bars. The horizontal dotted and dashed lines on the third panels mark the mean centroid lag and 1$\sigma$ uncertainty, respectively, for the entire emission line. The bottom panels show the peak value of the correlation coefficient between the bin flux in the line and continuum. Here the horizontal dotted lines mark the correlation coefficient $r_{max}$ between the entire emission line and the continuum, as calculated in Table~\ref{ccf-ab}.}
\label{hablag-vel}
\end{figure}
In order to obtain the velocity-delay map, we measured the lag as a function of velocity in several bins across the line profile. We divided both the \ha\ and \hb\ lines into ten bins of equal flux, and the width of these bins was no less than 1000--1500 km s$^{-1}$. For each bin we calculated light curves from the Balmer line fluxes. Then each of these light curves was cross-correlated with the continuum light curve following the same procedure as described in Section~\ref{fccf}. Figure~\ref{hablag-vel} shows hydrogen line profiles (mean and rms) subdivided into bins (two upper panels). The two middle panels demonstrate the lag measurements for each of the bins. The vertical error bars show $1\sigma$ uncertainties for the time lag, and the horizontal bars represent the bin width. The horizontal solid and dashed lines in the two middle panels show the mean BLR lag and associated errors as listed in Table~\ref{ccf-ab}. The bottom panels show the peak correlation coefficient between the bin flux in the line and continuum, $r_{max}$. Figure~\ref{hablag-vel} shows that
\begin{enumerate}
\renewcommand{\theenumi}{(\arabic{enumi})}
\item The mean and rms profiles of \hb\, and \ha\, are not symmetric with respect to zero
velocity. The centroid of the mean and rms profiles is shifted to the short-wave part of the line. The variable parts of \hb\ and \ha\ have two well-defined peaks, one of them is almost central (between 5 and 6 bins) and another is blue-shifted. In addition, there is a weaker peak in the red part of the line profile.
\item The time delay between the higher velocity gas in the BLR and the continuum is shorter
than the delay between the low velocity gas and the continuum. Such a behaviour is typical for
virialized gas motions.
\item The lag in the blue wing of the \hb\ line is greater than the lag in the red side of this line. The \ha\ velocity-resolved lags shows the same tendency. This is consistent with expectations from the infall model of gas motion. Thus, it is possible we have virialized motion combined with infall signatures.
\item The correlation coefficient of different segments of the lines is different. The bin corresponding to a radial velocity of $V_r \approx +1500\,km\,s^{-1}$ shows poor correlation with the
continuum variation, especially in \hb. This fact was earlier noted by \citet{Ser99}.
\end{enumerate}
\begin{figure*}
\includegraphics[width=170mm]{F07lag5-dtr.eps}
\caption{The same as in Figure~\ref{hablag-vel}, but for the \hb\ line for the five
time intervals.}
\label{hblag-5p}
\end{figure*}
\subsection{Velocity delay maps in the five time intervals}
We have computed the velocity-resolved time delays for the five time intervals given in Table \ref{time intervals}. For each subset we made the velocity-dependent cross-correlation analysis for the \hb\ line profile bins as described in the previous section. The \hb\ line was selected because its sampling is better. In Figure~\ref{hblag-5p} the mean and rms \hb\ profiles, the velocity-resolved time lag response, and the velocity-dependent peak correlation coefficient are shown for the five time intervals. Upon inspection of Figure~\ref{hblag-5p} it becomes clear that the mean and rms profiles are different among the five periods. The relative intensity in the blue peak and in the central peak changes very strongly: during the first interval the blue peak is higher then the red one. The opposite situation is seen in the fifth interval. The flux in continuum as well the flux in the \hb\ line systematically decreased from the second to the fourth time interval, as is seen in Figure~\ref{lc} and Figure~\ref{ccf-b}. The \hb\ rms profile shows two peaks in the first, second and third period, the flat top in the fourth period, and in the fifth period we see three peaks.
Figure~\ref{hablag-vel} demonstrates that the high-velocity gas in the wings exhibits a shorter lag than the low-velocity gas, supporting the virial nature of gas motion in BLR: the gas kinematics that is dominated by the central massive object.
However, the lag is slightly larger in the blue wing than in the red wing for all subintervals.
This is a signature of the infall gas motion. In the fifth period (2005--2008) the velocity delay map is more symmetric, but the seventh bin shows very small lag, as well as in the previous time interval. For 2005--2008 there is a poor correlation with the continuum for bins~7--10. A more detailed examination of the bin light curves for 2005--2008 (Figure~\ref{hblc05-08}) revealed that there is a trend for the \hb\ flux in bins 7--10, which is almost absent in the bins~1--6. Following the advice of our reviewer we removed the trend from the \hb\ light curves for bins 7--10. No more significant trends were found for other time intervals.
In Figure~\ref{hblag-5p} the detrended lags and correlation coefficients are shown by open circles. After detrending procedure, the lag--velocity dependence became more similar to the lag--velocity dependence for the first period, for which the difference in lag between the blue and red wings is largest.
\begin{figure}
\includegraphics[width=84mm]{F12lc05-08.eps}
\caption{Light curves in the continuum and \hb\, in 2005--2008 over the velocity range from $-6200$ to $+4500~km\,s^{-1}$ (top panels) as well as the light curves of the broad \hb\ line in bin~5 in the velocity range from $-980$ to $+25~km\,s^{-1}$, and in the bin~7 in the velocity range from 1000 to 2000~km\,s$^{-1}$ (bottom panels). The continuum fluxes are in units of $10^{-15}$~erg\,s$^{-1}$\,cm$^{-2}$\,\AA$^{-1}$, and the broad emission line fluxes are in units of $10^{-15}$~erg\,s$^{-1}$\,cm$^{-2}$.}
\label{hblc05-08}
\end{figure}
So, it is most likely that the BLR kinematic in Mrk~6 is a combination of the Keplerian gas motion and infall gas motion.
\section{Conclusion}
\label{sum}
We have reported our new results on the Mrk~6 nucleus from 1998--2008 observational data together with the previous results published by \citet{Ser99}. We found that
\begin{enumerate}
\renewcommand{\theenumi}{(\arabic{enumi})}
\item The flux of the Mrk~6 nucleus in 1992--2008 varied significantly in the continuum as well as in the \ha\ and \hb\ broad emission lines. The relative amplitude of the continuum flux variability is larger than in the hydrogen lines, and it is greater in the \hb\ line than in the \ha\ (see Table~\ref{stat}). It is typical for the most of Seyfert galaxies. This agrees with the predictions of \citet{Korista04} based on new photo-ionization calculations of the BLR-like gas.
\item We found the average time delay between the total \hb\ flux and the continuum flux at 5170~\AA\ to be $21.1 \pm 1.8$ days, and the time delay does not vary significant among individual time intervals. It seems that the size of the \hb\ emission region remains approximately the same over long time periods. The \ha\ flux responds to the changes in the $F5170$ continuum with a lag of $26.8^{+10.6}_{-4.8}$ days.
\item When the continuum flux varies, the photo-ionization models predict the existence of the relation between the BLR size and the luminosity. However, because the large uncertainties in the lag for individual time intervals we are unable to find such a relation. For the same reason, it is unable to obtain a dependence between the lag and line width, and it is impossible to check whether this dependence is consistent with the dependence $V\propto r^{-1/2}$ expected for the gravitationally dominated motion.
\item The \hb\ line width is larger than that of \ha. This is naturally explained by photo-ionization calculations \citep[e.g.,][]{Korista04}: the effective emission region of \hb\ is smaller and closer to the ionizing source than the effective region of \ha, and the gas velocities in \hb\ are higher.
\item By examining the velocity-resolved lags for the broad \hb\ and \ha\ lines, we found that the lag in the high-velocity wings are shorter than in the line core. This indicates virial motions of gas in the BLR. However, the lag is slightly larger in the blue wing than in the red wing for the entire time interval as well as for the individual periods considered in the present paper. This is a signature of the infall gas motion. Probably the BLR kinematic in the Mrk~6 nucleus is a combination of the Keplerian gas motion and infall gas motion.
\item Some profile segments often show poor correlation with the continuum flux. According to \citet{Gaskell10} this effect can arise because off-axis sources of ionizing continuum flux can appear, which might not make a detectable contribution to the total continuum flux variability, but they will have an influence on the line only over a narrow range of radial velocity in the BLR. If these local off-axis events will vary out of phase with the variability of the dominant source, the result will be to give a weak correlation between the continuum flux and the line flux in the narrow range of radial velocity.
\item We determined the black hole mass from the lag and line width measurements of the \hb\ and \ha\ lines. The mass was found to be $M_{BH} = (1.8\pm 0.2)\times 10^8\,M_{\odot}$ for the \hb\ line and slightly greater and less reliable from the \ha\ line. Under such a mass and the luminosity of $\lambda L_{\lambda}$(5100)=(2.51$\pm$0.38)$\times10^{43}$~erg\,s$^{-1}$, the Mrk~6 nucleus is located on the upper edge of the Mass-Luminosity diagram that corresponds to the Eddington ratio of about 0.01. This confirms the assumption \citep[e.g.,][]{Ser11} that there is anticorrelation between broad-line widths and Eddington luminosity ratio $L_{bol}/L_{Edd}$. The Mrk~6 position on the BLR~Size--Luminosity diagram does not contradict the fit $R_{BLR}\propto L^{0.5}$ determined by \citet{Bentz09a}.
\end{enumerate}
\section*{Acknowledgments}
We thank the anonymous reviewer for useful comments and suggestions. We also thank S. Nazarov and the staff of 2.6-m and 0.7-m telescope for help during our observations. SSG acknowledges the support to CrAO in the frame of the `CosmoMicroPhysics' Target Scientific Research Complex Programme of the National Academy of Sciences of Ukraine (2007--2012). VTD acknowledges the support of the Russian Foundation of Research (RFBR, project no.~09-02-01136a). The CrAO CCD cameras were purchased through the US Civilian Research and Development for Independent States of the Former Soviet Union (CRDF) awards UP1-2116 and UP1-2549-CR-03.
|
{
"timestamp": "2012-03-12T01:01:37",
"yymm": "1203",
"arxiv_id": "1203.2084",
"language": "en",
"url": "https://arxiv.org/abs/1203.2084"
}
|
\section{Introduction}
One of the features of the Standard Model (SM) of particle physics is the universality of the lepton couplings, i.e. the fact that the coupling of the $W^{\pm}$ to the leptons does not depend on their flavor. However the experimental results from LEP-II on this issue \cite{Heister:2004wr,Achard:2004zw,Abdallah:2003zm,Abbiendi:2007rs,Alcaraz:2006mx} showed a slight deviation from universality coming from the third family, giving \cite{Nakamura:2010zzi}:
\begin{equation} \label{eq:univvio}
R^W_{\tau\ell}=\frac{2 \, \mbox{BR} \left(W \rightarrow \tau \, \overline{\nu}_{\tau} \right)}{\mbox{BR} \left( W \rightarrow e \, \overline{\nu}_{e} \right) + \mbox{BR} \left(W \rightarrow \mu \, \overline{\nu}_{\mu} \right)}
= 1.055 (23),
\end{equation}
resulting in $2.4$ standard deviations\footnote{The result given in Eq.~\eqref{eq:univvio} is obtained from the PDG fit to the branching ratios of the W \cite{Nakamura:2010zzi}, that uses LEP2 and $p\bar{p}$ colliders data. It is worth mentioning that considering only LEP2 data the discrepancy grows to 2.8 $\sigma$ \cite{Alcaraz:2006mx} (2.6 $\sigma$ using only published data \cite{LEPNote}), all correlations included.} (all correlations included) from the SM prediction $R^W_{\tau\ell}|_{\mbox{\tiny SM}}= 0.999$ \cite{Kniehl:2000rb}, which uncertainty is negligible compared with the experimental error. Recalling also the following ratio:
\begin{equation}
\label{eq:3br}
R^W_{\mu e} = \mbox{BR}\left( W \rightarrow \mu \, \overline{\nu}_{\mu} \right) / \mbox{BR}\left( W \rightarrow e \, \overline{\nu}_{e} \right) = 0.983 (18),
\end{equation}
and the correspondent SM prediction $R^W_{\mu e}|_{\mbox{\tiny SM}}= 1.000$, it can be concluded that the two lightest families seem to attach to the universality principle. The confirmation or refutation of this measurement is obviously very important, since such a violation by the third family would be a clear indication of New Physics (NP) \cite{Pich:2008ni,Pich:2009zza}. However, it will not be easy for the LHC to reach such a precision in this observable, given the theoretical uncertainties associated to a hadronic machine. For this reason it is interesting to check indirectly this anomaly through its interplay with other related measurements.
\par
Precision electroweak observables (EWPO), as well as other precise low energy measurements, provide constraints on new models looking for deviations that could foresee the NP structure. We study in this article if it is possible to accommodate the apparent discrepancy on the $W\to\tau\bar{\nu}_\tau$ channel within the present situation provided by EWPO, where essentially no disagreements have been found. In particular, lepton universality has been tested successfully at the per-mil level in $Z\to\ell^+ \ell^-$ \cite{Nakamura:2010zzi} and $\tau\to\nu_\tau\ell\nu_\ell$ decays (see e.g. Table 3 in Ref.~\cite{Pich:2012sx}), what makes very challenging to find a NP explanation for the large anomaly shown in Eq.~\eqref{eq:univvio}. Just for the sake of illustration, we show the values obtained in leptonic Z decays \cite{Nakamura:2010zzi}:
\begin{eqnarray}
\label{eq:Zll}
\mbox{BR}\!\left( Z \!\rightarrow \mu^+ \mu^- \right) \!/ \mbox{BR}\!\left( Z \!\rightarrow e^+ e^- \right) \!&=&\! 1.001 (3), \nonumber \\
\mbox{BR}\!\left( Z \!\rightarrow \tau^+ \tau^- \right) \!/ \mbox{BR}\!\left( Z \!\rightarrow \mu^+ \mu^- \right) \!&=&\! 1.001 (3),
\end{eqnarray}
in good agreement with the SM predictions, $1.000$ and $0.998$ respectively.
\par
Instead of adhering to a specific model we will follow an Effective Field Theory approach, where NP is parameterized by a tower of higher-dimensional operators~\cite{Leung:1984ni,Buchmuller:1985jz,Grzadkowski:2010es}. All NP theories which spectrum does not contain new light mass physical states (in comparison to those of the SM), that are weakly coupled at the electroweak scale and invariant under the SM gauge symmetries reduce at lower energies to the same effective Lagrangian, feature that makes this EFT approach very appealing.
Guided by the above-mentioned experimental data on lepton universality, we will consider different frameworks where the New Physics does not affect operators involving first and second generation fermions. As we will explain, this can be implemented through the adoption of specific flavor symmetries.
\par
For the numerical analysis, we greatly benefit from Ref.~\cite{Han:2005pr}, where constraints on these effective operators were obtained via a global fit to precision electroweak data. We modify the associated fitting code to introduce additional observables and operators\footnote{The code can be freely downloaded at the web page {\tt http://ific.uv.es/lhcpheno/}.}. These fit procedures are a powerful tool to analyze the impact of current constraints on different models.
\par
We apply this method to study the possible NP effects in leptonic W decays allowed by electroweak precision data. We will emphasize the role that the leptonic tau decay and the exclusive channel $\tau^-\to\pi^-\nu_\tau$ play in constraining specific directions of the parameter space of our theory, and the need to include these observables in this kind of analyses. We will see that the observed departure from universality cannot be accommodated within the current experimental scenario under quite general assumptions. Thus in order to be able to explain the observed deviation from lepton universality as a genuine NP effect, it seems to be necessary to resort to a different description of NP that could involve the introduction of new light degrees of freedom or a strongly interacting sector.
\par
A closely related issue driven by the $W \ell \nu_{\ell}$ vertex is the ratio of widths involving the leptonic decays of pseudoscalar heavy mesons, $P \rightarrow \ell \, \overline{\nu}_{\ell}$. Accordingly, if any violation of universality is at work it also should be exposed in ratios of these decays into different charged leptons. Similarly, any modification of the SM coupling of $W$ with the tau lepton could show up, due to gauge symmetry, in the anomalous magnetic moment of the tau. We will comment how our results translate into these subjects.
\par
In the next section the EFT framework is introduced, along with different flavor symmetries and the relevant effective operators. In Sec.~\ref{sec:Wtaunu} we identify the operators that can generate a lepton universality violation in the third family, whereas in Sec.~\ref{sec:fit} we analyze through a global fit the bounds on these operators from EWPO and other low-energy measurements. Sec.~\ref{sec:heavymesons} is devoted to study the sensitivity of the leptonic decays of heavy mesons to the lepton universality violation, and Sec.~\ref{sec:conclusions} contains our conclusions. An Appendix collects several theoretical expressions not included in the main text.
\section{The Effective Field Theory framework}
\label{section:EFT}
Effective Field Theories embody the features, and particularly the dynamics, of the underlying theory. The astonishing performance of the SM suggests that whatever theory we find at higher energies has to reduce, upon integration of the relevant heavier degrees of freedom, to the key properties of the SM: symmetries and fields, that become its EFT. It is clear, though, that this approach breaks down if the underlying new physics contains physical states with mass $M \ll 1 \, \mbox{TeV}$, possibility that we do not consider in the present analysis. In this case the appropriate EFT should include that spectrum and its dynamics. In order to properly define our EFT setting we need moreover to assume that the new theory above the SM is weakly coupled at the weak scale, so that the gauge $SU(2)_L\times U(1)_Y$ symmetry is linearly realized.
\par The trail left in the procedure of integrating out heavier degrees of freedom is a Lagrangian with higher dimensional operators that respect its symmetry and content \cite{Leung:1984ni,Buchmuller:1985jz,Grzadkowski:2010es}:
\begin{equation} \label{eq:left}
{\cal L}_{\mbox{\tiny EFT}} = {\cal L}_{\mbox{\tiny SM}} + \frac{1}{\Lambda} \sum_a \, \widehat{\alpha}_a^{(5)} \, {\cal O}_a^{(5)} \, + \frac{1}{\Lambda^2} \sum_a \, \widehat{\alpha}_a^{(6)} \, {\cal O}_a^{(6)} ~+\ldots~,
\end{equation}
where ${\cal L}_{\mbox{\tiny SM}}$ is the SM Lagrangian, $\Lambda$ is the NP energy scale and ${\cal O}_a^{(n)}$ are $SU(2)_L\times U(1)_Y$ gauge-invariant operators of dimension $n$ built with SM fields (including the standard Higgs boson). Finally $\widehat{\alpha}_a^{(n)}$ are the dimensionless Wilson coefficients that carry the information of the underlying dynamics at the $\Lambda$ scale and are expected to be of ${\cal O}(1)$.
\par
The only gauge-invariant operator of dimension five violates lepton number, and thus it can be safely neglected under the assumption that the violation of that symmetry occurs at scales much higher than $\Lambda \sim 1$ TeV. Then the first order corrections to the SM predictions come from dimension-six operators. The contribution from these operators involve terms proportional to $v^2\slash\Lambda^2$, $v E \slash\Lambda^2$ and $E^2\slash\Lambda^2$, where $v \approx 174$ GeV is the vacuum expectation value of the Higgs field and E is the energy scale of the process considered. In order to be consistent with the truncation of the effective Lagrangian (\ref{eq:left}) we work at linear order in the above ratios, i.e. keeping only the contributions coming from the interference of the SM and dimension-six operators.
\par
In this article we consider the study of the apparent violation of universality in the couplings of W to leptons within the above EFT framework, with the goal of finding out if the observed deviation can be explained in terms of NP effects once constraints from precise electroweak observables are taken into account. Motivated by the data and for the sake of simplicity, we will assume two different flavor symmetries that we introduce in the next subsections.
\par
To set the stage for this discussion, we explain first the simpler case of $U(3)^5$ flavor symmetry.
In the absence of Yukawa couplings, the SM Lagrangian shows a $U(3)^5 = U(3)_{q}\times U(3)_{u}\times U(3)_{d}\times U(3)_{\ell}\times U(3)_{e}$ flavor symmetry,
corresponding to the independent rotation of each SM fermion field: the quark and lepton doublets $q$ and $\ell$ and the up-quark, down-quark and charged lepton
singlets $u$, $d$ and $e$. We can also decompose this symmetry group in the following way:
\begin{equation}
SU(3)^5\times U(1)_{L}\times U(1)_{B}\times U(1)_{Y}\times U(1)_{PQ}\times U(1)_{e}
\end{equation}
where the five global $U(1)$ symmetries can be identified with the total lepton and baryon number, the hypercharge, the Peccei-Quinn symmetry and a remaining global
symmetry that we choose to be the rotation of the charged lepton singlet. In the presence of Yukawa couplings this flavor symmetry breaks down to the subgroup $G=U(1)_{L}\times U(1)_{B}\times U(1)_{Y}$.
\par
Requiring that the higher dimensional operators respect the $U(3)^5$ flavor symmetry reduces significantly their number, suppresses undesired
Flavor Changing Neutral Current effects and leads to the Minimal Flavor Violation (MFV) framework after the introduction of the Yukawa spurions \cite{D'Ambrosio2002ex}.
The complete list of the twenty-one dimension-six $U(3)^5$ invariant operators can be found in Refs.~\cite{Han:2004az,Cirigliano2009wk}, where this flavor symmetry was assumed in the context of an EFT analysis of electroweak precision data. As an example we show here the three operators that do not contain fermions:
\begin{eqnarray}
\label{eq:0f}
O_{W\!B} &=&(h^\dagger \tau^a h) W^a_{\mu \nu} B^{\mu \nu}, \ \ \ O_h^3 = | h^\dagger D_\mu h|^2~, \nonumber \\
O_W &=& \epsilon_{abc} \, W^{a \nu}_{\mu} W^{b\lambda}_{\nu} W^{c \mu}_{\lambda}~,
\end{eqnarray}
where we follow, with minor modifications, the notation and conventions of Ref. \cite{Buchmuller:1985jz}: $h$ is the Higgs boson doublet; $\tau^a$ are the Pauli matrices; $W_{\mu \nu}^i = \partial_{\mu} W_{\nu}^i - \partial_{\nu} W_{\mu}^i + g \, \varepsilon_{ijk} W_{\mu}^j W_{\nu}^k$, $B_{\mu \nu} = \partial_{\mu} B_{\nu} - \partial_{\nu} B_{\mu}$ and the covariant derivative reads $D_{\mu} = \partial_{\mu} -i \frac{g}{2} \tau^i W_{\mu}^i-ig' Y B_{\mu}$, with hypercharge $Y(h)=1/2$.
\par
It is clear that in this special framework it is impossible to generate any departure from lepton universality, as the $U(3)^5$ symmetry allows only for flavor independent NP contributions. For this reason we will relax this symmetry group to smaller groups where the third family is singled out.
\subsection{$[U(2)\times U(1)]^5$ flavor symmetry
\label{sec:frameA}
Motivated by the experimental observations shown in Eqs.~\eqref{eq:univvio} and \eqref{eq:3br}, it is an interesting possibility to assume the flavor symmetry $[U(2)\times U(1)]^5$, that singularizes the third family with respect to the light ones, allowing for different NP contribution to the processes involving the heavy fermions: top, bottom and, in particular, $\tau$ and $\nu_\tau$.
\par
This framework was indeed studied in Ref.~\cite{Han:2005pr}, and we will use the same notation, in which $q_p$, $\ell_p$, $u_p$, $d_p$ and $e_p$ ($p=1,2$) represent only the two first generations of fermions, whereas $Q$, $L$, $t$, $b$ and $\tau$ represent the third family fields. The new notation makes clear which combinations of flavor indices are allowed by the flavor symmetry. The operators that do not involve fermions are the same as in the $U(3)^5$ case, whereas those involving one or two fermion bilinears split in several operators; for instance:
\begin{eqnarray}
O_{he} = i (h^\dagger D^\mu h)(\overline{e} \gamma_\mu e) \rightarrow
&&\!\!\!\! \! O_{he} = i (h^\dagger D^\mu h)(\overline{e} \gamma_\mu e) , \nonumber \\
&&\!\!\!\! \! O_{h\tau} = i (h^\dagger D^\mu h)(\overline{\tau} \gamma_\mu \tau).
\end{eqnarray}
The list of invariant operators is much longer than in the $U(3)^5$ symmetric case, but not all the operators affect the EWPO. For this reason, and following Ref.~\cite{Han:2005pr},
we do not include in our numerical analyses (i) operators involving top quarks; (ii) operators involving only third-generation fermions; or (iii) operators involving light quarks and third generation leptons\footnote{We noticed that the operator $O_{Lq}^3$ in Eq.~(\ref{eq:hanope}) can be strongly constrained by the experimental value of the $\tau \to \pi \nu_{\tau}$ process and it is consequently included in our analysis. Ref.~\cite{Han:2005pr}, not considering this observable, didn't include $O_{Lq}^3$.}. Moreover, motivated by the experimental result shown in Eq.~\eqref{eq:3br} and for the sake of simplicity we will assume that 2- and 4-fermion operators that only have light generation fermions can be neglected. In this way we are left with the following six operators with one fermion bilinear:
\begin{eqnarray}
&& O_{hf_1} = i (h^\dagger D^\mu h)(\overline{f}_1 \gamma_\mu f_1) + {\rm h.c.}~, \label{eq:12f} \\
&& O_{hf_2}^{1} = i (h^\dagger D^\mu h)(\overline{f}_2 \gamma_\mu f_2) + {\rm h.c.}~, \label{eq:22f} \\
&& O_{hf_2}^{3} = i \left( h^{\dagger} D_{\mu} \tau^i h \right) \left( \overline{f}_2 \, \gamma^{\mu} \, \tau^i \, f_2 \right) + {\rm h.c.} ~,
\label{eq:2f}
\end{eqnarray}
where $f_1=\tau,b$ and $f_2=L,Q$. We also have the following four-fermion operators \cite{Han:2005pr}:
\begin{eqnarray} \label{eq:hanope}
&&
O_{Lq}^{3} = (\overline{L} \gamma^\mu \tau^i L) (\overline{q} \gamma_\mu \tau^i q) ~ ,
O_{\ell Q}^{3}= (\overline{\ell} \gamma^\mu \tau^i \ell) (\overline{Q} \gamma_\mu \tau^i Q) ~,
\nonumber\\
&&
O_{\ell Q}^{1}= (\overline{\ell} \gamma^\mu \ell) (\overline{Q} \gamma_\mu Q) ~, ~~~~~
O_{Qe}=(\overline{Q} \gamma^\mu Q) (\overline{e} \gamma_\mu e) ~,
\nonumber\\
&&
O_{eb}=(\overline{e} \gamma^\mu e) (\overline{b} \gamma_\mu b) ~,\qquad
O_{\ell b}= (\overline{\ell} \gamma^\mu \ell) (\overline{b} \gamma_\mu b) ~,
\nonumber\\
&&
O_{\ell L}^{1}= (\overline{\ell} \gamma^\mu \ell) (\overline{L} \gamma_\mu L) ~,~~~~~~
O_{\ell L}^{3}= (\overline{\ell} \gamma^\mu \tau^i \ell) (\overline{L} \gamma_\mu \tau^i L) ~,
\nonumber\\
&&
O_{Le}= (\overline{L} \gamma^\mu L) (\overline{e} \gamma_\mu e) ~, ~~~~~~
O_{\ell\tau}= (\overline{\ell} \gamma^\mu \ell) (\overline{\tau} \gamma_\mu \tau) ~,
\nonumber\\
&& O_{e\tau}=(\overline{e} \gamma^\mu e) (\overline{\tau} \gamma_\mu \tau) ~.
\label{eq:4f}
\end{eqnarray}
\subsection{$U(2)^5$ flavor symmetry
In the limit of vanishing Yukawa couplings the SM Lagrangian is invariant under the $[U(2)\times U(1)]^5$ group symmetry considered in the previous section. We can work with a more realistic scenario keeping the third family Yukawas $\mathcal{L} = y_t \bar{Q} \widetilde{h} t + y_b \bar{Q} h b + y_\tau \bar{L} h \tau$ and neglecting only those of the two lightest generations. In this case the flavor symmetry breaks down to $U(2)^5\times U(1)^3$, that we will just call $U(2)^5$, since the three $U(1)$ subgroups are simply the Lepton and Baryon number and Hypercharge of the third generation\footnote{A recent analysis of the implications of current flavor data for the quark-sector component of this symmetry, i.e. $U(2)^3$, suitably broken by spurions \`a la MFV, can be found in Ref.~\cite{Barbieri:2011ci}.}.
Among the new operators that appear due to the reduction of the symmetry group, only the following four chirality-flipping operators will affect EWPO:
\begin{eqnarray}
\label{eq:operwlnu}
O_{\tau B}^{t} &=& \left( \overline{L} \, \sigma^{\mu \nu} \, \tau \right) h \, B_{\mu \nu} + {\rm h.c.} ~, \nonumber \\
O_{b B}^{t} &=& \left( \overline{Q} \, \sigma^{\mu \nu} \, b \right) h \, B_{\mu \nu} + {\rm h.c.} ~, \nonumber \\
O_{\tau W}^{t} &=& \left( \overline{L} \, \sigma^{\mu \nu} \, \tau^i \, \tau \right) h \, W_{\mu \nu}^i + {\rm h.c.} ~, \nonumber \\
O_{b W}^{t} &=& \left( \overline{Q} \, \sigma^{\mu \nu} \, \tau^i \, b \right) h \, W_{\mu \nu}^i + {\rm h.c.} ~.
\end{eqnarray}
Their chirality-flipping structure translates, in the processes of our interest here, into contributions proportional to the fermion masses, i.e. suppressed by the factor $m_f / v$ with respect to other NP contributions from dimension-six operators.
Given that we focus here on the $W \rightarrow \tau \, \overline{\nu}_{\tau}$ decay we will not consider in the following the operators $O_{bB}^{t}$ and $O_{bW}^{t}$.
\section{$W \rightarrow \tau \, \overline{\nu}_{\tau}$ decay in the EFT framework}
\label{sec:Wtaunu}
When the $U(2)^5$ flavor symmetry is assumed, the SM term and dimension-six operators contributing to the $W \rightarrow \tau \, \overline{\nu}_{\tau}$ decay are:
\begin{eqnarray}\label{eq:2ops}
{\cal L}_{\mbox{\tiny EFT}} &=& i \, \overline{L} D \! \! \! \! / L
+\frac{1}{\Lambda^2} \left\{ \widehat{\alpha}_{hL}^{3} {\cal O}_{hL}^{3} + \widehat{\alpha}_{\tau W}^{t}{\cal O}_{\tau W}^{t} + \mbox{h.c.} \right\} \\ \hspace*{-0.7cm}
\supset && \hspace*{-0.6cm} \frac{ g}{\sqrt2} \Big[ \Big(1 + 2 \alpha_{hL}^{3} \Big) \bar{\tau}_L \gamma^\mu \nu_\tau W^-_\mu + \frac{2}{g v}\alpha_{\tau W}^t \bar{\tau}_R \sigma^{\mu\nu} \nu_\tau W^-_{\mu\nu} \Big], \nonumber
\end{eqnarray}
where $W^-_\mu = (W^1_\mu + i W^2_\mu)\slash \sqrt2 $,
and we have introduced the normalized couplings $\alpha \equiv \frac{v^2}{\Lambda^2} \, \widehat{\alpha}$, that we assume to be real hereafter. Working at linear order in the $\alpha$ coefficients, the full decay width reads:
\begin{eqnarray} \label{eq:gammawlnu}
\Gamma \left( W \rightarrow \tau \,\overline{\nu}_\tau \right) & = & \frac{G_F \, M_W^3}{6 \, \sqrt{2} \, \pi} \left( 1- w_{\tau}^2 \right)^2 \\
&& \hspace{-2cm}\times \left\{ \left( 1 + 4 \alpha_{hL}^3 \right) \left(1+ \frac{w_{\tau}^2}{2} \right) + \, 6 \, \sqrt{2} \, w_{\tau} \, \alpha_{\tau W}^t \right. \biggr\} ~ , \nonumber
\end{eqnarray}
where $w_{\tau} = m_{\tau}/M_W$ and $G_F$ is the tree level Fermi coupling constant defined by $G_F \slash \sqrt2 = g^2\slash (8 M_W^2)$. The new contributions to the decay width have the following features:
\begin{itemize}
\item There are only two dimension-six operators contributing to this process: ${\cal O}_{hL}^{3}$ and ${\cal O}_{\tau W}^{t}$. This can be seen if the equations of motion are properly used to reduce the number of operators in the effective basis, as done in Ref.~\cite{Grzadkowski:2010es}, instead of using directly all the operators appearing in the original list of Ref.~\cite{Leung:1984ni,Buchmuller:1985jz}.
\item The lepton universality feature of the SM implies that $g_{\tau} = g$. The operator ${\cal O}_{hL}^3$ simply shifts the SM result in such a way that its effect can be encoded in the following redefinition:
\begin{eqnarray}
\label{eq:gtau}
g_\tau \equiv g \left( 1 + \delta g_\tau \right) = g \left( 1 + 2~ \alpha_{hL}^3 \right)~.
\end{eqnarray}
This operator is allowed in the two flavor symmetries that we consider.
\item The magnetic operator ${\cal O}_{\tau W}^t$ provides a new structure not present in the SM \cite{Bernabeu:1994wh,Rizzo:1997we,GonzalezSprinberg:2000mk}. Contrarily to ${\cal O}_{hL}^3$ this is a chirality flipping operator and it gives a contribution suppressed by $m_{\ell}/M_W$ due to the derivative dependence. Assuming the $[U(2)\times U(1)]^5$ flavor symmetry this term vanishes.
\end{itemize}
In what follows we will consider the universality ratios $R^W_{\ell\ell^\prime} = \Gamma(W\to\ell\nu_\ell) \slash \Gamma(W\to\ell^\prime\nu_\ell^\prime)$, instead of the simple decay rate,
in such a way that we do not have to worry about the NP corrections associated to the experimental determination of the Fermi constant $G_F$, since they cancel in the ratio.
\section{Fit procedure and results}
\label{sec:fit}
Once we have identified in the previous section the effective operators that can contribute to $R_{\tau\ell}^W$, generating a deviation from lepton universality, we study now the constraints that can be derived on these operators from EWPO and low-energy measurements.
\par
Looking for example at the experimental result \eqref{eq:Zll} it can be understood that one single operator will not be able to explain simultaneously the EWPO and the anomaly in the $W\tau\nu$ vertex shown in Eq.~\eqref{eq:univvio}, due to the gauge symmetry that connects W and Z bosons. However, when several operators are present one can have cancellations between them and a careful numerical analysis is needed.
\par
With that purpose we updated and modified the Mathematica code developed in Ref.~\cite{Han:2005pr}, that included electroweak observables at the Z line and at higher energies and other low energy measurements. In addition we include the leptonic tau decay and the exclusive channel $\tau \to \pi \nu_\tau$, that have an experimental error well below the 1\% level and a theoretical error under control. We consider also the anomalous magnetic moment of the tau lepton that, despite its very large experimental uncertainty, is able to constrain the magnetic operators poorly bounded by other observables. The associated formulas are collected in Appendix \ref{app:A} and the complete list of the observables used in our analysis can be found in Table \ref{tab:experiments}.
\begin{table}[tb]
\begin{tabular}{|l|l|l|l|}
\hline
Classification & Std. Notation & Measurement \\
\hline
Atomic parity &$Q_W(Cs)$ &Weak charge in Cs\\
violation & $Q_W(Tl)$ & Weak charge in Tl\\
\hline
DIS &$g_L^2,g_R^2$ &$\nu_\mu$-nucleon scattering (NuTeV)\\
&$R^\nu$ &$\nu_\mu$-nucleon scatt. (CDHS, CHARM)\\
&$\kappa$ &$\nu_\mu$-nucleon scatt. (CCFR)\\
&$g_V^{\nu e},g_A^{\nu e}$ &$\nu$-$e$ scatt. (CHARM II)\\
\hline
Z-pole &$\Gamma_Z$ &Total $Z$ width\\
&$\sigma_0$ &$e^+e^-$ hadronic cross section\\% at $Z$ pole\\
&$R_{f=e,\mu,\tau,b,c}^0$ &Ratios of decay rates \\
&$A_{FB}^{0,f=e,\mu,\tau,b,c}$ &FB asymmetries\\
&$A_{f=e,\mu,\tau,s,b,c}$ &Polarized asymmetries\\
&$\sin^2\theta_{eff}^{lept}$ &Hadronic charge asymmetry\\
\hline
LEPII &$\sigma_{f=q,c,b\mu,\tau}$ & Total cross sections for $e^+e^-\!\rightarrow\! f\overline f$\\
fermion &$A_{FB}^{f=c,b,\mu,\tau}$ & FB asymmetries for $e^+e^-\rightarrow f\overline f$\\
production &$d\sigma_e/d\cos\theta$ &$e^+e^-\rightarrow e^+e^-$ diff. cross section\\
\hline
W pair &$d\sigma_W/d\cos\theta$ &$e^+e^-\!\rightarrow\! W^+W^-$ diff. cross section\\
&$M_W$ &W mass \\
\hline
$V_{\mbox{\tiny{CKM}}}$ unitarity & $\Delta_{CKM}$ & $V_{ud}$ and $V_{us}$ extractions \cite{Cirigliano2009wk}\\
\hline
$\tau$ decays & $\tau\to \nu_\tau \ell \bar{\nu}_\ell$ & Leptonic $\tau$ decay ($\ell = e, \mu$) \cite{Nakamura:2010zzi}\\
& $\tau\to \nu_\tau \pi$ & Exclusive hadronic $\tau$ decay \cite{Nakamura:2010zzi} \\
\hline
Anomalous & $a_\tau$ & $e^+e^- \to e^+e^- \tau^+ \tau^-$ \\
magnetic & & cross section \cite{Abdallah:2003xd}\\
moment & & \\
\hline
\end{tabular}
\caption{\small{Measurements included in this analysis. See Ref.~\cite{Han:2004az} and references therein for detailed descriptions. References are shown only for the new observables.
\label{tab:experiments}}}
\end{table}
\par
We included in the program also the contribution to the different observables coming from the magnetic operators $O_{\tau W}^{t}$ and $O_{\tau B}^{t}$, not included in Ref.~\cite{Han:2005pr} since the $U(2)^5\times U(1)^5$ symmetry was assumed in that work. The formulas for the Z decay rate can be found in Appendix \ref{app:b}, whereas the formulas for $e^+ e^- \to \tau^+ \tau^-$ cross section have been taken from Ref.~\cite{GonzalezSprinberg:2000mk}.
\par
The leptonic decays of heavy pseudoscalar mesons ($B^\pm$, $D^\pm$, $D^\pm_{\mbox{\tiny S}}$) could in principle be considered in order to constrain NP effects in leptonic W decays, but they have not reached yet the necessary experimental precision: the relative error of the current data on the decays into tau are approximately $\mathcal{O}(6\%)$ for $D_{\mbox{\tiny S}}$ decays and $\mathcal{O}(20\%)$ for B decays, and some of the decays into muon and electron have not been seen yet, preventing a complete analysis of the lepton universality ratios. For these reasons, these observables have not been included in the fit. We will comment on them in Sec.~\ref{sec:heavymesons}.
\par
Concerning LHC measurements, the natural channels to analyze for the purpose of this paper are $pp\to \tau \bar{\nu} X$ and $pp\to \tau^+ \tau^-X$, where possible modifications of the $W\tau\nu$ vertex and its gauge counterpart $Z\tau^+\tau^-$ can be probed, but unfortunately there is no data available for these particular channels yet. On the theoretical side, the contribution to these processes coming from effective operators has been worked out in Refs.~\cite{Bhattacharya:2011qm,Cirigliano:2012} for first generation leptons. In any case, as we will see, the list of observables included in our fit is exhaustive enough to reach a solid answer to the possible lepton universality violations.
\par
With the above-mentioned observables $O^i$, we build a standard $\chi^2$ function as:
\begin{equation} \label{eq:chi2def}
\chi^2 \left(\boldsymbol{\alpha} \right) = \sum_i \left[ O_{\mbox{\tiny th}}^i \left(\boldsymbol{\alpha} \right) - O_{\mbox{\tiny exp}}^i \right]
\left[\sigma_{\mbox{\tiny O}}^2
\right]^{-1}_{ij} \left[ O_{\mbox{\tiny th}}^j \left(\boldsymbol{\alpha} \right) - O_{\mbox{\tiny exp}}^j \right] \, ,
\end{equation}
where the error matrix $\sigma^2_{\mbox{\tiny O}}$ includes the experimental error and the uncertainty on the SM prediction combined in quadrature. The theoretical value $O_{\mbox{\tiny th}}^i$ contains the up-to-date SM prediction and the contribution of higher dimensional operators through interference with SM vertices, i.e. linear in the $\alpha_a$ couplings.
\par
As a result of this fit we determine the value of the different Wilson coefficients $\alpha_a$, with their relative errors and the corresponding correlations, or in other words the bounds on the different NP effective operators. In particular we are interested in the bounds associated to the two operators that could generate a lepton universality violation in the W decay (see Eq.~\eqref{eq:2ops}), and finally in the determination of the universality ratio $R_{\tau \ell}^W$ extracted from our fit, to be compared with the experimental determination given in Eq. (\ref{eq:univvio}).
\subsection{(Semi)leptonic $\tau$ decays as precise electroweak observables}
\begin{figure}[t]
\begin{center}
\includegraphics[width=7.7cm]{Filipuzzi_Fig1.pdf}
\caption{\label{fig:TauConstrains}
Phenomenological constraints on the operators $O_{\ell L}^1$ and $O_{\ell L}^3$ from electroweak observables (red diagonal band) and leptonic tau decay rate (blu horizontal band). See Table \ref{tab:experiments} for the complete list of observables considered. The black ellipse is the 1$\sigma$ C.L. region when considering all observables together. }
\end{center}
\end{figure}
In a general analysis involving a big number of operators (free parameters in the fit) it is possible to encounter flat directions, i.e. directions in the parameter space that are not bounded by the experimental data. This means that some operators appear always in the same combination throughout all the observables considered in the fit and then only that combination can be constrained, and not each operator separately. In Ref.~\cite{Han:2005pr} four flat directions were identified in the particular fit we are using in this work. However, we show now how the addition of the leptonic tau decay to the list of EWPO included in the fit removes one of these flat directions.
\par
In the limit of $U(2)^5$ flavor symmetry the rate for the leptonic decay of the $\tau$ lepton reads\footnote{The operator corresponding to $\alpha_{h\ell}^3$ is defined in analogy with $O_{hL}^3$ in Eq.~\eqref{eq:2f}. This self-explanatory notation will be adopted hereafter for operators involving only light fermions. Although we neglect these operators in the subsequent numerical analysis we keep them in the analytic expressions for the sake of completeness.}:
\begin{eqnarray} \label{eq:taulepto}
\Gamma_{\tau\to\nu_\tau\ell\overline{\nu}_\ell} &=& \frac{G_F^2 m_\tau^5}{192 \, \pi^3} \left\{ \Big[1+ 4 \, \alpha_{h L}^{3} + 4 \, \alpha_{h \ell}^{3} - 4 \, \alpha_{\ell L}^{3} \Big] \times \right.\nonumber\\
&& \hspace{-1.8cm} \times f\!\left(\frac{m_\ell^2}{m_\tau^2}\right) \left. + 2 \, \sqrt{2} \, \alpha_{\tau W}^{t} \, \frac{m_\tau}{M_W} \, g\!\left(\frac{m_\ell^2}{m_\tau^2}\right) \right\} (1\!+\!\delta_{RC})~,
\end{eqnarray}
where $\ell=e,\mu$, $\delta_{RC}$ contains the radiative corrections to the SM contribution \cite{Pak:2008qt} and
\begin{eqnarray}
f(x) \!&=&\! 1 - 8x - 12 x^2 \ln{x} + 8x^3 - x^4, \\
g(x) \!&=&\! 1 - 6 x + 18 x^2 - 10 x^3 + 12 x^3 \ln{x} - 3 x^4 . \nonumber
\end{eqnarray}
\par
In order to show the constraining power of the tau decays let us consider the simple situation in which only the operators $O_{\ell L}^1$ and $O_{\ell L}^3$ are not vanishing. As shown in Fig.~\ref{fig:TauConstrains} the electroweak observables, and in particular the $e^+ e^- \to \tau^+ \tau^-$ cross section and the forward-backward asymmetry, are able to constrain only the combination $O_{\ell L}^1 + O_{\ell L}^3$. The inclusion of the the leptonic tau decay into the fit allows to reduce the one sigma C.L. region to the black ellipse. The two operators are then constrained at the $0.4\%$ and $0.2\%$ level, corresponding to an effective NP scale $\Lambda>2.7$ TeV and $\Lambda>4.1$ TeV (90$\%$ C.L.) respectively: very strong bounds that show the importance of leptonic tau decays as electroweak precision observable.
A similar role is played by the pionic $\tau$ decay, where experimental results and SM calculations are also below the per-mil level of precision. The expression for the $\tau\to\pi\nu_\tau$ decay rate within our $U(2)^5$ flavor symmetric EFT framework is the following:
\begin{eqnarray}
\Gamma_{\tau^- \to \pi^- \nu_{\tau}} &&= \frac{G_F^2 F_{\pi}^2}{8\pi} |V_{ud}|^2 m_\tau^3 \left( 1-\frac{M_\pi^2}{m_\tau^2}\right)^2 (1+\delta_{RC}') \nonumber\\
&& \times \left( 1 + 4~\alpha_{hL}^{3} + 4~\alpha_{hq}^{3} - 4~\alpha_{L q}^{3} \right) ~,
\end{eqnarray}
where $F_\pi$ denotes the pion decay constant and $\delta_{RC}'$ radiative corrections \cite{Guo:2010dv}. It is convenient to work once again with a normalized ratio, namely:
\begin{eqnarray} \label{eq:rtaupio}
R_{\tau/\pi}
\label{eq:Rtaupi}
&\equiv& \frac{\Gamma_{\tau^- \to \pi^- \nu_{\tau}}}{\Gamma_{\pi^- \to \mu^- \overline{\nu}_{\mu}}} \nonumber\\
&=& \frac{m_\tau^3}{2 \, m_\mu^2 M_\pi} \left( \frac{1- \frac{M_\pi^2}{m_\tau^2}}{ 1- \frac{m_\mu^2}{M_\pi^2}}\right)^2 \left( 1 + \delta_{\tau/\pi} \right) \nonumber\\
&& \hspace{-0.5cm} \times \left( 1 + 4\left(\alpha_{hL}^{3} - \alpha_{h\ell}^{3} \right) - 4\left(\alpha_{L q}^{3} - \alpha_{\ell q}^{3} \right) \right) .
\end{eqnarray}
where $\delta_{\tau/\pi}=0.0016(14)$ denotes the radiative corrections to the SM contributions \cite{Decker:1994dd}. As we can see, this observable represents another probe of the $\alpha_{hL}^3$ coefficient, and moreover it represents the only observable in our analysis sensitive to the $\alpha_{L q}^{3}$ coefficient. Comparing the experimental value of $R_{\tau/\pi}$ \cite{Nakamura:2010zzi,Schael:2005am} and its SM prediction we get a bound of $\Lambda > 3.1$ TeV (90$\%$ C. L.) on the NP effective scale for the four Wilson coefficients appearing in Eq.~\eqref{eq:Rtaupi}.
\subsection{$[U(2)\times U(1)]^5$ symmetric case: results}
In order to study if the $R_{\tau\ell}^W$ anomaly of Eq.~\eqref{eq:univvio} can be accommodated in our EFT framework as a genuine New Physics effect and not just a statistical fluctuation, we start with a single operator analysis where only the $\alpha_{hL}^3$ is present and all the observables of Table \ref{tab:experiments} are included. In this case we obtain the expected strong bound:
\begin{equation}\label{eq:Rtau}
R_{\tau\ell}^W = 0.9997 \pm 0.0015~,
\end{equation}
in good agreement with the SM prediction. As shown in Fig.~\ref{fig:SingleOperator}, the very precise measurements of leptonic $Z$ and $\tau$ decays dominate our fit, and makes impossible to accommodate the $R_{\tau\ell}^W$ anomaly
Once we include additional operators, things become less intuitive because cancelations between operators are possible, opening the possibility to explain the $R_{\tau\ell}^W$ anomaly and the leptonic Z and $\tau$ decays at the same time.
As a first global analysis, we assume the $[U(2)\times U(1)]^5$ flavor symmetry and we include the 17 operators given in Eqs.~\eqref{eq:0f} and (\ref{eq:12f}-\ref{eq:4f}). It is worth repeating that in order to simplify the discussion and given that the experimental data show no sign of NP related to the light families of fermions, we have assumed that the operators involving only light fermions can be neglected.
\par
Somehow surprisingly we find that even with so many operators, the constraint on $\alpha_{hL}^3$ is very strong, namely $- 3.6 \times 10^{-3} \leq \alpha_{hL}^{3} \leq - 0.5 \times 10^{-3} $ at $90\%$ C.L. Interestingly enough, this value is two sigmas away from zero, giving the following bound on the universality ratio:
\begin{equation}\label{eq:RtauBIS}
R_{\tau\ell}^W = 0.991 \pm 0.004 ~,
\end{equation}
where we quoted the error at 1$\sigma$ level in order to be comparable with the experimental result in Eq. (\ref{eq:univvio}). Thus we find the curious result that our fit is indeed able to accommodate a violation of lepton universality in the W decays, but \emph{in the opposite direction} than the direct experimental measurement. Somehow the small tensions present in the SM fit (see e.g. $\sigma_{had}^0$ in Fig.~\ref{fig:SingleOperator}) can be alleviated introducing some non-zero Wilson coefficients, being $\alpha_{hL}^3$ one of them. Obviously the inclusion of $R_{\tau\ell}^W$ will reduce this \lq\lq tension\rq\rq~moving the value of $\alpha_{hL}^3$ closer to zero ($- 3.2 \times 10^{-3} \leq \alpha_{hL}^{3} \leq - 0.08 \times 10^{-3} $ at $90\%$ C.L.).
\par
The conclusion is once more that we cannot accommodate the $R_{\tau\ell}^W$ along with our long list of precision observables, and thus we are forced to consider it a mere statistical fluctuation.
Unlike the single operator case where it could be naively expected, this represents a non-trivial result in a fit with seventeen free parameters.
\par
For the sake of completeness, let us mention that in a truly global $[U(2)\times U(1)]^5$ fit, where operators only involving light fermions (like e.g. ${\cal O}_{h\ell}^3$) are also included, the NP bounds become extremely weak and the current experimental value of $R_{\tau\ell}^W$ cannot be excluded anymore.
\begin{figure}[t]
\begin{center}
\includegraphics[width=8cm]{Filipuzzi_Fig2.pdf}
\caption{\label{fig:SingleOperator}
Bounds obtained for the NP coefficient $\alpha_{hL}^3$ from the different set of measurements included in our fit. Equivalently, these are the bounds on the deviation from lepton universality in the electroweak coupling to third generation leptons $g_\tau$ (see Eq.~\eqref{eq:gtau}). For comparison we also show the value obtained, using the experimental data in Eq. (\ref{eq:univvio}), from leptonic W decays (not included in our fit). $A_{FB}^{0,\tau}$ is the forward-backward asymmetry measured at LEP1 for tau pairs, $A_\tau$ includes the SLD measurement and the LEP1 total $\tau$ polarization and the LEP2 bound comes from $\tau$ pair cross sections and asymmetries. See PDG \cite{Nakamura:2010zzi}, chapter 10, for more details.}
\end{center}
\end{figure}
\subsection{$U(2)^5$ symmetric case: results}
Reducing the symmetry group to $U(2)^5$ introduces the chirality-flipping operators $O_{\tau W}^t$ and $O_{\tau B}^t$, offering additional NP contributions to the observables and higher cancellations between operators.
\par
From the associated global fit\footnote{We do not include in this $U(2)^5$-symmetry fit the leptonic polarization asymmetries $A_\ell$, since they have been extracted assuming only vector and axial-vector couplings.} with 19 free parameters, we get the following $90\%$ bounds on the two operators involved in the W decays: $- 3.7 \times 10^{-3} \leq \alpha_{hL}^3 \leq - 0.6 \times 10^{-3} $ and $0.04 \times 10^{-3} \leq \alpha_{\tau W}^{t} m_\tau / M_W \leq 5.0 \times 10^{-3}$, where we have explicitly shown the $m_\tau \slash M_W$ suppression that multiplies the $\alpha_{\tau W}^t$ coefficient in the observables. From these values we calculate the prediction for the universality ratio at 1$\sigma$ level:
\begin{equation}
R_{\tau\ell}^W = 1.01 \pm 0.01 ~.
\end{equation}
While the constraints on $\alpha_{hL}^3$ are very similar to the previous case, the presence of a second contribution from the magnetic operator increases the error (and the central value) of $R_{\tau\ell}^W$. This increase is however not enough to nicely accommodate the experimental value of $R_{\tau\ell}^W$ shown in Eq.~\eqref{eq:univvio}.
\section{Leptonic decays of heavy mesons}
\label{sec:heavymesons}
The leading SM contribution to the $P^- \rightarrow \ell^- \, \overline{\nu}_{\ell}$ decays is given by the W exchange and hence it is interesting to point out how these decays get modified by possible deviations from family universality in the $W\ell\nu$ coupling. In particular we are interested in the $D$, $D_{\mbox{\tiny S}}$ and $B$ decays because they are heavy enough to decay into the tau lepton. Although two dimension-six operators modify the vertex of the $W$ gauge boson with leptons, namely ${\cal O}_{h \ell}^3$ and ${\cal O}_{e W}^t$, only the first contributes to the leptonic decay of heavy mesons, due to the fact that the tensor coupling has no spin-0 component.
\par
In order to get rid of the hadronic uncertainties and the NP corrections to the Fermi constant or the CKM elements appearing in the individual decay widths, we will focus again on the ratio between the tau channel and a light lepton channel:
\begin{eqnarray}
R_{\tau \ell}^P
&=& \frac{\mbox{BR} \left( P \rightarrow \tau \, \overline{\nu}_{\tau} \right)}{\mbox{BR} \left( P \rightarrow \ell \, \overline{\nu}_{\ell} \right)}~,
\end{eqnarray}
where $\ell=e,\mu$. The effective Lagrangian that mediates these decays, including linear corrections in the $\alpha$ coefficients, can be found in Eq. (34) of \cite{Cirigliano2009wk}. Assuming the $U(2)^5$ flavor symmetry we find the following expressions for the ratios\footnote{In the $B$ decays we neglect the contribution from a new $U(2)^5$-invariant operator $O_{Qb\tau} = (\bar{L} \tau) (\bar{b}Q) + \mbox{h.c.}$, since this operator does not affect the EWPO included in our fit.}:
\begin{eqnarray} \label{eq:pratio}
R_{\tau \ell}^{D_{(s)}}
\!\!\!&=&\!\! \frac{h_{D_{(s)}}\!\!\left(m_{\tau}\right)}{h_{D_{(s)}}\!\!\left(m_{\ell}\right)} \left\{ 1 + 4 \left( \alpha_{hL}^3 \!-\! \alpha_{h\ell}^3 \right) - 4 \left( \alpha_{Lq}^3 \!-\! \alpha_{\ell q}^3 \right) \right\} ~,\nonumber\\
R_{\tau \ell}^{B}
\!\!&=&\!\! \frac{h_{B}\!\left(m_{\tau}\right)}{h_{B}\!\left(m_{\ell}\right)} \left\{ 1 + 4 \left( \alpha_{hL}^3 - \alpha_{h\ell}^3 \right) - 4 \left( \alpha_{LQ}^3 - \alpha_{\ell Q}^3 \right) \right\} ~,\nonumber \\
\end{eqnarray}
where $h_P\left(m\right) = m^2 \left( 1-m^2/M_{P}^2 \right)^2$.
\par
As expected, we find that the $\alpha_{hL}^3$ coefficient modifies these ratios. However, the bound on this coefficient from our analysis of EWPO and low-energy measurements is below the per-cent level (see Sec.~\ref{sec:fit}), a precision very far from current experimental results in these decays. The only ratio where we actually have a value, and not just an upper or lower limit, is $R_{\tau\mu}^{D_s} = 9.2(7)$ \cite{Nakamura:2010zzi}. We can compare this $\sim 8\%$ experimental error with the $\sim 0.7\%$ determination of the $R_{\tau/\pi}$ ratio, where exactly the same linear combination of NP couplings is probed, as shown in Eq.~\eqref{eq:Rtaupi}. This level of precision can actually be considered a benchmark sensitivity for future $D$ and $D_s$ meson experiments to become competitive in the NP search within our EFT framework
On the other hand the leptonic $B$ decays are interesting since they probe a different linear combination of NP coefficients, and therefore are complementary to other observables.
\section{Conclusions}
\label{sec:conclusions}
In the SM the coupling of leptons to the gauge bosons is flavor blind, a property that has been tested successfully in several different observables and experiments, sometimes even at the per-mil level of precision. The latest results from the LEP2 experiment in 2005 showed however a quite sizable deviation ($\sim5\%$) from universality in the $W\ell\nu_\ell$ coupling of more than two sigmas when comparing the third leptonic family with the two light ones, as shown in Eq.~\eqref{eq:univvio}.
\par
We have considered in this article the possibility that this deviation represents a real NP effect. We have performed an Effective Field Theory analysis where the NP effects are parameterized by a series of Wilson coefficients $\alpha_i$, that appear in the effective Lagrangian multiplying dimension-six operators. In order to reduce the number of unknown coefficients and motivated by the possible deviation from lepton universality in the $W\tau\nu_\tau$ vertex, we have assumed different flavor symmetries where the third family plays a special role.
\par
Within this framework we have analyzed if it is possible to accommodate the $R^W_{\tau\ell}$ anomaly of Eq.~\eqref{eq:univvio} as a real NP effect without spoiling the nice agreement between SM predictions and EWPO observables. As expected, it is not possible to do such a thing with just one effective operator at play, due mainly to the very precise $Z$ and $\tau$ leptonic decays, as nicely shown in Fig.~\ref{fig:SingleOperator}. More surprisingly we have found that EWPO are such strong constraints that not even in a global analysis where all the operators affecting the third family are present one can accommodate the $R^W_{\tau\ell}$ anomaly.
\par
Should this departure from universality be confirmed by new data, then our analysis disfavor the possibility of explaining it through a weakly coupled theory standing at the TeV scale, unless a quite non-trivial flavor structure occurs. Instead, it would be necessary to resort to a different description of NP that could involve the introduction of new light degrees of freedom or a strongly interacting sector with flavor dependent couplings to leptons. For example previous studies of this deviation from universality in $W$ decays have focused on the possibility that pair production of supersymmetric light charged Higgs bosons, almost degenerate with the $W$ and decaying largely into heavy fermions, could mimic $W \rightarrow \tau \, \overline{\nu}_{\tau}$ decays \cite{Dermisek:2008dq,Park:2006gk}. Modifications on the electroweak gauge group in order to singularize the third family have also been considered \cite{Li:2005dc}.
\par
Last but not least we have shown the importance of the current measurements in leptonic and semileptonic $\tau$ decays as New Physics constraining observables that probe new directions in the parameter space of our EFT framework, and we have analyzed the sensitivity of the leptonic decays of pseudoscalar mesons to the the violations of lepton universality.
\subsection*{Acknowledgements}
The authors wish to thank Z.~Han for his help in the use of the Mathematica code provided by Ref.~\cite{Han:2005pr}, and T.~Dorigo, A.~Pich and M.~Schmitt and for useful comments and discussions. A.~F. and J.~P. are partially supported by MEC (Spain) under grant FPA2007-60323, by MICINN (Spain) under grant FPA2011-23778, by the Spanish Consolider-Ingenio 2010 Programme CPAN (CSD2007-00042) and by CSIC under grant PII-200750I026. A.~F. is also partially supported by a FPU grant by MEC (Spain). J.~P. is also partially supported by Generalitat Valenciana under grant PROMETEO/2008/069. MGA is supported by the U.S. DOE contract DE-FG02-08ER41531 and by the Wisconsin Alumni Research Foundation.
|
{
"timestamp": "2012-03-12T01:01:43",
"yymm": "1203",
"arxiv_id": "1203.2092",
"language": "en",
"url": "https://arxiv.org/abs/1203.2092"
}
|
\section{Introduction}
Contemporary cosmological observations predict that we live in an accelerating universe \cite{riess,*perlm,*perlm2,*perlm3}. It remains to be understood, how to incorporate the observational predictions naturally in the standard frame work of cosmology. It is also predicted that $73 \%$ of the matter-energy content of the universe is due to something which has a negative pressure and commonly known as Dark Energy (DE). However, a tiny cosmological constant may be useful to address the observational data satisfactorily. Unfortunately while this solves the acceleration problem fairly well, it evokes other problems, e.g. the problem of fine tuning and cosmic coincidence problem etc. However, the above accelerating universe problem may be addressed in two ways: (i) considering modified theory of gravity, i.e. modifying the gravity sector or (ii) considering some unusual form of matter/energy so as to yield an accelerating phase in late universe, i.e. modification of the matter sector. In the former case a number of theories of gravity, for example with $f(R)$ \cite{carroll}, $f(T)$ \cite{uddin} or $f(G)$ \cite{linderfg} are proposed. A large number of other matter fields e.g. quintessence \cite{samipaddy}, tachyon \cite{gibb} etc. are considered to modify the gravitational sector.
Chaplygin gas (CG) was first introduced in 1904 in aerodynamics but recently it is being considered as one of the prospective candidate for dark energy. Although it contains a positive energy density it is referred as an exotic fluid due to its negative pressure. CG may be described by a complex scalar field originating from generalized Born-Infeld action. The equation of state for CG is given by:
\begin{equation}
\label{cgeos}
p=-\frac{A}{\rho},
\end{equation}
where $A$ is a positive constant. Later, a modified form of CG was considered \cite{billic, bentosen}. The equation of state for the modified CG is given by:
\begin{equation}
\label{gcgeos}
p=-\frac{A}{\rho^{\alpha}},
\end{equation}
where $0 \leq \alpha \leq 1$. This modified fluid is known as Generalized Chaplygin Gas (GCG). At high energy GCG behaves almost like a pressure less dust where as at low energy regime it behaves like a dark energy candidate its pressure being negative and almost constant. Thus GCG smoothly interpolates between a non-relativistic matter dominated phase in the early universe and a dark energy dominated phase in the late universe. This interesting property of GCG has motivated people to consider it as a candidate for unified dark matter and dark energy models on the other hand modification of the underlying theory of gravitation, however, can be thought of from a fundamentally different perspective. One of the interesting approaches is the theory of extra dimensions due to Kaluza \cite{kaluza} and Klein \cite{klein}. They used a $5D$ spacetime to unify gravity and electromagnetism. Kaluza-Klein (KK) model has been used in many literature for studying the models of cosmology as well as particle physics \cite{leekk, appelkk}. The cosmology of $5$-dimensional with pure geometry has a feature that properties of matter in $4$ dimension is induced by the $5$D geometry itself by virtue of what is known as Campbell's Theorem \cite{liucamp}. However, KK cosmology is also studied by including matter as well \cite{faraz, darabi}
Recently another very interesting approach in quantum gravity, namely the holographic principle \cite{susskind}, is used to understand the DE problem. According to this principle:{\it all information inside any spatial region can be observed, instead of its volume, on its boundary surface}. It was shown by Cohen. {\it et. al.} \cite{cohen} that quantum zero point energy of any system of size $L$ (its Infra Red (IR) cutoff) can not exceed the mass of a black hole of the same size. So, if $\rho_v$ denotes the quantum zero point energy density and $M_P$ be the reduced Planck Mass $L^3 \rho_v \leq L M_P^2$. A number of literature appeared \cite{cohen, li} where it was discussed how this Holographic principle could set up a proper ultraviolet cut off for the quantum zero point energy of the universe which in turn prevents one from grossly overestimating the value of $\Lambda$ from field theoretic consideration. Very recently the idea of Holographic Dark Energy (HDE) has been incorporated to build an interacting dark energy model in KK cosmology \cite{sharifkk}. In this letter, using the above theory we reconstruct the HDE model with GCG in the framework of KK cosmology. The letter is organized as follows: in section 2. field equations for KK cosmology are introduced. Then in section 3. we propose the Interactive Holographic GCG model in KK cosmology. The stability of the model is discussed in section 4. and finally in section 5 a brief discussion is presented.
\section{Field Equations of Kaluza-Klein Cosmology}
\label{feqkk}
The Einstein field equation is:
\begin{equation}
\label{efeq}
R_{AB}-\frac{1}{2}g_{AB}R=\kappa T_{AB}
\end{equation}
where $A$ and $B$ runs from $0$ to $4$, $R_{AB}$ is the Ricci tensor, $R$ is the Ricci scalar and $T_{AB}$ is the energy-momentum tensor.
The $5$-dimensional spacetime metric of Kaluza-Klein (KK) cosmology is given by \cite{ozel} :
\begin{eqnarray}
\label{kkmetric}
ds^2 &=& dt^2-a^2(t)\left[\frac{dr^2}{1-kr^2}+r^2(d\theta^2 \right. \nonumber \\
&& \left. + sin^2\theta d\phi^2\right)+\left(1-kr^2)d\psi^2\right],
\end{eqnarray}
where $a(t)$ denotes the scale factor and $k=0,1(-1)$ represents the curvature parameter for flat and closed(open) universe. We consider a cosmological model where KK universe is filled with a perfect fluid. The Einstein's field equation for the metric given by (\ref{kkmetric}) becomes:
\begin{equation}
\label{kkfeq1}
\rho=6\frac{\dot{a}^2}{a^2}+\frac{k}{a^2},
\end{equation}
\begin{equation}
\label{kkfeq2}
p=-3\frac{\ddot{a}}{a}-3\frac{\dot{a}^2}{a^2}-3\frac{k}{a^2}.
\end{equation}
For simplicity, we consider a flat universe, i.e. $k=0$. The above equations (eq. (\ref{kkfeq1}) and eq. (\ref{kkfeq2})) reduce to:
\begin{equation}
\label{kkfeq3}
\rho=6\frac{\dot{a}^2}{a^2},
\end{equation}
\begin{equation}
\label{kkfeq4}
p=-3\frac{\ddot{a}}{a}-3\frac{\dot{a}^2}{a^2}.
\end{equation}
The Hubble parameter is defined as $H=\frac{\dot{a}}{a}$. $T^{\mu\nu}_{;\nu}=0$ yields the continuity equation:
\begin{equation}
\label{kkcont1}
\dot{\rho}+4H(\rho+p)=0
\end{equation}
Using the equation of state $p=\omega\rho$, the equation of continuity becomes:
\begin{equation}
\label{kkcont2}
\dot{\rho}+4H\rho(1+\omega)=0
\end{equation}
We consider two kinds of fluids with total energy density $\rho=\rho_{\Lambda}+\rho_{m}$, $\rho_{\Lambda}$ corresponds to dark energy and $\rho_{m}$ is for matter inclusice of Cold Dark Matter (CDM) with an equation of state parameter $\omega_{m}=0$ hold separately. For non-interacting fluid, the conservation equations for $p_{\Lambda},\rho_{\Lambda}$ and $p_{m},\rho_{m}$. However, in the case of interacting dark energy models we consider the following equations admitted by the continuity equation :
\begin{equation}
\label{kkcontm}
\dot{\rho_m}+4H\rho(1+\omega_m)=Q
\end{equation}
\begin{equation}
\label{kkcontl}
\dot{\rho_{\Lambda}}+4H\rho(1+\omega_{\Lambda})=-Q
\end{equation}
where $Q$ denotes the interaction between dark energy and dark matter.
\section{Interacting Holographic GCG model}
\label{int GCG}
Consider the interaction quantity $Q$ to be of the form $Q=\Gamma \rho_{\Lambda}$ and denote the ratio of the energy densities with $r$, i.e. $r=\frac{\rho_m} {\rho_{\Lambda}}$. This gives the decay of GCG into CDM and $\Gamma$ is the decay rate. Following ref. \cite{setarehol} we define effective equation of state parameters as follows:
\begin{equation}
\label{efeosdef}
\omega^{eff}_{\Lambda}=\omega_{\Lambda}+\frac{\Gamma}{4H} \; \; and \; \; \omega^{eff}_{m}=-\frac{1}{r} \frac{\Gamma}{4H}
\end{equation}
The continuity equations are given by:
\begin{equation}
\label{kkcontm2}
\dot{\rho_m}+4H\rho(1+\omega^{eff}_m)=0
\end{equation}
\begin{equation}
\label{kkcontl2}
\dot{\rho_{\Lambda}}+4H\rho(1+\omega^{eff}_{\Lambda})=0
\end{equation}
In terms of Hubble parameter the Friedmann equation in flat KK universe can be rewritten as:
\begin{equation}
\label{frwkkh}
H^2=\frac{1}{6} \left(\rho_{\Lambda}+\rho_m \right),
\end{equation}
where we consider $M_p^2=1$. Density parameters are defined as:
\begin{equation}
\label{dpdef}
\Omega_m=\frac{\rho_m}{\rho_{cr}}, \Omega_{\Lambda}=\frac{\rho_{\Lambda}}{\rho_{cr}},
\end{equation}
where $\rho_{cr}=6H^2$. In terms of density parameters equation (\ref{frwkkh}) becomes:
\begin{equation}
\label{frwdp}
\Omega_m+\Omega_{\Lambda}=1
\end{equation}
Using eqs. (\ref{dpdef}) and (\ref{frwdp}) we obtain:
\begin{equation}
\label{rdp}
r=\frac{1-\Omega_{\Lambda}}{\Omega_{\Lambda}}
\end{equation}
We consider here GCG for describing interacting dark energy model (EOS is given by eq. (\ref{gcgeos})). For GCG in KK cosmology the energy density is:
\begin{equation}
\label{rhoa}
\rho_{\Lambda}=\left[ A+\frac{B}{a^{4(1+\alpha)}} \right]^\frac{1}{1+\alpha}
\end{equation}
The EOS parameters $\omega$ are given by:
\begin{equation}
\label{oml}
\omega_{\Lambda}=\frac{p_{\Lambda}}{\rho_{\Lambda}}=-\frac{A}{\rho^{\alpha+1}_{\Lambda}}=-\frac{A}{A+\frac{B}{a^{4(1+\alpha)}}},
\end{equation}
\begin{equation}
\label{omlef}
\omega^{eff}_{\Lambda}=-\frac{A}{A+\frac{B}{a^{4(1+\alpha)}}}+\frac{\Gamma}{4H}.
\end{equation}
Now we consider a holographic correspondence for GCG in KK cosmology. For that as shown in \cite{sharifkk} the energy density for flat KK universe :
\begin{equation}
\label{rhohol}
\rho_{\Lambda}=3c^2\pi^2L^2.
\end{equation}
The infrared cutoff of the universe $L$ in the flat KK universe is equal to the apparent horizon, which coincides with Hubble horizon \cite{cai}. So we can write as in \cite{sharifkk}:
\begin{equation}
\label{cutoff}
r_a=\frac{1}{H}=r_H=L
\end{equation}
The decay rate \cite{wang} is now given by ,:
\begin{equation}
\label{decay}
\Gamma=4b^2(1+r)H
\end{equation}
Using eq. (\ref{decay}) in eq. (\ref{omlef}) we obtain:
\begin{equation}
\label{omleffin}
\omega^{eff}_{\Lambda}=\frac{b^2-2}{1+\Omega_{\Lambda}}.
\end{equation}
The correspondence between holographic dark energy and GCG in KK model enables us to equate eq. (\ref{rhoa}) and eq. (\ref{rhohol}), which gives:
\begin{equation}
\label{A1}
\left[A+\frac{B}{a^{4(1+\alpha)}}\right]^{\frac{1}{1+\alpha}}=3c^2\pi^2L^2.
\end{equation}
Using eq. (\ref{rhohol}) and eq. (\ref{omlef}), we compute $A$, which is given by :
\begin{equation}
\label{afin}
A=\frac{b^2-2\Omega_{\Lambda}}{\Omega_{\Lambda}(1+\Omega_{\Lambda})}(3c^2\pi^2H^{-2})^{1+\alpha}
\end{equation}
Once again using eq. (\ref{afin}) in eq. (\ref{rhoa}), and comparing it with eq. (\ref{rhohol}) we determine $B$, which is given by:
\begin{equation}
\label{bfin}
B = \frac{\Omega_{\Lambda}^2-\Omega_{\Lambda}-b^2}{\Omega_{\Lambda}(1+\Omega_{\Lambda})}(3c^2\pi^2H^{-2}a^4)^{(1+\alpha)}.
\end{equation}
\section{Squared Speed Of Sound in Chaplygin Gas and Stability of the Model}
\begin{figure*}
\subfloat[Part 1][Variation of $\omega^{eff}_{\Lambda}$]{\includegraphics[width=0.4\textwidth]{efs.pdf} \label{oeff}} \qquad
\subfloat[Part 2][Variation of Sqared speed for GCG ($v_{\Lambda}$)]{\includegraphics[width=0.4\textwidth]{sqv.pdf} \label{sqvf}}\\
\caption[]{(Colour Online)Variation of (a) Effective dark energy equation of state and (b) squared speed for GCG is shown with $\Omega_{\Lambda}$For different choices of $b^2$ }
\end{figure*}
Stability of the GCG model may be analyzed determining the squared speed for GCG which is defined as:
\begin{equation}
\label{sqgcg}
v_g^2=\frac{dp_{\Lambda}}{d\rho_{\Lambda}}.
\end{equation}
The GCG model is unstable if $v_g^2 < 0$ \cite{setarehol}. For our model of holographic interacting GCG:
\begin{equation}
\label{sqgcgchol}
v_{\Lambda}^2=\frac{dp_{\Lambda}}{d\rho_{\Lambda}}=\frac{\dot{p}}{\dot{\rho}}.
\end{equation}
In this case $\dot{p}$ is given by:
\begin{equation}
\label{dotp}
\dot{p}=\dot{\omega^{eff}_{\Lambda}}\rho_{\Lambda}+\omega^{eff}_{\Lambda}\dot{\rho_{\Lambda}}
\end{equation}
where the over dot means differentiation with respect to time. So the expression for squared speed becomes:
\begin{equation}
\label{sqv}
v_{\Lambda}^2=\omega^{eff}_{\Lambda}+\dot{\omega^{eff}_{\Lambda}} \frac{\rho_{\Lambda}}{\dot{\rho_{\Lambda}}}.
\end{equation}
Using eq. (\ref{omleffin}) one would obtain:
\begin{equation}
\label{dotom}
\dot{\omega^{eff}_{\Lambda}}=-\frac{b^2-2}{(1+\Omega_{\Lambda})^2} \dot{\Omega_{\Lambda}},
\end{equation}
where $\dot{\Omega_{\Lambda}}$ is determined from eqs. (\ref{dpdef}) and (\ref{rhohol}). Using the relation $L=\frac{1}{H}$, we get:
\begin{equation}
\label{dotom2}
\dot{\Omega_{\Lambda}}=-2c^2\pi^2\frac{\dot{H}}{H^5}.
\end{equation}
Using the values for $\dot{\omega^{eff}_{\Lambda}}$ and $\dot{\Omega^{eff}_{\Lambda}}$ in eq. (\ref{sqv}), one obtains:
\begin{equation}
\label{sqvfin}
v_{\Lambda}^2=\omega^{eff}_{\Lambda}-\frac{c^2\pi^2(b^2-2)}{H^4(1+\Omega_{\Lambda})}=\omega^{eff}_{\Lambda}(1-2\Omega_{\Lambda}).
\end{equation}
Recent predictions from various observations puts a value on $\Omega_{\Lambda}\approx 0.73$, which will be used here. From eq. (\ref{sqvfin}) it would give $v_{\Lambda}^2=-0.46\omega^{eff}_{\Lambda}$. But $\omega^{eff} <0$ since it represents a form of dark energy, which ensures that $v_{\Lambda}^2>0$. Thus it is evident that our GCG model is stable at the present epoch.
\section{Discussion}
In this letter, a holographic dark energy (HDE) model is obtained considering interacting generalized Chaplygin Gas in KK Cosmology. Recently, Sharif and Khanum showed that interacting dark energy models can be proposed in the framework of compact KK cosmology. It is also shown that generalized second law of thermodynamics (GSLT) holds good at all time in the HDE model considered here. The above properties holds good in the case of interacting dark energy model described by GCG. It is seen from eq. (\ref{omleffin}) that the decay rate or the interaction term may be computed at the present epoch for a given $\omega^{eff}_{\Lambda}$. For example if we chose $\omega^{eff}_{\Lambda}=-1$ at the present epoch it corresponds to $b^2=0.27$. Naturally for $b^2<0.27$ the dark energy behaves like phantom. Since $\Omega_{\Lambda}$ is different for different epoch $b^2$ will take different values. Thus the dark energy might have evolved from a phantom phase at early epoch. This is evident from fig. (\ref{oeff}). It is to note that HDE model for Interacting GCG is unstable in standard GR (as shown by Setare in \cite{setarehol} ) but in KK cosmology a stable model for the same is possible at the present epoch. This is evident from fig. (\ref{sqvf}) where stability is studied by a plot of $v_{\Lambda^2}$ (squared speed) with $\Omega_{\Lambda}$ for various choices of $b^2$.The squared speed of sound in the case of GCG is not very much sensitive to $b^2$ values and becomes positive at around $\Omega_{\Lambda}\approx0.5$ for a range of $b^2$ values.
\begin{acknowledgement}
SG thanks CSIR for providing Senior Research Fellowship. BCP is thankful to IUCAA for providing visiting Associateship where the a part of the work is carried out. AS and RD are thankful to IRC, Physics Department, NBU for hospitality and extending facilities to complete the work.
\end{acknowledgement}
\bibliographystyle{spphys}
|
{
"timestamp": "2013-03-12T01:04:14",
"yymm": "1203",
"arxiv_id": "1203.2113",
"language": "en",
"url": "https://arxiv.org/abs/1203.2113"
}
|
\section{}
\section{Introduction}
Zero energy bound states in vortex cores of superconductors have been of much current interest in condensed matter physics.
Some classes of the vortex Majorana states, obeying non-Abelian statistics,
may serve as qubits for quantum computation. \cite{Kitaev,Nayak,Goldstein,BWT}
There are various theoretical proposals for realizing non-Abelian Majorana fermions in the core of vortices in topological superconductors,
e.g., a chiral $p$-wave superconductor, etc.\cite{RG,MR,Sau,Ivanov,FK,MTF,Alicea,TK2,Sato,Santos}
Besides such vortex zero modes, topological superconductor junction systems, in which the order parameter changes sharply in real space, possess generically
non-Abelian Majorana fermions.\cite{FK}
A useful method for charactering the existence of the zero energy bound states localized at point defects such as vortices or point intersections consisting of the junction interfaces
is the index theorem for an open infinite space,
derived by Callias \cite{Callias} and by Weinberg \cite{Weinberg}, and generalized by Niemi-Semenoff. \cite{NS1}
This theorem reveals the relationship between the zero energy bound states and the topology of background fields at large distance from the point defects.
In this paper, we investigate the index theorem for the heterostructure system involving the topological insulator (TI), mainly focusing on the superconductor-TI-ferromagnet insulator junctions.
We find that the number of Majorana zero modes is controlled not only by the phase winding of the superconducting gap, but also by non-topological massive bound states localized at the junctions.
The organization of this paper is as follows.
In Sec.\ref{sec2}, we first present our main results for the index of the superconductor-TI-ferromagnet insulator junctions,
and discuss its physical implications.
We give, in particular, a physical explanation on how non-topological massive bound states
affect the index for Majorana zero energy modes.
Our results are based on the celebrated Niemi-Semenoff index theorem.
To make this paper self-contained,
we briefly review the Niemi-Semenoff index theorem in Sec.\ref{sec3}.
In Sec.\ref{sec4}, we apply the Niemi-Semenoff index theorem to the superconductor-TI-ferromagnet insulator junctions,
and obtain the index theorem for topological heterostructure systems.
In Sec.\ref{sec5}, we also apply our results to
topological insulator-ferromagnetic insulator heterostructure systems.
We conclude in Sec.\ref{sec6} with some discussions.
\section{Setup and main results}
\label{sec2}
We consider the heterostructure system composed of an $s$-wave superconductor $\pi$-junction and ferromagnetic insulators
placed on a TI, as depicted in Fig. \ref{fig1}(a), and investigate the zero energy bound state localized at a point-like defect formed by the intersection
of the $\pi$-junction interface and the ferromagnetic domain wall.
The effective Bogoliubov-de Gennes Hamiltonian is written as
\begin{equation}
\begin{split}
\mathcal{H} &= -i v \tau_3 \sigma_j \partial_j + \Delta_1 \tau_1 + \Delta_2 \tau_2 + \bm{h} \cdot \bm{\sigma} -\mu \tau_3, \\
\end{split}
\label{eq11.5}
\end{equation}
where $j = 1,2 $, $\bm{\tau}= (\tau_1, \tau_2, \tau_3)$ and $\bm{\sigma} = (\sigma_1,\sigma_2, \sigma_3)$ are the Pauli matrices for the Nambu and spin space respectively,
$v$ is the velocity of the Dirac fermion,
$\Delta_1$ and $\Delta_2$ are the real and imaginary parts of the gap function, $\bm{h} \cdot \bm{\sigma}$ is the Zeeman term, and $\mu$ is the chemical potential.
Note that for the $\pi$-junction considered here, $\Delta_2=0$.
It is also assumed that the thickness of the superconducting film is sufficiently smaller than the penetration depth,
and hence $z$-dependence of $\bm{h}$ is negligible.
This system (\ref{eq11.5}) belongs to class D in the Altland-Zirnbauer symmetry classes\cite{Schnyder,Kitaev2},
and the vortex zero modes obeying non-Abelian statistics \cite{FK} are classified as the $\mathbb{Z}_2$ invariant.
The index theorem which is a main tool in this paper is applicable only to systems with chiral symmetry,
i.e. $\Pi\mathcal{H}\Pi^{\dag}=-\mathcal{H}$ is satisfied for a unitary operator $\Pi$.
This symmetry is, however, not preserved for (\ref{eq11.5}), because of $h_3\sigma_3$ and $-\mu\tau_3$ terms.
Nevertheless, as will be clarified below, the $\mathbb{Z}_2$ invariant of (\ref{eq11.5}) can be generically obtained from the index calculated for the case with $h_3=\mu=0$.
Thus, we first neglect these two terms to calculate the index.
We furthermore omit the $h_1\sigma_1$ term to simplify the analysis,
since this term does not affect the index of our system, as long as
$h_1$ is sufficiently small, and does not close the bulk energy gap.
Then, the Hamiltonian is reduces to that
with chiral symmetry (class BDI), $\tau_3 \sigma_3 \mathcal{H} \tau_3 \sigma_3 = -\mathcal{H}$,
\begin{equation}
\begin{split}
\mathcal{H} &= -i v \tau_3 \sigma_j \partial_j + \Delta_1 \tau_1
+ h_2 \sigma_2, \\
\end{split}
\label{eq12}
\end{equation}
and its ground state is classified by $\mathbb{Z}$. This enhanced topological number can be computed by the index theorem described in detail in the
next section.
Note here that particle-hole symmetry, $\tau_2 \sigma_2 \mathcal{H}^* \tau_2 \sigma_2 = -\mathcal{H}$, valid for (\ref{eq11.5}) as well as (\ref{eq12}),
ensures that the number of vortex zero modes are conserved modulo 2 even if the
neglected chiral-symmetry-breaking terms are switched on again. Thus, the $\mathbb{Z}_2$ invariant of (\ref{eq11.5})
can be derived from
the parity of the index of (\ref{eq12}).
\begin{figure}[!b]
\begin{center}
\includegraphics[width=\linewidth, trim=0 0cm 0 0cm]{Fig4_1.eps}
\end{center}
\caption{(a) The heterostructure geometry for an $s$-wave superconductor (SC) $\pi$-junction and a ferromagnetic insulator (FMI) on the surface of a topological insulator.
A filled circle at the interface is a point defect formed by the intersection of the $\pi$-junction and the ferromagnetic domain wall.
(b) The heterostructure geometry for a topological insulator-ferromagnetic insulator tri-junction. }
\label{fig1}
\end{figure}
Our central finding is that the heterostructure system composed of
an $s$-wave superconductor $\pi$-junction and ferromagnetic insulators on a topological insulator
(as shown in Fig.\ref{fig1}(a))
described by (\ref{eq12})
has the index:
\begin{equation}
\begin{split}
\mathrm{ind}\, \mathcal{H} =& \frac{1}{2} \left[ \mathrm{sign}(h_+) - \mathrm{sign}(h_-) \right] \\
&+ \mathrm{sign}(h_+) N_{x \rightarrow \infty} - \mathrm{sign}(h_-) N_{x \rightarrow -\infty},
\end{split}
\label{eq00}
\end{equation}
where $\mathrm{sgn}(h_{\pm})$ is the sign of the asymptotic Zeeman field $h_2(x \rightarrow \pm \infty,y)$,
and $N_{x \rightarrow \pm \infty}$ are integer numbers which count how many times the band inversion
occurs for massive bound states at the $\pi$-junction, as the Zeeman magnetic field increases from zero to
$h_2(x \rightarrow \pm \infty,y)$. (see Fig. \ref{fig0} and discussion given at the end of this section.)
Note that in Eq.(\ref{eq00}),
we take the origin of the $xy$ coordinate $(x,y)=(0,0)$ at the location of the point defect in Fig.\ref{fig1}(a).
It is also naturally assumed that the sign of $h_2(x \rightarrow \pm \infty,y)$ is independent of $y$.
Especially in the cases of the uniform asymptotic Zeeman field, $h(x \rightarrow \pm \infty, y) \equiv h_{\pm}$,
the index (\ref{eq00}) is simplified to
\begin{equation}
\begin{split}
\mathrm{ind}\, \mathcal{H} =& \frac{1}{2} \left[ \mathrm{sign}(h_+) - \mathrm{sign}(h_-) \right] \\
&+ \left[ \mathrm{sign}(h_+) \sum_{{E_{n}<|h_+|} } - \mathrm{sign}(h_-) \sum_{{E_{n}<|h_-|}} \right],
\end{split}
\label{eq0}
\end{equation}
where $h_{+}$ ($h_{-}$) is a Zeeman field at the $\pi$-junction interface for $x>0$ ($x<0$),
which induces mass gap of the one-dimensional gapless helical Majorana mode localized at the junction interface,
and $E_n (>0) $ denote the absolute value of the mass gaps of the one-dimensional massive modes localized at the junction interface.
The sum in (\ref{eq0}) is taken only for one part of the Kramer's pair.
As mentioned before, the sum in (\ref{eq0}) represents the number of times
the band inversion occurs for the massive bound states, as $h_{\pm}$ increase from zero to finite values.
Because of particle-hole symmetry, this counting can be expressed only by $E_n>0$, as shown in Eq. (\ref{eq0}).
The index (\ref{eq0}) (or (\ref{eq00})) expresses the number of zero energy Majorana bound states in
a point-like defect at the junction of a
chiral-symmetric superconductor (class BDI {in the Altland-Zirnbauer symmetry classes\cite{Schnyder,Kitaev2}}).
The index (\ref{eq0}) is interpreted as the phase winding of the superconducting gap $\Delta$ around the point defect.
In the case of the $\pi$-junction with a Zeeman field as shown in the FIG. \ref{fig1}(a),
the change of the phase of the gap function can be defined in the following way by using the
Teo-Kane's adiabatic argument. \cite{TK,QHZ}
\begin{figure}[!h]
\begin{center}
\includegraphics[width=\linewidth]{Fig8.eps}
\end{center}
\caption{(Color online) A schematic picture of the band inversion of massive bound states.
(a) Black lines denote the energy band of helical Majorana fermion, while red and blue lines denotes that of massive bound states.
Lower panels indicate the right-half part of the heterostructure geometry shown in FIG. \ref{fig1}(a). }
\label{fig0}
\end{figure}
Without a Zeeman field, the $\pi$-junction possesses a helical Majorana fermion localized at the junction interface (FIG. \ref{fig0}(a)).\cite{FK}
The Zeeman field from the ferromagnetic insulator lifts the Kramer's degeneracy, and induces a mass gap of the helical Majorana fermion (FIG. \ref{fig0}(b)).
Adiabatic deformation of the Hamiltonian without closing the energy gap enables us to introduce a nonzero imaginary part
of the superconducting gap
$\Delta_2$ at the junction interface.
In this process, the sign of $\Delta_2$ is determined by a Zeeman field such that $\mathrm{sign} (\Delta_2) = - \mathrm{sign}(h_2) $.
Hence, the phase shift is $- \pi \ \mathrm{sign}(h_2)$, which is described by the first term in (\ref{eq0}).
This contribution depends only on the sign of Zeeman field and does not depend on the detail of the junction interface.
The second term in (\ref{eq0}) is, on the other hand, a new contribution which was not discussed in previous literatures in the context of heterostructure systems
and depends on the detail of the junction interface.
The superconducting gap $\Delta_1$ changes its sign at the $\pi$-junction.
If the spatial variation of the magnitude of $\Delta_1$ in the vicinity of the junction is sufficiently slow,
there exist massive bound states localized at the junction, which come in Kramer's pairs with a mass gap $|E_1|$ (FIG. \ref{fig0}(a)).
The Zeeman field
parallel to $y$-axis shifts the mass gaps of the Kramer's pairs by $|E_1 \pm h_2|$, respectively(FIG. \ref{fig0}(b)).
(Here, we assume the Zeeman field is uniform.
If not, the mass gap of the bound states depends on the detail of Zeeman field. But the qualitative nature is unaffected.)
When $|h_2|$ reaches $|E_1|$, the energy gap at the junction interface is closed (FIG. \ref{fig0}(c)), and a band inversion occurs for $h_2 < - |E_1|$.
After band inversion, the junction interface structure
acquires the $- 2 \pi \ \mathrm{sign}(h_2)$ phase shift in addition to the $- \pi \ \mathrm{sign}(h_2)$ phase shift,
resulting in the total $- 3 \pi \ \mathrm{sign}(h_2)$ phase shift (FIG. \ref{fig0}(d)).
This additional $2 \pi$ phase production arises for each massive bound state with mass gap $|E_2|, |E_3|, \cdots$,
which describes the second term in (\ref{eq0}).
Therefore, massive bound states at the interface give rise to additional phase winding around the point defect formed by the intersection of the $\pi$-junction interface and the ferromagnetic domain wall shown in FIG. \ref{fig1}(a).
This new contribution from the second term of (\ref{eq0})
has an important implication for the class D heterostructure system.
As mentioned before, the class D system is characterized by the $\mathbb{Z}_2$ invariant for the Majorana zero modes,
which is exactly the parity of the index (\ref{eq0}) obtained by switching off chiral-symmetry-breaking terms.
Thus, the $\mathbb{Z}_2$ invariant of the class D heterostructure system
may be changed by the non-topological massive bound states.
This leads to a breakdown of topological protection of Majorana vortex modes when the second term of (\ref{eq0}) is
an odd integer.
\section{Niemi-Semenoff Index Theorem}
\label{sec3}
In this section, for the convenience of readers, we briefly review
the Niemi-Semenoff index theorem
which is used for the derivation of our results in the following sections.
The Niemi-Semenoff index theorem relates the number of zero energy modes in Dirac fermion systems to the geometrical structure of spatially varying mass terms.
In particular, the index is determined by the asymptotic behaviors of mass terms at open boundaries.
We consider the Dirac Hamiltonian with chiral symmetry in $d$-dimensional space with open boundaries,
the Hamiltonian of which is given by,
\begin{equation}
\begin{split}
\mathcal{H} &= -i \Gamma_i \partial_i + Q(\bm{x})
=\begin{pmatrix}
0 & \mathcal{D} \\
\mathcal{D}^{\dag} & 0
\end{pmatrix}, \\
\end{split}
\end{equation}
with $\mathcal{D} = - i \gamma_{i} \partial_{i} + K(\bm{x})$, for the basis that the $\Gamma$-matrices are represented as
\begin{equation}
\begin{split}
\Gamma_i =
\begin{pmatrix}
0 & \gamma_i \\
\gamma_i^{\dag} & 0
\end{pmatrix}, \
\Gamma_5 =
\begin{pmatrix}
1 & 0 \\
0 & -1
\end{pmatrix}.
\end{split}
\label{eq2}
\end{equation}
Here, the indices $i = 1,2,\dots, d$ are those for the spatial coordinates,
the $\gamma_{i}$ matrices are constant matrices that satisfy
$\gamma_i \gamma_j^{\dag} + \gamma_j \gamma_i^{\dag} = \gamma_i^{\dag} \gamma_j + \gamma_j^{\dag} \gamma_i= 2 \delta_{ij}$,
and $K(\bm{x})$ includes all background fields such as electromagnetic fields and the superconducting gap.
The index of the Hamiltonian is defined by
$\mathrm{ind}\,\mathcal{H} := \mathrm{dim\ ker} \ \mathcal{D}^{\dag}- \mathrm{dim\ ker} \ \mathcal{D}$,
which is the difference between the number of zero energy states of $\mathcal{H}$ with the opposite chirality.
We assume all background fields are asymptotically independent of the normal coordinate,
$\hat n_i(\bm{x}) \partial_i Q(x) \rightarrow 0 \ (|\bm{x}| \rightarrow \infty)$,
where $\hat n(\bm{x})$ is a unit vector normal to an open boundary at $|\bm{x}|\rightarrow \infty$.
It is known that $\mathrm{ind}\ \mathcal{H}$ is expressed as the sum of the volume integral of
the chiral anomaly and the surface integral of the chiral current, \cite{Weinberg,NS1}
\begin{equation}
\begin{split}
&\mathrm{ind} \ \mathcal{H} \\
&= \int d^d \bm{x} \ \mathrm{tr} \Braket{\bm{x}|\Gamma_5|\bm{x}} + \frac{1}{2} \oint d \hat S \ \mathrm{tr} \Braket{\bm{x} | i \hat \Gamma(\bm{x}) \Gamma_5 \mathcal{H}^{-1} | \bm{x} },
\end{split}
\label{eq5}
\end{equation}
where $d \hat S$ is the volume element of the boundary,
and $\hat \Gamma(\bm{x}) := \hat{n}_i(\bm{x}) \Gamma_i$.
The definition of the terms of r.h.s. in (\ref{eq5}) needs appropriate regularization.
In this paper, we symbolically use the expression of r.h.s. in (\ref{eq5}).
The first term in (\ref{eq5}) is the integrated chiral anomaly which is present only in even spatial dimensions.
When $d=2$, it is explicitly written in terms of the background field $Q(\bm{x})$ as \cite{Weinberg}
\begin{equation}
\begin{split}
\int d^2 \bm{x} \ \mathrm{tr} \Braket{\bm{x}|\Gamma_5|\bm{x}} = - \frac{1}{4 \pi} \int d^2 \bm{x} \mathrm{tr}\ i \Gamma_5 \Gamma_i \partial_i Q(\bm{x}).
\end{split}
\label{eq6}
\end{equation}
This formula will be used later. (see Eq.(\ref{eq14}) below)
The second term in (\ref{eq5}) is the boundary integral of the chiral current density normal to the boundary,
and this term can be rewritten as the spectral asymmetry constructed from the real part of the eigenvalues of a certain boundary operator $\mathcal{M}$ as shown below,
\begin{equation}
\begin{split}
&\mathrm{tr}\ \Braket{\bm{x} | i \hat \Gamma(x) \Gamma_5 \mathcal{H}^{-1} |\bm{x}} \\
&= \mathrm{tr}\ \Braket{\bm{x} | i \begin{pmatrix}
0 & \hat \gamma(\bm{x}) \\
\hat \gamma^{\dag}(\bm{x}) & 0
\end{pmatrix}
\begin{pmatrix}
1 & 0 \\
0 & -1
\end{pmatrix}
\begin{pmatrix}
0 & \mathcal{D} \\
\mathcal{D}^{\dag} & 0
\end{pmatrix}^{-1} |\bm{x}} \\
&= \mathrm{tr}\ \Braket{\bm{x} | \left( i \hat \gamma^{\dag}(\bm{x}) \mathcal{D} \right)^{-1} + \left[ \left( i \hat \gamma^{\dag}(\bm{x}) \mathcal{D} \right)^{-1} \right]^{\dag}|\bm{x}} .
\end{split}
\end{equation}
Here $\hat \gamma(x) = \hat{n}_i(x) \gamma_i$ are the normal components of $\gamma$ matrices,
and we used the cyclicity of trace.
The boundary operator $\mathcal{M}$ is defined by $i \hat \gamma^{\dag}(x) \mathcal{D} = \hat \partial + \mathcal{M}$,
\begin{equation}
\begin{split}
\mathcal{M} = \hat \gamma^{\dag}(\bm{x}) \gamma^T_i \partial_i + i \hat \gamma^{\dag}(\bm{x}) K(\bm{x}), \\
\end{split}
\label{eq8}
\end{equation}
where $\gamma^T_{i}(\bm{x})= \gamma_i - \hat \gamma(\bm{x}) \hat{n}_i(\bm{x})$ are the tangential components of $\gamma$ matrices,
and $\hat \partial = \hat n_i(\bm{x}) \partial_i$ is the directional derivative normal to the boundary.
We assume $\mathcal{H}$ does not possess zero modes at infinity, which corresponds to the absence of zero modes in $\mathcal{M}$.
Since $\mathcal{M}$ is independent of the coordinate normal to the boundary,
we can introduce the Fourier transformation for the normal coordinate:
\begin{equation}
\begin{split}
&\frac{1}{2} \oint d \hat S \mathrm{tr} \Braket{\bm{x} | i \hat \Gamma \Gamma_5 \mathcal{H}^{-1} | \bm{x} } \\
&= \frac{1}{2} \oint d \hat S \mathrm{tr} \Braket{\bm{x} | \frac{1}{\mathcal{M} + \hat \partial} + \frac{1}{\mathcal{M}^{\dag} - \hat \partial} | \bm{x} } \\
&= \frac{1}{4 \pi} \int_{-\infty}^{\infty} d \hat k \oint d \hat S \mathrm{tr} \Braket{\bm{x} | \frac{1}{\mathcal{M} + i \hat k} + \frac{1}{\mathcal{M}^{\dag} - i \hat k} | \bm{x} }. \\
\end{split}
\label{eq9_1}
\end{equation}
Introducing the eigenmodes $\mathcal{M} \phi = \lambda \phi$ and $\mathcal{M}^{\dag} \psi = \lambda^* \psi$,
we rewrite Eq. (\ref{eq9_1}) as
\begin{equation}
\begin{split}
&\frac{1}{4 \pi} \int_{-\infty}^{\infty} d \hat k \int d \lambda \ \rho(\lambda) \left( \frac{1}{\lambda + i \hat k} + \frac{1}{\lambda^* - i \hat k} \right)\\
&= \frac{1}{2} \int d \lambda \ \rho(\lambda) \mathrm{sign} \left[ \mathrm{Re}(\lambda) \right] \\
&=: \frac{1}{2} \eta \left( \mathrm{Re} \left( \mathcal{M} \right) \right),
\end{split}
\label{eq10}
\end{equation}
where $\rho(\lambda)$ is the spectral density of boundary operator $\mathcal{M}$.
This term is the spectral asymmetry constructed from the real part of the eigenvalues of $\mathcal{M}$.
Eventually, $\mathrm{ind}\,\mathcal{H}$ is written as \cite{NS1}
\begin{equation}
\begin{split}
\mathrm{ind}\, \mathcal{H}
= \int d^d x \ \mathrm{tr} \Braket{\bm{x}|\Gamma_5|\bm{x}} + \frac{1}{2} \eta \left( \mathrm{Re} \left( \mathcal{M} \right) \right).
\end{split}
\label{eq11}
\end{equation}
This is the Niemi-Semenoff index theorem for an open infinite space. \cite{NS1}
The integrand of the anomaly contribution is generally the total derivative.
Hence $\mathrm{ind} \ \mathcal{H}$ depends solely on the asymptotic behavior of background fields.
\section{Majorana zero modes at a point defect in superconductor-ferromagnet insulator heterostructure systems}
\label{sec4}
In this section,
we derive the index (\ref{eq0}) for the topological heterostructure system depicted in Fig. \ref{fig1}(a)
by applying the Niemi-Semenoff index theorem explained in the previous section.
For this purpose, we first obtain the boundary operator (\ref{eq8}) for the Hamiltonian (\ref{eq12}).
This is achieved by the following procedure.
By applying the unitary transformation,
\begin{equation}
\begin{split}
\Gamma_5 = \tau_3 \sigma_3 = \begin{pmatrix}
\sigma_3 & 0 \\
0 & -\sigma_3
\end{pmatrix}
\mapsto U \tau_3 \sigma_3 U^{\dag} = \begin{pmatrix}
1 & 0 \\
0 & -1
\end{pmatrix}
\end{split}
\end{equation}
with
\begin{equation}
\begin{split}
U = \begin{pmatrix}
1 & & & \\
& & & 1 \\
& & 1 & \\
& 1 & &
\end{pmatrix},
\end{split}
\end{equation}
the Hamiltonian (\ref{eq12}) is represented as
\begin{widetext}
\begin{equation}
\begin{split}
\mathcal{H} \mapsto
\begin{pmatrix}
0 & v ( \sigma_2 \partial_x - \sigma_1 \partial_y) + h_2 \sigma_2 + \Delta_1 \\
-v ( \sigma_2 \partial_x - \sigma_1 \partial_y) + h_2 \sigma_2 + \Delta_1 & 0
\end{pmatrix}.
\end{split}
\end{equation}
\end{widetext}
In this representation, $( \gamma_1 , \gamma_2 ) = (i\sigma_2, -i \sigma_1) $, and $K(x,y)= h_2 \sigma_2 + \Delta_1$.
Then, the boundary operator $\mathcal{M}$ defined by (\ref{eq8}) is
\begin{equation}
\begin{split}
\mathcal{M} = i v \sigma_3 \partial_T + \Delta_1 \sigma_T
+ h_T - i \hat h \sigma_3,
\end{split}
\label{eq13}
\end{equation}
where $\hat a = \hat n_i a_i$ and $a_T = n^T_i a_i$ are components of a vector $\bm{a} = (a_1,a_2)$
which are, respectively, normal and tangential to an open boundary at $|x|\rightarrow \infty $ or $|y|\rightarrow \infty$.
(We note that the origin of the $xy$ coordinate $(x,y)=(0,0)$ is taken at the position of the point defect in Fig. \ref{fig1}(a).)
From (\ref{eq6}), the anomaly part of the index is
\begin{equation}
\begin{split}
\int d^2 \bm{x} \mathrm{tr} \Braket{\bm{x}|\Gamma_5|\bm{x}}
&= \frac{1}{\pi} \int d^2 \bm{x} \epsilon_{i j} \partial_i h_j = \frac{1}{\pi} \oint d \bm{l} \cdot \bm{h}.
\end{split}
\label{eq14}
\end{equation}
To simplify the analysis, we consider the kink structure for the gap function $\Delta_1(y) = \Delta \tanh(y / \xi)$ with $\Delta>0$ at the interface of $\pi$-junction.
As we shall see momentarily, the parameter $\xi$ in the gap function describing the width of the kink gives rise to a crucial effect on the index.
Here, it is natural to assume that
the asymptotic value of the magnetic field,
$h_2(x,y) \rightarrow h_{\pm}(y)$ as $x \rightarrow \pm \infty$,
have nonzero values with definite signs, $\mathrm{sign}(h_{\pm})$,
near the interface of the $\pi$-junction.
On the other hand, the magnitudes of
$h_{\pm}(y) $ at $y \rightarrow \pm \infty$ does not affect the index of our system.
Thus, for simplicity, we assume $h_{\pm}(y) \rightarrow 0$ for $y \rightarrow \pm \infty$.
Now we calculate the spectral asymmetry $\eta(\mathrm{Re}(\mathcal{M}))$ for the heterostructure system
depicted in Fig. \ref{fig1}(a).
It follows from (\ref{eq13}) that the boundary operators $\mathcal{M}$ for $x \rightarrow \pm \infty$
and $y \rightarrow \pm \infty$
are given by,
\begin{align}
&\mathcal{M}_{x \rightarrow \infty}(y) = i v \sigma_3 \partial_y + \Delta_1(y) \sigma_2 + h_+(y), \label{boundary_1} \\
&\mathcal{M}_{x \rightarrow -\infty}(y) = -i v \sigma_3 \partial_y - \Delta_1(y) \sigma_2 - h_-(y), \label{boundary_2} \\
&\mathcal{M}_{y \rightarrow \infty}(x) = -i v \sigma_3 \partial_x - \Delta \sigma_1, \\
&\mathcal{M}_{y \rightarrow -\infty}(x) = i v \sigma_3 \partial_x - \Delta \sigma_1 .
\end{align}
The spectral asymmetry (\ref{eq10}) is the sum of the partial spectral asymmetries of four sides.
Each boundary operator is hermitian,
since the Zeeman field normal to the boundary in (\ref{eq13}) vanishes.
The spectral asymmetries for $\mathcal{M}_{y \rightarrow \pm \infty}$ are zero,
since the eigenvalues of $\mathcal{M}_{y \rightarrow \pm \infty}$ come in pairs $\pm \lambda$
due to ``chiral'' symmetry $\sigma_2 \mathcal{M}_{y \rightarrow \pm \infty} \sigma_2 = -\mathcal{M}_{y \rightarrow \pm \infty}$.
To calculate the spectral asymmetries for ${\cal M}_{x\rightarrow\pm\infty}$,
we exploit an approach developed by Lott: \cite{Lott,NS2}
Let $\mathcal{H_{\tau}}$ be a one-parameter family of Hamiltonians defined in $\tau \in [0, 1]$ which interpolate between a reference Hamiltonian ${\cal H}_0$ and ${\cal H}_1={\cal M}_{x\rightarrow\pm\infty}$.
In the calculation of the spectral asymmetry, we choose the reference Hamiltonian ${\cal H}_0$ for which
the spectral asymmetry $\eta(\mathcal{H}_0)$ is known.
The variance of the spectral asymmetry as a function of $\tau$ is composed of two parts: one is the
continuum part $\eta^c_{\tau}$ raised by the change of high energy continuum energy spectrum, and
the other is a discrete part arising from the spectral flow which changes by $\pm 2$ when a discrete eigenvalue $\lambda_n(\tau)$ crosses zero from
negative (positive) to positive (negative) energies: $\Delta \left[ \mathrm{sign}(\lambda_n(\tau)) \right] = \pm 2$.
Thus, we can write the spectral asymmetry in the form,
\begin{equation}
\begin{split}
\eta (\mathcal{H}) = \eta (\mathcal{H}_0) + \int d \tau \frac{d \eta^c_{\tau}}{d \tau} + 2 (\text{spectral flow}).
\end{split}
\label{eq20}
\end{equation}
Let us consider the spectral asymmetry for $x \rightarrow \infty$. In this case, the boundary operator ${\cal M}_{x\rightarrow\infty}$ is basically the Jackiw-Rebbi Hamiltonian.
Therefore, in the absence of a magnetic field, $h_+(y) \equiv 0$, $\mathcal{M}_{x \rightarrow \infty}$ possesses the Jackiw-Rebbi zero mode localized at the interface of $\pi$-junction,
$\phi_0(y)\propto {}^t (1,-1) e^{-\int^y \Delta_1(y')/v d y'}$.
When a magnetic field is switched on,
a finite value of $h_+(y)$ shifts the bound state energy from zero to a nonzero value with the same sign as $h_+$.
The spectral asymmetry for this boundary operator was previously computed by Lott \cite{Lott}.
Using the reference Hamiltonian given by
$\mathcal{H}_0 = i v \sigma_3 \partial_y + \Delta_1(y) \sigma_2 - \delta \mathrm{sign}(h_+) \sigma_1$
with a small positive constant $\delta$,
we obtain,
\begin{equation}
\begin{split}
&\frac{1}{2} \eta(\mathcal{M}_{x \rightarrow \infty}) \\
&= \frac{1}{2} \mathrm{sign} (h_+) - \frac{1}{\pi} \int d y h_+(y) + (\text{spectral flow}).
\end{split}
\label{eq21}
\end{equation}
The first term conforms to the fermion fractionalization in the Jackiw-Rebbi system.\cite{JR,GW}
The second term is the volume part of the variation of $\eta_{\tau}$, which cancels out the anomaly contribution of the index (\ref{eq14}).
The third term is the spectral flow contribution from $\mathcal{H}_0$ to $\mathcal{M}_{x \rightarrow \infty}$ which depends on the structure of $\Delta_1(y)$ and $h_+(y)$.
The spectral flow stems from the bound states localized at the interface of the $\pi$-junction.
Since the Lott's derivation of (\ref{eq21}) in ref.\onlinecite{Lott}
is highly technical, we give a more elementary derivation of (\ref{eq21}) in Appendix, which we believe is useful for readers.
If the finite value region of $h_+(y)$ is much longer than $\xi$, $h_+(y)$ is approximated as a constant chemical potential, $h_+(y) \rightarrow h_+$.
In this case, the $y$-dependent part of $\mathcal{M}_{x \rightarrow \infty}$, i.e., $i v \sigma_3 \partial_y + \Delta \tanh(y/\xi)$
is exactly solvable \cite{SK,Takayama}
The eigenvalues of this Hamiltonian are $E_0 = 0$,
\begin{equation}
\begin{split}
E_{n,\pm} = \pm \Delta \sqrt{\frac{n}{\nu} \left( 2-\frac{n}{\nu}\right)} , (n = 1,2,\dots,<\nu), \\
\end{split}
\label{eq22}
\end{equation}
and $E_{p,\pm} = \pm \sqrt{v^2 p^2 + \Delta^2} ,\ (p \in \mathbb{R})$, where $\nu = \xi \Delta/v = \xi/\xi_c$ is a ratio of $\xi$ to the coherence length of the superconducting state : $\xi_c = v/\Delta$.
$E_0$ and $E_{n,\pm}$ are the energy of the bound states localized at the interface of $\pi$-junction.
The Jackiw-Rebbi zero energy bound state exists for arbitrary $\xi$,
while the massive bound states exist only when $\xi > \xi_c$.
\begin{figure}[!]
\begin{center}
\includegraphics[width=0.9\linewidth, trim=0.5cm 2.0cm 0.5cm 2.3cm]{conductance_ferro_3.eps}
\end{center}
\caption{The $\tau$-dependent energy spectrum of a one parameter family of Hamiltonians $\mathcal{H}_{\tau}$:
$\mathcal{H}_{\tau} = (1-\tau) \mathcal{H}_0 + \tau \mathcal{H}$.
This figure shows the case of $\xi = 5 \xi_c$, $h_+ = 0.7 \Delta> 0$, and $\delta = 0.05 \Delta$.
}
\label{fig2}
\end{figure}
$h_+$ induces constant shifts to eigenvalues (\ref{eq22}),
and hence, the spectral flow is given by the bound states that cross zero between
$E_{n\pm} $ and $E_{n\pm} + h_+ $ as shown in Fig. \ref{fig2}:
\begin{equation}
\begin{split}
(\text{spectral flow}) = \mathrm{sign}(h_+) \sum_{{E_{n}<|h_+|} },
\end{split}
\label{eq23}
\end{equation}
where
$E_n=E_{n,+}$.
Note that due to the term $- \delta \mathrm{sign} (h_+) \sigma_1$ in the reference Hamiltonian $\mathcal{H}_0$,
the spectral flow from the Jackiw-Rebbi zero energy bound state is excluded in the sum (\ref{eq23}).
In a similar way, the spectral asymmetry of $\mathcal{M}_{x \rightarrow -\infty}$, $\frac{1}{2} \eta(\mathcal{M}_{x \rightarrow -\infty})$ can be calculated as
\begin{equation}
\begin{split}
- \frac{1}{2} \mathrm{sign} (h_-) +\frac{1}{\pi} \int d y h_-(y) - \mathrm{sign}(h_-) \sum_{{E_{n}<|h_-|}}.
\end{split}
\label{eq24}
\end{equation}
Using Eqs. (\ref{eq11}), (\ref{eq14}), (\ref{eq21}), (\ref{eq23}) and (\ref{eq24}) together,
we arrive at the formula (\ref{eq0}).
So far we have calculated the index for (\ref{eq12}).
The $\mathbb{Z}_2$ index $N$ for the vortex zero modes in class D heterostructure system (\ref{eq11.5}) is, as we have mentioned, given by $N = \mathrm{ind} \, \mathcal{H} \ (\mathrm{mod} \ 2)$.
Remarkably, the second term in (\ref{eq0}), which is basically the contribution from non-topological bound states of
the Jackiw-Rebbi Hamiltonian, can affect the $\mathbb{Z}_2$ index.
This indicates that the existence of the non-Abelian vortex zero modes depends on the kink-structure of the gap function parametrized by $\xi$.
Actually, in the case that the second term of (\ref{eq0}) is equal to an odd integer, which can indeed occur when $\xi > \xi_c$ (i.e. $\nu>1$),
the $\mathbb{Z}_2$ invariant for Majorana zero modes is changed,
resulting in the breakdown of the topological protection for the $\mathbb{Z}_2$ Majorana modes.
We now discuss the condition for which $\xi > \xi_c$ is realized.
Actually, to determine $\xi$ precisely, we need to solve the Bogoliubov-de-Gennes equation
for proper boundary conditions, which is out of the scope of this paper.
Instead of presenting such precise analysis, we here give a qualitative argument.
For the superconductor-ferromagnetic insulator junction as depicted in Fig. 1 (a), the gap function at the junction
is reduced by magnetic scattering at the interface between the superconductor and the ferromagnet.\cite{toku}
On the other hand, the dimension of the ferromagnet insulator along the $y$-axis denoted as $L_y$
plays the role of the characteristic length scale for the spatial variation of the exchange field along the $y$-axis.
Thus, when $L_y$ is sufficiently larger than the coherence length $\xi_c$,
we can neglect the spatial variation of the superconducting gap
raised by magnetic scattering near the interface, and hence it is expected that $\xi < \xi_c$ is satisfied, ensuring the topological protection of Majorana modes.
However, when $L_y$ is comparable to $\xi_c$, the spatially inhomogeneous
reduction of the superconducting gap due to magnetic scattering
crucially affects the magnitude of the parameter $\xi$.
In particular, when $L_y$ is slightly larger than $\xi$,
it may be possible that $\xi > \xi_c$ is realized, which leads to
the above-mentioned mechanism of the breakdown of $\mathbb{Z}_2$ nontriviality.
\section{Zero modes in line defects of topological insulator-ferromagnet insulator heterostructure junctions}
\label{sec5}
The index theorem (\ref{eq0}) is also applicable to
a topological insulator-ferromagnetic insulator tri-junction system, the setup of which is depicted in Fig. \ref{fig1} (b).
The two orbital effective Dirac model for this system is written as \cite{QHZ}
\begin{equation}
\begin{split}
\mathcal{H} = v k_z \mu_1 \sigma_3 -i v \mu_1 \sigma_j \partial_j + m(y) \mu_3 + h_2(x,y) \sigma_2 -\mu ,
\end{split}
\end{equation}
where $j = 1,2 $, $\bm{\mu}= (\mu_1, \mu_2, \mu_3)$ and $\bm{\sigma} = (\sigma_1,\sigma_2, \sigma_3)$ are the Pauli matrices for the orbital and the spin spaces, respectively,
$v$ is the velocity of the Dirac fermion, $m$ is the mass gap whose sign determines whether the system is in a topological $(m<0)$ or trivial $(m>0)$ phase, $\bm{h} \cdot \bm{\sigma}$ is a Zeeman term, and $\mu$ is the chemical potential.
We have assumed the translational invariance along the $z$-direction.
This system belongs to the class A, and the chiral gapless modes localized at line defects are classified as $\mathbb{Z}$.\cite{TK}
As in the case of the superconductor-ferromagnet insulator junction,
we, first, neglect chiral-symmetry breaking terms, putting $\mu=0$:
\begin{equation}
\begin{split}
\mathcal{H} &=v k_z \mu_1 \sigma_3 -i v \mu_1 \sigma_i \partial_i + m(y) \mu_3 + h_2(x,y) \sigma_2 \\
&=: v k_z \mu_1 \sigma_3 + \tilde{ \mathcal{H}}(x,y) ,
\end{split}
\end{equation}
Because of chiral symmetry $\{ \mu_1 \sigma_3 , \tilde{ \mathcal{H}}(x,y) \} = 0$,
the chiral zero bound states of $\tilde{ \mathcal{H}}(x,y)$ with chirality $\pm$ correspond to the chiral gapless modes with the energy dispersion
$\pm v k_z$ with chirality $\pm$.
Since $\tilde{ \mathcal{H}}(x,y)$ is of the same form as (\ref{eq12}), $\mathrm{ind} \,\tilde{ \mathcal{H}}$ is given by (\ref{eq0}),
but in this case, $E_n(>0)$ is the mass gap of the two-dimensional massive bound states localized at the surface of topological insulator.
The first contribution in (\ref{eq0}) agrees with the winding of the Axion vortex.\cite{QHZ}
The second contribution in (\ref{eq0}) correspond to the non-topological integer part of Axion field which depend on the microscopic structure of the interface between
the topological insulator and the trivial insulator.
The chiral gapless mode cannot be massive since backward scatterings are suppressed.
This index for chiral gapless modes in class A survives against any perturbations.
\section{conclusion and discussions}
\label{sec6}
Here, we remark a topological property for the index of heterostructure systems.
The index (\ref{eq0}) is stable against the continuous change of Hamiltonian $\mathcal{H}$
unless the boundary operator $\mathcal{M}$ have zero modes, i.e., unless the Hamiltonian $\mathcal{H}$ have no gapless modes at infinity.
In this sense, the index (\ref{eq0}) is topologically protected by the energy gap at the boundary.
This feature is similar to the topological order in bulk systems protected by a bulk energy gap.
In conclusion, we have shown that
the number of Majorana bound states in the $\pi$-junction-ferromagnet heterostructure systems is
affected
by massive bound state localized at the interface,
which has an important implication for topological protection of zero modes in class D systems.
This work was supported in part by the Grant-in-Aids for
Scientific Research from MEXT of Japan [Grants No. 23102714, No. 23540406, and No. 23103502(Innovation Areas ``Topological Quantum Phenomena'')],
and from the Japan Society for the Promotion of Science (Grant No. 21540378).
|
{
"timestamp": "2012-08-20T02:01:18",
"yymm": "1203",
"arxiv_id": "1203.2086",
"language": "en",
"url": "https://arxiv.org/abs/1203.2086"
}
|
\section{Introduction}
It has long been appreciated that interesting and
unique properties arise from the coupling between
the charge and magnetic degrees of freedom in materials.
Advances in our understanding of the physics regulating
these properties have given rise to systems with a
number of practical applications.
Prominent examples include multiferroic materials,
which exhibit simultaneous and cooperative ferroelectric
and magnetic ordering,\cite{multiferroic} as well as
giant\cite{gmr} and colossal\cite{cmr} magnetoresistance
materials, which exhibit substantial changes in their
electronic transport with the application of small magnetic
fields.
These and other {\sl magnetoelectric} (ME) {\sl effects}
--- coupling charge and magnetic degrees of freedom ---
provide unique functionalities and hold promise for novel
device applications.\cite{magnetoapp}
With the continuing drive to miniaturize electronic
devices, there is interest in better controlling or/and
enhancing the functionality of nanoscale systems; there
is particular interest in novel devices that utilize
phenomena inherent/unique to these nanometer
scales.\cite{nanodevices}
Here, we predict unique functionality in a nanoscale
device arising from the coupling of charge and magnetic
degrees of freedom in the ultimate miniaturization,
namely where devices are built
atom-by-atom.\cite{nanodevices,corralcomment}
We demonstrate that the interplay of spin-orbit coupling
and electronic scattering results in ME effects which
enable exquisite control at the atomic scale.
This control provides a powerful tool for manipulating
the response of quantum systems, adding desirable
functionalities not only for fundamental studies, but
also for possible future devices built in a
``bottom up" approach.
Our device system consists of a quantum corral (QC)
on a metal surface with spin-orbit coupling (SOC)
(e.g. Au(111) \cite{exptAu}), where the QC's wall
is made of magnetic atoms.
We demonstrate the possibility of controlling the
electronic properties of the QC by changing the
magnetization of the atoms forming the QC's wall;
we show that these ME effects allow one to control
the properties of systems placed inside the QC as
well as their electronic signatures.
This control provides powerful alternative tools for
probing and manipulating electronic properties at the
atomic scale.
The Hamiltonian for the system has the form
$\hat{H}$=$\hat{H}_{\rm QC}$+$\hat{H}_1$, where
$\hat{H}_{\rm QC}$ describes the QC, and $\hat{H}_1$
describes a system we place inside the QC, whose
properties will be controlled and/or probed (see below).
We describe the QC by the Hamiltonian
$\hat{H}_{\rm QC}$=$\hat{H}_0$+$\hat{V}$,
where $\hat{H}_0$ describes a two-dimensional electron gas
(2DEG) with SOC, and $\hat{V}$ is a scattering potential
describing the QC's wall.
The Hamiltonian for the 2DEG is
\begin{equation}
\hat{H}_0 = \frac{1}{2m^*} {\bf p}^2 + \lambda~ \hat{z}
\cdot \left( {\bf p} \times \overline{\sigma} \right)
\label{2DEGham}
\end{equation}
where ${\bf p}$ is the momentum operator of the 2DEG,
$\{ \sigma^{\mu} \}$ are the Pauli matrices, $m^*$ is
the electron's band mass, and $\lambda$ parameterizes
the SOC.
As described above, we are interested in the case where
the QC's wall is made of magnetic atoms ---
being interested in the system's low-energy properties,
we treat the atoms as a collection of $s$-wave
scatterers;\cite{scatteringbook}
we take
\begin{equation}
\hat{V} = \sum_i \left( V_0
+ \frac{J}{2} \overline{\tau}_i \cdot {\overline \sigma} \right)
\delta({\bf r} - {\bf r}_i) \ ,
\label{potential}
\end{equation}
where ${\bf r}$ is the position operator of the 2DEG,
$V_0$ describes the potential scattering,
$J$ is the exchange coupling between the
(magnetic) atoms and the 2DEG,
$\overline{\tau}_i$ is the spin operator of the
i$^{th}$ atom,
and the atoms are located at the positions
$\{ {\bf r}_i \}$. \cite{SI}
The form(s) of $H_1$ will be specified subsequently.
In this work, we will be interested in the case where
the atoms of the QC's wall are ferromagnetically (FM)
ordered --- we assume the atoms' moments are sufficiently
large and treat them as classical variables:
$(J/2)\langle \overline{\tau}_i \rangle$=${\bf M}$.
The physical quantity of interest is the electronic
local density of states (LDOS) in the QC, $A({\bf r},\omega)$.
[The differential conductance measured in scanning tunneling
microscopy is proportional to $A({\bf r},\omega)$.\cite{mahan}]
This is obtained from the QC's retarded Green's function (GF)
$G({\bf r},{\bf r}'; \omega)$ via
\begin{equation}
A({\bf r},\omega) = -\frac{1}{\pi}~ {\rm Im}\left\{
{\rm Tr}[G({\bf r}, {\bf r}; \omega)] \right\} \ ,
\end{equation}
where\cite{mahan,hewson}
\begin{equation}
G({\bf r},{\bf r}'; \omega)
= G_0({\bf r},{\bf r}'; \omega)
+ G_0({\bf r},{\bf R}_0; \omega) \hat{T}(\omega)
G_0({\bf R}_0,{\bf r}'; \omega) \ .
\label{fullgreens}
\end{equation}
In Eq.~\ref{fullgreens}, the $T$-matrix $\hat{T}(\omega)$ describes
the influence of $H_1$. Furthermore, $G_0({\bf r},{\bf r}'; \omega)$
is the bare GF of the QC, i.e. the GF in the absence of
$\hat{H}_1$.\cite{SI}
In what follows, we choose $(\pi \rho_0) V_0$=0.3 and
$(\pi \rho_0) |{\bf M}|$=0.5 ($\rho_0$=$m^*/2\pi$);\cite{SI}
we consider physically reasonable values of the parameters for the
2DEG:\cite{SOnanolett} a Fermi energy $E_F$=0.5eV, $m^*$=0.26$m_e$
($m_e$ is the bare electron mass), and
$\lambda$=4$\times$$10^{-11}$eV$\cdot$m.
The results we show are for an elliptical QC with 40 atoms, similar
to what has been realized experimentally:\cite{corralexp}
$(x/a)^2$+$(y/b)^2$=$R^2$ with $R$=57.22\AA, $a/b$=1.5,
and $(\pm c,0)$=$(\pm \sqrt{a^2 - b^2},0)$ being the ellipse's
foci;
we will also comment about results obtained for a circular QC.
It should be stressed the results we report are robust ---
changes in the relative size and phase of the ratio ${\bf M}/V_0$
have only quantitative effects on the results, leaving our overall
discussion and conclusions unaffected; furthermore, the
corral's geometry can, in fact, be tuned to enhance/optimize
the ME effects (depending on the parameters).
\begin{figure}[b]
\scalebox{0.72}{\includegraphics{figure1.eps} }
\vspace{-0.17in}
\caption{Spatial scan of the LDOS at $E_F$:
(a) ${\bf M}$=$|{\bf M}| \hat{y}$
(b) ${\bf M}$=$|{\bf M}| \hat{x}$.}
\label{fig:ellipsespace}
\end{figure}
We begin by discussing the electronic properties of the
QC (with $\hat{H}_1$=0).
Fig.~\ref{fig:ellipsespace} shows a spatial scan of the
LDOS at $E_F$ for different directions of ${\bf M}$.
Notice how the LDOS changes as one changes the direction
of ${\bf M}$ --- the magnetization of the QC's wall and
the SOC give rise to strong ME effects at the atomic scale.
These effects are due to the breaking of SU(2)
spin-rotation invariance by the SOC.
Furthermore, the differences in the LDOS for
${\bf M}$=$|{\bf M}| \hat{x}$ and
${\bf M}$=$|{\bf M}| \hat{y}$ in
the elliptical QC are due to the breaking of rotational
invariance --- for a circular QC, the DOS
for ${\bf M}$=$|{\bf M}| \hat{x}$
and ${\bf M}$=$|{\bf M}| \hat{y}$ are identical.
Fig.~\ref{fig:ellipseenergy} shows the energy dependence
of the DOS at particular points in the QC --- changing
the direction of ${\bf M}$ changes the energy dependence
of the DOS; indeed, by carefully choosing the position in
the QC, the changes can be quite pronounced
[see Fig.~\ref{fig:ellipseenergy}(b)].
For reference, the DOS with $|{\bf M}|$=0 is also shown,
as well as the DOS with $\lambda$=0 (in the inset).
\begin{figure}[t]
\scalebox{0.75}{\hspace{-0.31in} \includegraphics{figure2.eps}}
\vspace{-0.38in}
\caption{Energy dependence of the LDOS for different ${\bf M}$:
(a) ${\bf r}$=$(0,0)$ (b) ${\bf r}$=$(-c,0)$.
Inset: LDOS at $(-c,0)$ for $| {\bf M}|=0$ with and without SOC.}
\label{fig:ellipseenergy}
\end{figure}
The ME effects exhibited by this system could find utility
in a variety of applications that exploit the spatial and/or
energy dependence of the QC's LDOS.
To illustrate the utility of the spatial dependence
of the LDOS, we place a spin-1/2 magnetic impurity at
a position ${\bf R}_0$ inside the QC; we investigate
how its low-energy properties depend on its position.
We describe the magnetic impurity by the Hamiltonian
\begin{equation}
\hat{H}_{1} = J_K~ \overline{\tau} \cdot {\bf S}({\bf R}_0) \ ,
\label{sdhammy}
\end{equation}
where $\overline{\tau}$ is the impurity's spin operator,
${\bf S}({\bf R}_0)$ is the 2DEG's spin operator at the
position ${\bf R}_0$, and $J_K$ is the exchange coupling
between the impurity spin and the 2DEG. In what follows,
we take $J_K$$>$$0$.\cite{SI}
\begin{figure}[b]
\scalebox{0.69}{\hspace{-0.27in} \includegraphics{figure3.eps}}
\vspace{-0.38in}
\caption{Imaginary part of the $T$-matrix due to a
magnetic impurity for different ${\bf M}$:
(a) impurity at the origin ${\bf R}_0$=$(0,0)$
(b) impurity at the focus ${\bf R}_0$=$(-c,0)$.}
\label{fig:kondopeakellipse}
\end{figure}
Interestingly, this seemingly simple system exhibits
nontrivial behavior in the infrared, due to quantum
fluctuations --- a strongly correlated state arises, where
a cloud of conduction electrons forms a singlet with the
impurity.\cite{hewson}
This strongly correlated state manifests itself via a
resonance at (or near) the Fermi energy, referred to as
the Kondo resonance (KR).
This can be seen in Fig.~\ref{fig:kondopeakellipse}, where
the imaginary part of the impurity's $T$-matrix is shown and,
in particular, the KR appears.
Furthermore, the width of this KR represents the
dynamically generated scale characteristic of this strongly
correlated state, the Kondo temperature $T_K$.\cite{hewson}
\begin{figure}[t]
\scalebox{0.72}{\includegraphics{figure4.eps}}
\vspace{-0.17in}
\caption{Spatial dependence of the LDOS difference at $E_F$
due to a magnetic impurity,
$\delta A({\bf r}, E_F)$ (see text): (a) impurity at the
focus ${\bf R}_0$=$(c,0)$ (b) impurity at the origin
${\bf R}_0$=$(0,0)$.}
\label{fig:DOSkondo}
\end{figure}
The ME effects allow one to control the Kondo effect
exhibited by the magnetic impurity in the QC; more
generally, it allows one to control the magnetic
properties of an atom placed in the QC.
Indeed, Fig.~\ref{fig:kondopeakellipse} shows that the KR
can be controlled by changing (the direction of) ${\bf M}$.
Furthermore, we see that the change in the KR depends on
the position at which the impurity is placed\cite{uic}
--- one can control the impurity's properties in a desired
way by a judicious choice of its position.
It should also be noted that, besides impacting the
properties of the magnetic impurity, the QC is also
impacted by the magnetic impurity;
the impurity's influence on the QC depends on its
position.
This can be seen in Fig.~\ref{fig:DOSkondo}, where
a spatial scan of the quantity
$\delta A({\bf r},E_F)$=$A({\bf r},E_F)_{\hat{x}}$$-$$
A({\bf r},E_F)_{\hat{y}}$ --- the difference in the
LDOS (at $E_F$) between ${\bf M}$=$|{\bf M}| \hat{x}$
and ${\bf M}$=$|{\bf M}| \hat{y}$ --- is shown for
different positions of the magnetic impurity.
[Results obtained for a circular QC are qualitatively
similar to those obtained for an elliptical QC.]
\begin{figure}[t]
\scalebox{0.71}{
\hspace{-0.39in} \includegraphics{figure5.eps}}
\vspace{-0.42in}
\caption{Maps of
$|\partial_\omega \delta A({\bf r};\omega)|$
due to the vibrational mode for
${\bf M}$=$|{\bf M}| \hat{x}$
and ${\bf M}$=$|{\bf M}| \hat{y}$:
(a) and (b) at ${\bf r}$=$(c,0)$=${\bf R}_0$;
(c) and (d) at ${\bf r}$=$(-c,0)$, demonstrating the
filtering of the mirage signal.}
\label{fig:vibration}
\end{figure}
As we saw above, the ME effects allow manipulation of both
the spatial and energy dependence of the QC's LDOS.
As we now demonstrate, this change in the energy dependence
of the DOS can be used for signal filtering.
To this end, we place a molecule in the QC at the
focus $(c,0)$, such that a vibrational mode (VM) of this
molecule couples to the 2DEG of the QC; we investigate the
image or "mirage" of the VM at the other focus $(-c,0)$.
We describe the VM by\cite{holstein}
\begin{equation}
\hat{H}_{1} = \frac{1}{2} p^2
+ \frac{1}{2} \Omega^2 x^2 + g~x~ n({\bf R}_0) \ ,
\end{equation}
where $x$ ($p$) is the position (momentum) operator of
the VM, $\Omega$ is its characteristic frequency,
$n({\bf R}_0)$ is the 2DEG's density operator at the
position ${\bf R}_0$=$(c,0)$,
and $g$ describes the coupling between the VM and the 2DEG.
In what follows, we assume the coupling between the 2DEG
and the VM to be weak.\cite{SI}
The QC enables signals to be transmitted between
foci;\cite{corralexp} it also provides a
"cloak of invisibility", similar to what has been achieved
with electromagnetic fields,\cite{metamaterials} where
objects were made invisible within a certain frequency band.
More specifically, the electronic properties of the QC
govern the energy regime in which a signal transmitted
from one focus can be observed at the other focus;
in particular, signals within a certain frequency range
can be hidden from observation.\cite{manobalat}
This cloaking can be seen in Fig.~\ref{fig:vibration},
where a density plot of the energy derivative of the LDOS
$|\partial_\omega A({\bf r};\omega)|$ is shown --- even
when the VM is visible at the focus in which it is sitting
(Fig.~\ref{fig:vibration}(a) and (b)), the QC cloaks it
from observation at the other focus for $\Omega$ within
a certain range
(Fig.~\ref{fig:vibration}(c) and (d)).\cite{manobalat}
In this system, the ME effects allow the range over which
cloaking occurs to be controlled by changing the
orientation of ${\bf M}$ --- compare the results for
${\bf M}$=$|{\bf M}| \hat{x}$ and ${\bf M}$=$|{\bf M}|\hat{y}$.
Said in another way, the ME effects allow one to filter the
signal transmitted from one focus to the other.
\begin{figure}[t]
\scalebox{0.65}{\hspace{-0.41in} \includegraphics{fig6.eps}}
\vspace{-0.33in}
\caption{Evolution of the magnetization with the
external magnetic field ${\bf h}$: (a) $|{\bf h}|$=0
(b) $|{\bf h}|$=0.04$E_0$ (c) $|{\bf h}|$=0.12$E_0$
(d) $|{\bf h}|$=0.24$E_0$ where $E_0$=$\pi(\rho_0 J)^2$$E_F$.}
\label{fig:magnetization}
\end{figure}
Up to now, we have considered the system's electronic
properties, assuming the QC's wall to be FM ordered.
We have also considered the magnetic properties of the
wall and, in particular, the wall's magnetic ordering
tendencies;\cite{SI}
as the QC's wall is one-dimensional, fluctuations will
suppress ordering, and an external field ${\bf h}$ is
necessary to stabilize the order.
Fig.~\ref{fig:magnetization} shows results for the QC's
magnetization for various values of ${\bf h}$.
We see that the moments are disordered at zero field and, hence,
the magnetization is zero; a nonzero magnetization is obtained
as ${\bf h}$ is increased.
We have found that a FM aligned wall is obtained from readily
accessible magnetic fields ---
$|{\bf h}|$$\simeq$$0.2 \pi (\rho_0 J)^2 E_F$; for reasonable
values of the parameters, this gives
$|{\bf h}|$=${\cal O}$(1meV).
[It is worth noting such values are considerably lower than
Kondo temperatures that have been observed from atoms/molecules
on surfaces.\cite{kondosurface}]
Furthermore, we have found the geometry can, in fact, be
optimized, so that ferromagnetic ordering occurs at extremely
small values of ${\bf h}$.
This work demonstrates proof of principle of the functionality
afforded by ME effects at the atomic scale; indeed, we were
able to control the properties of systems placed inside the QC
as well as their electronic signals/signatures.
With a FM aligned wall, the ME effects allowed us to control
the magnetic properties of atoms placed inside the QC, as well
as to filter transmitted signals;
different magnetization patterns for the
wall,\cite{surfacemagnetchiral} as well as different QC
geometries, could provide further flexibility and control.
By placing several atoms/molecules inside the QC, one could
engineer devices where the ME effects allow the
entanglement\cite{entanglereview} between atoms/molecules to
be manipulated.
It should also be mentioned that the ME effects allow
manipulation of the properties of the QC's wall
--- the QC's wall itself provides a unique magnetic system
with interesting properties, which could also afford
means of transmitting and manipulating
information.\cite{surfacechain,surfaceRKKY}
EHK acknowledges the warm hospitality of the
Instituto de F\'isica T\'eorica (Madrid, Spain), where
most of this work was performed. SEU acknowledges
support from AvH Stiftung, and the hospitality of the
Dahlem Center for Complex Quantum Systems at FU-Berlin.
This work was supported by
NSF MWN/CIAM and PIRE grants (ATN and SEU),
the Spanish Grants TOQATA and QUAGATUA (JRL),
and the Spanish Grant FIS2009-11654 (JRL and EHK).
|
{
"timestamp": "2012-03-12T01:00:39",
"yymm": "1203",
"arxiv_id": "1203.1987",
"language": "en",
"url": "https://arxiv.org/abs/1203.1987"
}
|
\section{Introduction}
\label{sect:intro}
The concept of Axiomatic Quantum Field Theory has traditionally been explored only in Minkowski space: in particular, the Wightman axioms \cite{pctwightman} and the Haag-Kastler axioms \cite{haaglocalqf} outline ways of providing a set of axioms for a quantum field theory to obey. Over the past decade, advances have been made in the area of Axiomatic Quantum Field Theory in curved spacetimes. In particular, the work by Brunetti, Fredenhagen and Verch \cite{bfv} outlined a set of axioms, similar to the Haag-Kastler axioms for Quantum Field Theory on Minkowski space, that should be obeyed by any QFT that can be defined on curved spacetimes. The Haag-Kastler axiomatic framework is often described as \emph{Algebraic Quantum Field Theory}; the axioms lay out certain properties that should be held by any legitimate assignment of an algebra of observables to each arbitrary region of Minkowski space. Extending algebraic QFT to curved spacetime involves examining the ways one might amend these axioms to define the properties held by a suitable assignment of an algebra of observables to arbitrary regions of arbitrary spacetimes. In practice, though, to achieve meaningful results we have to apply some restrictions to the type of region and the type of spacetime we are allowed to choose. The axioms proposed in \cite{bfv} use the tools of category theory; the allowed regions in this case are open, globally hyperbolic subregions of globally hyperbolic spacetimes (definitions are given in section \ref{sect:lcdl}), and allow us to think of a particular quantum field theory as a functor between the category \Loc\ whose objects are globally hyperbolic spacetimes, and the category \Alg\ whose objects are at the very least $\ast$-algebras, but may possibly possess additional structure.
However, it turns out that these axioms alone allow for some rather undesirable pathological theories. In particular, some very recent work by Fewster and Verch \cite{dynloc} has shown that certain theories may satisfy the BFV axioms despite in some sense describing different physics depending on the spacetime to which it assigns the algebra of observables. The question of precisely what is meant by a theory representing the same physics in all possible spacetimes is still, by and large, an open one. While it is desirable to find a condition on theories that somehow formalises this property, this question is more easily answered by comparing theories with one other, and so the \emph{SPASs} (Same Physics in All Spacetimes) condition proposed in the aforementioned paper is a condition on \emph{classes} of theories. It is intended to be a necessary condition for such a class to comprise theories,
each of which represents the same physics in all spacetimes, according to some common definition of the term. In this paper we are concerned with the class of \emph{dynamically local} theories, which is shown in \cite{dynloc} to satisfy the SPASs condition.
The property of dynamical locality has other desirable consequences such as a no-go theorem for natural states, and dynamical locality has so far been demonstrated for some linear theories, including the minimally coupled massive Klein-Gordon field and the massless current algebra. However, it fails in the case of the minimally coupled massless Klein-Gordon field. It is therefore desirable to find further examples of well-known theories that can be constructed in a locally covariant way that either satisfy or violate dynamical locality.
We will prove in this paper that the nonminimally coupled Klein-Gordon scalar field is dynamically local in both the massive and massless case, and also that the extended algebra of noninteracting Wick polynomials can be shown to be dynamically local in the minimally coupled massive and conformally coupled massive cases; however, it fails to be dynamically local in the minimally coupled massless case.
\section{Local covariance and Dynamical Locality}
\label{sect:lcdl}
We are using the prescription in \cite{bfv} for the construction of locally covariant theories, in which a theory is considered to be functor from a category of spacetimes to a category of algebras. We must therefore first define the categories we will be using. We will follow the definitions and notation in \cite{dynloc} for the category of globally hyperbolic spacetimes. This category is denoted \Loc; its objects are quadruples $\Ms=\Mgot$ where $\M$ is a smooth paracompact orientable nonempty $n$-dimensional manifold with finitely many connected components, $\g$ is a smooth time-orientable metric for $\M$ with signature $+{}-\cdots-{}$, and $\mathfrak{o}$ and $\mathfrak{t}$ are choices of orientation and time-orientation respectively for $\M$. These spacetimes must also satisfy global hyperbolicity: there can be no closed causal curves in $\M$, and for each pair $p,q\in\M$ the intersection $J_\Ms^-(p)\cap J_\Ms^+(q)$ must be compact, where $J_\Ms^\pm(p)$ denotes the causal future ($+$) or past ($-$) of $p$ in $\M$.
An arrow of \Loc\ from an object $\Ns=(\N,\g_\N,\mathfrak{o}_\N,\mathfrak{t}_\N)$ to a second object $\Ms=(\M,\g_\M,\mathfrak{o}_\M,\mathfrak{t}_\M)$ is a smooth embedding $\psi:\N\hookrightarrow\M$ that is isometric (i.e.\ $\psi^\ast\g_\M=\g_\N$) and orientation- and time-orientation-preserving (i.e.\ $\psi^\ast\mathfrak{o}_\M=\mathfrak{o}_\N$, $\psi^\ast\mathfrak{t}_\M=\mathfrak{t}_\N$). It must also respect the causal structure: the image $\psi(\N)\subseteq\M$ must be \emph{causally convex} in $\M$, i.e.\ each causal curve in $\M$ with both endpoints lying within $\psi(\N)$ must be entirely contained within $\psi(\N)$.
A \emph{Cauchy surface} $\Sigma$ for a spacetime $\Ms$ is a subset of $\M$ that is intersected by every inextendible timelike curve in $\M$ exactly once. Clearly no Cauchy surface can have a timelike tangent at any point, but this definition does allow a Cauchy surface to have a null tangent; consequently we will refer to a Cauchy surface whose tangents are all spacelike as a \emph{spacelike Cauchy surface}. Global hyperbolicity of $\Ms$ is equivalent to $\Ms$ containing a smooth spacelike Cauchy surface \cite{bernalsanchez03}. An arrow in $\Loc$ whose image contains a Cauchy surface for its target is called a \emph{Cauchy arrow}. We may safely blur the distinction between a spacetime and its underlying manifold, so in the remainder of this paper we may use the same notation (e.g.\ $\Ms$) for both; for example, we will denote by $\Ccs(\Ms)$ the space of compactly supported smooth functions on the underlying manifold $\M$.
The category whose objects are candidates for the algebras of observables of a theory is denoted \Alg. The objects of \Alg\ are unital $\ast$-algebras, and the morphisms are unit-preserving $\ast$-monomorphisms.
\subsection{Locally covariant theories}
\label{sect:lc}
A \emph{locally covariant quantum field theory} is defined to be a covariant functor from \Loc\ to \Alg\ \cite{bfv}. That is, a theory $\A$ maps objects of \Loc\ to objects of \Alg, and arrows of \Loc\ to arrows of \Alg, such that:
\begin{itemize}
\item for any \Loc-arrow $\psi:\Ns\hookrightarrow\Ms$, the arrow $\A(\psi)$ has domain $\A(\Ns)$ and codomain
$\A(\Ms)$,
\item for any two \Loc-arrows $\psi_1:\textbf{\textit{O}}\hookrightarrow\Ns,$ $\psi_2:\Ns\hookrightarrow\Ms$,
we have $\A(\psi_2\circ\psi_1)=\A(\psi_2)\circ\A(\psi_1)$,
\item for any spacetime $\Ms$, we have $\A(\text{id}_\Ms)=\text{id}_{\A(\Ms)}$.
\end{itemize}
While this is the only property a theory needs to satisfy to be locally covariant, we generally wish to apply some further conditions on the theories we work with. In particular, there is no condition pertaining to causality in the basic definition of a locally covariant theory. A locally covariant theory $\A$ is said to be \emph{causal} if it has the following property: let $\psi_1:\Ns_1\hookrightarrow\Ms,\ \psi_2:\Ns_2\hookrightarrow\Ms$ be arrows in \Loc\ such that the images $\psi_1(\Ns_1)$ and $\psi_2(\Ns_2)$ are causally disjoint in $\Ms$. Then $[\A(\psi_1)A_1,\A(\psi_2)A_2]=0$ for any $A_1\in\A(\Ns_1)$, $A_2\in\A(\Ns_2)$.
We will also generally require our theories to satisfy the \emph{timeslice axiom}. Suppose $\psi:\Ns\hookrightarrow\Ms$ is an arrow in \Loc; a locally covariant theory $\A$ obeys the timeslice axiom if the \Alg-arrow $\A(\psi)$ is an isomorphism whenever the image of $\Ns$ in $\Ms$ under $\psi$ contains a Cauchy surface for $\Ms$ (alternatively, $\A(\psi)$ is an isomorphism whenever $\psi$ is a Cauchy arrow). The timeslice axiom allows us to define an automorphism of an algebra $\A(\Ms)$ called the relative Cauchy evolution, which is defined as follows.
For any spacetime $\Ms=\Mgot$, we define $\h\in\Ccs(T^0_2\M)$ to be a \emph{metric perturbation} if it is symmetric and the spacetime $\Ms[\h]=(\M,\g+\h,\mathfrak{o},\mathfrak{t}')$ is also an object in \Loc\ (where $\mathfrak{t}'$ is the unique choice of time-orientation that coincides with $\mathfrak{t}$ outside $\supp(\h)$). The set of all such metric perturbations on $\Ms$ is denoted $H(\Ms)$, and for any $O\subset\M$ we denote by $H(\Ms;O)$ all $\h\in H(\Ms)$ whose support lies within $O$.
Given some $\h\in H(\Ms)$, we pick globally hyperbolic subregions $\N^\pm$ of $\M$ such that each contains a Cauchy surface $\Sigma^\pm$ for $\Ms$, and such that $\N^\pm\subseteq\M\setminus J_\Ms^\mp(\supp(\h))$. Now, we can consider the spacetimes $\Ns^\pm=(\N^\pm,\g|_{\N^\pm},\mathfrak{o}|_{\N^\pm},\mathfrak{t}|_{\N^\pm})$ in their own right; each is a sub-spacetime of both $\Ms$ and $\Ms[\h]$, and we denote by $\iota^\pm,\iota^\pm[\h]$ the canonical embeddings of $\Ns^\pm$ respectively into $\Ms$ and $\Ms[\h]$. If a locally covariant theory $\A$ satisfies the timeslice axiom, then the arrows $\A(\iota^\pm)$ and $\A(\iota^\pm[\h])$ must be isomorphisms. It follows that we can form an automorphism $\rce_\Ms[\h]$ on $\A(\Ms)$ defined by
\[
\rce_\Ms[\h]=\A(\iota^-)\circ\A(\iota^-[\h])^{-1}\circ\A(\iota^+[\h])\circ\A(\iota^+)^{-1},
\]
called the \emph{relative Cauchy evolution} on $\Ms$ induced by $\h$. The relative Cauchy evolution can be shown to be independent of the choice of future and past subspacetimes $\Ns^\pm$ \cite[Prop.\ 3.3]{dynloc}.
\subsection{Dynamical locality}
\label{sect:dynloc}
It is natural to ask the question of whether the condition of local covariance, with the timeslice axiom, is enough to ensure that a theory is ``physically realistic''. As discussed before, the existence of certain pathological locally covariant theories has motivated the discussion in \cite{dynloc}, where the idea of the Same Physics in All Spacetimes (SPASs) is introduced as a condition on classes of theories that is claimed to be necessary for the theories to be considered physically realistic. A class of theories $T$ has the SPASs property if, whenever
\begin{itemize}
\item $\A,\mathscr{B}\in T$,
\item there exists a natural transformation $\zeta:\A\stackrel\cdot\longrightarrow \mathscr{B}$, and
\item there exists a globally hyperbolic spacetime $\Ms$ on which $\zeta_\Ms$ is an isomorphism,
\end{itemize}
then $\zeta$ is a natural isomorphism (i.e.\ $\zeta_\Ns$ is an isomorphism for each globally hyperbolic spacetime $\Ns$).
It is shown in \cite{dynloc} that one can construct a class $T$ of locally covariant causal theories that obey the timeslice axiom, but such that $T$ does not have the SPASs property. To this end, it is suggested that the additional axiom of \emph{dynamical locality}, defined below, is imposed. The class of dynamically local theories is a subclass of the locally covariant theories that obey the timeslice axiom, but it has the added advantage of satisfying the SPASs condition.
We first define the \emph{kinematic nets} and \emph{dynamical nets} of a locally covariant theory $\A$ obeying the timeslice axiom. Let $\Ms$ be a globally hyperbolic spacetime and $O$ be a globally hyperbolic open subregion of $\Ms$ with finitely many connected components, all of which are causally disjoint (we denote by $\mathscr{O}(\Ms)$ the set of possible such $O$). Clearly we can regard $\Ms|_O$ as a globally hyperbolic spacetime in its own right. We will denote the map embedding $\Ms|_O$ into $\Ms$ by $\iota_{\Ms;O}$. When we apply the functor $\A$ to $\Ms|_O$, we get the algebra $\A(\Ms|_O)$, which can be embedded in $\A(\Ms)$ by the map $\alpha^{\text{kin}}_{\Ms;O}$, defined to be the result of applying the same functor to $\iota_{\Ms;O}$. The \emph{kinematic net} is defined to be the map which assigns $O\mapsto\alpha^\text{kin}_{\Ms;O}$. The algebra obtained by applying $\A$ to the restriction $\Ms|_O$ is called the \emph{kinematic algebra} of $O$, denoted by $\Ak(\Ms;O)=\A(\Ms|_O)$.
Given such $\Ms$ and $O$, we can also define the \emph{dynamical net} as follows: given $O\in\mathscr{O}(\Ms)$, and compact $K\subset O$, we let
\[
\A^\bullet(\Ms;K)=\{A\in\A(\Ms):\rce_\Ms[\h]A=A\text{ for all }\h\in H(\Ms;K^\perp)\}.
\]
Here $K^\perp=\Ms\setminus J_\Ms(K)$ denotes the causal complement of a compact $K\subseteq\Ms$.
We then define the \emph{dynamical algebra} as
\beq
\label{eqn:dynalgdef}
\Ad(\Ms;O)=\bigvee_{K\in\mathscr{K}(\Ms;O)}\A^\bullet(\Ms;K),
\eeq
where $\mathscr{K}(\Ms;O)$ is the set of compact subsets of $\Ms$ with a \emph{multi-diamond} neighbourhood based in $O$. Here a multi-diamond is a finite union of causally disjoint diamonds, where we use the following definition from \cite{brunruzz}: a \emph{diamond} is a set $D_\Ms(B)$ such that there exists a spacelike Cauchy surface $\Sigma\subset\Ms$, and a chart $(U,\phi)$ of $\Sigma$, where $\phi(B)$ is a nonempty open ball in $\mathbb{R}^{n-1}$ with closure contained in $\phi(U)$, and $D_\Ms(B)$ denotes the domain of dependence of $B$.
The inclusion
\[
\alpha^{\text{dyn}}_{\Ms;O}:\Ad(\Ms;O)\hookrightarrow\A(\Ms)
\]
is unique (up to isomorphism), and we define the dynamical net to be the map which assigns $O\mapsto\alpha_{\Ms;O}^\text{dyn}$.
A theory is defined to be \emph{dynamically local} if for every globally hyperbolic spacetime $\Ms$ and nonempty $O\in\mathscr{O}(\Ms)$, we have $\Ak(\Ms;O)\cong\Ad(\Ms;O)$, or alternatively
\[
\alpha^\text{dyn}_{\Ms;O}\cong\alpha^\text{kin}_{\Ms;O}.
\]
This is equivalent to demanding that for all such $O,\Ms$ we have
\[
\alpha^\text{dyn}_{M;O}(\Ad(M;O))=\alpha^\text{kin}_{M;O}(\Ak(M;O)).
\]
For an additive theory, that is, one in which $\Ak(M;O)$ is generated by its subalgebras
corresponding to relatively compact subregions of $O$, \cite[Prop. 6.1]{dynloc} entails that we always have $\alpha^\text{kin}_{M;O}(\Ak(M;O))\subseteq\alpha^\text{dyn}_{M;O}(\Ad(M;O))$, and therefore it is sufficient for dynamical locality to show that
\beq
\label{eqn:dynloccond}
\alpha^\text{dyn}_{M;O}(\Ad(M;O))\subseteq\alpha^\text{kin}_{M;O}(\Ak(M;O)).
\eeq
This applies to all the theories we will study here.
\section{The Klein-Gordon Field and Wick Polynomials as LCTs}
\label{sect:const}
\subsection{Construction of the Klein-Gordon Theory}
\label{sect:kgfconst}
The Klein-Gordon operator on a spacetime $\Ms$ is denoted $P_\Ms =\Box_\g+\xi R_\g+m^2$. We call any solution $\phi\in\Cs(\Ms)$ to the field equation $P_\Ms \phi=0$ a \emph{classical solution} to the field equation. The coupling constant $\xi\in\mathbb{R}$ and the mass $m\geq0$ are held constant over all spacetimes. The Klein-Gordon operator has associated with it two unique continuous linear operators $E_\Ms ^\pm:\Ccs(\Ms)\to\Cs(\Ms)$ with the properties
\begin{align}
E_\Ms ^\pm P_\Ms f&=f=P_\Ms E_\Ms ^\pm f\label{eqn:funsol1}\\
\supp(E_\Ms ^\pm f)&\subseteq J_M^\pm(\supp(f))\label{eqn:funsol2}
\end{align}
for any $f\in\Ccs(\Ms)$ \cite{wald} (here we identify $\EM^\pm\PM f$ and $\PM\EM^\pm f$ with their preimage under the canonical embedding $\iota:\Ccs(\Ms)\hookrightarrow\Cs(\Ms)$). The operator $E_\Ms =E_\Ms ^--E_\Ms ^+$ is the \emph{(advanced-minus-retarded) fundamental solution} for the Klein-Gordon field on $\Ms$, and any classical solution $\phi$ with compact support on Cauchy surfaces is of the form $\phi=E_\Ms f$ for some $f\in\Ccs(\Ms)$. We denote by $E_\Ms (x,y)$ the antisymmetric bidistribution on test functions satisfying
\[
\int_\Ms dy\,E_\Ms(x,y)f(y)=(E_\Ms f)(x)
\]
for each $f\in\Ccs(\Ms)$. Furthermore, we denote
\[
E_\Ms (f,f')=\int_{\Ms}dx\,f(x)(\EM f')(x)=\int_{\Ms^{\times 2}}dx\,dy\,f(x)
\EM(x,y)f'(y),
\]
for $f,f'\in\Ccs(\Ms)$. Note that this entails
\beq
\label{eqn:Eantiprop}
\int_\Ms dx\,f(x)(\EM f')(x)=-\int_\Ms dx\,(\EM f)(x)f'(x).
\eeq
Given a fixed spacetime $\Ms$, we can construct the algebra of the Klein-Gordon quantum field theory as the unital $\ast$-algebra generated by elements $\Phi_\Ms (t)$, $t\in\Ccs(\Ms)$ satisfying the following four conditions:
\begin{subequations}
\begin{align}
&\text{The assignment }t\mapsto\Phi_\Ms (t)\text{ is linear},\label{eqn:kgcond1}\\
&\Phi_\Ms (t)^\ast=\Phi(\bar t),\label{eqn:kgcond2}\\
&[\Phi_\Ms (t),\Phi_\Ms (t')]=iE_\Ms (t,t')\id,\label{eqn:kgcond3}\\
&\Phi_\Ms (P_\Ms t)=0.\label{eqn:kgcond4}
\end{align}
\end{subequations}
While it can be observed that this algebra can be represented simply as a deformation of the symmetric tensor algebra $\Gamma_\odot(E_\Ms \Ccs(\Ms))$ (see e.g.\ \cite{dynloc2}), alternative ways of constructing this algebra can be seen in \cite{bf:qftcb,chilfred}. The following treatment is based on \cite{bf:qftcb}.
If we remove the condition \eqref{eqn:kgcond4}, then the algebra generated by the other three conditions is isomorphic to the unital $\ast$-algebra $\F(\Ms)$ comprising functionals on $\Cs(\Ms)$ of the form
\beq
\label{eqn:funcform}
F[f]=\sum_{n=0}^N\int_{\Ms^{\times n}}d^nx\,t_n(x_1,\ldots,x_n)f(x_1)\cdots f(x_n),
\eeq
where each $t_n$ is a totally symmetric finite sum of products of test functions in one variable:
\[
t_n(x_1,\ldots,x_n)=\mathbf{S}\sum_{j\text{ finite}}
\prod_{k=1}^n\varphi_{jk}(x_k)
\]
for some $\varphi_{jk}\in\Ccs(\Ms)$, where $\mathbf{S}$ denotes symmetrisation. We denote the set of all such $t_n$ as $\F^n(\Ms)$; we define $\F^0(\Ms)=\mathbb{C}$, and we may note that $\F^1(\Ms)=\Ccs(\Ms)$. We will use the shorthand notation
\[
t_n[f]=\int_{\Ms^{\times n}}d^nx\,t_n(x_1,\ldots,x_n)f(x_1)\cdots f(x_n).
\]
For each $F=\sum_{n=0}^Nt_n$ with $t_N\neq0$ we define $O(F)=N<\infty$.
The $k^\text{th}$ functional derivative of $F=\sum_{n=0}^Nt_n$ is given by
\beq
\label{eqn:kgfuncderiv1}
F^{(k)}[f](x_1,\ldots,x_k)=\sum_{n=k}^Nt^{(k)}_n[f](x_1,\ldots,x_k),
\eeq
where for $k\leq n$,
\beq
\label{eqn:kgfuncderiv2}
t^{(k)}_n[f](x_1,\ldots,x_k)=\frac{n!}{(n-k)!}\int_{\Ms^{\times(n-k)}}dx_{k+1}
\cdots dx_n\,t_n(x_1,\ldots,x_n)f(x_{k+1})\cdots f(x_n).
\eeq
For any $f\in\Ccs(\Ms)$, we may regard the functional derivative $F^{(k)}[f](x_1,\ldots,x_k)$ of an element $F\in\F(\Ms)$ as an element of $\F^k(\Ms)$ for $k\leq O(F)$. Addition in $\F(\Ms)$ is given by addition of functionals, and products of elements are defined by
\beq
\label{eqn:kgproddef}
(F\star F')[f]=\sum_{k=0}^{\text{min}(O(F),O(F'))}\frac{i^k}{2^kk!}E_\Ms^k\left(F^{(k)}[f],F'^{(k)}[f]\right),
\eeq
where for $t,t'\in\F^k(\Ms)$, we have
\beq
\label{eqn:kgproddef2}
E_\Ms^k(t,t')=\int_{\Ms^{\times(2k)}}d^kx\,d^ky\,t(x_1,\ldots,x_k)t'(y_1,\ldots,y_k)\prod_{j=1}^k E_\Ms(x_j,y_j),
\eeq
and for $\alpha,\beta\in\F^0(\Ms)=\mathbb{C}$,
\[
E_\Ms^0(\alpha,\beta)=\alpha\beta.
\]
The product \eqref{eqn:kgproddef} can be shown to be associative. The involution of $F=\sum_{n=0}^Nt_n\in\F(\Ms)$ is given by $F^\ast=\sum_{n=0}^N\bar t_n$, and the identity with respect to the $\star$ product is the constant functional $\id[f]\equiv 1$.
The algebra $\F(\Ms)$ is generated by elements satisfying conditions \eqref{eqn:kgcond1}--\eqref{eqn:kgcond3}, so it should be the case that we can recover the algebra $\A(\Ms)$ by reapplying condition \eqref{eqn:kgcond4}. The set $\J(\Ms)$, defined to be the set containing all elements $F\in\F(\Ms)$ satisfying $F[E_\Ms f]=0$ for all $f\in\Ccs(\Ms)$, is a two-sided $\ast$-ideal in $\F(\Ms)$ \cite{bf:qftcb};
on taking the quotient $\F(\Ms)/\J(\Ms)$ we obtain the algebra $\A(\Ms).$ We have the following result:
\begin{lemma}
\label{lem:kgfuncpolar}
Let $F=\sum_{n=0}^Nt_n\in\J(\Ms)$ for some spacetime $\Ms$, where $t_n\in\F^n(\Ms)$ for each $n=0,1,\ldots,N$. Then
\[
\tenpow{\EM} nt_n=0
\]
as a (nonlinear) functional on $\Ccs(\Ms)$ for all $n$.
\end{lemma}
\begin{proof}
If $F=\sum_{n=0}^N t_n\in\J(\Ms)$ with $t_n\in\F^n(\Ms)$, then for any $f\in\Ccs(\Ms)$ and $\kappa\in\mathbb{R}$ we have
\[
0=F[\EM(\kappa f)]=\sum_{n=0}^N\kappa^nt_n[\EM f].
\]
Consequently $t_n[\EM f]=0$ for each $n$, and so by \eqref{eqn:funcform}, and using the fact that for any $g,g'\in\Ccs(\Ms)$ we have $\int_\Ms dx \,g(x)\EM g'(x)=-\int_\Ms dx\,g'(x)\EM g(x)$, it follows that
\[
(\tenpow\EM nt_n)[f]=(-1)^nt_n[\EM f]=0.
\]
This holds for all $f\in\Ccs(\Ms)$, so the result follows.
\end{proof}
The ideal $\J(\Ms)$ generates an equivalence relation $\sM$; i.e.\ for any $F,F'\in\F(\Ms)$, $F\sM F'$ if and only if $F-F'\in\J(\Ms)$, or equivalently $F[\EM f]=F'[\EM f]$ for all $f\in\Ccs(\Ms)$. For any $F\in\F(\Ms)$, the equivalence class of $F$ under $\sM$ is denoted $[F]_\Ms$; the elements of the algebra $\A(\Ms)$ constitute the set of equivalence classes $[F]_\Ms$ with $F\in\F(\Ms)$.
Throughout this paper, we will wish to define the pullback of a \Loc-arrow $\psi:\Ns\hookrightarrow\Ms$ on a functional $F=\sum_{n=0}^Nt_n\in\F(\Ms)$, with $t_n\in\F^n(\Ms)$. Therefore, we denote
\[
\psi^\ast F=\sum_{n=0}^N(\psi^{\otimes n})^\ast t_n.
\]
In order to construct the Klein-Gordon QFT as a locally covariant theory, we must now define the action of the \Alg-arrow $\A(\psi)$ for an arbitrary \Loc-arrow $\psi:\Ns\hookrightarrow\Ms$. Given such a $\psi$, we first define a map
\begin{align*}
\F(\psi):\F(\Ns)&\to\F(\Ms)\\
F&\mapsto F\circ\psi^\ast.
\end{align*}
To see that $\F(\psi)F$ is indeed an element of $\F(\Ms)$ for any $F\in\F(\Ns)$, note that for $F=\sum_{n=0}^Nt_n,$ $t_n\in\F^n(\Ns)$, we have
\begin{align*}
(\F(\psi)F)[f]&=\sum_{n=0}^N\int_{\Ns^{\times n}}d^nx\,t_n(x_1,\ldots,x_n)f(\psi(x_1))\cdots f(\psi(x_n))\\
&=\sum_{n=0}^N\int_{\Ms^{\times n}}d^nx\,\psi_\ast t_n(x_1,\ldots,x_n)f(x_1)\cdots f(x_n)
\end{align*}
for any $f\in\Cs(\Ms)$, where the pushforward $\psi_\ast:\F^n(\Ns)\to\F^n(\Ms)$ is defined as
\[
\psi_\ast t_n(x_1,\ldots,x_n)=\begin{cases}t_n(\psi^{-1}(x_1),\ldots,\psi^{-1}(x_n)),&(x_1,\ldots,x_n)\in\psi(\Ns)^{\times n}\\
0,&\text{otherwise}.
\end{cases}
\]
Since $\psi^{-1}:\psi(\Ns)\to\Ns$ is a diffeomorphism, it follows that each $\psi_\ast t_n$ is an element of $\F^n(\Ms)$ as required. We define the action of $\psi_\ast$ on arbitrary $F\in\F(\Ns)$ by linearity, and note that $\psi^\ast\psi_\ast F=F$. For $F\in\F(\Ms)$, it also holds that $\psi_\ast\psi^\ast F=F$ if and only if the $n^\text{th}$ component of $F$ is supported in $\psi(\Ns)^{\times n}$ for $1\leq n\leq O(F)$. We may naturally define the push-forward on elements of $\Ccs(\Ns)$ by identifying it with the push-forward on $\F^1(\Ns)$.
We will now construct the map $\A(\psi):\A(\Ns)\to\A(\Ms)$ for a \Loc-arrow $\psi:\Ns\hookrightarrow\Ms$, and demonstrate that under this definition $\A$ becomes a covariant functor from \Loc\ to \Alg.
\begin{lemma}
\label{lem:kgfmor}
Let $\Ns,\Ms$ be
objects in \Loc, and $\psi:\Ns\hookrightarrow\Ms$ be a \Loc-arrow. Then, for any $F,F'\in\F(\Ns)$ we have
$F\sim_\Ns F'$ if and only if $\F(\psi)F\sM\F(\psi)F'$.
\end{lemma}
\begin{proof}
If $\F(\psi)F\sim_\Ms \F(\psi)F'$ then we have $(\F(\psi)F)[\EM g]=(\F(\psi)F')[\EM g]$ for every $g\in\Ccs(\Ms)$. Now, for every $f\in\Ccs(\Ns)$ it holds that $E_\Ns f=\psi^\ast\EM\psi_\ast f$; since $\psi_\ast f\in\Ccs(\Ms)$, it follows that
\[
F[E_\Ns f]=(\F(\psi)F)[\EM\psi_\ast f]=(\F(\psi)F')[\EM\psi_\ast f]=F'[E_\Ns f].
\]
Therefore $F\sim_\Ns F'$.
Now suppose that $F\sim_\Ns F'$. Since $O(F),$ $O(F')$ are finite, it follows that there is a compact region $K\subset\Ns$ with the property that the support of the $n^\text{th}$ components of both $F$ and $F'$ lie within $K^{\times n}$ for $1\leq n\leq\max(O(F),O(F'))$. Let $\Sigma_\Ns$ be a Cauchy surface for $\Ns$, and consider the intersection $S=J_\Ns(K)\cap\Sigma_\Ns$; for any classical solution $\EM f,$ $f\in\Ccs(\Ms)$, it will always be possible to pick a smooth pair of functions $(\varphi_f,\pi_f)$ on $\Sigma_\Ns$ which are compactly supported and coincide with the Cauchy data for $\psi^\ast\EM f$ on $S$ (even if $\psi(\Sigma_\Ns)$ cannot be extended to a Cauchy surface for $\Ms$). But since $(\varphi_f,\pi_f)$ are compactly supported they provide data for a solution $E_\Ns g$, for some $g\in\Ccs(\Ns)$. It then holds that $E_\Ns g$ must coincide with $\psi^\ast\EM f$ on the domain of determinacy of $S$; since this region contains $K$, it holds that $(E_\Ns g)|_K=(\psi^\ast\EM f)|_K$. It follows that
\begin{align*}
(\F(\psi)F)[\EM f]&=(\F(\psi)F)[\psi_\ast E_\Ns g]=F[E_\Ns g]\\
&=F'[E_\Ns g]=(\F(\psi)F')[\psi_\ast E_\Ns g]=(\F(\psi)F')[\EM f].
\end{align*}
Since the choice of $f\in\Ccs(\Ms)$ was arbitrary, we may conclude that $\F(\psi)F\sim_\Ms\F(\psi)F'$.
\end{proof}
\begin{lemma}
\label{lem:fstarmon}
Let $\Ns,\Ms$ be
objects in \Loc, and $\psi:\Ns\hookrightarrow\Ms$ be a \Loc-arrow. Then $\F(\psi)$ is a $\ast$-monomorphism.
\end{lemma}
\begin{proof}
Let $F\in\F(\Ns)$ and $f\in\Cs(\Ms)$. Writing $F=\sum_{n=0}^Nt_n$ with $t_n\in\F^n(\Ns)$, we have
\[
\F(\psi)(F^\ast)=\sum_{n=0}^N\psi_\ast \bar t_n=\sum_{n=0}^N\bar{\psi_\ast t_n}=(\F(\psi)F)^\ast.
\]
Now let $F,F'\in\F(\Ns)$; we have
\begin{align*}
\F(\psi)(F\star F')&=\sum_k\frac{i^k}{2^kk!}\F(\psi)
\left[E_\Ns^k\left(F^{(k)}[\,\cdot\,],F'^{(k)}[\,\cdot\,]\right)\right]\\
&=\sum_k\frac{i^k}{2^kk!}E_\Ns^k\left(F^{(k)}[\psi^\ast\,\cdot\,],F'^{(k)}[\psi^\ast\,\cdot\,]\right).
\end{align*}
But for any distributions $t,t'\in \F^k(\Ns)$, we have
\[
E_\Ns^k(t,t')=\EM^k(\psi_\ast t,\psi_\ast t'),
\]
and it is also easy to see that for any $F\in\F(\Ns)$, we have $F^{(k)}[\psi^\ast f]=\psi_\ast(\F(\psi)F)^{(k)}[f]$. It follows that
\begin{align*}
\F(\psi)(F\star F')&=\sum_k\frac{i^k}{2^kk!}\EM^k\left((\F(\psi)F)^{(k)}[\,\cdot\,],
(\F(\psi)F')^{(k)}[\,\cdot\,]\right)\\
&=(\F(\psi)F)\star(\F(\psi)F').
\end{align*}
It remains to show that $\F(\psi)$ is injective. If $F,F'\in\F(\Ns)$ with $F\neq F'$, there exists some $f\in\Ccs(\Ns)$ with $F[f]\neq F'[f]$; it follows that $(\F(\psi)F)[\psi_\ast f]\neq(\F(\psi)F')[\psi_\ast f]$, and therefore $\F(\psi)F\neq\F(\psi)F'$.
\end{proof}
The final result to prove for $\F(\psi)$ is that it is indeed a covariant functor:
\begin{lemma}
\label{lem:fisfunctor}
The map $\F:\Loc\to\Alg$ which maps an object $\Ms$ to $\F(\Ms)$ and an arrow $\psi$ to $\F(\psi)$ is a covariant functor.
\end{lemma}
\begin{proof}
Lemma \ref{lem:fstarmon} shows that for a \Loc-arrow $\psi:\Ns\hookrightarrow\Ms$, the map $\F(\psi)$ is indeed an arrow from $\F(\Ns)$ to $\F(\Ms)$. All that remains to prove is that $\F(\text{id}_\Ms)=\text{id}_{\F(\Ms)}$ for any spacetime $\Ms$, and that $\F(\psi_2)\circ\F(\psi_1)=\F(\psi_2\circ\psi_1)$ for any composable \Loc-arrows $\psi_1,\psi_2$. These result directly from the observations that $\text{id}_\Ms^\ast f=f$ for any $f\in\Cs(\Ms)$, and that $\psi_1^\ast\circ\psi_2^\ast=(\psi_2\circ\psi_1)^\ast$.
\end{proof}
We now define the map
\begin{align*}
\A(\psi):\A(\Ns)&\to\A(\Ms)\\
[F]_\Ns&\mapsto[\F(\psi)F]_\Ms.
\end{align*}
We can see from lemma \ref{lem:kgfmor} that this map is well defined, and indeed injective; it must also be a $\ast$-homomorphism, as a direct result of the properties of $\F$ proved in lemma \ref{lem:fstarmon}. We can therefore prove the following:
\begin{corollary}
\label{cor:aisfunctor}
The map $\A:\Loc\to\Alg$ which maps an object $\Ms$ to $\A(\Ms)$ and an arrow $\psi$ to $\A(\psi)$ is a covariant functor.
\end{corollary}
\begin{proof}
We have already shown that for any \Loc-arrow $\psi:\Ns\hookrightarrow\Ms$, the map $\A(\psi)$ is an \Alg-arrow from $\A(\Ns)$ to $\A(\Ms)$. The required properties for $\A$ to be a covariant functor follow directly from lemma \ref{lem:fisfunctor}.
\end{proof}
\begin{lemma}
\label{lem:aalgimage}
Let $\psi:\Ns\hookrightarrow\Ms$ be an arrow in \Loc. Then $A\in\A(\psi)(\A(\Ns))$ if and only if there exists $F\in\F(\Ms)$ such that $A=[F]_\Ms$, and $F[\EM f]=F[0]$ for every $f\in\Ccs(\Ms)$ such that $\supp(f)\cap J_\Ms(\Ns)=\emptyset$. Moreover, the theory $\A$ is causal.
\end{lemma}
\begin{proof}
Note that $\A(\psi)(\A(\Ns))$ comprises those elements $A\in\A(\Ms)$ that can be represented by those $F\in\F(\Ms)$ with $F=\sum_{n=0}^Nt_n,$ $t_n\in\F^n(\Ms)$, with the property that each $t_n$ can be written as $\psi_\ast t_n'$ for some $t_n'\in\F^n(\Ns)$. But these are precisely those $F=\sum_{n=0}^Nt_n$ for which $\supp(t_n)\subseteq\Ns^{\times n}$ for $n\geq1$, and so for such an $F$ we have $F[f]=F[0]$ for all $f\in\Ccs(\Ms)$ with $\supp(f)\cap\Ns=\emptyset$. Since $F\sim_\Ms F'$ if and only if $F[\EM f]=F'[\EM f]$ for all $f\in\Ccs(\Ms)$, it follows that $F$ represents an element of $\A(\psi)(\A(\Ns))$ if and only if $F[\EM f]=F[0]$ for all $f\in\Ccs(\Ms)$ with $\supp(f)\cap J_\Ms(\Ns)=\emptyset$.
Now suppose that $\Ns_1$ and $\Ns_2$ are spacetimes embedded in $\Ms$ by \Loc-arrows $\psi_1,\psi_2$ respectively, and that $\psi_1(\Ns_1)$ and $\psi_2(\Ns_2)$ are causally disjoint in $\Ms$. It follows that if $A_i\in\A(\psi_i)(\A(\Ns_i))$, $i=1,2$, we may pick $F_1,F_2\in\F(\Ms)$ such that $[F_i]_\Ms=A_i$, and that the $n^\text{th}$ component of $F_i$ is supported in $(\Ns_i)^{\times n}$. It is then clear from \eqref{eqn:kgproddef},\eqref{eqn:kgproddef2} that $(F\star F')[f]=F[f]F'[f]$ for any $f\in\Ccs(\Ms)$. It follows that $[A_1,A_2]=0$, and therefore the theory is causal.
\end{proof}
As a final note on this construction, we remark that a different construction of the Klein-Gordon scalar field theory is given in \cite{dynloc2}, where dynamical locality is proved in the massive minimally coupled case. The construction given above has the advantage that one is able to easily work with the elements of the algebra $\A(\Ms)$ themselves, rather than its generators only; this makes it easy to compute the relative Cauchy evolution for an arbitrary element directly. There is also a natural extension of this construction to the theory of Wick polynomials.
\subsection{Construction of the Theory of Wick Polynomials}
\label{sect:wpconst}
We can extend the construction of the Klein-Gordon theory to a larger theory containing the Wick polynomials. The general aim is to include in the algebras of functionals previously denoted $\F(\Ms)$ a greater range of distributions. The resulting enlarged theory will be denoted $\W$. The following construction follows \cite{bf:qftcb} and \cite{chilfred}.
We first need to establish the behaviour of the fundamental solution $\EM$ and the Klein-Gordon operator $\PM$ on distributions. For a distribution $t\in\mathcal{D}'(\Ms)$ (resp. $\mathcal{E}'(\Ms)$, i.e.\ compactly supported distributions), and arbitrary $f\in\Ccs(\Ms)$ (resp.\ $\Cs(\Ms)$), we simply define
\[
\left\langle\PM t,f\right\rangle=\left\langle t,\PM f\right\rangle.
\]
Since $\PM$ is a formally self-adjoint linear differential operator, the restriction of the map $\PM:\mathcal{D}'(\Ms)\to\mathcal{D}'(\Ms)$ to $\Cs(\Ms)$ is compatible with the previous definition of $\PM$ on smooth functions.
Now, analogously to the case for smooth functions, we now wish to construct maps $\bar{\EM^\pm}:\mathcal{E}'\to\mathcal{D}'$ satisfying
\begin{align}
\bar{\EM^\pm}\PM t&=t=\PM\bar{\EM^\pm} t\label{eqn:Eprop1dist}\\
\supp(\bar{\EM^\pm} t)&\subseteq J_\Ms^\pm(\supp(t)).\label{eqn:Eprop2dist}
\end{align}
We therefore let $\bar{\EM^\pm} t=(\EM^\mp)'t$: this expression is clearly a well-defined element of $\mathcal{D}'(\Ms)$ for any $t\in\mathcal{E}'(\Ms)$, and this definition ensures that \eqref{eqn:Eprop1dist} is satisfied. Moreover, we may see that \eqref{eqn:Eprop2dist} is satisfied by noting that for any $t\in\mathcal{E}'(\Ms)$, $f\in\Ccs(\Ms)$, we have $J_\Ms^\pm(\supp(t))\cap\supp(f)=\emptyset$ if and only if $\supp(t)\cap J^\mp_\Ms(\supp(f))=\emptyset$. We know that the maps $\EM^\pm$ satisfying \eqref{eqn:funsol1}, \eqref{eqn:funsol2} are unique, so the restrictions of $\bar{\EM^\pm}$ to $\Ccs(\Ms)$ must coincide with $\EM^\pm$.
As before, we let the fundamental solution $\EM:\mathcal{E}'(\Ms)\to\mathcal{D}'(\Ms)$ be defined by $\bar{\EM}=\bar{\EM^-}-\bar{\EM^+}$, and therefore $\bar{\EM}=-(\EM)'$, as would be expected from the relation \eqref{eqn:Eantiprop}.
From now on, we will drop the bar from the notation and simply write $\EM^{(\pm)} t$ for a distribution $t\in\mathcal{E}'(\Ms)$.
Recall that for any spacetime $\Ms$, the algebra of functionals $\F(\Ms)$ consists of elements of the form $F=\sum_{n=0}^Nt_n$, with each $t_n\in\F^n(\Ms)$ being a finite sum of finite products of test functions of one variable. We wish to include a much wider range of allowed distributions into the new theory $\W$, but we must apply enough restrictions to ensure that the resulting expressions are well defined. We might na\"{i}vely assume that we can use the same product as defined in \eqref{eqn:kgproddef} for distributions, but this is not the case. For example, consider two elements $t,t'\in\F^1(\Ms)$; we see that for any $f\in\Cs(\Ms)$,
\[
(t\star t')[f]=\int_{\Ms^{\times 2}}dx\,dy\,t(x)t'(y)\left(f(x)f(y)+\frac i2\EM(x,y)\right);
\]
again, for $t\in\mathcal{E}'(\Ms^{\times n})$ we use the notation
\[
t[f]=\left\langle t,\tenpow fn\right\rangle
=\int_{\Ms^{\times n}}d^{n-1}x\,t(x_1,\ldots,x_n)f(x_1)\cdots f(x_n),
\]
so for any $f\in\Ccs(\Ms)$ we have $t[\EM f]=(-1)^n(\tenpow\EM nt)[f].$
When $t$ and $t'$ are test functions the second term above is well defined, but pointwise products of distributions are not always so, and we require both a condition on the existence of such pointwise products and a deformation of the product to ensure that all the expressions are well defined. We can find a suitable condition for existence of pointwise products in \cite{hormander1}, namely \emph{H\"ormander's criterion}: If $t$ and $t'$ are distributions, then the pointwise product $t(x)t'(x)$ is a well-defined distribution if the set
\[
\{(x,k+k'):(x,k)\in WF(t),\ (x,k')\in WF(t')\}
\]
contains no element of the form $(x,0)$.
It is well known (see e.g.\ \cite{duistermaathormander}) that the wavefront set of the distribution $\EM(x,y)$ satisfies
\[
WF(\EM)\subset \bigcup_{\substack{x,y\in\Ms\\x\leftrightarrow y}}
(V^+_{\Ms;x}\times V^-_{\Ms;y})\cup (V^-_{\Ms;x}\times V^+_{\Ms;y}),
\]
where $V^\pm_{\Ms;x}\subset T^\ast_x\Ms$ is the forward/backward light cone at $x$, and $x\leftrightarrow y$ indicates that $x$ and $y$ are connected by a null geodesic. We denote by $V^\pm_\Ms$ the union $\bigcup_{x\in\Ms}V^\pm_{\Ms;x}$. We then define for $n\geq1$ (cf.\ \cite{chilfred})
\[
\T^n(\Ms)=\{t\in\mathcal{E}'(\Ms^{\times n}):t\text{ totally symmetric},\ WF(t)\cap \bar{(V^+_\Ms)^{\times n}\cup (V_\Ms^-)^{\times n}}=\emptyset\}.
\]
As before we also define $\T^0(\Ms)=\mathbb{C}$.
Such a definition ensures that the expression $\int_{\Ms^{\times2}}dx\,dy\,t(x)t'(y)\EM(x,y)$ for $t,t'\in\T^1(\Ms)$ is well defined (and more generally, that
\[
\int_{\Ms^{\times2}}dx_1\,dy\,t_n(x_1,\ldots,x_n)t'(y)\EM(x_1,y)
\]
for $t_n\in\T^n(\Ms)$, $t\in\T^1(\Ms)$ is always a well defined element of $\T^{n-1}(\Ms)$). Analogously to the previous case, we wish to define an algebra $\T(\Ms)$ comprising elements of the form
\beq
\label{eqn:thdef}
T=\sum_{n=0}^Nt_n
\eeq
with $t_n\in\T^n(\Ms)$. For any $f\in\Cs(\Ms)$ and $T$ of the above form we define the functional derivative $\T^{(k)}[f]$ in the same way as detailed in \eqref{eqn:kgfuncderiv1} and \eqref{eqn:kgfuncderiv2}. It is clear from the definition of $\T^k(\Ms)$ that the functional derivative $\T^{(k)}[f]$ is an element of $\T^k(\Ms)$.
It is shown in \cite{chilfred} that for any $t\in\T^n(\Ms)$, the wavefront set of $(\EM^\pm)_k t$ has the property that $WF((\EM^\pm)_k t) \cap\bar{(V_\Ms^+)^{\times n}\cup(V_\Ms^-)^{\times n}}=\emptyset$, where $(\EM^\pm)_k=\tenpow\id{k-1}\otimes\EM^\pm\otimes\tenpow\id{n-k}$. Since differential operators and multiplication by smooth functions cannot enlarge the wavefront set of a distribution, it follows that any element of $\mathcal{E}'(\Ms^{\times n})$ which is obtained via application of any such operators and $(\EM^\pm)_k$ on an element of $\T^n(\Ms)$ must itself be an element of $\T^n(\Ms)$.
Unfortunately, the restriction on elements of $\T^n(\Ms)$ alone does not solve the problem of ill-defined distributions. Note that for any $g\in\Ccs(\Ms)$, the distribution $t_2(x,y)=g(x)\delta(x-y)$ has empty wavefront set, and is therefore an element of $\T^2(\Ms)$; however
\[
(t_2\star t_2)[f]=\int_{\Ms^{\times 2}}dx\,dy\,t(x)t(y)\left(f(x)^2f(y)^2+2i
\EM(x,y)f(x)g(y)-\frac12\EM(x,y)^2\right),
\]
and the distribution $\EM(x,y)^2$ is ill-defined since it does not obey H\"ormander's criterion. A solution to this problem is given in \cite{bf:qftcb}: on each spacetime $\Ms$, it is possible to find symmetric distributions $H$ which satisfy the properties
\[
WF(\EM+2iH)=WF(\EM)\cap(V^+_\Ms\times V^-_\Ms)
\]
and
\beq
\label{eqn:hbisol}
H(\PM f,f')=0
\eeq
for all $f,f'\in\Ccs(\Ms)$.
There is no unique choice for $H$, and we denote by $\H(\Ms)$ the set of all such distributions. It follows that the distribution $(\EM+2iH)^k$ is well defined for any $k\geq1$ and $H\in\H(\Ms)$, and consequently we define a new product $\star_H$ that acts on distributions as
\[
(T\star_H T')[f]=\sum_{k=0}^{\text{min}(O(T),O(T'))}\frac{i^k}{2^kk!}E_{\Ms;H}^k\left(F^{(k)}[f],F'^{(k)}[f]\right),
\]
where for $t,t'\in\T^k(\Ms)$, we define
\[
E_{\Ms;H}^k(t,t')=\int_{\Ms^{\times(2k)}}d^kx\,d^ky\,t(x_1,\ldots,x_k)t'(y_1,\ldots,y_k)\prod_{j=1}^k
(E_\Ms(x_j,y_j)+2iH(x_j,y_j))
\]
for $k\geq1$. As before, we define $E_{\Ms;H}^0(\alpha,\beta)=\alpha\beta$. One can show that this product is still associative. We then denote by $\T_H(\Ms)$ the algebra comprising elements of the form given in \eqref{eqn:thdef} with product $\star_H$. Addition and involution on $\T_H(\Ms)$ are again given by addition and complex conjugation of distributions respectively.
It is possible to show that for any pair $H,H'\in\H(\Ms)$, the difference $H-H'$ is smooth \cite[Theorem 6]{bf:qftcb}, and also that the algebras $\T_H(\Ms)$ and $\T_{H'}(\Ms)$ are isomorphic; if we define the map
\begin{align}
\lambda_{H,H'}:\T_H(\Ms)&\to\T_{H'}(\Ms)\notag\\
T&\mapsto\sum_{n=0}^{\lfloor O(T)/2\rfloor}\frac1{n!}\left\langle
\tenpow{(H-H')}n,T^{(2n)}\right\rangle\label{eqn:lambdadef}
\end{align}
where for $t\in\T^{2n}(\Ms)$,
\beq
\left\langle\tenpow{(H-H')}n,t\right\rangle=\int_{\Ms^{\times(2n)}}d^{2n}x\,t(x_1,\ldots,x_{2n})\prod_{j=1}^n
(H(x_{2j-1},x_{2j})-H'(x_{2j-1},x_{2j})),
\label{eqn:anglefuncdef}
\eeq
then this is an isomorphism satisfying $\lambda_{H,H'}=\lambda_{H',H}^{-1}$, $\lambda_{H',H''}\circ\lambda_{H,H'}=\lambda_{H,H''}$ and
\[
T\star_{H}T'=\lambda_{H,H'}^{-1}(\lambda_{H,H'}(T)\star_{H'}\lambda_{H,H'}(T')).
\]
In exactly the same way that the set $\J(\Ms)$ is an ideal for $\F(\Ms)$, it also holds that the analogous set
\[
\JJ(\Ms)=\{T\in\T_H(\Ms):T[\EM f]=0\text{ for all }f\in\Ccs(\Ms)\}
\]
(which is independent of the choice of $H\in\H(\Ms)$) is an ideal for $\T_H(\Ms)$. We therefore define the algebra $\W_H(\Ms)=\T_H(\Ms)/\JJ(\Ms).$ Since the equivalence class of an element $T\in\T_H(\Ms)$ does not depend on $H$, we will denote it unambiguously by $[T]_\Ms$, and if $T-T'\in\JJ(\Ms)$ we will write $T\sim_\Ms T'$ as before. It follows from \eqref{eqn:hbisol} that $T\sim_\Ms T'$ if and only if $\lambda_{H,H'}T\sim_\Ms\lambda_{H,H'}T'$, so the isomorphism
\begin{align*}
\tl_{H,H'}:\W_H(\Ms)&\to\W_{H'}(\Ms)\\
[T]_{\Ms}&\mapsto[\lambda_{H,H'}T]_{\Ms}
\end{align*}
is well defined. We also note that the reasoning used to show lemma \ref{lem:kgfuncpolar} can be similarly used to show the corresponding result; that if $T\in\T_H(\Ms)$ can be written $T=\sum_{n=0}^Nt_n$ with $t_n\in\T^n(\Ms)$ for each $n$, then $T\in\JJ(\Ms)$ if and only if $t_0=0$ and
\beq
\label{eqn:wpfuncpolar}
\tenpow{\EM} n t_n=0
\eeq
for all $n=1,\ldots,N$.
Since there is no preferred method of uniquely specifying some $H\in\H(\Ms)$ for each spacetime $\Ms$, the above construction does not constitute a locally covariant theory, as we have not yet defined a unique algebra for each $\Ms$. We therefore wish to construct an algebra $\W(\Ms)$ which is independent of the choice of $H$. Again following \cite{bf:qftcb}, we do this by letting $\W(\Ms)$ comprise families of elements indexed by choice of $H\in\H(\Ms)$, as follows:
\[
\W(\Ms)=\{(W_H)_{H\in\H(\Ms)}:\tl_{H,H'}W_H=W_{H'}\text{ for all }H,H'\in\H(\Ms)\}.
\]
Given $W=(W_H)_{H\in\H(\Ms)}$, $W'=(W'_H)_{H\in\H(\Ms)}$, we define $(W+W')_H=W_H+W'_H$, $(W\star W')_H=W_H\star_H W'_H$ and $(W^\ast)_H=W_H^\ast$. These operations are clearly consistent with the compatibility condition $\tl_{H,H'}W_H=W_{H'}$. Since this condition also ensures that each family $W=(W_H)_{H\in\H(\Ms)}\in\W(\Ms)$ is completely defined by any single entry $W_H$, it follows that $\W(\Ms)\cong\W_H(\Ms)$ for any $H\in\H(\Ms)$.
Having given a prescription for defining $\W(\Ms)$, we must now find a suitable definition for the \Alg-arrow $\W(\psi)$ corresponding to a \Loc-arrow $\psi:\Ns\hookrightarrow\Ms.$ Throughout this section we will use the same notation as before for the definition of the pullback and pushforward of a \Loc-arrow on an arbitrary functional.
\begin{lemma}
Let $\Ns$, $\Ms$ be locally covariant theories, and let $\psi:\Ns\hookrightarrow\Ms$ be an arrow in \Loc. Then for any $H\in\H(\Ms),$ we have $\psi^\ast H\in\H(\Ns)$.
\end{lemma}
\begin{proof}
We have $WF(\phi^\ast t)\subseteq \phi^\ast WF(t)$ for any smooth $\phi:\Ns\to\Ms$ and distribution $t$ on $\Ms$ \cite[Theorem 2.5.11$'$]{hormanderfio1}. It is a clear consequence that we have equality whenever $\phi$ is a local diffeomorphism; this entails that when $\psi:\Ns\hookrightarrow\Ms$ is an arrow in \Loc, we have $WF(\psi^\ast T)=\psi^\ast WF(T)$ for any $T\in\mathcal{D}'(\Ms^{\times n})$.
Therefore
\[
WF(E_\Ns+2i\psi^\ast H)=\psi^\ast WF(\EM+2iH)=WF(E_\Ns)\cap(V_\Ns^+\times V_\Ns^-).
\]
Moreover, if $H(\PM f,f')=0$ for all $f,f'\in\Ccs(\Ms)$, it follows that $\psi^\ast H(P_\Ns f,f')=H(\PM\psi_\ast f,\psi_\ast f')=0$ for all $f,f'\in\Ccs(\Ns)$. Therefore $\psi^\ast H\in\H(\Ns)$.
\end{proof}
Note that for any \Loc-arrow $\psi:\Ns\hookrightarrow\Ms$, we may also say that $WF(\psi_\ast U)=\psi_\ast WF(U)$ for $U\in\mathcal{E}'(\Ns^{\times n})$.\footnote{We require compact support of $U$ here; if $U\in\mathcal{D}'(\Ns)$, then we might not have equality, although $(x_1,\ldots,x_n;k_1,\ldots,k_n)\in WF(\psi_\ast U)\setminus\psi_\ast WF(U)$ only if $x_k\in\partial(\psi(\Ns))$ for each $k$.}
Now, for any $H\in\H(\Ms)$ we define the map
\begin{align*}
\T_H(\psi):\T_{\psi^\ast H}(\Ns)&\to\T_H(\Ms)\\
T&\mapsto T\circ\psi^\ast.
\end{align*}
For any $T=\sum_{n=0}^Nt_n\in\T_{\psi^\ast H}(\Ns)$, $t_n\in\T^n(\Ns)$, we have
\[
(\T_H(\psi)T)[f]=\sum_{n=0}^N\psi_\ast t_n[f]
\]
as before, and since $t_n$ is compactly supported for each $n\geq1$, it follows that $WF(\psi_\ast t_n)= \psi_\ast WF(t_n)$. Thus $\T_H(\psi)T$ is an element of $\T_H(\Ms)$ as required. We can also use the same argument as for lemma \ref{lem:kgfmor} to see that for any $T,T'\in\T_{\psi^\ast H}(\Ns)$, it holds that $T\sim_\Ns T'$ if and only if $\T_H(\psi)T\sim_\Ms\T_H(\psi)T'$. Moreover, the result of lemma \ref{lem:fstarmon} extends directly to $\T_H(\psi)$, so it is indeed a $\ast$-monomorphism. Therefore the map
\begin{align}
\W_H(\psi):\W_{\psi^\ast H}(\Ns)&\to\W_H(\Ms)\notag\\
[T]_\Ns&\mapsto[\T_H(\psi)T]_\Ms\label{eqn:WHpsidef}
\end{align}
is a well-defined $\ast$-monomorphism. From this, we define the map $\W(\psi):\W(\Ns)\to\W(\Ms)$ by
\beq
(\W(\psi)W)_H=\W_H(\psi)W_{\psi^\ast H},
\label{eqn:Wpsidef}
\eeq
where $H\in\H(\Ms)$.
It is easy to show that this definition is consistent with the compatibility condition: i.e.\ $\tl_{H,H'}(\W(\psi)W)_H=(\W(\psi)W)_{H'}$ for all $H,H'\in\H(\Ms)$. We then have:
\begin{lemma}
The map $\W:\Loc\to\Alg$ which maps spacetimes $\Ms$ to $\W(\Ms)$ and \Loc-arrows $\psi$ to $\W(\psi)$ is a covariant functor.
\end{lemma}
\begin{proof}
It is trivial to show that for any spacetime $\Ms$, we have $\W(\text{id}_\Ms)=\text{id}_{\W(\Ms)}$. It remains to show that for any \Loc-arrows $\psi_1:\Ns_1\hookrightarrow\Ns_2,$ $\psi_2:\Ns_2\hookrightarrow\Ms$, it holds that $\W(\psi_2)\circ\W(\psi_1)=\W(\psi_2\circ\psi_1)$. For any $T\in\T_{\psi_1^\ast\psi_2^\ast H}(\Ns_1)$ and $\H\in\H(\Ms)$, we have
\[
\T_H(\psi_2)\T_{\psi^\ast_2H}(\psi_1)T=T\circ\psi_1^\ast\circ \psi_2^\ast=
\T_H(\psi_2\circ\psi_1)T.
\]
The desired result follows by \eqref{eqn:WHpsidef}, \eqref{eqn:Wpsidef}.
\end{proof}
The covariant functor $\W$ is thus a locally covariant theory which represents the extended algebra of Wick polynomials. We also have the corresponding result to lemma \ref{lem:aalgimage}:
\begin{lemma}
\label{lem:walgimage}
Let $\psi:\Ns\hookrightarrow\Ms$ be an arrow in \Loc. Then $W\in\W(\psi)(\W(\Ns))$ if and only if there exists $T\in\T_H(\Ms)$ such that $W_H=[T]_\Ms$ for some $H\in\H(\Ms)$, and $T[\EM f]=T[0]$ for every $f\in\Ccs(\Ms)$ such that $\supp(f)\cap J_\Ms(\Ns)=\emptyset$. Moreover, the theory $\W$ is causal.
\end{lemma}
\begin{proof}
$W\in\W(\psi)(\W(\Ms))$ if and only if we have $W_H\in\W_H(\psi)(\W_{\psi^\ast H}(\Ns))$ for some (and consequently every) $H\in\H(\Ms)$; the required results then follow using an analogous argument to that given in the proof of lemma \ref{lem:aalgimage}.
\end{proof}
\subsection{Spaces of smooth functions on spacetimes}
Before we consider the timeslice axiom and dynamical locality of the two theories, we discuss the following spaces of smooth functions on $\Ms$, in addition to $\Ccs(\Ms)$ and $\Cs(\Ms)$. We define
\begin{align*}
\Cs_s(\Ms)&=\{f\in\Cs(\Ms):\supp(f)\subseteq J_\Ms(K)
\text{ for some compact }K\subset\Ms\},\\
\Cs_{s,\pm}(\Ms)&=\{f\in\Cs_s(\Ms):\supp(f)\subseteq J_\Ms^\pm(K)
\text{ for some compact }K\subset\Ms\}.
\end{align*}
We also use the following notation for the canonical embeddings
\begin{align*}
\iota_{0,\pm}&:\Ccs(\Ms)\hookrightarrow\Cs_{s,\pm}(\Ms),\\
\iota_{\pm,s}&:\Cs_{s,\pm}(\Ms)\hookrightarrow\Cs_s(\Ms),\\
\iota_{s,\infty}&:\Cs_s(\Ms)\hookrightarrow\Cs(\Ms).
\end{align*}
We wish to demonstrate that there exist continuous maps $\hEM^\pm:\Ccs(\Ms)\to\Cs_{s,\pm}(\Ms)$ that satisfy $\EM^\pm=\iota_{s,\infty}\circ\iota_{\pm,s}\circ\hEM^\pm$. It is clear that for any $f\in\Ccs(\Ms)$, the function $\EM^\pm f$ lies within the range of $\iota_{s,\infty}\circ\iota_{\pm,s}$, we may unambiguously let $\hEM^\pm=(\iota_{s,\infty}\circ\iota_{\pm,\infty})^{-1}\circ\EM^\pm.$ To establish continuity we must first define the topologies on each of these spaces of functions. The spaces $\Cs(\Ms)$ and $\Ccs(\Ms)$ can be constructed as convex topological spaces, as follows \cite{waldmann,reedsimon}. A \emph{compact exhausting sequence} for $\Ms$ is a sequence $(K_n)_{n\in\mathbb{N}}$ of compact submanifolds of $\Ms$ such that $K_n\subset\mathring{K}_{n+1}$ for each $n$, and for every point $p\in\Ms$ there exists $N\in\mathbb{N}$ such that $p\in K_n$ for all $n>N$. Any space of smooth functions on a smooth manifold can be endowed with the $\Cs$ topology; we do not need to go into details here, except to say that the topology on $C^\infty(\Ms)$ is generated by seminorms $p_{K_n,k}$, $k,n\in\mathbb{N}$, where $(K_n)_{n\in\mathbb{N}}$ is a compact exhausting sequence for $\Ms$, and $p_{K_n,k}(f)$ is given by the supremum over $K_n$ of the norms of all covariant derivatives of $f$ of order no greater than $k$ (using a Riemannian metric to induce the norms of the derivatives). The $\Cs$ topology on a space of smooth functions on $\Ms$ is then defined as the subspace topology induced from $\Cs(\Ms)$. The topology of $\Ccs(\Ms)$, on the other hand, is constructed as an inductive limit of the topological spaces $\Cs_{K_n}$ (that is, the finest topology such that each embedding $\iota_n:\Cs_{K_n}(\Ms)\hookrightarrow\Ccs(\Ms)$ is continuous), where $(K_n)_{n\in\mathbb{N}}$ is again a compact exhausting sequence for $\Ms$, and $\Cs_K(\Ms)$ is the space $\{f\in\Cs(\Ms):\supp(f)\subseteq K\}$ endowed with the $\Cs$ topology. Now, for any inductive limit $X$ of locally convex spaces $(X_n)_{n\in\mathbb{N}}$, and locally convex space $Y$, a map $T:X\to Y$ is continuous if and only if each restriction $T|_{X_n}:X_N\to Y$ is continuous \cite[Theorem V.16]{reedsimon}. Since the space $\Cs_{K_n}(\Ms)$ inherits the subspace topology induced from $\Cs(\Ms)$, it follows that the embedding $\Ccs(\Ms)\hookrightarrow\Cs(\Ms)$ is continuous.
Now, for a spacetime $\Ms$ we wish to endow $\Cs_s(\Ms)$ and $\Cs_{s,\pm}(\Ms)$ with topologies in a similar way to that given for $\Ccs(\Ms)$ in \cite{waldmann,reedsimon}; starting with a compact exhausting sequence $(K_n)_{n\in\mathbb{N}}$ for $\Ms$, we consider the topological spaces $\Cs_{J_\Ms(K_n)}(\Ms)$ and $\Cs_{J_\Ms^\pm(K_n)}(\Ms)$ defined analogously to $\Cs_{K_n}(\Ms)$, and let $\Cs_s(\Ms)$ and $\Cs_{s,\pm}(\Ms)$ be the inductive limit of $\Cs_{J_\Ms(K_n)}(\Ms)$ and $\Cs_{J_\Ms^\pm(K_n)}(\Ms)$ respectively as $n\to\infty$. We then have:
\begin{lemma}
The embeddings $\iota_{0,\pm}$, $\iota_{\pm,s}$ and $\iota_{s,\infty}$ are all continuous in the relevant topologies.
\end{lemma}
\begin{proof}
For the sake of readable notation, we denote $X_n=\Cs_{K_n}(\Ms)$, $Y_n^\pm=\Cs_{J_\Ms^\pm(K_n)}(\Ms)$ and $Z_n=\Cs_{J_\Ms(K_n)}$. Firstly, we consider $\iota_{s,\infty}$: for any $n\in\mathbb{N}$, the space $Z_n$ is endowed with the subspace topology induced from $\Cs(\Ms)$, so the embedding must be continuous; therefore $\iota_{s,\infty}|_{Z_n}:Z_n\to\Cs(\Ms)$ is continuous for all $n$, as required for continuity of $\iota_{s,\infty}$. Now, for each $n$ we may factorise $\iota_{\pm,s}|_{Y_n}$ as the composition of the embeddings of $Y_n^\pm\hookrightarrow Z_n$ and $Z_n\hookrightarrow\Cs_s(\Ms)$; the former is continuous as $Y_n^\pm$ has the subspace topology induced from $Z_n$, and the latter is continuous by definition of $\Cs_s(\Ms)$. Therefore $\iota_{\pm,s}$ is continuous. Similarly, we may factorise $\iota_{0,\pm}|_{X_n}$ as the composition of the embeddings of $X_n\hookrightarrow Y_n^\pm$ and $Y_n^\pm\hookrightarrow\Cs_{s,\pm}(\Ms)$, both of which are continuous. Therefore $\iota_{0,\pm}$ is continuous.
\end{proof}
This also allows us to prove:
\begin{lemma}
The maps $\hEM^\pm:\Ccs(\Ms)\to\Cs_{s,\pm}(\Ms)$ are continuous.
\end{lemma}
\begin{proof}
We recall that if a topological space $Y$ is endowed with the subspace topology from a space $Z$, and the embedding is denoted $\iota:Y\hookrightarrow Z$, then a map $T:X\to Y$ is continuous if and only if $\iota\circ T$ is continuous. We note that $X_n=\Cs_{K_n}(\Ms)$ has the subspace topology induced from $\Cs(\Ms)$; since $\EM^\pm:\Ccs(\Ms)\to\Cs(\Ms)$ is continuous, it follows that the restrictions $\EM^\pm|_{X_n}:X_n\to\Cs(\Ms)$ are all continuous. Denoting the canonical embedding by $\iota_n:X_n\hookrightarrow\Cs(\Ms)$, it is clear that we may factorise $\EM^\pm|_{X_n}=\iota_n\circ\hEM^\pm|_{X_n}$, so each $\hEM^\pm|_{X_n}$ is continuous. Therefore $\hEM^\pm$ is continuous.
\end{proof}
We define
\begin{align*}
\hEM:\Ccs(\Ms)&\to\Cs_s(\Ms)\\
f&\mapsto\iota_{s,-}(\hEM^-f)-\iota_{s,+}(\hEM^+f),
\end{align*}
which is clearly continuous; we also define $\cEM:(\Cs_s(\Ms))'\to\mathcal{D}'(\Ms)$ by $\cEM=-(\hEM)'$. The map $\PM$ may be considered to act on elements of $\Cs_s(\Ms)$ and $\Cs_{s,\pm}$ in the obvious way, from which we see that strictly speaking $\PM\hEM^\pm f=\iota_{0,\pm}=\hEM^\pm\PM f$ for any $f\in\Ccs(\Ms)$.
We say that a distribution $t\in\mathcal{D}'(\Ms^{\times n})$ is \emph{time-compact} if there exist spacelike Cauchy surfaces $\Sigma^\pm\subset\Ms$ such that $\supp(t)\subseteq(J_\Ms^-(\Sigma^+)\cap J_\Ms^-(\Sigma^-))^{\times n}.$ Note that the action of a time-compact distribution $t$ is well-defined on $f\in\Cs_s(\Ms^{\times n})$, since the intersection $\supp(t)\cap\supp(f)$ is compact. Therefore any time-compact distribution can be considered to be an element of $(\Cs_s(\Ms^{\times n}))'$. We also say that a distribution $t$ is \emph{future-compact} if there exists a Cauchy surface $\Sigma\subset\Ms$ such that $\supp(t)\subseteq(J_\Ms^-(\Sigma))^{\times n}$, and \emph{past-compact} if there exists a Cauchy surface $\Sigma\subset\Ms$ such that $\supp(t)\subseteq(J_\Ms^+(\Sigma))^{\times n}$. We may similarly see that a future-/past-compact distribution can be considered to be an element of $(\Cs_{s,\pm}(\Ms^{\times n}))'$.
We then have the following result, which will be important later:
\begin{lemma}
\label{lem:eetapchi}
Let $u\in\mathcal{D}'(\Ms)$, with $u[\PM f]=0$ for all $f\in\Ccs(\Ms)$. Then there exists a distribution $t\in(\Cs_s(\Ms))'$ such that $u=\cEM t$.
\end{lemma}
\begin{proof}
Let $\Sigma^\pm$ be two disjoint Cauchy surfaces in $\Ms$ with $\Sigma^+\subset J_\Ms^+(\Sigma^-)$, and let $\cadv+\cret=1$ be a smooth partition of unity with $\cadv(x)=0,$ $\cret(x)=1$ for $x\in J_\Ms^+(\Sigma^+)$ and $\cadv(x)=1,$ $\cret(x)=0$ for $x\in J_\Ms^-(\Sigma^-)$. Now, let $\eta\in\Cs(\Ms)$ be time-compact, and defined such that $\eta(x)=1$ for all $x\in J_\Ms^-(\widetilde\Sigma^+)\cap J_\Ms^+(\widetilde\Sigma^-)$, where $\widetilde\Sigma^\pm\subset\Ms$ are further Cauchy surfaces disjoint from $\Sigma^\pm$ with $\Sigma^\pm\subset J_\Ms^\mp(\widetilde\Sigma^\pm)$. We define the map $\eta_s:\Cs_s(\Ms)\to\Ccs(\Ms)$ by the action of multiplication by $\eta$. We also consider $\chi^{\text{adv/ret}}:\Ccs(\Ms)\to\Ccs(\Ms)$ as defined by action of multiplication. The operator $\PM$ can be considered as an endomorphism acting on any of the spaces of functions we defined above; we may similarly consider it as an endomorphism on any of the dual spaces in question, by
\[
\langle \PM u,f\rangle=\langle u,\PM f\rangle.
\]
We may then show that $u=\cEM\eta_s'\PM(\cadv)' u$, as follows. Let $f\in\Ccs(\Ms)$ be arbitrary, then
\begin{align*}
(\cEM\eta_s'\PM(\cadv)' u)[f]&=-(\eta_s'\PM(\cadv)' u)[\hEM f]\\
&=(\eta_s'\PM(\cadv)' u)[\iota_{+,s}\hEM^+f]-(\eta_s'\PM(\cadv)' u)
[\iota_{-,s}\hEM^-f]\\
&=u[\cadv\PM\eta_s\iota_{+,s}\hEM^+f]-u[\cadv\PM\eta_s\iota_{-,s}
\hEM^-f].
\end{align*}
Since $u[\PM\eta_s\iota_{-,s}\hEM^-f]=0$, we may use $\cadv=\id-\cret$ to see that
\beq
(\cEM\eta_s'\PM(\cadv)'u)[f]=u[\cadv\PM\eta_s\iota_{+,s}\hEM^+f]
+u[\cret\PM\eta_s\iota_{-,s}\hEM^-f].\label{eqn:usplit}
\eeq
Now any $g\in\Cs_{s,+}(\Ms)$ can be split into a sum of three functions $g_-,$ $g_0$ and $g_+$, with the properties that $\supp(g_\pm)\subseteq J_\Ms^\pm(\Sigma^\pm)$ and $\supp(g_0)\subseteq J_\Ms^-(\widetilde\Sigma^+)\cap J_\Ms^+(\widetilde\Sigma^-)$.We may note that $\supp(g_-)$ and $\supp(g_0)$ are both compact, so we can consider $g_0$ and $g_-$ as elements of $\Ccs(\Ms)$, whereupon
\[
g=\iota_{0,+}g_-+\iota_{0,+}g_0+g_+.
\]
By construction, we have $\eta_s\iota_{+,s}\iota_{0,+}g_0=g_0$; the definition of $\cadv$ also shows that $\cadv T_1g_-=T_1g_-$ and $\cadv T_2g_+=0$ for any operators $T_1:\Ccs(\Ms)\to\Ccs(\Ms)$, $T_2:\Cs_{s,+}(\Ms)\to\Ccs(\Ms)$ such that $\supp(T_i f)\subseteq\supp(f)$ for all $f$, $i=1,2$. It follows directly from these that if we let $g=\hEM^+f$ and split as described, then
\begin{align*}
\cadv\PM\eta_s\iota_{+,s}\hEM^+f&=
\cadv\PM\eta_s\iota_{+,s}\iota_{0,+}g_-+
\cadv\PM\eta_s\iota_{+,s}\iota_{0,+}g_0+
\cadv\PM\eta_s\iota_{+,s}g_+\\
&=\PM\eta_s\iota_{+,s}\iota_{0,+}g_-+\cadv\PM g_0.
\end{align*}
But $f=(\iota_{0,+})^{-1}\PM\hEM^+f=\PM g_0+\PM g_-+(\iota_{0,+})^{-1}\PM g_+$, so by the properties of $\cadv$ we have
\begin{align*}
\cadv\PM\eta_s\iota_{+,s}\hEM^+f&=\PM\eta_s\iota_{+,s}\iota_{0,+}g_-
+\cadv f-\cadv\PM g_--\cadv(\iota_{0,+})^{-1}\PM g_+\\
&=\PM(\eta_s\iota_{+,s}\iota_{0,+}-\id)g_-+\cadv f.
\end{align*}
Since $u$ is a weak solution and $(\eta_s\iota_{+,s}\iota_{0,+}-\id)g_-$ is compactly supported, we have
\[
u[\cadv\PM\eta_s\iota_{+,s}\hEM^+f]=u[\cadv f].
\]
We may similarly conclude that $u[\cret\PM\eta_s\iota_{-,s}\hEM^-f]=u[\cret f].$ It follows from \eqref{eqn:usplit} that
\begin{align*}
(\cEM\eta_s'\PM(\cadv)'u)[f]&=u[\cadv f]+u[\cret f]\\
&=u[f].
\end{align*}
This proves the required result, and also gives us an explicit example of a distribution $t\in(\Cs_s(\Ms))'$ satisfying $u=\cEM t$.
\end{proof}
While we have been very careful with our definitions in this section, in the remainder of the paper we will not need to be so exact with our notation. Firstly, we make the observation that since any multiplication operator $\mu$ between spaces of smooth functions is formally self-adjoint, it makes sense to write $\mu't=\mu t$ for a distribution $t$ and formally regard $\mu t$ as the pointwise product of $t$ with the underlying function $\mu\in\Cs(\Ms)$. We will particularly use this convention when a distributional solution $u$ is of the form $u=\EM t$, where $t\in\mathcal{E}'(\Ms)$. Lemma \ref{lem:eetapchi} tells us that $\EM t=\cEM\eta_s'\PM(\cadv)'\EM t=\cEM\eta\PM\cadv\EM t$; however, regarding $\cadv\EM t$ as a pointwise product allows us to see that in fact the distribution $\PM\cadv\EM t$ must be supported within the region $J_\Ms^-(\Sigma^+)\cap J_\Ms^+(\Sigma^-)$ where $\cadv$ is non-constant, by the properties of $\PM$ and $\EM$. Moreover, the support of $\PM\cadv\EM t$ lies within $J_\Ms(\supp(t))$, which has compact intersection with $J_\Ms^-(\Sigma^+)\cap J_\Ms^+(\Sigma^-)$, so the support of $\PM\cadv\EM t$ is compact. Since $\eta=1$ everywhere within $\supp(\PM\cadv\EM t)$, we may suppress $\eta$ and instead regard $\PM\cadv\EM t$ itself as an element of $\mathcal{E}'(\Ms)$, writing
\beq
\label{eqn:epchidist}
\EM\PM\cadv\EM t=\EM t.
\eeq
Moreover $\PM\EM t=0$ for any $t\in\mathcal{E}'(\Ms)$, so we also have
\[
\EM\PM\cret\EM t=-\EM t.
\]
\section{The timeslice axiom and relative Cauchy evolution}
It is well known that both the Klein-Gordon theory \cite{bfv} and the enlarged algebra of Wick polynomials \cite{chilfred} obey the timeslice axiom. However, we will give a proof that the timeslice axiom holds in both cases, since the construction is different to that used in the aforementioned references, and since we require an explicit expression for the inverse maps $\A(\psi)^{-1}$ and $\W(\psi)^{-1}$ when $\psi:\Ns\hookrightarrow\Ms$ is a Cauchy arrow.
\subsection{The timeslice axiom for the Klein-Gordon Theory}
\label{sect:kgtimeslice}
In order to compute the relative Cauchy evolution for either $\A$ or $\W$, we must first demonstrate that they obey the timeslice axiom. It is worth asking first whether the theory $\F$ obeys the timeslice axiom; since the construction for $\F$ contains no condition relating to the field equation, we should not expect $\F$ to obey the axiom, and indeed this is the case: let $\Ns,\Ms$ be objects in \Loc, and $\psi:\Ns\hookrightarrow\Ms$ be a Cauchy arrow in \Loc. Suppose that $\psi(\Ns)\neq\Ms$; then, pick some nonzero $t\in\F^1(\Ms)$ whose support lies within $\Ms\setminus\psi(\Ns)$. Clearly $t\!\left[\bar t\right]\neq 0$, but as $\psi^\ast \bar t=0$, we have $(\F(\psi)F)\!\left[\bar t\right]=0$ for all $F\in\F(\Ns)$. Therefore $\F(\psi)$ is not surjective, and consequently cannot be invertible; hence $\F$ does not satisfy the timeslice axiom.
To demonstrate that $\A$, on the other hand, does obey the timeslice axiom, we use following lemma, which is proved in \cite{dimock} (and can also be seen to be a consequence of lemma \ref{lem:eetapchi}: see \eqref{eqn:epchidist}).
\begin{lemma}
\label{lem:epchi}
Let $\Sadv,\Sret$ be disjoint Cauchy surfaces in a globally hyperbolic spacetime $\Ms$, with $\Sret\subseteq J_\Ms^+(\Sadv)$, and let $\cadv+\cret=1$ be a smooth partition of unity on $\Ms$ with $\cadv(x)=0,\ \cret(x)=1$ for $x\in J_\Ms^+(\Sret)$ and $\cadv(x)=1,\ \cret(x)=0$ for $x\in J_\Ms^-(\Sadv)$. Then
\begin{align*}
\EM \PM\cadv \EM f&=\EM f,\\%\label{eqn:epchi1}\\
\EM \PM\cret \EM f&=-\EM
\end{align*}
for all $f\in\Ccs(\Ms)$. Moreover, $\PM\chi^\text{adv/ret}\EM f\in\Ccs(\Ms)$.
\end{lemma}
Defining $\zeta t=\PM\cadv \EM t$ for $t\in\F^1(\Ms)$, it follows directly that given $\Sadv,\Sret$ and $\cadv,\cret$ defined as above, for any $t\in\F^n(\Ms)$, $n\geq1$, we have
\[
\supp(\tenpow{\zeta}nt)\subseteq(J_\Ms^+(\Sadv)\cap J_\Ms^-(\Sret))^{\times n}\cap \supp(\tenpow\EM nt).
\]
Clearly $\tenpow\zeta n$ maps elements of $\F^n(\Ms)$ to elements of $\F^n(\Ms)$. We also note that by lemma \ref{lem:epchi}, we have
\begin{align}
\tenpow\zeta nt[\EM f]&=(-1)^n\tenpow{(\EM\zeta)}n t[f]\notag\\
&=(-1)^n\tenpow{\EM} n t[f]\notag\\
&=t[\EM f]\label{eqn:talphat}
\end{align}
for any $t\in\F^n(\Ms)$, $n\geq1$ and $f\in\Ccs(\Ms)$. It follows that if we define
\begin{align*}
\Zeta:\F(\Ms)&\to\F(\Ms)\\
\sum_{n=0}^Nt_n&\mapsto\sum_{n=0}^N\tenpow\zeta nt_n\qquad(t_n\in\F^n(\Ms)),
\end{align*}
we have $\Zeta F\sM F$ for all $F\in\F(\Ms)$.
\begin{lemma}
\label{lem:atimeslice}
The theory $\A$ obeys the timeslice axiom.
\end{lemma}
\begin{proof}
Suppose that $\psi$ is a \Loc-arrow from $\Ns$ to $\Ms$ with the property that $\psi(\Ns)$ contains a Cauchy surface for $\Ms$. We will always be able to find a second Cauchy surface for $\Ms$ in $\psi(\Ns)$ which is disjoint to the first; we denote the Cauchy surface to the past by $\Sadv$ and the one to the future by $\Sret$, and define the operator $\Zeta$ as above using these Cauchy surfaces for the construction; it follows that for any $F\in\F(\Ms)$, the $n^\text{th}$ component of $\Zeta F$ is supported in $\psi(\Ns)^{\times n}$ for each $n\geq1$. We then define
\begin{align}
\mathscr{G}(\psi):\F(\Ms)&\to\F(\Ns)\notag\\
F&\mapsto\psi^\ast\Zeta F.\label{eqn:Gmapdef}
\end{align}
For any $F\in\F(\Ms)$ and $f\in\Cs(\Ms)$, we then have
\[_{H\in\H(\Ms)}
(\F(\psi)\mathscr{G}(\psi)F)[f]=(\F(\psi)\psi^\ast\Zeta F)[f]=\psi^\ast\Zeta F[\psi^\ast f]=\psi_\ast\psi^\ast\Zeta F[ f].
\]
But since the $n^\text{th}$ component of $\Zeta F$ is supported in $\psi(\Ns)^{\times n}$, we have $\psi_\ast\psi^\ast\Zeta F=\Zeta F.$ Therefore $\F(\psi)\mathscr{G}(\psi)F=\Zeta F$. Now suppose that $F\in\F(\Ns)$ and $f\in\Cs(\Ns)$; then,
\[
(\mathscr{G}(\psi)\F(\psi)F)[f]=\mathscr{G}(\psi)(\psi_\ast F)[f]
=\psi^\ast\Zeta(\psi_\ast F)[f].
\]
Writing $F=\sum_{n=0}^Nt_n$, with $t_n\in\F^n(\Ns)$, we have
\[
\psi^\ast \Zeta(\psi_\ast F)=\sum_{n=0}^N\psi^\ast\tenpow\zeta n\psi_\ast t_n.
\]
But notice that for any $t\in\F^1(\Ms)$, $f\in\Ccs(\Ms)$, we have
\[
(\psi^\ast\zeta\psi_\ast t)[E_\Ns f]=(P_\Ns\psi^\ast(\cadv\EM\psi_\ast t))[E_\Ns f]
=(P_\Ns((\psi^\ast\cadv)E_\Ns t))[E_\Ns f]=t[E_\Ns f]
\]
by \eqref{eqn:talphat} and lemma \ref{lem:epchi}. We have therefore shown that $\F(\psi)\mathscr{G}(\psi)F\sM F$ for all $F\in\F(\Ms)$, and $\mathscr{G}(\psi)\F(\psi)F\sim_\Ns F$ for all $F\in\F(\Ns)$.
Next, we observe that if $F,F'\in\F(\Ms)$ with $F\sM F'$, then we have $\F(\psi)\mathscr{G}(\psi)F\sM\F(\psi)\mathscr{G}(\psi)F'$; by lemma \ref{lem:kgfmor} we then have $\mathscr{G}(\psi)F\sim_\Ns\mathscr{G}(\psi)F'$. This means that the map
\begin{align*}
\mathscr{B}(\psi):\A(\Ms)&\to\A(\Ns)\\
[F]_\Ms&\mapsto[\mathscr{G}(\psi)F]_\Ns
\end{align*}
is well defined, and we can conclude that $\mathscr{B}(\psi)\circ\A(\psi)=\text{id}_{\A(\Ns)},$ and $\A(\psi)\circ\mathscr{B}(\psi)=\text{id}_{\A(\Ms)}$. Therefore $\A(\psi)$ is invertible, and so $\A$ obeys the timeslice axiom.
\end{proof}
\subsection{The timeslice axiom for the theory of Wick Polynomials}
\label{sect:wptimeslice}
We now proceed to the timeslice axiom for $\W$, adapting the proof given for an equivalent construction in \cite{chilfred} for the construction used here.
Suppose that $\psi:\Ns\hookrightarrow\Ms$ is a Cauchy arrow in \Loc. We can then find two disjoint Cauchy surfaces $\Sadv,\Sret\subset\psi(\Ns)$ for $\Ms$ with $\Sret\subset J_\Ms^+(\Sadv)$. As before, we choose a smooth partition of unity $\cadv+\cret=1$ with $\cadv(x)=0,\ \cret(x)=1$ for $x\in J_\Ms^+(\Sret)$ and $\cadv(x)=1,\ \cret(x)=0$ for $x\in J_\Ms^-(\Sadv)$. It follows that if we again define $\zeta t=\PM\cadv\EM t$ for any $t\in\T^1(\Ms)$, and for any $H\in\H(\Ms)$, we let
\begin{align*}
Z:\T_H(\Ms)&\to\T_H(\Ms)\\
\sum_{n=0}^Nt_n&\mapsto\sum_{n=0}^N\tenpow\zeta nt_n\qquad (t_n\in\T^n(\Ms)),
\end{align*}
then by \eqref{eqn:epchidist}, $ZT\sim_\Ms T$ for all $T\in\T_H(\Ms)$, and $T$ is compactly supported in $\psi(\Ns)$. Moreover, since $Z$ is constructed from differential operators, multiplication by smooth functions and applications of $\EM^\pm$, we recall from our previous observation that $Z$ must indeed map elements of $\T_H(\Ms)$ to elements of $\T_H(\Ms)$.
Therefore, if we define
\begin{align}
\mathscr{S}_H(\psi):\T_H(\Ms)&\to\T_{\psi^\ast H}(\Ns)\notag\\
T&\mapsto \psi^\ast ZT,\label{eqn:tinvdef}
\end{align}
then the same argument as used in the proof of lemma \ref{lem:atimeslice} shows that $\T_H(\psi)\mathscr{S}_H(\psi)T\sim_\Ms T$ for all $T\in\T_H(\Ms)$ and $\mathscr{S}_H(\psi)\T_H(\psi)T\sim_\Ns T$ for all $T\in\T_{\psi^\ast H}(\Ns)$.
Now, if $\psi(\Ns)$ contains a Cauchy surface for $\Ms$ then for each $H\in\H(\Ns)$ there is precisely one $H'\in\H(\Ms)$ with $\psi^\ast H'=H$, as a result of the condition \eqref{eqn:hbisol}. We will denote this extension by $\psi_\bullet H$. Now suppose that $W=(W_H)_{H\in\H(\Ms)}\in\W(\Ms)$ with $W_H=[T_H]$, for some $T_H\in\T_H(\Ms)$ for each $H\in\H(\Ms)$. We then define
\begin{align}
\mathscr{U}_H(\psi):\W_H(\Ms)&\to\W_{\psi^\ast H}(\Ns)\notag\\
[T]_\Ms&\mapsto[\mathscr{S}_H(\psi)T]_\Ns.\label{eqn:winvdef}
\end{align}
This then gives us a map $\mathscr{U}(\psi):\W(\Ms)\to\W(\Ns)$ with the property that for any $H\in\H(\Ns)$, we have
\[
(\mathscr{U}(\psi)W)_H=\mathscr{U}_{\psi_\bullet H}(\psi)W_{\psi_\bullet H}.
\]
It is easy to show that $\W(\psi)\circ\mathscr{U}(\psi)=\text{id}_{\W(\Ms)}$, and $\mathscr{U}(\psi)\circ\W(\psi)=\text{id}_{\W(\Ns)}$. Therefore $\mathscr{U}(\psi)=\W(\psi)^{-1}$ and so $\W$ obeys the timeslice axiom.
\subsection{Relative Cauchy evolution for the Klein-Gordon Theory}
\label{sect:kgfrce}
In order to demonstrate (or rule out) dynamical locality for $\A$ or $\W$, we must first compute the relative Cauchy evolution of an arbitrary element; this has already been done for the scalar Klein-Gordon theory in \cite{bfv} for a different construction, and we will derive a similar expression in our formalism. We begin with the theory $\A$; we fix $\h\in H(\Ms)$ and choose two subspacetimes $\Ns^\pm\subseteq\Ms$, such that:
\begin{itemize}
\item each $\Ns^\pm$ is an object of \Loc, and their embeddings into $\Ms$ are arrows in \Loc,
\item each $\Ns^\pm$ contains two disjoint Cauchy surfaces $\Sadv_\pm,\Sret_\pm$ for $\Ms$ with the property that
$\Sadv_\pm\subseteq J_\Ms^-(\Sret_\pm)$,
\item each $\Ns^\pm$ is disjoint from the support of $\h$, and $\Ns^\pm\subseteq J^\pm_\Ms(\supp(\h))$.
\end{itemize}
We now choose two smooth partitions of unity $\cadv_\pm+\cret_\pm=1$ for $\Ms$, with the property that $\cadv_\pm(x)=1$, $\cret_\pm(x)=0$ for $x\in J_\Ms^-(\Sadv_\pm)$, and $\cadv_\pm(x)=0,$ $\cret_\pm(x)=1$ for $x\in J_\Ms^+(\Sret_\pm)$, and define
\begin{align*}
\zeta^\pm:\F^1(\Ms)&\to\F^1(\Ms)\\
t&\mapsto \PM\cadv_\pm\EM t.
\end{align*}
As before, we also let
\begin{align*}
\Zeta^\pm:\F(\Ms)&\to\F(\Ms)\\
\sum_{n=0}^N t_n&\mapsto\sum_{n=0}^N\tenpow{(\zeta^\pm)} n t_n\qquad(t_n\in\F^n(\Ms)).
\end{align*}
Additionally, we define
\begin{align*}
\zeta^\pm[\h]:\F^1(\Ms[\h])&\to\F^1(\Ms[\h])\\
t&\mapsto \PMh\cadv_\pm\EMh t,
\end{align*}
and define $\Zeta^\pm[\h]:\F(\Ms[\h])\to\F(\Ms[\h])$ in an analagous way to $Z^\pm$.
Now, if we denote by $\iota^\pm$, $\iota^\pm[\h]$ the embeddings of $\Ns^\pm$ into $\Ms$ and $\Ms[\h]$ respectively, it is clear that the \Alg-arrows $\A(\iota^\pm)$, $\A(\iota^\pm[\h])$ act as
\begin{align*}
\A(\iota^\pm)[F]_{\Ns^\pm}&=[\F(\iota^\pm)F]_\Ms,\\
\A(\iota^\pm[\h])[F]_{\Ns^\pm}&=[\F(\iota^\pm[\h])F]_{\Ms[\h]},
\end{align*}
and for any $F\in\F(\Ns^\pm)$, $f\in\Cs(\Ms)$ we have
\[
(\F(\iota^\pm)F)[f]=F\left[f|_{\Ns^\pm}\right]=(\F(\iota^\pm[\h])F)[f].
\]
Moreover, from lemma \ref{lem:atimeslice} we can see that the inverse arrows $\A(\iota^\pm)^{-1}$, $\A(\iota^\pm[\h])^{-1}$ act as
\begin{align*}
\A(\iota^\pm)^{-1}[F]_{\Ms}&=[\mathscr{G}(\iota^\pm)F]_{\Ns^\pm}\\
\A(\iota^\pm[\h])^{-1}[F]_{\Ms[\h]}&=[\mathscr{G}(\iota^\pm[\h])F]_{\Ns^\pm},
\end{align*}
where for any $f\in\Cs(\Ns^\pm)$, $F\in\F(\Ms)$ and $F'\in\F(\Ms[\h])$, we see from \eqref{eqn:Gmapdef} that
\begin{align*}
(\mathscr{G}(\iota^\pm)F)[f]&=(\Zeta^\pm F)[\iota^\pm_\ast f],\\
(\mathscr{G}(\iota^\pm[\h])F')[f]&=(\Zeta^\pm[\h]F')[\iota^\pm[\h]_\ast f].
\end{align*}
It follows that for any $A=[F]_\Ms\in\A(\Ms)$, we have
\begin{align*}
\rce_\Ms[\h]A&=\A(\iota^-)\A(\iota^-[\h])^{-1}\A(\iota^+[\h])
\A(\iota^+)^{-1}A\\
&=\left[\F(\iota^-)\mathscr{G}(\iota^-[\h])\F(\iota^+[\h])
\mathscr{G}(\iota^+)F\right]_{\Ms}.
\end{align*}
Now, for any $f\in\Cs(\Ms)$ and $F\in\F(\Ms)$ we have
\[
(\F(\iota^+[\h])\mathscr{G}(\iota^+)F)[f]=(\Zeta^+F)|_{\Ns^+}
\left[f|_{\Ns^+}\right],
\]
but since the range of $\Zeta^+$ is contained in $\iota^+(\Ns^+)$, it holds that
\[
\F(\iota^+[\h])\circ\mathscr{G}(\iota^+)=
\iota^+[\h]_\ast\circ(\iota^+)^\ast\circ\Zeta^+,
\]
and similarly
\[
\F(\iota^-)\circ\mathscr{G}(\iota^-[\h])=
\iota^-_\ast\circ\iota^-[\h]^\ast\circ\Zeta^-[\h].
\]
Explicitly, the relative Cauchy evolution of $A=[F]_\Ms$ is therefore given by $\rce_\Ms[\h]A=[B[\h]F]_\Ms,$ where
\begin{align*}
B[\h]:\F(\Ms)&\to\F(\Ms)\\
\sum_{n=0}^Nt_n&\mapsto\sum_{n=0}^N\tenpow{\beta[\h]}n t_n\qquad(t_n\in\F^n(\Ms)),
\end{align*}
and
\begin{align}
\beta[\h]:\F^1(\Ms)&\to\F^1(\Ms)\notag\\
t&\mapsto\PMh\cadv_-\EMh\PM\cadv_+\EM t.\label{eqn:betadef}
\end{align}
This definition is independent of the choice of $\cadv_\pm$, provided that the regions $\Ns^\pm$ in which they are non-constant lie strictly to the future/past of $\supp(\h)$.
\subsection{Relative Cauchy evolution for Wick Polynomials}
\label{sect:wprce}
We now calculate the relative Cauchy evolution of an element $W\in\W(\Ms)$ generated by a perturbation $\h\in H(\Ms)$. While the calculation is largely similar to the process for calculating the r.c.e.\ of an element of $\A(\Ms)$, there are some subtleties introduced by the need to specify an $H\in\H(\Ms)$ to form the algebras $\T_H(\Ms)$. We will proceed as before, fixing some $\h\in H(\Ms)$ and defining $\Ns^\pm$, $\Sadv_\pm,\Sret_\pm$ and $\iota^\pm$ and $\iota^\pm[\h]$ as in the previous subsection. The relative Cauchy evolution of an element $W\in\W(\Ms)$ is given by
\[
\rce_\Ms[\h]W=(\W(\iota^-)\circ\mathscr{U}(\iota^-[\h])\circ\W(\iota^+[\h])\circ\mathscr{U}(\iota^+))W.
\]
But when we calculate the component corresponding to $H\in\H(\Ms)$, we see that
\begin{align}
(\rce_\Ms[\h]W)_H&=\left(\W(\iota^-)\mathscr{U}(\iota^-[\h])\W(\iota^+[\h])\mathscr{U}(\iota^+)W\right)_H\notag\\
&=\W_H(\iota^-)\mathscr{U}_{\tilde H_\h}(\iota^-[\h])\W_{\tilde H_\h}(\iota^+[\h])\mathscr{U}_{\cH_\h}(\iota^+)W_{\cH_\h}\label{eqn:wprcedef}
\end{align}
where for any $H\in\H(\Ms)$, the distributions $\tilde H_\h\in\H(\Ms[\h])$ and $\cH_\h\in\H(\Ms)$ are defined by
\begin{align*}
\tilde H_\h&=\iota^-[\h]_\bullet(\iota^-)^\ast H\\
\cH_\h&=\iota^+_\bullet\iota^-[\h]^\ast\tilde H_\h.
\end{align*}
This definition is independent of the choice of $\Ns^\pm$, as a consequence of \eqref{eqn:hbisol}.
\begin{lemma}
\label{lem:Hdiffcomp}
Let $\Ms$ be a spacetime, and $\h\in H(\Ms)$ a metric perturbation on $\Ms$. Suppose that $H\in\H(\Ms),$ and let $\cH_\h,\Ns^\pm$ and $\cadv_\pm$ be defined as above. Then
\beq
\cH_\h=\tenpow{(\cEM(\eta_s^+)'\PMh(\cadv_+)'\check{E}_{\Ms[\h]}
(\eta_s^-)'\PM(\cadv_-)')}2H,\label{eqn:Hrcedef}
\eeq
where $\cadv_\pm:\Ccs(\Ms)\to\Ccs(\Ms)$ are the multiplication operators induced by the functions $\cadv_\pm\in\Cs(\Ms)$, and $\eta^\pm_s:\Cs_s(\Ms)\to\Ccs(\Ms)$ are defined as multiplication by time-compact smooth functions $\eta^\pm$ that are supported in $\Ns^\pm$, such that $\eta^\pm\equiv1$ in the region in which $\cadv_\pm$ is non-constant.
\end{lemma}
\begin{proof}
Since $H$ is a bisolution, we see from the proof of lemma \ref{lem:eetapchi} that
\[
\tenpow{(\cEM(\eta_s^\pm)'\PM(\cadv_\pm)')}2H=H.
\]
Since $\eta^\pm$ is supported in $\Ns^\pm$, it follows that $\tenpow{((\eta^\pm_s)'\PM(\cadv_\pm)')}2H$ is supported in $(\Ns^\pm)^{\times2}$, and therefore
\[
\tilde H_\h|_{\Ns^-}=H|_{\Ns^-}=\tenpow{{\check{E}_{\Ns^-}}}2\left.
\left(\tenpow{((\eta_s^-)'\PM(\cadv_-)')}2H\right)\right|_{\Ns^-}.
\]
Since the action of our multiplication operators does not depend on the metric of the underlying manifold, we may also consider them as maps on the corresponding function spaces on $\Ms[\h]$; since $\tilde H_\h$ is a bisolution on $\Ms[\h]$ and $\check{E}_{\Ms[\h]}|_{\Ns^-}=\check{E}_{\Ns^-}$, it follows that
\[
\tilde H_\h=\tenpow{(\check{E}_{\Ms[\h]}(\eta_s^-)'\PM(\cadv_-)')}2H.
\]
A similar argument yields $\cH_\h=\tenpow{(\cEM(\eta_s^+)'\PMh(\cadv_+)')}2\tilde H_\h,$ and so \eqref{eqn:Hrcedef} is satisfied.\footnote{Note that \eqref{eqn:Hrcedef} strongly resembles the action of the map $\beta[\h]$ defined in \eqref{eqn:betadef}, albeit with $\Ns^+$ and $\Ns^-$ interchanged; indeed, if we consider the subcategory of \Loc\ containing only Cauchy arrows, we can regard $\H$ as a functor from $\Loc$ to a suitable category of distribution spaces, with $\H(\psi)H=\psi_\bullet H$. This functor can be seen to be covariant; the resemblance remarked above can be explained by noting that we may define the relative Cauchy evolution of the functor $\H$ in the same way as for a locally covariant theory; this then satisfies $\rce_\Ms^{(\H)}[\h]\cH=H$.
}
\end{proof}
\begin{lemma}
\label{lem:Hdiffsupp}
Let $\Ms$ be a spacetime, with a metric perturbation $\h\in H(\Ms)$. Suppose also that $H\in\H(\Ms)$, and let $\cH_\h$ be defined as above. Then $\supp(H-\cH_\h)\subseteq(J_\Ms(\supp(\h))^{\times2}).$
\end{lemma}
\begin{proof}
Let $x\in\Ms$, with $x\notin J_\Ms^+(\supp(\h))$. Since $\supp(\h)$ is compact, we can find a choice for $\Ns^-$ with $x\in\Ns^-$. It follows that $H(x,y)=\tilde H_\h(x,y)$ for all $x\notin J_\Ms^+(\supp(\h))$. Similarly, if $x\notin J_\Ms^-(\supp(\h))$ then we can find a choice for $\Ns^+$ with $x\in\Ns^+$. Therefore $\cH_\h(x,y)=\tilde H_\h(x,y)$ for all $x\notin J_\Ms^-(\supp(\h))$. Consequently, if $x\in\supp(\h)^\perp$ then $H(x,y)=\cH_\h(x,y)$. The required result follows by symmetry of $H$.
\end{proof}
The coherency condition on elements of $\W(\Ms)$ tells us that \eqref{eqn:wprcedef} can be expressed as
\[
(\rce_\Ms[\h]W)_H=\W_H(\iota^-)\mathscr{U}_{\tilde H_\h}(\iota^-[\h])\W_{\tilde H_\h}(\iota^+[\h])
\mathscr{U}_{\cH_\h}(\iota^+)\tl_{H,\cH_\h}W_H.
\]
Explicitly, we can then see from \eqref{eqn:tinvdef},\eqref{eqn:winvdef} that the relative Cauchy evolution of an element $W=(W_H)_{H\in\H(\Ms)}\in\W(\Ms)$, where each $W_H$ can be represented by $T_H\in\T_H(\Ms)$, is given by
\[
(\rce_\Ms[\h]W)_H=[B_H[\h]\lambda_{H,\cH_\h}T_H]_\Ms,
\]
where
\begin{align*}
B_H[\h]:\T_{\cH}(\Ms)&\to\T_H(\Ms)\\
\sum_{n=0}^Nt_n&\mapsto\sum_{n=0}^N\tenpow{\beta[\h]}nt_n
\qquad(t_n\in\T^n(\Ms)),
\end{align*}
with
\begin{align*}
\beta[\h]:\T^1(\Ms)&\to\T^1(\Ms)\\
t&\mapsto \PMh\cadv_-\EMh\PM\cadv_+\EM t
\end{align*}
as before.
Before we proceed to the dynamical locality of $\A$ and $\W$ we will need the following results. The lemma is proved in appendix \ref{appx} (cf.\ \cite[Eqn.\ 8]{dynloc2}).
\begin{lemma}
\label{lem:rcederiv}
Let $\Ms$ be a spacetime and let $t\in\T_H^1(\Ms)$ for some $H\in\H(\Ms)$. For any $\h\in H(\Ms)$ and $f\in\Ccs(\Ms)$, we have
\[
\left.\dds (\beta[s\h]t)[\EM f]\right|_{s=0}=\int_\Ms dvol_\Ms\, h_{ab}T^{ab}[\EM t,\EM f],
\]
where
\begin{align*}
T^{ab}[u,\phi]=(\nabla^{(a}u)&(\nabla^{b)}\phi)-
\frac12g^{ab}(\nabla^cu)(\nabla_c\phi)\\
&+\frac12m^2g^{ab}u\phi+\xi(g^{ab}\Box_\g-
\nabla^a\nabla^b-G^{ab})(u\phi)
\end{align*}
for $u\in\EM\T^1(\Ms)$, $\phi\in\EM\Ccs(\Ms)$.
\end{lemma}
Note that the above expression is closely linked to the classical stress-energy tensor for the Klein-Gordon theory, which we may recover via $T^{ab}[\phi]=T^{ab}[\phi,\bar\phi]$ for a smooth classical solution $\phi$.
This result leads directly to the following:
\begin{corollary}
\label{cor:rcederivn}
Let $t_n\in\T^n_H(\Ms)$ for some $H\in\H(\Ms)$ and $f\in\Ccs(\Ms)$. Then
\[
\left.\dds\tenpow{(\beta[s\h])}nt_n[\EM f]\right|_{s=0}=n\int_\Ms dvol_\Ms\,h_{ab}T^{ab}\left[\EM\tau^n_f,\EM f\right],
\]
where
\beq
\label{eqn:taudef}\tau^n_f(x)=\int_{\Ms^{\times(n-1)}}d^{n-1}y\,
t_n(x,y_1,\ldots,y_{n-1})\EM f(y_1)\cdots\EM f(y_{n-1})
\eeq
for $n\geq2$, and $\tau^1_f(x)=t_1(x).$
\end{corollary}
Note that the previous two results also apply to the elements of $\F^1(\Ms)$ and $\F^n(\Ms)$ respectively, since we can consider any element of $\F^n(\Ms)$ as an element of $\T^n_H(\Ms)$ for any $H\in\H(\Ms)$.
\section{Dynamical Locality}
\label{sect:dynlockgfwp}
\subsection{Dynamical locality of the \texorpdfstring{$\xi\neq0$}{xi /= 0} Klein-Gordon theory}
\label{sect:kgfdl}
It has already been shown in \cite{dynloc2} that the Klein-Gordon theory is dynamically local in the case when $\xi=0$ and $m\neq0$, and that it is not dynamically local when $\xi=0$ and $m=0$. We wish to show that the Klein-Gordon theory $\A$ obeys the axiom of dynamical locality in the nonminimally coupled case, when $\xi\neq0$, for both $m=0$ and $m>0$. Therefore, we pick some spacetime $\Ms$ and $O\in\mathscr{O}(\Ms)$. The algebra $\A^{\text{kin}}(\Ms;O)$ is defined to be the algebra $\A(\Ms|_O)$; we recall from lemma \ref{lem:aalgimage} that for any \Loc-arrow $\psi:\Ns\hookrightarrow\Ms$, the algebra $\A(\psi)(\A(\Ns))$ comprises elements $A=[F]_\Ms$ such that $F[\EM f]=F[0]$ for every $f\in\Ccs(\Ms)$ such that $\supp(f)\cap J_\Ms(\Ns)=\emptyset$.
It follows that $F$ represents an element of $\alpha^\text{kin}_{\Ms;O}(\Ak(\Ms;O))$ if and only if $F[\EM f]=F[0]$ for all $f\in\Ccs(\Ms)$ with support lying in $O'=(\text{cl }O)^\perp$.
We can see from \eqref{eqn:dynalgdef} and \eqref{eqn:dynloccond} that if $\Ab(\Ms;K)\subseteq\alpha^\text{kin}_{\Ms;O}(\Ak(\Ms;O))$ for each spacetime $\Ms$, $O\in\mathscr{O}(\Ms)$ and $K\in\K(\Ms;O)$, then $\A$ obeys dynamical locality. Therefore, suppose that $A\in\Ab(\Ms;K)$; from the definition it follows that $\rce_\Ms[\h]A=A$ for all $\h\in H(\Ms;K^\perp)$. Now, suppose that $A$ is represented by a functional $F\in\F(\Ms)$. This means that $B[\h]F\sim_\Ms F$ for all $\h\in H(\Ms;K^\perp)$, and consequently $B[s\h]F-F\in \J(\Ms)$ for all $s\in\mathbb{R}$ sufficiently small that $s\h\in H(\Ms;K^\perp).$ Writing $F=\sum_{n=0}^Nt_n$, with each $t_n\in\F^n(\Ms)$, we can refer to lemma \ref{lem:kgfuncpolar} to see that for $n=1,\ldots,N$, we have
\beq
\label{eqn:kgdynloc1}
\left(\tenpow{(\beta[s\h])}nt_n\right)[\EM f]=t_n[\EM f]
\eeq
for all $f\in\Ccs(\Ms)$ and for all $\h\in H(\Ms;K^\perp)$.
Now, for each $n\geq1$ we differentiate \eqref{eqn:kgdynloc1} with respect to $s$ and set $s=0$; by corollary \ref{cor:rcederivn}, this yields
\[
\int_\Ms dvol_\Ms\,h_{ab}T^{ab}\left[\EM\tau^n_f,\EM f\right]=0
\]
for each $\h\in H(\Ms;K^\perp)$ and $f\in\Ccs(\Ms)$, where $\tau^n_f$ is defined as in \eqref{eqn:taudef}. It follows that for all $n\geq1$, we have
\[
T^{ab}[\EM\tau^n_f,\EM f](x)=0
\]
for all $x\in K^\perp$.
Now consider an arbitrary point $x\in K^\perp$, and a null geodesic $u:I\to K^\perp$, where $I\subset\mathbb{R}$ is an open interval containing 0 and $u(0)=x$. Since $u$ is a null geodesic, it satisfies both $u^au^bg_{ab}=0$ and $u^a\nabla_au^b=0$, where $u^a$ is the tangent vector to $u$. For each point $q$ on the geodesic we have $u_a(q)u_b(q)T^{ab}[\EM\tau^n_f,\EM f](q)=0$, and consequently for our chosen $x\in K^\perp$ we have
\[
(\nabla_u\EM\tau^n_f(x))(\nabla_u\EM f(x))+\xi\left(-\nabla_u^2-R_{ab}(x)u^au^b\right)\left((\EM\tau^n_f(x))
(\EM f(x))\right)=0.
\]
Note that this is equivalent to
\begin{align}
(1-2\xi)(\nabla_u\EM\tau^n_f(x))&(\nabla_u\EM f(x))-\xi R_{ab}u^au^b(\EM\tau^n_f(x))(\EM f(x))
\notag\\
&+\xi(\EM\tau^n_f(x)\nabla_u^2\EM f(x)+\EM f(x)\nabla_u^2\EM\tau^n_f(x))=0.
\label{eqn:kgst2}
\end{align}
It follows that for any $f\in\Ccs(\Ms)$ for which $\EM f(x)=0=\nabla_u\EM f(x)$ and $\nabla_u^2\EM f(x)\neq 0$,\footnote{Such a solution always exists; we may explicitly construct one as follows. We work in normal coordinates $q^a$ in a neighbourhood $S\ni x$ such that $x$ is at the origin, and the $q^0=0$ hyperplane is a subset of a spacelike Cauchy surface $\Sigma\subset\Ms$, and we take our null geodesic $u$ such that in coordinates, the tangent at $x$ is $u^a(x)=(1,1,0,\ldots,0).$ Then any solution $\psi$ is uniquely determined by its data $(\varphi,\pi)$ on $\Sigma$, where $\varphi(\underline{q})=\psi|_\Sigma(\underline{q})$ and $\pi(\underline{q})=(\nabla_0\psi)|_\Sigma(\underline{q})$. It is then easy to check that defining $\varphi(\underline{q})=(q^1)^2$, $\pi(\underline{q})=0$ for $\underline{q}\in\Sigma\cap S$ gives us a solution $\psi$ satisfying the above conditions.} we have $\EM\tau^n_f(x)=0$, as $\xi\neq0$.
In the case that $n=1$, we have $\EM\tau^1_f=\EM t_1$ for all $f$, so we immediately see that $\EM t_1(x)=0$ for all $x\in K^\perp$. Now, we look at the case where $n=2$. We have $\EM\tau^2_f(x)=\int_\Ms dy\,t_2(x,y)\EM f(y)$, which is linear in $f$. Let $f$ be chosen such that $\EM f(x)=0=\nabla_u\EM f(x)$ and $\nabla_u^2\EM f(x)\neq 0$; additionally, we choose $f'\in\Ccs(\Ms)$ such that $\supp(f')\subset \{x\}^\perp$. Then $\EM f+\EM f'=\EM f$ in an open neighbourhood of $x$, so
\[
\EM\tau^2_{f'}(x)=\EM\tau^2_{f+f'}(x)-\EM\tau^2_f(x)=0.
\]
It follows that for any $f'\in\Ccs(\Ms)$ supported outside $J_\Ms(x)$, we have
\[
\int_\Ms dy\,(\tenpow\EM2t_2)(x,y)f'(y)=-\EM\tau^2_{f'}(x)=0.
\]
Therefore $\tenpow{\EM}2t_2(x,y)=0$ whenever $x\in K^\perp$ and $y\in \{x\}^\perp$.
However, by the definition of $\F^2(\Ms)$, we have $\tenpow\EM2t_2(x,\cdot)\in\EM\Ccs(\Ms)$ for any fixed $x\in\Ms$, and it is therefore a smooth classical Klein-Gordon solution. If $\Sigma$ is a spacelike Cauchy surface containing $x$, then the data for $\tenpow\EM2t_2(x,\cdot)$ on $\Sigma$ is supported in $\{x\}$ for any $x\in K^\perp$ by the above result. But the data for a smooth solution is itself smooth, and therefore cannot be both nonzero and supported at a point. Consequently $\tenpow\EM2t_2(x,y)=0$ for any $(x,y)\in K^\perp\times\Ms$, and by symmetry we have $\supp(\tenpow\EM2 t_2)\subseteq J_\Ms(K)^{\times 2}$.
Now, consider the case where $n>2$. Suppose that we have $f,f_1$ such that $\EM f(x)=0=\nabla_u\EM f(x)$, $\EM f_1(x)=0=\nabla_u\EM f_1(x)$, and $\EM f(x)\neq0$. Then, for sufficiently small $\kappa$ we have $\EM\tau^n_{f+\kappa f_1}(x)=0$. Therefore, by symmetry of $t_n$ we have
\begin{align*}
\EM\tau^n_{f}(x)+(-1)^{n-1}(n-1)\kappa\int_{\Ms^{\times(n-1)}}d^{n-1}y\,\Big[(\tenpow\EM n &t_n)(x,y_1,\ldots,y_{n-1})\\
&f_1(y_1) f(y_2)\cdots f(y_{n-1})\Big]
+\mathcal{O}(\kappa^2)=0.
\end{align*}
Differentiating this expression with respect to $\kappa$ and setting $\kappa=0$, we have
\[
\int_{\Ms^{\times(n-1)}}d^{n-1}y\,(\tenpow\EM n t_n)(x,y_1,\ldots,y_{n-1}) f_1(y_1)f(y_2)\cdots f(y_{n-1})=0.
\]
We may repeat this argument to see that
\[
\int_{\Ms^{\times(n-1)}}d^{n-1}y\,(\tenpow\EM n t_n)(x,y_1,\ldots,y_{n-1})f_1(y_1)\cdots f_{n-1}(y_{n-1})=0
\]
for any $f_1,\ldots,f_{n-1}$ such that $\EM f_i(x)=0=\nabla_u\EM f_i(x)$, $i=1,\ldots,n-1$. It follows that for any $x_1\in K^\perp$, we have $\tenpow\EM n t_n(x_1,\ldots,x_n)=0$ whenever at least one of $x_2,\ldots,x_n$ lies in ${x_1}^\perp$. Fixing $x_1\in K^\perp$, we note that $\tenpow\EM nt_n(x_1,y_1,\ldots,y_{n-1})$ is a smooth Klein-Gordon $(n-1)$-solution; its data on a spacelike Cauchy surface $\Sigma\ni x$ is supported in $\{x\}^{\times(n-1)}.$ Consequently we must have $\tenpow\EM nt_n(x_1,y_1,\ldots,y_{n-1})=0$ for $x_1\in K^\perp,\ y_1,\ldots,y_{n-1}\in\Ms$ by smoothness. Therefore we have proved the following lemma:
\begin{lemma}
\label{lem:kgsuppprop}
Let $\Ms$ be a spacetime and let $t_n\in\F^n(\Ms)$, $n\geq1.$ If $O\in\mathscr{O}(\Ms)$, $K\in \mathscr{K}(\Ms;O)$ and $\left(\tenpow{(\beta[s\h])}nt_n\right)[\EM f]=t_n[\EM f]$ for all $f\in\Ccs(\Ms)$ and for all $\h\in H(\Ms;K^\perp)$, then
\[
\supp(\tenpow\EM nt_n)\subseteq J_\Ms(K)^{\times n}.
\]
\end{lemma}
From here we may prove the following result:
\begin{theorem}
The Klein-Gordon theory is dynamically local in the nonminimally coupled case, for all $m\geq0$.
\end{theorem}
\begin{proof}
Recall that for any spacetime $\Ms$ and $O\in\mathscr{O}(\Ms)$, the algebra $\alpha^{\text{kin}}_{\Ms;O}(\Ak(\Ms;O))$ comprises elements represented by functionals $F$ with the property that $F[\EM f]=F[0]$ for all $f\in\Ccs(\Ms)$ supported within $O'$. To demonstrate that the theory is dynamically local, it is sufficient to show that $\Ab(\Ms;K)\subseteq\alpha^{\text{kin}}_{\Ms;O}(\Ak(\Ms;O))$ for all $K\in\mathscr{K}(\Ms;O)$. Given such a $K$, and an element $A\in\Ab(\Ms;K)$ represented by $F=\sum_{n=0}^Nt_n$, with each $t_n\in\F^n(\Ms)$, we may see from \eqref{eqn:kgdynloc1} and lemma \ref{lem:kgsuppprop} that $\supp(\tenpow\EM nt_n)\subseteq J_\Ms(K)^{\times n}$ for each $n=1,\ldots,N$, and subsequently $t_n[\EM f]=0$ for all $f\in\Ccs(K^\perp)$. Therefore in particular we have $F[\EM f]=t_0=F[0]$ for all $f\in\Ccs(O')$, and so $F$ represents an element of $\alpha^{\text{kin}}_{\Ms;O}(\Ak(\Ms;O))$. Consequently the theory is dynamically local.
\end{proof}
\subsection{Dynamical Locality of the algebra of Wick Polynomials}
We now proceed to examine the cases in which we can demonstrate dynamical locality for the theory $\W$. We begin by looking at the minimally coupled massless case. The corresponding case for the Klein-Gordon theory is not dynamically local, and so one would not expect dynamical locality to hold here. Indeed, this is the case; when $\xi=m=0$, any constant function is a classical solution to the Klein-Gordon equation. Therefore, in any spacetime $\Ms$ with compact Cauchy surfaces, the function $\phi(x)=1$ is an element of $\EM\Ccs(\Ms)$. However, we have $ T^{ab}[\phi,\EM f]=0$; it follows that for any $t\in\T^1(\Ms)$ such that $\EM t\equiv1$, we have $t\in\Wd(\Ms;O)$ for any $O\in\mathscr{O}(\Ms)$. But it is also the case that if we pick $f\in\Ccs(O')$ with $\int_\Ms dx\,f(x)\neq0$, then $t[\EM f]\neq0$; therefore, $t\notin\Wk(\Ms;O)$.
We may, however, demonstrate dynamical locality in two cases. To do this, we need the following results:
\begin{lemma}
\label{lem:tsuppprop}
Let $\Ms$ be a spacetime with $O\in\mathscr{O}(\Ms)$ and $K\in\mathscr{K}(\Ms;O)$. Let $t_n\in\T^n(\Ms)$ for some $n\geq0$, and suppose that for all $f\in\Ccs(\Ms)$ and $\h\in H(\Ms;K^\perp)$ we have
\beq
\label{eqn:tintzero}
\int_\Ms dvol_\Ms\,h_{ab}T^{ab}[\EM\tau^n_f,\EM f]=0,
\eeq
where $\tau^n_f$ is defined as in \eqref{eqn:taudef}. Then, in the massive minimally coupled and massive conformally coupled cases, we have
$\supp(\tenpow\EM nt_n)\subseteq J_\Ms(K)^{\times n}$.
\end{lemma}
\begin{proof}
We will consider the massive minimally coupled case first, in which $m\neq0$ and $\xi=0$. Clearly $T^{ab}[\EM\tau^n_f,\EM f](x)=0$ for all $f\in\Ccs(\Ms)$ and $x\in K^\perp$; now, we fix $x\in K^\perp$ and pick some $f\in\Ccs(\Ms)$ such that $(\EM f)(x)\neq 0$. In the case where $\Ms$ has dimension 2, we note that
\[
0=g_{ab}T^{ab}[\EM\tau^n_f,\EM f](x)=m^2\EM\tau^n_f(x)\EM f(x),
\]
and consequently $\EM\tau^n_f(x)=0$ for any such $f$; in higher dimensions, we choose normal coordinates at $x$ oriented such that $\nabla_2\EM f(x)=\cdots=\nabla_{d-1}\EM f(x)=0$, and define $v_{ab}$ such that in these coordinates we have $v_{00}=1,\ v_{11}=-1$, and all other entries zero. It follows that $v_{ab}g^{ab}(x)=2$ and $v_{ab}\nabla^{(a}\EM\tau^n_f(x)\nabla^{b)}\EM f(x)=\nabla^a\EM\tau^n_f(x)\nabla_a\EM f(x)$, so that we have
\[
0=v_{ab}T^{ab}[\EM\tau^n_f,\EM f](x)=m^2\EM\tau^n_f(x)\EM f(x).
\]
Again, we may conclude that $\EM\tau^n_f(x)=0$ for any such $f$.
When $n=1$ we deduce immediately that $\EM t_1(x)=0$ for all $x\in K^\perp$. For $n=2$, we note that $\tau^2_f$ is linear in $f$, and as any $f\in\Ccs(\Ms)$ may be expressed as $f=f_1-f_2$ where $\EM f_1(x)\neq0\neq\EM f_2(x)$ we have $\EM\tau^2_f(x)=-\int_\Ms dy\,(\tenpow\EM2 t_2)(x,y)f(y)=0$ for all $f\in\Ccs(\Ms)$. Therefore $\tenpow\EM2 t_2(x,y)=0$ for all $x\in K^\perp$, and so $\supp(\tenpow\EM2 t_2)\subseteq J_\Ms(K)^{\times 2}$ by symmetry. For $n>2$, we pick $f\in\Ccs(\Ms)$ with $\EM f(x)\neq 0$ and let $f_1\in\Ccs(\Ms)$ be arbitrary; for sufficiently small $\kappa$ we have $\EM \tau^n_{f+\kappa f_1}(x)=0$. We differentiate this expression with respect to $\kappa$ and set $\kappa=0$, which yields
\[
\int_{\Ms^{\times(n-1)}}d^{n-1}y\,(\tenpow\EM n t_n)(x,y_1,\ldots,y_{n-1})f_1(y_1)f(y_2)\cdots f(y_{n-1})=0;
\]
we may then repeat this argument to see that
\[
\int_{\Ms^{\times(n-1)}}d^{n-1}y\,(\tenpow\EM n t_n)(x,y_1,\ldots,y_{n-1})f_1(y_1)f_2(y_2)\cdots f_{n-1}(y_{n-1})=0
\]
for any $f_1,\ldots,f_{n-1}\in\Ccs(\Ms)$. It follows that $\tenpow\EM n t_n(x,y_1,\ldots,y_{n-1})=0$ for all $x\in K^\perp$, and by symmetry we have $\supp(\tenpow\EM n t_n)\subseteq J_\Ms(K)^{\times n}$. This concludes the proof for the massive minimally coupled case.
In the massive conformally coupled case, where $m\neq0$ and $\xi=\frac{d-2}{4(d-1)}$, where $d$ is the dimension of $\Ms$, we have $g_{ab}T^{ab}[\phi_1,\phi_2]=m^2\phi_1\phi_2$ for any $\phi_1,\phi_2\in\EM\Ccs(\Ms)$. It follows that for all $x\in K^\perp$, we have $\EM\tau^n_f(x)\EM f(x)$ for all $f\in\Ccs(\Ms)$. We may use the same argument as above to show that $\supp(\tenpow\EM n t_n)\subseteq J_\Ms(K)^{\times n}.$
\end{proof}
\begin{lemma}
\label{lem:Esupport}
Let $t_n\in\T^n(\Ms)$, and suppose that $\supp(\tenpow\EM nt_n)\subseteq J_\Ms(K)^{\times n}$. Furthermore, let $S$ be any open neighbourhood of an arbitrary Cauchy surface $\Sigma\subset\Ms$. Then there exist $s,u_k\in\T^n(\Ms)$, $k=1,\ldots,n$, such that
\[
t_n=s+\sum_{k=1}^n(\PM)_ku_k,
\]
where we define $(\PM)_k=\tenpow\id{k-1}\otimes\PM\otimes\tenpow\id{n-k}$, and such that $\supp(s)\subseteq (J_\Ms(K)\cap S)^{\times n}$.
\end{lemma}
\begin{proof}
To prove this, we will need the result of lemma \ref{lem:Ekernel}: namely, that
\[
\ker\tenpow\EM n=\left\{\sum_{k=1}^n(\PM)_ku_k:u_k\in\T^n(\Ms)\right\}.
\]
Now, if $S$ is an open neighbourhood of a Cauchy surface, then we can find two disjoint Cauchy surfaces $\Sigma^\pm\subset S$ such that $\Sigma^+\subset J^+_\Ms(\Sigma^-)$. Let $\cadv+\cret=1$ be a smooth partition of unity such that $\cadv(x)=0,$ $\cret(x)=1$ for $x\in J_\Ms^+(\Sigma^+)$ and $\cadv(x)=1$, $\cret(x)=0$ for $x\in J_\Ms^-(\Sigma_-)$. We let $s=\tenpow{(\PM\cadv\EM)}nt_n$; by \eqref{eqn:epchidist} we have $\tenpow\EM ns=\tenpow\EM nt_n$, so by lemma \ref{lem:Ekernel} it follows that
\[
t_n-s=\sum_{k=1}^n(\PM)_ku_k
\]
for some $u_k\in\T^n(\Ms),$ $k=1,\ldots,n$. The required support properties of $s$ follow from the support of $\tenpow\EM nt_n$ and the fact that $\cadv$ is constant outside $S$.
\end{proof}
The above results allow us to prove the following:
\begin{theorem}
The theory $\W$ of Wick polynomials is dynamically local in the massive minimally coupled case and the massive conformally coupled case. The theory is not dynamically local in the massless minimally coupled case.
\end{theorem}
\begin{proof}We pick a spacetime $\Ms$, and some $O\in\mathscr{O}(\Ms)$; we will denote the dynamical and kinematic nets for $\W$ by $\wdyn_{\Ms;O}$, and $\wkin_{\Ms;O}$ respectively. We may then use a similar argument to that used above to see that $\wkin_{\Ms;O}(\W^\text{kin}(\Ms;O))$ comprises elements $W\in\W(\Ms)$ with $W=[W_H]_{H\in\H(\Ms)}$, where each $W_H\in\W_H(\Ms)$ can be represented by $T_H\in\T_H(\Ms)$ with the property that $T_H[\EM f]=T_H[0]$ for all $f\in\Ccs(O')$.
As already mentioned, for an additive theory, it is sufficient for dynamical locality to show that we have $\Wb(\Ms;K)\subseteq\wkin_{\Ms;O}(\W^\text{kin}(\Ms;O))$ for all $K\in\mathscr{K}(\Ms;O)$; we therefore pick some such $K$, and let $W\in\Wb(\Ms;K)$. Let $W=(W_H)_{H\in\H(\Ms)}$, and pick some fixed $H\in\H(\Ms)$; moreover, let $W_H\in\W_H(\Ms)$ be represented by a functional $T_H\in\T_H(\Ms)$. Since $\rce_\Ms[\h]W=W$ for all $\h\in H(\Ms;K^\perp)$ it follows that
\beq
\label{eqn:wpfunccond}
B_H[\h]\lambda_{H,\cH_\h}T_H\sim_\Ms T_H
\eeq
for all such $\h$. If $T_H=\sum_{n=0}^Nt_n$ with each $t_n\in\T^n(\Ms)$, then using \eqref{eqn:lambdadef}, interchanging sums and relabelling, we may write
\begin{align*}
\lambda_{H,\cH_\h}T_H&=\sum_{k=0}^{\lfloor n/2\rfloor}\frac{1}{k!}\sum_{n=0}^N\left\langle\tenpow{(H-\cH_\h)}k,
t_n^{(2k)}\right\rangle\\
&=\sum_{n=0}^N\sum_{k=0}^{\lfloor\frac{N-n}2\rfloor}\frac1{k!}\left\langle\tenpow{(H-\cH_\h)}k,t_{n+2k}^{(2k)}\right\rangle;
\end{align*}
the precise meaning of the notation here is given in \eqref{eqn:anglefuncdef}.
Note that in the second sum, the inner sum for each $n$ consists only of elements of $\T^n(\Ms)$; we write
\beq
\label{eqn:wptildedef}
\T^n(\Ms)\ni\tilde t_{n;\h}=\sum_{k=0}^{\lfloor\frac{N-n}2\rfloor}\frac1{k!}\left\langle\tenpow{(H-\cH_\h)}k,t_{n+2k}^{(2k)}\right\rangle
\eeq
for $n=0,\ldots,N$, and may express the condition \eqref{eqn:wpfunccond} as
\beq
\label{eqn:wpfunccond2}
\left(\tenpow{(\beta[\h])}n\tilde t_{n;\h}\right)[\EM f]=t_n[\EM f]
\eeq
for all $f\in\Ccs(\Ms)$ and for each $1\leq n\leq N$. We note that the $n=0$ term in \eqref{eqn:wpfunccond} requires
\beq
\label{eqn:wpfunccond0}
\sum_{k=1}^{\lfloor N/2\rfloor}\frac1{k!}\left\langle\tenpow{(H-\cH_\h)}k,t_{2k}^{(2k)}\right\rangle=0
\eeq
for all $\h\in H(\Ms;K^\perp).$
It follows from \eqref{eqn:wpfunccond2} that for $n\geq1$ we have
\[
\left.\frac{d}{ds}\left(\tenpow{(\beta[s\h])}n\tilde t_{n;s\h}\right)[\EM f]\right|_{s=0}=0
\]
for all $f\in\Ccs(\Ms)$ and $\h\in H(\Ms;K^\perp)$. But since $\beta[\mathbf{0}]=\id$ and $\tilde t_{n;\mathbf{0}}=t_n$, this is equivalent to
\beq
\label{eqn:sepwpdiffcond}
\left.\frac{d}{ds}\left(\tenpow{(\beta[s\h])}nt_n\right)[\EM f]\right|_{s=0}+\left.\frac{d}{ds}\tilde t_{n;s\h}[\EM f]\right|_{s=0}=0;
\eeq
by corollary \ref{cor:rcederivn}, we have
\beq
\left.\frac{d}{ds}\left(\tenpow{(\beta[s\h])}nt_n\right)[\EM f]\right|_{s=0}=n\int_\Ms dvol_\Ms\,h_{ab}T^{ab}[\EM \tau^n_f,\EM f],
\label{eqn:betaderivdef}
\eeq
where as before $\tau^n_f$ is defined according to \eqref{eqn:taudef}.
We now wish to show that in fact $\tilde t_{n;\h}\sim_\Ms t_n$ for all $\h\in H(\Ms;K^\perp)$ and $n\geq0$. To do so, we firstly note that by \eqref{eqn:wptildedef}, we automatically have $\tilde t_{N;\h}=t_N$ and $\tilde t_{N-1;\h}=t_{N-1}$ for all $\h\in H(\Ms)$. We may then proceed by descent, using the fact that $\tilde t_{n;\h}\sim_\Ms t_n$ for all $\h\in H(\Ms;K^\perp)$ if $\tilde t_{n+2k;\h}\sim_\Ms t_{n+2k}$ for all $k$ satisfying $2\leq2k\leq N-n$. This can be shown from the previous results, as follows.
If $\tilde t_{n+2k;\h}\sim_\Ms t_{n+2k}$ for $2\leq2k\leq N-n$, then (with $n$ replaced by $n+2k$) the second term on the left hand side of \eqref{eqn:sepwpdiffcond} vanishes, and so by \eqref{eqn:betaderivdef} we also have
\[
\int_{\Ms}dvol_\Ms\,h_{ab}T^{ab}[\EM \tau^{n+2k}_f,\EM f]=0
\]
for $2\leq2k\leq N-n$ and $\h\in H(\Ms;K^\perp)$. It follows from lemma \ref{lem:tsuppprop} that in the massive minimally coupled and massive conformally coupled theories, we have $\supp(\tenpow\EM{(n+2k)} t_{n+2k})\subseteq J_\Ms(K)^{\times(n+2k)}$. We may now use lemma \ref{lem:Esupport} to see that for any open neighbourhood $S$ of an arbitrary Cauchy surface, the distributions $t_{n+2k}$ may be written
\[
t_{n+2k}=s+\sum_{j=1}^{n+2k}(\PM)_ju_j
\]
where $s,u_j\in\T^{n+2k}(\Ms)$ and $\supp(s)\subseteq (J_\Ms(K)\cap S)^{\times(n+2k)}$. If we now fix some $\h\in H(\Ms,K^\perp)$ and choose $S$ such that $J_\Ms(\supp(\h))\cap J_\Ms(K)\cap S=\emptyset$, it follows that
\[
\left\langle\tenpow{(H-\cH_\h)}k,s^{(2k)}\right\rangle=0,
\]
recalling from lemma \ref{lem:Hdiffsupp} that $\supp(H-\cH_\h)\subseteq(J_\Ms(\supp(\h)))^{\times2}$. But this means that for all $f\in\Ccs(\Ms)$, we have
\[
\left\langle\tenpow{(H-\cH_\h)}k,t_{n+2k}^{(2k)}\right\rangle[\EM f]=
\sum_{j=1}^{n+2k}\left\langle\tenpow{(H-\cH_\h)}k,(\PM)_ju_j^{(2k)}
\right\rangle[\EM f]=0
\]
for $2\leq2k\leq N-n$, where we have used the fact that $(\PM\otimes\id)H=0=(\id\otimes\PM)H$ for any $H\in\H(\Ms)$. By \eqref{eqn:wptildedef}, we therefore have $\tilde t_{n;\h}\sim_\Ms t_n$.
As observed above, we certainly know that $\tilde t_{N;\h}\sim_\Ms t_N$ and $\tilde t_{N-1;\h}\sim_\Ms t_{N-1}$ for all $\h\in H(\Ms;K^\perp)$, and consequently by the above arguments $\tilde t_{N-2;\h}\sim_\Ms t_{N-2}$ and $\tilde t_{N-3;\h}\sim t_{N-3}$ for all $\h\in H(\Ms;K^\perp)$. We may continue this argument to see that in fact $\tilde t_{n;\h}\sim_\Ms t_n$ for all $n\geq0$ and $\h\in H(\Ms;K^\perp)$. Therefore \eqref{eqn:sepwpdiffcond} and \eqref{eqn:betaderivdef} tell us that \eqref{eqn:tintzero} is satisfied for all $n\geq1$; a final use of lemma \ref{lem:tsuppprop} tells us that $\supp(\tenpow\EM nt_n)\subseteq J_\Ms(K)^{\times n}$ for $n=1,\ldots,N$.
This firstly shows that the condition \eqref{eqn:wpfunccond0} is satisfied. More importantly, it shows that $t_n[\EM f]=0$ for all $f\in\Ccs(K^\perp)$ and $n\geq 1$, and therefore if $T$ represents an element of $\Wb(\Ms;K)$, then $T[\EM f]=t_0=T[0]$ for all $f\in\Ccs(K^\perp)$. If this is the case for all $K\in\mathscr{K}(\Ms;O)$ then $T$ represents an element of $\wkin_{\Ms;O}(\W^\text{kin}(\Ms;O))$, and so $\Wb(M;K)\subseteq\wkin_{\Ms;O}(\W^\text{kin}(\Ms;O))$ for all $K\subset O$. Therefore the massive minimally coupled and massive conformally coupled theories are dynamically local. We have already observed that the massless minimally coupled theory is not dynamically local.
\end{proof}
\subsubsection*{Acknowledgement} The author wishes to thank Chris Fewster for help, support and many useful conversations throughout the course of this work.
|
{
"timestamp": "2012-03-12T01:02:15",
"yymm": "1203",
"arxiv_id": "1203.2151",
"language": "en",
"url": "https://arxiv.org/abs/1203.2151"
}
|
\section{Introduction}
\label{sect:intro}
\par Gamma--ray bursts (GRBs) are cosmic explosions that release an extreme amount of energy in a very short time. As it was already noticed in the 1980s \citep[e.g.,][]{Norris1984} and became more obvious later, GRBs form two distinct populations \citep[e.g.,][]{Kouveliotou1993}: the short and long GRBs (SGRBs and LGRBs, respectively), defined at first approximation on the basis of the burst duration (SGRBs lasting less than $\sim 2\,\mathrm{s}$ in the observer frame), and likely corresponding to two different progenitors.
\par The merger of a double neutron star (NS-NS) or a neutron star -- black hole (NS-BH) binary system is currently the leading model for SGRBs. The events predicted by this model \citep[e.g.,][]{Eichler1989, Narayan1992, Nakar2007} are expected to have comparable time scale and energy release to those observed in SGRBs. In such systems, the delay between binary formation and merging is driven by the gravitational wave inspiral time, which is strongly dependent on the initial system separation. Some systems are thus expected to drift away from the star-forming regions in which they formed, before merging takes place. Simulations \citep{Belczynski2002, Belczynski2006} show that a large fraction of the merging events should take place in the outskirts or even outside the galaxies, in low density environments. A much faster evolutionary channel has been proposed \citep{BelczynskiKalogera2001, PernaBelczynski2002, Belczynski2006}, leading to merging in only $\sim 10^6-10^7\,\mathrm{yr}$, when most systems are still immersed in their star-forming regions. The above scenarios are based on ``primordial'' binaries, i.e., systems that were born as binaries. Alternatively, a sizeable fraction of NS-NS systems may form dynamically by binary exchange interactions in globular clusters during their core collapse \citep{Grindlay2006, Salvaterra2008}. The resulting time delay between star formation and merging would be dominated by the cluster core-collapse time and thus be comparable to the Hubble time \citep{Hopman2006}.
\par Merger scenario differs from the collapse of a massive star, which is believed to be associated with long duration GRBs (LGRBs). These two types of progenitors produce different outcomes when exploding as GRBs. At variance with LGRBs, we have no ``smoking gun'' (like supernova signatures) to identify the nature of the progenitors of SGRBs. However, SGRBs are not distinguished from LGRBs only by their duration, but also by other observed properties. If we consider the prompt emission, negligible spectral lag \citep{Norris2000,Norris2001} and hard spectra \citep{Kouveliotou1993} are common for SGRBs. As opposed to LGRBs, for which the isotropic equivalent gamma-ray energy, $E_{\gamma,\mathrm{iso}}$, is of the order of $10^{53}\,\mathrm{erg}$ and for which the host galaxies are typically dwarf galaxies with high star formation rate \citep{Fruchter2006, Savaglio2009}, SGRBs are typically less energetic ($E_{\gamma,\mathrm{iso}}$ is of the order of $10^{49} - 10^{51}\,\mathrm{erg}$), they occur in both early- and late-type galaxies with lower star formation rate and are associated with an old stellar population \citep{Nakar2007, Berger2009, Berger2011}. Furthermore, SGRBs have been found to be inconsistent with the $E_\mathrm{p,i}-E_\mathrm{iso}$ correlation (\citealt{Amati2006, Amati2007, Amati2008}).
\par The afterglows of SGRBs tend to be significantly fainter on average than those of LGRBs \citep{Kann2011}. This is believed to be a consequence of the energetics and the surrounding environment \citep{Nakar2007}. As shown in \citet{Campana2010}, a powerful tool to characterise the GRB environment is the study of their X-ray absorbing column densities. By a systematic analysis of LGRBs with known redshift promptly observed by the \textit{Swift} X-Ray Telescope (XRT), \citet{Campana2010} found clear evidence that LGRB X-ray afterglows are heavily absorbed and occur in dense environments, as expected in the context of a massive stellar progenitor.
\par In this paper we present a comprehensive analysis of the full sample of SGRBs with robust redshift determination, promptly observed\footnote{Within $150\,\mathrm{s}$ from the burst occurrence, and without the autonomous slew delay due to an observing constraint or due to a low merit value.} by the \textit{Swift} XRT up to January 2011. For all these events we derived the intrinsic X-ray column densities. Our findings are then compared to the results of \citet{Campana2010} obtained for LGRBs, with the aim of checking if the surrounding environment of these two classes of events is different and if this is perhaps related to the type of progenitor (as already discussed by \citealt{Salvaterra2010}), as well as to various SGRBs properties (redshift, duration, host galaxy offset and normalised host galaxy offset).
\par In Section \ref{sect:data}, we present the analysis of X-ray data taken from the \textit{Swift} XRT and describe how our sample was built. In Section \ref{sect:results}, we perform various analyses on our sample and discuss the results. Summary and conclusions are given in Section \ref{sect:conclusion}.
\par Throughout the paper we assume a standard cosmology with parameters: $H_0 = 71\,\mathrm{km\,s^{-1}\,Mpc^{-1}}$, $\Omega _\Lambda = 0.73$, $\Omega _\mathrm{M} = 0.27$.
\section{Data Analysis}
\label{sect:data}
\subsection{Sample selection}
\label{sect:sample_selection}
\par We collected the information on the $\mathrm{T}_{90}$\footnote{Time interval in which $90\%$ of the fluence in gamma-rays (in this case, in the $15-350\,\mathrm{keV}$ energy band) is detected.} in the observer's frame from the \textit{Swift}-BAT refined analysis GCN circulars\footnote{http://gcn.gsfc.nasa.gov/gcn3\_archive.html.}, together with the properties in gamma regime (spectral hardness and spectral lag) for all GRBs detected with the \textit{Swift} Burst Alert Telescope (BAT) until January 2011. We consider all GRBs classified as short in the Swift-BAT refined analysis GCN circulars$^3$, where additional properties in the gamma regime (apart from $\mathrm{T}_{90}$), such as the lack of a spectral lag and the hardness ratio are used to assess the short nature of a GRB. These criteria enable to include in our sample also SGRBs with an extended emission (EE), for which $\mathrm{T}_{90}$ can be well above $2\,\mathrm{s}$. We selected the events observed by the \textit{Swift} XRT and obtained a list of $60$ SGRBs.
\par To exclude any observational biases, we checked the time delay between the BAT trigger and the start of the observations by the XRT. We found that for $13$ SGRBs the XRT observations started with significant delay of hours or even days after the BAT trigger due to observing constrains. We eliminated these SGRBs and ended up with $47$ SGRBs. Among these, $6$ had no X-ray afterglow detected and $17$ had an X-ray afterglow too faint to perform any spectral analysis (see Section \ref{subsect:analysis}). Also these $23$ SGRBs have been excluded from our analyses.
\par For the remaining $24$ SGRBs (see Table \ref{tab:he_properties} for the complete list) we retrieved the redshift information from the literature. We used only redshifts that are robust, meaning that an optical afterglow (OA) was detected and found to lie within the host galaxy's light with a sub-arcsecond precision (so that the association between a host galaxy and a GRB is clear), and that the spectrum of the associated host galaxy was recorded. We obtained robust redshifts for $13$ SGRBs, mainly from the GCN Circulars Archive or from published papers. We put the remaining $11$ SGRBs to redshift $z=0$, to extend our analysis on SGRBs' intrinsic X-ray absorption with the obtained lower limits.
\par The sample of $13$ SGRBs with robust redshifts includes also $3$ GRBs for which the classification as a SGRB is still debated; GRB~060614 was a supernova-less GRB at $z=0.125$ down to very deep optical limits, with spectral lag typical for SGRBs, but with a $\mathrm{T}_{90}$ of $102\,\mathrm{s}$ and time-averaged spectral properties similar to LGRBs (\citealt{Gehrels2006, DellaValle2006, Mangano2007, Amati2007}). The possibility that this event is a SGRB with an EE was also suggested (\citealt{Fynbo2006, GalYam2006, Zhang2007}). GRB~090426 can be classified as a SGRB based on its $\mathrm{T}_{90}$, but the spectral and energy properties are more similar to those of LGRBs (\citealt{Levesque2009, Antonelli2009, Ukwatta2009, Thone2011}). Similarly, GRB~100724A can be classified as a SGRB based on its $\mathrm{T}_{90}$, but spectral lag and hardness of the spectrum point towards a LGRB classification \citep{Ukwatta2010}. Given the uncertainty in the classification, we do not consider also these $3$ GRBs in our analyses and thus end up with $10$ SGRBs with robust redshift determination.
\par Another selection criterion for SGRBs could be the inconsistency with the $E_\mathrm{p,i}-E_\mathrm{iso}$ correlation (\citealt{Amati2007}), in which case one additional GRB could be considered short: GRB~060505 (see also \citealt{Ofek2007, Thone2008, Mcbreen2008, Xu2009}). However, GRB~060505 was not observed promptly by the \textit{Swift} XRT and so we did not include it in our analyses.
\par For the sake of simplicity throughout the paper we name as Sample \MakeUppercase{\romannumeral 1} the sample of $10$ SGRBs with robust redshift determination (presented in Table \ref{tab:sample}), and as Sample \MakeUppercase{\romannumeral 2} the sample of $11$ SGRBs without robust redshift determination (presented in Table \ref{tab:sample_lowerlimits}).
\begin{table*}
\renewcommand{\arraystretch}{1.1}
\begin{center}
\tabcolsep 3pt
\begin{tabular}{ccccccccccccc}
\hline
&
&
&
&
\multicolumn{6}{c}{X-ray afterglow properties} &
\multicolumn{2}{c}{Host galaxy} &
Ref. \\
GRB &
$z$ &
$\mathrm{T}_{90}$ &
EE &
SL/LL &
$\mathrm{N_H}$ Gal. &
Exp. (interval) &
$\Gamma$ &
$\mathrm{N_H}(z)$ &
C-stat (dof) &
Offset &
Norm. offset &
\\
&
&
[s] &
&
&
[$10^{20}\,\mathrm{cm^{-2}}$] &
[$\mathrm{ks}$] ([$\mathrm{s}$]) &
&
[$10^{21}\,\mathrm{cm^{-2}}$] &
&
[$\mathrm{kpc}$] &
&
\\
\hline
050724 & $0.257$ & $3.0$ & yes & LL & $14.0$ & $22.0$ ($6000 - 10^5$) & $1.66_{-0.20}^{+0.15}$ & $2.1_{-1.3}^{+1.5}$ & $214.5 (279)$ & $2.54 \pm 0.08$ & $0.43 \pm 0.02$ & a) \\
051221A & $0.546$ & $1.4$ & no & LL & $5.7$ & $69.1$ ($6000-2 {\times} 10^5$) & $2.10_{-0.10}^{+0.14}$ & $2.2_{-1.1}^{+1.1}$ & $260.4 (307)$ & $1.53 \pm 0.31$ & $0.30 \pm 0.06$ & a) \\
061006 & $0.438$ & $130.0$ & yes & LL & $14.1$ & $20.6$ ($150-10^5$) & $1.84_{-0.27}^{+0.28}$ & $< 2.6^\mathrm{UL}$ & $104.1 (170)$ & $1.44 \pm 0.29$ & $0.40 \pm 0.08$ & a) \\
070714B & $0.923$ & $64.0$ & yes & LL & $6.4$ & $25.9$ ($450-6 {\times} 10^4$) & $1.97_{-0.16}^{+0.11}$ & $2.8_{-2.1}^{+2.1}$ & $283.6 (292)$ & $3.08 \pm 0.47$ & $0.78 \pm 0.12$ & a) \\
070724A & $0.457$ & $0.4$ & no & LL & $1.2$ & $0.024$ ($82-106$) & $1.51_{-0.23}^{+0.24}$ & $5.5_{-2.5}^{+2.8}$ & $161.9 (232)$ & $4.76 \pm 0.06$ & $1.12 \pm 0.02$ & a) \\
071227 & $0.381$ & $1.8$ & yes & LL & $1.3$ & $0.05$ ($90-140$) & $1.51_{-0.17}^{+0.17}$ & $2.8_{-1.1}^{+1.3}$ & $247.8 (330)$ & $14.8 \pm 0.3$ & $1.10 \pm 0.03$ & b) \\
080905A & $0.122$ & $1.0$ & no & / & $9.0$ & $0.9$ ($330-1230$) & $1.75_{-0.52}^{+0.56}$ & $< 3.4^\mathrm{UL}$ & $94.2 (76)$ & $18.1 \pm 0.4$ & / & a) \\
090510 & $0.903$ & $0.3$ & no & LL & $1.7$ & $0.14$ ($100-240$) & $1.79_{-0.15}^{+0.16}$ & $1.9_{-1.4}^{+1.5}$ & $284.1 (315)$ & $7.8 \pm 3.9$ & $1.29 \pm 0.65$ & c) \\
100117A & $0.915$ & $0.3$ & no & SL & $2.7$ & $0.05$ ($105-155$) & $1.44_{-0.22}^{+0.25}$ & $3.4_{-3.0}^{+4.4}$ & $156.6 (215)$ & $0.47 \pm 0.31$ & / & d) \\
100816A & $0.805$ & $2.9$ & yes & LL & $4.5$ & $16.1$ ($200-5 {\times} 10^4$) & $1.90_{-0.15}^{+0.13}$ & $2.3_{-1.5}^{+1.6}$ & $273.7 (309)$ & $8.2 \pm 2.3$ & $0.64 \pm 0.20$ & c) \\
\hline
\end{tabular}
\end{center}
\caption{Properties of $10$ SGRBs with robust redshifts (Sample \MakeUppercase{\romannumeral 1}). Columns are: GRB identifier, redshift, $\mathrm{T}_{90}$, SGRB with an extended emission (EE), short-lived (SL) or long-lived (LL) X-ray afterglow (see Section \ref{sl_ll}), Galactic X-ray absorbing column density, X-ray spectrum exposure time and time interval, photon index ($\Gamma$), intrinsic X-ray absorbing column density, Cash's C statistic of the spectral fit (with the corresponding degrees of freedom), host galaxy offset, normalised host galaxy offset, references. All SGRBs have an optical afterglow (OA) detected. Redshifts are obtained from \citet{Berger2009} and references therein, except for GRB~071227 \citep{DAvanzo2009}, GRB~080905A \citep{Rowlinson2010a}, GRB~090510 \citep{Mcbreen2010}, GRB~100117A \citep{Fong2011} and GRB~100816A \citep{Tanvir2010}. Errors are given at $90\%$ confidence level. $^\mathrm{UL}$ indicates upper limits. For GRB~070724A, GRB~071227, GRB~090510 and GRB~100117A we used WT mode spectra, while for others we used PC mode spectra in the specified time interval. For GRB~080905A we can not determine if it is SL or LL, due to the lack of the X-ray flux information around $10^4\,\mathrm{s}$ after the trigger. \newline References for host galaxy offsets: a) \citet{Church2011} and references therein; b) \citet{Fong2010, Berger2011}; c) Host galaxy offset from our inspection; d) \citet{Fong2011}.
}
\label{tab:sample}
\end{table*}
\subsection{Sample analysis}
\label{subsect:analysis}
\par In order to determine intrinsic absorption in X-rays for SGRBs, we performed a similar analysis to the one presented by \citet{Campana2010} for the LGRBs. We used the \textit{Swift} XRT GRB lightcurve repository \citep{Evans2009}. Because SGRBs have significantly fainter afterglows as opposed to LGRBs \citep{Kann2011}, it is sometimes impossible to get enough X-ray photons to perform any spectral analysis\footnote{We required at least $200$ counts with constant $0.3-1.5\,\mathrm{keV}$/$1.5-10\,\mathrm{keV}$ hardness ratio in the time interval that is specified in Table \ref{tab:sample} and Table \ref{tab:sample_lowerlimits} to perform spectral analysis and obtain the values of the intrinsic X-ray absorbing column density.}. In addition, typical X-ray afterglow light curves have a multi-component canonical shape and show strong variability and spectral evolution especially at early times. We try to avoid taking data from this epoch and therefore we loose the brightest part of the afterglow. Based on these facts, we used data mostly from photon counting (PC) mode, except for very dim afterglows, where we were forced to use also data from window timing (WT) mode to increase our statistics by gathering enough X-ray photons. We analysed data only in time epoch where the $0.3-1.5\,\mathrm{keV}$ to $1.5-10\,\mathrm{keV}$ hardness ratio is constant, in order to prevent spectral changes that can affect the X-ray column density determination.
\par We used the \textit{Swift} XRT GRB spectrum repository\footnote{http://www.swift.ac.uk/xrt\_spectra/.} to obtain the intrinsic X-ray absorbing column densities. We also checked that the spectral fits are consistent with the values reported on the repository. The spectra that we used were binned to have at least one count in each spectral bin, so that the C-statistic can be used for fitting. Using \texttt{XSPECv12.6} software \citep{Arnaud1996} we fitted the spectra with the combination of power-law behaviour and photoelectric absorption contributions (phabs$^\ast$zphabs$^\ast$powerlaw) in the $0.3-10\,\mathrm{keV}$ energy range. We fixed one contribution of photoelectric absorption to the value of the Galactic equivalent X-ray column density along the direction of a GRB using \citet{Kalberla2005}. Since there is an error associated with these values \citep{Wakker2011}, we added an uncertainty factor of $0.1 \times (1 + z)^{2.6}$ (i.e., $10\%$, propagated with the redshift) to the intrinsic error \citep{Watson2011}. The second contribution, shifted in energy to the rest-frame of the GRB using its redshift (for SGRBs from Sample \MakeUppercase{\romannumeral 2} we used $z=0$), was left as a free parameter to vary. The intrinsic X-ray absorbing column densities for our final sample are provided in Table \ref{tab:sample}.
\par Working on a sample of $10$ SGRBs observed with the Hubble Space Telescope, \citet{Fong2010} showed that while the host galaxy offsets (offset of a projected GRB optical afterglow location from the center of its host galaxy) of SGRBs are larger than those of LGRBs, the distribution of normalised host galaxy offsets (offset divided by the effective radius $R_e$ of the galaxy) is nearly identical due to the larger size of SGRBs' host galaxies. For each SGRB belonging to Sample \MakeUppercase{\romannumeral 1} we compared the derived X-ray column density with the value of the projected host galaxy offset and with the normalised host galaxy offset. Some SGRBs from Sample \MakeUppercase{\romannumeral 1} have their host galaxy and normalised host galaxy offset already presented in the sample of \citet{Fong2010}. However, for the sake of homogeneity, we computed independently the host galaxy effective radii for the SGRBs of Sample \MakeUppercase{\romannumeral 1}, using public archival and proprietary ground-based imaging data. All SGRBs' host galaxies were observed with the ESO-VLT, equipped with the FORS1/2 camera, with the exception of GRB~070714B (Gemini-N/GMOS data) and GRB~100816A (TNG/DOLORES data). Using the extended surface photometry tool of the GAIA\footnote{Graphical Astronomy and Image Analysis Tool, http://astro.dur.ac.uk/$\sim$pdraper/gaia/gaia.html} package, we obtained the surface brightness profiles of $8$ SGRBs' host galaxies. The obtained galaxy profiles were fitted with a S\'ersic model:
\begin{equation}
\label{sersic}
I(R) = I_e \, \mathrm{e}^{-b_n[\left({\frac{R}{R_e}}\right)^{1/n}-1]} \,,
\end{equation}
where $n$ is the index indicating the profile curvature ($n=1$ gives an exponential disk profile, $n=4$ is the de Vaucouleurs profile), $b_n$ is a dimensionless scale factor that can be approximated with $b_n~\sim~2n~-~\frac{1}{3}~+~\frac{4}{405n}~+~\frac{46}{25515n^2}$ \citep{Ciotti1999}, $R_e$ is the galaxy effective radius and $I_e$ is the intensity at $R=R_e$. For each fit, $n$, $R_e$ and $I_e$ were left free to vary. Host galaxy offsets and normalised host galaxy offsets for SGRBs of Sample \MakeUppercase{\romannumeral 1} are provided in Table \ref{tab:sample}.
\par To check for various correlations we used the Spearman's rank correlation test throughout the paper. The uncertainties in the correlation coefficients were estimated with a simple Monte Carlo simulation, by taking into account the true errors using
\begin{equation}
\label{eq:mc}
\mathrm{value^{simulation}} = \mathrm{value^{true}} + \epsilon \times \mathrm{error^{true}}\,,
\end{equation}
where $\epsilon$ is a random number drawn from a normal distribution with zero mean and unit variance. With this method we obtained $1000$ simulated correlation coefficients for each set of values, and estimated the uncertainty on the original correlation coefficient as the standard deviation of simulated coefficients. Together with the correlation coefficient and its uncertainty, the $p$-value is also given.
\section{Results and Discussion}
\label{sect:results}
\par We were primarily interested in the study of the intrinsic X-ray absorption, and its correlations with $\mathrm{T}_{90}$, redshift, and particularly with host galaxy offset and normalised host galaxy offset. We extended our analysis with the discussion on the properties in the gamma energy band, on the SGRBs with an extended emission and on the short-lived and long-lived X-ray afterglows (defined in light of the X-ray flux measured at $10^4\,\mathrm{s}$ after the burst; see Section \ref{sl_ll}). The study of these observables, which could be commonly obtained for SGRBs, provides the answer whether properties at high energies (such as $\mathrm{T}_{90}$, energetics and X-ray afterglow duration) are linked to the properties of the environment.
\subsection{Intrinsic X-ray absorption}
\label{sect:intrinsic_absorption}
\par We first derived the intrinsic X-ray absorbing column densities, $\mathrm{N_H}(z)$, for SGRBs belonging to Sample \MakeUppercase{\romannumeral 1}. We obtained a measure for $8$ of them, while for $2$ we obtained only upper limits (Table \ref{tab:sample}). Figure \ref{fig:nhdist} shows the distribution of $\mathrm{N_H}(z)$ for our whole sample. The distribution for the $8$ SGRBs with direct estimates of the intrinsic column density can be well fitted by a log Gaussian function, with a mean $\mathrm{log\,N_H} (z) = 21.4$ and a standard deviation $\sigma _{\mathrm{log\,N_H}(z)} = 0.1$ (reduced $\chi ^2$ of the fit is $0.4$ with $2$ d.o.f.). This shows that SGRBs are intrinsically absorbed.
\par Nevertheless, the mean $\mathrm{log\,N_H} (z)$ for SGRBs is on average lower than for LGRBs from \citet{Campana2010}, who reported a value of $21.9$ with a standard deviation of $0.5$. However, given the quoted error bars, the values are consistent. But it is worth to note that SGRBs from Sample \MakeUppercase{\romannumeral 1} span in redshift up to $z_\mathrm{max}^\mathrm{SGRBs} = 0.923$, so we are biased to lower redshifts when comparing $\mathrm{N_H}(z)$ values, since LGRBs from \citet{Campana2010} span in redshift up to $z_\mathrm{max}^\mathrm{LGRBs}=8.1$. To make a more appropriate comparison, we determined a mean $\mathrm{log\,N_H} (z)$ for LGRBs with $z < 0.923$ and $z > 0.923$ in the sample from \citet{Campana2010}, obtaining a mean $\mathrm{log\,N_H} (z<0.923) = 21.5$ with a standard deviation of $0.4$ (reduced $\chi ^2$ of the fit is $1.5$) and a mean $\mathrm{log\,N_H} (z>0.923) = 22.0$ with a standard deviation of $0.4$ (reduced $\chi ^2$ of the fit is $0.6$), respectively. Thus, by limiting the analysis to $z<0.923$, the mean $\mathrm{N_H}(z)$ of the LGRBs and the SGRBs from Sample \MakeUppercase{\romannumeral 1} do not differ significantly.
\begin{figure}
\begin{center}
\includegraphics[angle=-90,width=1\columnwidth]{hist_number_vs_NH_newerr.eps}
\end{center}
\caption{Intrinsic X-ray absorbing column densities ($\mathrm{N_H}(z)$) distribution for Sample \MakeUppercase{\romannumeral 1} (red histogram). Red dashed line and arrows represent $2$ SGRBs from Sample \MakeUppercase{\romannumeral 1} for which only upper limits were determined. Black arrows represent the lower limits of the $\mathrm{N_H}(z=0)$ value for $11$ SGRBs from Sample \MakeUppercase{\romannumeral 2} (we put them to $z=0$). Continuous red solid line is the log Gaussian fit for $8$ SGRBs from Sample \MakeUppercase{\romannumeral 1} without upper limits.}
\label{fig:nhdist}
\end{figure}
\par To further investigate if the $\mathrm{N_H}(z)$ distributions of SGRBs and LGRBs are different, we used the two-sample Kolmogorov-Smirnov (K-S) test. We tested the distributions of $8$ SGRBs from Sample \MakeUppercase{\romannumeral 1} and $85$ LGRBs from \citet{Campana2010}, obtaining the value of the K-S statistic $D_1 = 0.70$, with the corresponding probability that the two distributions are not drawn from different populations $P_1 = 7 \times 10^{-4}$. Using the subsample of $14$ LGRBs with $z<0.923$ instead, we obtained $D_2 = 0.52$ and $P_2 = 0.1$, indicating that the two distributions are likely drawn from the same population. After the submission of this work, \citet{Margutti2012} independently reached similar conclusions on the intrinsic X-ray column density distribution of SGRBs.
\begin{table}
\begin{center}
\tabcolsep 2.3pt
\begin{tabular}{cccccc}
\hline
GRB &
$\mathrm{T}_{90}$ &
OA &
SL/LL &
Exp. (interval) &
$\mathrm{N_H}(z=0)$
\\
&
[s] &
&
&
[$\mathrm{ks}$] ([$\mathrm{s}$]) &
[$10^{21}\,\mathrm{cm^{-2}}$]
\\
\hline
051227$^\mathrm{EE}$ & $8.0$ & yes & LL & $0.05$ ($115 - 160$) & $> 1.3$\\
060313 & $0.7$ & yes & LL & $22.7$ ($4000 - 7{\times} 10^4$) & $> 0.1 $\\
060801 & $0.5$ & no & SL & $13.3$ ($50 - 4{\times}10^4$) & $> 0.3$\\
080503$^\mathrm{EE}$ & $170.0$ & yes & SL & $0.06$ ($90 - 150$) & $> 0.5$\\
080702A & $0.5$ & no & SL & $8.4$ ($50 - 2{\times}10^4$) & $>1.7$ \\
080919 & $0.6$ & no & SL & $1.9$ ($100 - 2000$) & $> 1.7$\\
081226A & $0.4$ & no & SL & $13.9$ ($100 - 5{\times} 10^4$) & $> 3.5$\\
090515 & $0.04$ & yes & SL & $0.06$ ($80 - 140$) & $> 0.2$\\
090607 & $2.3$ & no & SL & $0.03$ ($103 - 133$) & $> 1.0$\\
100702A & $0.16$ & no & SL & $15.0$ ($250 - 5{\times} 10^4$) & $> 1.3$\\
101219A & $0.6$ & no & / & $0.85$ ($50 - 900$) & $> 0.9$\\
\hline
\end{tabular}
\end{center}
\caption{Properties and lower limits of the $\mathrm{N_H}$ values at $z=0$ for $11$ SGRBs without robust redshifts (Sample \MakeUppercase{\romannumeral 2}). Columns are: GRB identifier, $\mathrm{T}_{90}$, optical afterglow (OA) detection, short-lived (SL) or long-lived (LL) X-ray afterglow (see Section \ref{sl_ll}), X-ray spectrum exposure time and time interval, lower limit on the intrinsic X-ray absorbing column density assuming $z=0$. We used PC mode spectra in the specified time interval, except for GRB~051227, GRB~080503, GRB~090515 and GRB~090607 where we used WT mode spectra. GRB~051227 and GRB~080503 have an extended emission ($^\mathrm{EE}$). For GRB~101219A we can not determine if it is SL or LL, due to the lack of the X-ray flux information at $10^4\,\mathrm{s}$ after the trigger.}
\label{tab:sample_lowerlimits}
\end{table}
\par Further confirmation that SGRBs are intrinsically absorbed is given by examining the Sample \MakeUppercase{\romannumeral 2}. We noticed that even if we put these SGRBs to $z=0$ and thus obtain the lower limits on the X-ray absorption, we get significant absorption in excess of the Galactic value (Table \ref{tab:sample_lowerlimits}). These $11$ lower limits are marked as black arrows in Figure \ref{fig:nhdist}.
\par Having derived the intrinsic X-ray absorbing column densities, we checked if there is any dependence between $\mathrm{N_H}(z)$ and $\mathrm{T}_{90}$ and between $\mathrm{N_H}(z)$ and redshift, but there is no significant correlation. The Spearman's correlation coefficients are $\rho _\mathrm{T_{90}} ^{\mathrm{N_H}(z)} =-0.11 \pm 0.36$ ($p=0.80$) and $\rho _{z} ^{\mathrm{N_H}(z)} =0.14 \pm 0.35$ ($p=0.73$), respectively. The uncertainties on the correlation coefficients are quite large, but this is expected given the large uncertainties on the values of $\mathrm{N_H}(z)$.
\par We checked if $\mathrm{N_H}(z)$ correlates with the Galactic $\mathrm{N_H}$ or with photon index ($\Gamma$). We found no significant correlation, with the Spearman's correlation coefficients being $\rho _\mathrm{N_H(Gal.)} ^{\mathrm{N_H}(z)} =-0.47 \pm 0.32$ ($p=0.24$) and $\rho _\Gamma ^{\mathrm{N_H}(z)} =-0.52 \pm 0.36$ ($p=0.18$), respectively.
\subsection{Intrinsic X-ray absorption and host galaxy offset}
\par For SGRBs from Sample \MakeUppercase{\romannumeral 1} we can investigate if there is any correlation between $\mathrm{N_H}(z)$ and host galaxy offset. Figure \ref{fig:nhvsoffset} shows $\mathrm{N_H}(z)$ versus host galaxy offsets and $\mathrm{N_H}(z)$ versus normalised host galaxy offsets.
\begin{figure*}
\begin{center}
\includegraphics[angle=-90,width=0.49\textwidth]{NH_vs_offset_newerr.eps}
\includegraphics[angle=-90,width=0.49\textwidth]{NH_vs_norm_offset_newerr.eps}
\end{center}
\caption{Left: Intrinsic X-ray absorption versus host galaxy offset. Right: intrinsic X-ray absorption versus normalised host galaxy offset. Values are taken from Table \ref{tab:sample}. Filled points are SGRBs without an extended emission (EE), while empty points are those with an extended emission. Red points represent upper limits on $\mathrm{N_H}(z)$.}
\label{fig:nhvsoffset}
\end{figure*}
\par We tested the possible correlations between (normalised) host galaxy offsets and $\mathrm{N_H}(z)$ using the Spearman's rank correlation test for Sample \MakeUppercase{\romannumeral 1}, excluding the $2$ SGRBs that have only upper limits on $\mathrm{N_H}(z)$. For host galaxy offsets versus $\mathrm{N_H}(z)$, the Spearman's correlation coefficient is $\rho _\mathrm{offsets} ^{\mathrm{N_H}(z)} = -0.07 \pm 0.36$ ($p=0.87$). For normalised host galaxy offsets versus $\mathrm{N_H}(z)$, the Spearman's correlation coefficient is $\rho _\mathrm{norm. \, offsets} ^{\mathrm{N_H}(z)} = 0.20 \pm 0.35$ ($p=0.67$). If we exclude GRB~070724A and GRB~071227, because the lower limits on their normalised host galaxy offsets are larger than $1$ (this would indicate that a GRB occurs outside of its host galaxy, if characterised by the effective radius), the Spearman's correlation coefficient is $\rho _\mathrm{norm. \, offsets} ^{\mathrm{N_H }(z)} = -0.10 \pm 0.47$ ($p=0.95$). All values indicate that there is no significant correlation. The uncertainties on the correlation coefficients are large due to large uncertainties on the values of $\mathrm{N_H}(z)$ and (normalised) host galaxy offsets (see Figure \ref{fig:nhvsoffset}).
\par For normalised host galaxy offsets, which indeed better represent the relative location of a GRB inside its host galaxy, one would expect that (especially when smaller than $1$) they will be inversely correlated with $\mathrm{N_H}(z)$. This would indicate that more we go towards the edge of the galaxy, less intrinsic absorption we have, i.e., the environment gets less dense. We obtained no such result. The reason could be either the large errors or that here we are dealing with projected offsets, which do not give us any information about the position of a GRB in its host galaxy along the line of sight. For the same projected offset, a GRB can occur closer or further away from us in its host galaxy, resulting in less or more intervening host galaxy material along the line of sight. Such effect is of course less strong near the edges of the galaxies, as is likely the case of GRB~071227, for which an optical afterglow position falls at the very edge of the galaxy disk \citep{DAvanzo2009}.
\par Furthermore, we note that the above analysis is probably affected by some selection effect against large offsets (and likely lower $\mathrm{N_H}(z)$) because, in order to have events with robust redshift, SGRBs of Sample \MakeUppercase{\romannumeral 1} include only events with positional coincidence between the optical afterglow and the host galaxy light (see Section \ref{sect:sample_selection}).
\subsection{X-ray afterglow brightness and the properties in gamma regime}
\label{sect:heprop}
\par As pointed out in Sections \ref{sect:sample_selection} and \ref{sect:intrinsic_absorption}, $44$ SGRBs promptly observed by the {\it Swift} XRT can be divided into two classes, according to the brightness of their X-ray afterglow. Almost half of them have X-ray afterglow bright enough to perform X-ray spectroscopy (they have at least $200$ counts with constant $0.3-1.5\,\mathrm{keV}$/$1.5-10\,\mathrm{keV}$ hardness ratio in the specified time interval), while the other half have too faint X-ray afterglow, and thus no spectroscopic study could be performed. Moreover, among the ``faint'' SGRBs, six events (i.e., $14\%$ of the whole sample or $26\%$ of the ``faint'' SGRBs' sample) have no X-ray afterglow detected, in spite of the prompt {\it Swift} XRT follow-up. This is at variance with respect to the LGRBs, where for $\sim 95\%$ of the events promptly observed by the {\it Swift} XRT an X-ray afterglow is detected \citep{Evans2009}. Such a bimodality in the X-ray afterglow brightness can be explained by differences in the GRB energetics (with more energetic GRBs having brighter afterglows) or in the density of the circumburst medium (with GRB exploding in denser environments having brighter afterglows).
\par In order to check if these two classes of SGRBs show any significant difference in their prompt emission properties, we computed their distribution of fluence, 1-s peak photon flux and $\mathrm{T}_{90}$, measured by the {\it Swift} BAT, and compared them with a two-sample Kolmogorov-Smirnov (K-S) test. As can be seen in Table \ref{tab:he_properties}, SGRBs with brighter X-ray afterglows seem to have, on average, higher fluences, peak fluxes and longer durations. The probabilities associated with the K-S tests for each distribution is of the order of a few percent (see Table \ref{tab:kstest2}, upper part), likely suggesting that the two classes of SGRBs have different prompt emission properties. However, the presence of $8$ SGRBs with an extended emission (GRB~050724, GRB~051227, GRB~061006, GRB~070714B, GRB~071227, GRB~080123, GRB~080503 and GRB~100816A) introduce some biases in this study. In particular, SGRBs with an extended emission can spuriously increase the average value of fluence and $\mathrm{T}_{90}$. We thus repeated the K-S tests excluding the events showing an extended emission. We find that the associated probabilities increase, suggesting that the two classes of SGRBs do not show significant differences in their prompt emission properties (see Table \ref{tab:kstest2}, lower part).
\par The different brightness of their X-ray afterglows might thus be a consequence of the density of the environment around the GRB and being indicative of different progenitors. A possible theoretical explanation would be that we are dealing with two distinct NS-NS or NS-BH populations. The events showing brighter X-ray afterglows might be associated with primordial double compact object systems merging in a relatively short time (and thus occurring inside their host galaxies, in star forming environments; \citealt{PernaBelczynski2002}), while the events with fainter X-ray afterglows might be originated by double compact object systems which experienced a large natal kick or which are dynamically formed in globular clusters (associated with a low-density environments; see Section \ref{sect:intro}).
\par On the other hand, we note that the values of $\mathrm{T}_{90}$, fluence and 1-sec peak photon flux reported in Table \ref{tab:he_properties} are measured in the observer's frame. It has not been possible, due to the lack of redshift measurement for many SGRBs, to transform these values to the GRB's rest frame, especially for SGRBs with faint X-ray afterglow. This may again introduce some bias, especially for SGRBs that would happen at high redshift \citep{Berger2007}.
\begin{table}
\begin{center}
\tabcolsep 24pt
\begin{tabular}{ccc}
\hline
Distribution & $D$ & $P$ \\
\hline
\multicolumn{3}{c}{All SGRBs:} \\
$\mathrm{T}_{90}$ & $0.43$ & $0.02$ \\
Fluence & $0.43$ & $0.02$ \\
1-sec peak flux & $0.31$ & $0.20$ \\
& & \\
\multicolumn{3}{c}{SGRBs without an extended emission:} \\
$\mathrm{T}_{90}$ & $0.40$ & $0.09$ \\
Fluence & $0.30$ & $0.34$ \\
1-sec peak flux & $0.30$ & $0.36$ \\
\hline
\end{tabular}
\end{center}
\caption{Results of the two-sample Kolmogorov-Smirnov test from comparing $\mathrm{T}_{90}$, fluence and 1-sec peak photon flux distributions between ``bright'' (Sample \MakeUppercase{\romannumeral 1} and \MakeUppercase{\romannumeral 2}) and ``faint'' X-ray afterglow samples. Upper part are results for all SGRBs, while lower part are results for SGRB without an extended emission (see Table \ref{tab:he_properties}). $D$ represents the calculated value of the Kolmogorov-Smirnov statistic, and $P$ is the corresponding probability that two distributions are not drawn from different populations.}
\label{tab:kstest2}
\end{table}
\subsection{SGRBs with an extended emission}
\par To investigate if SGRBs with an EE from Sample \MakeUppercase{\romannumeral 1} lie on average closer to the centre of their host galaxies (as presented first by \citealt{Troja2008}), we plotted in Figure \ref{fig:normhostvst90} the $\mathrm{T}_{90}$ values versus host galaxy and normalised host galaxy offsets. The value of $\mathrm{T}_{90}$ somehow indicates the duration of a GRB, and SGRBs with an EE have on average larger $\mathrm{T}_{90}$ than SGRBs without an EE. Calculating the Spearman's rank correlation coefficient between $\mathrm{T}_{90}$ and host galaxy offsets, we obtain $\rho _\mathrm{offsets} ^\mathrm{EE} = -0.13 \pm 0.10$ ($p=0.73$). This shows that there is no clear anti-correlation between $\mathrm{T}_{90}$ and host galaxy offset. Similar conclusions were also presented by \citet{Fong2010}.
\par Extending this analysis, we checked if there is any correlation between $\mathrm{T}_{90}$ and normalised host galaxy offset (see Figure \ref{fig:normhostvst90}, bottom panel). In this case the data look somewhat less scattered. Calculating the Spearman's rank correlation coefficient (without GRB~100117A), we obtain $\rho _\mathrm{norm. \, offsets} ^\mathrm{EE} = -0.57 \pm 0.20$ ($p=0.15$). For normalised host galaxy offsets, there is a hint of anti-correlation with $\mathrm{T}_{90}$.
\par Physical explanation for such correlation is not entirely known at present. \citet{Troja2008} suggested that the anti-correlation between $\mathrm{T}_{90}$ and host galaxy offset may be due to different progenitors of SGRBs, arguing that SGRBs without an EE occur via NS-NS mergers. These have larger offsets from their host galaxy centre, while SGRBs with an EE occur via NS-BH mergers. Opposite to that, \citet{Church2011} argued that both BH-NS and NS-NS mergers have similar offset distributions, and \citet{PernaBelczynski2002} argued that different groups of NS-NS mergers exist, with a significant fraction of them merging well within their host galaxies. \citet{Norris2010} proposed that an EE, which is observed in $\sim 25\%$ of SGRBs discovered by the \textit{Swift} satellite, is a part of the prompt emission, probably not directly caused by the properties of the surrounding environment. Their explanation for SGRBs' dichotomy concerning an EE might lie in the physical properties of a compact binary merger (e.g., mass, angular momentum, etc.), while the progenitor type is only one. Alternatively, the temporally long-lasting soft tail observed in SGRBs could be originated by the afterglow emission and related to the density of the circumburst medium \citep{Bernardini2007, Caito2009, Caito2010, deBarros2011}.
\begin{figure}
\begin{center}
\includegraphics[angle=-90,width=1\columnwidth]{T90_vs_offsets_newerr.eps}
\end{center}
\caption{Host galaxy offset (top part, squares) and normalised host galaxy offset (bottom part, circles) versus $\mathrm{T}_{90}$ for SGRBs from Sample \MakeUppercase{\romannumeral 1}. Filled symbols represent SGRBs without an EE, while empty symbols represent SGRBs with an EE.}
\label{fig:normhostvst90}
\end{figure}
\subsection{Short-lived/Long-lived X-ray Afterglow}
\label{sl_ll}
\par \citet{Sakamoto2009} found that there exist two distinct classes of SGRBs based on the duration of the X-ray afterglow, namely short-lived (SL), for which the X-ray afterglow flux at $10^4\,\mathrm{s}$ after the trigger is less than $10^{-13}\,\mathrm{erg\,cm^{-2}\,s^{-1}}$, and long-lived (LL), for which the X-ray afterglow flux is more than $10^{-13}\,\mathrm{erg\,cm^{-2}\,s^{-1}}$. Based on the sample therein, \citet{Sakamoto2009} concluded that SL SGRBs show no extended emission, no optical afterglow and a large host galaxy offset.
\par We checked if this classification holds for SGRBs in our samples. Among them, there is GRB~100117A (Sample \MakeUppercase{\romannumeral 1}), which based on the X-ray light curve and according to the definition of \citet{Sakamoto2009} is SL, but it has an OA detected and a very small host galaxy offset. Besides that, in Sample \MakeUppercase{\romannumeral 2} there are GRB~090515 (\citealt{Rowlinson2010}), which is SL and shows no EE, but has an OA detected, and GRB~080503 (\citealt{Perley2009}), which is SL, but shows an EE and has an OA detected. Another possible candidate could be GRB~080905A, which has an OA detected, but we can not confirm if it is SL due to the lack of the X-ray flux information around $10^4\,\mathrm{s}$ after the trigger, although its light curve looks very similar to the light curves of other SL SGRBs.
\par We therefore conclude that there are short-lived SGRBs (GRB~080503, GRB~090515 and GRB~100117A), which have an OA detected and some also have an EE. According to these findings, the X-ray afterglow duration does not seem to be an unique indicator of a specific progenitor and/or an environment for SGRBs.
\section{Conclusion}
\label{sect:conclusion}
\par We presented here a systematic study of the environment of SGRBs performed through the measure of their intrinsic X-ray absorbing column densities. Our results show that there are possibly two distinct populations of SGRBs: half of them (Sample \MakeUppercase{\romannumeral 1} with robust redshifts and Sample \MakeUppercase{\romannumeral 2} without robust redshifts) show relatively bright X-ray afterglows and occur in dense environments, with X-ray column densities comparable to the one measured for LGRBs in the same redshift range. Another half of them have very faint or no X-ray afterglow at all. The properties in the gamma regime of these two classes of SGRBs do not differ significantly, possibly suggesting that the observed difference in their X-ray afterglow brightness is not due to the burst energetics and might be a consequence of the environment where they explode.
\par We found no correlation between the intrinsic X-ray absorption and the host galaxy offset or normalised host galaxy offset. This is perhaps not expected for normalised host galaxy offsets, however the results could be affected by the large uncertainties on the values of $\mathrm{N_H}(z)$ or by the fact that we are dealing with the projected offsets, which do not give us absolute position of a GRB inside its host galaxy along the line of sight.
\par We checked if SGRBs with an extended emission lie closer to the centre of their host galaxies. We found that there is no significant anti-correlation between duration ($\mathrm{T}_{90}$) and host galaxy offset, but that there is a hint of anti-correlation when using normalised host galaxy offsets instead.
\par We also tested if short-lived SGRBs from our samples have no extended emission, no optical afterglow and large host galaxy offset, as proposed by \citet{Sakamoto2009}. We found that this does not hold for all cases, since GRB~080503, GRB~090515 and GRB~100117A are SL, but have an OA detected. GRB~080503 has also an EE, while GRB~100117A has a very small host galaxy offset.
\par Finally, we want to stress that the sample of SGRBs with a robust redshift determination and host galaxy offset measured is quite small up to this date (Sample \MakeUppercase{\romannumeral 1} is about nine times smaller than the whole sample of LGRBs with known redshift from \citet{Campana2010}, and almost twice as small as the subsample of LGRBs with $z<0.923$), mainly due to their dim optical afterglows. Thus, the results might be affected by the size of the sample. Future data will show if the conclusions, drawn in this paper, also hold on a larger sample.
\section*{Acknowledgments}
This work made use of data supplied by the UK Swift Science Data Centre at the University of Leicester. We thank J.F. Graham for providing Gemini-N images of GRB 070714B. DK is grateful to the staff at INAF-OAB in Merate for their hospitality during his four-month visit, and acknowledges funding from the Slovene human resources development and scholarship fund (grant 11012-10/2011). AG acknowledges funding from the Slovenian Research Agency and from the Centre of Excellence for Space Sciences and Technologies SPACE-SI, an operation partly financed by the European Union, European Regional Development Fund and Republic of Slovenia, Ministry of Higher Education, Science and Technology. PDA, AM, SCampana, SCovino, MGB, SDV, RS and GT acknowledge the Italian Space Agency for
financial support through the project ASI I/004/11/0
|
{
"timestamp": "2012-05-31T02:02:40",
"yymm": "1203",
"arxiv_id": "1203.1864",
"language": "en",
"url": "https://arxiv.org/abs/1203.1864"
}
|
\section{Introduction}
\label{sec:Intro}
The well known Lady\v zenskaja-Babu\v ska-Brezzi (LBB) condition is a particular instance of the
so-called
discrete inf--sup condition which is necessary and sufficient for the well-posedness of discrete
saddle point problems
arising from discretization via Galerkin methods. If ${\bf X}_h$ denotes the discrete velocity space and
$M_h$ the discrete
pressure space, then the LBB condition for the Stokes problem states that there is a constant
$c$ independent of the discretization parameter $h$ such that
\begin{equation}
c \| q_h \|_{L^2} \leq \sup_{v_h \in {\bf X}_h}
\frac{ \int_\Omega (\DIV v_h)\, q_h}{\|v_h\|_{{\bf H}^1}},
\quad \forall q_h \in M_h.
\tag{LBB}
\label{eq:LBB}
\end{equation}
The reader is referred to \cite{GR86} for the basic theory on saddle point problems on Banach spaces
and
their numerical analysis. Simply put, this condition sets a structural restriction on the discrete
spaces so that
the continuous level property that the divergence operator is closed and surjective, see
\cite{MR82m:26014,MR1880723}, is preserved uniformly with respect to the discretization parameter.
In the literature the following condition, which we shall denote the generalized LBB condition, is
also assumed
\begin{equation}
c \| \GRAD q_h \|_{{\bf L}^2} \leq
\sup_{v_h \in {\bf X}_h}
\frac{ \int_\Omega (\DIV v_h)\, q_h }{ \| v_h \|_{{\bf L}^2} },
\quad \forall q_h \in M_h,
\tag{GLBB}
\label{eq:wLBB}
\end{equation}
here and throughout we assume $M_h \subset H^1(\Omega)$. By properly defining a discrete gradient
operator, the case of discontinuous pressure spaces can be analyzed with similar arguments to those
that we shall present. Condition \eqref{eq:wLBB}, for example, was used by Guermond
(\cite{MR2210084,MR2334774}) to show that approximate solutions to the three-dimensional Navier
Stokes equations constructed using the Faedo-Galerkin method
converge to a suitable, in the sense of Scheffer, weak solution.
On the basis of \eqref{eq:wLBB}, the same author has also built (\cite{MR2520170}) an
${\bf H}^s$-approximation theory for the Stokes problem, $0\leq s \leq1$.
Olshanski{\u\i}, in \cite{MR2833487}, under the assumption that the spaces satisfy \eqref{eq:wLBB}
carries out a multigrid analysis for the Stokes problem. Finally, Mardal et al.\@\xspace,
\cite{schoberlwinther}, use a weighted inf--sup condition to analyze preconditioning techniques
for singularly perturbed Stokes problems (see Section \ref{sec:section5} below).
It is not difficult to show that, on quasi-uniform meshes, \eqref{eq:wLBB} implies \eqref{eq:LBB}, see
\cite{MR2210084}. We include the proof of this result below for completeness.
The question that naturally arises is whether the converse holds. Recall that a well-known result of
Fortin \cite{BF91}
shows that the inf--sup condition \eqref{eq:LBB} is equivalent to the existence of a so-called Fortin
projection that is stable in ${{ \bf H}^1_0 (\Omega)}$. In this work, under the assumption that the mesh is
shape regular and quasi-uniform, we will show that \eqref{eq:wLBB} is equivalent
to the existence of a Fortin projection that has ${\bf L}^2$-approximation properties. Moreover,
when the domain is such that the solution to the Stokes problem
possesses ${\bf H}^2$-regularity, we will prove that
\eqref{eq:wLBB} is in fact equivalent to \eqref{eq:LBB}, again on quasi-uniform meshes.
The work by Girault and Scott (\cite{MR1961943}) must be mentioned
when dealing with the construction of Fortin projection operators with ${\bf L}^2$-approximation properties.
They have constructed such operators for many commonly
used inf--sup stable spaces, one notable exception being the lowest order Taylor-Hood element
in three dimensions.
However, \eqref{eq:wLBB} has been shown to hold for the lowest order Taylor-Hood
element directly \cite{MR2210084}.
Our results then can be applied to show that, \eqref{eq:wLBB} is satisfied
by almost all inf--sup stable finite element spaces, regardless of the smoothness of the domain.
This work is organized as follows.
Section~\ref{sec:prel} introduces the notation and assumptions we shall work with. Condition
\eqref{eq:wLBB} is discussed in Section~\ref{sec:wLBB}. In Section~\ref{sec:Equiv} we actually show the
equivalence of conditions \eqref{eq:LBB} and \eqref{eq:wLBB}, provided the domain is smooth enough.
A weighted inf--sup condition related to uniform preconditioning of the time-dependent Stokes problem
is presented in Section~\ref{sec:section5}, where we show that \eqref{eq:wLBB} implies it.
Some concluding remarks are provided in Section~\ref{sec:conclusion}.
\section{Preliminaries}
\label{sec:prel}
Throughout this work, we will denote by $\Omega \subset \mathbb R^d$ with $d=2$ or $3$ an open bounded
domain with Lipschitz boundary. If additional smoothness of the domain is needed, it will be specified
explicitly.
${{ L}^2 (\Omega)}$, ${{ H}^{1}(\Omega)}$ and ${{ H}^1_0 (\Omega)}$ denote, respectively, the usual Lebesgue and Sobolev spaces.
We denote by ${L^2_{\scriptscriptstyle\!\int\!=0} (\Omega)}$ the set of functions in ${{ L}^2 (\Omega)}$ with mean zero.
Vector valued spaces will be denoted by bold characters.
We introduce a conforming triangulation ${\mathcal T}_h$ of $\Omega$ which we assume shape-regular and
quasi-uniform in the sense of \cite{BF91}. The size of the cells in the triangulation is characterized by
$h>0$. We introduce finite dimensional spaces ${\bf X}_h \subset {{ \bf H}^1_0 (\Omega)}$ and
$M_h \subset {L^2_{\scriptscriptstyle\!\int\!=0} (\Omega)} \cap {{ H}^{1}(\Omega)}$ which are
constructed, for instance using finite elements,
on the triangulation ${\mathcal T}_h$. For these spaces, the inverse inequalities
\begin{equation}
\| v_h \|_{{\bf H}^1} \leq c h^{-1} \| v_h \|_{{\bf L}^2}, \quad \forall v_h \in {\bf X}_h,
\label{eq:invX}
\end{equation}
and
\begin{equation}
\| q_h \|_{H^1} \leq c h^{-1} \| q_h \|_{L^2}, \quad \forall q_h \in M_h,
\label{eq:invM}
\end{equation}
hold, see \cite{BF91}. Here and in what follows we denote by $c$ will a constant that is independent of $h$.
We shall denote by ${\mathcal C}_h : {{ \bf H}^1_0 (\Omega)} \rightarrow {\bf X}_h$ the so-called
Scott-Zhang interpolation operator (\cite{SZ90}) onto the velocity space and we recall that
\begin{equation}
\| v - {\mathcal C}_h v \|_{{\bf L}^2} + h \|{\mathcal C}_h v\|_{{\bf H}^1} \leq c h \| v \|_{{\bf H}^1}, \quad \forall v
\in {{ \bf H}^1_0 (\Omega)}.
\label{eq:SZprop}
\end{equation}
and
\begin{equation}
\| v - {\mathcal C}_h v\|_{{\bf H}^1} \leq c h \| v \|_{{\bf H}^2}, \quad \forall v \in {{ \bf H}^1_0 (\Omega)} \cap {{ \bf H}^2 (\Omega)}
\label{eq:SZh1}
\end{equation}
The Scott-Zhang interpolation operator onto the pressure space
${\mathcal I}_h: {L^2_{\scriptscriptstyle\!\int\!=0} (\Omega)} \rightarrow M_h$ can be defined analogously
and satisfies similar stability and approximation properties.
We shall denote by $\pi_h: {{ \bf L}^2 (\Omega)} \rightarrow {\bf X}_h$ the ${\bf L}^2$-projection onto ${\bf X}_h$ and
by $\Pi_0 : {{ L}^2 (\Omega)} \rightarrow {{ L}^2 (\Omega)}$ the $L^2$-projection operator onto the space
of piecewise constant functions, i.e.,\@\xspace
\[
\Pi_0 q = \sum_{T \in {\mathcal T}_h} \frac1{|T|}\left(\int_T q\right)\chi_T, \quad \forall q \in {{ L}^2 (\Omega)}.
\]
For one result below we shall require full ${\bf H}^2$-regularity of the solution to the Stokes problem:
\begin{assumption}\label{assumption1}
The domain $\Omega$ is such that for any $f\in {{ \bf L}^2 (\Omega)}$, the solution
$(\psi,\theta) \in {{ \bf H}^1_0 (\Omega)} \times {L^2_{\scriptscriptstyle\!\int\!=0} (\Omega)}$ to the Stokes problem
\begin{equation}
\label{eq:contstokes}
\begin{dcases}
-\LAP \psi + \GRAD \theta = f, & \text{in } \Omega, \\
\DIV \psi = 0, & \text{in } \Omega, \\
\psi = 0, & \text{on } \partial\Omega,
\end{dcases}
\end{equation}
satisfies the following estimate:
\begin{equation}
\| \psi \|_{{\bf H}^2} + \| \theta \|_{H^1} \leq c \| f \|_{{\bf L}^2}.
\label{eq:Cattabriga}
\end{equation}
\end{assumption}
Assumption~\ref{assumption1} is known to hold in two and three dimensions ($d=2,3$) whenever
$\Omega$ is convex or of class ${\mathcal C}^{1,1}$, see \cite[Theorem 6.3]{MR977489}.
By suitably defining a discrete gradient operator acting on the pressure space, the proofs for
discontinuous pressure spaces can be carried out with similar arguments.
We introduce the definition of a Fortin projection.
\begin{definition}
\label{def:Fortin}
An operator ${\mathcal F}_h : {{ \bf H}^1_0 (\Omega)} \rightarrow {\bf X}_h$ is called a Fortin projection if ${\mathcal F}_h^2 =
{\mathcal F}_h$ and
\begin{equation}
\int_\Omega \DIV(v-{\mathcal F}_h v)q_h = 0, \quad \forall v \in {{ \bf H}^1_0 (\Omega)}, \quad \forall q_h \in M_h.
\label{eq:Fortin}
\end{equation}
\end{definition}
We shall be interested in Fortin projections ${\mathcal F}_h$ that satisfy the condition:
\begin{equation}
\| {\mathcal F}_h v \|_{{\bf H}^1} \leq c \| v \|_{{\bf H}^1}, \quad \forall v \in {{ \bf H}^1_0 (\Omega)},
\tag{FH1}
\label{eq:fh1}
\end{equation}
or
\begin{equation}
\| v - {\mathcal F}_h v \|_{{\bf L}^2} \leq c h \| v\|_{{\bf H}^1}, \quad \forall v \in {{ \bf H}^1_0 (\Omega)}.
\tag{FL2}
\label{eq:fl2}
\end{equation}
Let us remark that the approximation property \eqref{eq:fl2} implies ${\bf H}^1$-stability.
\begin{lem}
\label{lem:fl2implfh1}
If an operator ${\mathcal F}_h : {{ \bf H}^1_0 (\Omega)} \rightarrow {\bf X}_h$ satisfies \eqref{eq:fl2} then it is ${\bf H}^1$-stable, i.e., \eqref{eq:fh1} is satisfied.
\end{lem}
\begin{proof}
The proof relies on the stability and approximation properties \eqref{eq:SZprop} of the Scott-Zhang operator
and on the inverse estimate \eqref{eq:invX}, for if $v \in {{ \bf H}^1_0 (\Omega)}$,
\begin{align*}
\| {\mathcal F}_h v \|_{{\bf H}^1} &\leq \| {\mathcal F}_h v - {\mathcal C}_h v \|_{{\bf H}^1} + c \| v \|_{{\bf H}^1}
\leq c h^{-1} \| {\mathcal F}_h v - {\mathcal C}_h v \|_{{\bf L}^2} + c \| v \|_{{\bf H}^1} \\
&\leq ch^{-1} \| v - {\mathcal F}_h v \|_{{\bf L}^2} + ch^{-1}\| v - {\mathcal C}_h v \|_{{\bf L}^2} + c \|v\|_{{\bf H}^1}.
\end{align*}
Conclude using the ${\bf L}^2$-approximation properties of the operators ${\mathcal F}_h$ and ${\mathcal C}_h$.
\end{proof}
\begin{rem}
Girault and Scott, \cite{MR1961943}, explicitly constructed a Fortin projection
that satisfies \eqref{eq:fh1} and \eqref{eq:fl2} for many
commonly used spaces. In fact, they showed that the approximation is local, i.e.,\@\xspace
\[
\| {\mathcal F}_h v - v \|_{{\bf L}^2(T)}+ h_T\| {\mathcal F}_h v -v\|_{{\bf H}^1(T)} \leq c h_T \| v
\|_{{\bf H}^1({\mathcal N}(T))}, \quad \forall v \in {{ \bf H}^1_0 (\Omega)}
\text{ and } \forall T \in {\mathcal T}_h,
\]
where ${\mathcal N}(T)$ is a patch containing $T$.
In particular, they have shown the existence of this projection for the Taylor-Hood elements in two
dimensions. In three dimensions they proved this result for all the Taylor-Hood elements except the
lowest order case.
\end{rem}
In this work we shall prove the implications
\[
\xymatrix{
\eqref{eq:LBB} \ar@{<=>}[r] \ar@{<=}[d]
&\exists {\mathcal F}_h \mbox{\, s.t.\,} \eqref{eq:Fortin} \text{ and } \eqref{eq:fh1} & \\
\eqref{eq:wLBB} \ar@{<=>}[r]
&\exists {\mathcal F}_h \mbox{\, s.t.\,} \eqref{eq:Fortin} \text{ and } \eqref{eq:fl2} & \eqref{eq:LBB} \text{ and
Assumption } \ref{assumption1} \ar@{=>}[l]
}
\]
thus showing that, in our setting, all these conditions are indeed equivalent.
The top equivalence is
well-known, see \cite{BF91,GR86,MR2050138}. The left implication is also known (see
\cite{MR2210084}), for completeness we show
this in Theorem~\ref{thm:wlbbimpllbb}. The bottom implications, although simple to prove, seem to be
new.
\section{The Generalized LBB Condition}
\label{sec:wLBB}
Let us begin by noticing that the generalized LBB condition \eqref{eq:wLBB} is actually a statement
about coercivity of the ${\bf L}^2$-projection on gradients of functions in the pressure space. Namely,
\eqref{eq:wLBB} is equivalent to
\begin{equation}\label{GLBBb}
\| \pi_h \GRAD q_h \|_{{\bf L}^2} \geq c \| \GRAD q_h \|_{{\bf L}^2}, \quad \forall q_h \in M_h.
\end{equation}
It is well known that \eqref{eq:wLBB} implies \eqref{eq:LBB}. For completeness we present the proof.
We
begin with a perturbation result.
\begin{lem}
\label{lem:verfurth}
There exists a constant $c$ independent of $h$ such that, for all $q_h \in M_h$, the following holds:
\[
c \| q_h \|_{L^2} \leq
\sup_{v_h \in {\bf X}_h} \frac{ \int_\Omega (\DIV v_h)\, q_h }{\| \GRAD v_h \|_{{\bf L}^2} }
+ h \| \GRAD q_h \|_{{\bf L}^2}.
\]
\end{lem}
\begin{proof}
The proof relies on the properties \eqref{eq:SZprop} of the Scott-Zhang interpolation operator ${\mathcal C}_h$,
\begin{align*}
c \|q_h\|_{L^2} &\leq \sup_{v \in {{ \bf H}^1_0 (\Omega)}} \frac{ \int_\Omega (\DIV v)\, \, q_h }{\| \GRAD v
\|_{{\bf L}^2} }
\leq
\sup_{ v \in {{ \bf H}^1_0 (\Omega)}} \frac{ \int_\Omega (\DIV\,{\mathcal C}_h v)\, q_h }{\| \GRAD ({\mathcal C}_h v)
\|_{{\bf L}^2} }
+
\sup_{ v \in {{ \bf H}^1_0 (\Omega)}}\frac{\int_\Omega\big(\DIV\left(v - {\mathcal C}_h v \right)\big)q_h }{\|\GRAD
v\|_{{\bf L}^2} }
\\ &\leq
\sup_{ v_h \in {\bf X}_h} \frac{ \int_\Omega (\DIV v_h) \, q_h }{\| \GRAD v_h \|_{{\bf L}^2} }
+ \sup_{ v \in {{ \bf H}^1_0 (\Omega)}} \frac{ \int_\Omega\left(v - {\mathcal C}_h v \right){\cdot}\GRAD q_h}{\|\GRAD
v\|_{{\bf L}^2}},
\end{align*}
conclude using \eqref{eq:SZprop}.
\end{proof}
On the basis of Lemma~\ref{lem:verfurth} we can readily show that \eqref{eq:wLBB} implies \eqref{eq:LBB}. Again,
this result is not new and we only include the proof for completeness.
\begin{thm}
\label{thm:wlbbimpllbb}
\eqref{eq:wLBB} implies \eqref{eq:LBB}.
\end{thm}
\begin{proof}
Since we assumed that $M_h \subset {L^2_{\scriptscriptstyle\!\int\!=0} (\Omega)} \cap {{ H}^{1}(\Omega)}$, the proof is straightforward:
\[
\sup_{ v_h \in {\bf X}_h} \frac{ \int_\Omega (\DIV v_h)\, q_h }{\|\GRAD v_h \|_{{\bf L}^2} }
=
\sup_{ v_h \in {\bf X}_h} \frac{ \int_\Omega v_h {\cdot} \GRAD q_h }{\|\GRAD v_h \|_{{\bf L}^2} }
\geq
\frac{ \int_\Omega \pi_h \GRAD q_h {\cdot}\GRAD q_h }{\|\GRAD \pi_h \GRAD q_h \|_{{\bf L}^2} }
=
\frac{ \|\pi_h \GRAD q_h \|_{{\bf L}^2}^2 }{ \| \GRAD \pi_h \GRAD q_h \|_{{\bf L}^2} }
\geq
c h \|\pi_h \GRAD q_h \|_{{\bf L}^2}
\]
where, in the last step, we used the inverse inequality \eqref{eq:invX}.
This, in conjunction with Lemma~\ref{lem:verfurth} and the characterization \eqref{GLBBb}, implies the result.
\end{proof}
Let us now show that the generalized LBB condition
\eqref{eq:wLBB} is equivalent to the existence of a Fortin operator satisfying \eqref{eq:fl2}.
We begin with a modification of a classical result.
\begin{lem}
\label{lem:tartarwithh}
For all $p\in {{ H}^{1}(\Omega)}$ there is $v\in{{ \bf H}^1_0 (\Omega)}$ such that
\[
\DIV v = p - \Pi_0 p, \qquad v|_{\partial T}=0 \quad \forall T\in {\mathcal T}_h,
\]
and
\[
\| v \|_{{\bf L}^2} \leq c \left( \sum_{T \in {\mathcal T}_h} h_T^4 \| \GRAD p \|_{{\bf L}^2(T)}^2 \right)^{1/2}.
\]
\end{lem}
\begin{proof}
Let $p\in {{ H}^{1}(\Omega)}$ and $T \in {\mathcal T}_h$. Clearly,
\[
\int_T p - \Pi_0 p = 0.
\]
A classical result (\cite{MR82m:26014,MR1846644,GR86,MR1880723}) implies that there is a
$v_T \in {\bf H}^1_0(T)$ with
$ \DIV v_T = p - \Pi_0 p$
in $T$ and
\begin{equation}
\| \GRAD v_T \|_{{\bf L}^2(T)} \leq c \| p - \Pi_0 p \|_{L^2(T)}.
\label{eq:saves}
\end{equation}
Given that the mesh is assumed to be shape regular, by mapping to the reference element it is seen
that
the constant in the last inequality does not depend on $T \in {\mathcal T}_h$.
Let $v \in {{ \bf H}^1_0 (\Omega)}$ be defined as $ v|_T = v_T$ for all $T$ in ${\mathcal T}_h$. By construction,
\[
\DIV v = p - \Pi_0 p, \quad \text{\ae in } \Omega.
\]
Moreover,
\[
\| v \|_{{\bf L}^2}^2 = \sum_{T \in {\mathcal T}_h} \| v \|_{{\bf L}^2(T)}^2
\leq c \sum_{T \in {\mathcal T}_h} h_T^2 \| \GRAD v \|_{{\bf L}^2(T)}^2
\leq c \sum_{T \in {\mathcal T}_h} h_T^2 \| p - \Pi_0 p \|_{L^2(T)}^2
\leq c \sum_{T \in {\mathcal T}_h} h_T^4 \| \GRAD p \|_{{\bf L}^2(T)}^2.
\]
The first equality is by definition; then we applied the Poincar\'e-Friedrichs inequality (since
$v|_T = v_T \in {\bf H}^1_0(T)$); next we used the properties of the function $v_T$ and the
approximation properties of the projector $\Pi_0$.
\end{proof}
With this result at hand we can prove the following.
\begin{thm}
\label{thm:fl2implwlbb}
If there exists a Fortin operator ${\mathcal F}_h$ that satisfies \eqref{eq:fl2}, then
\eqref{eq:wLBB} holds.
\end{thm}
\begin{proof}
Let $q_h \in M_h$. Using the properties of the operator $\Pi_0$ and
the local analogue of
the inverse inequality \eqref{eq:invM}, we get
\[
\| \GRAD q_h \|_{{\bf L}^2}^2
= \sum_{T \in {\mathcal T}_h} \left\| \GRAD \left(q_h - \Pi_0 q_h \right) \right\|_{{\bf L}^2(T)}^2
\leq \sum_{T \in {\mathcal T}_h} \frac1{h_T^2} \| q_h - \Pi_0 q_h \|_{{\bf L}^2(T)}^2
\leq \frac{c}{h^2} \| q_h - \Pi_0 q_h \|_{{\bf L}^2}^2.
\]
From Lemma~\ref{lem:tartarwithh} we know there exists $v \in {{ \bf H}^1_0 (\Omega)}$ with $\DIV v = q_h - \Pi_0 q_h$
and
\[
\| v \|_{{\bf L}^2} \leq c h^2 \| \GRAD q_h \|_{{\bf L}^2},
\]
hence
\[
\| \GRAD q_h \|_{{\bf L}^2}^2 \leq \frac{c}{h^2} \| q_h - \Pi_0 q_h \|_{L^2}^2
= \frac{c}{h^2} \int_\Omega (\DIV v) \, (q_h-\Pi_0 q_h)
= \frac{c}{h^2} \int_\Omega (\DIV v) \, q_h,
\]
where the last inequality follows from integration by parts over each $T$ and using the fact that $v|_{\partial T} = 0$ (see Lemma~\ref{lem:tartarwithh}).
Using the existence of the operator ${\mathcal F}_h$,
\[
\| \GRAD q_h \|_{{\bf L}^2}^2 \le \frac{c}{h^2} \int_\Omega (\DIV \,{\mathcal F}_h v) q_h
\leq \left( \sup_{ w_h \in {\bf X}_h} \frac{ \int_\Omega (\DIV w_h) \, q_h }{ \| w_h \|_{{\bf L}^2}
}\right)
\frac{c}{h^2}\| {\mathcal F}_h v \|_{{\bf L}^2}.
\]
It remains to show that
\[
\| {\mathcal F}_h v \|_{{\bf L}^2} \leq c h^2 \| \GRAD q_h \|_{{\bf L}^2}.
\]
For this purpose, we use the approximation property \eqref{eq:fl2} and
Lemma~\ref{lem:tartarwithh}
\[
\| {\mathcal F}_h v \|_{{\bf L}^2} \leq \| {\mathcal F}_h v - v\|_{{\bf L}^2} + \|v \|_{{\bf L}^2}
\leq ch \| \GRAD v \|_{{\bf L}^2} + c h^2 \|\GRAD q_h \|_{{\bf L}^2}
\leq ch^2 \| \GRAD q_h \|_{{\bf L}^2},
\]
where the last inequality holds because of \eqref{eq:saves}.
\end{proof}
The converse of Theorem~\ref{thm:fl2implwlbb} is given in the following.
\begin{thm}
\label{thm:wlbbimplfl2}
If \eqref{eq:wLBB} holds, then there exists a Fortin projector ${\mathcal F}_h$ that satisfies
\eqref{eq:fl2}.
\end{thm}
\begin{proof}
Let $v\in{{ \bf H}^1_0 (\Omega)}$. Define $(z_h, p_h) \in {\bf X}_h \times M_h$ as the solution of
\begin{equation}
\begin{dcases}
\int_\Omega z_h{\cdot} w_h -\int_\Omega p_h \DIV w_h = \int_\Omega v{\cdot} w_h, & \forall w_h \in {\bf X}_h, \\
\int_\Omega q_h \DIV z_h = \int_\Omega q_h \DIV v, & \forall q_h \in M_h.
\end{dcases}
\label{eq:l2stokes}
\end{equation}
Notice that \eqref{eq:wLBB} provides precisely necessary and sufficient conditions for this problem
to have a
unique solution.
Define ${\mathcal F}_h v := z_h$ we claim that this is indeed a Fortin projection that satisfies
\eqref{eq:fl2}.
By construction, \eqref{eq:Fortin} holds (see the second equation in \eqref{eq:l2stokes}). To show
that this is indeed a
projection, assume that $v=v_h \in {\bf X}_h$ in \eqref{eq:l2stokes}, setting $w_h = z_h - v_h$ we
readily obtain that
\[
\| z_h - v_h \|_{{\bf L}^2}^2 =0.
\]
It remains to show the approximation properties of this operator. We begin by noticing that
\eqref{eq:wLBB} implies
\begin{equation}
c \| \GRAD p_h \|_{{\bf L}^2} \leq \sup_{w_h \in {\bf X}_h} \frac{ \int_\Omega p_h \DIV w_h }{ \| w_h
\|_{{\bf L}^2}}
\leq \sup_{w_h \in {\bf X}_h} \frac{ \int_\Omega (v-{\mathcal F}_h v){\cdot} w_h }{ \| w_h \|_{{\bf L}^2}}
\leq \| v - {\mathcal F}_h v \|_{{\bf L}^2},
\label{eq:boundgradp}
\end{equation}
where we used \eqref{eq:l2stokes}. To obtain the approximation property \eqref{eq:fl2} we use the
Scott-Zhang interpolation operator ${\mathcal C}_h$,
\begin{align*}
\| {\mathcal F}_h v - v \|_{{\bf L}^2}^2 &= \int_\Omega ( {\mathcal C}_h v - v ){\cdot}( {\mathcal F}_h v - v )
+ \int_\Omega ( {\mathcal F}_h v - {\mathcal C}_h v ){\cdot}( {\mathcal F}_h v - v ) \\
& \leq
\| {\mathcal C}_h v - v \|_{{\bf L}^2}\| {\mathcal F}_h v - v \|_{{\bf L}^2}
+ \int_\Omega ( {\mathcal F}_h v - {\mathcal C}_h v ){\cdot}( {\mathcal F}_h v - v ).
\end{align*}
We bound the first term using the approximation property \eqref{eq:SZprop} of ${\mathcal C}_h$. To bound
the second
term we use problem \eqref{eq:l2stokes} with $w_h = {\mathcal F}_h v - {\mathcal C}_h v$, then
\[
\int_\Omega ( {\mathcal F}_h v - {\mathcal C}_h v ){\cdot}( {\mathcal F}_h v - v )
= \int_\Omega p_h \DIV( {\mathcal F}_h v - {\mathcal C}_h v )
= \int_\Omega p_h \DIV( v - {\mathcal C}_h v )
= -\int_\Omega \GRAD p_h {\cdot} ( v - {\mathcal C}_h v ),
\]
we conclude applying the Cauchy-Schwarz inequality and using \eqref{eq:boundgradp}.
\end{proof}
\section{Smooth Domains}
\label{sec:Equiv}
Here we show that, provided \eqref{eq:LBB} holds and, moreover,
the domain $\Omega$ is such that Assumption~\ref{assumption1} is satisfied, then \eqref{eq:fl2}
holds and hence \eqref{eq:wLBB} holds as well. This is shown in the following.
\begin{thm}
\label{thm:lbbimplwlbb}
Assume the domain $\Omega$ is such that the solution to
\eqref{eq:contstokes} possesses ${\bf H}^2$-elliptic regularity, i.e.,\@\xspace Assumption \ref{assumption1}
holds. Then \eqref{eq:LBB} implies that there is a Fortin operator
${\mathcal F}_h$ that satisfies \eqref{eq:fl2}.
\end{thm}
\begin{proof}
Let $v\in {{ \bf H}^1_0 (\Omega)}$. Define $(z_h,p_h) \in {\bf X}_h \times M_h$ as the solution to the discrete Stokes
problem
\begin{equation}
\begin{dcases}
\int_\Omega \GRAD z_h {:} \GRAD w_h -\int_\Omega p_h \DIV w_h = \int_\Omega \GRAD v {:} \GRAD w_h, & \forall w_h
\in {\bf X}_h, \\
\int_\Omega q_h \DIV z_h = \int_\Omega q_h \DIV v, & \forall q_h \in M_h,
\end{dcases}
\label{eq:stokes}
\end{equation}
where, in \eqref{eq:stokes}, the colon is used to denote the tensor product of matrices. Notice that \eqref{eq:LBB} implies that
this problem always has a unique solution.
Set ${\mathcal F}_h v := z_h$. Proceeding as in the proof of Theorem~\ref{thm:wlbbimplfl2} we see that this is
indeed a projection. Moreover, \eqref{eq:Fortin} holds by construction.
It remains to
show that \eqref{eq:fl2} is satisfied. To this end, analogously to the proof of
Theorem~\ref{thm:wlbbimplfl2},
we notice that \eqref{eq:LBB} implies
\[
\| p_h \|_{L^2} \leq c \| \GRAD ({\mathcal F}_h v - v )\|_{{\bf L}^2}.
\]
We now argue by duality. Let $\psi$ and $\phi$ solve
\eqref{eq:contstokes} with $f={\mathcal F}_h v- v$. Assumption \eqref{eq:Cattabriga} then implies
\begin{align*}
\| {\mathcal F}_h v - v \|_{{\bf L}^2}^2 &= \int_\Omega ({\mathcal F}_h v - v){\cdot}(-\LAP\psi + \GRAD \theta) \\
&= \int_\Omega \GRAD({\mathcal F}_h v -v ):\GRAD(\psi-{\mathcal C}_h \psi) -\int_\Omega(\theta - {\mathcal I}_h
\theta)\,\DIV({\mathcal F}_h v -v )\, \\
& \phantom{=}+ \int_\Omega \GRAD({\mathcal F}_h v -v ):\GRAD({\mathcal C}_h \psi) - \int_\Omega ({\mathcal I}_h \theta)\,\DIV({\mathcal F}_h v -v )\,
\end{align*}
Notice that since ${\mathcal I}_h \theta\in M_h$, $\int_\Omega ({\mathcal I}_h \theta)\,\DIV({\mathcal F}_h v -v )=0$.
Since $\DIV \psi = 0$,
using \eqref{eq:stokes}, the estimate for $p_h$, \eqref{eq:SZh1} and \eqref{eq:Cattabriga},
\[
\int_\Omega \GRAD({\mathcal F}_h v -v ):\GRAD({\mathcal C}_h \psi) = \int_\Omega p_h \DIV ({\mathcal C}_h \psi-\psi)
\leq ch \| v - {\mathcal F}_h v \|_{{\bf H}^1} \| v - {\mathcal F}_h v \|_{{\bf L}^2}.
\]
A direct application of of \eqref{eq:SZh1}, \eqref{eq:SZprop} and \eqref{eq:Cattabriga} allows us to
obtain the following estimates:
\[
\int_\Omega (\theta - {\mathcal I}_h \theta)\,\DIV({\mathcal F}_h v -v )
+ \int_\Omega \GRAD({\mathcal F}_h v -v ){:}\GRAD(\psi-{\mathcal C}_h \psi)
\leq
c h \| {\mathcal F}_h v - v \|_{{\bf L}^2}\| v \|_{{\bf H}^1}
\]
We conclude using a stability estimate for \eqref{eq:stokes}
\[
\| {\mathcal F}_h v - v \|_{{\bf L}^2} \leq c h \| {\mathcal F}_h v - v \|_{{\bf H}^1} \leq c h \| v \|_{{\bf H}^1},
\]
which, given \eqref{eq:LBB}, is uniform in $h$.
\end{proof}
\section{The Weighted LBB condition}
\label{sec:section5}
In relation to the construction of uniform preconditioners for
discretizations of the time dependent Stokes problem, Mardal, Sch\"oberl and Winther, \cite{schoberlwinther},
consider the following inf--sup condition,
\begin{equation}
c\| q_h \|_{H^1+\epsilon^{-1} L^2} \leq
\sup_{v_h \in {\bf X}_h} \frac{ \int_\Omega \DIV v_h q_h}{\|v_h\|_{{\bf L}^2 \cap \epsilon {\bf H}^1}},
\quad \forall q_h \in M_h.
\label{eq:winf-sup}
\end{equation}
where
\[
\| q \|_{H^1+\epsilon^{-1} L^2}^2= \inf_{q_1+q_2=q}
\left( \|q_1\|_{H^1}^2+ \epsilon^{-2} \|q_2\|_{L^2}^2 \right),
\]
and
\[
\|v\|_{{\bf L}^2 \cap \epsilon {\bf H}^1}^2 = \|v\|_{{\bf L}^2}^2 + \epsilon^2 \|v\|_{{\bf H}^1}^2.
\]
By constructing a Fortin projection operator that is ${\bf L}^2$-bounded they have showed, on quasi-uniform meshes,
that the inf--sup condition \eqref{eq:winf-sup} holds for the lowest order Taylor-Hood element in two dimension.
In addition, they proved the same result, on shape regular meshes, for the mini-element.
Here, we show that \eqref{eq:winf-sup} holds if we assume \eqref{eq:wLBB}. A simple consequence of this
result is that, on quasi-uniform meshes, \eqref{eq:winf-sup} holds for any order Taylor-Hood elements
in two and three dimensions.
\begin{thm}
\label{thm:wlbbimplweightlbb}
Let $\Omega$ be star shaped with respect to ball.
If the spaces ${\bf X}_h$ and $M_h$ are such that \eqref{eq:wLBB} is satisfied,
then the inf--sup condition \eqref{eq:winf-sup} holds with a constant that
does not depend on $\epsilon$ or $h$.
\end{thm}
\begin{proof}
We consider two cases: $\epsilon \ge h$ and $\epsilon < h$.
Given that the domain $\Omega$ is star shaped with respect to a ball,
we can conclude (\cite{schoberlwinther}) that the following
continuous inf--sup condition holds,
\begin{equation}\label{conweight}
c \| q \|_{H^1+\epsilon^{-1} L^2} \leq \sup_{v \in {{ \bf H}^1_0 (\Omega)}}
\frac{ \int_\Omega q\, \DIV v }{ \|v\|_{{\bf L}^2 \cap \epsilon {\bf H}^1}},
\quad \forall q \in {L^2_{\scriptscriptstyle\!\int\!=0} (\Omega)},
\end{equation}
with a constant $c$ independent of $\epsilon$.
We first assume that $\epsilon \ge h$. Using \eqref{conweight} for $q_h \in M_h$ we have,
\begin{align*}
c \| q_h \|_{H^1+\epsilon^{-1} L^2} &\leq
\sup_{v \in {{ \bf H}^1_0 (\Omega)} } \frac{ \int q_h\,\DIV v }{ \| v \|_{{\bf L}^2 \cap \epsilon {\bf H}^1 } } =
\sup_{v \in {{ \bf H}^1_0 (\Omega)} } \frac{ \int_\Omega q_h\,\DIV ({\mathcal F}_h v) }{\|{\mathcal F}_h v\|_{{\bf L}^2 \cap \epsilon {\bf H}^1}}
\frac{\|{\mathcal F}_h v\|_{{\bf L}^2 \cap \epsilon {\bf H}^1}}{\|v\|_{{\bf L}^2\cap \epsilon {\bf H}^1} } \\
&\leq \sup_{v_h \in {\bf X}_h} \frac{ \int_\Omega q_h\,\DIV v_h }
{\|v_h\|_{{\bf L}^2 \cap \epsilon {\bf H}^1} }
\sup_{ v \in {{ \bf H}^1_0 (\Omega)}} \frac{\|{\mathcal F}_h v\|_{{\bf L}^2 \cap \epsilon {\bf H}^1}}{\|v\|_{{\bf L}^2 \cap \epsilon {\bf H}^1}},
\end{align*}
where we used that, since \eqref{eq:wLBB} holds, Theorem~\ref{thm:wlbbimplfl2} shows that
there exists a Fortin operator ${\mathcal F}_h$ that satisfies
\eqref{eq:Fortin}. By Lemma~\ref{lem:fl2implfh1} and
the approximation properties \eqref{eq:fl2} of the Fortin operator,
\begin{align*}
\|{\mathcal F}_h v \|_{{\bf L}^2 \cap \epsilon {\bf H}^1} &\leq
c \left( \|{\mathcal F}_h v \|_{{\bf L}^2} + \epsilon \| {\mathcal F}_h v \|_{{\bf H}^1} \right) \leq
c \left( \| v \|_{{\bf L}^2} + \|v - {\mathcal F}_h v \|_{{\bf L}^2} + \epsilon \| v \|_{{\bf H}^1} \right) \\
&\leq c \left( \|v\|_{{\bf L}^2} + (\epsilon + h)\| v \|_{{\bf H}^1} \right)
\leq c \left( \|v\|_{{\bf L}^2} + 2\epsilon\| v \|_{{\bf H}^1} \right)
\leq c\| v \|_{{\bf L}^2 \cap \epsilon {\bf H}^1},
\end{align*}
where we used that $h \leq \epsilon$.
On the other hand, if $\epsilon < h$ we use $q_1 = q_h$ and $q_2 = 0$ in the definition of the weighted norm for the pressure space.
Condition \eqref{eq:wLBB} then implies
\[
\| q_h \|_{H^1+\epsilon^{-1} L^2} \leq
c \| \GRAD q_h \|_{{\bf L}^2} \leq c \sup_{v_h \in {\bf X}_h} \frac{ \int_\Omega q_h\,\DIV v_h }{ \| v_h \|_{{\bf L}^2} }
\leq c \sup_{v_h \in {\bf X}_h} \frac{ \int_\Omega q_h\, \DIV v_h }{ \| v_h \|_{{\bf L}^2 \cap \epsilon {\bf H}^1} }
\sup_{v_h \in {\bf X}_h} \frac{ \| v_h \|_{{\bf L}^2 \cap \epsilon {\bf H}^1} }{ \| v_h \|_{{\bf L}^2} }.
\]
By the inverse inequality \eqref{eq:invX},
\[
\frac{ \| v_h \|_{{\bf L}^2 \cap \epsilon {\bf H}^1} }{ \| v_h \|_{{\bf L}^2} } \leq
c\left( 1 + \epsilon h^{-1} \right).
\]
Conclude using that $\epsilon < h $.
\end{proof}
\section{Concluding Remarks}
\label{sec:conclusion}
There seems to be one main drawback to our methods of proof. Namely, all our results rely heavily on
the fact that we have a quasi-uniform mesh. However, at the present moment we do not know whether this
condition can be removed. Finally, it will be interesting to see if \eqref{eq:LBB} is in fact equivalent to
\eqref{eq:wLBB} on domains that do not satisfy the regularity assumption
\eqref{eq:Cattabriga} (e.g.\@\xspace non convex polyhedral domains).
On the other hand, it seems to us that condition \eqref{eq:wLBB} must be regarded as the most important one.
Our results show that, under the sole assumption that the mesh is quasi-uniform, this condition implies
the classical condition \eqref{eq:LBB} (Theorem~\ref{thm:wlbbimpllbb}). Moreover, as shown in
Theorem~\ref{thm:wlbbimplweightlbb}, this condition implies the weighted inf--sup condition \eqref{eq:winf-sup} on quasi-uniform meshes.
\bibliographystyle{plain}
|
{
"timestamp": "2012-03-09T02:04:12",
"yymm": "1203",
"arxiv_id": "1203.1870",
"language": "en",
"url": "https://arxiv.org/abs/1203.1870"
}
|
\section{Introduction}
The Standard Model (SM) is a successful theoretical framework for describing elementary particle interactions when confronted with experimental data.
However, recent observations~\cite{CDFnote:10584, Abazov:2011rq} by the CDF and D\O\, collaborations point to an anomalously large forward-backward asymmetry in $t\bar{t}$ production $\left(A_{FB}^{t\bar{t}}\right)$, significantly exceeding the SM prediction (see Ref.~\cite{Kamenik:2011wt} for a review).
Perhaps the most intriguing result is CDF's report~\cite{Aaltonen:2011kc} of a rise in $A_{FB}^{t\bar{t}}$ with the invariant mass of the $t\bar{t}$ pair $\left(M_{t\bar{t}}\right)$,
\begin{eqnarray}
A^{t\bar{t}}_\text{{low}} \equiv A_{FB}^{t\bar{t}}(M_{t\bar{t}} \le 450\, \text{GeV}) &= &(-11.6\pm15.3)\%~, \nonumber \\ A^{t\bar{t}}_{\text{high}} \equiv A_{FB}^{t\bar{t}}(M_{t\bar{t}} > 450 \,\text{GeV}) &= &(47.5\pm11.4)\% ~.\nonumber
\end{eqnarray}
In particular, $A^{t\bar{t}}_{\text{high}}$ is almost 3$\sigma$ away from SM prediction (all the relevant measurements, as well as the corresponding SM predictions are collected in Table \ref{tab:sum}).
This discrepancy invites a new physics (NP) explanation, and many models have been proposed to address the anomalously large $A_{FB}^{t\bar{t}}$. These models generally involve introducing new scalar ~\cite{Shu:2009xf,Dorsner:2009mq,Dorsner:2011ai,Nelson:2011us,Cheung:2009ch,Patel:2011eh,Blum:2011fa,Stone:2011dn,delaPuente:2011iu} or vector ~\cite{Barger:2010mw,Shelton:2011hq,Grinstein:2011yv,Grinstein:2011dz,Ligeti:2011vt,Tavares:2011zg,Bhattacherjee:2011nr} particles contributing to the $t\bar{t}$ production cross section in the $s$- and/or $t$-channel. While most of these models can easily raise the theoretical prediction for $A_{FB}^{t\bar{t}}$ to within 1$\sigma$ of the CDF measurement, it has proven to be extremely hard to address the central value of $A^{t\bar{t}}_{\text{high}}$ while being consistent with existing experimental constraints.
In this work we propose to explore another class of models involving new tensor (spin-2) particles around the weak scale with flavor-violating couplings to quarks. A simple effective field-theoretic (EFT) analysis reveals that the most general lowest-order couplings of a spin-2 state with quark bilinears are rather similar to the general-relativistic couplings of the graviton to energy/momentum. This leads to a strong energy dependence of the effects of virtual exchange of such states, which nicely agrees with the CDF observations. In particular, we show that this framework can accommodate all of the CDF measurements over a wide range of parameter space while being consistent with existing experimental bounds.
We treat the massive spin-2 particle as a low-energy signature of some unspecified UV physics. Among other possibilities, a low-energy effective theory of this type could arise from a theory of modified gravity~\cite{ArkaniHamed:1998rs, Antoniadis:1998ig,Randall:1999ee, Randall:1999vf,deRham:2010gu, deRham:2010kj} or could describe a spin-2 resonance of a strongly interacting sector not far above the weak scale~\cite{Morningstar:1997ff, Morningstar:1999rf,Chen:2005mg}.
The rest of the paper is organized as follows: in Sec.~\ref{sec:spin2} we describe the relevant couplings of the spin-2 state to quarks. The existing experimental constraints are examined in Sec.~\ref{sec:constraints}. We study parameter space for $t\bar{t}$ phenomenology and discuss our results in Sec.~\ref{sec:ttbar}. Finally, we conclude in Sec.~\ref{sec:con}.
\section{Low-Energy Effective Theory}
\label{sec:spin2}
Studies of low-energy effective field theories for massive spin-2 particles are motivated in various contexts. On one hand, such theories can be useful for describing spin-2 QCD resonances, such as glueballs, which become long-lived in the limit of large number of colors~\cite{Chen:2005mg}. On the other hand, massive spin-2 states frequently occur in the context of modified gravity. An incomplete list of examples from the latter category includes models with KK tower of gravitons, such as theories with large \cite{ArkaniHamed:1998rs, Antoniadis:1998ig} or warped \cite{Randall:1999ee, Randall:1999vf} extra dimensions, as well as the recently discovered class of purely four-dimensional, ghost-free models of massive gravity \cite{deRham:2010gu, deRham:2010kj}. Ratios of branching ratios to photons and to jets may be used to distinguish between the various possible underlying UV theories~\cite{Fok:2012zk}.
Regardless of the details of the UV theory, any consistent action for a complex, symmetric spin-2 field $h_{\mu\nu}$ with mass $M$ should reduce to the Fierz-Pauli \cite{Fierz:1939ix} form at the linearized level
\begin{equation}
\label{eq:lag}
\mathcal{L}_{FP} = -\frac{1}{2}h^{\dagger}_{\mu\nu}\left(\Box + M^2\right)h^{\mu\nu} + \frac{1}{2}\left.h^{\mu}_{\mu}\right.^{\dagger}\left(\Box + M^2\right)h^{\nu}_{\nu} - h^{\dagger}_{\mu\nu}\partial^{\mu}\partial^{\nu}h^{\rho}_{\rho} + h^{\dagger}_{\mu\nu}\partial^{\mu}\partial^{\rho}h_{\rho}^{\nu} + \text{h.c.}.
\end{equation}
Furthermore, if the field is of gravitational origin, its couplings to matter are usually constrained by the Equivalence Principle to be universal. However, RS-type models with the SM fields localized differently along the bulk are an important exception to this rule.
In the present work, we will be mostly interested in the implications of a massive, complex, spin-2 boson for the top quark forward-backward asymmetry. Resorting to an EFT approach, we will not make any assumptions about the precise origin of the spin-2 field. Among other possibilities, $h_{\mu\nu}$ could describe a bound state of some strongly coupled sector not far above the electroweak scale, or a non-universally interacting gravitational KK mode in some RS-like theory with complicated localization of matter. We will not attempt to construct any explicit model along these lines.
At low energies, the sector of the theory describing interactions of $h_{\mu\nu}$ with the quarks consists of operators of various dimension, suppressed by powers of some high scale, denoted by $f$ below. At the zero derivative order, there is a single coupling of a spin-2 field with a general quark bilinear,
\begin{equation}
\mathcal{L}_4 \supset \lambda_{ij} h^{\mu}_{\mu}\bar q_iq_j+\text{h.c.},
\label{0d}
\end{equation}
where $\lambda$ is a coupling constant, $\{i,j\}$ refer to quark flavor and the possible chirality of $q_i$ has been suppressed for simplicity; the quark fields correspond to the mass eigenstates after electroweak symmetry breaking. This interaction is similar in form to an ordinary Yukawa interaction of a SM singlet scalar; the only difference is in the spin-2 nature of the correlator
\begin{equation}
\langle \left.h^{\mu}_{\mu}\right.^{\dagger}\hspace{-1mm}(k) h^\nu_\nu(-k) \rangle\propto \frac{2-k^2/M^2-(k^2)^2/M^4}{k^2-M^2}.
\end{equation}
For the purposes of studying the $t\bar t$ forward-backward asymmetry however, this operator can be expected to lead to effects similar to those of a color-singlet scalar exchange. The consequences of the latter for the $t\bar{t}$ forward-backward asymmetry have been studied extensively, see for example~\cite{Shu:2009xf}, with the conclusion that it can not generate a large enough asymmetry. Moreover, such inter-generation couplings involving the top quark are constrained not to be too large from the same sign top production cross section at the LHC (see Section~\ref{sec:constraints}); we will thus ignore these operators below\footnote{If $h_{\mu\nu}$ is of gravitational/extra dimensional origin, $f$ represents the quantum gravity scale. Then the coupling of $h_{\mu\nu}$ to energy-momentum tensor leads to a natural suppression of the Yukawa coupling constant. In particular, $\lambda\sim m/f$, where $m$ is of the order of the mass of a heavier fermion present in the interaction.}.
At the one derivative order, the most general couplings of $h_{\mu\nu}$ to a fermion bilinear are given by the following expressions,
\begin{align}
\mathcal{L}_5 &\supset -\frac{1}{f} h_{\mu\nu}\left( S^{\mu\nu} + \eta^{\mu\nu} T\right) + \text{h.c.}, \\
\label{eq:1d}
S_{\mu\nu} &= \frac{i}{4}a^L_{ij}\bar{q}_{Li}\left(\gamma_{\mu}\partial_{\nu} + \gamma_{\nu}\partial_{\mu}\right)q_{Lj} + \frac{i}{4}b^L_{ij}\left(\partial_{\mu}\bar{q}_{Li}\gamma_{\nu}+ \partial_{\nu}\bar{q}_{Li}\gamma_{\mu}\right)q_{Lj} + \left(\text{L} \leftrightarrow \text{R}\right), \\
T &=\bar \lambda^{L}_{i j}\bar q_{Li}\, \slash \hspace{-0.23cm}{\partial} q_{Lj} + \left(\text{L} \leftrightarrow \text{R}\right),
\label{1d}
\end{align}
with arbitrary complex coefficients $\{a^{L,R},b^{L,R},\bar\lambda^{L,R}\}$. For external (on-shell) fermions, the interactions in \eqref{1d} can effectively be reduced via the Dirac equation to the non-derivative Yukawa couplings given in \eqref{0d}. Again, since we are interested in the leading order spin-2 effects on fermion scattering, we will ignore the operators in \eqref{1d} for the remainder of the paper.
\emph{A priori}, there are no constraints on the couplings $a^{L,R}$ and $b^{L,R}$. However, for a theory in which the spin-2 field is gravitational in nature, $S_{\mu\nu}$ should be related to the energy-momentum tensor. The fact that $h_{\mu\nu}$ interactions are non-universal, as well as couple different generations with each other does not rule out the possible gravitational interpretation of the theory - this can be accommodated e.g. in the framework of RS models with complicated matter localization along the bulk. In such cases the KK gravitons couple to quark \textit{flavor} eigenstates in a diagonal, albeit non-universal way. This, upon rotation to the mass basis, results in the following constraints on the couplings
\begin{equation}
a^{L,R}_{ij}=-b^{L,R}_{ij} \equiv g^{L,R}_{ij}.
\end{equation}
Although we remain completely agnostic about the origin of the spin-2 state, we will take these relations to hold in the analysis to follow; as we show below, restricting the parameter space in this way is already enough for generating the needed amount of asymmetry without running into conflict with other experimental bounds. Further constraints on the couplings come from the requirement of the invariance of the theory under the SM gauge group. We will not dwell on making the symmetry manifest, but will keep in mind that it implies some additional relations between the coupling constants.
The key observation to make at this point is that even if the spin-2 state is not associated with any gravitational dynamics, its couplings to fermions are quite similar to the coupling of the graviton to the energy/momentum. This leads to a large sensitivity of the effects associated with the virtual exchange of these states to the energy scales at hand. In particular, we will find that this fact fits with the observed pattern of an increase in $A_{FB}^{t\bar{t}}$ with $t\bar t$ invariant mass.
The range of validity of the low-energy effective theory is an important issue, since it determines the maximum energy to which the analysis performed below can be extrapolated. Given the complete ignorance of the UV completion of low-energy (massive) spin-2 theories, the best we can do is to make an educated guess of the relevant energy scale. For massless general relativity (GR) in 4D, the analysis of graviton loop corrections yields the following expansion parameter \cite{Donoghue:1994dn, Giudice:1998ck},
\begin{equation}
\alpha \simeq \left( \frac{E}{4\pi M_{Pl}} \right)^2,
\end{equation}
where $E$ represents the typical energy scale of a process under consideration, This is in complete analogy to what one finds for low-energy nonlinear sigma models, once $M_{Pl}$ is replaced by the pion decay constant, $f_\pi$.
The massive spin-2 representation of the Poincar\'e group propagates three additional (one helicity-0 and two helicity-1) degrees of freedom on top of the two helicity-2 modes of the massless theory. In a general nonlinear completion of Fierz-Pauli action, the strong coupling of the helicity-0 mode is usually responsible for the cutoff of the effective theory - in complete analogy to massive non-abelian theories, where the strong coupling of the longitudinal $W$-bosons leads to the necessity of UV completion at a scale $\sim 4\pi m_W/g$ (here $m_W$ and $g$ refer to the $W$-mass and the $SU(N)$ coupling constant). Due to the higher spin structure, non-linear sigma models involving the longitudinal modes of massive spin-2 theories are usually different in nature than those for spin-1\footnote{In particular, a longitudinal scalar mode of a massive spin-2 boson is more strongly coupled than that of a massive spin-1 particle. Mathematically this can be traced back to the piece in the massive spin-2 propagator which grows fastest with momentum, $$\langle h_{\mu\nu}h_{\alpha\beta}\rangle \supset \frac{k_\mu k_\nu k_\alpha k_\beta}{M^4 (k^2-M^2)} .$$}. This leads to a UV cutoff which is in general sensitive to the nonlinear completion. For example, in theories of massive (four-dimensional) GR with a graviton potential, the cutoff usually comes out to be a certain geometrical mean of the scales $M$ and $f$. A specific class of potentials which avoids the propagation of ghosts in the theory \cite{deRham:2010gu} leads to a sigma model with higher UV cutoff, compared to theories with a more general potential \cite{ArkaniHamed:2002sp}; some other possible UV/nonlinear completion (e.g. a completion beyong the potential, or one which relaxes the requirement of reproducing four-dimensional GR in the $M\to 0$ limit) can therefore be expected to yield yet more different sigma models. Another possibility is that new physics regulating the low-energy theory in the UV kicks in somewhat below the strong coupling scale\footnote{In extra dimensional examples, one might imagine the higher KK modes softening loop effects, thus providing a completion at intermediate energies - up to the fundamental quantum gravity scale.}, however it does not distort the spin-2 exchange effects up to higher energies.
As already emphasized, below we will not be concerned with the nature of the underlying UV/nonlinear theory and will expect the low-energy description to be valid up to scales somewhat above the scale $\bar f=min\{f/g_i,M\}$ where $g_i$ collectively denotes all cupling constants in \eqref{0d} - \eqref{1d}.
\section{Existing Experimental Constraints}
\label{sec:constraints}
We now turn our attention to the analysis of the existing experimental constraints on the massive spin-2 model considered in this work. Note that the main motivation of the present work is an illustration of strong energy-dependence of the effects of virtual spin-2 exchange and its possible relevance to observations that exhibit these effects. In this section we are concerned with preliminary estimates of the bounds on the spin-2 parameter space due to various experimental constraints -- just to show that the mechanism can be viable, or even robust. Of course, many additional studies need to be performed (e.g. a closer inspection of LHC bounds or studies of spin correlations of top quark daughters) for a complete phenomenological analysis. Wherever loop contributions are involved, we make the most conservative assumptions for their magnitude just to show that even with these overly restrictive assumptions, there still is a vast parameter space for addressing the $t\bar t$ forward-backward asymmetry. We leave a more detailed analysis of the experimental bounds for a future study.
Below we determine bounds from LEP, electroweak precision data, same-sign top-quark pair production, $B_d - \overline{B}_d$ mixing and dijet production at the Tevatron. As can be seen below, the effects of these constraints on the model are mild. We note however, that a recent analysis~\cite{Berger:2012nw} of the kinematic and dynamical aspects of the relationship between the asymmetries $A^{t\bar{t}}_{FB}$ and $A^{\ell}_{FB}$ measured by D\O\, favors new physics (NP) models that produce more right-handed than left-handed top quarks.
Even if the left-handed sector is taken to be suppressed, the right-handed sector can still accomodate the asymmetry, as we will show below. Such a suppression is further motivated by other experimental constraints, such as $B_d-\overline{B}_d$ mixing. It is also worth noting that the $t\bar{t}$ production cross section, Eqs.~\eqref{eq:T}-\eqref{eq:ST}, of the model at hand is symmetric under the exchange of the left- and right-handed couplings.
\begin{figure}
\includegraphics[width=0.4\textwidth]{lep.pdf}
\caption{One possible diagram for 4-jet production at LEP. The dashed line represents the spin-2 particle exchange.}
\label{fig:lep}
\end{figure}
\subsection{LEP Constraints}
The LEP constraints depend on how the massive spin-2 state couples to electrons and final state quarks. Since we are only interested in generating a large $A^{t\bar{t}}_{FB}$ at the moment, we can take $g_{ut}\sim\mathcal{O}(1)$ while allowing freedom for the couplings to leptons. In this scenario therefore, we do not anticipate any bounds from direct production at LEP. However, due to $SU(2)_W$ symmetry, $g^L_{\{d,b\}} = g^L_{\{u,t\}}$ where $\{u,t\}$ stands for any combination of $u$ and $t$, and a light spin-2 particle could lead to anomalous 4-jet events as shown in Fig.~\ref{fig:lep}. For examining this bound, we can implement the results in the literature (see e.g. Sec. IV.A. of Ref.~\cite{Grinstein:2011dz} and references therein) in our model. The amplitude for a 4-jet final state in the present model is suppressed by extra factors of $E^2/f^2$ compared to the case of a new scalar or vector field. Here $E$ denotes a relevant energy scale in the process.
\begin{comment}
\footnote{This is a conservative bound since we estimate the two derivatives by $s$. One of these derivatives acts on an external quark pair, and can be reduced to the heavier of the two quark masses by the equation of motion.}.
\end{comment}
The final operating energy of LEP II is 209 GeV and for the parameter space considered below $f$ is the highest scale in the processes at hand. Even if we conservatively take these suppression factors to be 1 therefore, the bound on the mass of the spin-2 particle from LEP is quite mild, $M \mbox{\raisebox{-.6ex}{~$\stackrel{>}{\sim}$~}} 100$ GeV.
\subsection{Electroweak Precision Tests}
Electroweak precision data (EWPD) can provide strong constraints on models that attempt to explain the $t\bar{t}$ forward-backward asymmetry~\cite{Grinstein:2011gq}. Corrections to EW precision observables due to the intermediate spin-2 state do not occur at tree level assuming it does not directly couple to the EW gauge bosons. At the one-loop level there is a contribution to the dimension-4 operator, $C_{Z\bar{u}u} Z_{\mu}\bar{u}\gamma^{\mu}u$, arising from the diagram in Fig. \ref{fig:zloop}. As shown in Ref.~\cite{Gresham:2012wc}, the most stringent constraint on $C_{Z\bar{u}u}$ comes from atomic parity violation experiments. The experimental and SM values for $Q_W$ in cesium atoms in the 2010 PDG~\cite{Nakamura:2010zzi} can be turned into a bound on the NP contribution to the coefficient of the dimension-4 operator, $\left|C^{NP}_{Z\bar{u}u}\right| < 1.3\cdot 10^{-3}$. An estimate of the spin-2 contribution to $Q_W(Cs)$, ignoring left-handed couplings, yields
\begin{equation}
\left|C^{NP}_{Z\bar{u}u}\right| \sim \frac{2e s_w}{3 c_w}\frac{\left|g^R_{ut}\right|^2\left(M^2 + m_t^2\right)}{16\pi^2 f^2} \Rightarrow
\frac{\left|g^R_{ut}\right|^2\left(M^2 + m_t^2\right)}{ f^2}\mbox{\raisebox{-.6ex}{~$\stackrel{<}{\sim}$~}} 2~.
\end{equation}
\begin{figure}
\includegraphics[width=0.3\textwidth]{zvertex.pdf}
\caption{The loop contributing to the EW precision observables such as $\Gamma_Z$ and $Q_W$.}
\label{fig:zloop}
\end{figure}
\subsection{Single-Top Production}
Single-top, spin-2 production via the reaction $u\,g \rightarrow t\,h_{\mu\nu}$ is PDF enhanced at the LHC and PDF suppressed at the Tevatron relative to spin-2 mediated $t\bar{t}$ production, which has a $q\bar{q}$ initial state. The phenomenology of spin-2 production can be classified into two categories. In the first case $h_{\mu\nu}$ is stable on collider time scales, which is predicted in large extra dimensions scenarios. The decay signature of this reaction - one b-tagged jet, one lepton, and high missing transverse energy (MET) - is not an event that is currently selected in single-top searches at the LHC. Single-top searches thus far always contain at least 2 jets or 2 leptons. The other scenario is that the spin-2 particle decays immediately upon production, which is the case in warped extra dimensions scenarios. In this case, bounds from single-top production can be avoided by making the branching ratio of the spin-2 particle into a $u\bar{u}$ pair small. As we will shown in Sec.~\ref{sec:dijet} the coupling $g_{uu}$, which controls the size of $Br(h_{\mu\nu} \rightarrow u\bar{u})$, is constrained to be small from dijet bounds.
\subsection{Same-Sign Top-Quark Pair Production}
The $t$-channel models aiming to explain the $A^{t\bar{t}}_{FB}$ asymmetry can be strongly constrained by limits on the same sign top quark pair production at the LHC. The ATLAS collaboration places limit on the production cross-section at $\sigma_{tt} \le 1.7$ pb~\cite{:2012bb}. In our model, the production arises from processes shown in Fig.~\ref{fig:tt}. These diagrams contribute to the coefficient of the effective 4-quark operator responsible for $tt$ pair production, $\mathcal{C}(\bar{t}_R\gamma^\mu u_R)(\bar{t}_R\gamma_\mu u_R)$. The CMS collaboration reports the bound on this coefficient of $\mathcal{C} \le 2.7\text{ TeV}^{-2}$~\cite{Chatrchyan:2011dk}.
\begin{table}[b]
\centering
\begin{tabular}{ |c || c | c | c | c | c | c | c |}
\hline
M [GeV] & 100 & 200 & 300 &400 & 500 & 600 & 700 \\
\hline
$\left|g^R_{tu}\right|$ &$0.05$ & $ 0.04$ & $ 0.03$ & $ 0.03$ & $ 0.03$ & $ 0.03$ & $ 0.02$ \\
\hline
\end{tabular}
\caption{95\% CL upper limit on $|g^R_{tu}|$ from a search for same-sign top-quark pair production by the ATLAS collaboration. $|g^R_{ut}|$ is fixed to be 1, and $f$ is set by requiring that $A^{t\bar t}_{\text{high}}=0.475$.}
\label{tab:tt_tree}
\end{table}
The tree-level diagram, Fig.~\ref{fig:tt_tree}, constrains the combination of couplings $g^R_{ut}(g^{R}_{tu})^\ast/f^2$. We have chosen to taken $g^R_{ut}=1$ in~\eqref{eq:1d} so that the tree-level cross section yields a bound on $g^R_{tu}$, which is given in Table \ref{tab:tt_tree}.
Since the spin-2 propagator contains pieces such as $\frac{k_\mu k_\nu k_\alpha k_\beta}{M^4 (k^2-M^2)}$, loops containing spin-2 particles are highly divergent. The most conservative estimate (i.e. neglecting the possibility of some derivatives acting on the external fermions to reduce the energy-dependence of the finite part of the loop integral) of the one-loop contribution to same-sign top quark production yields
\begin{equation}
\label{eq:sstop}
\mathcal{C} \sim \frac{\left|g^{R}_{ut}\right|^2 \left|g^{R}_{ii}\right|^2}{16\pi^2}\frac{\hat{s}}{f^4} \frac{\hat{s}^4}{M^8},
\end{equation}
where $i=u$, $t$ and $\hat{s}$ is the partonic center of mass energy. Since the PDFs drop significantly for the momentum fraction greater than 0.3, we can estimate $\hat{s} = 10\%$ of the LHC running energy, $\hat{s}=700$ GeV. For $f=350$ GeV, $M=500$ GeV and $g^{R}_{ut}=1$, we get $\mathcal{C} \approx 3 \left(g^{R}_{ii}\right)^2 \text{ TeV}^{-2}$. This leads to the bound $g^{R}_{ii}\lesssim 0.9$. Note that the bounds get weaker for a larger value of $f$ and/or $M$.
\begin{figure}
\subfloat[Tree-level diagram.]{\label{fig:tt_tree}\includegraphics[width=0.38\textwidth]{sstop_a.pdf}}\qquad
\subfloat[One of the possible 1-loop diagrams.]{\label{fig:tt_loop}\includegraphics[width=0.45\textwidth]{sstop_b.pdf}}
\caption{Feynman diagrams giving rise to the same sign top-quark pair production.}
\label{fig:tt}
\end{figure}
The bound from the non-derivative interactions in Eq.~\eqref{0d} can be easily obtained in a similar fashion. But since this coupling doesn't play much role in our analysis of $A_{FB}^{t\bar{t}}$, we will not dwell on the value of this bound. Put differently, we can take the coupling $\lambda_{ij}$ to be negligible while still producing a large $A_{FB}^{t\bar{t}}$.
Single-top, spin-2 production where the spin-2 particle immediately decays into $t\bar{u}$ or a $t\bar{t}$ pair can also contribute to same-sign top production. We aproximate $\sigma(u\,g \rightarrow t\,t\, j) \approx \sigma (u\,g \rightarrow t\,h_{\mu\nu})\times \left( Br(h_{\mu\nu} \rightarrow t\,\bar{t}) + Br(h_{\mu\nu} \rightarrow t\,\bar{u}) \right)$ as we are only interested in producing a quick estimate of the cross section. The ATLAS collaboration~\cite{:2012bb} imposes a cut, $|\eta| < 2.5$, when selecting lepton and jet candidates assoicated with same-sign top production. We imposed this cut when calculating the spin-2 contribution to this cross section. We neglect $Br(h_{\mu\nu} \rightarrow t\,\bar{u})$ since this branching ratio is proportional to $|g_{tu}|^2$, which is constrained to be small from same-sign top production in other channels. For $M = 400$ GeV and $f = 1$ TeV, the ATLAS bound on same-sign top production, $\sigma_{tt} < 1.7$ pb, yields the constraint, $|g_{ut}|^2 Br(h_{\mu\nu} \rightarrow t\,\bar{t}) \mbox{\raisebox{-.6ex}{~$\stackrel{<}{\sim}$~}} 0.97$.
\subsection{$B_d-\overline{B}_d$ Mixing}
Non-zero off-diagonal couplings can lead to flavor changing neutral currents (FCNCs).
Here we focus on $B_d-\overline{B}_d$ mixing as the most restrictive bounds on $g^{L}_{ut}$ are expected to come from this process. The spin-2 contributions to $B_d$ mixing can be described by the following four-quark operator
\begin{align}
\label{eq:ops}
Q_{1} &= \Big(\bar{d}_{L}\gamma_{\mu}b_{L}\Big) \Big(\bar{d}_{L}\gamma^{\mu}b_{L} \Big).
\end{align}
In general, other operators contribute to $B_d$ mixing as well. The coefficient of the operator in Eq.~\eqref{eq:ops} is constrained to be smaller than $\mathcal{O}(10^{-11})\text{ GeV}^{-2}$~\cite{Bona:2007vi}. These bounds constrain the couplings relevant for $A^{t\bar{t}}_{FB}$ in our model, in particular there is a constraint on $g^L_{ut}$.
This is due to the fact that by $SU(2)_L$ symmetry, $g^L_{db} = g^L_{ut}$.
The contribution to this operator arises at tree-level from Fig.~\ref{fig:bbbar}.
A quick dimensional analysis estimate reveals that the contribution from such a diagram is
\begin{figure}
\includegraphics[width=0.35\textwidth]{bbbar_tree.pdf}
\caption{A tree-level contribution to $B_d-\overline{B_d}$ mixing.}
\label{fig:bbbar}
\end{figure}
\begin{equation*}
\frac{g^L_{ut} \left(g^{L}_{tu}\right)^\ast} {f^2} \frac{m_b^2}{M^2} \approx 4g^L_{ut} \left(g^L_{tu}\right)^\ast 10^{-10}\, \text{GeV}^{-2},
\end{equation*}
where we estimate $m_b/M\sim10^{-2}$ and take $f=500$ GeV. Thus we see that for $g^L_{ut}(g^L_{tu})^\ast \lesssim \mathcal{O}(0.01)$, the constraints from $B_d-\overline{B}_d$ mixing can easily be satisfied. As in the case of same sign top-quark pair-production, the constraint gets weaker for larger values of $f$. Similar constraints hold for the right-handed couplings. However, there is no symmetry relating $g^R_{db}$ to $g^R_{ut}$. Thus there is more freedom available in the right-handed sector to address the top forward-backward asymmetry.
\subsection{Tevatron Dijet and Top Width Constraints}
\label{sec:dijet}
\begin{table}[b]
\centering
\begin{tabular}{ |c || c | c | c | c | c | c | c |}
\hline
$M$ [GeV] &300 & 400 & 500 &700 &900 &1100 &1300 \\ \hline
$\left|g^R_{uu}\right|$& 0.08 & 0.15& 0.16& 0.18& 0.17& 0.12& 0.09 \\
\hline
\end{tabular}
\caption{95\% CL upper limit on $|g^R_{uu}|$ from dijet constraints at the Tevatron. $|g^R_{ut}|$ is fixed to be 1, and $f$ is set by requiring that $A^{t\bar t}_{\text{high}}=0.475$.}
\label{tab:dijet}
\end{table}
As emphasized above, the derivative couplings of the spin-2 field to light quark pairs grow with energy, which can lead to strong constraints on the couplings at large invariant mass, $M_{jj}$. Following Ref.~\cite{Grinstein:2011dz}, we obtain the Tevatron dijet bounds from the CDF 95\% CL upper limits on the product of an RS graviton ($G^{\star}$) production cross section $\times$ its branching ratio to dijets ($\mathcal{B}$) $\times$ acceptance ($\mathcal{A}$); see Table 1 of Ref.~\cite{Aaltonen:2008dn}. The results with $\mathcal{B}\cdot\mathcal{A} = 1$ are collected in Table~\ref{tab:dijet}. The CDF analysis uses the RS parameter $k / \bar{M}_{Pl} = 0.1$, which translates into $ M \approx 0.383 {f}$ in the notation of this paper (with the convention that the largest dimensionless coupling is set to one). The dijet cross section in the present model is therefore related to the RS graviton cross section by $\sigma^{NP}_{jj} = C^4\,\sigma_{G^{\star}}$, where $ C = \left|g^R_{uu}\right| M / 0.383 f$. To obtain a meaningful bound, $f$ is taken to be of the value required to produce an asymmetry in the high mass bin of 47.5\% with $\left|g^R_{ut}\right| = 1$. $\sigma^{NP}_{jj}$ is estimated to be the $s$-channel NP cross section. $\sigma^{NP}_s$, see Eq.~\eqref{eq:S}, includes additional terms due to a finite top-quark mass; $F_i(x,\,0)$ should be used be for light quark production cross sections as opposed to $F_i(x,\,y)$. This is an overestimate of the actual dijet cross-section, which is produces a conservative estimate of the bound.
$g^R_{uu}$ does not play a role in generating a large asymmetry. However, $g^R_{uu}$ along with $g^R_{tt}$ are important for increasing the total cross section from the SM value up to what is measured at the Tevatron (see Table~\ref{tab:sum} for the relevant measurements and the corresponding SM predictions). A smaller value of $g^R_{uu}$ requires a larger value of $g^R_{tt}$ to produce the same cross section. A larger $g^R_{tt}$ and a smaller $g^R_{uu}$ could be expected in an RS model; localization of the top quark close to the IR brane leads to a large coupling, while the light quark couplings are relatively suppressed as they are localized in the bulk closer to the UV brane.
The dijet invariant mass and angular distributions measured the LHC may very well be more constraining than the Tevatron measurements. However, we do not consider LHC dijet data because these measurements only constrain the coupling $g_{uu}$ in this model. As previously noted, $g_{uu}$ plays no role in generating an asymmetry so constraining this coupling to be smaller does not affect the goal of this work. LHC dijet measurements will be important for constraining a more complete model, and we leave this work for a future publication.
When $M < m_t$, the non-standard top quark decay $\Gamma\left(t \rightarrow u\,h_{\mu\nu}\right)$ is allowed,
\begin{equation}
\Gamma_t^{NP} = \frac{\left|g_{ut} + g_{tu}\right|^2 m_t^7}{12 \pi f^2 M^4}\left(1 - \frac{M^2}{m_t^2}\right)^4\left(2 + 3\frac{M^2}{m_t^2}\right).
\end{equation}
The recent D\O\, measurement~\cite{Abazov:2012vd}, $\Gamma_t = \left(2.00^{+0.47}_{-0.43}\right)$ GeV, constrains how large $\left|g_{ut}\right| / f$ can be in the low mass region; see Figs.~\ref{fig:chi1} and~\ref{fig:chiC} for bounds on the spin-2 parameter space from $\Gamma_t$.
\section{Top Quark Forward-Backward Asymmetry}
\label{sec:ttbar}
\subsection{Calculation of the Cross Section}
\begin{figure}
\includegraphics[width=0.65\textwidth]{diagrams.pdf}
\caption{Most important spin-2 contributions to $t\bar t$ production at the Tevatron.}
\label{fig:hexchange}
\end{figure}
In the present section we study the effects of an intermediate massive spin-2 state on the top quark forward-backward asymmetry at the Tevatron. Calculations of the virtual exchange of a massive spin-2 graviton in the context of theories with large extra dimensions were done by Giudice \textit{et al}. in~\cite{ Giudice:1998ck}. However, the results of Ref.~\cite{ Giudice:1998ck} should be applied to collider phenomenology with care as they are only valid in the limit (in our notation) $f^2 \sim M^2 \gg \hat{s} \gg m_t^2$. Here we extend the results of~\cite{ Giudice:1998ck} by assuming there is no hierarchy between the aforementioned scales.
The weighted average (the average over initial spins and colors and the sum over final spins and colors) of the amplitude squared for the $q\bar{q} \rightarrow t\bar{t}$ scattering from a spin-2 $t$-channel exchange is
\begin{align}
\label{eq:T}
\langle\left|\mathcal{M}_t\right|^2\rangle &= \frac{\hat{s}^4}{128f^4\left(\left(\hat{t} - M^2\right)^2 + \Gamma^2 M^2\right)} \left[C_1^2\left(F_1 + \frac{m_t^2}{M^2}F_2+ \frac{m_t^4}{M^4}F_3\right)\right. \nonumber \\
&\left. + \frac{C_2^2}{18}\left(F_4 + \frac{m_t^2}{M^2}F_5 + \frac{m_t^4}{M^4}F_6 + \frac{m_t^6}{M^6}F_7 + \frac{m_t^8}{M^8}F_8\right)\right],
\end{align}
where $\hat{s},\,\hat{t}$ are the Mandelstam variables in the parton center-of-momentum frame, $q = \{u,\,c\}$ is assumed to be massless, and $\Gamma$ denotes the width of the spin-2 resonance. The functions $F_i$ are polynomials in $x \equiv \hat{t}/\hat{s}$ and $y \equiv m_t^2/\hat{s}$, and are defined in Appendix~\ref{sec:ap}. The $C_i$'s are combinations of couplings and are also given in Appendix~\ref{sec:ap}.
The interference of the $t$-channel spin-2 exchange with the SM leading order (LO) gluon exchange gives
\begin{equation}
\label{eq:Tint}
\langle2\,\text{Re}\left(\mathcal{M}_t\mathcal{M}_{SM}^{\ast}\right)\rangle = \frac{2\pi\alpha_s \hat{s}^2\left(\hat{t} - M^2 \right)}{27f^2\left(\left(\hat{t} - M^2\right)^2 + \Gamma^2 M^2\right)} \,C_2 \left(F_9 + \frac{m_t^2}{M^2}F_{10}+ \frac{m_t^4}{M^4}F_{11}\right),
\end{equation}
while the weighted average of the amplitude squared for the $q\bar{q} \rightarrow t\bar{t}$ scattering from a spin-2 $s$-channel exchange is
\begin{equation}
\label{eq:S}
\langle\left|\mathcal{M}_s\right|^2\rangle = \frac{\hat{s}^4}{128f^4\left(\left(\hat{s}- M^2\right)^2 + \Gamma^2 M^2\right)} \left(C_3\,F_{12} + C_4\,F_{13} + C_5\,F_{14} \right),
\end{equation}
where $q=\{u,\,d,\,s,\,c,\,b\}$. The exchange of any color-singlet particle in the $s$-channel can not interfere with color-octet gluon exchange in the SM. However, $s$-channel spin-2 exchange can interfere with the exchange of a spin-2 particle in the $t$-channel with $q=\{u,\,c\}$,
\begin{align}
\label{eq:ST}
\langle 2\,\text{Re}\left(\mathcal{M}_t\mathcal{M}_{s}^{\ast}\right)\rangle &= \frac{\hat{s}^4\left(\left(\hat{s} - M^2\right)\left(\hat{t} - M^2\right) + \Gamma^2 M^2\right)}{1152f^4\left(\left(\hat{s} - M^2\right)^2 + \Gamma^2 M^2\right)\left(\left(\hat{t} - M^2\right)^2 + \Gamma^2 M^2\right)} \\
&\times\left[C_6 \left(F_{15} + \frac{m_t^2}{M^2}F_{16}+ \frac{m_t^4}{M^4}F_{17}\right) + C_7 \left(F_{18} + \frac{m_t^2}{M^2}F_{19}+ \frac{m_t^4}{M^4}F_{20}\right)\right]. \nonumber
\end{align}
Our results are consistent with what was found in Ref. \cite{ Giudice:1998ck}\footnote{For example, $F_{12}(x,\,0) = G_4(x)$ and $F_4(x,0)/18 + F_{15}(x,0)/6 = G_{11}(x)$, where $G_i(x)$ are given in the appendix of~\cite{ Giudice:1998ck}.}.
\subsection{Tevatron Measurements and SM Predictions}
\label{sec:teva}
We start out by reviewing the recent observations of the anomalously large top quark forward-backward asymmetry, $A^{t\bar{t}}_{FB}$. The experimental evidence for contributions to $A^{t\bar{t}}_{FB}$ from physics beyond the SM is as follows. The CDF collaboration measured~\cite{CDFnote:10584} the asymmetry to be
$(20.0 \pm 7.0)\%$.
A recent D\O\, analysis~\cite{Abazov:2011rq} yielded the value
$A^{t\bar{t}}_{FB} = 19.6^{+6.2}_{-6.5}\%$,
in good agreement with the CDF measurement. D\O\, also reports a forward-backward asymmetry based on the rapidity of the leptons from top quark decays of $A^{\ell}_{FB} = (15.2 \pm 4.0)\%$ compared with the small SM value $(2.1 \pm 0.1)\%$ calculated using \texttt{MC@NLO}. All uncertainties have been added in quadrature. In addition, the CDF collaboration reports~\cite{Aaltonen:2011kc} that the asymmetry rises with the invariant mass of the $t\bar{t}$ system, with $A^{t\bar{t}}_{\text{high}} \equiv A^{t\bar{t}}_{FB}\left(M_{t\bar{t}} > 450\,\text{GeV}\right) = (47.5 \pm 11.4)\%$, and $A^{t\bar{t}}_{\text{low}} \equiv A^{t\bar{t}}_{FB}\left(M_{t\bar{t}} \le 450\,\text{GeV}\right) = -(11.6 \pm 15.3)\%$.
\begin{table}[b]
\centering
\begin{tabular}{ c | c c }
\hline \hline
Observable & Measurement & SM prediction~\cite{Manohar:2012rs} \\ \hline
$A^{t\bar{t}}_{FB}$ & $(20.0 \pm 4.7)\%$~\cite{Grinstein:2011dz} & $9.3^{+2.7}_{-2.5}\%$ \\
$A^{t\bar{t}}_{\text{high}}$ & $(47.5 \pm 11.4)\%$~\cite{Aaltonen:2011kc} & $14.1^{+3.2}_{-2.6}\%$ \\
$A^{t\bar{t}}_{\text{low}}$ & $-(11.6 \pm 15.3)\%$~\cite{Aaltonen:2011kc} & $5.4^{+0.9}_{-0.6}\%$ \\
$\sigma_{t\bar{t}}$ & $(8.5 \pm 0.9) \text{ pb}~\text{\cite{Aaltonen:2010hza}}$ & $6.59^{+0.24}_{-0.40}$ pb \\
\hline \hline
\end{tabular}
\caption{Measurements and predictions for observables in $t\bar{t}$ production at the Tevatron.}
\label{tab:sum}
\end{table}
Despite recent improvements in the SM calculations, the asymmetry in the high mass bin is still close to three standard deviations away from the SM value. The central value of a next-to-leading order plus next-to-next-to-leading logarithm (NLO+NNLL) QCD calculation of $A^{t\bar{t}}_{FB}$ is $7.3^{+1.1}_{-0.7}\, \%$~\cite{Ahrens:2011mw}. Recently calculated electroweak Sudakov (EWS) corrections enhance the QCD asymmetry by a factor of 1.041 to $7.7\%$~\cite{Manohar:2012rs}, while fixed order electroweak contributions add an additional $1.6\%$ to the asymmetry~\cite{Hollik:2011ps}.
The overlap between the EWS corrections and the fixed order EW contributions is estimated in Ref.~\cite{Manohar:2012rs} to be $\sim 0.5\%$. This yields a total SM prediction of $A^{SM}_{FB} = 9.2^{+2.8}_{-2.6}\,\%$. Similarly, combining the QCD predictions of Ref.~\cite{Ahrens:2011uf} (which use MSTW2008 PDFs~\cite{Martin:2009iq}) and EW effects calculated in Refs.~\cite{Manohar:2012rs, Hollik:2011ps}, the total SM prediction for $A^{SM}_{FB}$ in the low and high mass bins is $5.4^{+0.9}_{-0.6}\,\%$ and $14.1^{+3.2}_{-2.6}\,\%$ respectively. The $2.8\sigma$ deviation from the SM in the high mass bin may be taken as a signal of new physics (NP).
The total cross section for $t\bar{t}$ production was recently measured~\cite{Aaltonen:2010hza} by CDF to be $\sigma_{t\bar{t}}= (8.5\pm 0.9)$ pb. This measurement is consistent with the value reported~\cite{Abazov:2011mi} by D$\O$, $\sigma_{t\bar{t}}= 7.78^{+0.77}_{-0.64}$ pb. Cacciari \textit{et al}.~\cite{Cacciari:2011hy} calculated the total cross section at approximate NNLO QCD to be $6.722^{+0.243}_{-0.410}$ pb. The EWS correction factor for this observable is $\mathcal{R}_t = 0.98$~\cite{Manohar:2012rs}. For $M_{t\bar{t}} \le 450$ GeV the NLO+NNLL SM prediction for $\sigma^{SM}$ is 4.23 pb, while for $M_{t\bar{t}} > 450$ GeV $\sigma^{SM} = 2.40$ pb~\cite{Grinstein:2011dz}. $\mathcal{R}_t$ in these bins is 0.985 and 0.973 respectively\footnote{We thank Mike Trott for the computation of the EWS correction factors for $\sigma_{t\bar{t}}$ in the low and high mass bins.}. The measured and predicted values for $t\bar{t}$ observables at the Tevatron are summarized in Table~\ref{tab:sum}.
\subsection{New Physics Results}
Here we show that tree-level exchanges of a massive spin-2 particle can contribute significantly to $A^{t\bar{t}}_{FB}$, ameliorating the tension between measurements at the Tevatron and SM predictions. Following Ref.~\cite{Grinstein:2011dz}, we define a partonic level asymmetry,
\begin{equation}
A^{NP+SM}_{FB} = \frac{\sigma^{NP}_F - \sigma^{NP}_B}{\left(\sigma^{SM}\right)_{LO} + \sigma^{NP}} + A^{SM}_{FB} \frac{\sigma^{SM}}{\sigma^{SM} + \sigma^{NP}},
\end{equation}
which is to be compared against the binned partonic asymmetries reported in~\cite{Aaltonen:2011kc}. For later convenience, we define
\begin{equation}
A^{NP}_{FB} = \frac{\sigma^{NP}_F - \sigma^{NP}_B}{\left(\sigma^{SM}\right)_{LO} + \sigma^{NP}}
\end{equation}
as the NP contribution to the top-quark forward-backward asymmetry.
We use state-of-the-art predictions for the SM quantities $A^{SM}_{FB}$ and $\sigma^{SM}$, and LO predictions for the NP corrections. The partonic NP cross sections are convoluted into hadronic cross sections using NLO MSTW2008 parton distribution functions (PDFs)~\cite{Martin:2009iq}. The factorization and renormalization scales are taken to be $\mu = m_t = 173.1$ GeV, while $\alpha_s$ is set by the MSTW fit value: $\alpha_s\left(M_Z\right) = 0.12018$. The width of the spin-2 state is taken to be a tenth of its mass, $\Gamma = M / 10$. We take all of the couplings to be real, and ignore contributions from $F_{14}(x,y)$ as it is numerically negligible.
\begin{figure}
\centering
\includegraphics[width=0.7\textwidth]{graphics/totalAsym.pdf}
\caption{$A^{t\bar{t}}_{FB}$ vs. $f$. $\left|g^R_{ut}\right| = 1$ and all other couplings are zero. The thick, dashed line is the central value as measured at the Tevatron with $1\sigma$ error bands. The dotted line is the SM prediction. From left to right, $M$ decreases from 700 GeV to 100 GeV in steps of 100 GeV.}
\captionsetup{justification=raggedright}
\label{fig:1}
\end{figure}
The total asymmetry as a function of $f$ is shown in Fig.\ref{fig:1}. Here $\left|g^R_{ut}\right| = 1$ and we have set all other couplings to zero. The thick, dashed line is the central value as measured at the Tevatron with $1\sigma$ error bands. The dotted line is the SM prediction. From left to right (at the top of the plot), $M$ decreases from 700 GeV to 100 GeV in steps of 100 GeV. For intermediate values of $f$, the destructive interference with the SM exceeds the pure NP contribution, decreasing the asymmetry. As expected, the NP decouples for large $f$.
Fig.~\ref{fig:2} shows the effects of NP on $\sigma_{t\bar{t}}$ as a function of $f$. The cross section for $t$-channel NP is shown in Fig.~\ref{fig:T} with $\left|g^R_{ut}\right| = 1$ and all the other couplings set to zero. While Fig.~\ref{fig:S} shows the effect of $s$-channel NP as a function of $f$ with the only non-zero coupings being $\left|g^R_{uu}\right| = \left|g^R_{tt}\right| = 1$. In Fig.~\ref{fig:T}, $M$ again monotonically decreases from 700 GeV to 100 GeV from left to right in steps of 100 GeV. In Figure~\ref{fig:S} however, the ordering is not as simple, $M = \{100,\,200,\,300,\,400,\,700,\,500,\,600\}$ from left to right.
A global fit of the spin-2 model to the CDF measurements $A^{t\bar{t}}_{\text{high}},\, A^{t\bar{t}}_{\text{low}},\, \text{and}\,\sigma_{t\bar{t}}$ was performed using the method of least squares assuming the measurements are uncorrelated\footnote{Of course, the measurements actually are correlated, but this should not affect the conclusions we are able to draw from the fit in any qualitative way.}. The scale $f$ was fixed to be 1 TeV and $g^R_{ut},\, g^R_{uu},\, \text{and}\,g^R_{tt}$ were left as free parameters for a given $M$. The results of the fit are shown in Fig.~\ref{fig:chi}. The 1 and 2$\sigma$ confidence regions of allowed parameter space are shown in green and yellow respectively. The black line corresponds to $A^{t\bar{t}}_{\text{high}}= 47.5\%$. This is not necessarily the best-fit value. Experimentally disallowed parameter space due to the constraints from same-sign top production at the LHC, EWPD, and the width of the top quark are shown in blue, red, and brown respectively (see Section~\ref{sec:constraints} for a detailed discussion of experimental constraints on the model). The spin-2 model is able to hit the central value of the forward-backward asymmetry in the high mass bin in a large region of the parameter space.
\begin{figure}
\centering
\subfloat[$t$-channel NP]{\label{fig:T}\includegraphics[width=0.5\textwidth]{graphics/crosssectionT.pdf}}
\subfloat[$s$-channel NP]{\label{fig:S}\includegraphics[width=0.5\textwidth]{graphics/crosssectionS.pdf}}
\caption{$\sigma_{t\bar{t}}$ vs. $f$. $t$-channel NP is shown in Fig.~\ref{fig:T} with $\left|g^R_{ut}\right| = 1$ and all other couplings set to zero. Fig.~\ref{fig:S} shows $s$-channel NP with the only non-zero coupings being $\left|g^R_{uu}\right| = \left|g^R_{tt}\right| = 1$. In Fig.~\ref{fig:T}, $M$ decreases from 700 GeV to 100 GeV from left to right in steps of 100 GeV. In Figure~\ref{fig:S} however, the ordering is not as simple, $M = \{100,\,200,\,300,\,400,\,700,\,500,\,600\}$ from left to right.}
\captionsetup{justification=raggedright}
\label{fig:2}
\end{figure}
\begin{figure}
\centering
\subfloat[$g^R_{uu} = g^R_{tt} = 0$]{\label{fig:chi1}\includegraphics[width=0.5\textwidth]{graphics/chi1.pdf}}
\subfloat[$g^R_{uu} = g^R_{tt} = g^R_{ut} / 10 = 3 g^R_{tu}$]{\label{fig:chi4}\includegraphics[width=0.5\textwidth]{graphics/chi4.pdf}} \\
\subfloat[$g^R_{uu} = g^R_{tt} = g^R_{ut} / 2 = 15 g^R_{tu}$]{\label{fig:chi3}\includegraphics[width=0.5\textwidth]{graphics/chi3.pdf}}
\subfloat[$g^R_{uu} = g^R_{tt} = g^R_{ut} = 30 g^R_{tu}$]{\label{fig:chi2}\includegraphics[width=0.5\textwidth]{graphics/chi2.pdf}}
\caption{Results of a global fit of the spin-2 model to Tevatron observables. $A^{t\bar{t}}_{\text{high}}= 47.5\%$ is shown in black. The 1 and 2$\sigma$ confidence regions of allowed parameters are shown in green and yellow respectively. The blue, red, and brown regions are disfavored by constraints from same-sign top, EWPD, and the width of the top respectively.}
\captionsetup{justification=raggedright}
\label{fig:chi}
\end{figure}
\begin{figure}
\centering
\subfloat[$A^{t\bar{t}}_{FB}$]{\label{fig:asym350}\includegraphics[width=0.5\textwidth]{graphics/asym350.pdf}}
\subfloat[$d\sigma_{t\bar{t}}/dM_{t\bar{t}}$]{\label{fig:diff350}\includegraphics[width=0.5\textwidth]{graphics/diff350.pdf}}
\caption{Prediction from the spin-2 model for $A^{t\bar{t}}_{FB}$ and $d\sigma_{t\bar{t}}/dM_{t\bar{t}}$ with $M = 350$ GeV. The purple band represents the theoretical uncertainty from varying the factorization scale in the range $\mu = \{m_t / 2,\,2 m_t\}$. This example hits the central value of $A^{t\bar{t}}_{FB}$ in the high bin and is within $1\sigma$ of the central value in the low bin. Detector acceptance effects and the known increase in the measured value for $\sigma_{t\bar{t}}$ could account for the disagreement in the high mass bins for $d\sigma_{t\bar{t}}/dM_{t\bar{t}}$.}
\label{fig:4}
\end{figure}
Fig.~\ref{fig:asym350} shows the binned asymmetry predicted by the spin-2 model for $M = 350$ GeV, $\left|g^R_{ut}\right| / f = 2.36 /$TeV, and all other couplings set to zero. The CDF measurements with error bars are also shown. The purple band represents the theoretical uncertainty from varying the factorization scale in the range $\mu = \{m_t / 2,\,2 m_t\}$. This combination of parameters hits the central value of $A^{t\bar{t}}_{FB}$ in the high bin and is within $1\sigma$ of the central value in the low bin. The sum of the SM LO prediction plus the contribution from the spin-2 model with the same parameters as those used in Fig.~\ref{fig:asym350} for the binned differential cross section, $d\sigma_{t\bar{t}}/dM_{t\bar{t}}$, is shown in Fig.~\ref{fig:diff350}. Again, the purple band represents the uncertainty in the PDF factorization scale, and the CDF measurements, as reported in Ref.~\cite{Aaltonen:2009iz}, are also shown. The high bin values do not agree with the CDF measurements. However, we have not taken into account any detector acceptance effects. The deconvolution to the parton level done by CDF assumes the SM. As shown in Ref.~\cite{Grinstein:2011dz}, model-dependent acceptance effects can reduce the cross section by as much as a factor of $\sim 1/2$ in the high bins. Furthermore, the total cross section reported in Ref.~\cite{Aaltonen:2009iz} is $\sigma_{t\bar{t}} = 6.9 \pm 1.0$ pb, which is lower than the most recent measurements from both the CDF~\cite{Aaltonen:2010hza} and D\O ~\cite{Abazov:2011mi} collaborations. It is reasonable to assume that detector acceptance effects and the known increase in the measured value for $\sigma_{t\bar{t}}$ could account for the disagreement in the high mass bins for $d\sigma_{t\bar{t}}/dM_{t\bar{t}}$.
\subsection{LHC Measurements, Predictions, and Results}
There is no forward-backward asymmetry at the LHC because of its symmetric initial state, $pp$, as opposed to the $p\bar{p}$ initial state at the Tevatron. However, the same underlying physics that leads to $A_{FB}$ results in a \textit{charge asymmetry} at the LHC, which we define as
\begin{equation}
A^y_C = \frac{\sigma\left(\Delta y^2 > 0\right) - \sigma\left(\Delta y^2 < 0\right)}{\sigma\left(\Delta y^2 > 0\right) +\sigma\left(\Delta y^2 < 0\right)},
\end{equation}
where $\Delta y^2$ is the difference of the squares of rapidities of the top quark and anti-top quark, $\Delta y^2 = y_t^2 - y_{\bar{t}}^2$. The CMS collaboration reports~\cite{Chatrchyan:2011hk} the charge asymmetry of $A^y_C = \left(-1.3^{+4.0}_{-4.2}\right)\%$, which is consistent with the SM prediction $A^y_C = \left(1.15 \pm 0.06\right)\%$~\cite{Kuhn:2011ri}. The value of $A^y_C$ reported~\cite{ATLAS:2012an} by the ATLAS collaboration, $A^y_C = \left(-2.4 \pm 2.8\right)\%$, is consistent with the measurement of CMS.
\begin{comment}The spin-2 contribution to $A^{t\bar{t}}_{FB}$ must not spoil the agreement between the measured value of $\sigma_{t\bar{t}}$ at the LHC and the SM prediction.
\end{comment}
ATLAS recently measured~\cite{:2012kg} the top quark production cross section with $\sqrt{s} = 7$ TeV to be $\sigma_{t\bar{t}} = 176^{+17}_{-14}$ pb, which is consistent with the CMS observation $\sigma_{t\bar{t}} = \left(154 \pm 18\right)$ pb~\cite{Chatrchyan:2011yy}. A QCD prediction~\cite{Ahrens:2011mw} at approximate NNLO (using 1PI$_{\text{SCET}}$) yielded $\sigma_{t\bar{t}} = 155^{+11}_{-12}$ pb. The EWS correction factor~\cite{Manohar:2012rs}, $\mathcal{R}_t = 0.98$, is used to compute the full SM prediction, $\sigma_{t\bar{t}} = 152^{+11}_{-12}$ pb.
\begin{figure}
\centering
\includegraphics[width=0.7\textwidth]{graphics/asymC.pdf}
\caption{NP contributions to the inclusive charge asymmetry at the LHC and the forward-backward asymmetry at the Tevatron in the high mass bin. The pink band is the $1\sigma$ error of the measurement $A^{t\bar{t}}_{\text{high}}$, and the dashed line is the difference between the measured value of $A^{t\bar{t}}_{\text{high}}$ and the corresponding SM prediction. The blue and red curves are predictions of the spin-2 model for $M =100,\,200$ GeV respectively.}
\label{fig:asymC}
\end{figure}
Fig.~\ref{fig:asymC} shows $A^{y}_{C}$ at the LHC as a function of $A^{t\bar{t}}_{FB}$ at the Tevatron for the spin-2 model at hand. The pink band is the $1\sigma$ error of the measurement $A^{t\bar{t}}_{\text{high}}$, with the central value given by the dashed line. The blue and red curves are predictions of the spin-2 model for $M =100,\,200$ GeV respectively. These predictions are not within $1\sigma$ of both measurements simultaneously. This is a generic feature of any model that attempts to explain $A^{t\bar{t}}_{FB}$~\cite{AguilarSaavedra:2011hz,AguilarSaavedra:2011ug}.
If the only NP is a single spin-2 field, then the cutoff of the effective theory should be at least as large as the center-of-mass energy of the experiment. The most optimistic estimate of the cutoff is $\Lambda \approx 4\pi\bar{f}$. As argued in section~\ref{sec:spin2}, the cutoff is likely to be smaller than this estimate. Fixing $f$ to be 1 TeV, $g_{ut}$ should be less than $\mbox{\raisebox{-.6ex}{~$\stackrel{<}{\sim}$~}} 1.8$ if the effective theory is to be valid up to 7 TeV, as opposed to $g_{ut} \mbox{\raisebox{-.6ex}{~$\stackrel{<}{\sim}$~}} 6.4$ for $\sqrt{s} = 1.96$ TeV at the Tevatron. Adding heavier fields to the effective theory could raise the cutoff, as well as lead to a qualitatively different relation between the forward-backward and charge asymmetries measured at the Tevatron and LHC. Preliminary analysis has shown that interference between the virtual exchange of a lighter spin-2 particle and a heavier spin-2 state
could ameliorate the tension between the measurements of $A^{y}_{C}$ and $A^{t\bar{t}}_{FB}$.
However, a more comprehensive study of this effect is needed for quantitative predictions, and we leave this for a future study.
\begin{figure}
\centering
\includegraphics[width=0.7\textwidth]{graphics/chiC.pdf}
\caption{Fit to $A^{t\bar{t}}_{\text{high}}$ (CDF) and $\sigma_{t\bar{t}}$ (ATLAS). $f = 1$ TeV and $g^R_{ut}$ is a free parameter for a given $M$. The 1 and 2$\sigma$ confidence regions of allowed parameter space are shown in green and yellow respectively. The black line corresponds to $A^{t\bar{t}}_{\text{high}}= 47.5\%$. Experimentally disallowed parameter space due to constraints the width of the top quark is shown in brown.}
\label{fig:chiC}
\end{figure}
A global fit of the spin-2 model to the CDF measurement $A^{t\bar{t}}_{\text{high}}$ and the ATLAS measurement $\sigma_{t\bar{t}}$ was performed using the method of least squares. The scale $f$ was fixed to be 1 TeV and $g^R_{ut}$ was left as a free parameter for a given $M$. Again, $g_{ut}$ should be less than $\mbox{\raisebox{-.6ex}{~$\stackrel{<}{\sim}$~}} 1.8$ if effective theory is to be valid up to 7 TeV. The results of the fit are shown in Fig.~\ref{fig:chiC}. The 1 and 2$\sigma$ confidence regions of allowed parameter space are shown in green and yellow respectively. The black line corresponds to $A^{t\bar{t}}_{\text{high}}= 47.5\%$. This is not necessarily the best-fit value. Experimentally disallowed parameter space due to constraints the width of the top quark is shown in brown. As was the case with the fit to Tevatron observables, the spin-2 model is again able to hit the central value of the forward-backward asymmetry in the high mass bin in a large region of the parameter space.
There is an additional contribution to the $t\bar{t}$ production cross section at the LHC from single-top, spin-2 production where the spin-2 particle immediately decays into $u\bar{t}$. Again, we approximate the $2\rightarrow3$ cross section as $\sigma(u\,g \rightarrow t\,\bar{t}\,u) \approx \sigma(u\,g \rightarrow t\,h_{\mu\nu})\times Br(h_{\mu\nu} \rightarrow u\,\bar{t})$. We used the ATLAS collaboration's~\cite{:2012kg} cut, $|\eta| < 2.5$ when selecting muon and jet candidates associated with $t\bar{t}$ production, when we calculated the spin-2 contribution to this cross section. For $M = 200$ GeV and $f = 1$ TeV, requiring $\sigma_{t\bar{t}}$ to be within 1$\sigma$ of the measured value limits $|g_{ut}|$ to be less than 1.55 assuming $Br(h_{\mu\nu} \rightarrow u\bar{t}) = 1$. Alternatively, one may allow $|g_{ut}|$ to reach its maximum allowed value in the effective theory of approximately 1.8 by requiring $Br(h_{\mu\nu} \rightarrow u\bar{t}) \mbox{\raisebox{-.6ex}{~$\stackrel{<}{\sim}$~}} 0.9$ The contribution to $\sigma_{t\bar{t}}$ from this channel falls with the mass of the spin-2 particle such that theoretical considerations quickly become the dominant constraint on $|g_{ut}|$.
\subsection{Comments on Differential Measurements}
At the time this paper was submitted for publication, new measurements of the charge asymmetry at the LHC at the differential level were reported by the ATLAS~\cite{ATLAS:2012an} and CMS~\cite{CMS-PAS-TOP-11-030} collaborations. CMS also recently measured~\cite{CMS-PAS-TOP-11-013} the normalized, differential $t\bar{t}$ production cross section. By normalizing the differential cross section to the total cross section, certain systematic uncertainties and all normalization uncertainties cancel out, leading to a particularly precise measurement. In addition, the CDF collaboration updated~\cite{CDF-Note-10807} its analysis of the forward-backward asymmetry at the Tevatron to include the full Run II dataset. It is observed that $A^{t\bar{t}}_{FB}$ has an approximately linear dependence on both $M_{t\bar{t}}$ and $\Delta y$. In this section, we discuss the spin-2 model's predictions for these differential measurements.
Fig.~\ref{fig:update1} shows predictions of the spin-2 model for the forward-backward asymmetry at the Tevatron, the normalized differential top quark production cross section at the LHC, and the charge-asymmetry at the LHC. The blue, red, and green lines correspond to a spin-2 mass of $\{100,\,200,\,300\}$ GeV and a coupling $g_{ut}/f = \{0.85,\,1.38,1.84\}\,\text{TeV}^{-1}$. The dashed lines are the SM values for these observables. These calculations were made using \texttt{FeynRules}~\cite{Christensen:2008py} interfaced with \texttt{MadGraph 5}~\cite{Alwall:2011uj}.
\begin{figure}
\centering
\subfloat[$A^{t\bar{t}}_{FB}$ vs. $M_{t\bar{t}}$]{\label{fig:cdfAsymUpdate}\includegraphics[width=0.49\textwidth]{graphics/cdfAsymUpdate.png}}
\subfloat[$(d\sigma / dM_{t\bar{t}}) / \sigma$ vs. $M_{t\bar{t}}$]{\label{fig:cmsDiffXSection}\includegraphics[width=0.49\textwidth]{graphics/cmsDiffXSection.png}} \\
\subfloat[$A^{t\bar{t}}_{C}$ vs. $M_{t\bar{t}}$, ATLAS data]{\label{fig:atlasDiffAsym}\includegraphics[width=0.49\textwidth]{graphics/atlasDiffAsym.png}}
\subfloat[$A^{t\bar{t}}_{C}$ vs. $M_{t\bar{t}}$, CMS data]{\label{fig:cmsDiffAsym}\includegraphics[width=0.49\textwidth]{graphics/cmsDiffAsym.png}}
\caption{Predictions for forward-backward asymmetry at the Tevatron (upper left), normalized, differential cross section at the LHC (upper right), and charge asymmetry at the LHC (bottom). The blue, red, and green lines correspond to a spin-2 mass and coupling, $g_{ut}/f$, of (100 GeV, 0.85 TeV$^{-1}$), (200 GeV, 1.38 TeV$^{-1}$), and (300 GeV, 1.84 TeV$^{-1}$) respectively.}
\label{fig:update1}
\end{figure}
As was the case for the inclusive charge asymmetry, these predictions are not simultaneously within $1\sigma$ of both the differential charge asymmetry and the forward-backward asymmetry measurements. To the best of our knowledge, this is a generic feature of any model for $A^{t\bar{t}}_{FB}$ proposed before this paper was submitted for publication, see for example Refs.~\cite{AguilarSaavedra:2011hz,AguilarSaavedra:2011ug}. After this paper was submitted for publication, Drobnak \textit{et al}. discovered~\cite{Drobnak:2012cz} a class of models that can accommodate both measurements simultaneously. Based on the results of~\cite{Drobnak:2012cz}, a spin-2 field that is charged under certain representations of the SM gauge group may produce a large $A^{t\bar{t}}_{FB}$ and a negligible $A^{t\bar{t}}_{C}$. Verifying this claim is beyond the scope of this work, but it will be investigated in a future project.
The shape of the normalized, differential cross section predicted by the spin-2 model does not agree with the CMS measurement. However, as was the case for the differential cross section at the Tevatron, we have not taken into account any detector acceptance effects. As shown in Ref.~\cite{Grinstein:2011dz}, model-dependent acceptance effects can reduce the cross section by as much as a factor of $\sim 1/2$ in the high bins. No uncertainties from the choice of scale or PDFs have been included either. It is reasonable to assume that these unaccounted for effects could help to ameliorate some of the tension between the measured and predicted value of $(d\sigma/dM_{t\bar{t}})/\sigma$. Nevertheless, the shape of differential cross section constrains the spin-2 model's parameter space. The question becomes, how large of an asymmetry can generated once this constraint is taken into account.
\section{Conclusions}
\label{sec:con}
If it persists, the anomalously large top quark forward-backward asymmetry observed at the Tevatron is an indication of physics beyond the Standard Model. Since its appearance, many models involving new scalar, as well as vector particles around the weak scale have been proposed to address the anomaly. However, it has proven to be hard to raise the theoretical prediction to the central value of the CDF measurement in the high mass bin. We have shown that there is parameter space in this model, consistent with various experimental constraints that could accommodate the CDF measurement of $A_{FB}^{t\bar{t}}(M_{t\bar{t}}>450\text{GeV})$ of $47.5\%$. The peculiar derivative coupling of a spin-2 particle to fermions naturally leads to strong sensitivity of the asymmetry to the $t\bar t$ invariant mass. As a result, the picture of the top asymmetry increasing with energy observed by CDF naturally fits in this framework. If the observed $A_{FB}^{t\bar{t}}$ holds, it would be interesting to study the experimental bounds as well as the phenomenology of this model in more detail.
\begin{acknowledgements}
We thank Aneesh Manohar, Mike Trott, and David C. Stone for valuable discussions. This work has been supported in part by the U.S. Department of Energy under contract No. DOE-FG03-97ER40546.
\end{acknowledgements}
|
{
"timestamp": "2012-08-28T02:08:41",
"yymm": "1203",
"arxiv_id": "1203.2183",
"language": "en",
"url": "https://arxiv.org/abs/1203.2183"
}
|
\section{Introduction}
Higher order Fourier analysis is a notion which has many aspects and interpretations. The subject originates in a fundamental work by Gowers \cite{Gow},\cite{Gow2} in which he introduced a sequence of norms for functions on abelian groups and he used them to prove quantitative bounds for Szemer\'edi's theorem on arithmetic progressions \cite{Szem1} Since then many results were published towards a better understanding of the Gowers norms \cite{GrTao},\cite{GrTao2},\cite{GTZ},\cite{GowW},\cite{GowW2},\cite{GowW3},\cite{TZ},\cite{Sz1},\cite{Sz2},\cite{Sz3},\cite{Sz4},\cite{Sz5}
Common themes in all these works are the following four topics:
\medskip
\noindent{1.)~}{\it Inverse theorems for the Gowers norms.}
\noindent{2.)~}{\it Decompositions of functions into structured and random parts.}
\noindent{3.)~}{\it Counting structures in subsets and functions on abelian groups.}
\noindent{4.)~}{\it Connection to ergodic theory, nilmanifolds and nil sequences.}
\medskip
In the present paper we wish to contribute to all of these topics however we also put forward three other directions:
\medskip
\noindent{5.)~}{\it We develop an an algebraic interpretation of higher order Fourier analysis based on morphisms between structures that are generalizations of nilmanifolds.}
\noindent{6.)~}{\it We replace finite groups by arbitrary compact abelian groups.}
\noindent{7.)~}{\it We introduce limit objects for functions on abelian groups in the spirit of the graph limit theory.}
Important results on the fifth and sixth topics were also obtained by Host and Kra in the papers \cite{HKr2},\cite{HKr3}.
The paper \cite{HKr2} is the main motivation of \cite{NP} which is the corner stone of our approach.
\bigskip
\noindent{\bf Remark:}~~{\it Note that most the results in the present paper were obtained by the author in \cite{Sz1},\cite{Sz2},\cite{Sz3},\cite{Sz4}. However this paper together with \cite{NP} is a self contained account of the author's approach to higher order Fourier analysis. Many proofs are significantly different and more elementary than the discussion in the above four papers.
The material of \cite{Sz5} is not covered by this paper and it will be a part of another sequence of papers in the topic.}
\bigskip
To summarize the results in this paper we start with the definition of Gowers norms.
Let $f:A\rightarrow\mathbb{C}$ be a bounded measurable function on a compact abelian group $A$. Let $\Delta_t f$ be the function with $\Delta_tf(x)=f(x)\overline{f(x+t)}$.
With this notation
$$\|f\|_{U_k}=\Bigl(\int_{x,t_1,t_2,\dots,t_k\in A}\Delta_{t_1}\Delta_{t_2}\dots\Delta_{t_k}f(x)~d\mu^{k+1}\Bigr)^{2^{-k}}$$
where $\mu$ is the normalized Haar measure on $A$.
These norms satisfy the inequality $\|f\|_{U_k}\leq\|f\|_{U_{k+1}}$.
It is easy to verify that
\begin{equation}\label{u2norm}
\|f\|_{U_2}=\Bigl(\sum_{\chi\in\hat{A}}|\lambda_\chi|^4\Bigr)^{1/4}
\end{equation}
where $\lambda_\chi=(f,\chi)$ is the Fourier coefficient corresponding to the linear character $\chi$.
This formula explains the behaviour of the $U_2$ norm in terms of ordinary Fourier analysis.
However if $k\geq 3$, ordinary Fourier analysis does not seem to give a good understanding of the $U_k$ norm.
Small $U_2$ norm of a function $f$ with $|f|\leq 1$ is equivalent with the fact that $f$ is ``noise'' or ``quasi random'' from the ordinary Fourier analytic point of view. This means that all the Fourier coefficients have small absolute value. Such a noise however can have a higher order structure measured by one of the higher Gowers norms.
Isolating the structured part from the noise is a central topic in higher order Fourier analysis.
In $k$-th order Fourier analysis a function $f$ is considered to be quasi random if $\|f\|_{U_{k+1}}$ is small.
As we increase $k$, this notion of noise becomes stronger and stronger and so more and more functions are considered to be structured. Our goal is to describe the structures that arise this way.
The prototype of a decomposition theorem into structured and quasi random parts is Szemer\'edi's famous regularity lemma for graphs \cite{Szem2}.
The regularity lemma together with an appropriate counting lemma is a fundamental tool in combinatorics.
It is natural to expect that a similar regularization corresponding to the $U_{k+1}$ norm is helpful in additive combinatorics.
One can state the graph regularity lemma as a decomposition theorem for functions of the form $f:V\times V\rightarrow\mathbb{C}$ with $\|f\|\leq 1$.
Roughly speaking it says that $f=f_s+f_e+r_r$ where $f_r$ has small cut norm, $f_e$ has small $L^1$ norm and $f_s$ is of bounded complexity. (All the previous norms are normalized to give $1$ for the constant $1$ function.) We say that $f_r$ has complexity $m$ if there is a partition of $V$ into $m$ almost equal parts such that $f_s(x,y)$ depends only on the partition sets containing $x$ and $y$. This can also be formulated in a more algebraic way. A complexity $m$ function on $V\times V$ is the composition of $\phi:V\times V\rightarrow [m]\times [m]$ ({\bf algebraic part}) with another function $f:[m]\times [m]\rightarrow\mathbb{C}$ ({\bf analytic part}) where $[m]$ is the set of first $m$ natural numbers and $\phi$ preserves the product structure in the sense that $\psi=g\times g$ for some map $g:V\rightarrow [m]$.
In this language the requirement that the partition sets are of almost equal size translates to the condition that $\phi$ is close to be preserving the uniform measure.
Based on this one can expect that there is a regularity lemma corresponding to the $U_{k+1}$ norm of a similar form.
This means that a bounded (measurable) function $f$ on a finite (or more generally on a compact) abelian group is decomposable as
$f=f_s+f_e+f_r$ where $\|f_r\|_{U_{k+1}}$ is small, $\|f_e\|_1$ is small and $f_s$ can be obtained as the composition of $\phi:A\rightarrow N$ (algebraic part) and $g:N\rightarrow\mathbb{C}$ (analytic part) where $\phi$ is some kind of algebraic morphism preserving an appropriate structure on $A$.
The function $\phi$ would correspond to a regularity partition and $g$ would correspond to a function associating values with the partition sets.
The almost equality of the partition sets in Szemer\'edi's lemma should correspond to the fact that $\phi$ satisfies some approximative measure preserving property.
We show that this optimistic picture is almost exactly true with some new additional features.
Quite interestingly, if $k>1$, to formulate the regularity lemma for the $U_{k+1}$ norm we need to introduce new structures called $k$-step nilspaces. It turns out that $k$-step nilspaces are forming a category and the morphisms are suitable for the purpose of regularization.
Another interesting phenomenon is that geometry comes into the picture.
Topology and geometry does not play a direct role in stating the regularity lemma for graphs. (Note that a connection of Szemer\'edi's regularity lemma to topology was highlighted in \cite{LSz4}.) However in the abelian group case, even if we just regularize functions on finite abelian groups, compact geometric structures come up naturally as target spaces of the morphism $\phi$. To get a strong enough regularity lemma we will require that the values of the function $\phi:A\rightarrow N$ are so evenly distributed that we can basically say that $\phi$ (approximatly) reproduces the geometry of $N$ on the abelian group $A$. To guarantee that the composition $g\circ f$ respects this approximative geometry on $A$ we need to measure how much the function $g:N\rightarrow\mathbb{C}$ respects the geometry on $N$.
A possible way of doing it is to require that $g$ is continuous with bounded Lipschitz constant in some fixed metric on $N$. However the Lipschitz condition is not crucial in our approach. It can be replaced by almost any reasonable complexity notion. For example we can use an arbitrary ordering of an arbitrary countable $L^\infty$-dense set of continuous functions on $N$ and then we can require that $g$ is on this list with a bounded index.
\bigskip
To state our regularity lemma we will need the definition of nilspaces.
Nilspaces are common generalizations of abelian groups and nilmanifolds.
An abstract cube of dimension $n$ is the set $\{0,1\}^n$.
A cube of dimension $n$ in an abelian group $A$ is a function $f:\{0,1\}^n\rightarrow A$ which extends to an affine homomorphism (a homomorphism plus a translation) $f':\mathbb{Z}^n\rightarrow A$. Similarly, a morphism $\psi:\{0,1\}^n\rightarrow\{0,1\}^m$ between abstract cubes is a map which extends to an affine morphism from $\mathbb{Z}^n\rightarrow\mathbb{Z}^m$.
Roughly speaking, a nilspace is a structure in which cubes of every dimension are defined and they behave very similarly as cubes in abelian groups.
\begin{definition}[Nilspace axioms] A nilspace is a set $N$ and a collection $C^n(N)\subseteq N^{\{0,1\}^n}$ of functions (or cubes) of the form $f:\{0,1\}^n\rightarrow N$ such that the following axioms hold.
\begin{enumerate}
\item {\bf(Composition)} If $\psi:\{0,1\}^n\rightarrow\{0,1\}^m$ is a cube morphism and $f:\{0,1\}^m\rightarrow N$ is in $C^m(N)$ then the composition $\psi\circ f$ is in $C^n(N)$.
\item {\bf(Ergodictiry)} $C^1(N)=N^{\{0,1\}}$.
\item {\bf(Gluing)} If a map $f:\{0,1\}^n\setminus\{1^n\}\rightarrow N$ is in $C^{n-1}(N)$ restricted to each $n-1$ dimensional face containing $0^n$ then $f$ extends to the full cube as a map in $C^n(N)$.
\end{enumerate}
\end{definition}
\bigskip
If $N$ is a nilspace and in the third axiom the extension is unique for $n=k+1$ then we say that $N$ is a $k$-step nilspace.
If a space $N$ satisfies the first axiom (but the last two are not required) then we say that $N$ is a {\bf cubespace}.
A function $f:N_1\rightarrow N_2$ between two cubespaces is called a {\bf morphism} if $\phi\circ f$ is in $C^n(N_2)$ for every $n$ and function $\phi\in C^n(N_1)$.
The set of morphisms between $N_1$ and $N_2$ is denoted by $\mathrm{Hom}(N_1,N_2)$. With this notation $C^n(N)=\mathrm{Hom}(\{0,1\}^n,N)$.
We say that $N$ is a compact nilspace if $N$ has a compact, second countable, Hausdorff topology on it and $C^n(N)$ is a closed subset of $N^{\{0,1\}^n}$ for every $n$.
The nilspace axiom system is a variant of the Host-Kra axiom system for parallelepiped structures \cite{HKr2}. In \cite{HKr2} the two step case is analyzed and it is proved that the structures are tied to two nilpotent groups. A systematic analysis of $k$-step nilspaces (with a special emphasis on the compact case) was carried out in \cite{NP}. It will be important that the notion of Haar measure can be generalized for compact nilspaces. It was proved in \cite{NP} that compact nilspaces are inverse limits of finite dimensional ones and the connected components of a finite dimensional compact nilspace are nilmanifolds with cubes defined through a given filtration on the nilpotent group. It is crucial that a $k$-step compact nilspace $N$ can be built up using $k$ compact abelian groups $A_1,A_2,\dots,A_k$ as structure groups in a $k$-fold iterated abelian group bundle. The nilspace $N$ is finite dimensional if and only if all the structure groups are finite dimensional or equivalently: the dual groups $\hat{A_1},\hat{A_2},\dots,\hat{A_n}$ are all finitely generated.
It follows from the results in \cite{NP} that there are countably many finite dimensional $k$-step nilspaces up to isomorphism. An arbitrary ordering on them will be called a {\bf complexity notion}.
For every finite dimensional nilspace $N$ and natural number $n$ we fix a metrization of the weak convergence of probability measures on $C^n(N)$.
Let $M$ and $N$ be (at most) $k$-step compact nilspaces such that $N$ is finite dimensional. Let $\phi:M\rightarrow N$ be a continuous morphism and let us denote by $\phi_n:C^n(M)\rightarrow C^n(N)$ the map induced by $\phi$ using composition. The map $\phi$ is called $b$-balanced if the probability distribution of $\phi_n(x)$ for a random $x\in C^n(M)$ is at most $b$-far from the uniform distribution on $C^n(N)$ whenever $n\leq 1/b$.
Being well balanced expresses a very strong surjectivity property of morphisms which is for example useful in counting.
\begin{definition}[Nilspace-polynomials] Let $A$ be a compact abelian group. A function $f:A\rightarrow\mathbb{C}$ with $|f|\leq 1$ is called a $k$-degree, complexity $m$ and $b$-balanced nilspace-polynomial if
\begin{enumerate}
\item $f=\phi\circ g$ where $\phi:A\rightarrow N$ is a continuous morphism of $A$ into a finite dimensional compact nilspace $N$,
\item $N$ is of complexity at most $m$,
\item $\phi$ is $b$-balanced,
\item $g$ is continuous with Lipschitz constant $m$.
\end{enumerate}
\end{definition}
Note, that (as it will turn out) a nilspace-polynomial on a cyclic group is polynomial nilsequence with an extra periodicity property.
Now we are ready to state the decomposition theorem.
\begin{theorem}[Regularization]\label{reglem} Let $k$ be a fixed number and $F:\mathbb{R}^+\times\mathbb{N}\rightarrow\mathbb{R^+}$ be an arbitrary function. Then for every $\epsilon>0$ there is a number $n=n(\epsilon,F)$ such that for every measurable function $f:A\rightarrow\mathbb{C}$ on a compact abelian group $A$ with $|f|\leq 1$ there is a decomposition $f=f_s+f_e+f_r$ and number $m\leq n$ such that the following conditions hold.
\begin{enumerate}
\item $f_s$ is a degree $k$, complexity $m$ and $F(\epsilon,m)$-balanced nilspace-polynomial,
\item $\|f_e\|_1\leq\epsilon$,
\item $\|f_r\|_{U_{k+1}}\leq F(\epsilon,m)$~,~$|f_r|\leq 1$ and $|(f_r,f_s)|~,~|(f_r,f_e)|\leq F(\epsilon,m)$.
\end{enumerate}
\end{theorem}
\begin{remark} The Gowers norms can also be defined for functions on $k$-step compact nilspaces. It makes sense to generalize our results from abelian groups to nilspaces. Almost all the proofs are essentially the same. This shows that the (algebraic part) of $k$-th order Fourier analysis deals with continuous functions between $k$-step nilspaces.
\end{remark}
Note that various other conditions could be put on the list in theorem \ref{reglem}.
For example the proof shows that $f_s$ looks approximately like a projection of $f$ to a $\sigma$-algebra. This imposes strong restrictions on the value distribution of $f_s$ in terms of the value distribution of $f$.
Theorem \ref{reglem} implies inverse theorems for the Gowers norms.
It says that if $\|f\|_{U_{k+1}}$ is separated from $0$ then it correlates with a bounded complexity $k$-degree nilspace polynomial $\phi$.
(We can also require the function $\phi$ to be arbitrary well balanced in terms of its complexity but we omit this from the statement to keep it simple.)
\begin{theorem}[General inverse theorem for $U_{k+1}$]\label{invthem} Let us fix a natural number $k$. For every $\epsilon>0$ there is a number $n$ such that if $\|f\|_{U_{k+1}}\geq\epsilon$ for some measurable function $f:A\rightarrow\mathbb{C}$ on a compact abelian group $A$ with $|f|\leq 1$ then $(f,g)\geq\epsilon^{2^k}/2$ for some nilspace polynomial $g$ of degree $k$ and complexity at most $n$.
\end{theorem}
Note that this inverse theorem is exact in the sense that if $f$ correlates with a bounded complexity nilspace polynomial then its Gowers norm is separated from $0$.
\bigskip
A strengthening of the decomposition theorem \ref{reglem} and inverse theorem \ref{invthem} deals with the situation when the abelian groups are from special families.
For example we can restrict our attention to elementary abelian $p$-groups with a fixed prime $p$. Another interesting case is the set of cyclic groups or bounded rank abelian groups.
It also make sense to develop a theory for one particular infinite compact group like the circle group $\mathbb{R}/\mathbb{Z}$. (We will see in chapter \ref{chap:circle} that many features of higher order Fourier analysis become significantly simpler if we restrict it to the circle.)
It turns out that in restricted families of groups we get restrictions on the structure groups of the nilspaces that we have to use in our decomposition theorem.
To formulate these restrictions we need the next definition.
\begin{definition} Let $\mathfrak{A}$ be a family of compact abelian groups. We denote by $(\mathfrak{A})_k$ the set of finitely generated groups that arise as subgroups of $\hat{{\bf A}}_k$ where ${\bf A}$ is some ultra product of groups in $\mathfrak{A}$ and $\hat{{\bf A}}_k$ is the $k$-th order dual group of ${\bf A}$ in the sense of chapter \ref{chap:higherdual}.
\end{definition}
The following statements about $(\mathfrak{A})_k$ follow from lemma \ref{duexp} and lemma \ref{charzero}.
\begin{enumerate}
\item {\bf (Bounded exponent and characteristic $p$)}~If $\mathfrak{A}$ is the set of finite groups of exponent $n$ then $(\mathfrak{A})_k=\mathfrak{A}$ for every $k$. In particular if $n=p$ prime then $\mathfrak{A}$ and $(\mathfrak{A})_k$ are just the collection of finite dimensional vector spaces over the field with $p$ elements.
\item {\bf (Bounded rank)}~If $\mathfrak{A}$ is the set of finite abelian groups of rank at most $d$ then $(\mathfrak{A})_1$ is the collection of finitely generated abelian groups whose torsion part has rank at most $d$. If $k\geq 2$ then $(\mathfrak{A})_k$ contains only free abelian groups. The case $d=1$ is the case of cyclic groups.
\item {\bf (Characteristic $0$)}~If $\mathfrak{A}$ is a family of finite abelian groups in which for every natural number $n$ there are only finitely many groups with order divisible by $n$ then $(\mathfrak{A})_k$ contains only free abelian groups for every $k$.
\item {\bf (Tori)}~If $\mathfrak{A}$ contains only tori $(\mathbb{R}/\mathbb{Z})^n$ then $(\mathfrak{A})_k$ contains only free abelian groups for every $k$.
\end{enumerate}
\begin{definition} Let $\mathfrak{A}$ be a family of compact abelian groups. A $k$-step $\mathfrak{A}$-nilspace is a finite dimensional nilspace with structure groups $A_1,A_2,\dots,A_k$ such that $\hat{A_i}\in(\mathfrak{A})_i$ for every $1\leq i\leq k$.
A nilspace polynomial is called $\mathfrak{A}$-nilspace polynomial if the corresponding morphism goes into an $\mathfrak{A}$-nilspace.\end{definition}
Then we have the following.
\begin{theorem}[Regularization in special families]\label{restreg} Let $\mathfrak{A}$ be a set of compact abelian groups. Then Theorem \ref{reglem} restricted to functions on groups from $\mathfrak{A}$ is true with the stronger implication that the structured part $f_s$ is an $\mathfrak{A}$-nilspace polynomial.
\end{theorem}
\begin{theorem}[Specialized inverse theorem for $U_{k+1}$]\label{restinv} Let $\mathfrak{A}$ be a set of compact abelian groups. Then for functions on groups in $\mathfrak{A}$ theorem \ref{invthem} holds with $\mathfrak{A}$-nilspace polynomials.
\end{theorem}
This theorem shows in particular that if $\mathfrak{A}$ is the set of abelian groups in which the order of every element divides a fixed number $n$ (called groups of exponent $n$) then $\mathfrak{A}$-nilspaces (used in the regularization) are finite and all the structure groups have exponent $n$.
In the $0$ characteristic case $\mathfrak{A}$-nilspaces are $k$-step nilmanifold with a given filtration. This will help us to give a generalization of the Green-Tao-Ziegler theorem \cite{GTZ} for a multidimensional setting.
In the case of the circle group or more generally tori's, again we only get $k$-step nilmanifolds.
\bigskip
We highlight our results about counting and limit objects for function sequences.
Roughly speaking, counting deals with the density of given configurations in subsets or functions on compact abelian groups. We have two goals with counting. One is to show that our regularity lemma is well behaved with respect to counting and the second goal is
to show that function sequences in which the density of every fixed configuration converges have a nice limit object which is a measurable function on a nilspace. This fits well into the recently developed graph and hypergraph limit theories \cite{LSz1},\cite{BCLSV1},\cite{LSz3},\cite{LSz4},\cite{ESz}.
Counting in compact abelian groups has two different looking but equivalent interpretations. One is about evaluating certain integrals and the other is about the distribution of random samples from a function.
Let $f:A\rightarrow\mathbb{C}$ be a bounded function and the compact group $A$.
An integral of the form
$$\int_{x,y,z\in A} f(x+y)f(x+z)f(y+z)~d\mu^3$$
can be interpreted as the triangle density in the weighted graph $M_{x,y}=f(x+y)$.
Based on this connection, evaluating such integrals can be called counting in $f$.
Note that one might be interested in more complicated integrals like this:
$\int f(x+y+z)^5\overline{f(x+y)}~d\mu^3$
where conjugations and various powers appear.
It is clear that as long as the arguments are sums of different independent variables then all the above integrals can be obtained from knowing the seven dimensional distribution of
\begin{equation}\label{jointdis}
(f(x),f(y),f(z),f(x+y),f(x+z),f(y+z),f(x+y+z))\in\mathbb{C}^7
\end{equation}
where $x,y,z$ are randomly chosen elements form $A$ with respect to the Haar measure.
One can think of the above integrals as multi dimensional moments of the distribution in (\ref{jointdis}).
We will say that such a moment (or the integral itself) is simple if it does not contain higher powers. (We allow conjugation in simple moments.)
We will see that there is a slight, technical difference between dealing with simple moments and dealing with general moments.
Every moment can be represented as a colored (or weighted) hypergraph on the vertex set $\{1,2,\dots,n\}$ where $n$ is the number of variables and an edge $S\subseteq\{1,2,\dots,n\}$ represents the term $f(\sum_{i\in S}x_i)$ in the product.
The color of an edge tells the appropriate power and conjugation for the corresponding term.
The degree of a moment is the maximal size of an edge minus one in this hypergraph.
Let $\mathcal{M}$ denote the set of all simple moments and let $\mathcal{M}_k$ denote the collection of simple moments of degree at most $k$.
We will denote by $D_n(f)$ the joint distribution of $\{f(\sum_{i\in S} x_i)\}_{S\subset[n]}$ where $[n]=\{1,2,\dots,n\}$.
It is a crucial fact that all the moments and the distributions $D_n$ can also be evaluated for functions on compact nilspaces with a distinguished element $0$.
We call nilspaces with such an element ``rooted nilspaces''.
Let $N$ be a rooted nilspace. If we choose a random $n$-dimensional cube $c:\{0,1\}^n\rightarrow N$ in $C^n(N)$ with $f(0^n)=0$ then the joint distribution of the values $\{f(c(v))\}_{v\in\{0,1\}^n}$ gives the distribution $D_n(f)$.
We continue with an interesting example.
Let $C=\mathbb{R}/\mathbb{Z}$ the circle group and let $\chi(x)=e^{x2\pi i}$ defined on $C$. The dual group of $C$ is the cyclic group generated by the linear character $\chi$.
Let us consider the function sequence $f_n=\chi+\chi^n$ for $n\in\mathbb{N}$.
One can calculate that $\{f_n\}_{n=1}^\infty$ converges in the sense that for every $k\in\mathbb{N}$ and $M\in\mathcal{M}_k$ the values $\{M(f_i)\}_{i=1}^\infty$ converege. In fact for every $n$ the sequence $\{D_n(f_i)\}_{i=1}^\infty$ is a convergent sequence of distributions.
The natural question arises if there is a natural limit object for this function sequence. It turns out that there is no function on the circle which represents the limit however on the torus $C^2$ the function $f(x,y)=\chi(x)+\chi(y)$ has the property that $\lim_{i\rightarrow\infty}D_n(f_i)=D(f)$ holds for every $n$. It turns out that more complicated limit objects can arise. For example there are function sequences on the circle (or on finite abelian groups) which converge to functions on the Heisenberg nilmanifold.
Our next theorems provide limit objects for convergent function sequences.
\begin{theorem}[Limit object I.]\label{simplim} Assume that $\{f_i\}_{i=1}^\infty$ is a sequence of uniformly bounded measurable functions on the compact abelian groups $\{A_i\}_{i=1}^\infty$. Then if $\lim_{i\rightarrow\infty} M(f_i)$ exists for every $M\in\mathcal{M}$ then there is a measurable function (limit object) $g:N\rightarrow\mathbb{C}$ on a compact rooted nilspace $N$ such that $M(g)=\lim_{i\rightarrow\infty}M(f_i)$ for $M\in\mathcal{M}$.
\end{theorem}
\begin{corollary}[Limit object II.]\label{simplimcor} Let $k$ be a fixed natural number. Assume that $\{f_i\}_{i=1}^\infty$ is a sequence of uniformly bounded measurable functions on the compact abelian groups $\{A_i\}_{i=1}^\infty$. Then if $\lim_{i\rightarrow\infty} M(f_i)$ exists for every $M\in\mathcal{M}_k$ then there is a measurable function (limit object) $g:N\rightarrow\mathbb{C}$ on a compact $k$-step rooted nilspace $N$ such that $M(g)=\lim_{i\rightarrow\infty}M(f_i)$ for $M\in\mathcal{M}_k$.
\end{corollary}
Let $\mathcal{P}_r$ denote the space of Borel probability distributions supported on the set $\{x:|x|\leq r\}$ in $\mathbb{C}$.
\begin{theorem}[Limit object III.]\label{genlim} Assume that $\{f_i\}_{i=1}^\infty$ is a sequence of functions with $|f_i|\leq r$ on the compact abelian groups $\{A_i\}_{i=1}^\infty$. Then if $\lim_{i\rightarrow\infty} D_k(f_i)$ exists for every $k\in\mathbb{N}$ then there is a measurable function (limit object) $g:N\rightarrow\mathcal{P}_r$ on a compact rooted nilspace $N$ such that $D_k(g)=\lim_{i\rightarrow\infty}D_k(f_i)$ for $k\in\mathbb{N}$.
\end{theorem}
Let us observe that theorem \ref{genlim} implies the other two.
We devote the last part of the introduction to our main method and the simple to state theorem \ref{ultreg} on ultra product groups which implies almost everything in this paper.
Let $\{A_i\}_{i=1}^\infty$ be a sequence of compact abelian groups and let ${\bf A}$ be their ultra product.
Our strategy is to develop a theory for the Gowers norms on ${\bf A}$ and then by indirect arguments we translate it back to compact groups.
First of all note that $U_{k+1}$ is only a semi norm on ${\bf A}$. We prove in this paper that there is a unique maximal $\sigma$-algebra $\mathcal{F}_k$ on ${\bf A}$ such that $U_{k+1}$ is a norm on $L^\infty(\mathcal{F}_k)$ and $L^\infty(\mathcal{F}_k)$ is orthogonal to every function whose $U_{k+1}$ norm is zero.
It follows that every function $f\in L^\infty({\bf A})$ has a unique decomposition as $f_s+f_r$ where $\|f_r\|_{U_{k+1}}=0$ and $f_s$ is measurable in $\mathcal{F}_k$.
This shows that on the ultra product ${\bf A}$ it is simple to separate the structured part of $f$ from the random part.
The question remains how to describe the structured part in a meaningful way.
It turns out that to understand this we need to go beyond measure theory and use topology.
Note that the reason for this is not that the groups $A_i$ are already topological. Even if $\{A_i\}_{i=1}^\infty$ is a sequence of finite groups, topology will come into the picture in the same way.
We will make use of the fact that ${\bf A}$ has a natural $\sigma$-topology on it. A $\sigma$-topology is a weakening of ordinary topology where only countable unions of open sets are required to be open.
The structure of $\mathcal{F}_1$, which is tied to ordinary Fourier analysis, sheds light on how topology comes into the picture. It turns out that $\mathcal{F}_1$ can be characterized as the smallest $\sigma$-algebra in which all the continuous surjective homomorphisms $\phi:{\bf A}\rightarrow G$ are measurable where $G$ is a compact abelian group.
In other words the ordinary topological space $G$ appears as a factor of the $\sigma$-topology on ${\bf A}$.
The next theorem explains how nilspaces enter the whole topic:
\begin{theorem}[Characterization of $\mathcal{F}_{k+1}$.] The $\sigma$-algebra $\mathcal{F}_k$ is the smallest $\sigma$-algebra in which all continuous morphisms $\phi:{\bf A}\rightarrow N$ are measurable where $N$ is a compact $k$-step nilspace.
\end{theorem}
Another, stronger formulation of the previous theorem says that every separable $\sigma$-algebra in $\mathcal{F}_k$ is measurable in a $k$-step compact, Hausdorff nilspace factor of ${\bf A}$.
We will see later that we can also require a very strong measure preserving property for the nilspace factors $\phi:{\bf A}\rightarrow N$. This will also be crucial in the proofs.
As a corollary we have a very simple regularity lemma on the ultra product group ${\bf A}$.
\begin{theorem}[Ultra product regularity lemma]\label{ultreg} Let us fix a natural number $k$. Let $f\in L^\infty({\bf A})$ be a function. Then there is a unique (orthogonal) decomposition $f=f_s+f_r$ such that $\|f_r\|_{U_{k+1}}=0$ and $f_s$ is measurable in a $k$-step compact nilspace factor of ${\bf A}$.
\end{theorem}
\subsection{Nilmanifolds as nilspaces}\label{nilasnil}
In this chapter we outline the connection between nilmanifolds and nilspaces (for this topic see also \cite{NP}).
Let $F$ be a nilpotent Lie-group and let $F=F_0\geq F_1\geq F_2\geq...\geq F_k=\{1\}$ be a filtration in $F$ i.e. $[F,F_i]\leq F_{i+1}$ holds for every $0\leq i\leq k-1$. (Note that the existence of such a filtration implies that $F$ is at most $k$-nilpotent.)
Let us define the following cubic structure on $F$ which depends on the filtration $\{F_i\}_{i=1}^k$.
A map $f:\{0,1\}^n\rightarrow F$ is in $C^n(F)$ if it can be obtained from the constant $1$ map in a finite process where in each step we choose a natural number $1\leq i\leq k$ and an element $x\in N_i$ and then we multiply the value of $f$ on an $i+1$ co-dimensional face of $\{0,1\}^n$ by $x$.
It is not hard to show that $F$ together with this cubic structure is a $k$-step nilspace.
An alternative way of defining the same cubic structure gives the sets $C^n(N)$ directly through an equation system in $F^{2^n}$. For $n\in\mathbb{N}$ let $g_n:\{0,1\}^n\rightarrow\{1,2,\dots,2^n\}$ be the ordering such that $g_1(0)=1,g(1)=2$ and if $n>1$ then $g_n((v,0))=g_{n-1}(v)$ and $g_n((v,1))=2^n+1-g_{n-1}(v)$ where $v\in\{0,1\}^{n-1}$.
\begin{definition} Let $G$ be a group and $f:\{0,1\}^n\rightarrow G$ be some function. We say that $f$ satisfies the Gray code property if $$\prod_{i=1}^{2^n}f(g_n^{-1}(i))^{(-1)^i}=1.$$
\end{definition}
Let $C^n(F)$ be the collection of functions $f:\{0,1\}^n\rightarrow F$ such that if $d\leq k+1$ and $g:\{0,1\}^d\rightarrow \{0,1\}^n$ is a morphism then the map $f\circ g$ satisfies the Gray code property modulo $G_{d-1}$.
Note that it is enough to check the condition for morphisms $g:\{0,1\}^d\rightarrow\{0,1\}^n$ that are injective and the image is a $d$ dimensional face of $\{0,1\}^n$.
This definition of $C^n(F)$ shows that $C^n(F)\subset F^{\{0,1\}^n}$ is a closed set.
It will be crucial to describe morphisms from abelian groups (as nilspaces) into these nilspaces.
We will use the next two definitions by Leibman \cite{Lei},\cite{Lei2}.
\begin{definition}[Polynomial map between groups] A map $\phi$ of a group $G$ to a group $F$ is said to be polynomial of degree $k$ if it trivializes after $k+1$ consecutive applications of the operator $D_h,~h\in G$ defined by $D_h\phi(g)=\phi(g)^{-1}\phi(gh).$
\end{definition}
\begin{definition} Let $F$ be a $k$-nilpotent group with filtration $\mathcal{V}=\{F_i\}_{i=0}^k$ with $F=F_0$, $F_{i+1}\subseteq F_i$, $F_k=\{1\}$ and $[F_i,F]\subseteq F_{i+1}$ if $i<k$. A map $\phi:G\rightarrow F$ is a $\mathcal{V}$-polynomial if $\phi$ modulo $F_i$ is a polynomial of degree $i$.
\end{definition}
It is proved in \cite{Lei2} that $\mathcal{V}$ polynomials are closed under multiplication.
Let $A$ be an abelian group. The second definition of the nilspace structure on $F$ shows that nilspace morphisms from $A$ to $F$ are exactly the $\mathcal{V}$ polynomials.
\medskip
Let $\Gamma\leq F$ be a discrete co-compact subgroup in $F$.
We denote by $M$ the (left) coset space $\{g\Gamma\}_{g\in F}$. Manifolds of the form $M$ are called nil-manifolds.
Let $\pi:F\rightarrow M$ denote the projection $\pi(g)=g\Gamma$. We define the cubic structure on $M$ by $C^n(M)=\{\pi\circ c|c\in C^n(F)\}$. A simple calculation shows that $N$ together with this cubic structure is a $k$-step nilspace. However to guarantee that $C^n(M)\subset M^{2^n}$ is a closed set we also need that $F_i\cap\Gamma$ is co-compact in $F_i$ for every $0\leq i\leq k$. If this holds we will say that $\Gamma$ is co-compact in $\{F_i\}_{i=0}^k$.
Note that the structure groups of $M$ are the abelian groups $A_i=F_{i-1}\Gamma/F_i\Gamma$.
The next theorem is one of the main results in \cite{NP}.
\begin{theorem}\label{finitedimnil} Let $M$ be a compact $k$-step nilspace such that the structure groups $\{A_i\}_{i=1}^k$ are all finite dimensional tori. Then there is a nilpotent Lie-group $F$ with filtration $\mathcal{V}=\{F_i\}_{i=0}^k$ and a discrete subgrup $\Gamma\leq F$ which is co-compact in $\mathcal{V}$ such that $M$ is (topologically) isomorphic to the nilspace corresponding to $(\mathcal{V},\Gamma)$.
\end{theorem}
\subsection{A multidimensional generalization of the Green-Tao-Ziegler theorem.}\label{cyclic}
Our goal in this chapter is to relate nilspace polynomials on cyclic groups to Leibman type polynomials.
As a consequence we will obtain a new proof of the inverse theorem by Green, Tao and Ziegler for cyclic groups.
We will use the notation from chapter \ref{nilasnil}
\begin{lemma}\label{polyab} Let $A$ be an abelian group. Then the set of at most degree $k$ polynomials form $\mathbb{Z}^n$ to $A$ is generated by the functions of the form
\begin{equation}\label{abpoly}
f(x_1,x_2,\dots,x_n)=a\prod_{i=1}^n{{x_i}\choose{n_i}}
\end{equation}
where $a\in A$ and $\sum_{i=1}^n n_i\leq k$.
(We use additive notation here)
\end{lemma}
\begin{proof} We go by induction on $k$. The case $k=0$ is trivial. Assume that it is true for $k-1$. Let $g_1,g_2,\dots,g_n$ be the generators of $\mathbb{Z}^n$. If $\phi:\mathbb{Z}^n\rightarrow A$ is a polynomial map then $$\omega(y_1,y_2,\dots,y_k)=D_{y_1}D_{y_2}\dots D_{y_k}\phi$$ is a symmetric $k$-linear form on $\mathbb{Z}^n$. We claim that there is map $\phi'$ which is generated by the functions in (\ref{abpoly}) and whose $k$-linear form is equal to $\omega$. Let $f:\otimes^k(\mathbb{Z}^n)\rightarrow A$ be a homomorphism representing $\omega$. Then $f$ is generated by homomorphisms $h$ such that $h(g_{j_1}\otimes g_{j_2}\otimes\dots g_{j_k})=a$ for some indices $j_1,j_2,\dots,j_k$ (and any ordering of them) and take $0$ on any other tensor products of generators. It is enough to represent such an $h$ by a function of the form (\ref{abpoly}).
It is easy to see that if $n_i$ is the multiplicity of $i$ among the indices $\{j_r\}_{r=1}^k$ then (\ref{abpoly}) gives a polynomial whose multi linear form is represented by $h$.
The difference $\phi-\phi'$ has a trivial $k$-linear form which shows that it is a $k-1$ dimensional polynomial and then we use induction the generate $\phi-\phi'$.
\end{proof}
\begin{lemma}\label{felemelo} Let $\phi:\mathbb{Z}^n\rightarrow M$ be a morphism. Then there is a lift $\psi:\mathbb{Z}^n\rightarrow F$ such that $\psi$ is a $\mathcal{V}$-polynomial and $\psi$ composed with the projection $F\rightarrow M$ is equal to $\phi$.
\end{lemma}
\begin{proof} Using induction of $j$ we show the statement for maps whose image is in $F_{k-j}H$.
If $j=0$ then $\phi$ is a constant map and then the statement is trivial. Assume that we have the statement for $j-1$ and assume that the image of $\phi$ is in $F_{k-j}H$.
The cube preserving property of $\phi$ shows that $\phi$ composed with the factor map $F_{k-j}H\rightarrow F_{k-j}H/F_{k-j+1}H=A_{k-j}$ is a degree $k-j+1$ polynomial map $\phi_2$ of $\mathbb{Z}^n$ into the abelian group $A_{k-j}$.
We have from lemma \ref{polyab} that using multiplicative notation
\begin{equation}\label{polfor}
\phi_2(x_1,x_2,\dots,x_n)=\prod_{t=1}^m a_t^{f_t(x_1,x_2,\dots,x_n)}
\end{equation}
where $a_t\in A_{k-j}$ and $f_t$ is an integer valued polynomial of degree at most $k-j+1$ for every $t$. Let us choose elements $b_1,b_2,\dots,b_m$ in $F_{k-j}$ such that their images in $A_{k-j}$ are $a_1,a_2,\dots,a_m$.
Let us define the function $\alpha:\mathbb{Z}\rightarrow F$ given by the formula (\ref{polfor}) when $a_t$ is replaced by $b_t$. The map $\alpha$ is a $\mathcal{V}$-polynomial.
Since $\phi$ maps to the left cosets of
$H$ it makes sens to multiply $\phi$ by $\alpha^{-1}$ from the left. It is easy to see that the new map $\gamma=\alpha^{-1}\phi$ is a morphism of $\mathbb{Z}$ to $F_{k-j+1}H$ and thus by induction it can be lifted to a $\mathcal{V}$ polynomial $\delta$. Then we have that $\alpha\delta$ is a lift of $\phi$ to a polynomial map.
\end{proof}
\begin{corollary} If $A$ is a finite abelian group and $f:A\rightarrow M$ is a morphism then for every homomorphism $\beta:\mathbb{Z}^n\rightarrow A$ there is a degree $k$ polynomial map $\phi:\mathbb{Z}^n\rightarrow F$ such that $\phi$ composed with the factor map $F\rightarrow M$ is the same as $\beta$ composed with $f$.
\end{corollary}
\begin{definition}[$d$-dimensional polynomial nilsequence] Assume that $F$ is a connected $k$-nilpotent Lie group with filtration $\mathcal{V}$ and $\Gamma$ is a co-compact subgroup of $F$. Assume that $M$ is the left coset space of $\Gamma$ in $F$. Then a map $h:\mathbb{Z}^d\rightarrow\mathbb{C}$ is called a $d$-dimensional polynomial nilsequence (corresponding to $M$) if there is a polynomial map $\phi:\mathbb{Z}^d\rightarrow F$ of degree $k$ and a continuous Lipschitz function $g:M\rightarrow\mathbb{C}$ such that $h$ is the composition of $\phi$, the projection $F\rightarrow M$ and $g$.
The complexity of such a nilsequence is measured by the maximum of $c$ and the complexity of $N$.
\end{definition}
Let $\mathfrak{A}$ be a $0$-characteristic family of abelian groups. Then all the structure groups of $\mathfrak{A}$-nilspaces are tori. From theorem \ref{finitedimnil} and theorem \ref{restinv} we obtain the following consequence.
\begin{theorem}[polynomial nilsequence inverse theorem]\label{PNIT} Let $\mathfrak{A}$ be a $0$ characteristic family of finite abelian groups. Let us fix a natural number $k$. For every $\epsilon>0$ there is a number $n$ such that if $\|f\|_{U_{k+1}}\geq\epsilon$ for some measurable $f:A\rightarrow\mathbb{C}$ with $|f|\leq 1$ on $A\in\mathfrak{A}$ with $d$-generators $a_1,a_2,\dots,a_d$ then $(f,g)\geq\epsilon^{2^k}/2$ such that $g(n_1a_1+n_2a_2+\dots+n_da_d)=h(n_1,n_2,\dots,n_d)$ for some $d$-dimensional polynomial nilsequence $h$ of complexity at most $n$.
\end{theorem}
Note that the above theorem implies an interesting periodicity since the defining equation of $g$ is true for every $d$-tuple $n_1,n_2,\dots,n_d$ of integers.
Using theorem \ref{PNIT} we obtain the Green-Tao-Ziegler inverse theorems for functions $f:[N]\rightarrow\mathbb{C}$ with $|f|\leq 1$. Their point of view is that if we put the interval $[N]$ into a large enough cyclic group (say of size $m>N2^{k+1}$) then the normalized version of $\|f\|_{U_{k+1}}$ does not depend on the choice of $m$. The proper normalization is to divide with the $U_{k+1}$-norm of the characteristic function on $1_{[N]}$.
To use theorem \ref{PNIT} in this situation we need to make sure that $m$ is not too big and that it has only large prime divisors. This can be done by choosing a prime between $N2^{k+1}$ and $N2^{k+2}$.
Then we can apply theorem \ref{PNIT} for the family of cyclic groups of prime order which is clearly a $0$ characteristic family.
What we directly get is that $f$ correlates with a bounded complexity polynomial nil-sequence of degree $k$.
This seems to be weaker then the Green-Tao-Ziegler theorem because they obtain the correlation with a linear nil-sequence. However in the appendix of \cite{GTZ} it is pointed out that the two versions are equivalent.
Form theorem \ref{PNIT} we can also obtain a $d$-dimensional inverse theorem for functions of the form $f:[N]^d\rightarrow\mathbb{C}$ with $|f|\leq 1$. Here we use the family of $d$-th direct powers of cyclic groups with prime order.
\begin{theorem}[Multi dimensional inverse theorem] Let us fix two natural numbers $d,k>0$. Then for every $\epsilon>0$ there is a number $n$ such that for every function $f:[N]^d\rightarrow\mathbb{C}$ with $\|f\|_{U_{k+1}}\geq\epsilon$ there is a $d$-dimensional polynomial nil-sequence $h:\mathbb{Z}^d\rightarrow\mathbb{C}$ of complexity at most $n$ and degree $k$ such that $(f,h)\geq \epsilon^{2^k}/2$.
\end{theorem}
Note that $(f,h)$ is the scalar product normalized as as $(f,h)=N^{-d}\sum_{v\in [N]^d}(f(v)\overline{h(v)})$.
\subsection{An example involving the Heisenberg group}\label{heis}
In this chapter we discuss an example which highlights a difference between the nilseqence approach used in \cite{GTZ} and the nilspace-polynomial approach used in the present paper.
Let $e(x)=e^{x2\pi i}$. For an integer $1<t<m$ we introduce the function $f:\mathbb{Z}_m\rightarrow\mathbb{C}$ defined by $f(k)=\lambda^{k^2}$ where $\lambda=e(t/m^2)$ and $k=0,1,2,\dots,m-1$. Note that this function does not ``wrap around'' nicely like a more simple quadratic function of the form $k\mapsto\epsilon^{k^2}$ where $\epsilon$ is an $m$-th root of unity. This means that to define $f$ we need to choose explicit integers to represent the residue classes modulo $m$.
On the other hand it can be seen that $\|f\|_{U_3}$ is uniformly separated from $0$ so it has some quadratic structure.
In the nilsequence approach this function is not essentially different from the case of $k\mapsto\epsilon^{k^2}$.
However in our approach we are more sensitive about the periodicity issue since we want to establish $f$ through a very rigid algebraic morphism which uses the full group structure of $\mathbb{Z}_m$.
We will show that the quadratic structure of $f$ is tied to a nilspace morphism $\phi$ which maps $\mathbb{Z}_m$ into the Heisenberg nilmanifold.
The Heisenberg group $H$ is the group of three by three upper uni-triangular matrices with real entries.
Let $\Gamma\subset H$ be the set of integer matrices in $H$. It can be seen that $\Gamma$ is a co-compact subgroup. The left coset space $N=\{g\Gamma|g\in H\}$ of $\Gamma$ in $H$ is the Heisenberg nilmanifold.
Let $M\in H$ be the following matrix:
\[ \left( \begin{array}{ccc}
1 & 2t/m & t/m^2\\
0 & 1 & 1/m \\
0 & 0 & 1 \end{array} \right)\]
then
\[M^k= \left( \begin{array}{ccc}
1 & 2kt/m & k^2t/m^2\\
0 & 1 & k/m \\
0 & 0 & 1 \end{array} \right)\]
In particular $M^m$ is an integer matrix.
This implies that the map $\tau:k\rightarrow M^k\Gamma$ defines a periodic morphism from $\mathbb{Z}$ to $N$.
Since the period length is $m$, it defines a morphism $\phi:\mathbb{Z}_m\rightarrow N$.
Let $D$ be the set of elements in $H$ in which all entries are between $0$ and $1$. The set $D$ is a fundamental domain for $\Gamma$. We can define a function on $N$ by representing it on the fundamental domain.
Let $g:D\rightarrow\mathbb{C}$ be the function $A\rightarrow e(A_{1,3})$ where $A_{1,3}$ is the upper-right corner of the matrix $A$.
We compute $g(\tau(k))=g(M^k\Gamma)$ by multiplying $M^k$ back into the fundamental domain $D$.
Since $g(M^k\Gamma)$ is periodic we can assume that $0\leq k<m$.
Let us multiply $M^k$ from the right by
\[ \left( \begin{array}{ccc}
1 & -\lfloor 2kt/m\rfloor & -\lfloor k^2t/m^2\rfloor\\
0 & 1 & 0\\
0 & 0 & 1 \end{array} \right)\]
We get
\[ \left( \begin{array}{ccc}
1 & \{ 2kt/m\} & \{k^2t/m^2\}\\
0 & 1 & k/m\\
0 & 0 & 1 \end{array} \right)\in D\]
So the value of $g$ on $\tau(k)$ is $e(\{k^2t/m^2\})=e(k^2t/m^2)$.
\subsection{Higher order Fourier analysis on the cirlce}\label{chap:circle}
In this chapter we sketch a consequence of our results when specialized to the circle grouop $C=\mathbb{R}/\mathbb{Z}$.
Since the circle falls in to the $0$-characteristic case, theorem \ref{finitedimnil} shows that higher order Fourier analysis on the circle deals with continuous morphisms from $C$ to nilspaces that arise from nilmanifolds. We show that such morphisms arise from one parameter subgroups in nilpotent Lie-groups which periodically intersect a co-compact subgroup. It follows that Gowers norms have rather aesthetical inverse theorems on the circle.
Let us use the notation from chapter \ref{nilasnil}. We denote by $M$ the nilspace on $F/\Gamma$ corresponding to the filtration $\mathcal{V}$. Recall that $\pi:F\rightarrow M$ is the natural projection. We prove the following theorem.
\begin{theorem}\label{circhom} If $\phi:C\rightarrow M$ is a continuous nilspace morphism with $\phi(0)=\pi(1)$. Then there is a group homomorphism $f:\mathbb{R}\rightarrow F$ such that $f(1)\in\Gamma$ and $\phi(x\mathbb{Z})=f(x)\Gamma$.
\end{theorem}
Note that theorem \ref{circhom} implies that contintinuous morphisms from $C$ to $M$ (which are normalized in the way that $\phi(0)=\pi(1)$) are in a one to one correspondence with the elments of $\Gamma$. If $g\in\Gamma$ then basic Lie-group theory shows that there is a unique one parameter subgroup $f:\mathbb{R}\rightarrow F$ with $f(1)=g$.
A surprising consequence of theorem \ref{circhom} is that filtrations and polynomial maps become irrelevant for the description of morphisms of the circle. They can be characterized through classical group homomorphisms.
\begin{lemma}\label{lin} Let $f:C\rightarrow C$ be a continuous polynomial map. Then $f$ is linear. (Consequently we obtain that any continuous polynomial map $f:C\rightarrow C^n$ is linear.)
\end{lemma}
\begin{proof} In this proof we will think of $C$ as the complex unit circle. Assume by contradiction that $f$ is not linear. Then by repeatedly applying operators $D_h$ to $f$ we can get a non-linear quadratic function.
This means that it is enough to get a contradiction if $f$ is quadratic.
In this case $D_h(f)$ is a linear map that depends continuously on $h\in C$ in the $L_2$ norm. On the other hand $D_h(f)$ is a linear character times a complex number from the unit circle. The characters are orthogonal to each other and so the character corresponding to $D_h(f)$ has to be the same for every $h$. This is only possible if it is the trivial character but then $f$ is linear.
\end{proof}
\medskip
\noindent{\it Proof of theorem \ref{circhom}}~~We go by induction on $k$. If $k=0$ then the statement is trivial.
Assume that we have the statement for $k-1\geq 0$. By factoring out with the central subgroup $F_{k-1}\cap\Gamma$ we can assume that $F_{k-1}\cap\Gamma$ is trivial. Thus $F_{k-1}$ is a torus. Let $M_{k-1}$ denote the nilspace $(N/F_{k-1})/\Gamma$ and let $\varrho:M\rightarrow M_{k-1}$ denote the natural projection. Elements of $M_{k-1}$ are orbits of the action of $F_{k-1}$ on $M$. Since $\varrho$ is a continuous nilspace morphism we have that $\varrho\circ\phi$ is a continuous morphism. By our induction hypothesis there is a one parameter subgroup $f_1:\mathbb{R}\rightarrow F/F_{k-1}$ representing $\varrho\circ\phi$. Assume that $f_1(1)=gF_{k-1}$ for some $g\in\Gamma$. Let $f_2:\mathbb{R}\rightarrow F$ denote the unique one parameter subgroup with $f_2(1)=g$. We have that $f_2(x)F_{k-1}=f_1(x)$ holds for every $x\in\mathbb{R}$. We have that $\pi(f_2(x))$ is in the same $F_{k-1}$ orbit as $\phi(x)$. Let $h:\mathbb{R}\rightarrow F_{k-1}$ denote the unique function with $\phi(x)=h(x)\pi(f_2(x))$. It is easy to see from basic nilspace theory (see theorem \ref{bundec}) that $h$ is a polynomial map so by lemma \ref{lin} it is a homomorphism. From $\pi(f_2(1))=\pi(1)$ and $\phi(1)=\pi(1)$ we have that $h(1)=0$. Since $F_{k-1}$ is in the center of $F$ we have that $f(x)=\pi(x)f_2(x)$ is a homomorphism which satisfies the required conditions.
\medskip
We are ready to describe functions $f_s:C\rightarrow\mathbb{C}$ that are "structured" with respect to the $U_{k+1}$ norm.
Let $F$ be a $k$-nipontent Lie-group with a co-compact subgroup $\Gamma$ and let $h:F/\Gamma\rightarrow\mathbb{C}$ be a continuous function with $|h|\leq 1$. Let $g\in\Gamma$ be a fixed element and let $\tau:\mathbb{R}\rightarrow F$ be the unique one parameter subgroup with $\tau(1)=g$. Then we obtain a continuous function $f_s=h\circ\pi\circ\tau$ on $C=\mathbb{R}/\mathbb{Z}$.
Note that $f_s$ can be regarded as a continuous periodic nil-sequence.
Roughly speaking, if the complexity of $F/\Gamma$ and the Lipschitz constant of $h$ are both bounded then $f_s$ is a structured function in $k$-th order Fourier analysis. Theorem \ref{restinv} shows that if $\|f\|_{U_{k+1}}$ is separated from $0$ then $f$ correlates with such a structured function $f_s$.
Using this we also get an ``interval'' version of the Green-Tao-Ziegler theorem for the $U_{k+1}([0,1])$ norm.
Functions on the interval $[0,1]$ can be represented in a large enough Cyclic group say $\mathbb{R}/2^{k+1}\mathbb{Z}$.
We obtain that if $f:[0,1]\rightarrow\mathbb{C}$ is a measurable function with $|f|\leq 1$ and $\|f\|_{U_{k+1}}$ is separated from $0$ then $f$ correlates with a continuous bounded complexity nilsequence.
\section{Compact abelian groups, Gowers norms and nilspaces}
\subsection{$\sigma$-algebras of probability spaces}
Let $(\Omega,\mathcal{A},\mu)$ be a probability space. We will use the standard notation for normed spaces obtained from $\Omega$. For $L^p(\Omega,\mathcal{A},\mu)$ we use the short hand notation $L^p(\Omega)$ or $L^p(\mathcal{A})$. Since we never consider two different probability measures on the same $\sigma$-algebra the meaning of these abbreviations is always clear from the context. We denote by $L^p_u$ the unit balls in these normed spaces. In this paper we only use the values $p=2,\infty$.
We will often work with sub $\sigma$-algebras of $\mathcal{A}$ in a fixed probability space $(\Omega,\mathcal{A},\mu)$. For two sub $\sigma$-algebras $\mathcal{B},\mathcal{C}$ in $\mathcal{A}$ we denote by $\mathcal{B}\vee\mathcal{C}$ the $\sigma$-algebra generated by $\mathcal{B}$ and $\mathcal{C}$. The expression $\mathcal{B}\wedge\mathcal{C}$ denotes the intersection of $\mathcal{B}$ and $\mathcal{C}$. According to our definition a set $S$ is in $\mathcal{B}\wedge\mathcal{C}$ if there are measurable sets $B\in\mathcal{B}$ and $C\in\mathcal{C}$ such that ${\bf \mu}(S\triangle A)={\bf \mu}(S\triangle B)=0$.
If $\mathcal{B}$ is a sub $\sigma$-algebra in $\mathcal{A}$ and $f:\Omega_2\rightarrow \Omega$ is a measure preserving map for some probability space $\Omega_2$ then we denote by $\mathcal{B}\circ f$ the $\sigma$-algebra $\{f^{-1}(S)|S\in\mathcal{B}\}$.
Two sub $\sigma$-algebras $\mathcal{B}$ and $\mathcal{C}$ in $\mathcal{A}$ are called {\bf conditionally independent} if $\mathbb{E}(\mathbb{E}(f|\mathcal{C})|\mathcal{B})=\mathbb{E}(\mathbb{E}(f|\mathcal{B})|\mathcal{C})=\mathbb{E}(f|\mathcal{B}\wedge\mathcal{C})$ holds for an arbitrary bounded measurable function $f$. To prove that $\mathcal{B}$ and $\mathcal{C}$ are conditionally independent it is enough to check that $\mathbb{E}(f|\mathcal{C})=0$ whenever $f$ is measurable in $\mathcal{B}$ and $\mathbb{E}(f|\mathcal{B}\wedge\mathcal{C})=0$.
\begin{definition} Let $\{\mathcal{B}_i\}_{i=1}^n$ be a collection of sub $\sigma$-algebras in $\mathcal{A}$. Then we denote by $\mathcal{R}(\{\mathcal{B}_i\}_{i=1}^n)$ the set of functions of the form $f=\prod_{i=1}^nf_i$ where $f_i\in L^\infty_u(\mathcal{B}_i)$.
\end{definition}
We will use the following classical fact from measure theory.
\begin{lemma}\label{siggen} Let $\{\mathcal{B}_i\}_{i=1}^n$ be a collection of sub $\sigma$-algebras in $\mathcal{A}$ and let $\mathcal{B}=\bigvee_{i=1}^n\mathcal{B}_i$ be the $\sigma$-algebra generated by them. Then every function in $L^2(\mathcal{B})$ can be approximated with an arbitrary precision in $L^2$ by a finite linear combination of functions in $\mathcal{R}=\mathcal{R}(\{\mathcal{B}_i\}_{i=1}^\infty)$.
\end{lemma}
A $\sigma$-algebra $\mathcal{B}\subseteq\mathcal{A}$ is {\bf separable} if there is a countable subset $S\subset\mathcal{B}$ such that for every $\epsilon>0$ and $H\in\mathcal{B}$ there is a set $T\in S$ such that $\mu(H\triangle T)\leq\epsilon$. Every $\sigma$-algebra which is generated by countable many sets is separable.
We will use the following basic fact.
\begin{lemma}\label{sepsiggen} Let $\{\mathcal{B}_i\}_{i=1}^n$ be a collection of sub $\sigma$-algebras in $\mathcal{A}$ and let $\mathcal{C}\subseteq\bigvee_{i=1}^n\mathcal{B}_i$ be a separable $\sigma$-algebra. Then there are separable $\sigma$-algebras $\mathcal{B}'_i\subseteq\mathcal{B}$ for $1\leq i\leq n$ such that $\mathcal{C}\subseteq\bigvee_{i=1}^n\mathcal{B}'_i$.
\end{lemma}
We will need the following lemma on conditional independence.
\begin{lemma}\label{siggen2} Let $\mathcal{B}$ and $\mathcal{C}$ be two conditionally independent $\sigma$-algebras and let $\mathcal{B}_1$ be a sub $\sigma$-algebra of $\mathcal{B}$.
Then $(\mathcal{C}\vee\mathcal{B}_1)\wedge\mathcal{B}=(\mathcal{C}\wedge\mathcal{B})\vee\mathcal{B}_1$.
\end{lemma}
\begin{proof}
It is trivial that $(\mathcal{C}\vee\mathcal{B}_1)\wedge\mathcal{B}\supseteq(\mathcal{C}\wedge\mathcal{B})\vee\mathcal{B}_1$.
To see the other containment let $f\in L^\infty((\mathcal{C}\vee\mathcal{B}_1)\wedge\mathcal{B})$.
Using that $f\in L^\infty(\mathcal{C}\vee\mathcal{B}_1)$ we have by lemma \ref{siggen} that for an arbitrary small $\epsilon>0$ there is an approximation of $f$ in $L_2$ of the form $f'=\sum_{i=1}^n c_ib_i$
where $c_i\in L^\infty(\mathcal{C})$ and $b_i\in L^\infty(\mathcal{B}_1)$. Since $\|f-f'\|_2\leq\epsilon$ and $f\in L^\infty(\mathcal{B})$ we have that $$\epsilon\geq\|\mathbb{E}(f|\mathcal{B})-\mathbb{E}(f'|\mathcal{B})\|_2=\|f-\sum_{i=1}^n \mathbb{E}(c_i|\mathcal{B})b_i\|_2.$$
Using conditional independence we have that $E(c_i|\mathcal{B})$ is measurable in $\mathcal{B}\wedge\mathcal{C}$ for every $i$ and the whole sum is measurable in $(\mathcal{C}\wedge\mathcal{B})\vee\mathcal{B}_1$.
Using it for every $\epsilon$ the proof is complete.
\end{proof}
\subsection{Couplings of probability spaces}
Let $I$ be a finite index set and $\mathcal{U}=\{(\Omega_i,\mathfrak{S}_i,\mu_i)\}_{i\in I}$ be a system of probability spaces. If all the probability spaces in $\mathcal{U}$ are separable the we say that $\mathcal{U}$ is separable.
A {\bf coupling } of $\mathcal{U}$ is a probability space $(\Omega,\mathfrak{S},\mu)$ together with measure preserving transformations $\{\psi_i:\Omega\rightarrow\Omega_i\}_{i\in I}.$ This means that for every $i\in I$ and set $S\in\mathfrak{S}_i$ we have $\mu(\psi_i^{-1}(S))=\mu_i(S)$.
\begin{definition}\label{coupeq} Let $\{\psi_i:\Omega\rightarrow\Omega_i\}_{i\in I}$ and $\{\psi_i^*:\Omega^*\rightarrow\Omega_i\}_{i\in I}$ be two couplings of $\mathcal{U}$ on the spaces $(\Omega,\mathfrak{S},\mu)$ and $(\Omega^*,\mathfrak{S}^*,\mu^*)$.
We say that these two couplings are {\bf equivalent} if for every system of sets $\{S_i\in\mathfrak{S}_i\}_{i\in I}$ we have
\begin{equation}\label{cupconst}
\mu\Bigl(\bigcap_{i\in I}\psi_i^{-1}(S_i)\Bigr)=\mu^*\Bigl(\bigcap_{i\in I}{\psi_i^*}^{-1}(S_i)\Bigr).
\end{equation}
\end{definition}
\begin{remark} It is not hard to show that in the previous definition the equivalence of the two couplings imply that if $F$ is an arbitrary $m$ variable set formula (using intersection, union and complement) , $\{M_j\in\mathfrak{S}_{a_j}\}_{j=1}^m$ is a system a events, $M$ is the value of $F$ on $\{\psi_{a_i}^{-1}(M_i)\}_{i=1}^m$ and $M^*$ is the value of $F$ on $\{{\psi^*_{a_i}}^{-1}(M_i)\}_{i=1}^m$ then $\mu(M)=\mu^*(M^*)$.
In other words the two couplings are equivalent if they can't be distinguished using probabilities of events formulated with events in $\mathcal{U}$ in a fixed way.
\end{remark}
\begin{definition} Let $\prod_{i\in I}\mathfrak{S}_i$ denote the set of vectors $(S_i)_{i\in I}$ where $S_i\in\mathfrak{S}_i$. Let
\begin{equation}\label{couprep}
f:\prod_{i\in I}\mathfrak{S}_i\rightarrow [0,1]
\end{equation}
be a function. We say that $f$ represents the (equivalence class) of a coupling $\Psi=\{\psi_i:\Omega\rightarrow\Omega_i\}_{i\in I}$ if for $S=(S_i)_{i\in I}$ the value of $f(S)$ is equal to the left hand side of (\ref{cupconst}).
\end{definition}
The next lemma gives a simple answers to the following question: {\it For which functions $f$ is there a coupling $\Psi$ such that $f$ represents $\Psi$?}
\begin{lemma}\label{couprepl} A function $f$ of the form (\ref{couprep}) represents some coupling of $\mathcal{U}$ if and only if the following two conditions hold.
\begin{enumerate}
\item For every $j\in I$ and $A\in\mathfrak{S}_j$ if $S=(S_i)_{i\in I}$ denotes the vector with $S_i=\Omega_i$ for $i\neq j$ and $S_j=A$ then $f(S)=\mu_j(A)$.
\item $f$ is additive in every coordinate. This means that if in $S=(S_i)_{i\in I}$ the set $S_j$ is the disjoint union of $A$ and $B$ then $f(S)=f(T)+S(U)$ where $T$ (resp. $U$) is obtained from $S$ by replacing $S_j$ with $A$ (resp. $B$).
\end{enumerate}
\end{lemma}
Let ${\rm coup}(\mathcal{U})$ denote the set of equivalence classes of the possible couplings of the system $\mathcal{U}$.
Lemma \ref{couprepl} says that elements of ${\rm coup}(\mathcal{U})$ are in a one to one correspondence with functions of the form (\ref{couprep}) which satisfy the two algebraic conditions of the lemma.
We list a few basic concepts related to couplings.
\bigskip
\noindent{\bf Topology:}~We say that $\{\Psi_i\}_{i=1}^\infty$ is a convergent sequence in ${\rm coup}(\mathcal{U})$ if the representing functions $\{f_i\}_{i=1}^\infty$ converge for every fixed element $S\in\prod_{i\in I}\mathfrak{S}_i$. It is clear by lemma \ref{couprepl} that the pointwise limit of $\{f_i\}_{i=1}^\infty$ also represents a coupling. If the coupling is separable the we obtain a compact Hausdorff topological structure on ${\rm coup}(\mathcal{U})$.
\bigskip
\noindent{\bf Convexity:}~Let $\nu$ be any Borel probability measure on ${\rm coup}(\mathcal{U})$. Then the function $f$ defined by
$$f(S)=\int_{C\in {\rm coup}(\mathcal{U})} f_C(S)~d\nu$$ represents a coupling where $f_C$ is the function representing $C$. This shows that ${\rm coup}(\mathcal{U})$ is a convex set in the topological sense.
\bigskip
\noindent{\bf Complete dependence:}~We say that the coupling $\{\psi_i:\Omega\rightarrow\Omega_i\}_{i\in I}$ is completely dependent if for every $j\in I$ we have that $\mathfrak{S}_j\circ\psi_j=\vee_{i\neq j}\mathfrak{S}_i\circ\psi_i$. It is easy to see that this property depends only on the equivalence class of the coupling.
\bigskip
\noindent{\bf self-couplings and their sub-couplings:~}~Throughout this paper we will mostly study couplings when the probability spaces in $\mathcal{U}$ are all identical to a fixed space $(X,\mathcal{A},\nu)$. A coupling of such a system $\mathcal{U}$ is a system of measure preserving maps $\{\psi_i:\Omega\rightarrow X\}_{i\in I}$. Let us denote by ${\rm coup}(X,I)$ the set of self-couplings of $I$ copies of $X$. If $\phi:J\rightarrow I$ is a map then it induces a continuous map $\hat{\phi}:{\rm coup}(X,I)\rightarrow{\rm coup}(X,J)$ by $\hat{\phi}(\{\psi_i\}_{i\in I})=\{\psi_{\phi(j)}\}_{j\in J}$. If $\phi$ is injective and
$C\in{\rm coup}(X,I)$ then we call $\hat{\phi}(C)$ the sub-couplig of $C$ corresponding to $\phi$.
\bigskip
\noindent{\bf The multi linear form $\xi$:}~Let $G=\{g_i:\Omega_i\rightarrow\mathbb{C}\}_{i\in I}$ be a system of bounded measurable functions. Let $C=\{\psi_i:\Omega\rightarrow\Omega_i\}_{i\in I}$ be a coupling of $\mathcal{U}$.
Then $\xi$ is defined by
\begin{equation}\label{cupconst2}
\xi(C,G):=\mathbb{E}_{x\in\Omega}\Bigl(\prod_{i\in I} g_i(\psi_i(x))\Bigr).
\end{equation}
Notice that if $g_i$ is the characteristic function of a set $S_i\in\mathfrak{S}_i$ for every $i$ then the value in (\ref{cupconst2}) is equal to the value (on the left hand side) in (\ref{cupconst}) . This shows that $\xi$ determines the equivalence class of the coupling $C$.
\bigskip
\noindent{\bf Factor coupling and independence over a factor:}~Let $\mathfrak{S}_i'\subset\mathfrak{S}_i$ be a sub $\sigma$-algebras for $i\in I$. The coupling $\{\psi_i:\Omega\rightarrow\Omega_i\}_{i\in I}$ can be restricted to a coupling of the probability spaces $\{\Omega_i,\mathfrak{S}'_i,\mu_i\}_{i\in I}$. This will be called a factor coupling of the original one. We will say that the original coupling is independet over the factor given by $\{\mathfrak{S}_i'\}_{i\in I}$ if for every $j\in I$ and system of bounded measurable functions $G=\{g_i:\Omega_i\rightarrow\mathbb{C}\}_{i\in I}$ with $\mathbb{E}(g_j|\mathfrak{S}_j')=0$ we have $\xi(C,G)=0$. Equivalently, if $G=\{g_i:\Omega_i\rightarrow\mathbb{C}\}_{i\in I}$ is a system of bounded measurable functions and $G'=\{\mathbb{E}(g_i|\mathfrak{S}_i')\}_{i\in I}$ then $\xi(C,G)=\xi(C,G')$. This shows that if a coupling $C$ is independent over a certain factor $C'$, then the multi linear form $\xi(C,G)$ and thus $C$ itself is uniquely determined by $C'$.
\bigskip
The basic properties of $\xi$ are summarized in the next lemma.
\begin{lemma}\label{xiprop} The function $\xi$ satisfies the following properties.
\begin{enumerate}
\item For every $i\in I$ we have $\xi(C,G)\leq\|g_i\|_1\prod_{j\neq i}\|g_j\|_\infty$,
\item $\xi(G,C)$ is linear in each component $g_i$,
\item If $F=\{f_i:\Omega_i\rightarrow\mathbb{C}\}_{i\in I}$ is a system of measurable functions then
\begin{equation}\label{smallchange}
|\xi(G,C)-\xi(F,C)|\leq \Bigl(\sum_{i\in I}\|f_i-g_i\|_1\Bigr)\prod_{i=1}^n\max(\|f_i\|_\infty,\|g_i\|_\infty).
\end{equation}
\item for a fixed $G$ the value $\xi(C,G)$ depends only on the equivalence class of the coupling $C$,
\item for a fixed $G$ the function $C\rightarrow\xi(C,G)$ is continuous on ${\rm coup}(\mathcal{U})$.
\item If for every $i$ the space $\Omega_i$ is a compact, Hausdorff space and $\mathfrak{S}_i$ is the Borel $\sigma$-algebra then the functions of the form $C\rightarrow\xi(C,G)$ where $G$ is a system of continuous functions generate the topology on ${\rm coup}(\mathcal{U})$.
\end{enumerate}
\end{lemma}
\begin{proof} The first and second properties are trivial from the definition.
The third property follows from the first two by replacing each $g_i$ by $f_i$ in $n$ consecutive steps.
To see the fourth property observe that by the second property, if in $F=\{f_i:\Omega_i\rightarrow\mathbb{C}\}_{i\in I}$ every function is a step function then the function $\xi(F,C)$ is a linear combination of numbers of the form (\ref{cupconst}) and so $\xi(F,C)$ depends only on the equivalence class of $C$. Furthermore the function $\tau:C\rightarrow\xi(F,C)$ is a linear combination of continuous functions and so it is continuous. In the general case we can approximate every $g_i$ be a step function $f_i$ with $\|f_i\|_\infty\leq\|g_i\|_\infty$ such that $\|f_i-g_i\|_1\leq\epsilon$. Let $C'$ be an equivalent coupling with $C$. Then by (\ref{smallchange}) we have that both $|\xi(G,C)-\xi(F,C)|$ and $|\xi(G,C')-\xi(F,C')|$ are at most $|I|\epsilon\prod_{i\in I}\|g_i\|_\infty$. Since this is true for every $\epsilon>0$ and $\xi(F,C)=\xi(F,C')$ we have that $\xi(G,C)=\xi(G,C')$. The same argument shows that the function $\psi:C\rightarrow\xi(G,C)$ can be arbitrarily well approxiamted in $L^\infty$ by a continuous function $\tau$ and so $\psi$ is continuous.
The last property follows from inequality (\ref{smallchange}) and from the fact that every $\{0,1\}$ valued measurable function on $\Omega_i$ can be approximated with an arbitrary precision in $L^1$ by a continuous function of absolute value at most $1$.
\end{proof}
\subsection{Abelian groups, cubes and cubic couplings}\label{cubes}
Let $A$ be an abelian group. If $A$ is compact, Hausdorff and second countable then we use the short hand notion {\bf compact} for $A$. Compact abelian groups admit a unique, shift invariant probability measure called {\bf Haar measure}.
An affine version of $A$ is a set $X$ such that $A$ acts transitively and freely (fix point free) on $X$. This means that for every pair $x,y\in X$ there is a unique element $a\in A$ such that $x^a=y$. We can interpret $a$ as $y-x$, however $y+x$ does not have a natural interpretation. Affine abelian groups are very similar to abelian groups. If we fix an element in $x\in X$ then there is a natural bijection between $X$ and $A$ given by $a\longleftrightarrow x^a$. An {\bf affine homomorphism} from an abelian group $A$ to another abelian group $B$ is the composition of an ordinary homomorphism with a shift on $B$. Affine homomorphisms can be naturally defined between affine abelian groups making them a category.
\medskip
An {\bf abstract cube} of dimension $n$ is a set of the form $\{0,1\}^n$ (or more generally $\{0,1\}^I$ where $I$ is a set of size $n$). We denote by $0$ the all $0$ vector in $\{0,1\}^n$. A function $\phi:\{0,1\}^a\rightarrow\{0,1\}^b$ is called a cube morphism if it extends to an affine homomorphism $\phi':\mathbb{Z}^a\rightarrow\mathbb{Z}^b$.
Cube morphisms have a combinatorial description. They are maps $\phi:\{0,1\}^a\rightarrow\{0,1\}^b$ such that each coordinate function of $\phi(x_1,x_2,\dots,x_a)$ is one of $1$, $0$, $x_i$ and $1-x_i$ for some $1\leq i\leq a$.
An $n$-dimensional cube (or more precisely the morphism of a cube) in an abelian group $A$ is a map $c:\{0,1\}^n\rightarrow A$ which extends to an affine homomorphism $c':\mathbb{Z}^n\rightarrow A$.
Cubes can also be described through the formula
$$c(e_1,e_2,\dots,e_n)=x+\sum_{i=1}^n t_ie_i$$
where $x,t_1,t_2,\dots,t_n$ are elements in $A$. Finally
a map $c:\{0,1\}^n\rightarrow A$ is a cube if and only if for every morphism $\phi:\{0,1\}^2\rightarrow\{0,1\}^n$ we have that $$c(\phi(0,0))-c(\phi(1,0))-c(\phi(0,1))+c(\phi(1,1))=0.$$
Let $Q=\{0,1\}^n$. We denote by $\hom(Q,A)$ the set of morphisms of $Q$ into $A$. With respect to pont wise addition on $Q$ the set $\hom(Q,A)$ is an abelian group. Since every $n$-dimensional cube is uniquely determined by $x,t_1,t_2,\dots,t_n$ in the above formula we have that $\hom(Q,A)$ is isomorphic to the direct power $A^{n+1}$.
If $S$ is a subset in $Q$ and $f:S\rightarrow A$ is an arbitrary function then we denote by $\hom_f(Q,A)$ the set of maps $c\in\hom(Q,A)$ such that restriction of $c$ to $S$ is $f$.
Note that $\hom_f(Q,A)$ may be empty if $f$ does not extend to a morphism of the full cube.
If $\hom_f(Q,A)$ is not empty then it is a coset of the group $\hom_g(Q,A)$ where $g$ is the identically $0$ function on $S$.
In other words $\hom_f(Q,A)$ is an affine version of $\hom_g(Q,A)$.
If $A$ is compact then $\hom_g(Q,A)$ is also compact and so using its Haar measure we get a unique $\hom_g(Q,A)$ invariant probability space structure on $\hom_f(Q,A)$.
When it doesn't lead to confusion we will use the short hand notation $C^n(A)$ for $\hom(Q,A)$ and $C^n_f(A)$ for $\mathrm{Hom}_f(Q,A)$. If $S=\{0\}\subset Q$ and $f:S\rightarrow A$ is given by $f(0)=x$ then we use the short hand notation $C_x^n(A)$ for $\hom_f(Q,A)$.
It is clear that $C_0^n(A)$ is a subgroup of $C^n(A)$ which is isomorphic to $A^n$. If $x\neq 0$ then $C_x^n(A)$ is an affine version of $C_0^n(A)$.
It is a crucial idea in this paper to consider $C^n(A)$ (resp. $C^n_x(A)$) as a coupling of $2^n$ (resp. $2^n-1$) copies of $A$. For every $v\in\{0,1\}^n$ we define the map $\psi_v:C^n(A)\rightarrow A$ by $\psi_v(c)=c(v)$. These maps are surjective homomorphisms between compact abelian groups and so they are all measure preserving. Ovserve that if $v\neq 0$ then the restriction of $\psi_v$ to $C^n_x$ is a measure preserving map from $C^n_x$ to $A$.
The system of maps $\Psi^n=\{\psi_v\}_{v\in\{0,1\}^n}$ on $C^n(A)$ is a self coupling $A$ with index set $\{0,1\}^n$. Let $K_n=\{0,1\}^n\setminus\{0\}$ and let $\Psi^n_x$ be the restriction of the function system $\{\psi_v\}_{v\in K_n}$ to the probability space $C^n_x(A)$. Then $\Psi^n_x$ is a self coupling of $A$ with index set $K_n$.
It is very important to note that $\Psi^n_x$ is not a sub-coupling of $\Psi^n$ and it will depend on the choice of $x$.
Let $G=\{g_v\}_{v\in\{0,1\}^n}$ and $F=\{f_v\}_{v\in K_n}$ be systems of bounded measurable functions on $A$. Then the values of both $\xi(G,\Psi^n)$ and $\xi(F,\Psi^n_x)$ are crucial in this paper.
The following formulas follow directly from the definitions.
\begin{equation}\label{ginner} \xi(G,\Psi^n)=\mathbb{E}_{x,t_1,t_2,\dots,t_n}\prod_{v\in\{0,1\}^n}g_v(x+\sum_{i=1}^n v_it_i),
\end{equation}
\begin{equation}\label{corner}
\xi(F,\Psi_x^n)=\mathbb{E}_{t_1,t_2,\dots,t_n}\prod_{v\in K_n}f_v(x+\sum_{i=1}^n v_it_i).
\end{equation}
It turns out that certain calculations work out a bit nicer if we put conjugations on the terms in (\ref{ginner}) and (\ref{corner}) whose indices $v$ have an odd number of $1$'s. This motivates the next definitions.
Let us use the convention that if $t_v$ is a complex valued term which depends on an element $v\in\{0,1\}^n$ then $t_v^\star$ is $t_v$ if $v$ has an even number of $1$'s and is the conjugate of $t_v$ if $v$ has an odd number of $1$'s.
\begin{definition} Let $G=\{g_v\}_{v\in\{0,1\}^n}$ be $\{f_v\}_{v\in K_n}$ be function systems and $G^\star=\{g_v^\star\}_{v\in\{0,1\}^n}$ and $F^\star=\{f_v^\star\}_{v\in K_n}$ be their conjugated versions.
Then the {\bf Gowers inner product} $(G)$ of $G$ is defined by
$$(G)=\xi(G^\star,\Psi^n),$$
and the {\bf corner convolution} $[F]$ of $F$ is defined by
$$[F](x)=\xi(F^\star,\Psi^n_x).$$
By abusing the notation we will also define the convolution $[G]$. Let $G'$ be the function system obtained from $G$ by ignoring $g_0$. Then $[G]:=[G']$. We introduce the notations $$(G)^\times=\prod_{v\in\{0,1\}^n}g_v^\star\circ\psi_v~~~~~{\it and}~~~~~[F]^\times=\prod_{v\in K_n}f_v^\star\circ\psi_v.$$
If the function system $[F]$ (resp. $[G]$) is constant such that each member is equal to the same function $f$ (resp. $g$) then we use the short hand notations $$(f)_n=(F)~~,~~[g]_n=[G]~~,~~(f)_n^\times=(F)^\times~~,~~[g]_n^\times=[G]^\times.$$
\end{definition}
Convolutions of the form $[F]$ in the above definition will be also called $n$-th order convolutions if we need to emphasize the value $n$.
Let us observe that with the above notation we have the following equations.
\begin{equation}\label{coincon}
(G)=([G],\overline{g_0})=([G]^\times,\overline{g_0}\circ\psi_0),
\end{equation}
$$\mathbb{E}((G)^\times)=(G),$$
\begin{equation}\label{rankcon1}
\mathbb{E}([F]^\times|\psi_0)=[F]\circ\psi_0,
\end{equation}
\begin{equation}\label{rankcon2}
\mathbb{E}_{y\in C^k_0(A)}[F]^\times(z+y)=[F](\psi_0(z)).
\end{equation}
Note that (\ref{rankcon1}) and (\ref{rankcon2}) are the same equations written in a different form.
An easy way of seeing (\ref{rankcon1}) and (\ref{rankcon2}) is to write the elements of $C^k(A)$ as vectors $z=(x,t_1,t_2,\dots,t_k)$ as described in chapter \ref{cubes}. In this coordinate system $C^k_0(A)$ is the set of vectors of the form $(0,t_1,t_2,\dots,t_k)$. Then $\psi_0(z)=x$ and (\ref{corner}) shows that the value $[G](x)$ is the average of $g$ on the coset of $C^k_0(A)$ containing $z$.
\begin{remark}\label{convat} Let $w\in\{0,1\}^n$, $K=\{0,1\}^n\setminus\{w\}$. Let $F=\{f_v\}_{v\in K}$ be a function system. One can define the convolution $[F]$ in a similar way as above since our setup does not distinguish the $0$ vector in $\{0,1\}^n$. Let $\alpha:\{0,1\}^n\rightarrow\{0,1\}^n$ be an automorphism with $\alpha(0)=w$. Then define $[F]$ as the convolution of the function system $\{f_{\alpha(v)}\}_{v\in K_n}$. It is clear that it does not depend on the choice of $\alpha$.
\end{remark}
\begin{lemma} Let $A$ be a compact abelian group and $n$ be a natural number. Then the map $\gamma:x\rightarrow\Psi^n_x$ is a continuous map from $A$ to ${\rm coup}(\mathcal{U}_0)$.
\end{lemma}
\begin{proof} According to the last point in lemma \ref{xiprop} the topology on ${\rm coup}(\mathcal{U}_0)$ is generated by functions of the form $C\rightarrow \xi(F,C)$ where $F=\{f_v\}_{v\in K_n}$ is a system of continuous functions. This means that it is enough to check the continuity of the composition of $\gamma$ with such functions. This composition is the function $h:x\rightarrow\xi(F,\Psi^n_x)$. The formula (\ref{corner}) shows that the continuity of the functions $f_v$ imply the continuity of $h$.
\end{proof}
The previous lemma and the fourth point in lemma \ref{xiprop} imply the following corollary.
\begin{corollary} The function $[F]$ is continuous for an arbitrary system $F=\{f_v\}_{v\in K_n}$ of bounded measurable functions.
\end{corollary}
\subsection{Sub-couplings of cubic couplings}
Let $S\subset\{0,1\}^n$ be an arbitrary subset and $f:S\rightarrow A$ be an arbitrary function into a compact abelian group $A$.
Let $\phi:\{0,1\}^m\rightarrow\{0,1\}^n$ be a morphism and let $g$ denote the function $f\circ\phi$ on the set $\phi^{-1}(S)$. Then we denote by $\hat{\phi}:C^n_f(A)\rightarrow C^m_g(A)$ the map defined by $\hat{\phi}(c)=c\circ\phi$. The natural question arises: {\it Under what conditions is the map $\hat{\phi}$ measure preserving?}
First of all notice that $\hat{\phi}$ is a continuous morphism between affine compact abelian groups and thus $\hat{\phi}$ is measure preserving if and only if it is surjective.
The next lemma connects surjectivity of $\hat{\phi}$ with equivalence of couplings.
\begin{lemma}\label{cupis} Assume that for every $v\in\{0,1\}^n\setminus S$ the map $\psi_v:C^n_f(A)\rightarrow A$ is surjective and that $\hat{\phi}$ is surjective. Then the coupling $\{\psi_v\}_{v\in H}$ on $C^m_g(A)$ is equivalent with the coupling $\{\psi_{\phi(v)}\}_{v\in H}$ on $C^n_f(A)$ where $H=\{0,1\}^m\setminus\phi^{-1}(S)$.
\end{lemma}
\begin{proof} The statement is clear from the facts that $\psi_{\phi(v)}=\psi_v\circ\hat{\phi}$ for every $v$ in $H$ and that $\hat{\phi}$ is measure preserving.
\end{proof}
\begin{lemma}\label{cupis1} Let $0\leq k\leq n$ be integers and $S\subset\{0,1\}^n$ be the $k$-dimensional face in $\{0,1\}^n$ which consists of all vectors with $0$ in the last $n-k$ coordinates. Let $\tau:\{0,1\}^n\rightarrow\{0,1\}^{n-k}$ be the projection to the last $n-k$ coordinates. Assume that $f:S\rightarrow A$ is in $C^k(A)$ and $\phi:\{0,1\}^m\rightarrow\{0,1\}^n$ is a morphism such that $\tau\circ\phi$ is injective. Then $\hat{\phi}$ is surjective.
\end{lemma}
\begin{proof} Let $\tau'$ be the projection of $\{0,1\}^n$ to the first $k$ coordinates and let $z$ be the identically zero function on $S$. Since $C^n_f(A)=f\circ\tau'+C^n_z(A)$ we can assume without loss of generality that $f$ is identically $0$. It is enough to prove that $\hat{\phi}\circ\hat{\tau}:C^{n-k}_0(A)\rightarrow C_g^m(A)$ is surjective. This reduces the problem to the case when $k=0$. In this case $S$ is the $0$ vector and $\phi$ is an injective morphism.
By using appropriate automorphisms of $\{0,1\}^m$ and coordinate permutation of $\{0,1\}^n$ we can assume without loss of generality (using the injectivity of $\phi$) that $\phi(v_1,v_2,\dots,v_m)_i=v_i$ if $1\leq i\leq m$ (this can be obtained from the combinatorial description of morphisms).
We distinguish between two cases. In the first case $0$ is not in the range of $\phi$ and in the second case $\phi(0)=0$.
In the first case, since the image of $\phi$ does not contain the zero vector we can assume that either $\phi(v)_{m+1}=1$ or $\phi(v)_{m+1}=1-v_1$. Let $\tau_2:\{0,1\}^n\rightarrow\{0,1\}^{m+1}$ be the projection to the
first $m+1$ coordinates.
It is enough to show that $\hat{\phi}\circ\hat{\tau_2}$ is surjective. Using our parametrization, if $\phi(v)_{m+1}=1$ then for $t=(0,t_1,t_2,\dots,t_{m+1})$ in $C^{m+1}_0(A)$ we have that $\hat{\phi}(\hat{\tau}_2(t))=(t_{m+1},t_{m+1}+t_1,\dots,t_{m+1}+t_m)$ and if $\phi(v)_{m+1}=1-v_1$ the $\hat{\phi}(\hat{\tau}_2(t))=(t_{m+1},t_1,t_{m+1}+t_2,\dots,t_{m+1}+t_m)$. Both are surjective.
In the second case we denote by $\tau_2:\{0,1\}^n\rightarrow\{0,1\}^m$ the projection to the first $m$ coordinates. The map $\hat{\tau_2}:C^m_0(A)\rightarrow C^n_0(A)$ composed with $\hat{\phi}:C^n_0(A)\rightarrow C_0^m(A)$ is obviously bijective which completes the proof.
\end{proof}
\subsection{Gowers norms and corner convolutions}\label{chap:gc}
Let $A$ be a compact abelian group.
If $f:A\rightarrow\mathbb{C}$ then we define the function $\Delta_t f$ by $(\Delta_t f)(x)=f(x)\overline{f(x+t)}$.
The Gowers norm $\|f\|_{U_n}$ is defined by
\begin{equation}\label{gowersnorm}
\|f\|_{U_n}^{2^n}=\mathbb{E}_{x,t_1,t_2,\dots,t_n}\Delta_{t_1,t_2,\dots,t_n}f(x)=(f)_n
\end{equation}
for $f\in L^\infty(A)$.
The so-called Gowers-Cauchy-Schwartz inequality says that if $F=\{f_v\}_{v\in\{0,1\}^n}$ is a system of bounded measurable functions then
\begin{equation}\label{GCS}
|(F)|\leq\prod_{v\in\{0,1\}^n}\|f_v\|_{U_n}.
\end{equation}
\medskip
We continue with a basic trick which makes calculations with $(F)$ and $[F]$ easier.
Let $i\in [n]$ and let $Q\subset\{0,1\}^n$ be the set of vectors with $0$ in the $i$-th coordinate. Let $w\in\{0,1\}^n$ be the vector with $1$ at the $i$-th coordinate and $0$ everywhere else. For $t\in A$ we introduce $\delta_{i,t} F$ as the function system $\{f_v(x)\overline{f_{v+w}(x+t})\}_{v\in Q}$. If $F$ is a function system parametrized by $K_n$ then we define $\delta_{i,t} F$ by the previous formula such that $Q$ is repleced by $Q\setminus\{0\}$.
Then we have the following two equations
\begin{equation}\label{dimred}
(F)=\mathbb{E}_t((\delta_{i,t} F))~~~~{\rm and}~~~~[F](x)=\mathbb{E}_t(\overline{f}_w(x+t)\delta_{i,t}[F](x)).
\end{equation}
The equations in (\ref{dimred}) are useful because they reduce the dimension $n$ in the calculations and thus they can be used in proofs with inductions on $n$. The next lemma is an example for this.
\medskip
\begin{lemma}\label{cornineq} Let $F=\{f_v\}_{v\in K_n}$ be a system of bounded measurable functions on $A$. Then for every $j\in[n]$ we have that
$$|[F](x)|\leq\prod_{v\in K_n,v_j=0}\|f_v\|_\infty\prod_{v\in K_n,v_j=1}\|f_v\|_{U_n}.$$
\end{lemma}
\begin{proof} If $n=1$ then the statement is true with equality. If $n>1$ then by induction we assume that it is true for $n-1$. Without loss of generality (using symmetry) we can assume that $j\neq n$.
We have that $\delta_{n,t} F=\{g_v^t\}_{v\in K_{n-1}}$ where $g_v^t$ is the function $y\mapsto\overline{f}_{(v,1)}(y+t)f_{(v,0)}(y)$. Let $w=(0,0,\dots,0,1)\in\{0,1\}^n$.
By (\ref{dimred}), induction, Cauchy-Schwartz inequality and using the fact that $\|g_v^t\|_\infty\leq\|f_{(v,0)}\|_\infty\|f_{(v,1)}\|_\infty$ we get
$$|[F](x)|\leq\|f_w\|_2\mathbb{E}_t^{1/2}\Bigl(\prod_{v\in K_{n-1},v_j=0}\|g_v^t\|^2_\infty\prod_{v\in K_{n-1},v_j=1}\|g_v^t\|^2_{U_{n-1}}\Bigr)\leq$$
$$\|f_w\|_\infty\prod_{v\in K_{n-1},v_j=0}\|f_{(v,0)}\|_\infty\|f_{(v,1)}\|_\infty\mathbb{E}_t^{1/2}\Bigl(\prod_{v\in K_{n-1},v_j=1}\|g_v^t\|^2_{U_{n-1}}\Bigr)\leq$$
$$\prod_{v\in K_n,v_j=0}\|f_v\|_\infty\prod_{v\in K_{n-1},v_j=1}\Bigl(\mathbb{E}_t(\|g_v^t\|_{U_{n-1}}^{2^{n-1}})\Bigr)^{2^{1-n}}$$
Let $v\in K_{n-1}$ and let $H=\{h_z\}_{z\in \{0,1\}^n}$ be the function system defined by $h_z=f_{(v,0)}$ if $z_n=0$ and $h_z=f_{(v,1)}$ if $z_n=1$. Then by (\ref{GCS}), (\ref{dimred}) we get
$$\mathbb{E}_t(\|g_v^t\|_{U_{n-1}}^{2^{n-1}})=\mathbb{E}_t((\delta_{n,t}H))=(H)\leq\|f_{(v,0)}\|_{U_n}^{2^{n-1}}\|f_{(v,1)}\|_{U_n}^{2^{n-1}}$$
which completes the proof.
\end{proof}
\medskip
\begin{lemma} If $k\geq 1$ then $\|f\|_{U_k}\leq\|f\|_{2^{k-1}}$.
\end{lemma}
\begin{proof} We prove the statement by induction. If $k=1$ then $\|f\|_{U_1}=|\mathbb{E}(f)|\leq\|f\|_1$.
Let $f_t$ denote the function with $f_t(x)=f(x+t)$. We have by induction that
$$\|f\|_{U_{k+1}}^{2^{k+1}}=\mathbb{E}_{t}(\|f\overline{f_t}\|_{U_k}^{2^k})\leq\mathbb{E}_t(\|f\overline{f_t}\|_{2^{k-1}}^{2^k})\leq\mathbb{E}_{t,x}(|f(x)|^{2^k}|f(x+t)|^{2^k})=\|f\|_{2^k}^{2^{k+1}}.$$
\end{proof}
\begin{corollary}\label{l2becs} If $k\geq 2$ and $|f|\leq 1$ then $(f,f)=\|f\|_2^2\geq \|f\|_{U_k}^{2^{k-1}}$.
\end{corollary}
\begin{proof} If $|f|\leq 1$ then $\|f\|_{U_k}^{2^{k-1}}\leq\|f\|_{2^{k-1}}^{2^{k-1}}\leq\|f\|_2^2$.
\end{proof}
\subsection{Low rank approximation and products of convolutions}
\begin{lemma}\label{shiftap} Let $A$ be a compact abelian group, $B<A$ a compact subgroup, $\epsilon>0$ and $f:A\rightarrow\mathbb{C}$ be a measurable function with $\|f\|_\infty\leq 1$. Let furthermore $f_B(z)=\mathbb{E}_{y\in B}f(z+y)$. Then there are elements $a_1,a_2,\dots,a_n$ in $B$ with $n\leq 1+4/\epsilon^2$ such that the function $g(z)=\frac{1}{n}\sum_{i=1}^n f(z+a_n)$ satisfies $\|f_B-g\|_2\leq\epsilon$.
\end{lemma}
\begin{proof} Let $n$ be an integer with $4/\epsilon^2\leq n\leq 1+4/\epsilon^2$. Let $h(z,a_1,a_2,\dots,a_n)=\frac{1}{n}\sum_{i=1}^nf(z+a_n)$ and $r(z,a_1,a_2,\dots,a_n)=f_B(z)$ be functions defined on $A\times B^n$. For a fixed $z\in A$ let $Y_z$ denote the random variable $f(z+y)-f_B(z)$ where $y$ is chosen randomly from $B$. If $z\in A$ is fixed then the value of $h-r$ for a randomly chosen element $(a_1,a_2,\dots,a_n)$ in $B^n$ has the same distribution as the average of $n$ independent copies of $Y_z$ and so on this probability space ${\rm Var}(h-r)={\rm Var}(Y_z)/n\leq 4/n$.
By taking the average of this for every $z$ we get that
$\|h-r\|_2^2\leq 4/n$. Consequently there is a fixed vector $(a_1,a_2,\dots,a_n)$ in $B^n$ such that
the function $\mathbb{E}_z(h(z,a_1,a_2,\dots,a_n)-r(z,a_1,a_2,\dots,a_n))^2\leq 4/n\leq\epsilon^2$ which finishes the proof.
\end{proof}
\medskip
\begin{definition}\label{rankone} A {\bf rank one} function on $C^k(A)$ is a function of the form $[G]^\times$ where $G=\{g_v\}_{v\in K_k}$ is a function system in $L^\infty_u(A)$.
\end{definition}
Note that rank one functions are shift invariant on $C^k(A)$ and are closed under point wise multiplication.
The next lemma says that convolutions have low rank approximations when lifted to the space $C^k(A)$ with $\psi_0$.
\begin{lemma}[Low rank approximation]\label{lowrank} Let $F=\{f_v\}_{v\in K_k}$ be a system of functions in $L^\infty_u(A)$ and $\epsilon>0$. Then there is a function $g$ on $C^k(A)$ which is the average of at most $1+4/\epsilon^2$ rank one functions and $\|g-[F]\circ\psi_0\|_2\leq\epsilon$
\end{lemma}
\begin{proof} Using (\ref{rankcon2}) and lemma \ref{shiftap} we obtain that there exist elements $y_1,y_2,\dots,y_n$ in $C^k_0(A)$ with $n\leq 1+4/\epsilon^2$ such that the function $g(z)=\frac{1}{n}\sum_{i=1}^n [F]^\times(z+y_i)$ satisfies $\|[F]\circ\psi_0-g\|_2\leq\epsilon$. Since the functions $z\rightarrow [F]^\times(z+y_i)$ are all rank one functions the proof is complete.
\end{proof}
\begin{remark}\label{aptrans} It will be important that in lemma \ref{lowrank} the rank one functions occurring in the approximation use only shifted versions of functions from the system $F$.
\end{remark}
\begin{lemma}[product of convolutions]\label{prodconv} Let $F=\{f_v\}_{v\in K_k}$ and $G=\{g_v\}_{v\in K_k}$ be two systems in $L^\infty_u(A)$ and let $\epsilon>0$. Then there are function systems $H^i=\{h_v^i\}_{v\in K_k}$ in $L^\infty_u(A)$ for $i=1,2,\dots,n$ with $n\leq (1+64/\epsilon^2)^2$ such that
$$\|~[F][G]-\frac{1}{n}\sum_{i=1}^n[H^i]~\|_2\leq\epsilon.$$
\end{lemma}
\begin{proof} We use lemma \ref{lowrank} for both $F$ and $G$ with $\epsilon/4$. This way we obtain approximations $f$ and $g$ for $[F]\circ\psi_0$ and $[G]\circ\psi_0$ with $L^2$ error $\epsilon/4$ such that both $f$ and $g$ are the averages of at most $1+64/\epsilon^2$ functions of rank one. In particular $\|f\|_\infty,\|g\|_\infty\leq 1$.
Let us write $[F]\circ\psi_0=f+e_F$ and $[G]\circ\psi_0=g+e_G$ where $\|e_F\|_2,\|e_G\|_2\leq\epsilon/4$ and $\|e_F\|_\infty,\|e_G\|_\infty\leq 2$. Now we have
$$([F][G])\circ\psi_0=([F]\circ\psi_0)([G]\circ\psi_0)=fg+e_Fg+e_Gf+e_Fe_G.$$ By
$\|e_Gf\|_2,\|e_Fg\|_2\leq\epsilon/4$ and $\|e_Fe_G\|_2\leq\epsilon/2$ we get that $$\|([F][G])\circ\psi_0-fg\|_2\leq\epsilon.$$ The function $fg$ is the average of $n\leq (1+64/\epsilon^2)^2$ functions of rank one. Let us denote the corresponding function systems by $H^1,H^2,\dots,H^n$. By (\ref{rankcon1}) it follows that $$\mathbb{E}(fg|\psi_0)=\Bigl(\frac{1}{n}\sum_{i=1}^n[H^i]\Bigr)\circ\psi_0.$$ The function $([F][G])\circ\psi_0$ is already measurable in the $\sigma$-algebra generated by $\psi_0$ and so conditional expectation with respect to this $\sigma$ algebra leaves it invariant. Since conditional expectation is a contraction on $L^2$ and $\psi_0$ is measure preserving the proof is complete.
\end{proof}
\subsection{Higher degree cubes}
\begin{definition} Let $A$ be an Abelian group. A map $c:\{0,1\}^n\rightarrow A$ is a degree-$k$ cube if it extends to a degree-$k$ polynomial map $f:\mathbb{Z}^n\rightarrow B$.
\end{definition}
It can be seen that a function$\{0,1\}^n\rightarrow A$ is a degree-$k$ cube if and only if for every $k+1$ dimensional face $S\subset\{0,1\}^n$ we have $\sum_{v\in S}(-1)^{h(v)}c(v)=0$ where $h(v)=\sum_{i=1}^nv_i$.
If $k\leq-1$ then we define a degree-$k$ cube as the constant $0$ function on $\{0,1\}^n$.
For an integer $k\in\mathbb{Z}$ and abelian group $A$ we introduce the cubespace $\mathcal{D}_k(A)$ in which $C^n(\mathcal{D}_k(A))$ is the collection of degree-$k$ cubes of dimension $n$. It is easy to seet that if $k\geq 1$ then $\mathcal{D}_k(A)$ is a $k$-step nilspace. We regard $\mathcal{D}_k(A)$ as a degree-$k$ version of of $A$. In particular $\mathcal{D}_1(A)$ is the group $A$ with the usual cubic structure.
\begin{lemma}\label{dualker1} Let $n\in\mathbb{N},k\in\mathbb{Z}$. Let $A$ be a compact abelian group and let $f\in C^n(\mathcal{D}_{n-k-1}(\hat{A}))$. Then for every $c\in C^n(\mathcal{D}_k(A))$ we have that $\prod_{v\in\{0,1\}^n} \chi_v^\star(c(v))=1$ where $\chi_v=f(v)$.
\end{lemma}
\begin{proof} We go by induction on $n$. If $n=0$ then the statement is trivial.
Assume that $n>0$ and that the statement is true for $n-1$. Then by induction we have the product in the lemma is equal to $$\prod_{v\in Q}\Bigl(\chi_v\overline{\chi_{v+w}}(c(v))\Bigr)^\star\prod_{v\in Q}\Bigl(\chi_{v+w}(c(v)-c(v+w))\Bigr)^\star$$
where $Q=\{(v,0)|v\in\{0,1\}^{n-1}\}$~,~$w=(0,0,\dots,0,1)\in\{0,1\}^n$.
The first product satisfies the conditions with $n-1,k$ and the second one with $n-1,k-1$. We have by induction that both products are $1$.
\end{proof}
\begin{lemma}\label{dualker2} Let $n\in\mathbb{N},k\in\mathbb{Z}$. Let $A$ be a compact abelian group and let $f\in\hom(K_n,\mathcal{D}_{n-k-1}(\hat{A}))$. Then for every $c\in C^n_0(\mathcal{D}_k(A))$ we have that $\prod_{v\in\{0,1\}^n}\chi_v^\star(c(v))=1$ where $\chi_v=f(v)$.
\end{lemma}
\begin{proof} Let us extend $f$ to the full cube $\{0,1\}^n$ such that $\overline{f}(0)=\prod_{v\in K_n}\chi_v^\star$.
By lemma \ref{dualker1} it is enough to prove that this extension is in $C^n(\mathcal{D}_{n-k-1}(\hat{A}))$. If $k<0$ or $k>n$ then it is trivial. Assume that $0\leq k\leq n$ and
let $S$ be a face in $Q=\{0,1\}^n$ of dimension $n-k$. If $S\subset K_n$ then we have by our assumption that $\prod_{v\in S}f^\star(v)=1_A$. If $0\in S$ then $\prod_{v\in Q\setminus S}f^\star(v)=1_A$ since $Q\setminus S$ is a disjoint union of faces parallel to $S$. Then by $\prod_{v\in Q}f^\star(c)=1_A$ the proof is complete.
\end{proof}
\medskip
\subsection{Compact nilspaces}
The nilspace axioms were given in the introduction.
In this chapter we review some of the results from \cite{NP} which we use in this paper.
Let $A$ be an abelian group and $X$ be an arbitrary set. An $A$ bundle over $X$ is a set $B$ together with a free action of $A$ such that the orbits of $A$ are parametrized by the elements of $X$. This means that there is a projection map $\pi:B\rightarrow X$ such that every fibre is an $A$-orbit.
The action of $a\in A$ on $x\in B$ is denoted by $x+a$. Note that if $x,y\in B$ are in the same $A$ orbit then it makes sense to talk about the difference $x-y$ which is the unique element $a\in A$ with $y+a=x$. In other words the $A$ orbits can be regarded as affine copies of $A$.
A $k$-fold abelian bundle $X_k$ is a structure which is obtained from a one element set $X_0$ in $k$-steps in a way that in the $i$-th step we produce $X_i$ as an $A_i$ bundle over $X_{i-1}$.
The groups $A_i$ are the structure groups of the $k$-fold bundle. We call the spaces $X_i$ the $i$-th factors.
If all the structure groups $A_i$ and spaces $X_i$ are compact and the actions are continuous then the $k$-fold bundle admits a Borel probability measure which is built up from the Haar measures of the structure groups in a recursive way. Let $\pi_i$ denote the the projection from $X_k$ to $X_i$.
Assume that the measure $\mu_{k-1}$ is already defined on $X_{k-1}$ and $\mu_{k-1}^*$ denotes the measure defined by $\mu^*_{k-1}(\pi_{k-1}^{-1}(S))=\mu_{k-1}(S)$ for Borel sets in $X_{k-1}$. Then the measure $\mu_k$ is the unique measure on $X_k$ with the property $$\int_{X_k} f ~d\mu_k=\int_{x\in X_k}\int_{a\in A_k} f(x+a)~d\nu_k~d\mu^*_{k-1}$$ where $\nu_k$ is the Haar measure on $A_k$ and $f$ is a bounded Borel function on $X_k$.
\begin{definition} Let $N_k$ be a $k$-fold abelian bundle with factors $\{N_i\}_{i=1}^k$ and structure groups $\{A_i\}_{i=1}^k$. Let $\pi_i$ denote the projection of $N_k$ to $N_i$.
Assume that $N_k$ admits a cubespace structure with cube sets $\{C^n(N_k)\}_{n=1}^\infty$.
We say that $N_k$ is a {\bf $k$-degree bundle} if it satisfies the following conditions
\begin{enumerate}
\item $N_{k-1}$ is a $k-1$ degree bundle.
\item For every $n\in\mathbb{N}$ the set $C^n(N_k)$ is a $C^n(\mathcal{D}_k(A_k))$-bundle with the pointwise action over $C^n(N_{k-1})$. The projection of the bundle is given by the composition with $\pi_{k-1}$.
\end{enumerate}
\end{definition}
The next theorem form \cite{NP} says that $k$-degree bundles are the same as $k$-step nilspaces.
\begin{theorem}\label{bundec} Every $k$-degree bundle is a $k$-step nilspace and every $k$-step nilspace arises as a $k$-degree bundle.
\end{theorem}
It will be important that if $N$ is a $k$-step compact nilspace then since the set $C^n(N)$ admits a $k$-fold bundle structure it has a natural probability measure on it.
Furthermore we have that the maps $\psi_v:C^n\rightarrow N$ defined by $\psi_v(c)=c(v)$ are all measure preserving and thus we can define the expressions $[F],(F),[F]^\times,(F)^\times$ similarly as in case of abelian groups.
We have for example that $[F]$ is continuous for every bounded measurable function system on $N$.
We define $\|f\|_{U_n}$ by $\|f\|_{U_n}^{2^n}=(f)_n$ which is equal to $\mathbb{E}((f)_n^\times)$.
It turns out that if $n\leq k$ then $U_n$ is a semi-norm on $L^\infty(N)$ and it is a norm if $n\geq k+1$.
\begin{definition} Let $N$ be a $k$-step compact nilspace and $\chi\in\hat{A_k}$ be a linar character of the $k$-th structure group $A_k$. We denote by $W(\chi,N)$ the Hilbert space of functions $f\in L^2(N)$ such that $f(x+a)=f(x)\chi(a)$ holds for every $x\in N$ and $a\in A_k$.
\end{definition}
The next lemma is a direct consequence of theorem \ref{bundec}.
\begin{lemma}\label{nilspcharderiv} Assume that $N$ is a $k$-step compact nilspace, $\chi\in\hat{A}_k$ and $f\in W(\chi,N)$ is bounded. Then the function $(f)_{k+1}^\times$ (resp. $[f]_{k+1}^\times$ restricted to $C^{k+1}_x(N)$ for some $x\in N$) is the composition of a Borel function on $C^{k+1}(N_{k-1})$ (resp. $C^{k+1}_{\pi_{k-1}(x)}(N_{k-1})$~) and the projection $C^{k+1}(N)\rightarrow C^{k+1}(N_{k-1})$ induced by $\pi_{k-1}$.
Furthermore we have that $[f]_{k+1}\in V(\overline{\chi},N)$.
\end{lemma}
Another important fact about the spaces $W(\chi,N)$ is the following.
\begin{lemma}[Fourier decomposition on nilspaces]\label{nilfourdec} Let $N$ be a compact $k$-step nilspace. Then $$L^2(N)=\bigoplus_{\chi\in\hat{A_k}}W(\chi,N)$$ where the direct summands are orthogonal to each other. If $f:N\rightarrow\mathbb{C}$ is a bounded Borel measurable function then there is a unique decomposition
$f=\sum_{\chi\in\hat{A_k}}f_\chi$ into bounded functions $f_\chi\in W(\chi,N)$
converging in $L^2$.
\end{lemma}
\begin{proof} It is clear that the functions defined by $f_\chi(x)=\mathbb{E}_{b\in A_k}f(x+b)\overline{\chi(b)}$ satisfy the above equality.
\end{proof}
\begin{lemma}\label{nilspchar} Let $N$ be a $k$-step compact nilspace and $\chi\in\hat{A_k}$. Then there is a function $\phi\in W(\chi,N)$ such that $|\phi(x)|=1$ holds for every $x\in N$. Furthermore every function $f\in W(\chi,N)$ can be written as a product of $\phi$ and $h\circ\pi_{k-1}$ where $h$ is an $L^2$ function on the $k-1$ step factor of $N$.
\end{lemma}
\begin{proof} Let $f:N_{k-1}\rightarrow N_k$ be a function which chooses a Borel representative system for the fibres of $\pi_{k-1}$. Then we define $\phi(x)=\chi(x-f(\pi_{k-1}(x)))$. It is clear that $\phi$ satisfies the required condition.
\end{proof}
\begin{lemma}\label{gownilproj} Assume that $k\geq i\geq n-1\geq 0$ and $f\in L^\infty(N)$. Then $\|f\|_{U_n}=\|\mathbb{E}(f|\pi_i)\|_{U_n}$.
\end{lemma}
\begin{proof} We prove the statement by induction on $k-i$. If $k=i$ then there is nothing to prove. Assume that the statement is true for $i+1\leq k$. By $\mathbb{E}(f|\pi_i)=\mathbb{E}(\mathbb{E}(f|\pi_{i+1})\pi_i)$ we can assume that $f$ is measurable in the factor $\pi_{i+1}$. By abusing the notation we can assume that $f$ is defined on $N_{i+1}$ and we do the calculation inside $N_{i+1}$. We have that $C^{i+1}(N_{i+1})$ is a $C:=C^{i+1}(\mathcal{D}_{i+1}(A_{i+1}))$ bundle over $C^{i+1}(N_i)$. Recall that by definition the set $C$ is equal to the set of all functions $\{0,1\}^{i+1}\rightarrow A_{i+1}$. We have that $$\|f\|_{U_{i+1}}=\mathbb{E}((f)_{i+1}^\times)=\mathbb{E}(\mathbb{E}((f)_{i+1}^\times|\pi_i))=\mathbb{E}(\mathbb{E}_{c\in C}((f)_{i+1}^\times)^c)=\mathbb{E}((\mathbb{E}_{a\in A_{i+1}}f^a)_{i+1}^\times).$$ Since the right hand side is equal to $\|\mathbb{E}(f|\pi_i)\|_{U_{i+1}}$ the proof is complete.
\end{proof}
\medskip
Morphisms between compact nilspaces were defined in the introduction. We will also need a stronger notion of morphism which was defined in \cite{NP}.
\begin{definition}[Fibre surjective morphism] Let $N$ and $M$ be two $k$-step nilspaces. A morphism $\phi:N\rightarrow M$ is called fibre surjective if for every $0\leq i\leq k$ and $x\in N_i$ we have that $\phi(\pi_i^{-1}(x))=\pi_i^{-1}(y)$ for some element $y\in M_i$.
\end{definition}
Fibre surjective morphisms have the useful property (see \cite{NP}) that they are measure preserving in the very strong sense that the induced maps from $C^n_x(N)$ to $C^n_{\phi(x)}(M)$ are all measure preserving for arbitrary $x\in N$ and $n\in\mathbb{N}$. A crucial result from \cite{NP} says the following.
\begin{theorem}\label{inverselimit} If $N$ is a compact $k$-step nilspace then it is the inverse limit of finite dimensional $k$-step nilspaces where the maps used in the inverse system are all fibre surjective.
\end{theorem}
Note that a nilspace is called finite dimensional if all the structure groups $\{A_i\}_{i=1}^k$ are finite dimensional i.e. compact Abelian Lie-groups. This means that the dual groups $\{\hat{A_i}\}_{i=1}^k$ are all finitely generated. In particular finite nilspaces are $0$ dimensional.
\section{Ultra product groups and their factors}
\subsection{Ultra product spaces}
Let $\omega$ be a non principal ultra filter on the natural numbers.
Let $\{X_i,\mu_i\}_{i=1}^\infty$ be pairs where $X_i$ is a compact Hausdorff space and $\mu_i$ is a probability measure on the Borel sets of $X_i$.
We denote by ${\bf X}$ the ultra product space $\prod_{\omega}X_i$.
The space ${\bf X}$ has the following structures on it.
\medskip
\noindent{\bf Strongly open sets:} ~We call a subset of ${\bf X}$ strongly open if it is the ultra product of open sets $\{S_i\subset X_i\}_{i=1}^\infty$.
\medskip
\noindent{\bf Open sets:}~We say that $S\subset {\bf X}$ is open if it is a countable union of strongly open sets. Open sets on ${\bf X}$ form a $\sigma$-topology. This is similar to a topology but it has the weaker axiom that only countable unions of open sets are required to be open. It can be proved that ${\bf X}$ with this $\sigma$-topology is countably compact. This means that if ${\bf X}$ is covered by countably many open sets then there is a finite sub-system which covers ${\bf X}$.
\medskip
\noindent{\bf Borel sets:} A subset of ${\bf X}$ is called Borel if it is in the $\sigma$-algebra generated by strongly open sets. We denote by $\mathcal{A}({\bf X})$ the $\sigma$ algebra of Borel sets.
\medskip
\noindent{\bf Ultra limit measure:} If $S\subseteq {\bf X}$ is a strongly open set of the form $S=\prod_\omega S_i$ then we define $\mu(S)$ as $\lim_\omega\mu_i(S_i)$. It is well known that $\mu$ extends as a probability measure to the $\sigma$-algebra of Borel sets on ${\bf X}$.
\medskip
\noindent{\bf Ultra limit functions:} Let $T$ be a compact Hausdorff topological space. Let $\{f_i:X_i\rightarrow T\}_{i=1}^\infty$ be a sequence of Borel measurable functions. We define $f=\lim_\omega f_i$ as the function on ${\bf X}$ whose value on the equivalence class of $\{x_i\in X_i\}_{i=1}^\infty$ is $\lim_\omega f_i(x_i)$. Such functions will be called ultra limit functions. It is easy to see that ultra limit functions can always be modified on a $0$ measure set that they becomes measurable in the Borel $\sigma$-algebra on ${\bf X}$. This means that ultra limit functions are automatically measurable in the completion of the Borel $\sigma$-algebra.
\medskip
\noindent{\bf Measurable functions:} It is an important fact (see \cite{ESz}) that every bounded measurable function on ${\bf X}$ is almost everywhere equal to some ultra limit function $f=\lim_\omega f_i$.
\medskip
\noindent{\bf Continuity:} A function $f:{\bf X}\rightarrow T$ from ${\bf X}$ to a topological space $T$ is called continuous if $f^{-1}(U)$ is open in ${\bf X}$ for every open set in $T$. If $T$ is a compact Hausdorff topological space then $f$ is continuous if and only if it is the ultra limit of continuous functions $f_i:X_i\rightarrow T$. Furthermore the image of ${\bf X}$ in a compact Hausdorff space $T$ under a continuous map is compact.
\medskip
The fact that an ultra product space has only a $\sigma$-topology and not a topology might be upsetting for the first look. However one can look at ${\bf X}$ as a space which is glued together form many ``ordinary'' topological spaces. These topological spaces appear as quotients of ${\bf X}$.
\begin{definition} A {\bf topological factor} of ${\bf X}$ is a surjective continuous map ${\bf X}\rightarrow T$ to a countably based, Hausdorff topological space $T$.
\end{definition}
Note that the compactness of $T$ follows automatically from the fact that ${\bf X}$ is countably compact. One can equivalently define topological factors through equivalence relations $\sim$ on ${\bf X}$ such that the collection of open sets on ${\bf X}$ that are unions of equivalence classes form a countably based, Hausdorff topological space.
\subsection{Ultra product groups and the correspondence principle}
Let $\{A_i\}_{i=1}^\infty$ be a sequence of compact abelian groups. Let $({\bf A},\mathcal{A},{\bf \mu})$ be the triple where ${\bf A}=\prod_\omega A_i$, $\mathcal{A}$ is the Borel $\sigma$ algebra on ${\bf A}$ and ${\bf \mu}$ is the ultra limit of the normalized Haar measures on $A_i$.
To avoid the situation when ${\bf A}$ is finite we will assume that for every $n\in\mathbb{N}$ the set $\{i:|A_i|>n\}$ is in the ultra filter $\omega$.
Note that ${\bf A}$ is an abelian group which is similar to an ordinary compact group in the sense that it admits the shift invariant probability measure ${\bf \mu}$ defined on the Borel $\sigma$-algebra $\mathcal{A}$.
\medskip
\noindent{\bf Associated structures:}~In this paper we will often work with algebraic structures associated with ${\bf A}$. The most typical one is the cube space $C^k({\bf A})$. In all of our cases there is a commutativity between taking ultra products and taking the associated structure. For example there is a natural bijection between $C^k({\bf A})$ and $\prod_\omega C^k(A_i)$.
\medskip
\noindent{\bf Topology and $\sigma$-algebra:}~An important source of differences between compact groups and their ultra products is the different behavior of their Borel $\sigma$-algebras. As an abstract set, the ultra product $\prod_\omega A_i\times A_i$ is in a natural bijection with $\prod_\omega A_i\times\prod_\omega A_i={\bf A}\times{\bf A}$.
However the Borel $\sigma$-algebra on $\prod_\omega A_i\times A_i$ is bigger than the product $\mathcal{A}\otimes\mathcal{A}$. (The same thing is true for the $\sigma$-topology defined on them.)
\medskip
\noindent{\bf Shift invariant $\sigma$-algebras:}~A sub $\sigma$-algebra $\mathcal{B}\subset\mathcal{A}$ is called {\bf shift invariant} if $S\in\mathcal{B}$ implies that $S+t\in\mathcal{B}$ holds for every $t\in{\bf A}$. It is clear that $\mathcal{A}$ and the measure ${\bf \mu}$ are both shift invariant.
\subsection{Higher order Fourier $\sigma$-algebras}
Let $f=\lim_\omega f_i$ be a bounded measurable function on the ultra product group ${\bf A}$.
The general correspondence principle for ultra product groups yields that $\|f\|_{U_k}=\lim_\omega\|f_i\|_{U_k}$ holds for every natural number $k$.
Since the ultra limit of positive numbers is non-negative it follows that $U_k$ is a semi norm on $L^\infty({\bf A})$.
We show in this chapter that for every $k\in\mathbb{N}$ there is a unique largest $\sigma$-algebra $\mathcal{F}_k\subset\mathcal{A}$ such that $U_{k+1}$ is a norm on $L^\infty(\mathcal{F}_k)$ and $L^\infty(\mathcal{F}_k)$ is orthogonal to every function with zero $U_{k+1}$ norm. Roughly speaking, a function is measurable in $\mathcal{F}_k$ if it is purely ``structured'' in $k$-th order Fourier analysis.
One of many advantages of the ultra product framework is that there is clear distinction between $k$-th order noise and $k$-th order structure on ${\bf A}$. Functions that have zero $U_{k+1}$ norm are considered to be ``pure noise'' in the $k$-th order setting.
\begin{definition} Let $\mathcal{B}\subseteq\mathcal{A}$ be a $\sigma$-algebra. Then we denote by $[\mathcal{B}]_k$,$[\mathcal{B}]_k^\times$ and $(\mathcal{B})_k^\times$ the $\sigma$-algebra generated by the functions $[F],[F]^\times$ and $(F)^\times$ where $F$ runs through all the function systems $\{f_v\}_{v\in\{0,1\}^k}$ in $L^\infty(\mathcal{B})$.
\end{definition}
We have that $$[\mathcal{B}]^\times_k=\bigvee_{v\in K_k}\mathcal{B}\circ\psi_v~~~{\rm and}~~~(\mathcal{B})^\times_k=\bigvee_{v\in\{0,1\}^k}\mathcal{B}\circ\psi_v.$$
\begin{lemma}\label{convcontlift} Let $\mathcal{B}\subset\mathcal{A}$~,~$k\in\mathbb{N}$ and let $\mathcal{C}\subset\mathcal{A}$ be the unique $\sigma$-algebra with $\mathcal{C}\circ\psi_0=\mathcal{A}\circ\psi_0\wedge[\mathcal{B}]_k^\times$. Then $\mathcal{C}\subset[\mathcal{B}]_k$.
\end{lemma}
\begin{proof} Let $f\in L^\infty(\mathcal{C})$. We have by lemma \ref{siggen} that for every $\epsilon>0$ the function $f\circ\psi_0$ has an epproximation of the form $g=\sum_{i=1}^t[F^i]^\times$ with $L^2$ error at most $\epsilon$ where each $F^i$ is a function systme $\{f^i_v\}_{v\in K_k}$ in $L^\infty(\mathcal{B})$. Then $g'=\sum_{i=1}^t[F^i]$ satisfies $g'\circ\psi_0=\mathbb{E}(g|\psi_0)$ and thus by $\epsilon\geq\|g-f\circ\psi_0\|_2\geq\|\mathbb{E}(g|\psi_0)-f\circ\psi_0\|_2=\|g'-f\|_2$ we have that $f$ has an arbitrary precise approximation in $L^2([\mathcal{B}]_k)$.
\end{proof}
\begin{definition} Let $\mathcal{F}_k$ be the $\sigma$-algebra $[\mathcal{A}]_{k+1}$. We say that $\mathcal{F}_k$ is the {\bf $k$-th order Fourier $\sigma$-algebra}.
\end{definition}
\begin{theorem}[Properties of $\mathcal{F}_k$]\label{propfk} Let $k$ be a natural number. Then
\begin{enumerate}
\item On the space $C^{k+1}({\bf A})$ we have
$\mathcal{F}_k\circ\psi_0=\mathcal{A}\circ\psi_0\wedge [\mathcal{A}]_{k+1}^\times,$
\item $\|f\|_{U_{k+1}}=0$ holds if and only if $\mathbb{E}(f|\mathcal{F}_k)=0$,
\item $U_{k+1}$ is a norm on $L^\infty(\mathcal{F}_k)$,
\item $\mathcal{F}_k$ is shift invariant,
\item On the space $C^{k+1}({\bf A})$ we have
$\mathcal{F}_k\circ\psi_0=\mathcal{A}\circ\psi_0\wedge [\mathcal{F}_k]_{k+1}^\times,$
\item $\mathcal{F}_0$ is the trivial $\sigma$-algebra and $\mathcal{F}_0\subset\mathcal{F}_1\subset\mathcal{F}_2\subset\dots$ is an increasing chain.
\end{enumerate}
\end{theorem}
\begin{proof}
Let $\mathcal{B}$ denote the $\sigma$-algebra whose pre-image under $\psi_0$ is equal to $\mathcal{A}\circ\psi_0\wedge[\mathcal{A}]_{k+1}^\times$. The first statement in the theorem says that $\mathcal{B}=\mathcal{F}_k$. By lemma \ref{convcontlift} we have that $\mathcal{B}\subseteq\mathcal{F}_k$.
For the statement $\mathcal{F}_k\subseteq\mathcal{B}$ it is enough to prove that $[F]\circ\psi_0$ is measurable in $[\mathcal{A}]_{k+1}^\times$ for every function system $F=\{f_v\}_{v\in K_{k+1}}$ in $L^\infty_u(\mathcal{A})$. This follows immediately from lemma \ref{lowrank}.
Let $R$ be the set of rank one functions on $C^{k+1}({\bf A})$. We claim that $\mathbb{E}(f|\mathcal{F}_k)=0$ holds if and only if $f$ is orthogonal to every function in $R$.
One direction is trivial since $L^\infty(\mathcal{F}_k)$ contains $R$. Assume that $f$ is orthogonal to $R$. Let $G=\{g_v\}_{v\in K_{k+1}}$ be a function system in $L^\infty_u({\bf A})$ and let $g=[G]^\times$ be the corresponding rank one function. We have by (\ref{coincon}) that $0=(f,[G])=(f\circ \psi_0,g)$. This shows that $f\circ\psi_0$ is orthogonal to every rank one function and thus $\mathbb{E}(f\circ\psi_0|\mathcal{G})=0$. Using $\mathcal{F}_k\circ\psi_0\subseteq\mathcal{G}$ from the first part of the theorem we obtain $$0=\mathbb{E}(f\circ\psi_0|\mathcal{F}_k\circ\psi_0)=\mathbb{E}(f|\mathcal{F}_k)\circ\psi_0$$ showing that $\mathbb{E}(f|\mathcal{F}_k)=0$.
By (\ref{GCS}) and (\ref{coincon}) we have that $\|f\|_{U_{k+1}}=0$ if and only if $f$ is orthogonal to $R$. This proves the second statement. The third statement follows form the fact that if $f\in L^\infty(\mathcal{F}_k)$ then $f=\mathbb{E}(f|\mathcal{F}_k)$.
Statement four follows from the fact that convolution of shifts of functions (with a fix element $t\in{\bf A}$) is the shift of the convolution. Thus the generating system of $\mathcal{F}_k$ is shift invariant.
For the fifth statement let $\mathcal{B}'\subset\mathcal{A}$ be the $\sigma$-algebra such that $\mathcal{B}'\circ\psi_0=\mathcal{A}\circ\psi_0\wedge[\mathcal{F}_k]_{k+1}^\times$. Our goal is to show that $\mathcal{F}_k=\mathcal{B}'$. It is clear from the first statement of the theorem that $\mathcal{B}'\subseteq\mathcal{F}_k$. To show the other inclusion it is enough to show that every convolution in $R$ is contained in $\mathcal{B}'$. Let $F=\{f_v\}_{v\in K_{k+1}}$ be a function system in $L^\infty_u({\bf A})$. Let $G=\{g_v:=\mathbb{E}(f_v|\mathcal{F}_k)\}_{v\in K_{k+1}}$. By the second part of the theorem we get $\|f_v-g_v\|_{U_{k+1}}=0$ for every $v\in K_{k+1}$. Using the fact that convolution is linear in the components and lemma \ref{cornineq} we conclude that in a process, where we step by step replace the terms $f_v$ by $g_v$ in the system $F$, the convolution doesn't change. It follows that $[G]=[F]$. Now remark \ref{aptrans} and lemma \ref{lowrank} show that $[F]\circ\psi_0=[G]\circ\psi_0$ can be approximated by rank one functions using only translates of the functions $g_v$. Since the functions $g_v$ are all measurable in $\mathcal{F}_k$ and $\mathcal{F}_k$ is shift invariant we obtain that the function $[F]\circ\psi_0$ is measurable in $[\mathcal{F}_k]_{k+1}^\times$. Thus we get that $[F]$ is measurable in $\mathcal{B}'$.
The last statement follows from the fact that every convolution of order $k$ is also a convolution of order $k+1$. this can be seen by using constant one functions in a function system on $K_{k+1}$ outside of a $k$ dimensional face containing the $0$ vector.
\end{proof}
\subsection{Identities for convolutions}
\begin{lemma}\label{fkconvfk} Let $n,k$ be natural numbers and let $F=\{f_v\}_{v\in K_n}$ be a system of functions in $L^\infty(\mathcal{F}_k)$. Then the function $[F]$ is measurable in $\mathcal{F}_k$.
\end{lemma}
\begin{proof} If $n\leq k+1$ then the claim is clear by the definition of $\mathcal{F}_k$. Assume that $n>k+1$. By theorem \ref{propfk} the statement is equivalent with the fact that $[F]$ is orthogonal to any function $g\in L^\infty({\bf A})$ with $\|g\|_{U_{k+1}}=0$. Let $g_v=f_v^\star$ for $v\in K_n$. Then $[F](x)$ can be calculated by the formula (\ref{corner}). It is clear that for every fixed $t_{k+2},t_{k+3},\dots,t_n$ in ${\bf A}$ the expected value of the expression in (\ref{corner}) according to $t_1,t_2,\dots,t_{k+2}$ is a $k+1$-th order convolution of functions in $\mathcal{F}_k$ and thus it is orthogonal to $g$. Then the non-standard version of Fubini's theorem \cite{ESz} finishes the proof.
\end{proof}
\begin{lemma}\label{szorzat} If $f\in L^\infty(\mathcal{F}_k)$ and $g\in L^\infty({\bf A})$ satisfies $\|g\|_{U_{k+1}}=0$ the $\|fg\|_{U_{k+1}}=0$.
\end{lemma}
\begin{proof} Using theorem \ref{propfk} we have $\mathbb{E}(fg|\mathcal{F}_k)=f\mathbb{E}(g|\mathcal{F}_k)=0$ and thus $\|fg\|_{U_{k+1}}=0$.
\end{proof}
\begin{lemma}\label{szomszed} Let $F=\{f_v\}_{v\in \{0,1\}^{k+1}}$ be a function system in $L^\infty(\mathcal{A})$ such that for some pair of two neighbouring vertices $w_1,w_2\in \{0,1\}^{k+1}$ we have $f_{w_1}\in L^\infty(\mathcal{F}_{k-1})$ and $\|f_{w_2}\|_{U_k}=0$. then $(F)=0$.
\end{lemma}
\begin{proof} Assume that $w_1$ and $w_2$ differ at the $i$-th coordinate. Then by lemma \ref{szorzat} we have that for every $t\in{\bf A}$ the system $\delta_{i,t} F$ has a function (obtained from $f_{w_1}$ and $f_{w_2}$) with zero $U_k$-norm. Then by (\ref{GCS}) we have that $(\delta_{i,t}(F))=0$. and thus (\ref{dimred}) finishes the proof.
\end{proof}
The next two lemmas are useful consequences of lemma \ref{szomszed}.
\begin{lemma}\label{twofunct} Let $f,g$ be $L_\infty(\mathcal{A})$ functions such that $f=f_1+f_2$ where $f_1=\mathbb{E}(f|\mathcal{F}_{k-1})$ and $g=g_1+g_2$ where $g_1=\mathbb{E}(g|\mathcal{F}_{k-1})$. Then
\begin{equation}\label{twofun} \mathbb{E}_t\Bigl(\|f(x)\overline{g(x+t)}\|^{2^k}_{U_k}\Bigr)=\mathbb{E}_t\Bigl(\|f_1(x)\overline{g_1(x+t)}\|^{2^k}_{U_k}\Bigl)+\mathbb{E}_t\Bigr(\|f_2(x)\overline{g_2(x+t)}\|^{2^k}_{U_k}\Bigr).
\end{equation}
\end{lemma}
\begin{proof} Let $H=\{h_v\}_{v\in\{0,1\}^{k+1}}$ be the function system such that $h_w=f$ if the last coordinate of $w$ is zero and $h_w=g$ if the last coordinate is one. It is clear by (\ref{dimred}) that the left hand side of (\ref{twofun}) is equal to $(H)$. By the linearity of $(H)$ in each coordinate we can decompose it into $2^{2^{k+1}}$ terms using $f=f_1+f_2$ and $g=g_1+g_2$. Then lemma \ref{szomszed} shows that only the two terms representing the right hand side of (\ref{twofun}) are not necessarily zero.
\end{proof}
\begin{lemma}\label{vetites1} Let $F=\{f_v\}_{v\in K_{k+1}}$ be a function system in $L^\infty({\bf A})$. Furthermore let $G=\{g_v\}_{v\in K_{k+1}}$ with $g_v=\mathbb{E}(f_v|\mathcal{F}_{k-1})$. Then $[G]=\mathbb{E}([F]~|~\mathcal{F}_{k-1})$.
\end{lemma}
\begin{proof} Since $[G]$ is measurable in $\mathcal{F}_{k-1}$ by lemma \ref{fkconvfk}, it is enough to prove that for an arbitrary $h\in L^\infty(\mathcal{F}_{k-1})$ we have $([G],h)=([F],h)$. Let us write $f_v=g_v+r_v$ for every $v\in K_{k+1}$ and notice that $\|r_v\|_{U_{k-1}}=0$. Let $F'=\{f'_v\}_{v\in\{0,1\}^{k+1}}$ be the function system with $f'_v=f_v$ if $v\neq 0$ and $f'_0=\overline{h}$. Then by (\ref{coincon}) we have that $([F],h)=(F')$.
The linearity of the Gowers inner product implies that we can decompose $(F')$ into $2^{2^{k+1}-1}$ terms according to the decompositions $f_v=g_v+r_v$. By lemma \ref{szomszed} we have that the only non zero term is the one where we use $g_v$ at every place. This term is equal to $([G],h)$.
\end{proof}
\medskip
In the rest of the chapter we study conditions that force convolutions of the form $[F]$ to be $0$. For example lemma \ref{cornineq} implies that if any function in the function system $F=\{f_v\}_{v\in K_n}$ has zero $U_n$-norm then $[F]$ is zero. We will need some notation.
For an element $v\in\{0,1\}^n$ we introduce the height $h(v)$ of $v$ as the coordinate sum of $v$. For $v,w\in\{0,1\}^n$ we say that $v\leq w$ if $w_i=0$ implies $v_i=0$ for every $i\in [n]$. In other words $v\leq w$ if ${\rm supp}(v)\subseteq{\rm supp}(w)$. A simplicial set $S\in\{0,1\}^n$ is a set such that $w\in S$ and $v\leq w$ implies $v\in S$. For $v\in S$ the degree $d(v)$ is defined as $\max\{h(w)|w\in S,v\leq w\}$. A maximal element $v$ in $S$ is an element with $h(v)=d(v)$.
A maximal face of $S$ is a set of the form $\{v|v\leq w\}$ where $w\in S$ is maximal.
The hight of $S$ is the maximum of the heights of its elements.
For a number $i\in [n]$ let us define the projection $p_i:\{0,1\}^n\rightarrow\{0,1\}^{n-1}$ given by deleting the $i$-th coordinate. It is clear that the image of a simplicial set $S$ under the projection $p_i$ is again simplicial.
\begin{lemma}\label{simpzero} Let $S\subset\{0,1\}^n$ be a simplicial set and let $s,u\in S$ be two distinct elements. Let $K=\{0,1\}^n\setminus\{u\}$ and $k=d(s)$. Let $F=\{f_v\}_{v\in K}$ be a function system in $L^\infty({\bf A})$ such that $\|f_s\|_{U_k}=0$ and $f_v=1_{{\bf A}}$ if $v\in K\setminus S$.
Then the convolution $[F]$ (taken at $u$) is identically $0$.
\end{lemma}
\begin{proof}
For the definition of $[F]$ in this case see remark \ref{convat}. If $h(s)=n$ or $h(u)=n$ then $S=\{0,1\}^n$ and thus $k=n$. In this this case lemma \ref{cornineq} shows that $[F]=0$. We prove the statement by induction on $n-h(s)$. The case $n-h(s)=0$ is now proved. We can assume that $h(s),h(u)<n$.
Using the fact that neither of $s$ and $u$ is the all $1$ vector we get that there is a coordinate $r\in [n]$ such that $s_r=0$ and the vector $s'$ obtained from $s$ by changing the $r$-th coordinate to $1$ satisfies $s'\neq u$. Let us decompose $f_{s'}$ as $f_{s'}=g_1+g_2$ where $g_1=\mathbb{E}(f_{s'}|\mathcal{F}_{k-1})$and $\|g_2\|_{U_k}=0$. Similarly we introduce two function systems $F_1,F_2$ where $F_i$ is obtained from $F$ by replacing $f_{s'}$ by $g_i$.
By linearity of convolution we get that $[F]=[F_1]+[F_2]$. If $s'\notin S$ then $f_{s'}=1_{{\bf A}}$~,~$g_2=0$ and so $[F_2]=0$. If $s'\in S$ then $d(s')=d(s)=k$ and so by induction $[F_2]=0$. It remains to show that $[F_1]=0$.
We use our induction step for the function system $\delta_{r,t} F_1$. Notice that by lemma \ref{szorzat} the function $f_s(x)\overline{f_{s'}(x+t)}$ of coordinate $p_r(s)$ in $\delta_{r,t} F_1$ has zero $U_k$ norm. It is clear that in the complement of $p_r(S)$ every function in $\delta_{r,t} F_1$ is $1_{{\bf A}}$. Since $h(p_r(s))=h(s)$ and $d(p_r(s))\leq d(s)=k$ we have by induction that $[\delta_{r,t} F_1]=0$. By (\ref{dimred}) we obtain that $[F_1]=0$.
\end{proof}
\begin{corollary}\label{simpzerocor} Let $S\subset\{0,1\}^n$ be a simplicial set of hight at most $k$, and let $\{f_v\}_{v\in\{0,1\}^n}$ be a function system in $L^\infty({\bf A})$ such that $f_v=1_{{\bf A}}$ if $v\in\{0,1\}^n\setminus S$. Let $G=\{g_v:=\mathbb{E}(f_v|\mathcal{F}_{k-1})\}_{v\in\{0,1\}^n}$. Then $(F)=(G)$.
\end{corollary}
\begin{proof} By the multi linearity of $(F)$ it is enough to prove that if $\|f_v\|_{U_k}=0$ holds for some $v\in S$ then $(F)=0$. (Then the statement follows by decomposing each $f_v$ as $g_v+(f_v-g_v)$ where $\|f_v-g_v\|_{U_k}=0$.)
Let $u\neq v$ be some element in $S$. We have by (\ref{coincon}) that $(F)$ is the scalar product of the convolution of $F$ taken at $u$ with $\overline{f_u}$. Then lemma \ref{simpzero} finishes the proof.
\end{proof}
\subsection{Higher order dual groups and Fourier decompositions}
Fourier analysis on a compact abelian group relies on the fact that $L^2(A)$ is the orthogonal sum of one dimensional shift invariant subspaces. This decomposition is unique and the one dimensional subspaces are forming an abelian group $\hat{A}$ under point wise multiplication. The group $\hat{A}$ is a discrete ablelian group called the {\bf dual group} of $A$. Each one dimensional shift invariant subspace is generated by a continuous homomorphism form $A$ to the unit circle in the complex plane. Consequently $\hat{A}$ is also the group of linear characters under pointwise multiplication.
The decomposition
\begin{equation}\label{mdecomp1}
L^2(A)=\bigoplus_{\chi\in\hat{A}}W_\chi
\end{equation}
give rise to the Fourier decomposition
\begin{equation}\label{fdecomp1}
f=\sum_{\chi\in\hat{A}}f_\chi
\end{equation}
converging in $L_2$ where $f_\chi$ is the projection of $f$ to $W_\chi$.
\begin{remark} Let $X$ be an affine version of $A$ (see chapter \ref{cubes}). Then then the same decomposition as (\ref{mdecomp1}) holds for $L^2(X)$ where the one dimensional subspaces are again indexed by $\hat{A}$. This shows that the Fourier decomposition (\ref{fdecomp1}) can be uniquely defined on $X$ despite of the fact that linear characters are not uniquely defined on $X$ (they depend on a constant multiplicative factor).
\end{remark}
In this chapter we study similar decompositions in $L^2({\bf A})$ for an ultra product group ${\bf A}$. We will see that a new interesting phenomenon emerges in the ultra product setting which is a crucial part of our approach to higher order Fourier analysis.
Let $\hat{{\bf A}}$ denote the set of one dimensional shift invariant subspaces of $L^2({\bf A})$. The surprising fact is that $L^2({\bf A})$ is not generated by the spaces in $\hat{{\bf A}}$ and thus ordinary Fourier analysis is not enough to treat an arbitrary measurable function on ${\bf A}$. In fact it turns out that the space spanned by the spaces in $\hat{{\bf A}}$ is exactly $L^2(\mathcal{F}_1({\bf A}))$ where $\mathcal{F}_1$ is the first order Fourier $\sigma$-algebra on ${\bf A}$.
This means that the use of ordinary Fourier analysis is restricted to functions that are measurable in $\mathcal{F}_1$.
We will need higher order generalizations of $\hat{{\bf A}}$ to define the analogy of (\ref{mdecomp1}) and (\ref{fdecomp1}) for functions that are measurable in $\mathcal{F}_k$.
\begin{definition} A {\bf module of order $k$} is a closed subspace $W\subset L^2({\bf A})$ such that if $f\in L^\infty(\mathcal{F}_{k-1}({\bf A}))$ then $fW\subseteq W$ (using pointwise multiplication). For every set of elements $\{\phi_i\}_{i\in I}$ in $L^2({\bf A})$ there is a unique smallest module $W$of order $k$ containing all of them. We say that $W$ is generated by the system $\{\phi_i\}_{i\in I}$. The {\bf rank} of $W$ is the smallest cardinality of a generating system.
\end{definition}
Note that modules of order one are just linear subspaces of $L^2({\bf A})$. In this case rank is equal to the dimension of the subspace. Using the above definition we arrive to our main definition.
\begin{definition} A {\bf $k$-th order character} of ${\bf A}$ is a function $\phi:{\bf A}\rightarrow\mathbb{C}$ of absolute value one such that $\Delta_t\phi$ is measurable in $\mathcal{F}_{k-1}$ for every $t\in{\bf A}$. The {\bf $k$-th order dual group} $\hat{{\bf A}}_k$ of ${\bf A}$ is the set of $k$-th order rank one modules generated by $k$-th order characters.
\end{definition}
Note that the definition implies that every element of $\hat{{\bf A}}_k$ is a shift invariant rank one module of order $k$. We will see later that $\hat{{\bf A}}_k$ could be equivalently defined as the set of shift invariant rank one modules of order $k$.
To justify the name ``higher order dual group'' we need to give a group structure to it. It is basically the point wise multiplication but we need to define it carefully. The product of two $L^2$ functions is not necessary in $L^2$. For this reason we define the product of $W_1,W_2\in\hat{{\bf A}}_k$ as the $L_2$ closure of the set of products $f_1f_2$ where $f_1$ and $f_2$ are bounded functions from $W_1$ and $W_2$.
It is clear that $\hat{{\bf A}}_k$ becomes an abelian group with this multiplication where the inverse of an element $W$ is obtained by conjugating the elements in $W$.
Note that $\hat{{\bf A}}_k$ is isomorphic to the group of $k$-th order characters factored out by the group of functions in $L^\infty(\mathcal{F}_{k-1}({\bf A}))$ of absolute value one.
\begin{lemma}\label{erosort} Let $\mathcal{B}\subseteq\mathcal{A}({\bf A})$ be a shift invariant $\sigma$-algebra. Let $\phi:{\bf A}\rightarrow\mathbb{C}$ be a function with $|\phi|=1$ such that $\Delta_t\phi$ is measurable in $\mathcal{B}$ for every $t\in{\bf A}$. Then either $\mathbb{E}(\phi|\mathcal{B})$ is constant $0$ or $\phi\in L^\infty(\mathcal{B})$. In particular if $\phi$ is a $k$-th order character which is not in the trivial module then $\mathbb{E}(\phi|\mathcal{F}_{k-1})=0$.
\end{lemma}
\begin{proof}
Let $f=\mathbb{E}(\phi|\mathcal{B})$. Then for every fixed $t$ we have $$f(x+t)=\mathbb{E}(\phi(x+t)|\mathcal{B})=\mathbb{E}(\phi\overline{\Delta_t\phi}|\mathcal{B})=f\overline{\Delta_t\phi}$$
and thus $f\overline\phi$ is translation invariant. We obtain that $f=c\phi$ for some constant $c$. Using that $f$ is the projection of $\phi$ to $L^2(\mathcal{B})$ we get that either $f=0$ or $f=\phi$.
\end{proof}
\begin{lemma}\label{kisk} Every $k$-th order character (and thus every module in $\hat{{\bf A}}_k$) is in $L^2(\mathcal{F}_k)$.
\end{lemma}
\begin{proof} Let $\phi$ be a $k$-th order character. We have by (\ref{dimred}) that $$\mathbb{E}_t(\|\Delta_t\phi\|_{U_k}^{2^k})=\|\phi\|_{U_{k+1}}^{2^{k+1}}.$$
Using that $\Delta_t\phi$ is measurable in $\mathcal{F}_{k-1}$ we get by theorem \ref{propfk} that $\|\Delta_t\phi\|_{U_k}>0$ holds for every $t$ and thus by the above formula $\|\phi\|_{U_{k+1}}>0$. This means by theorem \ref{propfk} that $\mathbb{E}(\phi|\mathcal{F}_k)\neq 0$. Since $\phi$ is also a $k+1$-th order character we obtain by lemma \ref{erosort} that $\phi$ has to be in the trivial module and so $\phi$ is measurable in $\mathcal{F}_k$.
\end{proof}
\begin{lemma}\label{charort} Every two distinct modules $W_1,W_2$ in $\hat{{\bf A}}_k$ are orthogonal to each other.
\end{lemma}
\begin{proof} Let $g_1=f_1\phi_1\in W_1$ and $g_2=f_2\phi_2\in W_2$ be two elements where $\phi_1,\phi_2$ are $k$-th order characters and $f_1,f_2\in L^\infty(\mathcal{F}_{k-1}({\bf A}))$. We have that $$(g_1,g_2)=\mathbb{E}(\mathbb{E}(g_1\overline{g_2}|\mathcal{F}_{k-1}))=\mathbb{E}(f_1f_2\mathbb{E}(\phi_1\overline{\phi_2}|\mathcal{F}_{k-1})).$$
By lemma \ref{erosort} the right hand side is $0$. Since such elements $g_1$ and $g_2$ are $L^2$ dense in $W_1$ and $W_2$ which completes the proof.
\end{proof}
An important consequence of our main result, theorem \ref{main} is the following.
\begin{theorem}[Higher order Fourier decomposition]\label{hofdecomp} For every $1\leq k\in\mathbb{N}$ we have that
$$L^2(\mathcal{F}_k({\bf A}))=\bigoplus_{W\in\hat{{\bf A}}_k}W$$
and so every function $f\in L^2(\mathcal{F}_k({\bf A}))$ has a unique decomposition
$$f=\sum_{W\in\hat{{\bf A}}_k}f_W$$ converging in $L^2$ where $f_W$ is the projection of $f$ to the modul $W$.
\end{theorem}
In the rest of this chapter we focus on the measure theoretic properties of higher order characters.
The simplest examples for $k$-th order characters are functions $\phi:{\bf A}\rightarrow\mathcal{C}$ such that
$$\Delta_{t_1,t_2,\dots,t_{k+1}}\phi(x)=1$$
for every $t_1,t_2,\dots,t_{k+1},x$ in ${\bf A}$ or equivalently: $(\phi)_{k+1}^\times=1$.
Such functions could be be called {\bf pure} characters.
Unfortunately for $k>1$ there are groups ${\bf A}$ on which not every modul in $\hat{{\bf A}}_k$ can be represented by a pure character.
This justifies the next definition.
\begin{definition}[Locally pure characters] Let $\mathcal{B}\subseteq\mathcal{A}({\bf A})$ be any $\sigma$ algebra.
We denote by $[\mathcal{B},k]^*$ the set of functions $\phi:{\bf A}\rightarrow\mathbb{C}$ of absolute value $1$ such that $(\phi)^\times_{k+1}$ is measurable in $(\mathcal{B})_{k+1}^\times$.
\end{definition}
In case $\mathcal{B}$ is the trivial $\sigma$-algebra then $[\mathcal{B},k]^*$ is just the set of pure characters with $(\phi)_{k+1}^\times=1$.
\begin{lemma}\label{pureprop} Let $\mathcal{B}\subseteq\mathcal{A}({\bf A})$ be a $\sigma$-algebra. Then: \begin{enumerate}
\item $[\mathcal{B},k]^*$ is an Abelian group with respect to point wise multiplication.
\item $[\mathcal{B},0]^*$ is the set of $\mathcal{B}$ measurable functions $f:{\bf A}\rightarrow \mathbb{C}$ of absolute value $1$.
\item $[\mathcal{B},k]^*\subseteq [\mathcal{B},k+1]^*$
\item If $\mathcal{B}$ is shift invariant and $\phi\in[\mathcal{B},k]^*$ then $\Delta_t\phi\in[\mathcal{B},k-1]^*$ for every $t\in{\bf A}$
\end{enumerate}
\end{lemma}
\begin{proof} The first three properties are trivial. We show the last statement. Let $f=f(x,t_1,t_2,\dots,t_{k+1})$ be a bounded function on $C^{k+1}({\bf A})$ using the parametrization $C^{k+1}\simeq{\bf A}^{k+2}$ introduced in chapter \ref{cubes} such that $f$ is measurable in $(\mathcal{B})_{k+1}^\times$. We claim that for almost every fixed value of $t_k$ the restriction of $f$ defined on $C^k({\bf A})$ is measurable in $(\mathcal{B})_k^\times$.
Using the shift invariance of $\mathcal{B}$ the statement is obviously true for functions in $\mathcal{R}(\{\mathcal{B}\circ\psi_v\}_{v\in\{0,1\}^{k+1}})$ (see notation in lemma \ref{siggen}). Then lemma \ref{siggen} shows the general case.
By applying the claim for $(\phi)_{k+1}^\times$ we get that for almost every $t\in{\bf A}$ the function $(\Delta_t\phi)_k^\times$ is measurable in $(\mathcal{B})_k^\times$ and thus for such $t$'s we have that $\Delta_t\phi\in [\mathcal{B},k-1]^*$. Now let $t\in{\bf A}$ be an arbitrary fixed element. By the previous result both $\Delta_{t'}\phi$ and $\Delta_{t-t'}\phi$ are in $[\mathcal{B},k-1]^*$ for almost every $t'$ and by fixing one such element we obtain that $\Delta_t\phi(x)=\Delta_{t'}\phi(x)\Delta_{t-t'}\phi(x+t')$ is in $[\mathcal{B},k-1]^*$.
\end{proof}
\begin{lemma}\label{charmes} Every element in $[\mathcal{B},k]^*$ is measurable in the $\sigma$-algebra $\mathcal{F}_k\vee\mathcal{B}$.
\end{lemma}
\begin{proof} Assume that $f\in[\mathcal{B},k]^*$. Let $\mathcal{C}=[\mathcal{A}]_{k+1}^\times$.
Let $g_1=[f]_{k+1}^\times$ and $g_2=f\circ\psi_0/g_1$ on $C^{k+1}({\bf A})$.
We have that $g_1$ is measurable in $\mathcal{C}\vee\mathcal{B}\circ\psi_0$ and $g_2$ is measurable in $\mathcal{C}$. This means that $f\circ\psi_0=g_1g_2$ is measurable in $\mathcal{C}\vee\mathcal{B}\circ\psi_0$. On the other hand $f\circ\psi_0$ is measurable in $\mathcal{A}\circ\psi_0$ and thus it is measurable in
$(\mathcal{C}\vee\mathcal{B}\circ\psi_0)\wedge\mathcal{A}\circ\psi_0.$
We claim that $\mathcal{A}\circ\psi_0$ is conditionally independent from $\mathcal{C}$. By theorem \ref{propfk} we have that $\mathcal{A}\circ\psi_0\wedge\mathcal{C}=\mathcal{F}_k\circ\psi_0$. Assume that $h\in L^\infty({\bf A})$ is orthogonal to $L^2(\mathcal{F}_k)$. Then $\|h\|_{U_{k+1}}=0$ and thus by (\ref{GCS}) we have that $h\circ\psi_0$ is orthogonal to every element in $\mathcal{R}(\{\mathcal{A}\circ\psi_v\}_{v\in\{0,1\}^{k+1}})$. Lemma \ref{siggen} shows that $h\circ\psi_0$ is orthogonal to $L^2(\mathcal{C})$ which is needed for conditional independence.
Now lemma \ref{siggen2} implies that $f\circ\psi_0$ is measurable in
$(\mathcal{C}\wedge\mathcal{A}\circ\psi_0)\vee\mathcal{B}\circ\psi_0$ which by theorem \ref{propfk} is equal to $\mathcal{F}_k\circ\psi_0\vee\mathcal{B}\circ\psi_0$.
\end{proof}
\begin{lemma}\label{charsep} If $\phi$ is a $k$-th order character then $\phi\in[\mathcal{F}_{k-1},k]^*$.
\end{lemma}
\begin{proof} It is clear that for every $t\in C^{k+1}({\bf A})$ we have that $\Delta_t (\phi)_{k+1}^\times$ is measurable in $\mathcal{B}=(\mathcal{F}_{k-1})_{k+1}^\times$. Using that $\mathcal{B}$ is shift invariant and lemma \ref{erosort} we obtain that either $(\phi)_{k+1}^\times$ is measurable in $\mathcal{B}$ or $E((\phi)_{k+1}^\times|\mathcal{B})=0$.
However the second possibility is impossible since the integral of $(\phi)_{k+1}^\times$ is equal to the $k+1$-th Gowers norm of $\phi$ which is positive because $\phi\in\mathcal{F}_k$.
\end{proof}
\begin{lemma}\label{charchar} A function $\phi:{\bf A}\rightarrow\mathbb{C}$ is a $k$-th order character if and only if $\phi\in[\mathcal{F}_{k-1},k]^*$.
\end{lemma}
\begin{proof} Lemma \ref{charsep} shows one implication. To see the other implication assume that $\phi\in[\mathcal{F}_{k-1},k]^*$. Then if $t\in{\bf A}$ is arbitrary we have by lemma \ref{pureprop} that $\Delta_t\phi\in [\mathcal{F}_{k-1},k-1]^*$. By lemma \ref{charmes} we obtain that $\Delta_t\phi$ is measurable in $\mathcal{F}_{k-1}$ showing that $\phi$ is a $k$-th order character.
\end{proof}
\begin{lemma}\label{charconv} Let $n,k\in\mathbb{N}^+$. Let $\beta:K_n\rightarrow\hat{{\bf A}}_k$ be a map such that $\beta\notin\hom(K_n,\mathcal{D}_{n-k-1}(\hat{{\bf A}}_k))$ and let $F=\{f_v\}_{v\in K_n}$ be a system of bounded functions on ${\bf A}$ such that $f_v$ is in $\beta(v)$. Then $[F]$ is identically $0$.
\end{lemma}
\begin{proof} We go by induction on $n$. If $n=1$ then lemma \ref{erosort} shows the claim. Assume that the statement holds for $n-1\geq 1$.
Let $S\subset K_n$ be a face such that $\sum_{v\in S}\beta(v)\neq 0$ and assume that $i\in [n]$ is a direction parallel to $S$. Then $\delta_{i,t} F$ satisfies the condition for $n-1$ and thus by induction we have that $[\delta_{i,t} F]$ is identically $0$. The statement for $n$ follows from (\ref{dimred}).
\end{proof}
\subsection{Topological nilspace factors of ultra product groups}
\bigskip
\begin{definition} A continuous surjective function $\gamma:{\bf A}\rightarrow T$ into a compact Hausdorff space $T$ is a {\bf $k$-step nilspace factor} of ${\bf A}$ if the cubespace structure on $T$ (obtained by composing cubes in ${\bf A}$ with $f$) is a $k$-step nilspace. Equivalently, a $k$-step nilspace factor of ${\bf A}$ is given by a continuous nilspace morphism $\gamma:{\bf A}\rightarrow N$ into a compact $k$-step nilspace such that the induced maps $\gamma^n:C^n({\bf A})\rightarrow C^n(N)$ are surjective for every $n$.
\end{definition}
Note that nilspace factors of ultra product groups are automatically compact nilspaces.
Throughout this chapter $N$ is a $k$-step nilspace with structure groups $A_1,A_2,\dots,A_k$ and $i$-step factors $\pi_i:N\rightarrow N_i$ where $0\leq i\leq k$.
We will need the following lemma.
\begin{lemma}\label{homok} Let $\gamma:{\bf A}\rightarrow N$ be a $k$-step nilspace factor of ${\bf A}$. Then $\gamma$ is measurable in $\mathcal{F}_k({\bf A})$ and there are homomorphism $\tau_k:A_i\rightarrow\hat{{\bf A}}_i$ for $1\leq i\leq k$ such that if $\chi\in\hat{A_i}$ and $f\in W(\chi,N_i)$ then $f\circ\pi_i\circ\gamma$ is in the module $\tau_i(\chi)$.
\end{lemma}
\begin{proof} By lemma \ref{kisk} and lemma \ref{nilfourdec} the calim that $\gamma$ is measurable in $\mathcal{F}_k$ follows from the existence of the map $\tau_k$. By induction on $k$ we can assume that the maps $\{\tau_i\}_{i=1}^{k-1}$ exist as required by the lemma. To see the existence of $\tau_k$ let $\chi\in\hat{A}_k$ be an arbitrary character and let $\phi\in V(\chi,N)$ be a function of absolute value $1$ (guaranteed by lemma \ref{nilspchar}). We have that $(\phi)_{k+1}^\times\circ\gamma^{k+1}=(\phi\circ\gamma)_{k+1}^\times$.
From our induction hypothesis, lemma \ref{nilspcharderiv}, lemma \ref{charchar} we obtain that $\phi\circ\gamma$ is a $k$-th order character that we denote by $\tau_k({\bf A})$. Then lemma \ref{nilspchar} and our induction assumption will guarantee that $\tau_k(\chi)$ depends only on $\chi$ and thus $\tau_k:\hat{A}_k\rightarrow\hat{{\bf A}}_k$ is well defined.
The fact that $\tau_k$ is a homomorphism is clear from the definitions.
\end{proof}
We introduce the following four properties for nilspace factors.
\medskip
\noindent{\bf 1.)~measure preserving:}~We say that the nilspace factor $\xi:{\bf A}\rightarrow N$ is measure preserving if all the maps $\gamma^n:C^n({\bf A})\rightarrow C^n(N)$ are measure preserving.
\medskip
\noindent{\bf 2.)~Rooted measure preserving:}~We say that $\gamma$ is rooted measure preserving if for every $a\in{\bf A}$ and natural number $n\in\mathbb{N}$ the map $\gamma^n_a:C^n_a({\bf A})\rightarrow C^n_{\gamma(a)}(N)$ induced by $\gamma$ is measure preserving.
\medskip
\noindent{\bf 3.)~Character preserving:}~The homomorphisms $\tau_i:\hat{A}_k\rightarrow\hat{{\bf A}}_k$ defined in lemma \ref{homok} are all injective.
\medskip
\noindent{\bf 4.)~Factor consistent:}~We say that $\gamma:{\bf A}\rightarrow N$ is factor consistent if for every $1\leq i\leq k$ and bounded measurable function $f:N\rightarrow\mathbb{C}$ the function $\mathbb{E}(f\circ\gamma|\mathcal{F}_i({\bf A}))$ is measurable in the $\sigma$-algebra of $\pi_i\circ\gamma$.
\medskip
\begin{theorem}\label{charpres} Let $\gamma:{\bf A}\rightarrow N$ be a $k$-step nilspace factor. Then the following statements are equivalent.
\begin{enumerate}
\item $\gamma$ is character preserving,
\item $\gamma$ is rooted measure preserving,
\item $\gamma$ is measure preserving,
\item $\gamma$ is factor consistent.
\end{enumerate}
\end{theorem}
\begin{definition} A nilspace factor $\gamma$ satisfying the equivalent conditions in theorem \ref{charpres} is called a {\bf strong} nilspace factor.
\end{definition}
\bigskip
\noindent{\it Proof of theorem \ref{charpres}:} We prove the statement by induction on $k$. If $k$ is $0$ then everything is trivial. Assume that the statement is true for $k-1$. This means that the factor $\gamma_{k-1}$ satisfies all the conditions simultaneously.
\noindent~$(1)\Rightarrow (2)$:~~
Let $x\in{\bf A}$ be an arbitrary element and let $y=\gamma(x)$.
Let $\nu$ denote the probability distribution on $C_y^n(N)$ obtained by composing the uniform distribution on $C_x^n({\bf A})$ by $\gamma$. By theorem \ref{bundec} $C_y^n(N)$ is a $C^n_0(\mathcal{D}_k(A_k))$-bundle over $C^n_{\pi_{k-1}(y)}(N_{k-1})$. Thus by induction it is enough to show that $\nu$ is invariant under the natural action of $C_0^n(\mathcal{D}_k(A_k))$ on $C_y^n(N)$.
This invariance can be proved by showing for a function system $U$ which linearly spans an $L^1$-dense set in $L^\infty(C_y^n(N))$ that for every $r\in C_0^n(\mathcal{D}_k(A_k))$ and $u\in U$ the equation $\mathbb{E}_\nu(u)=\mathbb{E}_\nu(u^r)$ holds. (The shift $u^r$ of $u$ is defined as the function satisfying $u^r(y)=u(y+r)$.)
We define $U$ as the collection of all functions $[F]^\times$
on $C_y^n(N)$ where $\{\chi_v\in\hat{A_k}\}_{v\in K_n}$ is a system of characters and $f_v\in W{\chi_v,N)}$ is a continuous function for every $v\in K_n$.
The set $U$ is closed under multiplication and contains a separating system of continuous functions. It follows from the Stone-Weierstrass theorem that every function in $L^\infty(C_y^n(N))$ can be approximated by some finite linear combination of elements from $U$.
We have for $r\in C_0^n(\mathcal{D}_k(A_k))$ that
\begin{equation}\label{chmult}
([F]^\times)^r=[F]^\times\prod_{v\in K_n}\chi_v(r_v).
\end{equation}
where $r_v=\psi_v(r)$ is the component of $r$ at $v$.
There are two cases.
In the first case the function $\beta: v\rightarrow \chi_v$ is not in $\hom(K_n,\mathcal{D}_{n-k-1}(\hat{A_k}))$. In this case by the character preserving property of $\gamma$ and by lemma \ref{charconv} we get that $\mathbb{E}_\nu([F]^\times)=\mathbb{E}_\mu([F\circ\gamma]^\times)=0$ and so by (\ref{chmult}) we have $\mathbb{E}_\nu([F]^\times)=0=\mathbb{E}_\nu(([F]^\times)^r)$.
In the second case $\beta\in\hom(K_n,\mathcal{D}_{n-k-1}(\hat{A}_k))$. Then we have by lemma \ref{dualker2} that $\prod_{v\in K_n}\chi_v(r_v)=1$.
\bigskip
\noindent$(2)\Rightarrow(3):$~~ Notice that a random element in $C_x^{n+1}({\bf A})$ (resp. $C_{\gamma(x)}^{n+1}(N)$) restricted to an $n$ dimensional face of $\{0,1\}^{n+1}$ not containing $0^{n+1}$ is $C^n({\bf A})$ (resp. $C^n(N)$) with the uniform distribution.
\bigskip
\noindent$(3)\Rightarrow(4):$~~ Let $f:N\rightarrow\mathbb{C}$ be anarbitrary measurable function and let $f_1=\mathbb{E}(f|\pi_i)~,~f_2=f-f_1$. We have by lemma \ref{homok} that $f_1\circ\gamma$ is measurable in $\mathcal{F}_i$. By lemma \ref{gownilproj} we have that $\|f_2\|_{U_{i+1}}=\|\mathbb{E}(f_2|\pi_i)\|_{U_{i+1}}=0$. By the measure preserving property of $\gamma$ we get that $$0=\|f_2\|_{U_{i+1}}^{2^{i+1}}=\mathbb{E}((f_2)^\times)=\mathbb{E}((f_2\circ\gamma)^\times)=\|f_2\circ\gamma\|_{U_{i+1}}^{2^{i+1}}.$$ It follows from theorem \ref{propfk} that $f_1\circ\gamma=\mathbb{E}(f\circ\gamma|\mathcal{F}_i)$.
\bigskip
\noindent$(4)\Rightarrow(1):$~~ Let $f:N\rightarrow\mathbb{C}$ be a function in $W(\chi,N)$ of absolute value $1$.
We show that if $\chi$ is non-trivial then $f\circ\gamma$ is non-trivial in $\hat{{\bf A}}_k$. This is equivalent with saying that $f\circ\gamma$ is not measurable in $\mathcal{F}_{k-1}({\bf A})$.
Assume by contradiction that $g=f\circ\gamma$ is measurable in $\mathcal{F}_{k-1}$. We have by factor consistency that there is a function $h:N_{k-1}\rightarrow\mathbb{C}$ such that $f\circ\gamma=h\circ\pi_{k-1}\circ\gamma$ almost everywhere on ${\bf A}$. Our induction hypothesis implies that the factor $\pi_{k-1}\circ\gamma$ is rooted measure preserving. This means by lemma \ref{nilspcharderiv} that for every fix $x\in{\bf A}$ we have $$[f\circ\gamma]_{k+1}(x)=\mathbb{E}([f\circ\gamma]_{k+1}^\times(x))=\mathbb{E}([f]_{k+1}^\times(\gamma(x)))=[f]_{k+1}(\gamma(x)).$$
By $([f\circ\gamma]_{k+1},\overline{f})=\|f\circ\gamma\|^{2^{k+1}}_{U_{k+1}}\neq 0$ we have that $[f\circ\gamma]_{k+1}$ is not identically $0$.
Furthermore by the induction hypothesis we have that
$$[h]_{k+1}(\pi_{k-1}(\gamma(x)))=[h\circ\pi_{k-1}\circ\gamma]_{k+1}(x)=[f\circ\gamma]_{k+1}(x).$$
It follwos that $[h\circ\pi_{k-1}]_{k+1}=[f]_{k+1}$ holds on $N$. This is a contradiction since by lemma \ref{nilspcharderiv} we have that $[f]_{k+1}\in W(\overline{\chi},N)$ and so it does not factor through $\pi_{k-1}$.
\medskip
\begin{lemma}\label{strongcirc} Let $\gamma:{\bf A}\rightarrow N$ be a strong $k$-step nilspace factor and let $\phi:N\rightarrow M$ be a fibre surjective morphism. Then $\phi\circ\gamma$ is a strong nilspace factor.
\end{lemma}
\begin{proof} Using the fact that $\phi$ induces measure preserving maps from $C^n(N)$ to $C^n(M)$ it follows that $\phi\circ\gamma$ is measure preserving. Then theorem \ref{charpres} shows the claim.
\end{proof}
\section{The main theorem}
We are ready to state the main theorem of this paper which is our crucial tool to describe higher order Fourier analysis.
The result is a decomposition theorem of an arbitrary bounded measurable function on ${\bf A}$ into a $k$-th ordered structured part and a $k$-th order random part where randomness is measured by the Gowers norm $U_{k+1}$.
\begin{theorem}[Main theorem]\label{main} Let $f:{\bf A}\rightarrow\mathbb{C}$ be an arbitrary bounded measurable function and $k\in\mathbb{N}$ be a natural number. Then there is a strong $k$-step nilspace factor $\gamma:{\bf A}\rightarrow N$ and a Borel measurable function $h:N\rightarrow\mathbb{C}$ such that $\|f-h\circ\gamma\|_{U_{k+1}}=0$.
\end{theorem}
\begin{remark} Note that in the theorem \ref{main} the decomposition $f=f_s+f_r$ with $f_s=h\circ\gamma$ and $\|f_r\|_{U_{k+1}}=0$ is unique since $f_s=\mathbb{E}(f|\mathcal{F}_k)$. This follows from the fact that $f_s$ is measurable in $\mathcal{F}_k$ and $f_r$ is orthogonal to $L^2(\mathcal{F}_k)$.
\end{remark}
An equivalent formulation of the main theorem is the following version of it.
\begin{theorem}[Second version of the main theorem]\label{main2} A function $f:{\bf A}\rightarrow\mathbb{C}$ is measurable in $\mathcal{F}_k$ if and only if $f=h\circ\gamma$ for some strong $k$-step nilspace factor $\gamma:{\bf A}\rightarrow N$ and Borel measurable function $h:N\rightarrow\mathbb{C}$.
\end{theorem}
It is easy to see that in this statement we don't need to assume that $f$ is bounded. Various strengthenings of the main theorem can also be formulated. These statements have essentially the same proof.
\begin{theorem}[Third version of the main theorem]\label{main3} For every separable $\sigma$-algebra $\mathcal{G}\subset\mathcal{F}_k$ there is a separable $k$-step nilspace factor $\gamma:{\bf A}\rightarrow N$ such that $\mathcal{G}$ is contained in the $\sigma$-algebra generated by $\gamma$.
\end{theorem}
\begin{remark}[Affine invariance] In theorem \ref{main3} one can also assume that the topological factor $\gamma$ is invariant under a countable set of prescribed invertible affine transformations of the form $\alpha:{\bf A}\rightarrow{\bf A}$ defined by $\alpha(x)=nx+a,~n\in\mathbb{N},a\in{\bf A}$. This helps in connecting our results with classical ergodic theory.
\end{remark}
\bigskip
\subsection{Outline of the proof}\label{sketch}
In this chapter we list the major components of the proof of theorem \ref{main3}.
\bigskip
\noindent{\bf Step 1. (Finding the $\sigma$-algabra)}~Before finding $\gamma$ we construct a weaker object namely the $\sigma$-algebra generated by $\gamma$. The idea is to mimic the properties of $\mathcal{F}_k$ by a separable sub $\sigma$-algabra containing $\mathcal{G}$.
\begin{definition} Let $\mathcal{B}\subset\mathcal{A}$ be a $\sigma$-algebra. We say that $\mathcal{B}$ is a {\bf nil $\sigma$-algebra} (of order $k$) if $\mathcal{B}=[\mathcal{B}]_{k+1}$ and for every $2\leq i\leq k+1$ we have that on $C^i({\bf A})$
\begin{equation}\label{rhsnil}
[\mathcal{B}]_i\circ\psi_0=\mathcal{A}\circ\psi_0\wedge[[\mathcal{B}]_i]_i^\times
\end{equation}
\end{definition}
With the help of the next lemma we extend $\mathcal{G}$ into a separable nil $\sigma$-algebra $\mathcal{B}$.
\begin{lemma}\label{embednil} Every separable sigma algebra in $\mathcal{F}_k$ is contained in a separable nil $\sigma$-algebra of order $k$.
\end{lemma}
In the rest of the proof it will be enough to show that each separable nil $\sigma$-algebras is generated by a strong nilspace factor.
\bigskip
\noindent{\bf Step 2.~(Topologization)}~In this step we construct a topological factor from a separable nil $\sigma$-algebra $\mathcal{B}\subset\mathcal{F}_k$. Let $\gamma:{\bf A}\rightarrow N$ denote the topological factor generated by all convolutions $[F]$ where $\{f_v\}_{v\in K_{k+1}}$ is a function system in $L^\infty(\mathcal{B})$.
We say that a function $f$ is continuous (resp. a set $S\subseteq{\bf A}$ is open) in $\gamma$ if $f=h\circ\gamma$ (resp. $S=\gamma^{-1}(S')$) for some continuous function $h$ (resp. open set $S'$) on $N$.
Note that $N$ has an inherited cubic structure $\{C^n(N)\}_{n=0}^\infty$ which arises by composing cubes in ${\bf A}$ with $\gamma$.
It will be a useful point of view that $\gamma$ is the topological factor generated by the single map $x\rightarrow \mathcal{B}\circ\Psi^{k+1}_x$ in the coupling topology.
To prove theorem \ref{main3} it is enough to show the following proposition.
\begin{proposition}[topologization]\label{topologization}If $\mathcal{B}$ is a separable nil $\sigma$-algebra of order $k$ then the corresponding topological factor $\gamma:{\bf A}\rightarrow N$ (generated by $k+1$-th order convolutions) is a $k$-step strong nilspace factor.
\end{proposition}
An important observation is that the first two nilspace axioms automatically hold in $N$.
The proof of proposition \ref{topologization} deals with the checking of the third nilspace axiom.
(It will be clear that $\gamma$ is factor consistent and thus it gives a strong nilspace factor.)
We fix a natural number $k$ and by induction we assume that proposition \ref{topologization} is true for $k-1$.
Note that the statement is trivial for $k=0$. Let us introduce the notation $\mathcal{B}_i:=[\mathcal{B}]_{i+1}=\mathcal{B}\cap\mathcal{F}_i$ for $1\leq i\leq k+1$.
It will be crucial that, by induction, the topological factor $\gamma_{k-1}:{\bf A}\rightarrow N_{k-1}$ corresponding to $\mathcal{B}_{k-1}$ is a strong $k-1$ step nilspace factor. Since $\gamma_{k-1}$ generates a courser topology than $\gamma$ we have a natural projection $\pi_{k-1}:N\rightarrow N_{k-1}$.
\bigskip
\noindent{\bf Step 3.~(Local properties of $\mathcal{B}$)}~ We prove the following measure theoretic analogy of the unique gluing aximom in $k$-step nilspaces.
\begin{lemma}\label{compdep} Let $\mathcal{B}$ be a nil-$\sigma$-algebra of order $k$ and $x\in{\bf A}$. Then the coupling $\mathcal{B}\circ\Psi^{k+1}_x$ is completely dependent.
\end{lemma}
\bigskip
\noindent{\bf Step 4.~(Convolutions of open sets)}~For a set $S\subset {\bf A}$ let ${\rm cl}(S)$ denote the closure of $S$ in the topolgy generated by $\gamma$. We prove the next topological statement which is a preparation for the proof of the unique gluing axiom for $N$.
\begin{lemma}\label{uclos} Let $\varrho:K_{k+1}\rightarrow{\bf A}$ be some function. Then there is at most one element $z
\in N$ with the following property. For every system of $\gamma$-open neighborhoods $U(v)$ of $\varrho(v)$ where $v$ runs through $K_{k+1}$ we have that $\gamma^{-1}(z)\subset{\rm cl}({\rm supp}([F]))$ where $F=\{1_{U(v)}\}_{v\in K_{k+1}}$.
\end{lemma}
Note that the only application of lemma \ref{compdep} is in the proof of lemma \ref{uclos}.
These two lemmas together form the most technical part of the proof.
\bigskip
\noindent{\bf Step 5.~(Support of measure)}~To prove the unique gluing axiom for $\gamma$ we have to check that if $\varrho:K_{k+1}\rightarrow{\bf A}$ is a morphism then in lemma \ref{uclos} the set ${\rm cl}({\rm supp}([F]))$ is not empty.
This will follow by analyzing the support of the measure in the topology generated by $\gamma$ on $C^n({\bf A})$ and $C_x^n({\bf A})$.
\begin{definition}[Positive cubes]\label{defposcube} Let $n\in\mathbb{N},x\in{\bf A}$ arbitrary. We say that $c\in C^n({\bf A})$ (resp $c\in C^n_x({\bf A})$) is positive if it is in the support of the uniform measure on $C^n({\bf A})$ (resp. $C^n_x({\bf A})$) with respect to the topology generated by $\gamma$.
\end{definition}
Note that it is not clear from the above definition that if $c\in C^n_x({\bf A})$ is positive then $c$ is also positive in the space $C^n({\bf A})$. To avoid confusion we will always emphasize the space in which $c$ is positive.
The main result in this part of the proof is that every cube is positive.
\subsection{Nil $\sigma$-algabras}
\begin{lemma}\label{weaknilprop} Let $\mathcal{B}\subset\mathcal{A}$ be a $\sigma$-algebra such that $[\mathcal{B}]_n\subseteq\mathcal{B}$. Then for every $1\leq i\leq j\leq n$ we have the following statements.
\begin{enumerate}
\item For every $f\in L^\infty(\mathcal{B})$ we have $\mathbb{E}(f|[\mathcal{B}]_i)=\mathbb{E}(f|\mathcal{F}_{i-1})$,
\item $[\mathcal{B}]_i=\mathcal{B}\cap\mathcal{F}_{i-1}$~,
\item $[[\mathcal{B}]_i]_j=[\mathcal{B}]_j$.
\end{enumerate}
\end{lemma}
\begin{proof} Let $f'=\mathbb{E}(f|\mathcal{F}_{i-1})$ and Let $g=f-\mathbb{E}(f'|[\mathcal{B}]_i)$. Using the fact that $[\mathcal{B}]_i\subseteq\mathcal{F}_{i-1}$ we obtain that if $h\in L^\infty([\mathcal{B}]_i)$ then $$(g,h)=(\mathbb{E}(g|\mathcal{F}_{i-1}),h)=(f'-\mathbb{E}(f'|[\mathcal{B}]_i),h)=0.$$ We have by $[g]_i\in L^\infty([\mathcal{B}]_i)$ that $\|g\|_{U_i}^{2^i}=(g,\overline{[g]})=0$. It follows that $0=\mathbb{E}(g|\mathcal{F}_{i-1})$ and thus $f'=\mathbb{E}(f'|[\mathcal{B}]_i)$ implying that $f'\in L^\infty([\mathcal{B}]_i)$. Using again that $[\mathcal{B}]_i\subseteq\mathcal{F}_{i-1}$ the proof of the first statement is complete. The second statement follows immediately from the first one.
To see the third statement observe that $[[\mathcal{B}]_i]_i\subseteq [\mathcal{B}]_i$ and thus the second statement applied for $[\mathcal{B}]_i$ we obtain that $[[\mathcal{B}]_i]_j=[\mathcal{B}]_i\cap\mathcal{F}_{j-1}=(\mathcal{B}\cap\mathcal{F}_{i-1})\cap\mathcal{F}_{j-1}
=\mathcal{B}\cap\mathcal{F}_{j-1}=[\mathcal{B}]_j$.
\end{proof}
\medskip
An immediate corollary of lemma \ref{weaknilprop} is the following,
\begin{corollary}\label{nilsubnil} If $\mathcal{B}\subset\mathcal{A}$ is a nil $\sigma$-algebra of order $k$ and $1\leq i\leq k+1$ then $[\mathcal{B}]_{i+1}=\mathcal{B}\cap\mathcal{F}_i$ is a nil $\sigma$-algebra of order $i$.
\end{corollary}
\medskip
\noindent{\it Proof of lemma \ref{embednil}}~Starting with a separable $\sigma$-algebra $\mathcal{G}\subset\mathcal{F}_k$ we construct an increasing sequence of separable $\sigma$-algebras $\mathcal{G}=\mathcal{G}_0\subset\mathcal{G}_1\subset\mathcal{G}_2\subset\dots$ in $\mathcal{F}_k$ in the following way. Assume that $\mathcal{G}_n$ is already constructed.
Theorem \ref{propfk} together with lemma \ref{sepsiggen} imply that for every $2\leq i\leq k+1$ there is a separable $\sigma$-algebra $\mathcal{D}_i\subset\mathcal{F}_{i-1}$ such that $[\mathcal{G}_n]_i\circ\psi_0\subset [[\mathcal{D}_i]_i]_i^\times$. We define $\mathcal{G}_{n+1}=(\vee_{i=2}^{k+1}\mathcal{D}_i)\vee\mathcal{G}_n\vee[\mathcal{G}_n]_{k+1}$. Let $\mathcal{B}=\vee_{n=1}^\infty\mathcal{G}_n$. We claim that $\mathcal{B}$ is a nil $\sigma$-algebra.
Since $\{\mathcal{G}_i\}_{i=0}^\infty$ is a chain we have that for every set $S$ in $\mathcal{B}$ and $\epsilon>0$ there is an index $j$ and set $S'\in\mathcal{G}_j$ such that ${\bf \mu}(S\triangle S')\leq\epsilon$. Furthermore for every function in $f\in L^\infty_u(\mathcal{B})$ and $\epsilon>0$ there is an index $j$ and function $f'\in L_u^\infty(\mathcal{G}_j)$ such that $\|f-f'\|_2\leq\epsilon$. We say that $S'$ (resp. $f'$) is a finite index $\epsilon$-approximation of $S$ (resp. $f$).
Let $F=\{f_v\}_{v\in K_{k+1}}$ be a function system in $L^\infty_u(\mathcal{B})$. Then the convolution $[F]$ can be approximated arbitrarily well by a convolution of finite index $\epsilon$-approximations of the function system. On the other hand such convolutions are contained in some memeber of the chain $\{\mathcal{G}_n\}_{n=1}^\infty$. It follows that $[\mathcal{B}]_{k+1}\subseteq\mathcal{B}$. Since $\mathcal{B}\subset\mathcal{F}_k$ we have by lemma \ref{weaknilprop} that $[\mathcal{B}]_{k+1}=\mathcal{B}\cap\mathcal{F}_k=\mathcal{B}$.
Let $\mathcal{C}_i$ be the unique sigma algebra on ${\bf A}$ with $\mathcal{C}_i\circ\psi_0=\mathcal{A}\circ\psi_0\cap[[\mathcal{B}]_i]_i^\times$.
Similarly to the case of convolutions, by considering finite index approximations, we have for $2\leq i\leq k+1$ that $[\mathcal{B}]_i
\subseteq\mathcal{C}_i$. On the other hand we have by lemma \ref{convcontlift} that $\mathcal{C}_i\subset[\mathcal{B}]_i$.
\subsection{Local properties of $\mathcal{B}$}
In this chapter we prove lemma \ref{compdep}. By induction we assume that the statement is true for $k-1$. We start by proving the following statement using the induction hypothesis.
\medskip
\noindent{\bf Claim 1.}~~{\it Assume that $S_1,S_2\subset\{0,1\}^n$ are simplicial sets such that $S_1$ has dimension at most $k$. Let $w\in S_2\setminus S_1$. On $C^n_x({\bf A})$ let
$$\mathcal{D}_1=\bigvee_{v\in (S_1\cup S_2)\setminus\{0,w\}}\mathcal{B}\circ\psi_v~~~~~~{\it and}~~~~~~\mathcal{D}_2=\bigvee_{v\in S_2\setminus\{0,w\}}\mathcal{B}\circ\psi_v.$$
Assume that on $C^n_x({\bf A})$ that the $\sigma$-algebra $\mathcal{B}\circ\psi_w$ is contained in $\mathcal{D}_1$ then it is also contained in $\mathcal{D}_2$.}
\medskip
We prove the statement by induction on the size of $S_1\setminus S_2$. If $S_1\subset S_2$ then there is nothing to prove. Assume that $S_1\nsubseteq S_2$ and $u$ is a maximal element of $S_1$ which is not contained in $S_2$. Then $d=d(u)=h(u)\leq k$.
On the space $C^n_x({\bf A})$ let $$\mathcal{D}_3=\bigvee_ {(S_1\cup S_2)\setminus\{0,u\}}\mathcal{B}\circ\psi_v~~~~~~{\rm and}~~~~~~\mathcal{D}_4=\bigvee_ {(S_1\cup S_2)\setminus\{0,u,w\}}\mathcal{B}\circ\psi_v.$$
Our induction hypothesis of the lemma on $k$ guarantees that $\mathcal{B}_{d-1}\circ\psi_u$ is contained in $\bigvee_{0\neq v<u}\mathcal{B}_{d-1}\circ\psi_v$ which is contained in $\mathcal{D}_4$.
Assume that $f\in L^\infty(\mathcal{B})$ has the property that $\mathbb{E}(f|\mathcal{B}_{d-1})=0$. Then by lemma \ref{weaknilprop} we have that $\mathbb{E}(f|\mathcal{F}_{d-1})=0$ and thus $\|f\|_{U_d}=0$. It follows by lemma \ref{simpzero} that $f\circ\psi_u$ is orthogonal to the $\sigma$-algebra $\mathcal{D}_3$.
This means that $\mathcal{B}\circ\psi_u$ and $\mathcal{D}_3$ are conditionally independent. Furthermore $\mathcal{B}\circ\psi_0\wedge\mathcal{D}_3=\mathcal{B}_{d-1}\circ\psi_u\subset\mathcal{D}_4$.
We get by using lemma \ref{siggen2} that $$\mathcal{B}\circ\psi_w\subset\mathcal{D}_1\wedge\mathcal{D}_3=(\mathcal{B}\circ\psi_u\vee\mathcal{D}_4)\wedge\mathcal{G}_3=(\mathcal{B}\circ\psi_u \wedge\mathcal{D}_3)\vee\mathcal{D}_4=\mathcal{D}_4.$$
Since $S_1\setminus\{u\}$ is a simplicial set, by induction we have that $\mathcal{B}\circ\psi_w$ is contained in $\mathcal{D}_2$.
\bigskip
Now we switch to the proof of the lemma. Let $z\in K_{k+1}$ be arbitrary. Our goal is to show that on $C^{k+1}_x({\bf A})$ the $\sigma$-algebra $\mathcal{B}\circ\psi_z$ is generated by the system $\{\mathcal{B}\circ\psi_v\}_{z\neq v\in K_{k+1}}$. Let $T=\{0,1\}^{[k+1]\times[2]}$.
We will need the following special subsets of $T$.
$$S_1=\{v~|~v_{i,1}=0~{\rm for}~1\leq i\leq k+1\} ~~,~~S_2=\{v~|~v_{i,2}=0~{\rm for}~1\leq i\leq k+1\},$$
$$S_3=\{v~|~v_{k+1,2}=0~,~h(v)\leq k~,~v_{i,1}v_{i,2}=0~{\rm for}~1\leq i\leq k\},$$
$$S_4=\{v~|~v_{k+1,2}=0~,~v_{i,1}v_{i,2}=0~{\rm for}~1\leq i\leq k\}.$$
It is clear that all the sets $S_1,S_2,S_3,S_4$ are simplicial.
Without loss of generality we can assume that the last coordinate of $z$ is $1$.
Let $w$ be the vector such that $w_{i,1}=0$ and $w_{i,2}=z_i$ for $1\leq i\leq k+1$.
On $\hom_{0\mapsto x}(T,{\bf A})$ let $\mathcal{G}_1=\bigvee_{v\in S_1\setminus\{w,0\}}\mathcal{B}\circ\psi_v$ and for $2\leq i\leq 4$ let
$\mathcal{G}_i=\bigvee_{v\in S_i\setminus\{0\}}\mathcal{B}\circ\psi_v$.
The statement of the lemma is equivalent with the fact that $\mathcal{B}\circ\psi_w$ is contained in $\mathcal{G}$.
\medskip
\noindent{\bf Claim 2.}~$\mathcal{B}\circ\psi_w\subseteq \mathcal{G}_1\vee\mathcal{G}_4$
\medskip
Let $\phi:\{0,1\}^{k+1}\rightarrow T$ be defined such that $\phi(v)_{i,1}=0~,~\phi(v)_{i,2}=v_i$ if $1\leq i\leq k$ and $\phi(v)_{k+1,1}=1-v_{k+1}~,~\phi(v)_{k+1,2}=v_{k+1}$. We have that $\phi(z)=w$. Let $K=\{0,1\}^{k+1}\setminus\{z\}$. If $v\in K$ then $\phi(v)\in S_1\cup S_4$ and thus $\mathcal{B}\circ\psi_{\phi(v)}\in\mathcal{G}_1\vee\mathcal{G}_4$.
Since by lemma \ref{cupis1} and lemma \ref{cupis} the coupling $\{\psi_{\phi(v)}\}_{v\in\{0,1\}^{k+1}}$ is the same as $\Psi^{k+1}$ by (\ref{rhsnil}) applied for $i=k+1$ we get that $\mathcal{B}\circ\psi_w$ is generated by $\bigvee_{v\in K}\mathcal{B}\circ\psi_{\phi(v)}$. This shows the claim.
\medskip
\noindent{\bf Calim 3.}~ $\mathcal{G}_2\vee\mathcal{G}_3=\mathcal{G}_4$.
\medskip
The containment $\subseteq$ is trivial. For a vector $v$ in $T$ let $h^*(v)=\sum_{i=1}^{k+1}v_{i,2}$.
We prove by induction on $h^*(v)$ that if $v\in S_4$ then $\mathcal{B}\circ\psi_v$ is in $\mathcal{G}_2\vee\mathcal{G}_3$. If $h^*(v)=0$ then $v\in S_2$ and the statement is trivial.
Assume that the statement holds for every $v\in S_4$ with $h^*(v)\leq n-1$ (where $n\geq 1$) and that $h^*(b)=n$ for some $b\in S_4$.
If $h(b)\leq k$ then $b\in S_3$ and the statement is trivial. We can assume that $h(b)=k+1$.
This means that $b_{i,1}+b_{i,2}=1$ for every $1\leq i\leq k+1$. Let $\phi:\{0,1\}^{k+1}\rightarrow S_4$ be defined such that $\phi(v)_{i,1}=v_i~,~\phi(v)_{i,2}=0$ if $b_{i,2}=0$ and $\phi(v)_{i,1}=1-v_i~,~\phi(v)_{i,2}=v_i$ if $b_{i,2}=1$. It is clear that $\phi$ is an injective cube morphism and that the image of $\phi$ does not contain $0$. Let $K=\{0,1\}^{k+1}\setminus\{1\}$. For every $v\in K$ we have that either $h^*(\phi(v))<n$ or $\phi(v)\in S_3$. Using the induction hypothesis, in both cases $\mathcal{B}\circ\psi_{\phi(v)}$ is in $\mathcal{G}_2\vee\mathcal{G}_3$. Since by lemma \ref{cupis1} and lemma \ref{cupis} the coupling $\{\psi_{\phi(v)}\}_{v\in\{0,1\}^{k+1}}$ is the same as $\Psi^{k+1}$ we obtain by (\ref{rhsnil}) and $b=\phi(1)$ the $\mathcal{B}\circ\psi_b$ is generated by $\bigvee_{ v\in K}\mathcal{B}\circ\psi_{\phi(v)}$ which is conatined in $\mathcal{G}_2\vee\mathcal{G}_3$. This finishes the proof of the claim.
\medskip
By claim 2. and claim 3. we obtain that $\mathcal{B}\circ\psi_w\subseteq\mathcal{G}_1\vee\mathcal{G}_2\vee\mathcal{G}_3$. Now claim 1. shows that $\mathcal{G}_3$ can be omitted and thus $\mathcal{B}\circ\psi_w\subseteq\mathcal{G}_1\vee\mathcal{G}_2$. On the other hand since $\mathcal{G}_2$ is independent from $\mathcal{G}_5=\mathcal{B}\circ\psi_w\vee\mathcal{G}_1$ we obtain from lemma \ref{siggen2} that $$\mathcal{B}\circ\psi_w\subset (\mathcal{G}_2\vee\mathcal{G}_1)\wedge\mathcal{G}_5=
(\mathcal{G}_2\wedge\mathcal{G}_5)\vee\mathcal{G}_1=\mathcal{G}_1.$$
\subsection{Convolutions of open sets}\label{conopen}
In this chapter we prove lemma \ref{uclos}.
Assume that there exists such an element $z$ and that $z=\gamma(x)$ for some $x\in{\bf A}$.
The statement of the lemma is equivalent with saying that the coupling $\mathcal{B}\circ\Psi_x^{k+1}$ is uniquely determined by the couplings $\mathcal{B}\circ\Psi_{\varrho(v)}^{k+1}$ where $v$ runs through $K_{k+1}$.
Our strategy is to put all these couplings into one big coupling $\Upsilon$ as sub-couplings and then we do the calculations in $\Upsilon$. The first part of the proof deals with the construction of $\Upsilon$.
\medskip
\noindent{\bf Construction of $\Upsilon$:}~~For every $v\in K_{k+1}$ let us choose a decreasing sequence $\{U_i(v)\}_{i=1}^\infty$ of $\gamma$ neighborhoods of $\varrho(v)$ which forms a neighborhood basis.
Let $F_i=\{1_{U_i(v)}\}_{v\in K_{k+1}}$ and $\{x_i\}_{i=1}^\infty$ be a sequence such that $\lim_{i\rightarrow\infty}\gamma(x_i)=z$ and $x_i\in{\rm supp}([F_i])$.
Let $T=\{0,1\}^{[k+1]\times[2]}$ and let $\tilde{T}$ be the set of vectors $v$ in $T$ with $v_{i,1}v_{i,2}=0$ for $1\leq i\leq k+1$. Let $S=\{v~|~v_{i,1}=0~{\rm for}~1\leq i\leq k+1\}$. Let $t\in T$ be the vector with $t_{i,1}=0,t_{i,2}=1$ for $1\leq i\leq k+1$. Let $\tau:\{0,1\}^{k+1}\rightarrow S$ be the map such that $\tau(v)_{i,1}=0$ and $\tau(v)_{i,2}=1-v_i$. In particular $\tau(0)=t$.
Let $Q_i$ be the probability space of cubes $c\in\hom_{t\mapsto x_i}(T,{\bf A})$ conditioned on the event that $c(\tau(v))\in U_i(v)$ holds for every $v\in K_{k+1}$. The fact that $[F_i](x_i)\neq 0$ guarantees that we condition on a positive probability event. We will use $Q_i$ to define a coupling $\Upsilon_i$ on copies of $\mathcal{B}$ indexed by $T\setminus S$.
We will see that the system $\{\psi_v\}_{v\in T\setminus S}$ restricted to $Q_i$ is a coupling of copies of $({\bf A},\mathcal{A},\mu)$ and $\Upsilon_i$ will be defined as the factor coupling according to $\mathcal{B}\subset\mathcal{A}$.
In order to show this we establish $\Upsilon_i$ as a convex combination of couplings.
Let $Q_i'$ be the probability space of cubes $c\in\hom(S,{\bf A})$ with $c(t)=x_i$ conditioned on the event that $c(\tau(v))\in U_i(v)$. Let $r:Q_i\rightarrow Q_i'$ be the restriction map to $S$. It is clear that $r$ is measure preserving and the preimage $r^{-1}(c)$ for $c\in Q_i'$ is the set $\hom_c(T,{\bf A})$. Since the restriction of $\{\psi_v\}_{v\in T\setminus S}$ to each set $\hom_c(T,{\bf A})$ is a coupling of copies of $({\bf A},\mathcal{A},\mu)$ the restriction of $\{\psi_v\}_{v\in T\setminus S}$ to $Q_i$ is the convex combination of these couplings with distribution given by $Q_i'$.
The final step is to obtain $\Upsilon$ as the limit of some convergent subsequence from $\{\Upsilon_i\}_{i=1}^\infty$ in the coupling topology.
\medskip
We will need the following notation. For every $w\in\{0,1\}^{k+1}$ let $\phi_w:\{0,1\}^{k+1}\rightarrow \tilde{T}$ such that $\phi_w(v)_{i,1}=v_i$ and $\phi_w(v)_{i,2}=(1-w_i)(1-v_i)$ holds for $i\in [k+1]$. It is clear that for every fixed $w$ the map $\phi_w$ is a cube morphism. For $1\leq i\leq k+1$ we denote by $T_i\subset T$ the set of vectors $v\in T$ whose coordinate sum $h(v)$ is at most $i$ and let $\tilde{T}_i=T_i\cap\tilde{T}$. We can think of $T_i$ as the $i$-dimensional frame of $T$.
\medskip
\noindent{\bf Claim 1.}~~{\it The coupling $\Upsilon$ has the following two properties.
\begin{enumerate}
\item The sub coupling of $\Upsilon$ induced by $\phi_w:K_{k+1}\rightarrow T\setminus S$ is equal to $\mathcal{B}\circ\Psi_{\varrho(w)}^{k+1}$ for every $w\in K_{k+1}$ and is equal to $\mathcal{B}\circ\Psi_x^{k+1}$ if $w=0$.
\item Let $\Upsilon^i$ denote the sub-coupling of $\Upsilon$ induced by $T_i\setminus S\rightarrow T\setminus S$. Then $\Upsilon_i$ is independent over its $\mathcal{B}_{i-1}$ factor for $1\leq i\leq k+1$.
\end{enumerate}
}
\medskip
The first property follows from lemma \ref{cupis1}, lemma \ref{cupis} and the way $\Upsilon_i$ is obtained as a convex combination of couplings.
To see the second property this let $\Upsilon^i_j$ denote the subcoupling of $\Upsilon_j$ induced by the map $T_i\setminus S\rightarrow T\setminus S$. It is obviously enough to show that $\Upsilon^i_j$ is independent over its $\mathcal{B}_{i-1}$ factor.
Let $F=\{f_v\}_{v\in T_i\setminus S}$ be a bounded $\mathcal{B}$ measurable function system on ${\bf A}$ and assume that
$\mathbb{E}(f_{v'}|\mathcal{B}_{i-1})=0$ for some $v'\in T_i\setminus S$. By lemma \ref{weaknilprop} we obtain that $\mathbb{E}(f_{v'}|\mathcal{F}_{i-1})=0$ and so $\|f_{v'}\|_{U_i}=0$. We have to show that $\xi(\Upsilon^i_j,F)=0$.
Let $G=\{g_v\}_{v\in T\setminus\{t\}}$ be the extended function system defined as follows. If $v\in T_i\setminus S$ then $g_v=f_v$, if $v\in T\setminus (T_i\cup S)$ then $g_v=1$ and if $v\in {S\setminus\{t\}}$ then $g_v$ is the characteristic function of $U_i(\tau^{-1}(v))$. By definition we have that $\xi(\Upsilon^i_j,F)=[G^\star](x_i)$ where the convolution is taken at $t$. Using that $S\cup T_i$ is simplicial and that $d(v')\leq i$ we get by lemma \ref{simpzero} that $[G^\star](x_i)=0$.
\medskip
\noindent{\bf Claim 2.}~~{\it There is at most one self coupling $\Theta$ of $\mathcal{B}$ with index set $\tilde{T}\setminus S$ with the following two properties.
\begin{enumerate}
\item The sub coupling of $\Theta$ induced by $\phi_w:K_{k+1}\rightarrow \tilde{T}\setminus S$ is equal to $\mathcal{B}\circ\Psi_{\varrho(w)}^{k+1}$ whenever $w\neq 0$,
\item Let $\Theta^i$ denote the sub-coupling of $\Theta$ induced by $\tilde{T}_i\setminus S\rightarrow \tilde{T}\setminus S$. $\Theta_i$ is independent over its $\mathcal{B}_{i-1}$ factor for $1\leq i\leq k+1$.
\end{enumerate}
}
Assume that there is such a coupling $\Theta=\{\theta_v\}_{v\in\tilde{T}\setminus S}$. We prove by induction on $i$ that by the conditions on the claim the coupling $\Theta_i$ is uniquely determined. For $i=1$ the coupling $\Theta_i$ is independent and so it is a unique object. Assume that the uniqueness is verified for $i-1$ and $i>1$. The second condition implies that it is enough to prove the uniqueness of the $\mathcal{B}_{i-1}$ factor of $\Theta_i$. We use a second induction to show this.
For a vector $v$ in $T$ let $h^*(v)=\sum_{i=1}^{k+1}v_{i,2}$ and let $\tilde{T}_{i,n}=\tilde{T}_{i-1}\cup\{z|h^*(z)\leq n~,~z\in\tilde{T}_i\}$.
We prove by induction on $n=h^*(v)$ that the $\mathcal{B}_{i-1}$ factor of the sub-coupling on $\tilde{T}_{i,n}$ is uniquely determined by the $\mathcal{B}_{i-1}$ factor of the sub-coupling on $\tilde{T}_{i,n-1}$. If $n=0$ then from $v\in \tilde{T}_i\setminus \tilde{T}_{i-1}$ we have that $h(v)=i$ and every vector $v'< v$ is in $\tilde{T}_{i-1}$. Since by lemma \ref{compdep} the sub-coupling of $\mathcal{B}_{i-1}$ on $\{v'|v'\leq v,v'\neq 0\}$ is completely dependent and is isomorphic to $\mathcal{B}_{i-1}\circ\Psi_{\varrho(1)}^i$ the statement is clear.
Assume by induction that the statement holds for $n-1$ and $h^*(v)=n$. Let $v',w\in\{0,1\}^{k+1}$ be the vectors with $v'_i=v_{i,1}+v_{i,2}$ and $w_i=1-v_{i,2}$. Let $Q=\{z|z\neq 0~,~z\leq v'\}\subset\{0,1\}^{k+1}$ and let $\alpha:Q\rightarrow\tilde{T}$ be the restriction on $\phi_w$ to $Q$. It is clear that every element in the image of $\alpha$ which is not equal to $v$ is either in $\tilde{T}_{i-1}$ or has $h^*$ value at most $n-1$. By the first property of $\Theta$ we have that the restriction of $\Theta$ to the image of $\alpha$ is $\Psi_{\varrho(w)}^i$. Using lemma \ref{compdep} and our induction step we obtain the induction statement for $v$.
\medskip
Let $\tilde{\Upsilon}$ be the sub-coupling of $\Upsilon$ on $\tilde{T}$. We get from our two claims that $\tilde{\Upsilon}$ is uniquely determined by the function $\gamma\circ\varrho$. Since $\mathcal{B}\circ\Psi_x^{k+1}$ is sub-coupling of $\tilde{\Upsilon}$ we get that $\gamma(x)$ is uniquely determined and so the proof of the lemma is complete.
\medskip
\subsection{Support of measure}
In this chapter we prove statements related to definition \ref{defposcube}.
Notice that $c\in C^n_x({\bf A})$ if and only for every system of open sets $\{U(v)\}_{v\in K_n}$ with $c(v)\in U(v)$ we have that if $F=\{1_{U(v)}\}_{v\in K_n}$ then $[F](x)>0$. General properties of supports of measures on compact spaces imply the next lemma.
\begin{lemma}\label{ascubepos} Let $x\in{\bf A}~,~n\in\mathbb{N}$ be arbitrary. Then almost every $c\in C^n({\bf A})$ ($c\in C^n_x({\bf A})$) is positive. Furthermore positive cubes in $C^n({\bf A})$ (resp. $C_x^n({\bf A})$) form a closed set in the topology generted by $\gamma$.
\end{lemma}
\begin{lemma}\label{opcontconv} Let $x\in{\bf A}$. Then for every $\gamma$-open set $U$ containing $x$ there is a system of $\gamma$-open sets $\{U(v)\}_{v\in K_{k+1}}$ such that if $F=\{1_{U(v)}\}_{v\in K_{k+1}}$ then $x\in{\rm supp}([F])\subseteq U$.
\end{lemma}
\begin{proof} Let $c\in C^{k+1}_x$ be positive. For every $i\in\mathbb{N}$ let $\{U_i(v)\}_{v\in K_{k+1}}$ be a system of open sets such that $\{U_i(v)\}_{i=1}^\infty$ is a descending neighborhood basis for $c(v)$. Let $F_i=\{1_{U_i(v)}\}_{v\in K_{k+1}}$. Assume by contradiction that $({\bf A}\setminus U)\cap{\rm supp}([F_i])\neq\emptyset$. Then since $B_i=({\bf A}\setminus U)\cap{\rm cl}({\rm supp}([F_i]))$ is a descending chain of $\gamma$-closed sets we have that there is an element $y\in\cap_{i=1}^\infty B_i$. Lemma \ref{uclos} implies that $\gamma(x)=\gamma(y)$ which contradicts the fact that $x$ and $y$ are separated by the $\gamma$-open set $U$.
\end{proof}
\begin{definition} Let $x\in{\bf A}$. We say that $f:{\bf A}\rightarrow\mathbb{R}$ is a convolution neighborhood of $x$ if $f(x)\neq 0$ and $f=[F]$ where $F=\{f_v\}_{v\in K_{k+1}}$ is a function system consisting of $0-1$ valued $\mathcal{B}$-measurable functions.
\end{definition}
Note that the values of a convolution neighborhood $f$ of $x$ are non-negative and by lemma \ref{opcontconv} for every open neighborhood $U$ of $x$ there is a convolution neighborhood $f$ of $x$ such that ${\rm supp}(f)\subseteq U$.
\begin{lemma}\label{convnproj} Let $f$ be a convolution neighborhood of $x\in{\bf A}$. Then there is a $\gamma_{k-1}$ continuous non-negative function $g$ such that $g=\mathbb{E}(f|\mathcal{F}_{k-1})$ (almost everywhere) and $g(x)>0$.
\end{lemma}
\begin{proof} Assume that $f=[F]$ with a $0-1$ valued $\mathcal{B}$ measurable function system $F=\{f_v\}_{v\in K_{k+1}}$. Let $G=\{g_v\}_{v\in K_{k+1}}$ where $g_v=\mathbb{E}(f_v|\mathcal{F}_{k-1})$ and $g=[G]$. Note that by lemma \ref{weaknilprop} each $g_v$ is measurable in $\mathcal{B}_{k-1}$. Then by lemma \ref{vetites1} we have that $g=\mathbb{E}(f|\mathcal{F}_{k-1})$. Furthermore, since $g_v$ is almost surely positive on $f^{-1}_v(1)$ we have that $[G](x)>0$. Finally, using our induction hypothesis that $\gamma_{k-1}$ is a strong nilspace factor we get that $g=[G']\circ\gamma$ where $G'=\{g'_v\}_{v\in K_{k+1}}$ is a function system on $N_{k-1}$ such that $g_v=g'_v\circ\gamma_{k-1}$. We obtain that $g$ is $\gamma_{k-1}$-continuous.
\end{proof}
\begin{lemma}\label{openpos} Every non empty $\gamma$-open set has positive measure.
\end{lemma}
\begin{proof} We prove the statement by induction on $k$. For $k=0$ it is clear and we assume that it is true for $\gamma_{k-1}$. Let $U$ be a non-empty $\gamma$-open set, $x\in U$ and $f$ be a convolution neighborhood of $x$ such that ${\rm supp}(f)\subseteq U$. It is enough to show that $\mathbb{E}(f)>0$. On the other hand the function $g$ satisfying the conditions of lemma \ref{convnproj} has the property that $\mathbb{E}(g)=\mathbb{E}(f)$. Furthermore since $g$ is not identically $0$ and non-negative we have by our induction hypothesis that $\mathbb{E}(g)>0$.
\end{proof}
\begin{lemma}\label{pospreshef} Let $c\in C^n({\bf A})$ be a positive cube and let $\phi:\{0,1\}^m\rightarrow\{0,1\}^n$ be an injective cube morphism. Then $c_2=c\circ\phi\in C^m({\bf A})$ is also a positive cube.
\end{lemma}
\begin{proof} Let $\{U(v)\}_{v\in\{0,1\}^m}$ be a system of $\gamma$-open sets with $c_2(v)\in U(v)$. Let $\{W(v)\}_{v\in\{0,1\}^n}$ be defined such that $W(v)=U(\phi^{-1}(v))$ if $v\in{\rm im}(\phi)$ and $W(v)={\bf A}$ otherwise. It is clear by lemma \ref{cupis1} and lemma \ref{cupis} that $\cap_{v\in\{0,1\}^m}\psi_v^{-1}(U(v))$ has the same measure as $\cap_{v\in\{0,1\}^n}\psi_v^{-1}(W(v))$ and thus by the positivity of $c$ we obtain the positivity of $c_2$.
\end{proof}
\subsection{Every cube is positive}
The main result of this chapter is the following.
\begin{proposition}\label{cubepos} Every cube in ${\bf A}$ is positive with respect to $\gamma$.
\end{proposition}
In this chapter we assume by induction on $k$ that proposition \ref{cubepos} is true for $\gamma_{k-1}$.
We will need the following lemmas.
\begin{lemma}\label{uniqclos1} Let $c_1,c_2\in C^{k+1}({\bf A})$ be two positive cubes such that $\gamma\circ c_1$ agrees with $\gamma\circ c_2$ on $K_{k+1}$. Then $\gamma\circ c_1=\gamma\circ c_2$.
\end{lemma}
\begin{proof} Assume by contradiction that $\gamma(c_1(0))\neq \gamma(c_2(0))$. Let $U$ be a $\gamma$-open set containing $c_1(0)$ such that $c_2(0)\notin{\rm cl}(U)$. Let furthermore $\{U(v)\}_{v\in K_{k+1}}$ be a system of open sets with $c_1(v)\in U(v)$ for every $v\in K_{k+1}$ such that ${\rm supp}([F])\subseteq U$ holds for the function system $F=\{1_{U(v)}\}_{v\in K_{k+1}}$. The existence of such a system of open sets is guaranteed by lemma \ref{opcontconv}. Notice that by the condition of the lemma, $c_2(v)\in U(v)$ holds for every $v\in K_{k+1}$. Let us define $U(0)$ as the complement of ${\rm cl}(U)$. Then we have that the measure of $\cap_{v\in\{0,1\}^{k+1}}\psi_v^{-1}(U(v))$ is equal to the integral of $[F]$ on $U(0)$ and thus it is $0$. This contradicts the positivity of $c_2$.
\end{proof}
\begin{lemma}\label{simpglpos} let $S\subset\{0,1\}^n$ be a simplicial set of hight at most $k$. Let $q:S\rightarrow {\bf A}$ be a map such that the restriction of $q$ to any maximal face composed with $\gamma_{k-1}$ is a cube in $N_{k-1}$. Then there is a positive cube $c\in C^n({\bf A})$ such that $\gamma\circ c$ restricted to $S$ is equal to $\gamma\circ q$.
\end{lemma}
\begin{proof} Using the fact that positive cubes are forming a closed set, it is enough to show the following. Let $\{U(v)\}_{v\in S}$ be an arbitrary system of $\gamma$-open sets such that $q(v)\in U(v)$ holds for every $v\in S$. Then there is a positive cube $c_2\in C^n({\bf A})$ such that $c_2(v)\in U(v)$ for every $v\in S$.
To see this we choose a function system $F=\{f_v\}_{v\in S}$ such that $f_v$ is a convolution neighborhood of $q(v)$ and ${\rm supp}(f_v)\in U(v)$ for every $v\in S$. Let $F'=\{f_v\}_{v\in\{0,1\}^n}$ be the function system where for $v\in \{0,1\}^n\setminus S$ the function $f_v$ is identically $1$. Let $G=\{g_v\}_{v\in\{0,1\}^n}$ be the function such that $g_v$ is the continuous projection of $f_v$ guaranteed by lemma \ref{convnproj}. Note that if $v\in\{0,1\}^n\setminus S$ then $g_v$ is also constant $1$.
Let $c_3:\{0,1\}^n\rightarrow{\bf A}$ be a cube in $C^n({\bf A})$ such that $\gamma_{k-1}\circ c_3=\gamma_{k-1}\circ q$ on $S$. The existence of $c_3$ follows from the assumption that $\gamma_{k-1}$ defines a nilspace factor: It is an easy consequence of the nilspace axioms (see \cite{NP}) that morphisms of simplicial sets in $\{0,1\}^n$ into nilspaces can always be extended to the full cube.
We have that $c_3(v)\in{\rm supp}(g_v)$ holds for every $v\in\{0,1\}^m$. Since $c_3$ is positive in $\gamma_{k-1}$ we have that $(G)>0$ holds. On the other hand by corollary \ref{simpzerocor} we have that $(F')=(G)$ and thus the measure of $T=\cap_{v\in S}\psi_v^{-1}(U(v))$ on $C^n({\bf A})$ is positive. By lemma \ref{ascubepos} we have that $T$ contains a positive cube.
\end{proof}
\medskip
We are ready to prove proposition \ref{cubepos}. Let $c\in C^{k+1}({\bf A})$ be an arbitrary cube.
Let $T,S$ be defined as in chapter \ref{conopen}. Let $\phi:\{0,1\}^{k+1}\rightarrow S$ be defined by $\phi(v)_{i,1}=0$ and $\phi(v)_{i,2}=v_i$ and for $w\in\{0,1\}^{k+1}$ let $\phi_w:\{0,1\}^{k+1}\rightarrow T$ be defined by $\phi_w(v)_{i,1}=v_i$ and $\phi_w(v)_{i,2}=w_i(1-v_i)$. Let $\hat{\phi}_w:\mathrm{Hom}_{c\circ\phi^{-1}}(T,A)\rightarrow C_{c(w)}^{k+1}({\bf A})$ be the map given by composing elements from $\mathrm{Hom}_{c\circ\phi^{-1}}(T,A)$ by $\phi_w$. By lemma \ref{cupis1} we obtain that $\hat{\phi}_w$ is measure preserving. This fact combined with lemma \ref{ascubepos} implies that for almost every element in $c_2\in\mathrm{Hom}_{c\circ\phi^{-1}}(T,A)$ we have that for every $w\in\{0,1\}^{k+1}$ the cube $\hat{\phi}_w(c_2)$ is positive in $\Psi_{c(w)}^{k+1}$. Let $c_2$ be a fixed cube with this property.
Let $T_2\subset T$ be the subset of elements $v$ such that $v_{i,1}v_{i,2}=0$ holds for every $1\leq i\leq k+1$ and let $T_3\subset T_2$ be the set of vectors in $T_2$ in which the coordinate sum is at most $k$. Both $T_2$ and $T_3$ are simplicial sets.
We define the bijection $\tau:T_2\rightarrow T_2$ by $\tau(v)_{i,1}=1-v_{i,1}-v_{i,2}$ and $\tau(v)_{i,2}=v_{i,2}$. Let $c_3=c_2\circ\tau$. The maximal faces of $T_2$ are the images of the maps $\tau^{-1}\circ\psi_w$. It follows that $c_3$ restricted to every maximal face is a positive cube.
The restriction of $c_3$ to $T_3$ satisfies the conditions of lemma \ref{simpglpos} and thus there is a positive cube $c_4\in\mathrm{Hom}(T,{\bf A})$ such that the restriction of $c_4$ to $T_3$ is equal to the restriction of $c_3$ to $T_3$.
The restriction of $c_4$ to every maximal face of $T_2$ is positive and thus by lemma \ref{uniqclos1} we have that $c_4$ restricted to $T_2$ is equal to $c_3$. Then $c=c_3\circ\phi_1$ is positive by lemma \ref{pospreshef}
\medskip
\subsection{Verifying the nilspace structure}
In this chapter we finish the proof of proposition \ref{topologization}.
It is clear that $N$ satisfies the first two axioms. We focus on the last axiom. Let $K=\{0,1\}^n\setminus\{1^n\}$. Let $q:K\rightarrow{\bf A}$ be a map such that the restriction of $q$ to every maximal face of $K$ composed with $\gamma$ is a cube in $N$.
In order to verify the third nilpsace axiom we have to show that there is a cube $c\in C^n({\bf A})$ such that $\gamma\circ c=\gamma\circ q$ on $K$.
Let $S$ be the set of vectors in $\{0,1\}^n$ of hight at most $k$. It is clear that the restriction of $q$ to $S$ satisfies the conditions of lemma \ref{simpglpos}. Using the lemma we get that there is a positive cube $c\in C^n({\bf A})$ such that $\gamma\circ c$ is equal to $\gamma\circ q$ on the set $S$.
We claim that $\gamma\circ c$ is equal to $\gamma\circ q$ on $K$.
Let $F$ be an arbitrary maximal face in $K$. The condition on $q$ guarantees that there is a cube $c_2:F\rightarrow{\bf A}$ such that $\gamma\circ c_2=\gamma\circ q$ on $F$. In particular $\gamma\circ c_2=\gamma\circ c$ holds on $F\cap S$.
The claim is equivalent with $\gamma\circ c_2=\gamma\circ c$ on $F$.
We prove this by contradiction. Let $v\in F$ be an element of minimal hight for which $\gamma\circ c_2\neq\gamma\circ c$. Then there is a face $F'\subseteq F$ of dimension $k+1$ whose maximal element is $v$. Since $\gamma\circ c_2=\gamma\circ c$ holds on $F'\setminus\{v\}$ lemma \ref{uniqclos1} together with proposition \ref{cubepos} shows the contradiction.
It is clear from lemma \ref{weaknilprop} that $\gamma$ is factor consistent. By theorem \ref{charpres} we obtain that $\gamma$ is a strong nilspace factor.
\medskip
\section{Higher order dual groups}\label{chap:higherdual}
The goal of this part of the paper is to analyze the structure of $\hat{{\bf A}}_k$. We start with the prof of theorem \ref{hofdecomp}.
\medskip
\noindent{\it Proof of theorem \ref{hofdecomp}}.~~Let $f:{\bf A}\rightarrow\mathbb{C}$ be an arbitrary bounded function measurable in $\mathcal{F}_k$. Then according to theorem \ref{main} there is a strong $k$-step nilspace factor $\gamma:{\bf A}\rightarrow N$ and a Borel measurable function $h:N\rightarrow\mathbb{C}$ such that $f$ is equal to $h\circ\gamma$. Using lemma \ref{nilfourdec} and lemma \ref{nilspchar} we can decompose $h$ as $\sum_{\chi\in\hat{A_k}}g_\chi h_\chi$ (converging in $L^2$) where $h_\chi\in W(\chi,N)$~,~$|h_\chi|=1$ and $g_\chi$ is measurable in the $k-1$ step factor of $N$. We obtain that $f=\sum_{\chi\in\hat{A_k}}(g_\chi\circ\gamma)( h_\chi\circ\gamma)$. The terms $g_\chi\circ\gamma$ are measurable in $\mathcal{F}_{k-1}$ and the terms $h_\chi\circ\gamma$ are $k$-th order characters by lemma \ref{homok}. It follows that $L^2(\mathcal{F}_k)$ is spanned by the modules in $\hat{A}_k$ and thus by lemma \ref{charort} the proof is complete.
\medskip
\begin{lemma}\label{highconv} Let $f,g$ be two functions in $L^\infty(\mathcal{F}_k)$. For $a\in\hat{{\bf A}}_k$ let us denote the component of $f$ and $g$ in $a$ by $f_a$ and $g_a$. Then the component of $fg$ in $c\in\hat{{\bf A}}_k$ is equal to
$$\sum_{ab=c}f_ag_b$$
where the above sum has only countable many non zero term and the sum is convergent in $L^2$.
\end{lemma}
\begin{proof} First of all we observe that if $g$ is contained in a single rank one module then
the $k$-th order decomposition of $fg$ is $\sum_{a\in\hat{{\bf A}}_k} f_ag$ since it converges in $L^2$ and the terms $f_ag$ are from distinct rank one modules.
From this observation we also get the statement if $g$ has finitely many non zero components.
If $g$ has infinitely many components then for an arbitrary $\epsilon$ we can approximate $g$ with precision $\epsilon$ in $L^2$ by a sub-sum $g_\epsilon$ of its components . Then $fg=fg_\epsilon+f(g-g_\epsilon)$. Here the $\|f(g-g_\epsilon)\|_2\leq \|f\|_\infty\epsilon$. So as $\epsilon$ goes to $0$ the $L^2$ error we make also goes to $0$.
\end{proof}
\begin{definition} Let $f$ be a function in $L^2(\mathcal{F}_k)$. We say that the $k$-th dual-support $S_k(f)\subseteq\hat{{\bf A}}_k$ of $f$ is the set of rank one modules that are not orthogonal to $f$.
It is clear that $S_k(f)$ is a countable set.
\end{definition}
The next two statement follows from lemma \ref{highconv}.
\begin{lemma}\label{prodsup} If $f,g\in L^\infty(\mathcal{F}_k)$ then $S_k(fg)\subseteq S_k(f)S_k(g)$ and $S_k(f+g)\subseteq S_k(f)\cup S_k(g)$. Furthermore if $t\in{\bf A}$ is fixed then the function $f_t(x)=f(x+t)$ satisfies $S_k(f_t)=S_k(f)$.
\end{lemma}
For a subgroup $T\leq\hat{{\bf A}}_k$ we denote by $\mathfrak{H}_k(T)$ the collection of sets $U$ that are measurable in $\mathcal{F}_k$ and $S(1_U)\subset T$. It follows from lemma \ref{prodsup} that $\mathfrak{H}_k(T)$ is a shift invariant $\sigma$-algebra such that $\mathcal{F}_{k-1}\subseteq\mathfrak{H}_k(T)\subseteq\mathcal{F}_k$. If $\mathcal{B}\subset\mathcal{F}_k$ is a separable $\sigma$-algebra and $\{U_i\}_{i=1}^\infty$ is a generating system of $\mathcal{B}$ then for the countable group $T$ generated by $\{S_k(1_{U_i})\}_{i=1}^\infty$ we have that $\mathcal{B}\subseteq\mathfrak{H}_k(T)$.
\begin{lemma}\label{secder} If $\phi$ is a $k$-th order character then there is a countable subgroup $T<\hat{{\bf A}}_{k-1}$ such that $\Delta_{t_1,t_2}\phi$ is measurable in $\mathfrak{H}_{k-1}(T)$ for every pair $t_1,t_2\in{\bf A}$.
\end{lemma}
\begin{proof} Lemma \ref{charsep} and lemma \ref{sepsiggen} implies that there is a separable $\sigma$-algebra $\mathcal{B}\subset\mathcal{F}_{k-1}$ such that $\phi\in [\mathcal{B},k]^*$. Let $T\subset\hat{{\bf A}}_{k-1}$ be a countable subgroup such that $\mathcal{B}\subseteq\mathfrak{H}_{k-1}(T)$.
Then $\phi\in[\mathfrak{H}_{k-1}(T),k]^*$. Lemma \ref{pureprop} implies that $\Delta_{t_1,t_2}\phi\in[\mathfrak{H}_{k-1}(T),k-2]^*$ for every $t_1,t_2\in{\bf A}$.
Using lemma \ref{charmes} we obtain that $\Delta_{t_1,t_2}\phi$ is measurable in $\mathfrak{H}_{k-1}(T)\vee\mathcal{F}_{k-2}=\mathfrak{H}_{k-1}(T)$.
\end{proof}
\begin{lemma}\label{nozer} Let $f,g$ be $L^\infty(\mathcal{A})$ functions such that non of $\mathbb{E}(f|\mathcal{F}_{k-1})$ and $\mathbb{E}(g|\mathcal{F}_{k-1})$ is the $0$ function. Then $\mathbb{E}_t\Bigl(\|f(x)\overline{g(x+t)}\|_{U_k}^{2^k}\Bigr)>0$.
\end{lemma}
\begin{proof} The support of both $f_1=\mathbb{E}(f|\mathcal{F}_{k-1})$ and $g_1=\mathbb{E}(g|\mathcal{F}_{k-1})$ has positive measure. This means that for a positive measure set of $t$'s the supports of $f_1$ and $g_1(x+t)$ intersect each other in a positive measure set. (By Fubini's theorem, the expected value of the measure of the intersection is the product of the measures of the supports.) Since $U_k$ is a norm on $L^\infty(\mathcal{F}_{k-1})$ we get that in (\ref{twofun}) the right hand side is not $0$. By lemma \ref{twofunct} the proof is complete.
\end{proof}
\begin{proposition}\label{fixtype} Let $\phi$ be a $k$-th order character. Then there is a countable subgroup $T\leq\hat{{\bf A}}_{k-1}$ such that $S_{k-1}(\Delta_t\phi)S_{k-1}^{-1}(\Delta_t\phi)\subseteq T$ for every fixed $t\in {\bf A}$.
\end{proposition}
\begin{proof} Let $T$ be the subgroup of $\hat{{\bf A}}_{k-1}$ guaranteed by lemma \ref{secder}. We have that $S_{k-1}(\Delta_{t_1,t_2}\phi)\subseteq T$ for every $t_1,t_2$. Let $t_1\in{\bf A}$ be an arbitrary fixed element and let $\Delta_{t_1}\phi=f_1+f_2+\dots$ be the unique $k-1$-th order Fourier decomposition of $\Delta_{t_1}\phi$ into non zero functions. Assume that $\lambda_i\in\hat{{\bf A}}_{k-1}$ is the module containing $f_i$ for every $i$. We have to show that $\lambda_i\lambda_j^{-1}\in T$ for every pair of indices $i,j$.
Let us choose a $k-1$-th order character $\phi_i$ from every module $\lambda_i$ and let $g_i$ denote $(\Delta_{t_1}\phi)\overline{\phi_i}$.
We have that $\mathbb{E}(g_i|\mathcal{F}_{k-2})$ is not $0$.
This means by lemma \ref{nozer} and theorem \ref{propfk} that for a positive measure of $t_2$'s $\mathbb{E}(g_i(x)\overline{g_j(x+t_2)}|\mathcal{F}_{k-2})$ is not the $0$ function. On the other hand $g_i(x)\overline{g_j(x+t_2)}=(\Delta_{t_1,t_2}\phi(x))\overline{\phi_i(x)}\phi_j(x+t_2)$.
Here $\overline{\phi_i(x)}\phi_j(x+t_2)$ is an element from the module $\lambda_j\lambda_i^{-1}$.
If $\mathbb{E}(g_i(x)\overline{g_j(x+t_2)}|\mathcal{F}_{k-2})$ is not $0$ for some $t_2$ then the $\lambda_i\lambda_j^{-1}$ component of $\Delta_{t_1,t_2}\phi$ is not zero. It shows that $\lambda_i\lambda_j^{-1}\in T$.
\end{proof}
\begin{lemma}\label{charhom} For every $k$-th order character $\phi$ there is a countable subgroup $T\subset\hat{{\bf A}}_{k-1}$ and a homomorphism $h:{\bf A}\rightarrow\hat{{\bf A}}_{k-1}/T$ such that $S_{k-1}(\Delta_t\phi)$ is contained in the coset $h(t)$.
\end{lemma}
\begin{proof} Proposition \ref{fixtype} implies that there is a countable subgroup $T\subset\hat{{\bf A}}_{k-1}$ such that $S_{k-1}(\Delta_t\phi)$ is contained in a coset of $T$ for every element $t\in{\bf A}$. We denote this coset by $h(t)$. We have to show that $h$ is a homomorphism. This follwos from
$\Delta_{t_1+t_2}\phi(x)=\Delta_{t_2}\phi(x+t_1)\Delta_{t_1}\phi(x)$
together with lemma \ref{prodsup}.
\end{proof}
\begin{lemma}\label{countriv} Let $k\geq 2$ and $\phi$ be a $k$-th order character such that there is a countable subgroup $T\subseteq \hat{{\bf A}}_{k-1}$ with the property that $\Delta_t\phi$ is measurable in $\mathfrak{H}_{k-1}(T)$ for every $t\in{\bf A}$. Then $\phi$ is measurable in $\mathcal{F}_{k-1}$ (or in other words $\phi$ represents the trivial module).
\end{lemma}
\begin{proof} For $a\in T$ let $Q_a=\{t|t\in{\bf A},a\in S_{k-1}(\Delta_t\phi)\}$. First we claim that $Q_a$ is measurable. Indeed, if $\phi'$ is a fixed character representing $a$ then $Q_a$ is the set of $t$'s for which the measurable function $t\mapsto\|\overline{\phi'}\Delta_t\phi\|_{U_{k-1}}$ is not zero. Using $\sigma$-additivity and $\cup_{a\in T}Q_a={\bf A}$ we obtain that there is a fixed $a\in T$ such that $Q_a$ has positive measure. Assume that $\phi'$ represents $a$ and let $f(x)=\overline{\phi'(x)}\phi(x)$. Let $H=\{h_v\}_{v\in\{0,1\}^k}$ be the function system with $h_{(w,0)}=f$ and $h_{(w,1)}=\phi$ for $w\in\{0,1\}^{k-1}$. Then we have by (\ref{dimred}) that $$(H)=\mathbb{E}_t(\|f(x)\overline{\phi(x+t)}\|^{2^{k-1}}_{U_{k-1}})=\mathbb{E}_t(\|\overline{\phi'}\Delta_t\phi\|^{2^{k-1}}_{U_{k-1}})>0.$$ By (\ref{GCS}) we obtain that $\|\phi\|_{U_k}>0$ and so lemma \ref{erosort} implies that $\phi$ represent the trivial module.
\end{proof}
\subsection{Various Hom-sets}
In this chapter we use multiplicative notation for Abelian groups. For two Abelian groups $A_1$ and $A_2$ we denote by $\hom(A_1,A_2)$ the set of all homomorphism from $A_1$ to $A_2$. The set $\hom(A_1,A_2)$ is an Abelian group with respect to the point wise multiplication. Let $\aleph_0(A_2)$ denote the set of countable subgroups in $A_2$. The groups $\hom(A_1,A_2/T)$ where $T\in\aleph_0(A_2)$ are forming a direct system with the natural homomorphisms $\hom(A_1,A_2/T_1)\rightarrow\hom(A_1,A_2/T_2)$ defined when $T_1\subseteq T_2$.
\begin{definition} $\hom^*(A_1,A_2)$ is the direct limit of the direct system $$\{\hom(A_1,A_2/T)\}_{T\in\aleph_0(A_2)}$$
with the homomorphisms induced by embeddings on $\aleph_0(A_2)$.
\end{definition}
We describe the elements of $\hom^*(A_1,A_2)$.
Let $H(A_1,A_2)$ be the disjoint union of all the sets $\hom(A_1,A_2/T)$ where $T$ runs through the countable subgroups of $A_2$. If $h_1\in\hom(A_1,A_2/T_1)$ and $h_2\in\hom(A_1,A_2/T_2)$ are two elements in $H(A_1,A_2)$ then we say that $h_1$ and $h_2$ are equivalent if there is a countable subgroup $T_3$ of $A_2$ containing both $T_1$ and $T_2$ such that $h_1$ composed with $A_2/T_1\rightarrow A_2/T_3$ is the same as $h_2$ composed with $A_2/T_2\rightarrow A_2/T_3$.
The equivalence classes in $H(A_1,A_2)$ are forming an Abelian group that is the same as $\hom^*(A_1,A_2)$.
For two abelian groups let $\hom^c(A_1,A_2)$ denote the set of homomorphisms whose image is countable. We denote by $\hom^0(A_1,A_2)$ the factor $\hom(A_1,A_2)/\hom^c(A_1,A_2)$. If $T$ is a countable subgroup of $A_2$ then there is a natural embedding of $\hom^0(A_1,A_2/T)$ into $\hom^*(A_1,A_2)$ in the following way.
The set $\hom(A_1,A_2/T)$ is a subset of $H(A_1,A_2)$. It is easy to see that $\phi_1,\phi_2\in\hom(A_1,A_2/T)$ are equivalent if and only if they are contained in the same coset of $\hom^c(A_1,A_2)$. From the definitions it follows that
\begin{equation}\label{homstarcup} \hom^*(A_1,A_2)=\bigcup_{T\in\aleph_0(A_2)} \hom^0(A_1,A_2/T).
\end{equation}
From (\ref{homstarcup}) we obtain the next lemma.
\begin{lemma}\label{exp} If $A_1$ is of exponent $n$ then so is $\hom^*(A_1,A_2)$.
\end{lemma}
Note that an abelian group is said to be of exponent $n$ if the $n$-th power of every element is the identity.
It is easy to see that If $p$ is a prime number and $A_2$ is of exponent $p$ then $\hom^*(A_1,A_2)=\hom^0(A_1,A_2)$.
\begin{definition} We say that an Abelian group is essentially torsion free if there are at most countably many finite order elements in it.
\end{definition}
\begin{lemma}\label{estors} If $A$ is essentially torsion free then $A/T$ is essentially torsion free whenever $T\in\aleph_0(A)$.
\end{lemma}
\begin{proof} Assume by contradiction that there are uncountably many finite order elements in $A/T$. Then there is a natural number $n$ and element $t\in T$ such that the set
$S=\{x~|~x\in A,~x^n=t\}$ is uncountable. Then for a fixed element $y\in S$ the set $Sy^{-1}$ is an uncountable set of finite order elements in $A$ which is a contradiction.
\end{proof}
\begin{lemma}\label{nullfree} If $A_2$ is essentially torsion free then $\hom^0(A_1,A_2)$ is torsion free.
\end{lemma}
\begin{proof} Assume by contradiction that there is an element $\tau\in\hom(A_1,A_2)$ and $n\in\mathbb{N}$ such that $\tau(A_1)$ is uncountable but $\tau^n(A_1)$ is countable.
Similarly to the proof of lemma \ref{estors} this means that there is a fixed element $t\in\tau^n(A_1)$ whose pre image under the map $x\mapsto x^n$ is uncountable which is a contradiction.
\end{proof}
\begin{lemma}\label{estfree} If $A_2$ is essentially torsion free then $\hom^*(A_1,A_2)$ is torsion free.
\end{lemma}
\begin{proof} By (\ref{homstarcup}) it is enough to prove that for every $T\in\aleph_0(A_2)$ the group $\hom^0(A_1,A_2/T)$ is torsion free. Lemma \ref{estors} implies that $A_2/T$ is essentially torsion free. Lemma \ref{nullfree} finishes the proof.
\end{proof}
\subsection{On the structure of the higher order dual groups}
We return to the structure of $\hat{{\bf A}}_k$.
Let us start with $\hat{{\bf A}}_1$.
\begin{lemma}\label{firstdual} The group $\hat{{\bf A}}_1$ is isomorphic to $\prod_\omega\hat{A_i}$.
\end{lemma}
\begin{proof}
We have that ${\bf A}$ is the ultra product of a sequence $\{A_i\}_{i=1}^\infty$ of compact abelian groups.
Let $H\subseteq\hat{{\bf A}}_1$ denote the set of those characters that are ultra limits of characters on $\{A_i\}_{i=1}^\infty$.
Let $\lambda_i:A_i\rightarrow\mathbb{C}$ and $\chi_i:A_i\rightarrow\mathbb{C}$ be two sequences of linear characters. Let furthermore $\lambda$ be the ultra limit of $\{\lambda_i\}_{i=1}^\infty$ and $\chi$ be the ultra limit of $\{\chi_i\}_{i=1}^\infty$.
If $\lambda_i$ differs from $\chi_i$ on an index set which is in the ultra filter then they are orthogonal at this index set and so they are orthogonal in the limit. This implies that $\lambda=\chi$ if and only if the sequences $\{\lambda_i\}_{i=1}^\infty$ and $\{\mu_i\}_{i=1}^\infty$ agree on a set from the ultra filter. In other words $H$ is isomorphic to the ultra product of the dual groups of $\prod_\omega\hat{A_i}$.
We show that $H=\hat{{\bf A}}_1$. Assume by contradiction that $H$ is strictly smaller than ${\bf A}_1$. Then there is a character $\phi\in\hat{{\bf A}}_1$ which is orthogonal to every character in $H$.
We have that $\phi=\lim_\omega\phi_i$ where $\phi_i:A_i\rightarrow\mathbb{C}$ is a measurable function of absolute value $1$. For $i\in\mathbb{N}$ let $a_i$ denote the $L^\infty$ norm of the Fourier transform of $\phi_i$.
We have that $\lim_\omega a_i=0$. Equation (\ref{u2norm}) implies that $\|\phi_i\|_{U_2}\leq \sqrt{a_i}$ and thus $\|\phi\|_{U_2}=\lim_\omega\|\phi_i\|_{U_2}=0$ which is a contradiction by lemma \ref{kisk}.
\end{proof}
\medskip
By lemma \ref{charhom} every $k$-th order character $\phi$ induces a homomorphism from ${\bf A}$ to $\hat{{\bf A}}_{k-1}/T$ for some countable subgroup. This homomorphism represents an element in $\hom^*({\bf A},\hat{{\bf A}}_{k-1})$. We denote this element by $q_k(\phi)$. If $q_k(\phi_1)=q_k(\phi_2)$ then Lemma \ref{countriv} shows that if $k\geq 2$ then $\phi_1$ and $\phi_2$ belong to the same rank one module.
This implies the following theorem.
\begin{theorem}\label{dualemb} If $k\geq 2$ then $q_k:\hat{{\bf A}}_k\rightarrow\hom^*({\bf A},\hat{{\bf A}}_{k-1})$ is an injective homomorphism.
\end{theorem}
Note that if $k=1$ then $\hat{{\bf A}}_1$ is embedded into $\hom({\bf A},\hat{{\bf A}}_0)$ where $\hat{{\bf A}}_0$ is defined as the complex unit circle with multiplication.
The next theorem follows immediately from theorem \ref{dualemb}
\begin{theorem}[Structure of the dual groups]\label{dualstruct} $\hat{{\bf A}}_k$ is isomorphic to a subgroup in
$$\hom^*({\bf A},\hom^*({\bf A},\dots,\hom^*({\bf A},\hat{{\bf A}}_1))\dots)$$
where the number of $\hom^*$-s is $k-1$,
\end{theorem}
\begin{proof} The proof follows directly from Lemma \ref{dualemb} and the fact that $\hom^*(A_1,A_2)\subseteq \hom^*(A_1,A_3)$ whenever $A_2\subseteq A_3$.
\end{proof}
Theorem \ref{dualstruct} has the next two useful consequences.
\begin{lemma}\label{duexp} Let $e$ be a natural number and assume that the groups $\{A_i\}_{i=1}^\infty$ have exponent $e$. Then $\hat{{\bf A}}_k$ has exponent $e$ for every $k\geq 1$.
\end{lemma}
\begin{proof} We have by lemma \ref{firstdual} that $\hat{{\bf A}}_1$ has exponent $e$. Then lemma \ref{exp} and theorem \ref{dualstruct} finish the proof.
\end{proof}
\begin{lemma}\label{charzero} Let $\{A_i\}_{i=1}^\infty$ be a sequence of groups such that for every natural number $n>1$ there are only finitely many indices $i$ such that $\hat{A_i}$ has an element of order $n$. Then $\hat{{\bf A}}_k$ is torsion free for every $k\geq 1$.
\end{lemma}
\begin{proof} We have by lemma \ref{firstdual} that $\hat{{\bf A}}_1$ is torsion free. Then lemma \ref{estfree} and theorem \ref{dualstruct} finish the proof.
\end{proof}
\section{Consequences of the main theorem}
\subsection{Regularization, inverse theorem and special families of groups}
\bigskip
We prove theorem \ref{reglem}, theorem \ref{invthem}, theorem \ref{restreg} and theorem \ref{restinv}.
\noindent{\it Proof of theorem \ref{reglem}}~We proceed by contradiction. Let us fix $k$ and $F$. Assume that the statement fails for some $\epsilon>0$.
This means that there is a sequence of measurable functions $\{f_i\}_{i=1}^\infty$ on the compact abelian groups $\{A_i\}_{i=1}^\infty$ with $|f_i|\leq 1$ such that $f_i$ does not satisfy the statement with $\epsilon$ and $n=i$. Let $\omega$ be a fixed non-principal ultra filter and ${\bf A}=\prod_\omega A_i$.
We denote by $f$ the ultra limit of $\{f_i\}_{i=1}^\infty$.
By theorem \ref{main} we have that $f=f_s+f_r$ where $\|f_r\|_{U_{k+1}}=0$ and $f_s=\tilde{\gamma}\circ g$ for some strong nilspace factor $\tilde{\gamma}:{\bf A}\rightarrow N$ and measurable function $g:N\rightarrow\mathbb{C}$.
Now we use theorem \ref{inverselimit} which says that $N$ is an inverse limit of finite dimensional nilspaces $\{N_i\}_{i=1}^\infty$ in such a way that the projections from $N$ to $N_i$ are all fibre surjective.
Let $\mathcal{N}_i$ denote the $\sigma$-algebra generated by the projection to $N_i$. Then we have that
$g=\lim_{i\rightarrow\infty}\mathbb{E}(g|\mathcal{N}_i)$ in $L^1$.
It follows that there is an index $j$ such that
$g_j=\mathbb{E}(g|\mathcal{N}_j)$ satisfies $\|g-g_j\|_1\leq\epsilon/3$.
Let $\gamma:{\bf A}\rightarrow N_j$ be the composition of $\tilde{\gamma}$ with the fibre surjective map $N\rightarrow N_j$.
We have by lemma \ref{strongcirc} that $\gamma$ is a strong nilspace factor of ${\bf A}$.
Furthermore there is a Lipschitz function $h:N_j\rightarrow\mathbb{C}$ with $|h|\leq 1$ and Lipschitz constant $c$ such that $\|g_j-h\|_1\leq\epsilon/3$.
Using that $\gamma$ is a strong nilspace factor we have that $q=\gamma\circ h$ satisfies that $\|f_s-q\|\leq 2\epsilon/3$.
Let $f_e=f_s-q$.
The function $\gamma$ is a continuous function so there is a sequence of continuous functions $\{\gamma_i':A_i\rightarrow N_j\}_{i=1}^\infty$ such that $\lim_{\omega}\gamma'_i=\gamma$. It is easy to see that $\gamma'_i$ is an approximate morphism with error tending to $0$.
It follows from \cite{NP} that it can be corrected to a morphism $\gamma_i$ (if $i$ is sufficiently big) such that the maximum point wise distance of $\gamma_i$ and $\gamma_i'$ goes to $0$. As a consequence we have that $\lim_\omega\gamma_i=\gamma$.
Let $f^i_s=\gamma_i\circ h$, and let $f^i_r$ be a sequence of measurable functions with $\lim_\omega f^i_r=f_r$.
We set $f^i_e=f_i-f^i_s-f^i_r$. It is clear that $\lim_\omega f^i_s=q$ and $\lim_\omega f^i_e=f_e$.
We also have that $\lim_\omega\|f^i_r\|_{U_{k+1}}=\|f_r\|_{U_{k+1}}=0$, $\lim_\omega (f^i_r,f^i_s)=(f_r,q)=0$ and $\lim_\omega(f^i_r,f^i_e)=(f_r,f_e)=0$.
Let $m$ be the maximum of the complexity of $N_i$ and $c$.
There is an index set $S$ in $\omega$ such that
\begin{enumerate}
\item $\|f^i_r\|_{U_{k+1}}\leq F(\epsilon,m)$,
\item $\|f^i_e\|_1\leq \epsilon$,
\item $|(f^i_r,f^i_s)|~,~|(f^i_r,f^i_e)|\leq F(\epsilon,m)$,
\item $\gamma_i$ is at most $F(\epsilon,m)$ balanced,
\end{enumerate}
hold simultaneously on $S$.
Note that $\gamma$ itself is $0$ balanced.
This is a contradiction.
\bigskip
\noindent{\it Proof of theorem \ref{restreg}}~In the proof of theorem \ref{reglem} the nilspace $N_j$ that we construct is a character preserving factor of ${\bf A}$. This means that the $i$-th structure group of $N_j$ is embedded into $\hat{{\bf A}}_i$. This shows that $N_j$ is a $\mathfrak{A}$-nilspace.
\bigskip
\noindent{\it Proof of theorem \ref{invthem} and theorem \ref{restinv}}~It is clear that if we apply theorem \ref{reglem} with $\epsilon_2>0$ and function $F(a,b)=a/b$ then in the decomposition $f=f_s+f_e+f_r$ the scalar product $(f,f_s)$ is arbitrarily close to $(f_s,f_s)$ and $\|f_s\|_{U_{k+1}}$ is arbitrarily close to $\|f\|_{U_{k+1}}$ if $\epsilon_2$ is small enough (depending only on $\epsilon$). This means by corollary \ref{l2becs} that $(f_s,f_s)\geq2\epsilon^{2^k}/3$ holds if $\epsilon_2$ is small and also $(f,f_s)\geq\epsilon^{2^k}/2$ holds simultaneously.
The inverse theorem is special families follows in the same way from theorem \ref{restreg}.
\subsection{Limit objects for convergent function sequences}
We start the chapter with an important observation.
\begin{lemma} Let $f\in L^\infty({\bf A})$ and $M\in\mathcal{M}_i$ be a simple moment of degree $i$. Then $M(f)=M(\mathbb{E}(f|\mathcal{F}_i))$.
\end{lemma}
\begin{proof}\label{momentproj} Let $F=\{f_v\}_{v\in K_{i+1}}$ be a function system such that each $f_v$ is one of $1_{{\bf A}},f,\overline{f}$ and let $F'=\{\mathbb{E}(f_v|\mathcal{F}_i)\}_{v\in K_{i+1}}$. Observe that for each $M\in\mathcal{M}_i$ there is a fuction system of this form such that $[F](0)=M(f)$. Then by the multilinearity of convolutions and lemma \ref{cornineq} we have that $[F](0)=[F'](0)=M(\mathbb{E}(f|\mathcal{F}_i))$.
\end{proof}
We continue with a few technical notions. We denote the $\sigma$-algebra $\vee_{i=1}^\infty\mathcal{F}_i$ by $\mathcal{F}$. We say that a $\sigma$-algebra $\mathcal{B}$ is a nil $\sigma$-algebra of infinite order if $\mathcal{B}=\vee_{i=1}^\infty[\mathcal{B}]_i$ and (\ref{rhsnil}) holds for every $i\in\mathbb{N}$.
Lemma \ref{weaknilprop} shows that in this case $[\mathcal{B}]_i=\mathcal{B}\cap\mathcal{F}_i$ for every $i\in\mathbb{N}$ and that $[\mathcal{B}]_i$ is a nil $\sigma$-algebra of order $i-1$ for every $i\in\mathbb{N}$.
The proof of lemma \ref{embednil} with minor modifications shows that every separable sub $\sigma$-algebra in $\mathcal{F}$ is contained in a separable nil $\sigma$-algebra of infinite order.
First we prove theorem \ref{simplim} and corollary \ref{simplimcor}.
\bigskip
Let $\{f_i:A_i\rightarrow\mathbb{C}\}$ be a sequence of functions with $|f_i|\leq r$.
Let ${\bf A}$ be the ultra product of $\{A_i\}_{i=1}^\infty$ and $f$ be the ultra limit of $\{f_i\}_{i=1}^\infty$.
We have that $M(f)=\lim_\omega M(f_i)=\lim M(f_i)$ for every moment $M$.
Let $g$ denote the projection of $f$ to the $\sigma$-algebra $\mathcal{F}$ and let $g_i=\mathbb{E}(f|\mathcal{F}_i)$.
Using that $\mathbb{E}(f|\mathcal{F}_i)=\mathbb{E}(g|\mathcal{F}_i)=g_i$ holds for every $i\in\mathbb{N}$ we get by lemma \ref{momentproj} that $M(g)=M(f)=M(g_i)$ holds for every moment $M\in\mathcal{M}_i$.
Let $\mathcal{B}$ be a separable nil $\sigma$-algebra of infinite order such that $g$ is measurable in $\mathcal{B}$ and let $\mathcal{B}_i=[\mathcal{B}]_{i+1}=\mathcal{B}\cap\mathcal{F}_i$.
Using theorem \ref{main} we can create a sequence $\gamma_i:{\bf A}\rightarrow N_i$ of nilspace factors generating the $\sigma$-algebra $\mathcal{B}_i$ in a way that these factors form an inverse system.
Let $\gamma:{\bf A}\rightarrow N$ be the inverse limit of these factors.
Since all the functions $g_i$ are measurable in the $\sigma$-algebra generated by $\gamma$ the function $g$ is also measurable in it. This means that there is a function $h:N\rightarrow\mathbb{C}$ such that $\gamma\circ h=g$.
Using the rooted measure preserving property of the factors $\gamma_i$ we have $M(g)=M(h)$ for every simple moment $M$.
This completes the proof of theorem \ref{simplim}. For corollary \ref{simplimcor} let $h_i=\mathbb{E}(h|N_i)$. It is clear that $h_i\circ\gamma=g_i$ and so $M(h_i)=M(g_i)=M(f)$ holds for every $M\in\mathcal{M}_i$.
\bigskip
The proof of theorem \ref{genlim} goes in a very similar way. The only difference is that we project all the functions $f^a\overline{f^b}$ with $a,b\in\mathbb{N}$ to $\mathcal{F}$ (resp. $\mathcal{F}_i$). The resulting function system $g^{a,b}$ (resp. $g^{a,b}_i$) at almost every point $x$ describes the complex moments of a probability distribution on the complex disc of radius $r$. Then we chose the separable nil $\sigma$-algebra $\mathcal{B}\subset\mathcal{F}$ so that each $g^{a,b}$ is measurable in $\mathcal{B}$. The rest of the proof is essentially the same.
\vskip 0.2in
\noindent
Bal\'azs Szegedy
\noindent
University of Toronto, Department of Mathematics,
\noindent
St George St. 40, Toronto, ON, M5R 2E4, Canada
|
{
"timestamp": "2012-03-13T01:01:17",
"yymm": "1203",
"arxiv_id": "1203.2260",
"language": "en",
"url": "https://arxiv.org/abs/1203.2260"
}
|
\section{Introduction}\label{sec1}
The screening effect is the geostatistical term for the phenomenon of
nearby observations tending to reduce the influence of more distant
observations when using kriging (optimal linear prediction) for spatial
interpolation [\citet{JouHui78}, \citet{ChiDel99}]. This
phenomenon is often invoked as a justification for ignoring more
distant observations when using kriging [\citet{MemMou07},
\citet{Eme09}]. Only in some very limited special cases is the
effect exact in the sense\vadjust{\goodbreak} that the more distant observations make no
contribution to the kriging predictor, so it is natural to use
asymptotics as a way to study the screening effect.
Let us set some notation. Write $x\cdot y$ for the inner product of
commensurate vectors $x$ and $y$. Suppose $Z$ is a mean square
continuous, stationary, mean 0 Gaussian process on $\R^d$ with
autocovariance function $K(x) = E\{Z(x)Z(0)\}$ and spectral density
$f$, so that $K(x) = \int_{\R^d} e^{i\omega\cdot x} f(\omega) \,d\omega$.
When the mean is assumed known to be 0, kriging is often called simple
kriging. Throughout this work, we assume that the problem of interest
is to predict $Z(0)$. For $S\subset\R^d$, write $Z(S)$ for the vector
of observations (in some order) of $Z$ on~$S$, and define $e(S)$ to be
the error of the best linear predictor, or BLP, of $Z(0)$ based on
$Z(S)$. Let $N_\ep$ and $F_\ep$ be two classes of sets indexed by the
parameter $\ep>0$, with $N_\ep$ representing observations near 0 and
$F_\ep$ more distant observations. We will say that $N_\ep$
asymptotically screens out the effect of $F_\ep$ if
\begin{equation}\label{main}
\lim_{\ep\downarrow0} \frac{Ee(N_\ep\cup F_\ep)^2}
{Ee(N_\ep)^2} = 1.
\end{equation}
\citet{Ste02} argues that a useful asymptotic approach is to let
the smallest distance from the observations to the predictand
tend to 0 as $\ep\downarrow0$.
Specifically,
\citet{Ste02} proves (\ref{main}) when, essentially, for some
$x_0\in
\R^d$
not in the integer lattice, $F_\ep$ is all
points of the form $\ep(x_0+J)$ for $J$ in the integer lattice, $N_\ep$
is the restriction of $F_\ep$ to some fixed region with 0 in its interior
and $f$ is regularly varying at infinity
[\citet{BinGolTeu87}]
in every direction with a common index of variation.
The methods used in \citet{Ste02} make strong use of the gridded
nature of the observations and are not applicable here.
Furthermore, requiring
$f$ to be regularly varying at infinity with common index of variation in
all directions excludes
models for spatial-temporal phenomena that
exhibit a different degree of smoothness in space than in time.
Section \ref{sec4} provides further discussion of these issues.
Ramm (\citeyear{Ram05}), Chapter 5, takes a different approach to
studying an asymptotic
screening effect by considering a process observed with white noise
everywhere in some domain and letting the variance of the white noise
tend to 0.
In this work, we take a closer look at how the set where $Z$ is observed
affects whether an asymptotic screening effect holds.
We will take the sets $N_\ep$ and $F_\ep$ to have a particular form
that simplifies the asymptotic analysis.
Suppose $x_1,\ldots,x_n$ are distinct nonzero elements of~$\R^d$,
$y_1,\ldots,y_m$ are
distinct elements of $\R^d$ and $y_0\in\R^d$ is nonzero.
For the rest of this work, let $N_\ep= \{\ep x_1,\ldots,\ep x_n\}$
and $F_\ep=\{y_0+\ep y_1,\ldots,y_0+\ep y_m\}$.
Section \ref{sec2} explores when (\ref{main}) holds through a series of examples
leading to a~broad conjecture under
a key assumption on the spectral density $f$ of the random field:
for every $R<\infty$,
\begin{equation}\label{f-cond}
\lim_{\omega\to\infty} \sup_{|\nu|<R} \biggl| \frac{f(\omega
+\nu)}{f(\omega)}
-1\biggr| = 0.
\end{equation}
The examples will demonstrate that one generally needs a further
condition on $N_\ep$ depending on the mean square differentiability
properties of the process. For nondifferentiable processes, no further
assumptions on $N_\ep$ may be needed. Indeed, for nondifferentiable
processes on $\R$, Theorem \ref{thm1} in Section~\ref{sec3} has
(\ref{main}) as its conclusion under (\ref{f-cond}) and a mild
additional condition on $f$. For nondifferentiable processes on~$\R^2$,
if one restricts the cardinality of $N_\ep$ to~1 and of $F_\ep$ to 2
(and sets $y_2=0$), then Theorem \ref{thm2} proves (\ref{main}) under~(\ref{f-cond}) without any additional conditions on $f$.
Mat\'ern models [\citet{Ste99N1}] appear in both the examples
and the proof of Theorem \ref{thm1}.
Define $\mathcal{K}_\nu$ to be the modified Bessel function of the second
kind of order $\nu$ [\citet{autokey8}].
The Mat\'ern model on $\R^d$ has autocovariance function
$\phi(\alpha|x|)^\nu\mathcal{K}_\nu(\alpha|x|)$
for positive $\phi,\alpha$ and $\nu$.
The parameter $\nu$ controls the smoothness of the process: $Z$ has $m$
mean square derivatives in any direction if and only if $ \nu> m$.
The corresponding spectral density equals
$\phi(\alpha^2+|\omega|^2)^{-\nu-d/2}$ times a constant depending
on $\alpha,\nu$ and $d$.
All Mat\'ern models satisfy (\ref{f-cond}).
\section{Examples}\label{sec2}
This section studies a number of examples to gain some insight into
the conditions on $f$ and $N_\ep$ that are needed in order for
(\ref{main}) to hold.
The derivations of these results are elementary but not necessarily easy.
Rather than give detailed derivations of all of them, I will outline
derivations in a few of the more difficult examples in Section \ref{sec51}.
\begin{figure}[b]
\includegraphics{909f01.eps}
\caption{Prediction problem for triangular autocovariance function.
Prediction site \mbox{($+$ sign)}, nearby observation (solid circle)
and distant observations (open circles).}
\label{fig1}
\end{figure}
To see why a condition like (\ref{f-cond}) is needed, let us first
consider an example on $\R$
addressed in \citet{SteHan89} and Stein (\citeyear{Ste99N1}),
pages \mbox{67--69}.
Suppose $n=1$, $x_1=1$, $m=2$, $y_0=1$, $y_1=0$ and $y_2=1$; see Figure
\ref{fig1}.
Consider $K(x)= e^{-|x|}$,
a Mat\'ern model with smoothness parameter $\frac{1}{2}$.
The corresponding process is mean square continuous but is not mean
square differentiable, and it is easy to show $Ee(N_\ep)^2\sim2\ep$
as $\ep\downarrow0$.
This process is Markov, so that $Ee(N_\ep\cup F_\ep)^2=Ee(N_\ep)^2$
for all $\ep<1$ and (\ref{main}) holds trivially.
Next consider $K(x) = (1-|x|)^+$ (where the superscipt $+$ indicates positive
part), for which $f(\omega) = \frac{1-\cos\omega}{\pi\omega^2}$,\vspace*{1pt} which
does not satisfy (\ref{f-cond}).
\citet{SteHan89}, page 180, give the BLP based on $Z(N_\ep\cup
F_\ep)$,
from which it is not difficult to show that $Ee(N_\ep)^2\sim2\ep$, just
like for $K(x)= e^{-|x|}$,
but $Ee(N_\ep\cup F_\ep)^2\sim\frac{3}{2}\ep$ as $\ep\downarrow
0$ so
that $Ee(N_\ep\cup F_\ep)^2/Ee(N_\ep)^2\to
\frac{3}{4}$ as $\ep\downarrow0$.
The choice of $y_0=1$ is critical here: for $y_0\ne1$ but positive
(keeping $x_1=1,y_1=0,y_2=1$),
$Ee(N_\ep\cup F_\ep)^2/Ee(N_\ep)^2\to1$ as $\ep\downarrow0$.
The anomaly for $y_0=1$ is related\vadjust{\goodbreak} to the lack of differentiability
of $K(x)$ at $x=1$, which is in turn related to the oscillations at
high frequencies in $f$.
See \citet{Ste05} for further discussion on the relationship of the
differentiability of $K$ away from the origin and the high-frequency behavior
of $f$.
Proposition 1 in \citet{Ste05} provides a second example showing
why a condition like (\ref{f-cond}) is needed to have a screening
effect. The following special case of this result suffices to
illustrate the point. Suppose\vspace*{1pt} $Z$ is a~stationary process
on $\R^2$ with autocovariance function $K(s,t) = e^{-|s|-|t|}$ for
$s,t\in\R$. The corresponding spectral density $f(\omega_1,\omega_2)$
is proportional to $\frac {1}{(1+\omega_1)^2 (1+\omega_2^2)}$, which
does not satisfy~(\ref{f-cond}). Consider the situation pictured in
Figure \ref{figtensor}, for which $x_1=(0,1)$, $y_0=(1,0)$, $y_1=(0,1)$
and $y_2=(0,0)$. Then using either direct calculation or Proposition 1
in \citet{Ste05}, $\lim_{\ep\downarrow0} Ee(N_\ep\cup F_\ep)^2/
Ee(N_\ep)^2 = 1-e^{-2}$.\vspace*{2pt}
\begin{figure}
\includegraphics{909f02.eps}
\caption{Prediction problem for autocovariance function $K(s,t) =
e^{-|s|-|t|}$. Symbols as in Figure~\protect\ref{fig1}.}
\label{figtensor}
\end{figure}
\begin{figure}[b]
\includegraphics{909f03.eps}
\caption{Prediction problems for Mat\'ern model with $\nu=\frac{3}{2}$
on $\R$. Symbols as in Figure \protect\ref{fig1}.}
\label{fig2}
\end{figure}
The remaining examples all consider $f$ satisfying (\ref{f-cond}). To
see why an additional condition on $N_\ep$ is needed for (\ref{main})
to hold for differentiable processes, consider a Mat\'ern model with
smoothness parameter $\frac{3}{2}\dvtx K(x) =e^{-|x|}(1+|x|)$, for
which the corresponding process is exactly once mean square
differentiable. For $N_\ep= \{\ep\}$, $F_\ep=\{1\}$ (top
plot\vspace*{1pt} in Figure \ref{fig2}), straightforward calculations
yield $Ee(N_\ep)^2\sim \ep^2$ and $Ee(N_\ep\cup
F_\ep)^2\sim\frac{e^2-5}{e^2-4}\ep^2$ as $\ep\downarrow0$ so
$Ee(N_\ep\cup F_\ep)^2/Ee(N_\ep)^2\to \frac{e^2-5}{e^2-4}$ as
$\ep\downarrow0$. Unlike the triangular case, there is nothing special
about $y_0=1$ here and the more general result for $y_0>0$ is
$Ee(N_\ep\cup F_\ep)^2/Ee(N_\ep)^2\to1-y_0^2/(e^{2y_0}-1-2y_0-y_0^2)$.
The reason the limit is less than 1 is not because there is anything
unusual about $f$, but rather that $N_\ep$ is inadequate. Specifically,
since $Z(0) = Z(\ep) - \ep Z'(0) + o_p(\ep)$ and
$\cov\{Z(\ep),Z'(0)\}\to0$ as $\ep\downarrow0$, it is apparent that
having even a somewhat informative predictor for $Z'(0)$\vspace*{-1pt} would provide
useful information about~$Z(0)$ not contained in $Z(\ep)$. In fact, as
$\ep\downarrow0$, it is possible to show that $\hat{Z'(0)} =
\frac{e}{e^2-4}Z(\ep)-\frac{2}{e^2-4}Z(1)$\vspace*{-1pt} is an asymptotically optimal
predictor of $Z'(0)$ based on $(Z(\ep),Z(1))$ and, in turn, that
$Z(\ep)-\ep\hat{Z'(0)}$ is an asymptotically optimal predictor of
$Z(0)$ based on $(Z(\ep),Z(1))$. A screening effect does hold if $2\ep$
is added to $N_\ep$ (bottom plot of Figure \ref{fig2}). Then it is
possible to show that $Ee(N_\ep\cup F_\ep)^2\sim Ee(N_\ep)^2
\sim\frac{8}{3}\ep^3$\vspace*{2pt} as $\ep\downarrow0$, so (\ref{main}) is
true. Furthermore, as $\ep\downarrow0$, $2Z(2\ep)-Z(\ep) =
Z(\ep) - \ep[\{Z(2\ep)-Z(\ep)\}/\ep]$ is an asymptotically optimal
predictor of $Z(0)$ based on $Z(N_\ep\cup F_\ep)$ and
$\{Z(2\ep)-Z(\ep)\}/\ep$ is a~consistent predictor of $Z'(0)$. A
reasonable conjecture for a process on $\R$ with exactly~$p$ mean
square derivatives whose spectral density satisfies (\ref{f-cond}) is
that any distinct $x_1,\ldots,x_n$ with $n>p$ suffices to make
(\ref{main}) true.
It is helpful to consider this problem in the spectral domain.
We need some further notation to proceed.
For nonnegative-valued functions $a$ and~$b$ defined on a common domain
$D$, write $a(x) \ll b(x)$ if there exists finite~$C$ such that
$a(x) \le Cb(x)$ for all $x\in D$ and, for $x\in\R$,
$a(x) \ll b(x)$ as $x\downarrow x_0$ if, for some $c>0$, $a(x) \ll b(x)$
for $D=(x_0,x_0+c)$.
Write $a(x) \asymp b(x)$ if $a(x) \ll b(x)$ and $b(x) \ll a(x)$ and
define $a(x) \asymp b(x)$ as $x\downarrow0$ if $a(x) \ll b(x)$ as
$x\downarrow x_0$ and $b(x) \ll a(x)$ as $x\downarrow x_0$.
For a complex-valued function~$g$ and a nonnegative function $f$ defined
on a domain $D$ (always~$\R^d$ here),
define $\|g\|_f = \sqrt{\int_D |g(x)|^2f(x) \,dx}$.
To each random variable of the form $\sum_{j=1}^n \lambda_j Z(s_j)$
there is a corresponding
function $\sum_{j=1}^n \lambda_j e^{i\omega\cdot s_j}$,
and the mapping is an isometric isomorphism in the sense that
$E\{\sum_{j=1}^n \lambda_j Z(s_j)\}^2 = \int_{\R^d} |
{\sum_{j=1}^n \lambda_j e^{i\omega\cdot s_j}}|^2 f(\omega)
\,d\omega$.
Write $\sum_{j=1}^n \phi_{j\ep} Z(\ep x_j)$ for the BLP of $Z(0)$ based
on $Z(N_\ep)$ and
$\phi_\ep(\omega) = \sum_j \phi_{j\ep}
e^{i\ep\omega\cdot x_j}$ for the corresponding function.
If we set $\eta_\ep(\omega)=1-\phi_\ep(\omega)$, then
$Ee(N_\ep)^2= \|\eta_\ep\|^2_f$.
For any $A\subset\R^d$, call $\int_A |\eta_\ep(\omega)|^2
f(\omega) \,d\omega/\|\eta_\ep\|^2_f$ the fraction of
$Ee(N_\ep)^2$ attributable to the set of frequencies $A$.
Write $b(r)$ for the ball of radius $r$ centered at the origin.
For the scenario in Figure \ref{fig2}(a),
for any fixed $\omega_0>0$, as $\ep\downarrow0$,
\begin{equation}\label{lowfreq}
\frac{\int_{b(\omega_0)} |\eta_\ep(\omega)|^2 f(\omega)
\,d\omega}
{\|\eta_\ep\|_f^2} \sim\frac{2}{\pi}\biggl\{\tan^{-1} \omega_0
-\frac{\omega_0}
{1+\omega_0^2}\biggr\} > 0
\end{equation}
so that an asymptotically nonnegligible fraction of $Ee(N_\ep)^2$ is
attributable to a fixed range of frequencies. Similar to the definition
of $\eta_\ep$, let $\psi_\ep$ be the function corresponding to
$e(N_\ep\cup F_\ep)$, so that $Ee(N_\ep\cup F_\ep)^2 =
\|\psi_\ep\|_f^2$. Then (\ref{lowfreq}) allows $Z(1)$ to improve the
prediction nonnegligibly by making $|\psi_\ep(\omega)|^2
/|\eta_\ep(\omega)|^2$ substantially\vadjust{\goodbreak} smaller than 1 in a neighborhood
of the origin. In contrast, for the scenario in Figure \ref{fig2}(b),
$\int_{b(\omega_0)} |\eta_\ep(\omega)|^2 f(\omega) \,d\omega\ll\ep^4$
as $\ep \downarrow0$ for any fixed $\omega_0$, so that
$\int_{b(\omega_0)} |\eta_\ep(\omega)|^2 f(\omega) \,d\omega
\ll\ep\|\eta_\ep\|_f^2$ as $\ep\downarrow0$. In this\vspace*{1pt} case,
making $|\psi_\ep(\omega)|^2 /|\eta_\ep(\omega)|^2$ substantially
smaller than 1 in a~neighborhood of the origin cannot yield a~nonnegligible asymptotic impact on the mean squared prediction error.
Thus, $\int_{b(\omega_0)^c} |\psi_\ep(\omega)|^2 f(\omega)
\,d\omega/\break\int_{b(\omega_0)^c} |\eta_\ep(\omega)|^2 f(\omega)
\,d\omega$ must\vspace*{1pt} be bounded by some constant less than 1 as
$\ep\downarrow0$ for all $\omega_0$ for (\ref{main}) not to hold. The
fact that $f$ is well behaved at high frequencies [i.e., satisfies
(\ref{f-cond})] effectively precludes this possibility so that
(\ref{main}) holds. This line of reasoning forms the basis of the proof
of Theorem \ref{thm1}; see Section \ref{sec52}.
It is interesting to reconsider the two cases pictured in
Figure \ref{fig2},
for a process that is\vspace*{1pt} not quite mean square differentiable:
$K(x) = |x|\mathcal{K}_1(|x|)$, a Mat\'ern model with
smoothness parameter 1, for which $K(x) = 1 + \frac{1}{2}x^2\log(
\frac{1}{2}|x|)+\frac{1}{4}(2\gamma-1)x^2+O({x^4\log}|x|)$ as
$x\to0$
with $\gamma$ being Euler's constant.
The corresponding spectral density $f$ is proportional to
$(1+\omega^2)^{-3/2}$.
Since the process has no mean square derivatives, I conjecture that
(\ref{main}) should hold for any nonempty $N_\ep$.
For the scenario in Figure \ref{fig2}(a),
$Ee(N_\ep)^2\sim-\ep^2\log\ep$ and, for fixed
$\omega_0>0$,
\[
\int_{b(\omega_0)} |\eta_\ep(\omega)|^2 f(\omega) \,d\omega
\asymp\int_{b(\omega_0)} \frac{|1-e^{i\ep\omega}|^2 + \{K(\ep
)-1\}^2}
{(1+\omega^2)^{3/2}}\,d\omega\asymp\ep^2
\]
as $\ep\downarrow0$. Thus, the fraction of the $Ee(N_e)^2$ attributable
to $b(\omega_0)$ tends to~0 as $\ep\downarrow0$, although at only a
logarithmic rate. Not coincidentally, direct calculation shows that for
$F_\ep= \{1\}$, (\ref{main}) holds and I would expect it to hold for
more general~$F_\ep$. In fact, Theorem \ref{thm2} in Section \ref{sec3}
applies in this case and it follows that (\ref{main}) holds when
$F_\ep$ has two points (and $y_2=0$).
Next consider some settings for the Mat\'ern model with
$\nu=\frac{3}{2}$ on $\R^2$. Figure \ref{fig3}(a) shows a situation in
\begin{figure}
\includegraphics{909f04.eps}
\caption{Prediction problems for isotropic Mat\'ern model on $\R^2$ with
$\nu=\frac{3}{2}$. Symbols as in Figure~\protect\ref{fig1}.}
\label{fig3}
\end{figure}
which there are two nearby observations in the vertical direction from
the origin and two distant observations in
the horizontal direction. One might imagine that because the nearby
observations provide no information about how the process varies in the
horizontal direction, the distant observations might provide
nonneglible new information about $Z(0)$. However, Section \ref{sec51}
demonstrates that (\ref{main}) does hold in this case. The next two
examples are related to the one-dimensional examples considered in
Figure \ref{fig2} for a Mat\'ern model with $\nu=\frac{3}{2}$. Write
$Z_{i,j}$ for the $ij$th partial derivative of $Z$. In Figure
\ref{fig3}(b), $N_\ep$ has three observations, but they are collinear
along a line that does not go through the origin and it is possible to
show that the BLP of $Z_{1,0}(0,0)$ based on $Z(N_\ep)$ has
asymptotically negligible correlation with $Z_{1,0}(0,0)$ as
$\ep\downarrow0$. As a consequence, the asymptotic results are
identical to what we had in Figure \ref{fig2}(a):
$Ee(N_\ep)^2\sim\ep^2$ and $Ee(N_\ep\cup
F_\ep)^2\sim\frac{e^2-5}{e^2-4}\ep^2$ as $\ep\downarrow0$. If $N_\ep$
has three points arranged\vadjust{\goodbreak} as in Figure~\ref{fig3}(c), then
$\{Z(\ep,0)-\frac{1}{2}Z(2\ep,\ep)-\frac{1}{2}Z(2\ep,-\ep)\} /\ep$ is a
consistent predictor of $Z_{1,0}(0,0)$ and (\ref{main}) holds; see
Section \ref{sec51}.
Now consider a model satisfying (\ref{f-cond}) for which the process
is not
equally differentiable in all directions.
\citet{Ste05} gives an example of such a model.
Specifically, consider a space--time model on $\R^3\times\R$ with spectral
density $\{(1+|\omega_1|^2)^2+\omega_2^2\}^{-2}$,
$(\omega_1,\omega_2)\in\R^3\times\R$.
Writing $\erfc$ for the complementary error function,
the corresponding autocovariance function $K$ is \citet{Ste05}
\begin{eqnarray}\label{stein2005}
K(x,t)
& = & \frac{1}{16} \pi^2 e^{|x|}\erfc\biggl(|t|^{1/2}+\frac{|x|}{
2|t|^{1/2}}\biggr)\biggl(1-|x|+\frac{4t^2}{|x|}\biggr)
\nonumber\\
& &{} + \frac{1}{16} \pi^2 e^{-|x|}\erfc\biggl(|t|^{1/2}-\frac{|x|}{
2|t|^{1/2}}
\biggr)\biggl(1+|x|-\frac{4t^2}{|x|}\biggr)
\\
& &{} + \frac{1}{4}\pi^{3/2}|t|^{1/2}\exp\biggl(-|t|-\frac{|x|^2}
{4|t|}\biggr)\nonumber
\end{eqnarray}
for $x\ne0$ and $t\ne0$.
For $x=0$ or $t=0$, we can define $K$ by continuity.
For $t=0$, we get
$K(x,0)=\frac{1}{8}\pi^2e^{-|x|}(1+|x|)$, the Mat\'ern model with
$\nu=\frac{3}{2}$, so
the corresponding process is exactly once mean square differentiable in any
spatial direction.
\citet{Ste05} shows that
$K(0,t)=\frac{1}{8}\pi^2-\frac{2}{3}\pi^{3/2}|t|^{3/2}+O(t^2)$
as $t\to0$ so that $K(0,t)$ is not twice differentiable
in $t$ at $t=0$, and the corresponding process is not mean square
differentiable in time.
For (\ref{stein2005}), let us again consider the setting in Figure
\ref{fig3}(c)
with the horizontal axis corresponding to the first spatial coordinate
and the vertical axis corresponding to time.
It now turns out that the two points in $N_\ep$ off of the horizontal
axis contribute negligibly to the BLP whether or not $F_\ep$ is included.
The problem is that the lack of differentiability of $Z$ in the vertical
direction implies that the BLP of $Z_{1,0}(0,0)$ based on $Z(N_\ep)$
has asymptotic correlation 0 with $Z_{1,0}(0,0)$.
Consequently, the asymptotic results are the same as in Figure \ref
{fig2}(a) for
$K(x) = e^{-|x|}(1+|x|)$; that is,
$Ee(N_\ep\cup F_\ep)^2/Ee(N_\ep)^2\to
\frac{e^2-5}{e^2-4}$ as $\ep\downarrow0$
(Section \ref{sec51}).
Figure \ref{fig4} displays two other settings we now consider for $K$
as in (\ref{stein2005}). In Figure~\ref{fig4}(a), we have $Ee(N_\ep\cup
F_\ep)^2/Ee(N_\ep)^2\to 1$ as $\ep\downarrow0$ and, in Figure~\ref{fig4}(b),
$Ee(N_\ep\cup F_\ep)^2/Ee(N_\ep)^2\to
\frac{e^2-5}{e^2-4}$ as $\ep\downarrow0$. These two cases show that it
is possible to have sets $N_\ep\subset \tilde N_\ep$ yet have that
(\ref{main}) holds for the pair of sets $(N_\ep,F_\ep)$ but not
$(\tilde N_\ep, F_\ep)$, further complicating any search for a general
result that applies to processes that are not equally smooth in all
directions.\looseness=1
\begin{figure}
\includegraphics{909f05.eps}
\caption{Prediction problems for autocovariance function given in
(\protect\ref{stein2005}). Horizontal axes are first spatial coordinate
and vertical axes are time. Symbols as in Figure \protect\ref{fig1}.}
\label{fig4}
\vspace*{3pt}
\end{figure}
These examples demonstrate that any general theorem that encompasses all
of them will need a condition on $N_\ep$ that depends on $f$.
The following conjecture is in accord with all of the examples
presented here:
\begin{conjecture}\label{conjec1}
Suppose $f$ satisfies (\ref{f-cond}) and the following assumption:
\renewcommand{\theassumption}{A}
\begin{assumption}\label{assumpA}
for $j=1,\ldots,n$, all
mean square derivatives of $Z$ at the origin in the direction $x_j$
can be predicted based on $Z(N_\ep)$ with mean squared error tending
to 0
as $\ep\downarrow0$.
\end{assumption}
Then for all $r>0$,
\[
\lim_{\ep\downarrow0} \frac{Ee\{N_\ep\cup b(r)^c\}^2}
{Ee\{N_\ep\}^2} = 1.
\]
\end{conjecture}
Note that here I have expanded the set of distant observations to
include all locations more than $r$ from the origin, which simplifies
the statement of the result although undoubtedly complicates its proof
(assuming it is true). It is somewhat unsatisfying to have the
condition on $N_\ep$ given in terms of properties of predictors of
derivatives of $Z$ rather than some purely geometric condition, but I
see no way to accommodate the examples treated here for $K$ as in
(\ref{stein2005}) without a condition something like Assumption
\ref{assumpA}. Verifying whether Assumption \ref{assumpA} holds in any
particular setting may require a fair amount of work, although for
$N_\ep$ of fixed and finite cardinality as we consider here, it should
generally be possible to make this determination. Note that if all mean
square derivatives of $Z$ at the origin can be consistently predicted
based on $Z(N_\ep)$ as $\ep\downarrow0$, then Assumption \ref{assumpA}
holds for any $\tilde N_\ep= \{\ep s_1,\ldots,\ep s_\ell\}$ with
$\{x_1,\ldots,x_n\}\subset\{ s_1,\ldots, s_\ell\}$.
In all of the examples for which (\ref{main}) holds,
\begin{equation}\label{low-freq}
\lim_{\ep\downarrow0}
\frac{\int_{b(\omega_0)}|\eta_\ep(\omega)|^2f(\omega) \,d\omega}
{\|\eta_\ep\|_f^2}= 0
\end{equation}
for all $\omega_0>0$, and I suspect that Assumption \ref{assumpA} is equivalent to
(\ref{low-freq}).
Examining the proof of Theorem \ref{thm1} in Section \ref{sec52}
[see (\ref{term1})], one sees that (\ref{low-freq})
is essential to making the proof work.
\section{Theorems}\label{sec3}
I do not know how to prove Conjecture \ref{conjec1} in anything like its full generality.
Assuming $Z$ is not differentiable in any direction simplifies matters
considerably, because Assumption \ref{assumpA} then holds for any~nonemp\-ty~$N_\ep$.
Theorem~\ref{thm1} considers nondifferentiable processes on $\R$
and Theorem~\ref{thm2} nondifferentiable processes on $\R^2$.
\begin{theorem}\label{thm1}
Suppose, for $d=1$ and some $\alpha\in(0,2)$,
\begin{equation}\label{fasym}
f(\omega) \asymp(1+|\omega|)^{-\alpha-1},
\end{equation}
and $f$ satisfies (\ref{f-cond}).
Then (\ref{main}) holds.
\end{theorem}
Condition (\ref{fasym}) is stronger than necessary to guarantee $Z$
is not differentiable.
Because part of the proof is to show that the low frequencies do not
matter in the limit, (\ref{fasym}) can likely be weakened
to hold only for all $\omega$ sufficiently large.
Removing~(\ref{fasym}) entirely would be more difficult.\vadjust{\goodbreak}
The next theorem applies to nondifferentiable processes
in $\R^2$ and does not require any conditions on $f$ beyond (\ref{f-cond}).
However, it does restrict $N_\ep$ to have only one point and $F_\ep$
to have
two.
The theorem also assumes $y_2=0$, but this restriction does not
meaningfully detract from the content of the result and, in any case,
could be removed at the cost of a somewhat messier proof.
Extending the result to $\R^d$ is straightforward,
but taking $d>3$ is pointless
in this setting because any 4 points in $\R^d$ fall on a three-dimensional
hyperplane, and even taking $d=3$ provides no new insight beyond what is
learned from the two-dimensional setting.
\begin{theorem}\label{thm2}
Suppose $Z$ has spectral density $f$ satisfying (\ref{f-cond}) and
that~$Z$ is not mean square differentiable in any direction.
In addition, suppose $N_\ep= \{\ep x_1\}$ and $F_\ep=\{y_0,y_0+\ep
y_1\}$,
where $x_1,y_0$ and $y_1$ are all nonzero.
Then (\ref{main}) holds.
\end{theorem}
Note that the example
referred to in Figure \ref{figtensor} satisfies the conditions
on~$N_\ep$ and $F_\ep$ in Theorem \ref{thm2}, and the process is not mean square
differentiable in any direction, but $f$ does not satisfy (\ref{f-cond}).
As we have seen,
(\ref{main}) does not hold in this setting, so that Theorem \ref{thm2} would be false
if we removed (\ref{f-cond}).
Throughout this work we assume that $Z$ has a known mean 0.
It is common in practice to assume that $Z$ has an unknown constant
mean $\mu$ and then predict $Z(0)$ by what is called the ordinary
kriging predictor, which is just an example of
the best linear unbiased predictor [Stein (\citeyear{Ste99N1})].
In all of the examples considered in Section \ref{sec2}, for which (\ref{main}) holds
for simple kriging, it still holds for ordinary kriging.
Furthermore, Theorems \ref{thm1} and \ref{thm2} can be easily shown to hold for ordinary kriging
by proving that, under the conditions of the theorems, the ordinary
kriging predictor based on $N_\ep$ is asymptotically optimal relative
to the simple kriging predictor (see the ends of each proof in Section \ref{sec5}).
Thus, if Conjecture \ref{conjec1} holds for simple kriging, then I would expect it
also holds for ordinary kriging.
\section{Discussion}\label{sec4}
The space--time process on $\R^3\times\R$ considered in Section
\ref{sec3}
with spectral density $\{(1+|\omega_1|^2)^2+\omega_2^2\}^{-2}$,
$(\omega_1,\omega_2)\in\R^3\times\R$, is an example of a process
with a different degree of differentiability in time than in space.
It is a special case of the stochastic fractional heat equations
studied by \citet{KelLeoRui05}, which are in
turn a special case of a class of space--time processes suggested in
\citet{Ste05} whose spectral densities are of the form
\begin{equation}\label{doubly-matern}
f(\omega_1,\omega_2) = \{c_1(a_1^2+|\omega_1|^2)^{\alpha_1}+
c_2(a_2^2+|\omega_2|^2)^{\alpha_2}\}^{-\nu}
\end{equation}
for\vspace*{1pt} $\omega_1\in\R^{d_1}$, $\omega_2\in\R^{d_2}$, $\nu>
\frac {d_1}{2\alpha_1} +\frac{d_2}{2\alpha_2}$ and
$c_1,c_2,\alpha_1,\alpha_2$ and $a_1^2+a_2^2$ positive to ensure $f$ is
integrable. Because of the superficial similarity of this model to the
Mat\'ern model, we might call it doubly Mat\'ern. All spectral
densities of the form (\ref{doubly-matern}) satisfy (\ref{f-cond}) and
thus, I conjecture, satisfy an asymptotic screening effect whenever
Assumption \ref{assumpA} applies to $N_\ep$. At the same time, by
adjusting the parameters $\alpha_1,\alpha_2$ and $\nu$, we can obtain
processes with any desired degree of differentiability in time and any
separate degree of differentiability in space [\citet{Ste05}].
Note that $f$ of the form (\ref{doubly-matern}) satisfies the
conditions of Theorem \ref{thm2} when $d_1=d_2=1$,
$2\nu\le\frac{3}{\alpha_1}+\frac {1}{\alpha_2}$ and
$2\nu\le\frac{1}{\alpha_1}+\frac{3}{\alpha_2}$, the last two conditions
being necessary and sufficient to make $Z$ not mean square
differentiable in any direction. \citet{autokey15} derives some
results for the covariance structure when $a_1=a_2=0$ and $\alpha_2=1$.
Despite its flexibility, model (\ref{doubly-matern}) is still
restrictive in some ways, in
particular in exhibiting what \citet{Gne02} calls full symmetry,
due to the fact that $f(\omega_1,\omega_2)=f(\omega_1,-\omega_2)$, and
hence the corresponding process has the same covariance structure
with time running backwards as it does with time running forward.
Thus, for example, this model is unsuitable for processes with a
dominant direction of advection.
\citet{Ste05} discusses possible approaches to extending this
model to
allow for asymmetries.
As noted in Section \ref{sec3}, (\ref{low-freq}), which says that only an
asymptotically negligible fraction of $Ee(N_\ep)^2$
can be attributed to some fixed frequency range, is crucial to obtaining
a screening effect.
This same property was also the key idea in \citet{Ste99N2} to obtaining
explicit results on the asymptotic efficiency of predictors based on
an incorrect spectral density having similar behavior to the correct spectral
density at high frequencies.
The high-frequency behavior of a Gaussian process is also crucial
to estimation of the covariance structure [\citet{Ste99N1}], and
misspecification
of this high-frequency behavior can lead to poor behavior of estimates,
particularly if likelihood-based methods are used [\citet{Ste99N1},
Chapter~6,
and \citet{Ste08}].
As statisticians strive to advance the statistical analysis of
spatial-temporal processes, they should pay close attention to the
spectral behavior of the models they use.
In particular, models that do not satisfy (\ref{f-cond}) should be
used with caution.
\section{Proofs}\label{sec5}
\subsection{Examples}\label{sec51}
For a random vector $Y$, write $\cov(Y)$ for the covariance matrix of $Y$,
write 0 for a column vector of zeroes whose length is apparent from
context and denote transposes by primes.
The following result simplifies the calculations for several of the examples.
\begin{lemma}\label{lem1}
$\!\!\!$If there exists $a(\ep)\,{>}\,0$, $\delta_\ep\,{\in}\,\R^{n+m}$ and $\Delta
_\ep$ an
\mbox{$(n\,{+}\,m)\,{\times}\,(n\,{+}\,m)$} matrix such that
\begin{equation}\label{lemma-limit}
\lim_{\ep\downarrow0}
\cov\pmatrix{
a(\ep)\{Z(0) - \delta_\ep\cdot Z(N_\ep\cup F_\ep)\} \cr
\Delta_\ep Z(N_\ep\cup F_\ep)}
= \pmatrix{
k & 0' \cr
0 & K
}
\end{equation}
for some $k>0$ and $K$ positive definite, then
\[
\lim_{\ep\downarrow0}
\frac{E\{Z(0) - \delta_\ep\cdot Z(\ep)\}^2}{Ee(N_\ep\cup F_\ep
)^2} = 1.
\]
\end{lemma}
For (\ref{lemma-limit}) to hold, $\tilde e_\ep= Z(0) - \delta_\ep\cdot
Z(N_\ep\cup F_\ep)$ must satisfy $E e(N_\ep\cup F_\ep)^2/\break E\tilde
e_\ep^2 \to1$ as $\ep\downarrow0$. To prove the lemma, note that
(\ref{lemma-limit}) and $K$ positive definite imply $\cov\{ \Delta_\ep
Z(\ep)\}$ is positive definite for all $\ep$ sufficiently small. Thus,
for all~$\ep$ sufficiently small, the BLP of $Z(0)$ based on
$Z(N_\ep\cup F_\ep)$ is the same as the BLP of $Z(0)$ based on
$\Delta_\ep Z(N_\ep\cup F_\ep)$. Since matrix inverse is a continuous
function in some neighborhood of $K$, using basic results on BLPs
[e.g., Stein (\citeyear{Ste99N1}), Section~1.2],
\begin{eqnarray*}
& & a(\ep)^2 Ee(N_\ep\cup F_\ep)^2 \\
& &\qquad = a(\ep)^2\bigl(\var\tilde e_\ep
-\cov\{\tilde e_\ep,Z(N_\ep\cup F_\ep)'\Delta'_\ep\} \\
& &\qquad\quad\hspace*{37pt}\hspace*{25.6pt}{} \times
[\cov\{\Delta_\ep Z(N_\ep\cup F_\ep)\}]^{-1}
\cov\{\Delta_\ep Z(N_\ep\cup F_\ep),\tilde e_\ep
\}\bigr) \\
& &\qquad = \var\{a(\ep)\tilde e_\ep\}
-\cov\{a(\ep)\tilde e_\ep,Z(N_\ep\cup F_\ep)'\Delta'_\ep\} \\
& &\qquad\quad{} \times
[\cov\{\Delta_\ep Z(N_\ep\cup F_\ep)\}]^{-1}
\cov\{\Delta_\ep Z(N_\ep\cup F_\ep),a(\ep)\tilde e_\ep
\} \\
& &\qquad \to k - 0' K^{-1} 0
\end{eqnarray*}
as $\ep\downarrow0$, and the lemma follows.
To apply Lemma \ref{lem1} to the setting in Figure \ref{fig3}(a) with
$K(x)=e^{-|x|}(1+|x|)$, it suffices to show
\begin{eqnarray*}
& & \lim_{\ep\downarrow0}
\cov\pmatrix{
\ep^{-3/2}\{Z(0,0) - 2 Z(0,\ep)+Z(0,2\ep)\} \cr
Z(0,\ep) \cr
\ep^{-1}\{Z(0,2\ep)-Z(0,\ep)\} \cr
Z(1,0) \cr
\ep^{-1}\{Z(1+\ep,0)-Z(1,0)\}}
\\[3pt]
& &\qquad = \pmatrix{
\frac{8}{3} & 0 & 0 & 0 & 0 \cr
0 & 1 & 0 & 2e^{-1} & -e^{-1} \cr
0 & 0 & 1 & 0 & 0 \cr
0 & 2e^{-1} & 0 & 1 & 0 \cr
0 & -e^{-1} & 0 & 0 & 1}.
\end{eqnarray*}
To show, for example, that $\cov[ \ep^{-1}\{Z(0,2\ep)-Z(0,\ep)\},
\ep^{-1}\{Z(1+\ep,0)-Z(1,0)\}]\to0$, define the function $\tilde K$
on $[0,\infty)$ by $\tilde K(r) = e^{-r}(1+r)$, which has bounded derivatives
of all orders on $[0,\infty)$.
Then using a Taylor series,
\begin{eqnarray*}
\hspace*{-2pt}& & \cov\{Z(0,2\ep)-Z(0,\ep),Z(1+\ep,0)-Z(1,0)\} \\
\hspace*{-2pt}& &\quad = K\bigl(\sqrt{(1+\ep)^2 +4\ep^2}\bigr) - K\bigl(\sqrt
{(1+\ep)^2
+\ep^2}\bigr) \\
\hspace*{-2pt}& &\qquad{} - K\bigl(\sqrt{1+4\ep^2}\bigr) + K\bigl(\sqrt{1+\ep
^2}\bigr) \\
\hspace*{-2pt}& &\quad = K'(1+\ep)\bigl\{\sqrt{(1+\ep)^2 + 4\ep^2}-
\sqrt{(1+\ep)^2 + \ep^2}\bigr\} \\
\hspace*{-2pt}& &\qquad{} -K'(1)\bigl\{\sqrt{1+4\ep^2}-\sqrt{1+\ep^2}\bigr\} +
O(\ep^4) \\
\hspace*{-2pt}& &\quad = K'(1+\ep)(1+\ep)\biggl\{\frac{2\ep^2}{(1+\ep)^2}-\frac{\ep^2}
{2(1+\ep)^2}\biggr\} - K'(1)\biggl(2\ep^2-\frac{1}{2}\ep^2\biggr) +
O(\ep^4) \\
\hspace*{-2pt}& &\quad \ll\ep^3,
\end{eqnarray*}
and $\cov[ \ep^{-1}\{Z(0,2\ep)-Z(0,\ep)\},
\ep^{-1}\{Z(1+\ep,0)-Z(1,0)\}]\to0$ follows.
Lemma \ref{lem1} can be applied to the setting in Figure \ref{fig3}(c)
with $K(x)=e^{-|x|}(1+|x|)$ by showing
\begin{eqnarray*}
& & \lim_{\ep\downarrow0}
\cov\pmatrix{
\ep^{-3/2}\bigl\{Z(0,0) - 2
Z(\ep,0)+\frac{1}{2}Z(2\ep,\ep)+\frac{1}{2}Z(2\ep,-\ep)\bigr\} \vspace*{2pt}\cr
Z(\ep,0) \cr
\ep^{-1}\{2Z(\ep,0)-Z(2\ep,\ep)-Z(2\ep,-\ep)\} \cr
Z(1,0)}
\\
& &\qquad = \pmatrix{
\frac{1}{3}\bigl(10\sqrt{5}-8\sqrt{2}\bigr) & 0 & 0 & 0 \cr
0 & 1 & 0 & 2e^{-1} \cr
0 & 0 & 4 & -2e^{-1} \cr
0 & 2e^{-1} & -2e^{-1} & 1}.
\end{eqnarray*}
Specifically, it is not necessary to consider $Z(2\ep,\ep)$
and $Z(2\ep,-\ep)$ separately: by symmetry,
the BLP of $Z(0,0)$ based on $Z(N_\ep\cup F_\ep)$
depends on $Z(2\ep,\ep)$ and $Z(2\ep,-\ep)$ only
through $Z(2\ep,\ep)+Z(2\ep,-\ep)$.
As a final example, let us apply Lemma \ref{lem1} to the setting in Figure
\ref{fig3}(c) with $K$ given by (\ref{stein2005}). Again by symmetry, we
can restrict to predictors that depend on $Z(2\ep,\ep)$ and
$Z(2\ep,-\ep)$ only through $Z(2\ep,\ep)+Z(2\ep,-\ep)$. For $a$ and $b$
fixed and positive, using a Taylor series and
\[
\erfc(x) = 1 -\frac{2}{\sqrt{\pi}}\biggl(x- \frac{1}{3}x^3\biggr)
+ O(|x|^5)
\]
as $x\to0$ [\citet{autokey8}, page 162], it is possible to show
\[
K(a\ep,b\ep) = \tfrac{1}{8}\pi^2 - \tfrac{2}{3}(\pi b\ep)^{3/2} +
O(\ep^2)
\]
as $\ep\downarrow0$.
This result also holds when $a$ or $b$ equals 0.
It follows that
\begin{eqnarray*}
& & \lim_{\ep\downarrow0}
\cov\pmatrix{
\ep^{-1}\{Z(0,0) - Z(\ep,0)\} \cr
Z(\ep,0) \cr
\ep^{-3/4}\{2Z(\ep,0)-Z(2\ep,\ep)-Z(2\ep,-\ep)\}}
\\[3pt]
& &\qquad = \pmatrix{
\frac{1}{8}\pi^2 & 0 & 0 \vspace*{2pt}\cr
0 & \frac{1}{8}\pi^2 & 0 \vspace*{2pt}\cr
0 & 0 & \frac{8}{3}\bigl(2-\sqrt{2}\bigr)\pi^{3/2}}
\end{eqnarray*}
so that $Z(\ep,0)$ is an asymptotically optimal predictor of $Z(0,0)$
based on~$N_\ep$.
Furthermore, for $c_1=2/(e^2-4)$ and $c_2=-e/(e^2-4)$,
\begin{eqnarray*}
& & \lim_{\ep\downarrow0}
\cov\pmatrix{
\ep^{-1}\{Z(0,0) - (1+c_1\ep)Z(\ep,0)-c_2\ep Z(1,0)\} \cr
Z(\ep,0) \cr
Z(1,0) \cr
\ep^{-3/4}\{2Z(\ep,0)-Z(2\ep,\ep)-Z(2\ep,-\ep)\}}
\\[3pt]
& &\qquad = \pmatrix{
\dfrac{1}{8}\pi^2\dfrac{e^2-5}{e^2-4} & 0 & 0 & 0 \vspace*{2pt}\cr
0 & \dfrac{1}{8}\pi^2 & \dfrac{\pi^2}{4e} & 0 \vspace*{2pt}\cr
0 & \dfrac{\pi^2}{4e} & \dfrac{1}{8}\pi^2 & 0 \vspace*{2pt}\cr
0 & 0 & 0 & \dfrac{8}{3}\bigl(2-\sqrt{2}\bigr)\pi^{3/2}},
\end{eqnarray*}
and the conditions of Lemma \ref{lem1} are satisfied.
\subsection{\texorpdfstring{Proof of Theorem \protect\ref{thm1}}{Proof of Theorem 1}}\label{sec52}
Theorem 3.1 in \citet{Xia08} implies
\begin{equation}\label{rate}
\|\eta_\ep\|_f^2=Ee(N_\ep)^2 \asymp\ep^\alpha
\end{equation}
as $\ep\downarrow0$.
Let us use (\ref{rate}) to show that ${\sum_{j=1}^n} |\phi_{j\ep}|$
is bounded in
$\ep$ as $\ep\downarrow0$.
If we define $M_\ep= \max(1,\sum_{j=1}^n |\phi_{j\ep}|)$
and $x_0=0$, we can
write $\eta_\ep(\omega)$ in the form $M_\ep\sum_{j=0}^n \mu_{j\ep
} e^{i\ep\omega
x_j}$ for appropriate $\mu_{j\ep}$'s, where, by construction,
$|\mu_{j\ep}|\le1$ for all $j$ and $\ep$.
Thus, if we can show $M_\ep$ bounded, then
${\sum_{j=1}^n }|\phi_{j\ep}|$ is also bounded.
By (\ref{fasym}), there exists $0<C_1\le C_2<\infty$ such that
\begin{equation}\label{f-bounds}
\frac{C_1}{(1+|\omega|)^{\alpha+1}}\le f(\omega) \le\frac
{C_2}{(1+|\omega|)^{\alpha+1}}
\end{equation}
for all $\omega$.
Thus, making the change of variables $\nu= \ep\omega$ in the second
step,
\begin{eqnarray}\label{bound1}
Ee(N_\ep)^2 & \ge&
C_1M_\ep^2 \int_{-\infty}^{\infty}\Biggl| \sum_{j=0}^n \mu_{j\ep}
e^{i\ep\omega x_j}\Biggr|^2 (1+|\omega|)^{-\alpha-1} \,d\omega
\nonumber\\
& = & C_1M_\ep^2 \ep^\alpha\int_{-\infty}^{\infty}\Biggl| \sum_{j=0}^n
\mu_{j\ep} e^{i\nu x_j}\Biggr|^2 (\ep+|\nu|)^{-\alpha-1} \,d\nu
\\
& \ge& C_1M_\ep^2 \biggl(\frac{1}{2}\ep\biggr)^\alpha\int_1^\infty
\Biggl| \sum_{j=0}^n \mu_{j\ep} e^{i\nu x_j}\Biggr|^2 \nu
^{-\alpha-1} \,d\nu\nonumber
\end{eqnarray}
for all $\ep<1$.
Suppose $M_\ep$ is unbounded.
Then there exists a sequence $\{\ep(k)\}$ tending to 0
such that $M_{\ep(k)}\to\infty$.
Because the $\mu_{j\ep}$'s are bounded,
there exists $ (\mu_0,\ldots,\mu_n)\in\R^{n+1}$ and
a subsequence of $\{\ep(k)\}$, call it $\{\ep(k_\ell)\}$, along
which\vadjust{\goodbreak}
$(\mu_{0\ep(k_\ell)},\ldots,\mu_{n\ep(k_\ell)}) \to(\mu
_0,\ldots,\mu_n)$
as $\ell\to\infty$.
Since $\alpha>0$, by dominated convergence, it follows that
\[
\int_1^\infty
\Biggl| \sum_{j=0}^n \mu_{j\ep(k_\ell)} e^{i\nu x_j}\Biggr|^2 \nu
^{-\alpha-1}
\,d\nu\to\int_1^\infty
\Biggl| \sum_{j=0}^n \mu_{j} e^{i\nu x_j}\Biggr|^2 \nu^{-\alpha-1}
\,d\nu> 0
\]
as $\ell\to\infty$, which, together with (\ref{bound1}),
contradicts (\ref{rate}), so $M_\ep$ and
${\sum_{j=1}^n }|\phi_{j\ep}|$ must be bounded as $\ep\downarrow0$.
Now consider the behavior of $\eta_\ep$ at low frequencies.
Define $p_{\ep} = 1/\sum_j \phi_{j\ep}$ and
$\tilde\eta_\ep(\omega) = 1 - p_{\ep}\sum_{j=1}^n
\phi_{j\ep}e^{i\ep\omega x_j}$.
By (\ref{fasym}) and (\ref{rate}), $\int_0^1 |\eta_\ep(\omega)|^2
\,d\omega\ll\ep^\alpha$ and, writing $\mathrm{Re}$ for real part,
$|\eta_\ep(\omega)|^2 \ge\{\operatorname{Re} \eta_\ep(\omega)\}^2 =
(1-p_\ep)^2 +O(\ep^2)$ uniformly for $\omega\in[0,1]$.
It follows that
\begin{equation}\label{Phirate}
(p_{\ep} - 1)^2 \ll\ep^\alpha
\end{equation}
as $\ep\downarrow0$.
Using $|e^{ix}-1| \le|x|$ for all $x\in\R$,
for $\beta\in[0,1]$ and $\alpha\in(0,2)$,
\begin{equation}\label{phibound}
\int_{b(\ep^{-\beta})} | \tilde\eta_\ep(\omega)|^2 f(\omega)
\,d\omega\ll\ep^2\int_0^{\ep^{-\beta}} \frac{\omega
^2}{1+\omega^{\alpha+1}}
\,d\omega\ll\ep^{2-\beta(2-\alpha)}
\end{equation}
as $\ep\downarrow0$.
Because $\sum_{j=1}^n \phi_{j\ep} Z(\ep x_j)$ is the BLP of $Z(0)$,
$\|\eta_\ep\|^2_f\le\|\tilde\eta_\ep\|^2_f$, so that
\begin{eqnarray}\label{lowfreqbd}
\int_{b(\ep^{-\beta})} | \eta_\ep(\omega)|^2 f(\omega)
\,d\omega& \le& \int_{b(\ep^{-\beta})} | \tilde\eta_\ep(\omega)|^2
f(\omega) \,d\omega\nonumber\\[-8pt]\\[-8pt]
& &{} + \int_{b(\ep^{-\beta})^c}\{
| \tilde\eta_\ep(\omega)|^2 - | \eta_\ep(\omega)|^2\}
f(\omega) \,d\omega.\nonumber
\end{eqnarray}
Straightforward algebra shows
\begin{eqnarray}\label{eta-bound}
& & | \tilde\eta_\ep(\omega)|^2 - | \eta_\ep(\omega)|^2
\nonumber\\
& &\qquad = (p_\ep^2-1)|\phi_\ep(\omega)|^2-2(p_\ep-1)\operatorname{Re}
\phi_\ep(\omega) \\
& &\qquad = 2(p_\ep-1)^2|\phi_\ep(\omega)|^2+2(p_\ep-1)[
|\phi_\ep(\omega)|^2-\operatorname{Re} \phi_\ep(\omega)].\nonumber
\end{eqnarray}
The boundedness of the $\phi_{j\ep}$'s in $\ep$ implies
$| |\phi_\ep(\omega)|^2-p_\ep^{-2}|^2 \ll\min(1,\ep
^2\omega^2)$
and $|{\operatorname{Re} \phi_\ep}(\omega) - p_\ep^{-1}|\ll
\min(1,\ep^2\omega^2)$, and it follows that
\[
\bigl| |\phi_\ep(\omega)|^2-\operatorname{Re} \phi_\ep(\omega)\bigr|
\ll|p_\ep-1| + \min(1,\ep^2\omega^2)
\]
as $\ep\downarrow0$,
which, together with (\ref{Phirate}) and (\ref{eta-bound}), yields
\[
| \tilde\eta_\ep(\omega)|^2 - | \eta_\ep(\omega)|^2 \ll\ep
^\alpha+
\ep^{\alpha/2}\min(1,\ep^2\omega^2)
\]
as $\ep\downarrow0$. Thus,
\begin{eqnarray}\label{eta-tilde}
& & \int_{b(\ep^{-\beta})^c} \{| \tilde\eta_\ep(\omega)|^2 -
| \eta_\ep(\omega)|^2 \}f(\omega) \,d\omega\nonumber\\
& &\qquad \ll
\int_{\ep^{-\beta}}^{\ep^{-1}}
\frac{\ep^\alpha+\ep^{2+\alpha/2}\omega^2}{\omega^{\alpha
+1}}\,d\omega+
\int_{\ep^{-1}}^\infty\frac{\ep^{\alpha/2}}{\omega^{\alpha+1}}
\,d\omega
\\
& &\qquad \ll\ep^{3\alpha/2} + \ep^{\alpha(\beta+1)}\nonumber
\end{eqnarray}
as $\ep\downarrow0$.
Combining this bound with (\ref{phibound}) and
(\ref{lowfreqbd}) implies that for all $\beta\in[0,1]$,
\begin{equation}\label{lowfreqbd2}
\int_{b(\ep^{-\beta})} | \eta_\ep(\omega)|^2 f(\omega)
\,d\omega\ll\ep^{2-\beta(2-\alpha)} + \ep^{3\alpha/2} +
\ep^{\alpha(\beta+1)}
\end{equation}
as $\ep\downarrow0$.
Note that the bound in (\ref{lowfreqbd2}) is
$o(\ep^\alpha)$ as $\ep\downarrow0$
for all $\alpha\in(0,2)$ and $\beta\in(0,1)$.
Let $\Lambda_\ep=
(\la_{1\ep},\ldots,\la_{m\ep})$, and
assume $\Lambda_\ep\ne0$ hereafter, as the case $\Lambda_\ep=0$ is
trivial to
handle.
We next show
the correlation of $e(N_\ep)$ and $\Lambda_\ep\cdot Z(F_\ep)$ is
asymptotically negligible.
Defining $\la_\ep(\omega) = \sum_{j=1}^m \la_{j\ep} e^{i\ep
\omega y_j}$,
\begin{eqnarray}\label{corr1}
& &
\corr\{ e(N_\ep), \Lambda_\ep\cdot Z(F_\ep)
\} \nonumber\\
& &\qquad = \frac{\int_{b(\ep^{-\beta})} \eta_\ep(\omega) e^{-i\omega y_0}
\overline{\la_{\ep} (\omega)}f(\omega) \,d\omega}{\|\eta_\ep\|_f
\|\la_\ep\|_f}\nonumber\\[-8pt]\\[-8pt]
&&\qquad\quad{}+ \frac{\int_{b(\ep^{-\beta})^c} \eta_\ep(\omega) e^{-i\omega y_0}
\overline{\la_{\ep} (\omega)}f(\omega) \,d\omega}{\|\eta_\ep\|_f
\|\la_\ep\|_f} \nonumber\\
& &\qquad \stackrel{\Delta}{=} I_1+I_2.\nonumber
\end{eqnarray}
Using the Cauchy--Schwarz inequality and (\ref{lowfreqbd2}), for
$\beta\in(0,1)$,
\begin{equation}\label{term1}
I_1 \le
\frac{\sqrt{\int_{b(\ep^{-\beta})}|\eta_\ep(\omega)|^2
f(\omega) \,d\omega}}{\|\eta_\ep\|_f}\to0
\end{equation}
as $\ep\downarrow0$, uniformly in $\Lambda_\ep$.
Next,
define $R_k = 2\pi k/y_0$ and $k_\ep= \lfloor y_0\ep^{-\beta}/(2\pi
)\rfloor$.
Then $R_{k_\ep}\le\ep^{-\beta}$ and
\begin{eqnarray}\label{corr2}
& & \biggl|\int_{b(\ep^{-\beta})^c} \eta_\ep(\omega) e^{-i\omega y_0}
\overline{\la_{\ep} (\omega)}f(\omega) \,d\omega\biggr| \nonumber
\\
& &\qquad \le2\sum_{k=k_\ep}^\infty\biggl|\int_{R_k}^{R_{k+1}}
\eta_\ep(\omega) e^{-i\omega y_0}
\overline{\la_{\ep} (\omega)}f(\omega) \,d\omega\biggr| \nonumber
\\
& &\qquad \le2\sum_{k=k_\ep}^\infty f(R_k)\biggl|\int_{R_k}^{R_{k+1}}
\eta_\ep(\omega) e^{-i\omega y_0}
\overline{\la_{\ep} (\omega)} \,d\omega\biggr|
\\
& &\qquad\quad{} + 2\sum_{k=k_\ep}^\infty\int_{R_k}^{R_{k+1}}
|\eta_\ep(\omega)
\la_{\ep} (\omega)|
|f(\omega)-f(R_k)| \,d\omega
\nonumber\\
& &\qquad \stackrel{\Delta}{=} I_3+I_4.\nonumber
\end{eqnarray}
For $\omega\in(R_k,R_{k+1}]$, by (\ref{f-cond}),
there exist constants $c_k\to0$ as $k\to\infty$ such that
\begin{equation}\label{fbound}
|f(\omega)-f(R_k)| \le
c_k \min\{f(R_k),f(\omega)\},
\end{equation}
so
\begin{eqnarray}\label{corr3}
I_4 & \le& 2\sum_{k=k_\ep}^\infty c_k \int_{R_k}^{R_{k+1}} |\eta
_\ep(\omega)
\la_{\ep} (\omega)| f(\omega) \,d\omega\nonumber\\
& \le& 2 \sup_{k\ge k_\ep} c_k \int_{R_{k_\ep}}^\infty|\eta_\ep
(\omega)
\la_{\ep} (\omega)| f(\omega) \,d\omega\\
& \le& \sup_{k\ge k_\ep} c_k \|\eta_\ep\|_f
\|\la_\ep\|_f,\nonumber
\end{eqnarray}
the last step by the Cauchy--Schwarz inequality.
Now consider $I_3$ in (\ref{corr2}).
Defining $\theta_\ep(\omega) = \eta_\ep(\omega) \overline{\la
_\ep(\omega)}$,
we can write $\theta_\ep$ in the form
$\sum_{j=1}^{m(n+1)}\theta_{j\ep}e^{i\ep\omega z_j}$.
For $M_\ep= \max(1,{\sum_{j=1}^n} |\phi_{j\ep}|)$,
let $M = \limsup_{\ep\downarrow0} M_\ep$, which we showed is finite.
Then, setting $L_\ep={\sum_{j=1}^m }| \la_{j\ep}|$,
it is easy to show that ${\sum_{j=1}^{m(n+1)}}|\theta_{j\ep}| \le
(2M+1)L_\ep$ for all $\ep$ sufficiently small.
Integrating by parts,
\begin{eqnarray*}
\int_{R_k}^{R_{k+1}}\theta_\ep(\omega)e^{-i\omega y_0} \,d\omega
&=& \frac{e^{-iR_k y_0}}{iy_0}\{\theta_\ep(R_k)-\theta_\ep
(R_{k+1})\}\\
&&{}+\frac{1}{iy_0}\int_{R_k}^{R_{k+1}} e^{-i\omega y_0} \theta_\ep
'(\omega)
\,d\omega.
\end{eqnarray*}
Defining $\check{z} = {\max_j} |z_j|$, we have
$|\theta_\ep(R_k)-\theta_\ep(R_{k+1})| \le{\sum_j }|\theta_{j\ep}|
|1-e^{i2\pi\ep z_j/y_0}|\le2\pi(2M+1)\check{z}L_\ep\ep/y_0$
and $|\theta_\ep'(\omega)|\le
(2M+1)\check{z}L_\ep\ep$ for all $\ep$ sufficiently small, so that
\begin{equation}\label{intbound}
\biggl|\int_{R_k}^{R_{k+1}}\theta_\ep(\omega)e^{-i\omega y_0}
\,d\omega\biggr|
\le\frac{4\pi}{y_0^2}(2M+1)\check{z}L_\ep\ep
\end{equation}
for all $\ep$ sufficiently small.
Setting $\beta=\frac{1}{2}$,
inequalities (\ref{f-bounds}) and (\ref{intbound}) imply
\begin{eqnarray}\label{bound2}
I_3 & \le& 2C_2(2M+1)\check{z}L_\ep\sum_{k=k_\ep}^\infty\frac
{\ep}{k^{\alpha+1}}
\nonumber\\[-8pt]\\[-8pt]
& \le& 2\alpha^{-1}C_2(2M+1)\check{z}L_\ep\biggl(\frac
{8}{y_0}\biggr)^\alpha
\ep^{\alpha/2+1}
\nonumber
\end{eqnarray}
for all $\ep$ sufficiently small.
Similarly to (\ref{rate}), it is possible to show
$L_\ep\ep^{\alpha/2}\ll\|\la_\ep\|_f$ as $\ep\downarrow0$, so that
by (\ref{rate}) and (\ref{bound2}),
\begin{equation}\label{corr4}
\frac{I_3}
{\|\eta_\ep\|_f\|\la_\ep\|_f} \ll\ep^{1-\alpha/2}
\end{equation}
as $\ep\downarrow0$
uniformly in $\Lambda_\ep$.
Since $\alpha<2$, this bound tends to 0 uniformly in~$\Lambda_\ep$.
Applying (\ref{corr3}) and (\ref{corr4}) to (\ref{corr2})
yields $I_2$ [defined in (\ref{corr1})] tending to 0
as $\ep\downarrow0$ uniformly in~$\Lambda_\ep$, which
together with (\ref{corr1}) and (\ref{term1}), implies
\begin{equation}\label{corr5}
\lim_{\ep\downarrow0} \sup_{\Lambda_\ep}
|{\corr}\{ e(N_\ep),
\Lambda_\ep\cdot Z(F_\ep)\}| = 0.\vadjust{\goodbreak}
\end{equation}
To finish the proof, it suffices to prove $e(N_\ep)$ is asymptotically
uncorrelated with all linear combinations of $Z(N_\ep\cup F_\ep)$.
Specifically, defining $\Xi_\ep= (\xi_{1\ep},\ldots,\xi_{n\ep})$,
if we can show
\begin{equation}\label{corrlim}
\lim_{\ep\downarrow0}\sup_{\Lambda_\ep,\Xi_\ep}|{\corr}
\{ e(N_\ep),
\Lambda_\ep\cdot Z(F_\ep) - \Xi_\ep\cdot Z(N_\ep) \}|=0,
\end{equation}
then the theorem follows since
\[
\frac{Ee(N_\ep\cup F_\ep)^2}
{Ee(N_\ep)^2} = 1 - \sup_{\Lambda_\ep,\Xi_\ep}
\corr\{ e(N_\ep), \Lambda_\ep\cdot Z(F_\ep) - \Xi_\ep\cdot
Z(N_\ep)
\}^2.
\]
Because $e(N_\ep)$ is the error of a BLP based on $N_\ep$,
$\corr\{ e(N_\ep), \Xi_\ep\cdot Z(N_\ep)\}=0$
for all $\Xi_\ep$.
Thus, (\ref{corrlim}) follows from (\ref{corr5}) if
\begin{equation}\label{varbound}
\var\{\Lambda_\ep\cdot Z(F_\ep)\} \ll
\var\{ \Lambda_\ep\cdot Z(F_\ep) - \Xi_\ep\cdot Z(N_\ep
)\}
\end{equation}
uniformly in $\Lambda_\ep$ and $\Xi_\ep$.
There is nothing to prove if $\Xi_\ep=0$, so assume \mbox{$\Xi_\ep\ne0$}
hereafter.
Consider the Mat\'ern spectral
density $f_\alpha(\omega) = (1+\omega^2)^{-(\alpha+1)/2}$,
for which the corresponding autocovariance function is $K_\alpha
(x)\,{=}\,c_\alpha
|x|^{\alpha/2}\mathcal{K}_{\alpha/2}(|x|)$,
where $c_\alpha=
\pi^{1/2}/\{2^{\alpha/2-1}\Gamma((\alpha+1)/2)\}$ [\citet{Ste99N1},
page 31].
I will
write the subscript $\alpha$ to indicate quantities such as
variances calculated under $K_\alpha$.
Since, by~(\ref{fasym}), $f(\omega)\asymp f_\alpha(\omega)$,
(\ref{varbound}) is equivalent to
\begin{equation}
\var_\alpha\{\Lambda_\ep\cdot Z(F_\ep)\}
\ll\var_\alpha\{ \Lambda_\ep\cdot Z(F_\ep)
- \Xi_\ep\cdot Z(N_\ep)\}
\end{equation}
uniformly in $\Lambda_\ep$ and $\Xi_\ep$, which is in turn
equivalent to
\begin{equation}\label{corrbound1}
\limsup_{\ep\downarrow0}
\sup_{\Lambda_\ep,\Xi_\ep}| {\corr_\alpha}\{\Lambda_\ep
\cdot
Z(F_\ep),\Xi_\ep\cdot Z(N_\ep)\}|<1.
\end{equation}
Define $\la_{\cdot\ep}\,{=}\,\sum_{j=1}^m \la_{j\ep}$, $\tilde\la_{j\ep}
\,{=}\,\la_{j\ep}-\frac{1}{m}\la_{\cdot\ep}$, $\tilde L_\ep\,{=}\,\sum_{j=1}^m
|\tilde\la_{j\ep}|$ and $\tilde\Lambda_\ep\,{=}\,
(\tilde\la_{1\ep},\ldots,\allowbreak\tilde\la_{m\ep})$. Using\vspace*{1pt}
the series expansion for $K_\alpha$ [\citet{Ste99N1}, (15) page
32] and setting $b_\alpha=\pi/\{\Gamma(\alpha+1)\sin(\frac{1}{2}\pi
\alpha)\}$ and $S_\alpha(\ep) = -\sum_{j,k=1}^m
\tilde\la_{j\ep}\tilde\la_{k\ep} |y_j-y_k|^\alpha$,
\[
\var_\alpha\{\tilde\Lambda_\ep\cdot Z(F_\ep)\}
- b_\alpha\ep^\alpha S_\alpha(\ep) \ll\ep^2\tilde L_\ep^2.
\]
Now $S_\alpha(\ep)$ is nonnegative because $\sum_{j=1}^m \tilde
\lambda_{j\ep}
=0$, and $|x|^\alpha$ is a valid variogram for $\alpha\in(0,2)$
[\citet{Ste99N1}, page 37].
Furthermore, if $\tilde L_\ep\ne0$, $S_\alpha(\ep)/\tilde L_\ep^2$
is trivially bounded from above.
It is also uniformly bounded from below: if $S_\alpha(\ep)/\tilde
L_\ep^2$
tends to a limit along any
sequence of $\ep$ values, then there is a further subsequence along which
$\tilde\Lambda_\ep/\tilde L_\ep$
converges to some $\tilde\Lambda= (\tilde\la_{1},\ldots,\tilde\la
_{m})\ne0$
and, along this subsequence, by dominated convergence,
\[
\frac{b_\alpha S_\alpha(\ep)}{\tilde L_\ep^2}
\to\int_{-\infty}^\infty\Biggl|
\sum_{j=1}^m \tilde\lambda_j e^{i\omega x_j}\Biggr|^2 |\omega
|^{-\alpha-1}
\,d\omega> 0.
\]
Thus, no subsequence of $S_\alpha(\ep)/\tilde L_\ep^2$ can have 0 as
its limit
and
\begin{equation}\label{vartilde}
\var_\alpha\{\tilde\Lambda_\ep\cdot Z(F_\ep)\} \asymp\ep^\alpha
\tilde
L_\ep^2,
\end{equation}
which holds even if $\tilde L_\ep=0$.
Again using the series expansion for $K_\alpha$,\break
$|{\cov_\alpha}\{Z(y_0+\ep y_1),\tilde\Lambda_\ep\cdot
Z(F_\ep)\}| \ll\ep^\alpha\tilde L_\ep$,
so that
$\corr_\alpha\{Z(y_0+\ep y_1),\tilde\Lambda_\ep\cdot\break
Z(F_\ep)\} \to0$
uniformly in $\tilde\Lambda_\ep\ne0$.
Thus,
\begin{equation}\label{var1}
\var_\alpha\{\Lambda_\ep\cdot Z(F_\ep)\} \sim\la_{\cdot\ep
}^2K_\alpha(0) +
\var_\alpha\{\tilde\Lambda_\ep\cdot Z(F_\ep)\}
\end{equation}
as $\ep\downarrow0$, uniformly in $\Lambda_\ep$.
Results similar to (\ref{vartilde}) and (\ref{var1})
apply to $\Xi_\ep\cdot Z(N_\ep)$.\vspace*{1pt}
Next, define
$\xi_{\cdot\ep} = \sum_{j=1}^n \xi_{j\ep}$, $\tilde\xi_{j\ep}
=\xi_{j\ep}-\frac{1}{n}\xi_{\cdot\ep}$, $\tilde X_\ep= {\sum_{j=1}^n}
|\tilde\xi_{j\ep}|$ and $\tilde\Xi_\ep=
(\tilde\xi_{1\ep},\ldots,\tilde\xi_{n\ep})$ and
consider
\begin{eqnarray}\label{covexp}\quad
& &
\cov_\alpha\{\Lambda_\ep\cdot Z(F_\ep),\Xi_\ep\cdot Z(N_\ep)\}
\nonumber\\
& &\qquad = \la_{\cdot\ep}\xi_{\cdot\ep}K_\alpha(y_0+\ep y_1 - \ep x_1)
+ \la_{\cdot\ep}\cov_\alpha\{Z(y_0+\ep y_1),\tilde\Xi_\ep\cdot
Z(N_\ep)\}
\\
& &\qquad\quad{} + \xi_{\cdot\ep}\cov_\alpha\{Z(\ep x_1),\tilde\Lambda
_\ep\cdot
Z(F_\ep)\} + \cov_\alpha\{\tilde\Lambda_\ep\cdot Z(F_\ep),\tilde
\Xi_\ep\cdot
Z(N_\ep)\}.
\nonumber
\end{eqnarray}
Since $K_\alpha$ has a bounded second derivative outside of a
neighborhood of
the origin, it is straightforward to obtain the following bounds:\vspace*{2pt}
\begin{eqnarray*}
|\la_{\cdot\ep}\xi_{\cdot\ep}K_\alpha(y_0+\ep y_1 - \ep x_1) -
\la_{\cdot\ep}\xi_{\cdot\ep}K_\alpha(y_0)| &\ll&\ep|\la_{\cdot
\ep}||
\xi_{\cdot\ep}|,
\\
|\la_{\cdot\ep}\cov_\alpha\{Z(y_0+\ep y_1),\tilde\Xi_\ep\cdot
Z(N_\ep)\}|
&\ll&\ep|\la_{\cdot\ep}|\tilde X_\ep,
\\
|\xi_{\cdot\ep}\cov_\alpha\{Z(\ep x_1),\tilde\Lambda_\ep\cdot
Z(F_\ep)\}| &\ll&\ep|\xi_{\cdot\ep}|\tilde L_\ep\vspace*{2pt}
\end{eqnarray*}
and
\[
|{\cov_\alpha}\{\tilde\Lambda_\ep\cdot Z(F_\ep),\tilde\Xi_\ep
\cdot
Z(N_\ep)\}| \ll\ep^2\tilde L_\ep\tilde X_\ep
\]
as $\ep\downarrow0$.
Applying these bounds to (\ref{covexp}) gives
\begin{eqnarray}\label{b5}
& &
|{\cov_\alpha}\{\Lambda_\ep\cdot Z(F_\ep),\Xi_\ep\cdot Z(N_\ep)\} -
\la_{\cdot\ep}\xi_{\cdot\ep}K_\alpha(y_0)|
\nonumber\\[-8pt]\\[-8pt]
& &\qquad \ll\ep|\la_{\cdot\ep}||
\xi_{\cdot\ep}| + \ep|\la_{\cdot\ep}|\tilde X_\ep+ \ep|\xi
_{\cdot\ep}|\tilde
L_\ep+ \ep^2\tilde L_\ep\tilde X_\ep.
\nonumber
\end{eqnarray}
Now, from (\ref{var1}),
\begin{eqnarray}\label{Kalpha}\quad
& & \frac{|\la_{\cdot\ep}\xi_{\cdot\ep}K_\alpha(y_0)|}{\sqrt
{\var_\alpha\{
\Lambda_\ep\cdot Z(F_\ep)\}\var_\alpha\{\Xi_\ep\cdot Z(N_\ep)\}}}
\nonumber\\
& &\qquad \sim
\frac{|\la_{\cdot\ep}\xi_{\cdot\ep}K_\alpha(y_0)|}{\sqrt{\la
_{\cdot\ep}^2
K_\alpha(0)+\var_\alpha\{\tilde\Lambda_\ep\cdot Z(F_\ep)\}}
\sqrt{\xi_{\cdot\ep}^2 K_\alpha(0)+\var_\alpha\{\tilde\Xi_\ep
\cdot
Z(N_\ep)\}}}
\\
& &\qquad \le\frac{K_\alpha(y_0)}{K_\alpha(0)},
\nonumber
\end{eqnarray}
which is in $(0,1)$ for all $y_0\ne0$.
And, since $\alpha<2$,
\begin{eqnarray*}
\frac{\ep|\la_{\cdot\ep}||
\xi_{\cdot\ep}| + \ep|\la_{\cdot\ep}|\tilde X_\ep+ \ep|\xi
_{\cdot\ep}|\tilde
L_\ep+ \ep^2\tilde L_\ep\tilde X_\ep}{\sqrt{\la_{\cdot\ep
}^2+\ep^\alpha
\tilde L_\ep^2}\sqrt{\xi_{\cdot\ep}^2+\ep^\alpha
\tilde X_\ep^2}} \to0,\vadjust{\goodbreak}
\end{eqnarray*}
which, together with (\ref{var1}), (\ref{b5}) and (\ref{Kalpha}),
proves (\ref{corrbound1}) and hence (\ref{corrlim}) and the theorem.
To prove that Theorem \ref{thm1} also applies to ordinary kriging, note
that by setting $\beta=\frac{1}{2}$, (\ref{phibound}) and (\ref{eta-tilde})
together with (\ref{rate}) imply $\|\tilde\eta_\ep\|_f^2\sim\|\eta
_\ep\|_f^2$
as $\ep\downarrow0$.
Since $\tilde\eta_\ep$ corresponds to the error of a linear unbiased
predictor under the constant mean model, we have that the mean squared
error of the ordinary kriging predictor based on $Z(N_\ep)$ is at
least $\|\eta_\ep\|_f^2$ and at most $\|\tilde\eta_\ep\|_f^2$, so that
if (\ref{main}) holds for the simple kriging predictor it also holds
for the
ordinary kriging predictor.
\subsection{\texorpdfstring{Proof of Theorem \protect\ref{thm2}}{Proof of Theorem 2}}\label{sec53}
Restricting $N_\ep$ to one point and $F_\ep$ to 2 allows us to make
use of
Lemma \ref{lem1} to prove (\ref{main}).
Setting $y_2=0$ simplifies the calculations without changing any essential
details.
Specifically, defining $V(x) = K(0)-K(x)$, we will show that
\begin{eqnarray}\label{t2vec}\quad
W'_\ep& = & (W_{\ep1},W_{\ep2},W_{\ep3},W_{\ep4}) \nonumber\\[-8pt]\\[-8pt]
& = &
\biggl( \frac{Z(0)-Z(\ep x_1)}{\sqrt{V(\ep x_1)}},Z(\ep
x_1),Z(y_0+\ep y_1),
\frac{Z(y_0)-Z(y_0+\ep y_1)}{\sqrt{V(\ep y_1)}}\biggr)'
\nonumber\hspace*{-25pt}
\end{eqnarray}
has limiting covariance matrix of the form given in (\ref{lemma-limit}),
from which Theorem~\ref{thm2} readily follows.
Let us consider the easier parts of the proof first.
Independent of $\ep$,
the variances of the elements of $W_\ep$ are $1,K(0),K(0)$ and 1, respectively.
Since~$Z$ has a spectral density, $K$ is continuous and $|K(y)| < K(0)$
for all $y\ne0$.
Thus, $\cov(W_{\ep2},W_{\ep3}) \to K(y_0)$ as $\ep\downarrow0$, and
the $2\times2$ matrix with $K(0)$ on the diagonals and $K(y_0)$
elsewhere is
positive definite.
Thus, it suffices to show that the other offdiagonal elements of the
covariance matrix of $W_\ep$ tend to $0$ as $\ep\downarrow0$.
First, $\cov(W_{\ep1},W_{\ep2}) = \frac{1}{2}\sqrt{V(\ep x_1)/K(0)}
\to0$ as $\ep\downarrow0$.
Similarly, $\cov(W_{\ep3},W_{\ep4})\to0$ as $\ep\downarrow0$.
Now consider $\cov(W_{\ep1},W_{\ep3})$.
We have
\[
\cov\{Z(0)-Z(\ep x_1),Z(y_0+\ep y_1)\} =
\int_{\R^2} e^{-i\omega\cdot(y_0+\ep y_1)}(1-e^{i\ep\omega\cdot
x_1})f(\omega)
\,d\omega,
\]
so that for $D(T) = \{\omega\dvtx |\omega\cdot x_1| \le T \}$,
\begin{eqnarray}\label{corr13}
\cov(W_{\ep1},W_{\ep3})^2
&\le&\frac{\{\int_{\R^2} |1-e^{i\ep\omega\cdot x_1}
| f(\omega)\,d\omega\}^2}{K(0)\int_{\R^2} |1-e^{i\ep
\omega\cdot x_1}
|^2 f(\omega)\,d\omega}\nonumber\\[-0.5pt]
&\le&\frac{2\{\int_{D(T)} |1-e^{i\ep\omega\cdot x_1}
| f(\omega)\,d\omega\}^2}{K(0)\int_{D(T)}
|1-e^{i\ep\omega\cdot x_1}
|^2 f(\omega)\,d\omega} \\[-0.5pt]
&&{}+ \frac{2\{\int_{D(T)^c}
|1-e^{i\ep\omega\cdot
x_1} | f(\omega)\,d\omega\}^2}{K(0)\int_{D(T)^c}
|1-e^{i\ep\omega\cdot x_1} |^2 f(\omega)\,d\omega}
\nonumber
\end{eqnarray}
for all $T$ sufficiently large (to guarantee $\int_{D(T)}
|1-e^{i\ep\omega\cdot x_1}
|^2 f(\omega)\,d\omega> 0$).
Because $\ep^{-1}|1-e^{i\ep\omega\cdot x_1} | \le|\omega
\cdot x_1|$
and $\ep^{-1}|1-e^{i\ep\omega\cdot x_1} | \to|\omega
\cdot x_1|$
as $\ep\downarrow0$, by dominated convergence,
\begin{equation}
\lim_{\ep\downarrow0}\label{eplim1}
\frac{\{\int_{D(T)} |1-e^{i\ep\omega\cdot x_1}
| f(\omega)\,d\omega\}^2}{\int_{D(T)}
|1-e^{i\ep\omega\cdot x_1} |^2 f(\omega)\,d\omega}
= \frac{\{\int_{D(T)}|\omega\cdot x_1|f(\omega)\,d\omega\}^2}
{\int_{D(T)} |\omega\cdot x_1|^2 f(\omega)\,d\omega}.
\end{equation}
By the Cauchy--Schwarz inequality,
\begin{eqnarray}\label{eplim2}
&&\frac{\{\int_{D(T)^c}
|1-e^{i\ep\omega\cdot
x_1} | f(\omega) \,d\omega\}^2}{\int_{D(T)^c}
|1-e^{i\ep\omega\cdot x_1} |^2 f(\omega) \,d\omega}\nonumber\\[-0.5pt]
&&\qquad \le \frac{\int_{D(T)^c}
|1-e^{i\ep\omega\cdot x_1} |^2 f(\omega) \,d\omega
\int_{D(T)^c} f(\omega) \,d\omega}{\int_{D(T)^c}
|1-e^{i\ep\omega\cdot x_1}|^2 f(\omega) \,d\omega}
\\[-0.5pt]
&&\qquad = \int_{D(T)^c} f(\omega)\,d\omega.
\nonumber
\end{eqnarray}
From (\ref{corr13})--(\ref{eplim2}), we will have $\cov(W_{\ep
1},W_{\ep
3})\to0$ as $\ep\downarrow0$ if the right-hand sides of (\ref
{eplim1}) and
(\ref{eplim2}) tend to 0 as $T\to\infty$.
The integrability of $f$ implies $\int_{D(T)^c} f(\omega)\,d\omega\to
0$ as
$T\to\infty$, so consider\vspace*{1pt} the right-hand side of (\ref{eplim1}).
Let~$A$ be the $2\times2$ matrix with first row given by $x_1$,
orthogonal\vspace*{1pt} rows and determinant of 1 and set $v = (v_1, v_2)' = A\omega$.
Define $\bar f(v_1) = \int_{-\infty}^{\infty} f( A^{-1} v
) \,dv_2$.
Up to a linear rescaling, $\bar f$ is
the spectral density of the process $Z$ along the $x_1$ direction,
so it is integrable.
In addition, because $Z$ is not mean square differentiable in
any direction, $\int_0^\infty v_1^2\bar f(v_1) \,dv_1 = \infty$.
Then for any even function~$g$, $\int_{D(T)}g(\omega\cdot
x_1)f(\omega)
\,d\omega= 2\int_0^T g(v_1)\bar f(v_1) \,dv_1$, so that for $0<S<T$,
\begin{eqnarray}\label{0ST}
\frac{\{\int_{D(T)}|\omega\cdot x_1|f(\omega)\,d\omega\}^2}
{\int_{D(T)} |\omega\cdot x_1|^2 f(\omega)\,d\omega}
& = & \frac{\{ \int_0^T v_1\bar f(v_1) \,dv\}^2}
{ \int_0^T v_1^2\bar f(v_1) \,dv_1} \nonumber\\[-8pt]\\[-8pt]
& = & \frac{\{ \int_0^S v_1\bar f(v_1) \,dv_1 +
\int_S^T v_1\bar f(v_1) \,dv_1\}^2}
{ \int_0^T v_1^2\bar f(v_1) \,dv_1}.
\nonumber
\end{eqnarray}
If we can show that
\begin{equation}\label{lim0ST}
\lim_{S\to\infty}\lim_{T\to\infty} \frac{\{ \int_0^S
v_1\bar f(v_1)
\,dv_1
+ \int_S^T v_1\bar f(v_1) \,dv_1\}^2} { \int_0^T v_1^2\bar
f(v_1)
\,dv_1} = 0,
\end{equation}
then the right-hand side of (\ref{eplim1}) will tend to 0 as $T\to
\infty$.
To prove (\ref{lim0ST}), expand the square in the numerator and consider
each term separately.
First, by the Cauchy--Schwarz inequality,
\[
\lim_{T\to\infty}
\frac{\{ \int_0^S v_1\bar f(v_1) \,dv_1\}^2}
{ \int_0^T v_1^2\bar f(v_1) \,dv_1}
= \lim_{T\to\infty}\frac{ \int_0^S v_1^2\bar f(v_1) \,dv_1 \int_0^S
\bar f(v_1) \,dv_1}
{ \int_0^T v_1^2\bar f(v_1) \,dv_1} = 0.
\]
Again by the Cauchy--Schwarz inequality,
\[
\frac{\{
\int_S^T v_1\bar f(v_1) \,dv_1\}^2}
{ \int_0^T v_1^2\bar f(v_1) \,dv_1}
\le\frac{ \int_S^T v_1^2\bar f(v_1) \,dv_1 \int_S^T \bar f(v_1) \,dv_1}
{ \int_0^T v_1^2\bar f(v_1) \,dv_1} \\
\le\int_S^T \bar f(v_1) \,dv_1,
\]
which tends to 0 when one takes $\lim_{S\to\infty}\lim_{T\to\infty
}$ since
$\bar f$ is integrable.
Finally,
\begin{eqnarray*}
\frac{ \int_0^S v_1\bar f(v_1) \,dv_1 \int_S^T v_1\bar f(v_1) \,dv_1}
{\int_0^T v_1^2\bar f(v_1) \,dv_1}
&\le& \frac{ \int_0^S v_1\bar f(v_1) \,dv_1 \int_S^T v_1\bar
f(v_1)
\,dv_1} { \int_S^T v_1^2\bar f(v_1) \,dv_1} \\
&\le& \frac{ \int_0^S v_1\bar f(v_1) \,dv_1}{S}\cdot
\frac{\int_S^T v_1\bar f(v_1) \,dv_1} {\int_S^T v_1\bar f(v_1)
\,dv_1} \\
&\le& \frac{1}{S}\int_0^{S^{1/2}} v_1\bar f(v_1) \,dv_1 +
\frac{1}{S}\int_{S^{1/2}}^S v_1\bar f(v_1) \,dv_1 \\
&\le& \frac{1}{S^{1/2}} \int_0^{S^{1/2}}\bar f(v_1) \,dv_1
+ \int_{S^{1/2}}^S \bar f(v_1) \,dv_1,
\end{eqnarray*}
which tends to 0 as $S\to\infty$, and (\ref{lim0ST}) follows.
Thus, $\cov(W_{\ep1},W_{\ep3})\to0$ as $\ep\downarrow0$.
Similarly, $\cov(W_{\ep2},W_{\ep4})\to0$ as $\ep\downarrow0$.
We will need the following lemma to handle $\cov(W_{\ep1},W_{\ep4})$:
\begin{lemma}\label{lem2}
If $Z$ is not mean square differentiable in the direction $x$, then
\[
\lim_{\ep\downarrow0}\frac{\ep^2}{V(\ep x)} = 0.
\]
\end{lemma}
To prove the lemma, first note that the assumption on $Z$ is equivalent
to
\begin{equation}\label{no2}
\int_{\R^2} |\omega\cdot x|^2 f(\omega) \,d\omega= \infty.
\end{equation}
If $\limsup_{\ep\downarrow0}\frac{\ep^2}{V(\ep x)} > 0$, then there
must exist some sequence $\ep_n\downarrow0$ along which
$\lim_{n\to\infty}\frac{V(\ep_n x)}{\ep_n^2} = C$ for some finite
$C$, or
\[
\lim_{n\to\infty}\int_{\R^2} \frac{|1-e^{i\ep_n\omega\cdot
x}|^2}{\ep_n^2}
f(\omega) \,d\omega= C.
\]
But for any finite $T$, by dominated convergence,
\begin{eqnarray*}
C & = & \lim_{n\to\infty}\int_{\R^2} \frac{|1-e^{i\ep_n\omega
\cdot x}|^2}
{\ep_n^2} f(\omega) \,d\omega\\
& \ge& \lim_{n\to\infty}\int_{|\omega|<T} \frac{|1-e^{i\ep
_n\omega\cdot
x}|^2}{\ep_n^2} f(\omega) \,d\omega\\
& = & \int_{|\omega|<T} |\omega\cdot x|^2 f(\omega) \,d\omega
\end{eqnarray*}
for all $T$, which contradicts (\ref{no2}), and the lemma is proven.
Consider
\begin{eqnarray*}
& & \cov\{Z(0)-Z(\ep x_1),Z(y_0)-Z(y_0+\ep y_1)\} \\
& &\qquad = \int_{\R^2} e^{-i\omega\cdot y_0}(1-e^{i\ep\omega\cdot
x_1})
(1-e^{-i\ep\omega\cdot y_1}) f(\omega) \,d\omega.
\end{eqnarray*}
Define $f_1(\omega) = \min(f(\omega),1)$, and write $\cov_1$ to indicate
covariances calculated under the spectral density $f_1$.
Then (\ref{f-cond}) and $f$ integrable
imply that $f(\omega) = f_1(\omega)$ outside some bounded set, and it easily
follows that
\begin{eqnarray}\label{f1}
& & \cov\{Z(0)-Z(\ep x_1),Z(y_0)-Z(y_0+\ep y_1)\} \nonumber\\[-8pt]\\[-8pt]
& &\qquad = \cov_1\{Z(0)-Z(\ep x_1),Z(y_0)-Z(y_0+\ep y_1)\} + O(\ep^2).
\nonumber
\end{eqnarray}
Because $Z$ is not mean square differentiable in any direction, Lemma
\ref{lem2}
implies the $O(\ep^2)$ remainder in (\ref{f1}) makes no
contribution to $\lim_{\ep\downarrow0}\cov(W_{\ep1},W_{\ep4})$.
We proceed by rotating coordinates so that one of the frequency axes points
in the direction of $y_0$.
Specifically,
let $B$ be the $2\times2$ orthogonal matrix with determinant 1 and
first row equal to $y_0$ and set $\tau= (\tau_1, \tau_2)' = B\omega$.
Then, defining $H_\ep(\tau_1,\tau_2) =
(1-e^{i\ep(B^{-1}\tau)\cdot x_1})
(1-e^{-i\ep(B^{-1}\tau)\cdot y_1})$,
\begin{eqnarray*}
& & \cov_1\{Z(0)-Z(\ep x_1),Z(y_0)-Z(y_0+\ep y_1)\} \\
& &\qquad = \int_{\R^2} e^{-i\tau_1}H_\ep(\tau_1,\tau_2)
f_1(B^{-1}\tau) \,d\tau\\
& &\qquad = \int_\R\sum_{k=-\infty}^{\infty} \int_{2\pi k}^{2\pi(k+1)}
e^{-i\tau_1}H_\ep(\tau_1,\tau_2)
f_1(B^{-1}\tau) \,d\tau_1
\,d\tau_2.
\end{eqnarray*}
Define the function $g$
on $\R^2$ by, for $2\pi k\le\tau_1 < 2\pi(k+1)$, $g(B^{-1}\tau) =
1 -
f_1(B^{-1}(2\pi k, \tau_2)')/f_1(B^{-1}\tau)$ if
$f_1(B^{-1}\tau) >0$ and 0 otherwise.
We have
\begin{eqnarray}\label{splitk}
& & \cov_1\{Z(0)-Z(\ep x_1),Z(y_0)-Z(y_0+\ep y_1)\} \nonumber\\
& &\qquad = \int_\R\sum_{k=-\infty}^{\infty} f_1\biggl( B^{-1}\pmatrix
{2\pi k\cr
\tau_2}\biggr) \int_{2\pi k}^{2\pi(k+1)}
e^{-i\tau_1}H_\ep(\tau_1,\tau_2) \,d\tau_1
\,d\tau_2\\
& &\qquad\quad{} +
\int_{\R^2} e^{-i\omega\cdot y_0}(1-e^{i\ep\omega\cdot
x_1})
(1-e^{-i\ep\omega\cdot y_1}) f_1(\omega)g(\omega)
\,d\omega.
\nonumber
\end{eqnarray}
By (\ref{f-cond}), $g(\omega)\to0$ as $\omega\to\infty$.
Thus, given $\delta>0$, we can find $T<\infty$ such that $g(\omega
)<\delta$
for $|\omega|>T$.
Then
\begin{eqnarray*}
& &
\biggl| \int_{\R^2} e^{-i\omega\cdot y_0}(1-e^{i\ep\omega
\cdot x_1})
(1-e^{-i\ep\omega\cdot y_1}) f_1(\omega)g(\omega)
\,d\omega\biggr|
\\
& &\qquad \le\ep^2 \int_{|\omega|\le T} |\omega\cdot x_1||\omega\cdot y_1|
f_1(\omega)|g(\omega)| \,d\omega\\
& &\qquad\quad{} + 4\delta\int_{|\omega|>T}
|1-e^{i\ep\omega\cdot x_1}|
|1-e^{-i\ep\omega\cdot y_1}| f_1(\omega) \,d\omega.
\end{eqnarray*}
By the Cauchy--Schwarz inequality and $f_1\le f$, $\int_{|\omega|>T}
|1-e^{i\ep\omega\cdot x_1}|
|1-\break e^{-i\ep\omega\cdot y_1}|\times f_1(\omega) \,d\omega\le
\sqrt{V(\ep x_1)V(\ep y_1)}$,
which, together with Lemma \ref{lem2}, implies
\begin{equation}\label{4delta}
\limsup_{\ep\downarrow0}
\frac{|
\int_{\R^2} e^{-i\omega\cdot y_0}(1-e^{i\ep\omega\cdot
x_1})
(1-e^{-i\ep\omega\cdot y_1}) f_1(\omega)g(\omega)
\,d\omega|}
{\sqrt{V(\ep x_1)V(\ep y_1)}} \le4\delta.
\end{equation}
Since $\delta$ is arbitrary, this $\limsup$ must in fact be 0.
Now return to the the first term on the right-hand side of (\ref{splitk}).
Integrating by parts,
\begin{eqnarray*}
\int_{2\pi k}^{2\pi(k+1)} e^{-i\tau_1}H_\ep(\tau_1,\tau_2)
\,d\tau_1
&=& iH_\ep\bigl(2\pi(k+1),\tau_2\bigr)- iH_\ep(2\pi k,\tau_2)\\
&&{} -
i\int_{2\pi k}^{2\pi(k+1)} e^{-i\tau_1}\,\frac{\partial}{\partial
\tau_1}
H_\ep(\tau_1,\tau_2) \,d\tau_1.
\end{eqnarray*}
There exists finite $C$ independent of $\ep$ and $\tau$ such that
\[
\biggl| \frac{\partial}{\partial\tau_1}
H_\ep(\tau_1,\tau_2)\biggr| \le C\ep\bigl\{\bigl|1-e^{i\ep
(B^{-1}\tau)\cdot
x_1}\bigr|+\bigl|1-e^{-i\ep(B^{-1}\tau)\cdot y_1}\bigr|\bigr\},
\]
which implies
\begin{eqnarray}\label{Hbound}
& & \biggl|\int_{2\pi k}^{2\pi(k+1)} e^{-i\tau_1}H_\ep(\tau_1,\tau
_2) \,d\tau_1\biggr|
\nonumber\\[-8pt]\\[-8pt]
& &\qquad \le4\pi C\ep
\bigl\{\bigl|1-e^{i\ep(B^{-1}\tau)\cdot
x_1}\bigr|+\bigl|1-e^{-i\ep(B^{-1}\tau)\cdot y_1}\bigr|\bigr\}.
\nonumber
\end{eqnarray}
We can choose $T$ finite so that if
$2\pi k\le\tau_1 \le2\pi(k+1)$, then $f_1( B^{-1}(2\pi k,\allowbreak \tau
_2)')
\le2 f( B^{-1}\tau)$ whenever $|\tau| > T$.
Applying this result and (\ref{Hbound})
to the first term on the right-hand side of
(\ref{splitk}) and changing variables back to $\omega= B^{-1}\tau$,
we get
\begin{eqnarray}\label{laststep}
& & \Biggl| \int_\R\sum_{k=-\infty}^{\infty} f_1\biggl( B^{-1}
\pmatrix{2\pi k\cr\tau_2}\biggr) \int_{2\pi k}^{2\pi(k+1)}
e^{-i\tau_1}H_\ep(\tau_1,\tau_2)
\,d\tau_1 \,d\tau_2\Biggr| \nonumber\\
& &\qquad \le8\pi C\ep\int_{\R^2} f(\omega)
\{|1-e^{i\ep\omega\cdot
x_1}|+|1-e^{-i\ep\omega\cdot y_1}|\} \,d\omega
+O(\ep^2)
\\
& &\qquad \le8\pi C\ep\bigl\{\sqrt{V(\ep x_1)}+\sqrt{V(\ep y_1)}\bigr\}
\sqrt{\int_{\R^2} f(\omega) \,d\omega} + O(\ep^2),
\nonumber
\end{eqnarray}
where the last step uses the Cauchy--Schwarz inequality.
From Lemma \ref{lem2} and~(\ref{laststep}), it follows that
\[
\limsup_{\ep\downarrow0}
\frac{| {\int_\R}
\sum_{k=-\infty}^{\infty} f_1( B^{-1}{2\pi
k\choose\tau_2}) \int_{2\pi k}^{2\pi(k+1)}
e^{-i\tau_1}H_\ep(\tau_1,\tau_2)
\,d\tau_1 \,d\tau_2|}
{\sqrt{V(\ep x_1)V(\ep y_1)}} = 0.
\]
Together with (\ref{splitk}) and (\ref{4delta}), this limit implies
\[
\limsup_{\ep\downarrow0}
\frac{\cov_1 \{Z(0)-Z(\ep x_1),Z(y_0)-Z(y_0+\ep y_1)\}}
{\sqrt{V(\ep x_1)V(\ep y_1)}} = 0,
\]
which together with (\ref{f1}) and Lemma \ref{lem2}, implies
$\lim_{\ep\downarrow0} \cov\{W_{\ep1},W_{\ep4}\} = 0$.
Theorem \ref{thm2} applies to ordinary kriging as well.
Specifically, $Z(\ep x_1)$ is an asymptotically
optimal linear predictor of $Z(0)$ based on $Z(N_\ep\cup F_\ep)$
when the mean of $Z$ is assumed to be 0,
so since it
is a linear unbiased predictor when the mean is an unknown
constant, $Z(\ep x_1)$
must also be asymptotically optimal with respect to this
more restricted class of predictors.
\section*{Acknowledgment}
The author thanks Steven Lalley for help with the proof of Theorem
\ref{thm2}.
|
{
"timestamp": "2012-03-09T02:03:00",
"yymm": "1203",
"arxiv_id": "1203.1801",
"language": "en",
"url": "https://arxiv.org/abs/1203.1801"
}
|
\section{Introduction}
Sensitivity and resolution (both spectral and angular) are the main limiting factors
in observational Astronomy. In the case of angular (i.e., spatial) resolution,
the strong limitation that will always affect the observations, regardless of
the quality of our instruments, is the {\em diffraction limit}. When an
instrument
is diffraction-limited, its response to a plane wave (i.e., to a point source
located at infinity) is the so-called {\em Point Spread Function} (PSF), which
has a width related to the smallest angular scale that can be resolved with the
instrument.
It is well-known that the diffraction limit decreases with both an increasing
observing frequency and an increasing aperture of the instrument. Hence, the only
way to achieve a higher angular resolution at a given frequency is to increase the
instrument aperture.
In this sense, the {\em aperture synthesis}, which is a technique related to
astronomical interferometry (see, e.g., Thomson, Moran \& Swenson \cite{TMS}),
presently seems to be almost the only way to further increase the angular
resolution currently achieved at any wavelength.
But there is a crucial difference in interferometric observations, compared to
those obtained with other techniques. When aperture synthesis is performed,
an interferometer does not directly observe the structure of a source, but samples
a fractions of its Fourier transform (the so-called {\em visibilities}). In other
words, the space from which an interferometer takes measurements is not the sky
itself, but the Fourier transform of its intensity distribution over
the whole field of view.
This special characteristic of interferometers strongly affects how these devices
behave when we observe sources of sizes well below the diffraction limit, as
we will see in the following sections.
It is possible, of course, to compute the inverse Fourier transform of a set of
visibilities and (try to) recover the intensity distribution of the observed sources
in the sky. When combined with certain
deconvolution algorithms, this approach of {\em imaging} a set of visibilities may
be very useful if we are dealing with relatively extended sources. However, if the sources
observed are very compact (relative to the diffraction limit of the interferometer),
important and uncontrollable effects may arise in the imaging of the source intensity
profiles, either coming from the (non-linear) deconvolution algorithms and/or from the
gridding (pixelation) in the sky plane. These effects may result in strong biases
in the estimate of source sizes, based in measurements performed in the sky plane.
The over-resolution power of an interferometer is a function of the baseline
sensitivity, and may play an important role in the analysis of data coming from
future ultra-sensitive interferometric arrays (like the square kilometer array, SKA,
or the Atacama large millimeter array, ALMA).
In this research note, we review several well-known aspects related to the
effect of source compactness in visibility space, showing that it is possible
to find out, from the observed visibilities, information on the size of sources
much smaller than the diffraction limit achieved in the aperture synthesis.
We also estimate the maximum theoretical over-resolution power of an
interferometer and discuss a statistical test to estimate upper bounds to the size
of ultra-compact sources observed with high-sensitivity interferometers.
In Lobanov (\cite{Lobanov}), the reader will find a detailed discussion of the
resolution limits obtained for specific shapes of the brightness distribution.
Here, we extend the discussion to a more general case of super-resolution that
can be achieved with interferometers.
\section{Compact sources in visibility space}
The size of a source slightly smaller than the diffraction limit of an
interferometer leaves a very clear fingerprint in visibility space, although
its effect on the sky plane (after all the imaging and deconvolution
steps) may be much less clear.
For instance, if the size of a source is similar to the full width at half
maximum (FWHM) of the PSF, the radial profile in the visibility amplitudes
will decrease with baseline length falling down to $\sim$0.5 times the maximum
visibility amplitude at the maximum projected baseline.
Hence, a source with a size of the order of the diffraction limit maps into an amplitude
profile in visibility space that can be well detected and characterized by an
interferometer.
If the size of the source decreases, the visibility amplitude in the
longest baseline increases; in the limit case when the size of the source
tends to zero, the visibility amplitude in the longest baseline tends to be
as large as that in the shortest baseline.
It is common, indeed, to compute the amplitude ratio between the visibilities in the
shortest baseline and those in the longest baseline as a quantitative representation
of the {\em degree of compactness} of the observed sources (e.g., Kovalev et al.
\cite{Kovalev}; Lobanov \cite{Lobanov}).
\begin{figure}
\centering
\includegraphics[width=8cm]{./paperfig1c.eps}
\caption{Source size (in units of the FWHM of the synthesized beam) as
a function of the ratio of the visibility amplitude in the longest
baseline to that in the shortest baseline.}
\label{fig2}
\end{figure}
In Fig. \ref{fig2}, we show the size of a source as a function of the visibility
amplitude in the longest baseline divided by that in the shortest baseline. The size shown
in Fig. \ref{fig2} is given in units of the FWHM of the synthesized beam (we assume throughout
this paper that {\em uniform weighting} is applied in the gridding of the visibilities, prior
to the Fourier inversion; see Thomson, Moran, \& Swenson \cite{TMS} for more details).
If the sensitivity of an interferometer allows us to detect a small decrease in the visibility
amplitudes at the longest baselines, we are able to obtain information on the size of sources
much smaller than the diffraction limit of
the interferometer (i.e., much smaller than the FWHM of the PSF). The over-resolution
power of an interferometer is, hence, dependent on the sensitivity of the observations,
and can be {\em arbitrarily large}.
Figure \ref{fig2} has been computed using a different intensity profiles for the observed
source. It is obvious that the use of different source shapes (e.g., a
Gaussian profile or a ring-like source) in a fit to the visibilities, results in different
size estimates for the same dataset.
Hence, if the structure of the observed source is similar to the model
used in the fit to the visibilities (i.e., if we have a good a priori information on the real
shape of the source), we can obtain precise estimates
of sizes much smaller than the FWHM of the synthesized beam.
Indeed, the fact that the diffraction limit can be largely extrapolated by model fitting is
well known since a long time, because in these cases the data are fitted with a simple
model (i.e., with a small number of parameters), in contrast to the image-synthesis approach,
where the super-resolution capabilities are much more limited due to the larger parameter
space of the model (i.e., the image pixels). Therefore,
it is very difficult to obtain, from any data analysis based on the sky plane, size estimates of
compact sources with a precision similar to that achieved in Fourier space. In the former case,
the gridding of the images (i.e, the pixelation of the PSF), together with
the particulars of the deconvolution algorithms, may smear out the fine details in the intensity
profiles that encode the information on the structures of the underlying compact sources. On the
other hand, compact sources on the sky are seen as very extended structures in Fourier
space. Thus, a direct analysis of the visibilities (e.g., as described in Pearson \cite{Pearson})
is the optimum way to work with data coming from compact sources, since the effects of gridding
in Fourier space will always be negligible.
\section{Maximum theoretical over-resolution power of an interferometer}
\label{MaxTheo}
As it is described in the previous section, the maximum over-resolution power of an
interferometer (i.e., the minimum size of a source, in units of the FWHM of the synthesized
beam, that can be resolved) depends on how precisely we can measure lower visibility
amplitudes at the longest baselines. Hence, an interferometer with an arbitrarily
large sensitivity will accordingly have an arbitrarily large over-resolution power.
However, real interferometers have finite sensitivities, which depend on several
factors (e.g., observing frequency, bandwidth, source coordinates, weather
conditions,...). We can therefore ask the question of what is the minimum size
of a source (relative to the diffraction limit of an interferometer) that still allows
us to extract a size information from the observed visibilities.
In the extreme case of a very compact (and/or weak) source, such that it is not possible
to estimate a statistically-significant lowering in the
visibility amplitudes at the longest baselines (because of the noise contribution to the
visibilties), the only meaningful statistical analysis that can still be applied to the data
is to estimate an {\em upper limit} to the source size by means of hypothesis testing.
Let us observe a source and assume the null hypothesis, $H_0$, that it
has no structure at all (i.e., the source is point-like, so the visibility amplitudes are
constant through the whole Fourier space). The likelihood, $\Lambda_0$, corresponding to a
point-like model source fitted to the visibilities is
\begin{equation}
\Lambda_0 \propto \exp{\left( -\chi_0^2 \right)} = \exp{\left( - \sum_{j,k}^{N} { F^{jk}(V_j-S_0)(V_k-S_0) } \right)}
\label{chinull}
\end{equation}
\noindent where $N$ is the number of visibilities, $V_j = A_j\exp{(i\phi_j)}$ is the $j$-th
visibility, $S_0$ is the maximum-likelihood (ML) estimate of the source flux density (it can
be shown to be equal to the real part of the weighted visibility average, $\langle V \rangle$),
and $F$ is the Fisher matrix of the visibilities (i.e., the inverse of the their
covariance matrix), i.e.
$$ F = C^{-1} \mathrm{~~~with~~~} C_{j,k} = \rho_{jk} \sigma_j \sigma_k.$$
In this equation, $\rho_{jk}$ is the correlation coefficient between the $j$-th and
$k$-th visibility, and $\sigma_j$ is the uncertainty of the $j$-th visibility.
Equation \ref{chinull} is generic and accounts for any correlation in the
visibilities (e.g., possible global or antenna-dependent amplitude biases).
It can be shown that $\log{\Lambda_0}$ follows a $\chi^2$ distribution (close to
its maximum) with a number of degrees of freedom equal to the
rank of matrix $F$ minus unity. In the particular case when there is no correlation
between visibilities\footnote{Indeed, this also holds when there {\em is} correlation,
unless in the pathological cases when the covariance matrix may be degenerate.},
the number of degrees of freedom equals $N-1$.
Let us model the visibilities with a function, $V^m(S,q\,\theta)$, that corresponds
to the model of a (symmetric) source of size $\theta$ and flux density $S$ (the model
amplitude, $V^m$, depends on the product of $\theta$ times the distance in Fourier space,
$q$). Hence, the visibility $V_j$ is modelled by the amplitude $V^m_j = V^m(S,q_j\,\theta)$.
The likelihood of this new modelling, $\Lambda_m$, is also given by
Eq. \ref{chinull}, but changing $S_0$ by $V^m$. In addition, the distribution of
$\log{\Lambda_m}$ (close to its maximum value) also follows a $\chi^2$ distribution, but
with one degree of freedom less than that of $\log{\Lambda_0}$.
Let us ask the question of what is the maximum value of $\theta$ (we call it
$\theta_M$) corresponding to a value of the log-likelihood of $V^m$ that
is compatible, by chance, with the parent distribution of the log-likelihood of a
point source. We will estimate $\theta_M$ by computing a critical probability for the
hypothesis that both quantities come from the same parent distribution (this is, indeed,
our null hypothesis, $H_0$). Critical probabilities of 5\% and 0.3\% will be used in our h
ypothesis testing (these values correspond
to the 2$\sigma$ and 3$\sigma$ cutoffs of a Gaussian distribution, respectively). The value of
$\theta_M$ estimated in this way will be the maximum size that the observed source may have,
such that the interferometer could have measured the observed visibilities with a chance given
by the critical probability of $H_0$.
The log-likelihood ratio (in our case, the difference of chi-squares; see Mood, Franklin, \&
Duan \cite{Estat}) between the model of a point source and that of a source with visibilities
modelled by $V^m$
follows a $\chi^2$ distribution with one degree of freedom, as long as $N$ is large
(e.g., Wilks \cite{Wilks}). Hence, $H_0$ will not be discarded as long as
\begin{equation}
\log{\Lambda_m} - \log{\Lambda_0} < \lambda_c/2,
\label{Ratio1}
\end{equation}
\noindent where $\lambda_c$ is the value of the log-likelihood corresponding to the critical
probability of the null hypothesis, assuming a $\chi^2$ distribution with one degree of freedom.
$\lambda_c$ takes the values 3.84 or 8.81 (for a 5\% and a 0.3\% probability cuttoff of $H_0$,
respectively). Working out Eq. \ref{Ratio1},
we arrive to
\begin{equation}
\sum_{jk}^N{\left((V^m_j - 2\,V_j)V^m_k+2\,S_0V_k -S_0^2 \right)F^{jk}} = \frac{\lambda_c}{2}.
\label{Carnati1}
\end{equation}
We can simplify Eq. \ref{Carnati1} in the special case when the off-diagonal elements in
the covariance matrix are small (so the visibilities are nearly independent). On the one
hand, the standard deviation, $\sigma$, of the weighted visibility average is
$ 1/\sigma^2 = \sum_j{F_{jj}} = \sum_{j}{1/\sigma_j^2}$; on the other hand, the weighted
average of any visibility-related quantity, $A$, is
$\langle A \rangle = \sigma^2 \sum_j{F_{jj}A_j}$. Hence, if $F_{jk} \sim 0$ for $j \neq k$
(i.e., if both matrices, $F$ and $C = F^{-1}$ are diagonal),
we have
\begin{equation}
\langle \left(V^m\right)^2 + S_0^2 - 2 V V^m \rangle \sim \langle \left(S_0 - V^m \right)^2 \rangle = \frac{\lambda_c \sigma^2}{2},
\label{Carnati1.5}
\end{equation}
\noindent i.e., the critical probability in our hypothesis testing depends on the
weighted average of the quadratic difference between the two models being compared (a
point source with flux density $ S_0 = \langle V \rangle$ and an extended source with
a model given by $V^m$). In this equation, it is assumed that
$\langle V V^m \rangle \sim \langle S_0 V^m \rangle$. Indeed,
$$\langle V V^m \rangle = \langle S_0 V^m \rangle + \langle (V-S_0) V^m \rangle. $$
Since the source is compact in terms of the diffraction limit of the interferometer
(otherwise, this hypothesis testing would not be meaningful), the dependence of
$V^m$ on the distance in Fourier space, $q$, will be small (i.e., the best-fitting
model $V^m$ will almost be constant for the whole set of observations).
As long as the decrease in $V^m$ with $q$ is smaller than the standard deviation of the
visibilities (which holds if there is no hint of a decrease of visibility amplitudes
with $q$), the quantity $\langle (V-S_0) V^m \rangle$ will always approach zero
(notice that $\langle (V-S_0) \rangle = 0$).
Hence, from Eq. \ref{Carnati1.5}, we can finally write
\begin{equation}
\langle \left( 1 - \frac{V^m}{\langle V \rangle} \right)^2 \rangle = \frac{\lambda_c}{2(\mathrm{SNR})^2}
\label{Carnati2}
\end{equation}
\noindent where SNR is the signal-to-noise ratio of the weighted visibility
average (i.e., $\langle V \rangle /\sigma$). If the covariance matrix is far from diagonal,
Eq. \ref{Carnati2} will not apply, since the off-diagonal elements in Eq. \ref{Carnati1}
would be added to the left-hand side of Eq. \ref{Carnati2}.
However, it can be shown that if all the off-diagonal elements of $F_{jk}$ are roughly
equal and $\langle V \rangle \sim \langle V^m \rangle \sim S_0$ (which is true if both
models, $V^m$ and a point source $S_0$, satisfactorily fit to the data) then the combined
effect of the off-diagonal elements in Eq. \ref{Carnati1} cancels out, and
Eq. \ref{Carnati2} still applies.
This restriction for the $F$ matrix (and hence for the correlation matrix) can be interpreted
in the following way. A distribution of visibilities following a covariance matrix with
roughly equal off-diagonal elements implies that
all the antennas in the interferometer should have similar sensitivities, and the baselines
related to each one of them should cover the full range of distances in Fourier space.
If these conditions are fulfilled, it is always possible to remove any bias in the antenna
gains, and produce a set of visibilities with a roughly equal covariance between them.
If, for instance, the gain at one antenna was biased, the visibilities of
the baselines of all the other antennas would have
different amplitudes in similar regions of the Fourier space, hence allowing us to correct
for that bias by means of amplitude self-calibration. However, if the array was sparse,
there might be antenna-related amplitude biases affecting visibilities at disjoint regions
of Fourier space, thus preventing the correction for these biases using {\em closeby}
measurements from baselines of other antennas.
Interferometers with a sparse distribution of elements may thus produce sets of visibilities
with different covariances (stronger at closer regions in
Fourier space), thus making it difficult to estimate $\theta_M$ correctly (unless in the
very unlikely cases when the {\em whole} matrix $F$ is known!). This is
the case of the NRAO Very Long Baseline Array (VLBA), where the antennas at Mauna Kea and
St. Croix only appear in the longest baselines; lower visibility amplitudes at these
baselines could be thus related to biased antenna gains, instead of source structure.
The new interferometric
arrays, made of many similar elements with a smooth spatial distribution (as ALMA or the SKA)
almost fulfill the condition of homogeneous Fourier coverage described here (i.e., there are
almost no antennas exclusively appearing at long or short baselines), and are hence
very robust for the over-resolution of compact sources well below
their diffraction limits.
Equation \ref{Carnati2} allows us to estimate the value of $\theta_M$ from a given
distribution of baseline lengths, $q_j$, and for a given SNR in the weighted visibility average.
Let us now assume that we have a very large number of visibilities
and the sampling of baseline lengths, $q_j$, is quasi-continuous. We then have
\begin{equation}
\int_0^Q{n(q)\,(1-\frac{1}{\langle V \rangle}V_m(S,q\,\theta_M))^2\,dq} = \frac{\lambda_c}{2(\mathrm{SNR})^2},
\label{cond3}
\end{equation}
\noindent where $n(q)$ is the (normalized) density of visibilities at a distance $q$ in
Fourier space
and $Q$ is the maximum baseline length of
the interferometer.
Usually, $n(q)$ is large for small values of $q$ and decreases with increasing $q$ (i.e., the number
of short baselines is usually larger than the number of long baselines)\footnote{This statement may
not hold in special cases where the array consists of a few distant compact subarrays, whose elements
are considered as independent parts of the interferometer.}.
The effect of $n(q)$ on $\theta_M$ is such that an interferometer with a large number of
long baselines has a higher over-resolution power than another interferometer with a lower
number of long baselines, {\em even} if the maximum baseline length, $Q$, is the same for
both interferometers. The over-resolution power can also be increased if we decrease
the right-hand sides of Eqs. \ref{Carnati2} and \ref{cond3}. This can be achieved by increasing the
sensitivity of the antennas and/or the observing time, {\em even} if the maximum baseline length
is unchanged.
\begin{figure}
\centering
\includegraphics[width=8cm]{./paperfig3b.eps}
\caption{Minimum detectable size of a source (in units of the FWHM of the synthesized
beam) as a function of the source model (for a constant density of baseline lengths).}
\label{fig3}
\end{figure}
We show in Fig. \ref{fig3} the value of $\theta_M$ (in units of the FWHM of the synthesized
beam) corresponding to different array sensitivities
(i.e., different SNR in the visibility average) and source models (i.e.,
Gaussian, uniform-disk, sphere, and ring).
He have used a baseline-length distribution with constant density (a constant $n(q)$).
In all cases, the over-resolution power of the interferometer can be very well
approximated by the following expression
\begin{equation}
\theta_M = \beta\,\left( \frac{\lambda_c}{2(\mathrm{SNR})^2} \right)^{1/4} \times \mathrm{FWHM},
\label{PhenoEq}
\end{equation}
\noindent where $\beta$ slightly depends on the shape of $n(q)$ and the intensity profile of
the source model. It usually takes values in the range 0.5--1.0 (it is larger for steeper $n(q)$
and/or for source intensity profiles with higher intensities at smaller scales). This
equation is very similar to Eq. A.9 in Lobanov (\cite{Lobanov}), although we notice that
in the more general
case, $\theta_M$ should be solved directly from Eq. \ref{Carnati2} (or even from Eq. \ref{Carnati1},
if the array was sparse and/or the covariance matrix was far from homogeneous or diagonal).
We notice that $\theta_M$ in Fig. \ref{fig3} can be interpreted in two different ways; either as
the maximum possible size of a source that generates visibilities compatible with a point-like
source or as the {\em true} minimum size of a source that can still be resolved by the
interferometer. Any of these two (equivalent) interpretations of $\theta_M$ lead us to the
conclusion that an interferometer is capable of resolving structures well below the mere
diffraction limit achieved in the aperture synthesis.
In the cases of ultra-sensitive interferometric arrays like the SKA, where dynamic
ranges of even $10^6$ will be eventually achieved in the images,
the over-resolution power in observations
of strong and compact sources can be very large. As an example, a dedicated
observation with the SKA (let us assume 200 antennas) during one hour (with an integration
time of 2 seconds), and an SNR of 100 for each visibility, results in a minimum resolvable
size of only $\sim$$2\times10^{-3}$ times the FWHM of the synthesized beam (Eq. \ref{PhenoEq}).
As a result, and depending on the
observing frequency, the over-resolution power of the SKA would allow us to study details of
sources at angular scales down to a few $\mu$as. This is, indeed, a resolution higher than
the diffraction limit achieved with the current VLBI arrays (i.e., using much longer baselines).
\section{Summary}
We have reviewed the effects of source compactness in interferometric observations.
The analysis of visibilities in Fourier space allows us to estimate sizes
of very compact sources (much smaller than the diffraction limit achieved in the aperture
synthesis). As the sensitivity
of the interferometer increases, the minimum size of the sources that can still
be resolved decreases (i.e., the over-resolution power of the interferometer increases). In this
sense, the analysis of observations of very compact sources in Fourier space is
more reliable (and robust) than alternative analyses based on synthesized
images of the sky intensity distribution (and affected by beam gridding, deconvolution biases, etc.).
We study the case of extremely compact sources observed with an intereferometer of finite
sensitivity. If the source is such compact and/or weak that it is not possible to detect
structure in the visibilities, we describe a test of hypothesis to set a strong
upper limit to the size of the source.
We also compute the minimum possible size of a source whose structure can still be
resolved by an interferometer (i.e., the maximum theoretical over-resolution power of an
interferometer, computed from Eq. \ref{Carnati2} and approximated in Eq. \ref{PhenoEq}).
The over-resolution power depends on the number of visibilities, the array sensitivity,
and the spatial distribution of the baselines, and increases if 1) the
number of long baselines increases (i.e., not necessarily the {\em maximum} baseline
length, but only the {\em number} of long baselines relative to the number of short
baselines); 2) the observing time increases; and/or 3) the array sensitivity increases.
\begin{acknowledgements}
The authors are thankful to J.M. Anderson for discussion and to the anonymous
referee for his/her useful comments.
MAPT acknowledges support through grant AYA2006-14986-C02-01 (MEC), and
grants FQM-1747 and TIC-126 (CICE, Junta de Andaluc\'ia).
\end{acknowledgements}
|
{
"timestamp": "2012-03-12T01:01:31",
"yymm": "1203",
"arxiv_id": "1203.2071",
"language": "en",
"url": "https://arxiv.org/abs/1203.2071"
}
|
\section{Introduction}
Humans are inherently capable of determining whether one word pair is more
semantically related than another. For example, given the word pairs
{\em honey}--{\em bee} and {\em paper}--{\em car}, one can easily identify
the former pair to be more semantically related than the latter. This, however, is not
true for machines. A lot of work has been done in automating the
process in the last fifteen years. While some approaches do better
than others and have been applied to solving practical problems,
none has matched human judgment.
Typically, automated systems assign a score of {\bf semantic relatedness} to
a given pair of words ({\bf target words}) calculated from a {\bf relatedness measure}. The absolute
score is usually irrelevant on its own. For example, a relatedness
score of 0.7 between {\em a} and {\em b}, in a possible range of 0 to
1, does not imply that {\em a} and {\em b} are more related than the
average word pair. However, given that the semantic relatedness of
{\em c} and {\em d} is 0.6, the system can conclude that {\em a} and
{\em b} are more related than {\em c} and {\em d}. Thus even though
the absolute score given by a relatedness measure is not of much
significance, it is important that the measure give a higher score to
word pairs which humans think are more related and comparatively lower
scores to word pairs that are less related. This ability to mimic
human judgment of semantic relatedness has been used in numerous
applications such as automated spelling correction, word sense
disambiguation, thesaurus creation, information retrieval, text
summarization, and identifying discourse structure.
Existing measures of semantic relatedness rely either on
ontologies and semantic networks or just raw text.
\inlinecite{Budanitsky99}, \inlinecite{BudanitskyH00} and \inlinecite{PatwardhanBT03} do an
extensive survey and comparison of the various
WordNet-based measures. Measures that use just raw text,
known as the {\bf distributional measures}, have been described
individually (for example, in \inlinecite{SchutzeP97}, \inlinecite{Hindle90},
\inlinecite{Lin98C}, \inlinecite{PereiraTL93}, etc) but have not been
extensively compared among each other.
This paper focuses on distributional measures and analyzes
their strengths and limitations. Particular attention is paid to the different
kinds of distributional measures and their components.
New measures are suggested that overcome some of
their drawbacks. Characteristics of WordNet-based and distributional
measures are contrasted and finally, future research directions are
suggested which
may determine a better understanding of semantic relatedness.
\section{Background}
\subsection{Co-occurrences}
Words that occur within a certain window of a target word are
called the {\bf co-occurrences} of the word. The window size may be a
few words on either side, the complete sentence, a paragraph or
the entire document. Consider the sentence below:
\begin{center}
{\tt the plane flew through a cloud}
\end{center}
\noindent If we consider the window size to be the complete sentence,
{\em flew} co-occurs with {\em the, plane, through, a}
and {\em cloud}. The set of words that co-occur with a word
constitute the context of the word. They are used in tasks such as
information retrieval, word sense disambiguation, and semantic
relatedness.
\pagebreak
\subsection{Word Association Ratio}
Given two events $x$ and $y$ with probabilities $P(x)$ and $P(y)$,
their {\bf pointwise mutual information}~\cite{Fano61}\endnote{In their respective papers, Robert Fano as well
as Ken Church and Patrick Hanks refer to pointwise mutual information as mutual
information.}, PMI for short, or just
$I$, is defined as follows:
\begin{equation}
\label{eq:MI}
I(x,y) = \log_{2}\frac{P(x,y)}{P(x)P(y)}
\end{equation}
\noindent $P(x,y)$ is the joint probability of $x$ and $y$. If $I(x,y)$
evaluates to be close to zero, i,e, $P(x,y) \approx P(x) \times P(y)$,
then it means that events $x$ and $y$ occur together just as often
as is expected from their individual probabilities.
If $I(x,y) \gg 0$, it implies that $x$ and $y$
occur together more often than would be expected from their individual
probabilities and hence have a strong correlation.
\inlinecite{ChurchH89}\endnotemark[\value{endnote}] introduce {\bf word association ratio}, which is similar
to pointwise mutual information. If $x$ and $y$ are words with probabilities
$P(x)$ and $P(y)$ (estimated by corpus counts), their
association ratio is defined to be the same as in (\ref{eq:MI}),
except that $P(x,y)$ stands for the probability that $x$ appears,
within a certain window, before $y$. It should be noted that
$P(x,y)$ is no longer symmetric ($P(x,y) \neq P(y,x)$) as $P(x,y)$ and $P(y,x)$ represent
two different events.
If two words have a word association ratio
close to zero then they do not share an interesting relationship but
if $I(w_1, w_2) \gg 0$, then $w_2$ follows $w_1$ (within a certain window)
more often than chance and the words $w_1$ and $w_2$ are strong co-occurrences.
\textcolor{Black}{Theoretically, word association ratio may yield negative values
(word pair occurs less frequently than expected by random chance)
but \inlinecite{ChurchH89} show that it is hard to accurately predict
negative word association ratios with confidence. Systems which
use word association ratio may be adversely affected by this. A common
approach to counter this is to equate the negative association values
to 0 (for example, \inlinecite{Lin98C}). This usually means that the system will
ignore such words.}
A problem with PMI in general (which is inherited by word association ratio)
is that low frequency events get higher scores than expected. \inlinecite{PantelL02}
try to overcome this by multiplying the PMI value with a correction factor.
Although, Pantel and Lin give the correction factor
for word association ratio using syntactically related co-occurring words,
a more generic form applicable for pointwise mutual information is as shown below:
\begin{equation}
\label{eq:PMICor}
I_{\text{\itshape corrected}}(x,y) = \log_{2}\frac{P(x,y)}{P(x)P(y)} \times \frac{\min (\text{\itshape freq}(x),\text{\itshape freq}(y))}
{\min (\text{\itshape freq}(x),\text{\itshape freq}(y)) + 1}
\end{equation}
\noindent \textcolor{Black}{The correction factor is large (close to 1) if both the events occur
a large number of times and small (close to 0) if any of the two events
occurs very few times.}
\subsection{Relatedness vs Similarity}
A closely related concept to semantic relatedness is {\bf semantic
similarity}. While there is some overlap in their meanings and they
may be used interchangeably in certain contexts, it is important to be
aware of their distinction. \inlinecite{BudanitskyH00} and \inlinecite{BudanitskyH04}
point out that semantic similarity is used when
similar entities such as {\em apples} and {\em
bananas} or {\em table} and {\em furniture} are compared.
These entities are close to each other in an is-a hierarchy.
For example, {\em apples} and {\em bananas} are hyponyms of
{\em fruit} and {\em table} is a hyponym of {\em furniture}.
However, even dissimilar
entities may be semantically related, for example, {\em door} and {\em knob}, {\em tree} and {\em shade}, or
{\em gym} and {\em weights}. In this case the two entities are
not similar per se, but are related by some relationship. This relationship
may be one of the classical relationships such as meronymy (is part of) as in {\em door--knob}
or a non-classical one as in {\em tree--shade} and {\em gym--weights}.
Thus two entities are semantically related if they are semantically similar (close together
in the is-a hierarchy) or share any other classical or
non-classical relationships. As \inlinecite{BudanitskyH04} point out,
semantic similarity is a subset of semantic relatedness.
The concept of {\bf semantic distance} has traditionally been used
in the context of both semantic relatedness and semantic similarity.
In the former context, it represents the inverse of semantic relatedness,
while in the latter, it is the inverse of semantic similarity. In this paper
as well, we shall continue to use the term for both concepts with
the confidence that the context will disambiguate the intended
meaning.
\subsection{The Distributional Hypothesis}
Given a text corpus, individual words have more or less differing contexts
around them. The context of a word is composed of words co-occurring with
it within a certain window around it.
Distributional measures use statistics acquired from a large text corpora
to determine how similar
the contexts of two words are.
These measures are also used as proxies to measures of semantic similarity
as words
found in similar contexts tend to be semantically similar. This is known as
the {\bf distributional hypothesis} (\inlinecite{Firth57} and \inlinecite{Harris68})
and such measures have traditionally
been referred to as measures of {\bf distributional similarity}.
The hypothesis makes intuitive sense as \inlinecite{BudanitskyH04} point out.
If two words have many co-occurring
words then similar things are being said about both of them and
so they are likely to be semantically similar. Conversely, if two words are
semantically similar
then they are likely to be used in a similar fashion in text and thus
end up with many common co-occurrences. For example, the semantically
similar {\em bug} and {\em insect} are expected to have a number of common
co-occurring words such as {\em crawl, squash, small, woods}, and so on,
in a large enough text corpus.
Like measures of distributional similarity there exist measures of
what we will call {\bf distributional relatedness} (\inlinecite{SchutzeP97} and \inlinecite{YoshidaYK03}).
These measures use raw text
and co-occurrence information to determine semantic relatedness between
two words. The distributional hypothesis mentioned earlier is generic
enough to be the basis for both distributional similarity and
distributional relatedness. We propose more specific hypotheses that
demarcate the two.
\begin{quote}
{\bf Hypothesis of distributional similarity:} \\
{\bf Distributionally similar} words tend to be semantically similar,
where two words ($w_1$ and $w_2$, say) are said to be distributionally similar if they have
many common co-occurring words and these co-occurring words are each
related to $w_1$ and $w_2$ by the same syntactic relation.
\end{quote}
\begin{quote}
{\bf Hypothesis of distributional relatedness:} \\
{\bf Distributionally related} words tend to be semantically related,
where two words ($w_1$ and $w_2$, say) are said to distributionally related if they have
many common co-occurring words and this set of co-occurring words is not
restricted to only those that are related to $w_1$ and $w_2$ by
the same syntactic relation.
\end{quote}
The two hypotheses are based on the fact that semantically similar
words belong to the same broad part of speech (noun, verb, etc)
and are thus each syntactically related to most common co-occurring
words by the same syntactic relation. Further, the more two
words are semantically related,
the more common co-occurring words they have.
Consider the semantically related word pair {\em doctor--operate}.
In a large enough body of text, the two words are likely to
have the following common co-occurring words: {\em patient, scalpel,
surgery, recuperate}, and so on. All these words
will be used by a measure of distributional relatedness and the pair
will be assigned a high score. However, a measure
of distributional similarity will not use any of these co-occurring words
(and likely no other, for that matter) as they are not related to
the target words by the same syntactic relation. The word
{\em doctor} is almost always used as a noun while {\em operate}
is a verb. Thus {\em doctor} and {\em operate} will get a very
low score of distributional similarity. The word pair {\em doctor--nurse},
on the other hand, will get a high score of distributional relatedness and
distributed similarity.
Thus an important characteristic of any
distributional measure is whether it is a measure of distributional similarity or more
generally that of distributional relatedness.
It should be noted that a measure of distributional similarity
will provide a high score for certain closely
related but dissimilar words belonging to the same thematic role.
For example, {\em homeless} and {\em drunk} which refer to dissimilar
concepts but share a non-classical relationship of association
({\em homeless} and {\em drunk} tend to occur together in text)
will likely get a high score as they belong to the same part of
speech (adjective) and may have many common co-occurring words
such as {\em beggar, person, helped,} and so on, related
by the same syntactic relation.
This is a limitation of the current measures of distributional
similarity and the impact of the limitation on the
ability of the measures to mimic semantic similarity is worth determining.
The relevant literature uses the term {\bf distance} as inverse of
distributional similarity. In order to clearly distinguish between
semantic distance, this paper will refer to the inverse of
distributional similarity as {\bf distributional distance}.
Like semantic distance, distributional distance will also be
used as the inverse of distributional relatedness, and the context
should help disambiguate the intended meaning.
\subsection{Relatedness of Words and Concepts}
Measures of semantic relatedness and similarity are applied to
particular concepts (or particular senses of the words); for example,
one may determine the semantic relatedness of
{\em bank} in the {\em financial institution} sense and {\em
interest} in the {\em interest rate} sense. Distributional
measures, on the other hand, usually assign scores to word pairs irrespective
of the nature of their polysemy (how many senses they have)
or the particular senses they have been used in.
Distributional measures will need a much more knowledge
rich source (for example large amounts of sense-tagged corpora) than raw text
to assign scores to word-sense pairs.
\subsection{Evaluation}
The presence of a large number of relatedness measures necessitates
a suitable evaluation to determine which methods come closest
to the human notions of relatedness and to determine how good they
each are. There exist two modes of evaluation. The first
involves the creation of two ranked lists of certain word pairs.
One list is created using a relatedness measure while the other
is ranked by humans.
The correlation of the
two rankings is indicative of how closely the measure mimics
human judgment of relatedness. \inlinecite{RubensteinG65} were the first to
conduct quantitative experiments with human subjects
who were asked to rate 65 word pairs on a scale of 0.0 to 4.0 as per
their relatedness. The word pairs chosen ranged from very
similar and almost synonymous to unrelated. \inlinecite{MillerC91} also
conducted a similar study on 30 word pairs taken from the Rubenstein-Goodenough
pairs. However, lack of large amounts of data from human subject experimentation
limits the quality of this mode of evaluation.
The second and a more indirect way of evaluating measures of
semantic relatedness is by the performance
of natural language tasks that use them,
for example, automatic spelling correction, word sense
disambiguation, estimation of unseen bigram \textcolor{Black}{(not found in training data)}
probabilities, and so on.
\section{Distributional Measures}
\subsection{Spatial Metrics}
A popular technique to determine distributional relatedness between two words
is to map them to points in a multidimensional space such that the distance
between the two points is an indicator of distributional and thereby semantic distance
between them.
Large co-occurrence matrices pertaining to each word, which store
the set of words that co-occur with it within a certain
window size, are created from a text corpus.
Consider a multidimensional space where
the number of dimensions is equal to size of vocabulary. A word
$w_1$ can be represented by a point in this space such that
the vector $\vec w_1 $ from the origin to this point has
equal positive components in all dimensions corresponding to
words that co-occur with $w_1$. Similarly, vector $\vec w_2$
can be created for word $w_2$. This section describes three distributional distance metrics
that quantify the distance between $\vec w_1 $ and $\vec w_2$.
\subsubsection{Cosine}
The {\bf cosine} method (denoted by $\text{\itshape Cos}$) is one of the earliest distributional measures.
Given two words $w_1$ and $w_2$, the cosine measure calculates
the cosine of the angle between $\vec w_1 $ and $\vec w_2 $.
If a large number of words co-occur with both $w_1$ and $w_2$,
$\vec w_1 $ and $\vec w_2$ will have a small angle between
them, the cosine will be large, and we get a large
relatedness value between them. \textcolor{Black}{The cosine measure gives
scores in the range from $0$ (unrelated) to $1$ (maximally related). }
\begin{equation}
\text{\itshape Cos}(w_1,w_2) = \frac{\vec w_1 . \vec w_2}{\mid \vec w_1 \mid \times \mid \vec w_2 \mid}
\end{equation}
\noindent A limitation of the cosine method in its original form is that all
co-occurring words are treated the same, irrespective of
how often they co-occurred with $w_1$ and $w_2$.
A popular variation \textcolor{Black}{(\inlinecite{YoshidaYK03}, \inlinecite{Lee99}, and \inlinecite{SchutzeP97})} that
incorporates this information is stated below:
\begin{equation}
\label{eq:Cos}
\text{\itshape Cos}(w_1,w_2) = \frac{\sum_{w \in C(w_1) \cup C(w_2)} \left( P(w|w_1) \times P(w|w_2) \right) }
{\sqrt{ \sum_{w \in C(w_1)} P(w|w_1)^2 } \times \sqrt{ \sum_{w \in C(w_2)} P(w|w_2)^2 } }
\end{equation}
\noindent $C(x)$ is the set of words that co-occur (within a certain window)
with the word $x$ in a corpus.
\noindent \textcolor{Black}{$P(x|y)$ is the probability that a particular co-occurrence is composed of
$x$ and $y$, given that word $y$ is one of the words in the co-occurrence pair.}
It can be approximated by simple corpus counts.
Once again, the formula is the cosine of the angle between the word vectors
$\vec w_1 $ and $\vec w_2$ but the word vectors incorporate the strength of association
of the co-occurring words with the target words. The component of $\vec x$ in a
dimension (corresponding to word $y$, say) is equal to the strength of association
of $y$ with $x$. Thus the vectors corresponding to two words are closer together,
and thereby get a high distributional relatedness score, if they share many co-occurring words
and the co-occurring words have more or less the same strength of association
with the two target words. In the above formula conditional probability
of the co-occurring words given the target words is used as the strength
of association.
The cosine is used, among others, by \inlinecite{SchutzeP97} and
\inlinecite{YoshidaYK03}, who suggest methods of automatically generating thesauri from
text corpora. \inlinecite{SchutzeP97} use the
Tipster category B corpus~\cite{Tipster} (450,000 unique terms)
and the {\em Wall Street Journal} to create a large but sparse
co-occurrence matrix of 3,000 medium-frequency words (frequency
rank between 2,000 and 5,000). Latent semantic indexing and
single-value decomposition (see \inlinecite{SchutzeP97} for details) are used to reduce the dimensionality
of the matrix and get for each term a word vector of its 20 strongest
co-occurrences. The cosine of a word vector (say $\vec w_1$) with each of the other
word vectors is calculated and the top scores along with
the words whose vector generated the top scores is noted. These
words form the thesaurus entries for $w_1$.
\inlinecite{YoshidaYK03} believe
that words that are closely related for one person may be
distant for another. They use
around 40,000 HTML documents to generate personalized thesauri
for six different people. Documents used to create the thesaurus
for a person are retrieved from the subject's home page and a web crawler which
accesses linked documents. The authors also suggest a root-mean-squared
method to determine the similarity of two different thesaurus
entries for the same word.
\subsubsection{Manhattan and Euclidean Distances}
Distance between two points (words) in multidimensional space can
be calculated using the {\bf Manhattan distance} a.k.a.\@ {\bf $L_1$ norm}
(denoted by $L_1$)
or {\bf Euclidean distance} a.k.a.\@ {\bf $L_2$ norm} (denoted by $L_2$).
In the Manhattan distance~(\ref{eq:L1}) \textcolor{Black}{(\inlinecite{DaganLP97}, \inlinecite{DaganLP99},
and \inlinecite{Lee99})}, the disparity in strength of association of
$w_1$ and $w_2$ with each word that they co-occur with, is summed.
The more the disparity in association, the more is the distributional distance between the two words.
The Euclidean distance~(\ref{eq:L2}) \textcolor{Black}{(\inlinecite{Lee99})} employs the root mean squared of the
disparity in association to get the final distributional distance.
\textcolor{Black}{Both $L_1$ norm and $L_2$ norm give values in the range 0 (zero
distance or maximally related) and infinity (maximally distant or unrelated).}
\begin{eqnarray}
\label{eq:L1}
L_1(w_1,w_2)& =& \sum_{w \in C(w_1) \cup C(w_2)} \mid P(w|w_1) - P(w|w_2) \mid \\
\label{eq:L2}
L_2(w_1,w_2)& =& \sqrt{\sum_{w \in C(w_1) \cup C(w_2)} \left(P\left(w|w_1\right) - P\left(w|w_2\right)\right)^2 }
\end{eqnarray}
\noindent The above formulae use conditional probability
of the co-occurring words given the target words as the strength
of association.
The distributional relatedness of words may be found by taking the reciprocal of the
distributional distance or similar suitable method.
\textcolor{Black}{\inlinecite{Lee99} compared the ability of all three spatial metrics to
determine the probability of an unseen (not found in training data) word
pair. The measures in order of their performance (from better to worse) were: $L_1$ norm,
cosine, and $L_2$ norm.}
\textcolor{Black}{\inlinecite{Weeds03} determined the correlation of word pair ranking as per
a handful of distributional measures with human rankings (Miller and Charles
word pairs~\inlinecite{MillerC91}). Using verb-object pairs from the {\em British
National Corpus (BNC)},
she found the correlation of $L_1$ norm with human rankings to be 0.39.}
\subsection{Set Operations}
\textcolor{Black}{ Distributional measures, as discussed earlier, aim to determine semantic
similarity (or relatedness) using words that co-occur with
the target words.
The problem can be transformed to finding the similarity of
two sets ($W_1$ and $W_2$, say), where each set has as its members the co-occurring words of the
two target words ($w_1$ or $w_2$), respectively.
One can now use set operations such as {\bf Jaccard} and {\bf Dice coefficient}
to determine the similarity of the two sets and thereby, the semantic
similarity of the target words.}
\begin{eqnarray}
\label{eq:Jaccard}
\text{\itshape Jaccard}(w_1,w_2)& =& \frac{\left|W_1 \cap W_2\right|}{|W_1 \cup W_2|} \\
\label{eq:Dice}
\text{\itshape Dice}(w_1,w_2)& =& \frac{2 \times \left|W_1 \cap W_2\right|}{|W_1| + |W_2|}
\end{eqnarray}
\noindent Both measures give
scores in the range from $0$ (unrelated) to $1$ (maximally related)
and will rank
word pairs identically.
\pagebreak
\newdisplay{theorem}{Theorem}
\begin{theorem}
\textcolor{Black}{If the similarity of word pair
one is less than the similarity of word pair two, as determined by
the Jaccard coefficient, then the similarity of word pair one will
be less than the similarity of word pair two, as determined by
the Dice Coefficient.}
\end{theorem}
\begin{pf*}{Proof}
Let $x$ be the number of co-occurrences common to
word pair one and $y$ the number of words that co-occur with just
one of the two words in word pair one.\\
Let $l$ and $m$ be the
corresponding values for word pair two.\\
Therefore,
\begin{eqnarray}
\text{\itshape Jaccard}(\text{\itshape pair one})& =& \frac{x}{x+y} \\
\text{\itshape Dice}(\text{\itshape pair one})& =& \frac{2x}{2x+y} \\
\text{\itshape Jaccard}(\text{\itshape pair two})& =& \frac{l}{l+m} \\
\text{\itshape Dice}(\text{\itshape pair two})& =& \frac{2l}{2l+m}
\end{eqnarray}
\noindent Given,
\begin{eqnarray}
\text{\itshape Jaccard}(\text{\itshape pair one})& <& \text{\itshape Jaccard}(\text{\itshape pair two})\\
\Rightarrow \quad \frac{x}{x+y}& <& \frac{l}{l+m}\\
\Rightarrow \quad xl+xm& <& xl+yl\\
\label{eq:condition}
\Rightarrow \quad xm& <& yl
\end{eqnarray}
To prove,
\begin{eqnarray}
\text{\itshape Dice}(\text{\itshape pair one})& <& \text{\itshape Dice}(\text{\itshape pair two})\\
\text{or,}\quad \frac{2x}{2x+y}& <& \frac{2l}{2l+m}\\
\text{or,}\quad 4xl+2xm& <& 4xl+2yl\\
\text{or,}\quad xm& <& yl
\end{eqnarray}
which is true (from (\ref{eq:condition})). \qed
\end{pf*}
Thus, in terms of measuring distributional similarity/relatedness,
Jaccard and Dice coefficients are identical. \inlinecite{Lee99} shows that the Jaccard
coefficient
performs better than $L_1$ norm in an unseen bigram probability
estimation task.
\subsubsection{Pseudo-Fuzzy Metrics}
\textcolor{Black}{Simple set operations as stated above do not consider the strength of
association of the co-occurring word with the target words.
The strength of association can be incorporated into the metrics
by considering the co-occurrence sets to be {\bf pseudo-fuzzy}.
The degree of membership of each word in the pseudo-fuzzy set corresponding
to a target word is its strength of association
with the target word. We call the sets pseudo-fuzzy (and not fuzzy)
because the range of membership values is now dependent on the measure of association
used --- conditional probability: 0 to 1, PMI (ignoring negative values\endnote{It
is hard to accurately predict
negative word association ratios with confidence (\inlinecite{ChurchH89}).}): 0 to infinity.
Even though conditional probability has a range from 0 to 1 like a
standard fuzzy set membership function, the conditional probabilities
of all the words with respect to a particular target word sum up to $1$.
This need not be (and usually is not) true of the membership values for a regular fuzzy set.}
Use of conditional probability (denoted by CP) as the strength of association and application
of {\bf Jaccard} and {\bf Dice coefficient} on the pseudo-fuzzy set results in the following
formulae.
Similar to the case of regular sets, it can easily be shown
that the Dice and Jaccard coefficients of pseudo-fuzzy sets also rank
word pairs identically.
\begin{eqnarray}
\label{eq:JaccardCP}
\text{\itshape Jaccard}^{\text{\em CP}}(w_1,w_2)& =& \frac{\sum_{w \in C(w_1) \cup C(w_2)} \min (P(w|w_1),P(w|w_2))}
{ \sum_{w \in C(w_1) \cup C(w_2)} \max (P(w|w_1),P(w|w_2))}\\
\label{eq:DiceCP}
\text{\itshape Dice}^{\text{\em CP}}(w_1,w_2)& =& \frac{2 \times \sum_{w \in C(w_1) \cup C(w_2)} \min (P(w|w_1),P(w|w_2))}
{ \sum_{w \in C(w_1)} P(w|w_1) + \sum_{w \in C(w_2)} P(w|w_2)} \\
&=& \frac{2 \times \sum_{w \in C(w_1) \cup C(w_2)} \min (P(w|w_1),P(w|w_2))}{1 + 1} \\
\label{eq:DiceSimple}
&=& \sum_{w \in C(w_1) \cup C(w_2)} \min (P(w|w_1),P(w|w_2))
\end{eqnarray}
\noindent \textcolor{Black}{Observe that the special nature of the membership
function forces the Dice coefficient to equate to simplified form
(\ref{eq:DiceSimple}) which
is also the numerator of the Jaccard coefficient. Since Dice and Jaccard
are identical in terms of ranking word pairs, use of this simplified form
is computationally optimal if one decides to use the Dice or Jaccard coefficient
with conditional probability as the strength of association.}
\textcolor{Black}{\inlinecite{DaganMM95} use a weighted version of the Jaccard coefficient
on pseudo-fuzzy sets with PMI as the strength of association. They
do not provide quantitative comparison with other distributional measures
and do not
derive their measure as shown above. Viewing co-occurrence information as pseudo-fuzzy
sets enabling the use of any of the numerous set operations to determine distributional
similarity is a novel approach. Part of our future research is to determine how well
such measures fare compared to the others.
}
\subsection{Mutual Information--Based Measures}
\inlinecite{Hindle90} was one of the first to factor the strength
of association of co-occurring words into a distributional similarity measure. The hypothesis
is that the more similar the association of co-occurring words
with the two target words, the more semantically similar they are.
Hindle\endnote{In their respective papers, Donald Hindle and
Dekang Lin refer to pointwise mutual information as mutual
information.} used pointwise mutual information (PMI) as the strength of association.
Consider the nouns $n_j$ and
$n_k$ that exist as objects of verb $v_i$ in different
instances within a text corpus. Hindle used formula
(\ref{eq:Hindle1}) to determine the distributional similarity of $n_j$ and $n_k$
solely from their occurrences as object of $v_i$. \textcolor{Black}{The minimum of
the two PMIs captures the similarity in the strength of association of $v_i$
with each of the two nouns. Note that in case of negative PMI values,
the maximum function captures the PMI which is lower in absolute value.}
\begin{equation}
\label{eq:Hindle1}
\text{\itshape Hin}_{\text{\itshape obj}} (v_i, n_j, n_k) = \left\{ \begin{array}{l} \min(I(v_i,n_j), I(v_i,n_k)),\\ \qquad \quad \text{if}\; I(v_i,n_j) > 0\; \text{and}\; I(v_i,n_k) > 0 \\
\mid \max(I(v_i, n_j), I(v_i, n_k))\mid,\\ \qquad \quad \text{if}\; I(v_i, n_j) < 0\; \text{and}\; I(v_i, n_k) < 0 \\
0, \quad \quad \text{otherwise} \end{array} \right.
\end{equation}
\noindent $I(n, v)$ stands for the PMI (word association ratio, to be more precise)
between the words $n$ and $v$. Hindle used an analogous formula to calculate
the distributional similarity ($Hin_{subj}$) using the subject-verb relation. The overall
distributional similarity between any two nouns is calculated by the formula (\ref{eq:Hindle2}).
\begin{equation}
\label{eq:Hindle2}
\text{\itshape Hin}(n_1,n_2) = \sum_{i = 0}^{N} \left( \text{\itshape Hin}_{\text {\itshape obj}}(v_i, n_1, n_2) + \text{\itshape Hin}_{\text{\itshape subj}}(v_i, n_1, n_2) \right)
\end{equation}
\noindent The measure
gives similarity scores from 0 (maximally dissimilar) to infinity (maximally similar).
Note that in Hindle's measure, the set of co-occurring words used is restricted
to include only those words that have the same syntactic relation with both
target words (either verb-object or verb-subject). This is therefore a measure
of distributional similarity and not distributional relatedness. \textcolor{Black}{A form of
Hindle's measure where all co-occurring words are used, making it a measure of
distributional relatedness, is shown below:}
\begin{equation}
\label{eq:Hindle3}
\text{\itshape Hin}_{\text{\itshape{rel}}}(w_1,w_2) = \sum_{w \in C(w)} \left\{ \begin{array}{l} \min(I(w,w_1), I(w,w_2)),\\ \qquad \quad \text{if}\; I(w,w_1) > 0\; \text{and}\; I(w,w_2) > 0 \\
\mid \max(I(w, w_1), I(w, w_2))\mid,\\ \qquad \quad \text{if}\; I(w, w_1) < 0\; \text{and}\; I(w, w_2) < 0 \\
0, \quad \quad \text{otherwise} \end{array} \right.
\end{equation}
\noindent{$C(x)$ is the set of words that co-occur with word $x$.}
\inlinecite{Lin98C} suggests a different measure derived from his
information theoretic definition of similarity \cite{Lin98B}. Further,
he uses a broad set of syntactic relations apart from subject-verb and
verb-object relations and shows that using multiple relations is beneficial even by
Hindle's measure. He first extracts triples of the form $(x,r,y)$ from
the partially parsed text, where the word $x$ is related to $y$ by the
syntactic relation $r$. If $I(x,r,y)$ is the information contained in
the proposition: the triple $(x,r,y)$ occurred a constant $c$ times,
then Lin defines the distributional similarity between two words, $w_1$ and $w_2$, as follows:
\begin{equation}
\label{eq:LinCorpus}
\text{\itshape Lin}(w_1,w_2) = \frac{\sum_{(r,w)\, \in\, T(w_{1})\, \cap\, T(w_{2})} \left(I(w_{1},r,w) + I(w_{2},r,w)\right)}
{{\sum_{(r,w')\, \in\, T(w_1)} I(w_1,r,w') + \sum_{(r,w'')\, \in\, T(w_2)} I(w_2,r,w'')}}
\end{equation}
\noindent $T(x)$ is the set of all word pairs $(r,y)$ such that the pointwise mutual information
$I(x,r,y)$, is positive. Note that this is different from \inlinecite{Hindle90}
where even the cases of negative PMI were also considered. \textcolor{Black}{As mentioned earlier,
\inlinecite{ChurchH89} show that it is hard to accurately predict
negative word association ratios with confidence. Thus, co-occurrence pairs with negative
PMI are ignored. The measure gives similarity scores from 0 (maximally dissimilar) to 1 (maximally similar).}
Lin's measure distinguishes itself from that of Hindle in two respects.
Firstly, he normalizes the distributional similarity between two words ($w_1$ and $w_2$) determined by their
PMI with common co-occurring words by the total PMI of
$w_1$ and $w_2$ with the rest of the related words. This is a
significant improvement as now high PMI of the target words
with shared co-occurring words does not guarantee a high distributional similarity score.
As an additional requirement, the target words must have low PMI
with words they do not both co-occur with.
The second difference in the two formulae is that Hindle uses a minimum of
the PMI between each of the target words and the shared
co-occurring word, while Lin uses the sum. Taking
the sum has the drawback of not penalizing for a mismatch in strength
of co-occurrence, as long as $w_1$ and $w_2$ both co-occur with a word.
We suggest a new measure of distributional similarity
(denoted by $\text{\itshape Saif}$) which counters this but keeps the
normalizing factor of Lin's measure:
\begin{equation}
\label{eq:Saif}
\text{\itshape Saif}(w_1,w_2) = \frac{2 \times \sum_{(r,w)\, \in\, T(w_{1})\, \cap\, T(w_{2})} \min(I(w_{1},r,w), I(w_{2},r,w))}
{\sum_{(r,w') \in T(w_1)} I(w_1,r,w') + \sum_{(r,w'') \in T(w_2)} I(w_2,r,w'')}
\end{equation}
\noindent The multiplication by two is done to get scores in the range of
0 to 1 (note that the sum in Lin's formula was replaced by a min). The
multiplication has no effect on the relative ranking of word pairs
by their similarities. Also notice that like Hindle's measure, both Lin's
and mine are measures of distributional similarity.
\inlinecite{Hindle90} used a portion of the {\em Associated Press} news stories
(6 million words) to classify the nouns into semantically related
classes. \inlinecite{Lin98C} used his measure to generate a thesaurus
from a 64-million-word corpus of the {\em Wall Street Journal, San Jose Mercury}
and {\em AP Newswire}. He also provides a
framework for evaluating automatically generated thesauri by comparing
them with WordNet-based and Roget-based thesauri. He shows that the thesaurus
created with his measure is closer to the WordNet and Roget-based thesauri than
that of Hindle.
\subsubsection{Mutual Information--Based Spatial and Fuzzy Metrics}
Variations of the spatial metrics (equations (\ref{eq:Cos}), (\ref{eq:L1}), and
(\ref{eq:L2})) that use pointwise mutual information instead
of conditional probability as the strength of association are possible.
Following are the formulae for mutual information--based spatial metrics.
\begin{eqnarray}
\label{eq:MI-based spatial metrics}
\text{\itshape Cos}^{\text {\itshape MI}}(w_1,w_2)& =& \frac{\sum_{w \in C(w_1) \cup C(w_2)} \left( I\left(w,w_1\right) \times I\left(w,w_2\right)\right)}
{\sqrt{\sum_{w \in C(w_1)} I(w,w_1)^2} \times \sqrt{\sum_{w \in C(w_2)} I(w,w_2)^2} } \\
\label{eq:L1 MI}
L_1^{\text {\itshape MI}}(w_1,w_2)& =& \sum_{w \in C(w_1) \cup C(w_2)} \mid I(w,w_1) - I(w,w_2) \mid \\
\label{eq:L2 MI}
L_2^{\text {\itshape MI}}(w_1,w_2)& =& \sqrt{\sum_{w \in C(w_1) \cup C(w_2)} (I(w,w_1) - I(w,w_2))^2}
\end{eqnarray}
\noindent Use of pointwise mutual information as the strength of association in the fuzzy
metrics (see equations (\ref{eq:DiceCP}) and (\ref{eq:JaccardCP})) discussed earlier
results in the following:
\begin{eqnarray}
\label{eq:JaccardMI}
\text{\itshape Jaccard}^{\text{\em MI}}(w_1,w_2)& =& \frac{\sum_{w \in C(w_1) \cup C(w_2)} \min (I(w,w_1),I(w,w_2))}
{ \sum_{w \in C(w_1) \cup C(w_2)} \max (I(w,w_1),I(w,w_2))} \\
\label{eq:DiceMI}
\text{\itshape Dice}^{\text{\em MI}}(w_1,w_2)& =& \frac{2 \times \sum_{w \in C(w_1) \cup C(w_2)} \min (I(w,w_1),I(w,w_2))}
{ \sum_{w \in C(w_1)} I(w,w_1) + \sum_{w \in C(w_2)} I(w,w_2)}
\end{eqnarray}
\noindent Observe that $\text{\em Saif}(w_1,w_2)$ (equation (\ref{eq:Saif})) equates to $\text{\em Dice}_{\text{\em MI}}(w_1,w_2)$
if the restriction to use only positive pointwise mutual information, is lifted.
\subsection{Relative Entropy--Based Measures}
\subsubsection{Kullback-Leibler divergence}
Given two probability mass functions
$p(x)$ and $q(x)$, their {\bf relative entropy} ($D(p\Vert q)$) is:
\begin{equation}
D(p\Vert q) = \sum_{x \in X} p(x) \log \frac{p(x)}{q(x)} \hspace{1in} \text {for } q(x) \ne 0
\end{equation}
\noindent Intuitively, if $p(x)$ is the accurate probability mass function corresponding
to a random variable $X$, $D(p\Vert q)$ is the information lost on
approximating $p(x)$ by $q(x)$. In other words, $D(p\Vert q)$ is
indicative of how different the two distributions are. Relative
entropy is also called the {\bf Kullback-Leibler divergence} or the
{\bf Kullback-Leibler distance} (denoted by $\text{\itshape KLD}$).
\inlinecite{PereiraTL93} and \inlinecite{DaganPL94} point
out that words have probabilistic distributions with respect to neighboring
syntactically related words. For example, there exists a certain probabilistic
distribution ($d_1 (P(v|n_1))$, say) of a particular noun $n_1$ being the object of any verb.
This distribution can be estimated by corpus counts of parsed or chunked text.
Let $d_2$ ($P(v|n_2)$) be the corresponding distribution for noun $n_2$.
These distributions ($d_1$ and $d_2$) define the contexts of the two nouns ($n_1$
and $n_2$, respectively). As per the distributional hypothesis~\cite{Harris68},
the more these contexts are similar, the more are $n_1$ and $n_2$ semantically similar.
Thus the Kullback-Leibler distance between the two distributions
is indicative of the semantic distance between the nouns $n_1$ and $n_2$.
\begin{equation}
\begin{array}{rcll}
\text{\itshape KLD}(n_1,n_2)& =& D(d_1\Vert d_2) & \\
& =& \sum_{v \in \text {Vb}} P(v|n_1) \log \frac{P(v|n_1)}{P(v|n_2)} & \text {for } P(v|n_2) \ne 0 \\
& =& \sum_{v \in \text {Vb'}(n_1) \cup \text {Vb'}(n_2)} P(v|n_1) \log \frac{P(v|n_1)}{P(v|n_2)} & \text {for } P(v|n_2) \ne 0
\end{array}
\end{equation}
\noindent where $\text {\itshape Vb}$ is the set of all verbs and $\text{\itshape Vb'}(x)$
is the set of verbs that have $x$ as the object.
The distributional similarity is determined by taking the reciprocal of the
Kullback-Leibler distance or similar suitable method.
Note that the set of co-occurring words used is restricted
to include only verbs that each have the same syntactic relation (verb-object) with both
target nouns. This is therefore a measure
of distributional similarity and not distributional relatedness.
It should be noted that the verb-object relationship is not inherent to the
measure and that one or more of any other syntactic relations may be used.
The distributional relatedness may even be determined using all words
co-occurring with the target words. Thus a more generic expression of the
Kullback-Leibler divergence is as follows:
\begin{equation}
\begin{array}{rcll}
\label{eq:KLD}
\text{\itshape KLD}(w_1,w_2)& =& D(d_1\Vert d_2) & \\
& =& \sum_{w \in V} P(w|w_1) \log \frac{P(w|w_1)}{P(w|w_2)} & \text {for } P(w|w_2) \ne 0 \\
& =& \sum_{w \in C(w_1) \cup C(w_2)} P(w|w_1) \log \frac{P(w|w_1)}{P(w|w_2)} & \text {for } P(w|w_2) \ne 0
\end{array}
\end{equation}
\noindent $V$ is the vocabulary (all the words found in a corpus).
$C(x)$, as mentioned earlier, is the set of words occurring (within
a certain window) with word $x$. The inverse of the distributional
distance calculated above yields the distributional relatedness of $w_1$
and $w_2$.
\pagebreak
It should be noted that the Kullback-Leibler distance is not symmetric, that
is, the distance from $w_1$ to $w_2$ is not necessarily, and even not likely,
the same as the distance from $w_2$ to $w_1$. This asymmetry is
counter-intuitive to the general notion of semantic similarity of words, although
\inlinecite{Weeds03} has argued in favor of asymmetric measures.
Further, it is very likely that there be instances such that $P(w_1|v)$
is greater than 0 for a particular verb $v$, while due to data sparseness
or grammatical and semantic constraints,
the training data has no sentence where $v$ has the object $w_2$. This makes
$P(w_2|v)$ equal to 0 and the ratio of the two probabilities
infinite. Kullback-Leibler divergence is not defined in such cases but approximations
may be made by considering smoothed values for the denominator.
\inlinecite{PereiraTL93} use relative entropy to create
clusters of nouns from verb-object pairs corresponding
to a thousand most frequent nouns in the {\itshape Grolier's Encyclopedia}, June 1991
version (10 million words).
\inlinecite{DaganPL94} use Kullback-Leibler distance to estimate
the probabilities of bigrams that were not seen in a text corpus. They point
out that a significant number of possible bigrams are not seen in any
given text corpus. The probabilities of such bigrams may be determined
by taking a weighted average of the probabilities of bigrams composed
of distributionally similar words. Use of Kullback-Leibler distance as the semantic
distance metric yielded a 20\% improvement in perplexity on the {\em Wall
Street Journal} and dictation corpora provided by ARPA's HLT program~\cite{Paul91}.
The use of distributionally similar words
to estimate unseen bigram probabilities will likely lead to erroneous results
in case of less-preferred and strongly-preferred collocations (word pairs).
\inlinecite{DianaH02} point out that even though words like {\em task} and {\em job}
are semantically very similar, the collocations they form with other words
may have varying degrees of usage. While {\em daunting task} is a
strongly-preferred collocation, {\em daunting job} is rarely used. Thus
using the probability of one bigram to estimate that of another will
not be beneficial in such cases.
\subsubsection{$\alpha$ Skew Divergence}
{\bf $\alpha$ skew divergence} ({\em ASD}) is a slight modification of the Kullback-Leibler
divergence, that obviates the need for smoothed probabilities. It has the
following formula:
\begin{equation}
\label{eq:alpha}
\text{\itshape ASD}(w_1,w_2) = \sum_{w \in C(w_1) \cup C(w_2)} P(w|w_1) \log \frac{P(w|w_1)}{\alpha P(w|w_2) + (1 - \alpha) P(w|w_1)}
\end{equation}
\noindent $\alpha$ is a parameter that may be varied but is usually set to $0.99$.
Note that the denominator within the logarithm is never zero with a non-zero
numerator. Also, the measure retains the asymmetric nature of the
Kullback-Leibler divergence.
\inlinecite{Lee01} shows that $\alpha$ skew divergence performs better than
Kullback-Leibler divergence in estimating word co-occurrence probabilities. \inlinecite{Weeds03}
achieves a correlation of \textcolor{Black}{$0.48$ and $0.26$ with human judgment on the Miller and Charles
word pairs using $ASD(w_1,w_2)$ and $ASD(w_2,w_1)$, respectively.}
\subsubsection{Jensen-Shannon Divergence}
A relative entropy--based measure that overcomes the drawback of asymmetry in
Kullback-Leibler divergence is the {\bf Jensen-Shannon divergence}
a.k.a.\@ {\bf total divergence to the average} a.k.a.\@ {\bf information radius}.
It is denoted by $\text{\itshape JSD}$ and has the following formula:
\begin{eqnarray}
\label{eq:JSD}
\text{\itshape JSD}(w_1,w_2)& =& D\left(d_1 \Vert \frac{1}{2}(d_1 + d_2)\right) + D\left(d_2 \Vert \frac{1}{2}(d_1 + d_2)\right) \\
& =& \sum_{w \in C(w_1) \cup C(w_2)} \Bigg( P(w|w_1) \log \frac{P(w|w_1)}{\frac{1}{2}\left(P(w|w_1) + P(w|w_2)\right)} + \nonumber\\
& & \qquad \qquad P(w|w_2) \log \frac{P(w|w_2)}{\frac{1}{2}\left(P(w|w_1) + P(w|w_2)\right)} \Bigg)
\end{eqnarray}
\noindent The Jensen-Shannon divergence is the sum of Kullback-Leibler divergence between each of the individual
distributions $d_1$ and $d_2$ with the average distribution ($\frac{d_1 + d_2}{2}$).
Further, it can be shown that the Jensen-Shannon divergence avoids the problem
of zero denominator as in Kullback-Leibler divergence. The Jensen-Shannon divergence
is therefore always well defined and, like {\bf $\alpha$ skew divergence}, obviates the need for smoothed estimates.
\textcolor{Black}{The Kullback-Leibler divergence, $\alpha$ Skew Divergence, and Jensen-Shannon divergence
all give distributional distance scores from 0 (maximally similar/related) to infinity (completely dissimilar/unrelated).}
\subsection{Co-occurrence Retrieval Models}
The distributional measures suggested by \inlinecite{Weeds03} are based on the notion of
substitutability. The more appropriate it is to substitute word $w_1$
in place of word $w_2$ in a suitable natural language task,
the more semantically similar they are. The natural language task she focuses
on is {\bf co-occurrence retrieval} \textcolor{Black}{(the retrieval of words that co-occur with a target word
from text)} and depending on the definition of
{\em appropriate} she suggests six different distributional measures
called the {\bf co-occurrence retrieval models (CRMs)}.
Let $N_1$ be the set of co-occurrences of $w_1$ \textcolor{Black}{retrieved from a text corpus}
and $N_2$ that of $w_2$. In
order to determine how appropriate it is to
substitute $w_1$ in place of $w_2$ we have to decide how important
it is to get as many co-occurrences as possible listed in $N_2$ ({\bf recall}, denoted by $R$)
and how important it is to not get co-occurrences not listed in $N_2$ ({\bf precision}, denoted by $P$).
Thus Weeds' distributional
measures have a precision component and a recall component. The final
score is a weighted sum of the precision, recall and standard $F$ measure
(see equation~(\ref{eq:CRMfinal})\endnote{$P$ is
short for $P(w_1,w_2)$, while $R$ is short for $R(w_1,w_2)$. The abbreviations
are made due to space constraints and to improve readability.}).
The weights determine the importance of precision and recall and are
determined empirically.
If precision and recall are equally important, then we get a symmetric measure
which gives the same scores to the distributional similarity of $w_1$ with $w_2$ and
$w_2$ with $w_1$. Otherwise, we get an asymmetric measure which
assigns different similarities to the two cases.
As substitutability is defined as a measure of distributional similarity,
metrics such as precision and recall which quantify how good the substitution is,
are used to calculate the distributional similarity.
\begin{equation}
\label{eq:CRMfinal}
CRM(w_1,w_2) = \gamma \Biggl[ \frac{2 \times P \times R}{P + R} \Biggr] + (1 - \gamma) \Biggl[ \beta [ P ] + (1 - \beta) [R] \Biggr]
\end{equation}
\noindent $\gamma$ and $\beta$ are tuned parameters that lie between 0 and 1.
Weeds argues that the asymmetry in substitutability is intuitive
as in many cases
it may be okay to substitute a word, say {\em dog}, with another, say {\em animal}, but the
reverse is not likely to be acceptable as often. Since
substitutability is a measure of semantic similarity, she believes that
distributional similarity between two words should reflect this property
as well. Hence, like the Kullback-Leibler divergence, all her
distributional similarity models are inherently asymmetric.
A word's co-occurrence information may be specified by the set of
co-occurring words alone, or by specifying
the strength of co-occurrences, as well. This strength may be captured
by a suitable measure of word association such as
conditional probability or pointwise mutual information between the
co-occurring words and the target words.
Also, the difference
in the strength of co-occurrence may or may not be used
to penalize the substitutability of one word for another.
\inlinecite{Weeds03} provides six distinct formulae for precision
and recall, depending on the the strength of co-occurrence
and penalty for differences in strength of association.
The precision (or recall) can be considered as the product of a
core precision (or recall) formula (denoted by $core$) and a penalty function (denoted by $penalty$).
The CRMs that use simple counts of the common co-occurrences
in $N_1$ and $N_2$ and not the strength of associations
as core precision and recall values are called {\bf type-based
CRMs} (denoted by the superscript {\em type}).
The CRMs that use conditional probabilities of the common co-occurrences
in $N_1$ and $N_2$ with the target words
as core precision and recall values are called {\bf token-based
CRMs} (denoted by the superscript {\em token}).
The CRMs that use pointwise mutual information of the common co-occurrences
in $N_1$ and $N_2$ with the target words
as core precision and recall values are called {\bf mutual information--based
CRMs} (denoted by the superscript {\em mi}). The core precision
and recall formulae for type, token and mutual information--based
CRMs are listed below:
\begin{eqnarray}
\text{core}_P^{type\ {}\ {}}(w_1,w_2)& =& \frac{\mid C(w_1) \cap C(w_2) \mid}{\mid C(w_1) \mid} \\
\text{core}_R^{type\ {}\ {}}(w_1,w_2)& =& \frac{\mid C(w_1) \cap C(w_2) \mid}{\mid C(w_2) \mid} \\
\text{core}_P^{token\/}(w_1,w_2)& =& \sum_{w \in C(w_1) \cap C(w_2)} P(w|w_1) \\
\text{core}_R^{token\/}(w_1,w_2)& =& \sum_{w \in C(w_1) \cap C(w_2)} P(w|w_2) \\
\text{core}_P^{mi\ {}\ {}\ {}}(w_1,w_2)& =& \frac{\sum_{w \in C(w_1) \cap C(w_2)} I(w,w_1)}{\sum_{w \in C(w_1)} I(w,w_1)} \\
\text{core}_R^{mi\ {}\ {}\ {}}(w_1,w_2)& =& \frac{\sum_{w \in C(w_1) \cap C(w_2)} I(w,w_2)}{\sum_{w \in C(w_2)} I(w,w_2)}
\end{eqnarray}
\noindent The CRMs that do not
penalize difference in strength of co-occurrence are called
{\bf additive CRMs} (denoted by the subscript {\em add}).
The CRMs that do penalize are called {\bf difference-weighted CRMs}
(subscript {\em dw}). The penalty is
a conditional probability--based function (\ref{eq:penalty P type}, \ref{eq:penalty R type}) for the token- and type-based
CRMs, and a mutual information--based function (\ref{eq:penalty P mi}, \ref{eq:penalty R mi}) for the mutual information--based
CRM.
\begin{eqnarray}
\label{eq:penalty P type}
penalty_{P}^{type} = penalty_{P}^{token}& =& \frac{\min(P(w|w_1),P(w|w_2))}{P(w|w_1)} \\
\label{eq:penalty R type}
penalty_{R}^{type} = penalty_{R}^{token}& =& \frac{\min(P(w|w_1),P(w|w_2))}{P(w|w_2)} \\
\label{eq:penalty P mi}
penalty_{P}^{mi}& = &\frac{\min(I(w,w_1),I(w,w_2))}{I(w,w_1)} \\
\label{eq:penalty R mi}
penalty_{R}^{mi}& =& \frac{\min(I(w,w_1),I(w,w_2))}{I(w,w_2)}
\end{eqnarray}
The precision and recall of additive and
difference-weighted CRMs is listed in the appendix.
\inlinecite{Weeds03} extracted verb-object pairs of 2,000 nouns from the {\em British
National Corpus (BNC)}. The verbs related to the target words
by the verb-object relation were used. Thus each of the co-occurring
verbs is related to the target nouns by the same syntactic relation
and therefore the measures capture distributional similarity,
not relatedness.
Correlation with human judgment (Miller and Charles word pairs) showed that
difference-weighted (0.61) and additive mutual information--based measures
(0.62) performed far better than the rest of the CRMs.
\section{Discussion and Analysis of Distributional Measures}
The previous section described numerous distributional measures.
Variations of the measures are possible depending
on certain general properties of a distributional measure.
This section discusses a few of the important properties along
with an analysis of their effect in assigning semantic relatedness.
\subsection{Simple Co-occurrences vs Syntactically Related Words}
\inlinecite{Harris68}, one of the early proponents of the distributional hypothesis,
used syntactically related words to represent the context of a word.
However, the strength of association of any word appearing in the
context of the target words may be used to determine their distributional similarity.
\inlinecite{DaganLP97}, \inlinecite{Lee99}, and \inlinecite{Weeds03}
represent the context of a noun with verbs whose object it is (single
syntactic relation),
\inlinecite{Hindle90} represents the context of a noun with verbs with which
it shares the verb-object or subject-verb relation, while
\inlinecite{Lin98C} uses words related to a noun by any of the many pre-decided syntactic relations
to determine distributional similarity.
\inlinecite{SchutzeP97} and \inlinecite{YoshidaYK03} use all co-occurring words in a
pre-decided window size.
Although \inlinecite{Lin98C} shows that the use of multiple syntactic
relations is more beneficial as compared to just one,
there exist no published results on whether using only syntactically
related words (as compared to all co-occurrences) improves or worsens
the quality of semantic similarity assignment.
Use of syntactically related words entails the
requirement of chunking or parsing the data. Once the data is
suitably parsed, the computational cost of such methods is
lower as distributional similarity is determined with much fewer words.
\subsubsection{Use of Multiple Syntactic Relations}
\inlinecite{Lin98C} used a subset of words that co-occurred with the target
words to determine their distributional similarity. Only those co-occurrences that
are syntactically related (by any of the pre-decided list of relations)
to the target words are chosen.
Once this restricted set of co-occurrences is determined, distributional similarity
is determined by formula~(\ref{eq:LinCorpus}) shown earlier. Observe that
the formula does not distinguish between the co-occurrences related by
different syntactic relations.
An alternative
is to calculate a distributional similarity value using each of the syntactic
relations individually and then determine the overall distributional similarity from
these results. The overall distributional similarity may be as simple as the average similarity (see (\ref{eq:Rel_Avg}))
or the maximum (see (\ref{eq:Rel_Max})) of individual similarity results.
Distributional similarity so calculated is justified in the following
two paragraphs, respectively.
\begin{eqnarray}
\label{eq:Rel_Avg}
\text{\itshape Sim}_{\text{\itshape Overall\_Avg}}(w_1,w_2)& =& \frac{1}{N} \big( \text{\itshape Sim}_{r1}(w_1,w_2) +
\text{\itshape Sim}_{r2}(w_1,w_2) +\nonumber\\
& & \qquad \ldots + \text{\itshape Sim}_{rN}(w_1,w_2) \big)\\
\label{eq:Rel_Max}
\text{\itshape Sim}_{\text{\itshape Overall\_Max}}(w_1,w_2)& =& \max (\text{\itshape Sim}_{r1}(w_1,w_2),
\text{\itshape Sim}_{r2}(w_1,w_2), \nonumber\\
& & \qquad \ldots,
\text{\itshape Sim}_{rN}(w_1,w_2))
\end{eqnarray}
\noindent where $N$ is the total number of syntactic relations considered and,
\begin{equation}
\text{\itshape Sim}_{\text{\itshape ri}}(w_1,w_2) = \frac{\sum_{(ri,w)\, \in\, T(w_{1})\, \cap\, T(w_{2})} (I(w_{1},ri,w) + I(w_{2},ri,w))}
{{\sum_{(ri,w')\, \in\, T(w_1)} I(w_1,ri,w') + \sum_{(ri,w')\, \in\, T(w_2)} I(w_2,ri,w')}}
\end{equation}
\noindent where {\em ri} is a particular syntactic relation.
Consider the scenario where word $w'$ has a strong word association ratio
(large MI value) with $w_1$ but does not co-occur with $w_2$. The large
MI value is added to the denominator as per Lin's measure~(\ref{eq:LinCorpus}). This
results in a low distributional similarity value. However, a number of words are considered
semantically related
even though there exist words ({\bf exclusive co-occurrences}, say) that have strong word association ratios
with one or the other target word but not both. A mark of semantically related
words is the presence of a number of
common co-occurring words with whom they are both strongly associated. One or few
strong co-occurrences of a target word that
do not co-occur with the other target word do not imply that the target words are
semantically unrelated.
For example, consider the rather similar pair of nouns {\em bananas} and {\em mangoes}.
The adjective {\em juicy} is likely to have a large association ratio with {\em mangoes}
but not so with {\em bananas}.
The large MI value of {\em mangoes} and {\em juicy}
may lead to an excessively low distributional similarity value as per Lin's measure~(\ref{eq:LinCorpus}).
Averaging the different distributional similarity values (as in (\ref{eq:Rel_Avg})) calculated from individual syntactic
relations instead of Lin's original
method moderates the strongly negative effect of such exclusive co-occurrences
by restricting it to a particular syntactic relation (in this case, adjective-noun).
It should be noted that the disparity in the strength of association of {\em mangoes}
and {\em juicy} versus {\em banana} and {\em juicy}, is useful in bringing
out the differences between {\em mango} and {\em banana} which may be used
to determine that {\em mango} and {\em orange} are more semantically related
than {\em mango} and {\em banana}. However, as pointed out earlier, we do not
want a strong co-occurrence to have an adverse affect on the estimation of
distributional similarity in all other cases.
Taking the maximum of individual distributional similarity values~(\ref{eq:Rel_Max}) takes the aforementioned idea
one step ahead and is grounded in the following hypothesis:
\begin{quote}
Different
syntactic relations are accurate predictors of the semantic similarity for different
pairs of words.
\end{quote}
\noindent For example, fruits tend to have strong word
associations with adjectives like {\em sweet, bitter, ripe} and {\em juicy},
and low association values with verbs that they are related to by the subject-verb
relation. For example, consider the sentences:
\begin{center}
{\tt the ripe mango fell to the ground} \\
{\tt the ripe plum fell to the ground}
\end{center}
\noindent The words {\em ripe} and {\em mango} are related by the adjective-noun
relation and are likely to have a large value of association. On the
other hand, {\em mango} and {\em fell} which are the subject and verb,
respectively, are likely to have a low measure of association because
almost anything can fall.
The adjective-noun relation is thus expected to yield a higher distributional similarity value
than the subject-verb relation. Employing equation~(\ref{eq:Rel_Max})
in this case
will mean that co-occurrences related to the target words by the adjective-noun
relation will be used to determine the distributional similarity while all other co-occurrences
will be ignored. Thus only the relation
that has the strongest associated co-occurrences is used to determine the
distributional similarity as these co-occurrences are expected to be the best predictors
of semantic similarity.
A measure where other sets of co-occurrences,
which are weak predictors of semantic similarity, are allowed to influence the result may
cause more harm than benefit. Flipping the argument on its head,
target words predicted to be strongly distributionally similar by two
or more syntactic relations should be assigned higher distributional
similarity values than in the case of just one. Using the maximum
method will loose out on this information.
Part of our future work will be to determine if calculating individual
similarity values from different syntactic relations and then arriving at
the final similarity is closer to human judgment or not. Also, as
pointed out, both the average or maximum approaches have their advantages
and disadvantages. It will be interesting to determine which method
gives semantic similarity values closer to the human notion of semantic
similarity.
\subsection{Compositionality}
The various measures of distributional similarity may be divided into two
kinds as per their composition. In certain measures each co-occurring
word contributes to some {\em finite calculable} distributional distance between
the target words.
The final score of distributional distance is
the sum of these contributions.
We will call such measures {\bf compositional measures}.
The relative entropy--based measures,
$L_1$ norm and $L_2$ norm fall in this category.
On the other hand,
the cosine measure along with
Hindle's and Lin's mutual information--based measures
belong to the category of what we call {\bf non-compositional} measures.
Each co-occurring word shared by both target
words contributes a score to the numerator and the denominator.
Words that co-occur with just one of the two target words
contribute scores only to the denominator. The ratio is calculated once all
co-occurring words are considered.
Thus the distributional distance contributed
by individual co-occurrences is not calculable and the final semantic
distance cannot be broken down into compositional distances contributed
by each of the co-occurrences.
It must be noted that it is not clear as to which of the two kinds of measures
(compositional or non-compositional)
resembles human judgment more closely and how much they differ
in their ranking of word pairs. Our future work aims to determine this.
\subsubsection{Primary Compositional Measures}
The compositional measures of distributional similarity (or relatedness) capture the contribution
to distance between the target words ($w_1$ and $w_2$) due to a co-occurring word by three
primary mathematical manipulations of the co-occurrence distributions ($d_1$ and $d_2$):
the {\bf difference}, denoted by $\text{\itshape Dif}$ (as in $L_1$ norm), {\bf division},
denoted by $\text{\itshape Div}$ (as in the relative entropy--based
measures) and {\bf product}, denoted by $\text{\itshape Pdt}$ (as in the conditional probability or mutual information--based cosine method).
We will call the three types of compositional measures {\bf primary compositional
measures (PCM)}. Their form is depicted below:
\begin{eqnarray}
\label{eq:diff}
{\text{\itshape Dif}}& =& \sum_{w \in C(w_1) \cup C(w_2)} \left| P(w|w_1) - P(w|w_2) \right| \\
\label{eq:div}
{\text{\itshape Div}}& =& \sum_{w \in C(w_1) \cup C(w_2)} \left| \log \frac{P(w|w_1)}{P(w|w_2)} \right| \\
\label{eq:pdt}
{\text{\itshape Pdt}}& =& \sum_{w \in C(w_1) \cup C(w_2)} \frac{P(w|w_1) \times P(w|w_2)}{\text{\itshape Scaling Factor}}
\end{eqnarray}
\noindent Observe that by taking absolute values in expressions (\ref{eq:diff}) and
(\ref{eq:div}), the variation in the distributions for different co-occurring words
has an additive affect and not one of cancellation. This corresponds to our
distributional hypothesis --- the more the disparity in distributions, the more is
the semantic distance between the target words. The product form (\ref{eq:pdt}) also
achieves this and is based on the theorem:
\begin{quote}
The product of any two numbers will
always be less than or equal to the square of their average.
\end{quote}
In other words,
the more two numbers are close to each other in value, the higher is the ratio of
their product to a suitable
scaling factor (for example, the square of their average).
Note that the difference and division measures
give higher values when there is large disparity between the strength
of association of co-occurring words with the target words. They are therefore
measures of distributional distance and not distributional similarity.
The product method gives higher values when the strengths of association
are closer, and is a measure of distributional relatedness.
Although all three methods seem intuitive, each produces different distributional similarity
values and more importantly, given a set of word pairs, each is likely
to rank them differently. For example, consider the division and difference
expressions applied to word pairs ($w_1$, $w_2$) and ($w_3$, $w_4$). For simplicity, let there be
just one word $w'$ in the context of all the words. Given:
\begin{eqnarray*}
P(w'|w_1) = 0.91 \\
P(w'|w_2) = 0.80 \\
P(w'|w_3) = 0.60 \\
P(w'|w_4) = 0.50
\end{eqnarray*}
\noindent The distributional distance between word pairs as per the difference PCM:
\begin{eqnarray*}
\text{\itshape Dif\/}(w_1, w_2)& =& | 0.91 - 0.8 | = 0.11 \\
\text{\itshape Dif\/}(w_3, w_4)& =& | 0.6 - 0.5 | = 0.1
\end{eqnarray*}
\noindent The distributional distance between word pairs as per the division PCM:
\begin{eqnarray*}
\text{\itshape Div}(w_1, w_2)& =& \left| \log \frac{0.91}{0.8} \right| \; = 0.056 \\
\text{\itshape Div}(w_3, w_4)& =& \left| \log \frac{0.6}{0.5} \right| \; = 0.079
\end{eqnarray*}
\noindent Observe that for the same set of co-occurrence probabilities, the difference-based
measure ranks the ($w_3, w_4$) pair more distributionally similar (lower distributional distance), while the division-based measure
gives lower distributional similarity values for word pairs having large
co-occurrence probabilities. This behavior is not intuitive and it remains
to be seen, by experimentation, as to which of the three, difference, division
or product, yields distributional similarity measures closest to human notions
of semantic similarity.
The $L_1$ norm is a basic implementation of the difference
method. A simple product-based measure of distributional similarity is as proposed below:
\begin{equation}
\label{eq:prod}
{\text{\itshape Pdt}}^{\text{\itshape Avg}}(w_1,w_2) = \sum_{w \in C(w_1) \cup C(w_2)} \frac{P(w|w_1) \times P(w|w_2)}{(\frac{1}{2}(P(w|w_1) + P(w|w_2)))^2}
\end{equation}
\noindent The scaling factor used is the square of the average probability.
It can be proved that if the sum of two variables is equal to a constant ($k$, say).
Their values must be equal to $k/2$ in order to get the largest product.
Now, let $k$ be equal to the sum of $P(w|w_1)/(P(w|w_1) + P(w|w_2))$ and
$P(w|w_2)/(P(w|w_1) + P(w|w_2))$. This sum will always be equal to $1$ and hence
the product ($Z$) will be largest only when the two numbers are equal i,e, $P(w|w_1)$
is equal to $P(w|w_2)$. In other words, the farther $P(w|w_1)$ and $P(w|w_2)$
are from their average, the smaller is the product $Z$. Therefore, the measure
gives high scores for low disparity in strengths of co-occurrence and low
scores otherwise.
The incorporation of $2$ in the scaling factor results in a measure that ranges between $0$ and
$1$.
The relative entropy--based methods use a weighted division method.
Observe that
both Kullback-Leibler divergence \textcolor{Black}{(formula repeated here for convenience --- equation (\ref{eq:KLDII}))}
and Jensen-Shannon divergence do not take absolute
values of the division of co-occurrence probabilities. \textcolor{Black}{This will mean that
if $P(w|w_1) > P(w|w_2)$,
the logarithm of their ratio will be positive
and if $P(w|w_1) < P(w|w_2)$, the logarithm will be a negative number.}
Therefore, there will be a cancellation of contributions to distributional distance by
words that have higher co-occurrence probability with respect to $w_1$
and words that have a higher co-occurrence probability with respect to $w_2$.
\textcolor{Black}{Observe however that the weight $P(w|w_1)$ multiplied to the logarithm means that in general
the positive logarithm values receive higher weight than the negative ones, resulting
in a net positive score. Therefore, with no absolute value of the logarithm, as in the KLD,
the weight plays a crucial role.}
A modified Kullback-Leibler divergence ($D^{\text{\itshape Abs}}$) which incorporates
the absolute value is suggested in equation (\ref{eq:KLDAbs}):
\begin{eqnarray}
\label{eq:KLDII}
& &{\text{\itshape KLD}}(w_1,w_2) = D(d_1\Vert d_2) = \sum_{w \in C(w_1) \cup C(w_2)} P(w|w_1) \log \frac{P(w|w_1)}{P(w|w_2)}\\
\label{eq:KLDAbs}
& &{\text{\itshape KLD}}^{\text{\itshape Abs}}(w_1,w_2) = D^{\text{\itshape Abs}}(d_1\Vert d_2) = \sum_{w \in C(w_1) \cup C(w_2)} P(w|w_1) \left| \log \frac{P(w|w_1)}{P(w|w_2)} \right|\nonumber\\
& &
\end{eqnarray}
\noindent The updated Jensen-Shannon divergence measure will remain the same as in
equation (\ref{eq:JSD}), except that it is a manipulation of $D^{\text{\itshape Abs}}$
and not the original Kullback-Leibler divergence (relative entropy).
\begin{equation}
\label{eq:IRadAb}
{\text{\itshape JSD}}^{\text{\itshape Abs}}(w_1,w_2) = D^{\text{\itshape Abs}}(d_1 \Vert \frac{1}{2}(d_1 + d_2)) + D^{\text{\itshape Abs}}(d_2 \Vert \frac{1}{2}(d_1 + d_2))
\end{equation}
\noindent \textcolor{Black}{Note that once the absolute value of the logarithm is taken, it no longer makes
much sense to use an asymmetric weight ($P(w|w_1)$) as in the KLD or as necessary to use
a weight at all.} Equation~(\ref{eq:UnwKLD})
shows a simple
division-based measure. It is an
unweighted form of ${\text{\itshape KLD}}^{\text{\itshape Abs}}(w_1,w_2)$ and so we will call it
${\text{\em KLD}}_{\text {\em Unw}}^{\text{\itshape Abs}}$.
\begin{eqnarray}
\label{eq:UnwKLD}
\text{\itshape KLD}_{\text{\itshape Unw}}^{\text{\itshape Abs}}(w_1,w_2) = \text{\itshape Div}(w_1,w_2)
= \sum_{w \in C(w_1) \cup C(w_2)} \left| \log \frac{P(w|w_1)}{P(w|w_2)} \right|
\end{eqnarray}
\noindent Experimental evaluation of these suggested modifications of Kullback-Leibler divergence and
informations radius is part of future work.
\subsubsection{Weighting the PCMs}
The performance of the primary compositional measures may be improved
by adding suitable weights to the distributional distance contributed by each co-occurrence.
The idea is that some co-occurrences may be better indicators of semantic
distance than others. Usually, a formulation of the strength of association of the co-occurring
word with the target words is used as weight, the hypothesis being that
a strong co-occurrence is likely to be strong indicator of semantic distance.
Weighting the primary compositional measures results in some of the existing
measures. For example, as pointed out earlier, the Kullback-Leibler divergence
is a weighted form of the division measure (not considering the absolute
value). Here, the conditional probability of a co-occurring word with respect
to the first word ($P(w|w_1)$) is used as the weight. Since the weight is
dependent on the first word and not the other, we have asymmetry. A more
symmetric weight could be the average of the conditional probabilities
between the co-occurring word and each of the two target words. A symmetrically
weighted division PCM ${\text{\itshape Saif}}^{\text{\itshape Div}}_{\text{\itshape AvgWt}}$
is shown below:
\begin{eqnarray}
{\text{\itshape Saif}}^{\text{\itshape Div}}_{\text{\itshape AvgWt}}(w_1,w_2)& =& \sum_{w \in C(w_1) \cup C(w_2)} \frac{1}{2}\left(P(w|w_1) + P(w|w_2)\right) \left| \log \frac{P(w|w_1)}{P(w|w_2)} \right|\nonumber\\
& &
\end{eqnarray}
\noindent We can have corresponding, symmetric weighted Jensen-Shannon divergence and
$\alpha$ skew divergence. $L_2$ norm is a weighted version of the $L_1$ norm,
the weight being: $P(w|w_1) - P(w|w_2)$.
A simple product measure with
weights is shown below:
\begin{eqnarray}
\text{\em Pdt}_{\text{\em AvgWt}}^{\text{\em Avg}}& =& \sum_{w \in C(w_1) \cup C(w_2)} \frac{1}{2}(P(w|w_1) + P(w|w_2))
\frac{P(w|w_1) \times P(w|w_2)}{(\frac{1}{2}(P(w|w_1) + P(w|w_2)))^2} \nonumber\\
\label{eq:prodWtd}
& =& \sum_{w \in C(w_1) \cup C(w_2)} \frac{P(w|w_1) \times P(w|w_2)}{\frac{1}{2}(P(w|w_1) + P(w|w_2))}
\end{eqnarray}
\textcolor{Black}{A better weight (which is also symmetric) may be chosen given
the following hypothesis:}
\begin{quote}
\textcolor{Black}{The stronger the association of a co-occurring word with a target word,
the better indicator of semantic properties of the target word it is.}
\end{quote}
\noindent \textcolor{Black}{The co-occurring word is likely to have different
strengths of associations with the two target words. Taking the maximum
of the two as the weight (\inlinecite{DaganMM95}) will mean that more weight is
given to a co-occurring word if it has high strength of association with
any of the two target words. As \inlinecite{DaganMM95} point out, there is strong evidence for dissimilarity
if the strength of association with the other target word is much lower than the maximum,
and strong evidence of similarity if the strength of association with both
target words is more or less the same. Equation~(\ref{eq:SaifWtdDiv}) is a
weighted division PCM that captures this intuition.}
\begin{eqnarray}
\label{eq:SaifWtdDiv}
& &{\text{\itshape Saif}}^{\text{\itshape Div}}_{\text{\itshape MaxWt}}(w_1,w_2)\nonumber\\
& &= \sum_{w \in C(w_1) \cup C(w_2)}
\frac{\max \left(P(w|w_1),P(w|w_2)\right)}
{\sum_{w' \in C(w_1) \cup C(w_2)} \max \left(P(w'|w_1),P(w'|w_2)\right)}
\left| \log \frac{P(w|w_1)}{P(w|w_2)} \right|\nonumber\\
& &
\end{eqnarray}
\noindent \textcolor{Black}{Similarly weighted product and difference measures may
be created. Both ${\text{\itshape Saif}}^{\text{\itshape Div}}_{\text{\itshape MaxWt}}$ and
${\text{\itshape Saif}}^{\text{\itshape Div}}_{\text{\itshape AvgWt}}$
give distributional distance scores from 0 (maximally similar/related) to infinity (completely dissimilar/unrelated).}
It would be interesting to note the effect of weighting on these measures
and also to determine which weight factor is more suitable.
\subsection{Measure of Association}
As mentioned earlier, distributional measures use
the disparity in association of the target words with their co-occurring words
to determine relatedness. ~\inlinecite{Lin98C} and ~\inlinecite{Hindle90} use
pointwise mutual information as the measure of association. The mutual information--based
CRMs of \inlinecite{Weeds03}
also use the same. All other measures studied in this paper
use simple conditional probability of the co-occurring words, given the
target word. It should be noted that replacing the strength of association
in a measure with another can result in a different distributional measure.
For example, the mutual information--based spatial and
fuzzy metrics discussed earlier. Lin's measure (\ref{eq:LinCorpus}) using conditional probability
(CP) is shown below:
\begin{equation}
\text{\itshape Lin}^{\text{\itshape CP}}(w_1,w_2) = \frac{\sum_{(r,w)\, \in\, T(w_1)\, \cap\, T(w_2)} (P(w|w_1) + P(w|w_2))}
{{\sum_{(r,w')\, \in\, T(w_1)} P(w'|w_1) + \sum_{(r,w')\, \in\, T(w_2)} P(w'|w_2)}}
\end{equation}
Of course, in case of certain measures, for example the division-based
primary compositional measures, use of pointwise mutual information
and conditional probability is equivalent.
\begin{eqnarray}
{\text{\itshape Div}}^{{\text{\itshape MI}}}(w_1,w_2)& =& \sum_{w \in C(w_1) \cup C(w_2)} \left| \log \frac{\frac{P(w,w_1)}{P(w)P(w_1)}}{\frac{P(w,w_2)}{P(w)P(w_2)}} \right| \\
& =& \sum_{w \in C(w_1) \cup C(w_2)} \left| \log \frac{P(w|w_1)}{P(w|w_2)} \right| \\
& =& Div(w_1,w_2)
\end{eqnarray}
\inlinecite{Weeds03} shows that her mutual information--based CRMs exhibit higher correlation
with human judgment on the Miller and Charles word pairs compared
to the ones that use conditional probability. It remains to be seen
if other measures follow the same pattern.
\subsection{Predictors of Semantic Relatedness}
Given a pair of target words, the vocabulary may be divided into
three sets: (1) the set of words that co-occur with both target
words (common); (2) words that co-occur with exactly one of the two target
words (exclusive); (3) words that do not co-occur with either of the two
target words.
~\inlinecite{Hindle90}
uses evidence only from words that co-occur with both target words
to determine the distributional similarity. All the other measures discussed in this paper
so far, use words that co-occur with just one target word, as well.
One can argue that the more there are common co-occurrences between
two words, the more they are related. For example, {\em drink} and
{\em sip} may be considered related as they have a number of common
co-occurrences such as {\em water, tea} and so on. Similarly,
{\em drink} and {\em chess} can be deemed unrelated as words that
co-occur with one, do not with the other. For example, {\em water}
and {\em tea} do not usually co-occur with {\em chess}, while {\em
castle} and {\em move} are not found close to {\em drink}. Measures
that use all co-occurrences (common and exclusive) tap into
this intuitive notion.
However, certain strong exclusive co-occurrences can adversely
effect the measure.
\textcolor{Black}{Consider the classic {\em strong tea}
vs {\em powerful tea} example (\inlinecite{Halliday66}).} The words {\em
strong} and {\em powerful} are semantically very related.
However, the word {\em coffee} is likely to
co-occur with {\em strong} but not with {\em powerful}. Further,
{\em strong} and {\em coffee} can be expected to have a large
value of association as given by a suitable measure, say PMI.
This large PMI value, if used in the distributional relatedness formula, can
greatly reduce the final value. Thus it is not clear if
the benefit of using all co-occurrences is outweighed by the
drawback pointed out.
A further advantage of using only common co-occurrences is that
the Kullback-Leibler divergence can now be used without the
need of smoothed probabilities.
\begin{equation}
\text{\em KLD}_{\text{\em Com}}(w_1,w_2) = \sum_{w \in C(w_1) \cap C(w_2)} P(w|w_1) \log \frac{P(w|w_1)}{P(w|w_2)}
\end{equation}
\noindent Observe that we are taking the intersection of the
set of co-occurring words instead of union as in the original
formula (\ref{eq:KLD}).
\subsection{Capitalizing on Asymmetry}
Given a hypernym-hyponym pair ({\em automobile-car}, say)
asymmetric distributional measures such as the Kullback-Leibler divergence,
$\alpha$ skew divergence and the CRMs generate different
values as the distributional similarity of $w_1$ with $w_2$ as compared to $w_2$
with $w_1$. Usually, if $w_1$ is a more generic concept than $w_2$,
the measures find $w_1$ to be more distributionally similar to $w_2$ than
the other way round. \inlinecite{Weeds03} argues that this behavior is intuitive as
it is more often okay to substitute a generic concept in place of a specific one
than vice versa, and substitutability is a indicator of semantic similarity.
On the other hand, in most cases the notion of asymmetric semantic similarity
is counter-intuitive, and possibly detrimental. Further, in case
two words share a hypernym-hyponym relation, they are likely to be
highly semantically similar. Thus given two words, it may make sense to always choose the
higher of the two distributional similarity values suggested by an asymmetric measure
as the final distributional similarity between the two. This way an asymmetric measure (${\text{\em Sim}}_{\text{\em Asym}}$)
can easily be converted into a symmetric one (${\text{\em Sim}}_{\text{\em Asym}}$),
while still capitalizing
on the asymmetry to generate more suitable distributional similarity values for
hypernym-hyponym word pairs. Equation~(\ref{eq:SymMax}) states the formula for the
proposed conversion. A specific implementation on the KL divergence
formula is given in equation~(\ref{eq:KLDMax})
\begin{eqnarray}
\label{eq:SymMax}
\text{\em Sim}_{\text{\em Max}}(w_1, w_2)& =& \max (Sim_{Asym}(w_1,w_2), Sim_{Asym}(w_2,w_1)) \\
\label{eq:KLDMax}
\text{\em KLD}_{\text{\em Max}}(w_1, w_2)& =& \max (\text{\em KLD}(w_1,w_2), \text{\em KLD}(w_2,w_1))
\end{eqnarray}
Another method to convert an asymmetric measure of distributional
similarity (or relatedness) into a symmetric one is by taking the average (formula~\ref{eq:SymAvg})
of the two possible similarity values. A specific implementation on the KL divergence
formula is given in equation~(\ref{eq:KLDAvg})
\begin{eqnarray}
\label{eq:SymAvg}
& &\; \; \text{\em Sim}_{\text{\em Avg}}(w_1, w_2) = \frac{1}{2} (Sim_{Asym}(w_1,w_2) + Sim_{Asym}(w_2,w_1)) \\
\label{eq:KLDAvg}
& &\text{\em KLD}_{\text{\em Avg}}(w_1, w_2) = \frac{1}{2} (\text{\em KLD}(w_1,w_2) + \text{\em KLD}(w_2,w_1)) \\
& & = \frac{1}{2} \sum_{w \in C(w_1) \cup C(w_2)} \left(P(w|w_1) \log \frac{P(w|w_1)}{P(w|w_2)} + P(w|w_2) \log \frac{P(w|w_2)}{P(w|w_1)}\right) \\
& & = \frac{1}{2} \sum_{w \in C(w_1) \cup C(w_2)} \left(P(w|w_1) \log \frac{P(w|w_1)}{P(w|w_2)} - P(w|w_2) \log \frac{P(w|w_1)}{P(w|w_2)} \right)\\
& & = \frac{1}{2} \sum_{w \in C(w_1) \cup C(w_2)} \left(P(w|w_1) - P(w|w_2)\right) \log \frac{P(w|w_1)}{P(w|w_2)}
\end{eqnarray}
Determining the effectiveness of such conversions of existing asymmetric
measures is part of our future work.
\subsection{How CRMs Fit}
The CRMs suggested by \inlinecite{Weeds03} are the first distributional measures
to be evaluated by comparing ranked word pairs
with those ranked by humans (Miller and Charles word pairs).
At first glance the CRMs may look quite distinct from the rest
of the distributional measures studied so far, owing to their
rather complex formulae and multiple optimizing parameters.
However, setting the parameters to certain standard values
equates a few of the CRMs to other measures. The difference-weighted
token-based CRM suggested by Weeds has identical values for precision and recall.
She proves that the precision (or recall) is inversely
related to the $L_1$ norm measure. This seemingly odd result of equating a distributional distance
measure with a precision (or recall) value makes sense due to the following ---
as substitutability is defined as a measure of distributional similarity,
metrics such as precision and recall which quantify how good the substitution is,
reflect the distributional similarity and are inversely related to distributional
distance.
Thus setting $\gamma = 0$ and $\beta = 1$ or $0$,
causes the CRM to behave like the $L_1$ norm.
Further, as
shown below, setting $\gamma = 1$ (in other words, taking
the $F$ measure) makes the difference-weighted
mutual information--based CRM identical to the mutual information--based
Dice coefficient (\ref{eq:DiceMI}). Following\endnotemark[\value{endnote}]
is a proof of the same.
The precision and recall of the difference-weighted MI-based CRMs are
repeated here (equations (\ref{eq:PdwMI2}) and (\ref{eq:RdwMI2})) for convenience.
\begin{eqnarray}
\label{eq:PdwMI2}
P_{dw}^{MI}(w_1,w_2) = \frac{\sum_{w \in C(w_1) \cap C(w_2)} \min(I(w,w_1),I(w,w_2))}{\sum_{w \in C(w_1)} I(w,w_1)} \\
\label{eq:RdwMI2}
R_{dw}^{MI}(w_1,w_2) = \frac{\sum_{w \in C(w_1) \cap C(w_2)} \min(I(w,w_1),I(w,w_2))}{\sum_{w \in C(w_2)} I(w,w_2)}
\end{eqnarray}
\pagebreak
\begin{theorem}
\textcolor{Black}{The difference-weighted mutual information--based CRM equates to the mutual information--based
Dice coefficient if its parameter $\gamma$ is set to $1$.
}
\end{theorem}
\begin{pf*}{Proof}
\begin{eqnarray*}
Sim_{dw}^{MI}(w_1,w_2)& =& \gamma \Biggl[ \frac{2 \times P \times R}{P + R} \Biggr] + (1 - \gamma) \Biggl[ \beta [ P ] + (1 - \beta) [R] \Biggr] \\
& & \\
& =& 1 \Biggl[ \frac{2 \times P \times R}{P + R} \Biggr] + (1 - 1) \Biggl[ \beta [ P ] + (1 - \beta) [R] \Biggr] \\
& & \\
& =& \frac{2 \times P \times R}{P + R}
\end{eqnarray*}
On substituting values for $P$ and $R$ from equations~(\ref{eq:PdwMI2}) and (\ref{eq:RdwMI2}):
\begin{eqnarray*}
& &Sim_{dw}^{MI}(w_1,w_2)\\
& & \\
& &= \frac{2 \times \left( \frac{\sum_{w \in C(w_1) \cap C(w_2)} \min(I(w,w_1),I(w,w_2))}{\sum_{w \in C(w_1)} I(w,w_1)}\right) \times \left( \frac{\sum_{w \in C(w_1) \cap C(w_2)} \min(I(w,w_1),I(w,w_2))}{\sum_{w \in C(w_2)} I(w,w_2)}\right)}{\left(\frac{\sum_{w \in C(w_1) \cap C(w_2)} \min(I(w,w_1),I(w,w_2))}{\sum_{w \in C(w_1)} I(w,w_1)}\right) + \left( \frac{\sum_{w \in C(w_1) \cap C(w_2)} \min(I(w,w_1),I(w,w_2))}{\sum_{w \in C(w_2)} I(w,w_2)}\right)} \\
& & \\
& &= \frac{2 \times \left( \frac{\left(\sum_{w \in C(w_1) \cap C(w_2)} \min(I(w,w_1),I(w,w_2)) \right)^2} {\left(\sum_{w \in C(w_1)} I(w,w_1)\right)\left(\sum_{w \in C(w_2)} I(w,w_2)\right)} \right)} {\frac{\left( \sum_{w \in C(w_1) \cap C(w_2)} \min(I(w,w_1),I(w,w_2)) \right) \left( \sum_{w \in C(w_1)} I(w,w_1) + \sum_{w \in C(w_2)} I(w,w_2) \right)} {\left(\sum_{w \in C(w_1)} I(w,w_1)\right)\left(\sum_{w \in C(w_2)} I(w,w_2)\right)}} \\
& & \\
& &= \frac{2 \times \sum_{w \in C(w_1) \cap C(w_2)} \min(I(w,w_1),I(w,w_2)} {\sum_{w \in C(w_1)} I(w,w_1) + \sum_{w \in C(w_2)} I(w,w_2)} \\
& & \\
& &= \text{\itshape Dice}^{\text{\em MI}}(w_1,w_2)
\end{eqnarray*}
\qed
\end{pf*}
\subsection{Hit and Miss Co-occurrences}
\textcolor{Black}{Lastly, we examine two kinds of co-occurrences that pose a challenge
to existing distributional measures: (1) Word pairs that occur together
less number of times than what would be expected by chance.
Measures like PMI cannot predict their association values with confidence
and as pointed out earlier this is countered by ignoring them completely.
This means that the system misses out on evidence from
this set of co-occurrence pairs. (2) Co-occurrence pairs formed by a
word with target words that are near synonyms.
\inlinecite{DianaH02}
point out that
near synonyms (for example, {\itshape hidden} and {\itshape concealed})
may form strong and anti-collocations, respectively, with the same co-occurring word
(for example,
{\itshape agenda}). All distributional measures that use strength of
association to determine semantic relatedness will consider the large
discrepancy in strength of association as evidence of unrelatedness.
Therefore, these co-occurrence
pairs, which are not ignored (unlike the previous ones),
will negatively impact the ability of distributional measures to predict semantic
relatedness of near synonyms.
It should be noted that we cannot eliminate such co-occurrences in a
straightforward manner simply because we are not aware apriori if the
target words are near synonyms.
It would be interesting to determine
the precise quantitative effect of such co-occurrences on the performance
of distributional measures.
}
\subsection{Summarizing the Distributional Measures}
In the last two sections we have seen numerous distributional measures.
Tables~\ref{tab:distributional measures of distance - I}, \ref{tab:distributional measures of distance - II},
\ref{tab:distributional measures of similarity - I}, and \ref{tab:distributional measures of similarity - II}
listed in the appendix
summarize their properties.
\section{Semantic Network and Ontology-Based Measures}
Creation of electronically available ontologies and semantic
networks like WordNet has allowed their use to help solve numerous
natural language problems including the measurement of semantic
distance between two words.
\inlinecite{Budanitsky99},
\inlinecite{BudanitskyH00} and \inlinecite{PatwardhanBT03}
have done an extensive survey of the various WordNet-based measures,
their
comparisons with human judgment on selected word pairs, and
their efficacy in applications such as spelling correction
and word sense disambiguation. Hence, this paper provides
just a brief summary of the major WordNet-based measures
of similarity and focuses on their comparison with
distributional ones.
One of the earliest and simplest measures is the \inlinecite{RadaMBB89}
{\bf edge counting} method. The shortest path in the network
between the two target words ({\bf target path}) is determined.
The more edges there are
between two words, the more distant they are. Elegant as it
may be, the measure
relies on the unlikely assumption that all the network edges
correspond to identical semantic distance between the nodes
they connect. Nodes in a network may be connected by numerous
relations such as hyponymy, meronymy and so on.
Edge counts apart, \inlinecite{HirstS98} take into account
the fact that if the target path consists of edges that belong
to a number of such relations, the target words are likely
more distant. The idea is that if we start from a particular node
({\bf base word})
and take a path via a particular relation (say, hyponymy),
to a certain extent the words reached will be quite related
to the base word. However, if during the way we take edges
belonging to different relations (other than hyponymy), very
soon we may reach words that are unrelated. Hirst and St-Onge's
measure of semantic relatedness is listed below:
\begin{equation}
\text{\itshape HS}(c_1,c_2) = C - \text{\itshape path length} - k \times d
\end{equation}
\noindent where $c_1$ and $c_2$ are the target concepts/words.
And, $d$ is the number of times an edge corresponding to a different
relation than that of the preceding edge is taken.
$C$ and $k$ are empirically determined constants.
\inlinecite{LeacockC98}
used just one relation (hyponymy) and modified the path length
formula to reflect the fact that edges lower down in the {\em is-a}
hierarchy correspond to smaller semantic distance than the ones
higher up. For example, {\em sports car} and {\em car} (low in the hierarchy)
are much
more similar than {\em transport} and {\em instrumentation} (higher
up in the hierarchy) even
though both pairs of words are separated by exactly one edge in the
{\em is-a} hierarchy of WordNet.
\begin{equation}
{\text{\itshape LC}}(c_1,c_2) = -\log \frac{{\text{\itshape len}}(c_1,c_2)}{2D}
\end{equation}
\noindent where $D$ is the depth in the taxonomy.
\inlinecite{Resnik95} suggested a measure that used corpus statistics
along with the knowledge obtained from a semantic network.
The measure is based on the notion that the semantic similarity of two
words may be determined from the word that represents their
similarity (the {\bf lowest common subsumer} or {\bf lowest super-ordinate (lso)}).
The more the information
contained in this node, the more similar the two words are.
Observe that using information content (IC) has the effect of inherently
scaling the semantic similarity measure by depth of the taxonomy. Usually, the lower
the lowest super-ordinate, the lower is the probability of occurrence of
the lso and the concepts subsumed by it, and hence, the higher is its
information content.
\begin{equation}
{\text{\itshape Res}}(c_1,c_2) = -\log {\text{\itshape p}}(lso(c_1,c_2))
\end{equation}
\noindent As per the formula, given a particular lowest super-ordinate, the exact positions
of the target words below it in the hierarchy do not have any effect
on the semantic similarity. Intuitively, we would expect that word pairs closer
to the lso are more similar than those that are distant. \inlinecite{JiangC97} and \inlinecite{Lin97}
incorporate this notion into their measures which
are arithmetic variations of the same terms.
\textcolor{Black}{The \inlinecite{JiangC97} measure (denoted by {\itshape JC}\/) determines how dissimilar each target concept
is from the lso ($IC(c_1) - IC(lso)$ and $IC(c_2) - IC(lso)$). The final semantic
distance between the two concepts is
then taken to be the sum of these differences (see~\inlinecite{Budanitsky99} for more
details).
\inlinecite{Lin97} points out that the lso is what is common between the two
target concepts and that its information content is the
common information between the two concepts. Lin's formula (denoted by {\itshape Lin}\/) can thus be thought of
as taking the Dice coefficient of the information in the two concepts.
}
\begin{eqnarray}
{\text{\itshape JC}}(c_1,c_2)& =& 2\log p({\text{\itshape lso}}(c_1,c_2))
- (\log(p(c_1)) + (\log(p(c_2))) \\
{\text{\itshape Lin}}(c_1,c_2)& =& \frac{2 \times \log p({\text{\itshape lso}}(c_1,c_2))}
{\log(p(c_1)) + (\log(p(c_2))}
\end{eqnarray}
\noindent \inlinecite{BudanitskyH00} show that the Jiang-Conrath measure
has the highest correlation (0.850) with the Miller and Charles word pairs
and performs better than all other measures considered in a spelling correction task. \inlinecite{PatwardhanBT03}
get similar results using the measure for word sense disambiguation (especially of nouns).
\section{Comparison of Distributional and Ontology-Based Measures}
Distributional and ontology-based measures use distinct sources
of knowledge to achieve the same goal --- the ability to mimic
human judgment of semantic relatedness. Owing to the difference in
methodology, many interesting comparisons may be made. The
next few subsections aim at bringing them to light.
\subsection{Knowledge Source versus Similarity Measure}
Ontologies are much more expensive resources than raw data, which
is freely available. Creating an ontology requires human experts,
is time intensive and rather brittle to changes in language.
Once created, updating an ontology is again expensive and there
is usually a lag between the current state of language usage/comprehension
and the semantic network representing it. Further, the complexity
of human languages makes creation of even a near perfect semantic
network of its concepts impossible. Thus in many ways the
ontology-based measures are as good as the networks on which
they are based. On the other hand, large corpora, trillions of
words in size, may now be collected by a simple web crawler.
Large corpora of more formal writing are also available (for
example, the {\em Wall Street Journal} or the {\em American Printing
House for the Blind (APHB)} corpus). Therefore,
using an appropriate distributional measure that best captures
the semantic similarity--predicting information, plays a much more vital role
in case of distributional measures.
As ontologies are a rich source of information where the various
concepts are linked together by powerful relations such as
hyponymy and meronymy, the ontological measures likely
correctly identify target words related by edges
that belong to just one relation as very similar. However,
data sparseness may force distributional measures to assign
low similarity values to clearly related word pairs. Assigning
appropriate semantic similarity values when target words are connected by
different relational edges poses a major challenge to
ontological measures.
\subsection{Domain-Specific Semantic Similarity}
So far, this paper has talked about {\bf universal similarity measures}.
Given a word pair, the measures each give just one similarity value.
However, two words may be very semantically similar in a certain domain but
not so much in another. For example, the word pair {\em space} and {\em time}
are closely related in the domain of quantum mechanics but not so much
in most others. Ontologies have been made for specific domains, which
may be used to determine semantic similarity specific to these domains. However,
the number of such ontologies is very limited. On the other hand, large amounts of
corpora specific to particular domains are much easier to collect,
allowing a widespread use of distributional {\em domain-specific} similarity.
\subsection{Associated Words}
Certain word pairs have a special relation with each other. For example, {\em
strawberry} and {\em cream}, {\em doctor} and {\em scalpel}, and so on.
These words are not similar physically or in
properties, but {\em strawberries} are usually eaten with {\em cream} and
a {\em doctor} uses a {\em scalpel} to make an incision. An ontology-based
measure will correctly identify the amount of
semantic relatedness only if such relations are inherent in the ontology.
For example, if the agent-instrument relation does not link
concepts in a semantic network (as in WordNet), the ontology-based
measures will not identify {\em doctor} and
{\em scalpel} as related.
Of the various distributional measures discussed, the ones
that use simple co-occurrences capture such
semantic relatedness, as words that tend to occur together are likely
to have large set of common co-occurring words. Measures (e.g., \inlinecite{Lin98C}, \inlinecite{Hindle90})
that consider a word $w$ to be a shared co-occurrence only if $w$
is related to both target words by the same syntactic relation,
will not find such words related, simply because such words
that tend to occur in the same sentence are likely to have
different thematic roles and thus different syntactic relations with
common co-occurring words.
\subsection{Multi-faceted Concepts}
The various senses of a word represent distinct concepts. Each of
these concepts can usually be described by a number of attributes
or features. These attributes may be physical descriptions like
color, shape and composition or function, purpose and role.
Two words are adjudged similar if they share a number of such
attributes and if the strength of the shared attributes is high.
By strength we mean
how strongly an attribute helps define the words. The more
prominent a shared feature, the more similar the two words are.
Further, it is possible that words $w_1$ and $w_2$ are related
as they share a certain set of attributes, while $w_2$ and $w_3$
are related because they share a different set of attributes.
Thus $w_1$ and $w_3$ are likely not as related as $w_1$ and $w_2$,
or $w_2$ and $w_3$. For example, the physical {\em key} is closely related
to the abstract {\em password}, as they are both {\em means of getting access}.
{\em Password} is closely related to {\em encryption} as they
both pertain to {\em data security}.
However, the physical {\em key} has little to do with {\em encryption}
and the two are not so much related. Thus semantic relatedness is not
necessarily transitive and may be a function of a subset of relevant
attributes, not necessarily all.
Hierarchies in an ontology are built by repetitive division of
concepts as per their attributes. The order in which these
attributes are used to create the tree structure can result
in dramatically different hierarchies. For example, consider
a scenario depicted in figure \ref{fig:hierarchy}, where the attributes
$a_1$ and $a_2$ are used in
different orders to create different hierarchies of the words
$w_1, w_2, w_3$ and $w_4$. Notice that while $w_1$ and
$w_2$ are closer to each other than $w_1$ and $w_3$ in
hierarchy-1, it is the other way round in hierarchy-2.
Thus variations in the order of use of attributes for
creating the hierarchy can result in different sets of
words being close to each other.
It should be noted that
real-world semantic hierarchies are created by well formed
methodologies and hence the order of attributes used
to create the hierarchy is not arbitrary. That said, there
is room for variation and further, once a particular
hierarchy is chosen, it captures certain semantic relations
in its structure, while others are lost.
\begin{figure}
\centerline{\includegraphics[height=4.5cm, width=12cm]{hierarchy.eps}}
\caption{Hierarchy Variations.}
\label{fig:hierarchy}
\end{figure}
In general, ontology-based measures of similarity
capitalize on the property that words that occur close to
each other in a hierarchy share a lot of attributes and are
therefore similar.
However, they usually rely on a fixed hierarchy.
Word pairs that would be closer in variations of the
hierarchy are not considered. Thus ontology-based measures
are likely to wrongly assign a low semantic similarity value to
such word pairs. For example, consider {\em key} and
{\em password}. They are both {\em means to gain access}
but the is-a hierarchy in WordNet lists them in completely
different branches of the network (figure~\ref{fig:WNet}).
The attribute determining whether the word refers to a
physical entity or an abstraction is used first to classify
the words and hence {\em key} and {\em password} fall into
different branches at the top of the hierarchy itself.
Thus an ontology-based measure is likely to find them unrelated.
Distributional measures are not bound by a fixed hierarchy
and have a better chance at appropriately identifying the
semantic similarity of such word pairs. It would be worth determining
the extent to which this is true.
\begin{figure}[htbp]
\begin{center}
\includegraphics[height=5cm, width=10cm]{wordnet.eps}
\end{center}
\caption{{\em key} and {\em password} in the `is-a' hierarchy of WordNet}
\label{fig:WNet}
\end{figure}
\subsection{Evaluation and Complementarity}
Ontology-based and distributional measures of similarity
have each been individually shown to be reasonable quantifiers of
semantic similarity. WordNet-based
measures have been used for applications such as spelling correction
and word sense disambiguation, while distributional measures have
primarily been used for estimating probabilities of unseen bigrams.
Exhaustive comparisons of
WordNet-based measures with each other (e.g., \inlinecite{Budanitsky99},
\inlinecite{BudanitskyH00} and \inlinecite{PatwardhanBT03}) have found that
the Jiang-Conrath measure performs better than the rest.
\inlinecite{DaganPL94} perform experiments with a few relative entropy--based
measures and find that Jensen-Shannon divergence
is slightly better than Kullback-Leibler divergence and $L_1$ norm in
estimating bigram probabilities of unseen words and in a pseudo--word
sense disambiguation experiment.
However, the various distributional measures have not been used
to rank the Miller-Charles or Rubenstein-Goodenough word pairs,
for which estimates of human judgment of semantic relatedness are available.
Experiments to this end will also enable a comparison of the
distributional measures with the ontology-based measures
for which this data is already available.
Similar to the case of various ontological
measures, it is worth determining which distributional measure
is closest to human notion of semantic similarity
and how well the distributional measures, which rely just
on raw data, fare against, the more expensive and knowledge rich, ontology-based
measures.
Since the two kinds of measures rely on different knowledge
sources, there is a likelihood that distributional measures
more accurately identify the semantic similarity of a certain subset of word pairs,
while the ontological methods do so for a different subset. One of
the more important problems in the field is to quantify this
complementarity. It should be noted that since a similarity measure is
evaluated by comparison of ranked word
pairs and not by the similarity values alone, capitalizing on the
complementarity by creating a combined semantic similarity predictor is a
much harder problem.
\section{Conclusions}
The paper has provided a detailed analysis of various corpus-based
distributional measures and compared them with
measures based on ontologies and semantic networks. Merits
and limitations of the various measures were listed. New measures
that are likely to overcome the drawbacks of present distributional
measures were suggested. Specifically, a distributional measure that
keeps the best of \inlinecite{Hindle90} and \inlinecite{Lin98C}, overcoming
their respective drawbacks, was proposed. Variations of
Kullback-Leibler divergence and Jensen-Shannon divergence that better
capture the disparity in co-occurrence probabilities were
suggested. A simple technique to convert asymmetric measures into
symmetric ones was suggested. Novel approaches are described to determine
distributional similarity by better utilization of co-occurring words related by different
syntactic relations.
The paper identified significant research problems that need to
be answered through experimentation. This will help better understand how statistics
from raw data may be manipulated to determine appropriate
similarity values between two words. For example, whether the use of syntactically
related, rather than plain co-occurrences, significantly improves the
measure? Or, are simple co-occurrences just as useful?
What kinds of co-occurrences (common, or exclusive, as well)
should be used to determine distributional relatedness?
Is pointwise mutual information or conditional probability
a more suitable measure of association to be used in the various distributional
measures?
Do compositional or non-compositional distributional measures
produce more intuitive semantic similarity
values?
Which mathematical operation (difference, division, or
product) of the co-occurrence distributions yields
values that are closest to human judgment, in case of compositional
measures? A direct evaluation of the distributional measures (other than
$L_1$ norm, $\alpha$ skew divergence and the CRMs for
which these results exist) by their
correlation with the Miller-Charles and Rubenstein-Goodenough word
pairs will provide better insight into their relative abilities
and will enable a comparison with WordNet-based measures for which the
correlation coefficients are already available.
Lastly, the paper pointed out that even though ontological measures
are likely to perform better as they rely on a much richer knowledge
source, distributional measures have certain distinct advantages.
For example, they can easily provide domain-specific similarity values
for a large number of domains, their ability to determine similarity
of contextually associated word pairs more appropriately,
and the flexibility to identify multi-faceted concepts as
related from appropriate commonalities that may not be
explicitly encoded in a semantic network. Thus it is very
likely that ontological measures are better at predicting
semantic similarity for certain word pairs, while the distributional
measures do so for others. To identify the extent of this
complementarity and a suitable combined
methodology to assign semantic similarity, remain significant problems
in the field.
A significant challenge in achieving this is how to reconcile
the nature of the two kinds of measures --- while ontological
measures predict the semantic similarity of two concepts (or word senses),
distributional measures do so for two words. One of the problems
we intend to pursue is the development of a methodology that enables
the use of distributional measures to predict semantic similarity
of concepts, with no or little sense-tagged data.
|
{
"timestamp": "2012-03-09T02:04:26",
"yymm": "1203",
"arxiv_id": "1203.1889",
"language": "en",
"url": "https://arxiv.org/abs/1203.1889"
}
|
\section{Introduction}
The kinematic characterization of high-z galaxy populations is a key observational element to distinguish between different galaxy evolutionary scenarios, since it may help to find out the fraction of rotating disks and mergers at different cosmic epochs. This is a key observational input to constrain the relative role of major mergers vs. steady cool gas accretion in shaping galaxies, which still is under discussion (e.g. \citealt{genzel01, T08, dekel09, FS09, lemoine09, lemoine10, Bou11, Epi11}).
With the aid of integral field spectroscopy (IFS) we can resolve the kinematic status and internal processes at work within galaxies and better understand the role of the dominant mechanisms involved at early epochs. For instance, \cite{FS09} classified the SINS sample (i.e., 63 galaxies with $1.3< z<2.6$) concluding that these objects can be classified as: `rotation-dominated', interacting or merging systems and `dispersion-dominated' (i.e., dominated by large amounts of random motions). They found these types in a similar fraction (i.e., 1/3 each). They found a small fraction of galaxies showing signs of major merger events (i.e., \citealt{FS06, Law09, genzel06, genzel08}) therefore suggesting that the majority of the star forming galaxies at these redshifts is fed by gas via continuous cold flows along streams (including minor mergers).
At intermediate redshift (z $\sim$ 0.6), spatially resolved kinematics has revealed a large fraction of chaotic velocity fields, supporting that most local spirals were rebuilt after a major merger since z = 1 (\citealt{hammer05, puech10}). At similar redshifts, \cite{Flores06} and \cite{Yang08} investigated the velocity fields of galaxies with IFS, and derived that only 35\% of their sources can be classified as rotating disks while the rest show perturbed or complex kinematics.
Some local galaxy populations are particularly relevant for the study of galaxy evolution. This is the case of nearby Luminous and Ultraluminous Infrared Galaxies (LIRGs, L$_{IR}$ = [8-1000 $\mu$m] =10$^{11}$ - 10$^{12}$ L$_\odot$, and ULIRGs, L$_{IR} >$ 10$^{12}$ L$_\odot$, respectively). Despite these objects are relatively rare in the local universe, they are much more numerous at high-z and are responsible of a significant fraction of the whole past star formation beyond z $\sim$ 1 (e.g., \citealt{LF05, PG05, PPG08}). These objects exhibit a large range of interacting/merging properties, from mostly isolated discs for low luminosity LIRGs (e.g. log L$_{IR}$ $\leq$ 11.3, \citealt{A04}) to a majority of merger remnants for ULIRGs (e.g. log L$_{IR}$ $\geq$ 12, \citealt{V02}). Local (U)LIRGs were initially considered as the local counterpart of high-z (U)LIRGs like those discovered by Spitzer and the more luminous submillimiter galaxies (SMGs; e.g. \citealt{B02, T06}). The study of local (U)LIRGs is also important to better understand the star formation history in the universe and it can provide an opportunity to correlate the properties of local and high-z populations. Moreover, large scale turbulences, tidal tails and outflows can be studied in detail allowing a good characterization of these objects. However, there are some discussion about the similarities between local and distant (U)LIRGs (e.g., \citealt{RJ11} and references therein).
Recently several authors have analyzed the velocity fields and velocity dispersion maps of different galaxy samples with the aim of discerning merging and non-merging systems based on their kinematic properties using the $kinemetry$ methodology (\citealt{K06}). \cite{jesseit07} and \cite{kron07} have investigated the distortions in the velocity fields of simulated interacting $disc/merger$ galaxies at different redshifts in the range $z = 0-1$, while \cite{S08} (hereafter, S08) provide a straightforward mean of classifying $z\sim2$ galaxies, observed with SINFONI/VLT, as $disks$ or $mergers$ based on the asymmetries of their velocity field and velocity dispersion maps. They compare and distinguish two kinematic classes characterized by recent major merger events ({i.e., \it mergers}) and those without signs of interacting or recent merger activity (i.e., {\it disks}).
Similar works based on IFS data (i.e., OSIRIS/Keck, \citealt{BZ09, Gon10}) are focused on nearby objects (i.e., supercompact UV-Luminous Galaxies (ScUVLGs) and Lyman Break Analogs (LBAs)). The morphology and kinematics of LBAs projected at high redshift show similarities with compact and dispersion-dominated $z\sim2$ galaxies (\citealt{BZ09}). This suggests that galaxy interaction and/or mergers could also be driving the dynamics of dispersion-dominated $z\sim2$ galaxies.
In this paper we present new spatially resolved kinematics of four local LIRGs obtained with the VLT/VIMOS IFU as part of a larger project to characterize the properties of (U)LIRGs on the basis of optical and infrared Integral Field Spectroscopy (\citealt{Co05, A08, MI10, RZ11} and references therein). The fact that two of our LIRGs are post-coalescence mergers will offer the opportunity to explore the potential of the $kinemetry$ method when analyzing the velocity fields and velocity dispersion maps in moderately disturbed and partially relaxed systems. The paper is structured as follows. In Section 2, we present the sample selection as well as some details about the observations, data reductions, line fitting and velocity map construction. Section 3 is devoted to the description of the main kinematic properties of the individual objects. In Section 4, the $kinemetry$ analysis and its potential to distinguish $discs$ and $mergers$ are presented and discussed with the aid of different asymmetry planes. Finally, the main results and conclusions are summarized in Section 5.
Throughout the paper we will consider H$_0$ = 70 km/s/Mpc, $\Omega_M$ = 0.3 and $\Omega_\Lambda$ = 0.7.
\section{The VIMOS sample, observations, data reduction, line fitting and map construction}
\subsection{The sample and morphological classification}
For the present analysis we have selected 4 LIRG galaxies at a similar distance ($\sim$ 70 Mpc): two of them (i.e., IRAS F11255-4120, IRAS F10567-4310) are morphologically classified as {\it class 0} objects and two (i.e., IRAS F04315-0840, IRAS F21453-3511) as {\it class 2} objects according to \cite{A08} and \cite{RZ10} (hereafter, Paper I and Paper III respectively). In this classification scheme {\it class 0} is defined as an object with relatively symmetric morphology, appearing to be isolated with no evidence for strong past or ongoing interaction (hereafter, $disks$), while {\it class 2} is an object with a morphology suggesting a post-coalescence merging phase, with a single nucleus or two distinct nuclei at a projected distance D $<$ 1.5 kpc (hereafter, {\it post-coalescence mergers}). We have chosen these four systems at the same distance as they will allow us to discuss their relative kinematic properties avoiding linear resolution dependency effects. In table \ref{table_sample} we present the main properties of the sample: note that the systems classified as post-coalescence mergers have higher L$_{IR}$ than the disks.
\begin{table*}
\centering
\caption{General properties of the LIRGs sub-sample.}
\label{table_sample}
\begin{scriptsize}
\begin{tabular}{c c c c c c c c c c c} \\
\hline\hline\noalign{\smallskip}
ID1 & ID2 & $\alpha$ & $\delta$ & $z$ & D& scale & log L$_{IR}$ & Class & References \\
IRAS & Other & (J2000)& (J2000) & & (Mpc) & (pc/arcsec) & (L$_{\odot}$) & & \\
(1) & (2) & (3)& (4) &(5)&(6) & (7) & (8) & (9) & (10) \\
\hline\noalign{\smallskip}
F11255-4120 & ESO 319-G022 & 11:27:54.1 & -41:36:52 & 0.016351 & 70.9 & 333 & 11.04 & 0 & 1 \\
F10567-4310 & ESO 264-G057 & 10:59:01.8 & -43:26:26 & 0.017199 & 74.6 & 350 & 11.07 & 0 & 1 \\
F04315-0840 & NGC 1614 & 04:33:59.8 & -08:34:44 & 0.015983 & 69.1 & 325 & 11.69 & 2 & 1, 2 \\
F21453-3511 & NGC 7130 & 21:48:19.5 & -34:57:05 & 0.016151 & 70.0 & 329 & 11.41 & 2 & 1, 2 \\
\hline\hline
\end{tabular}
\end{scriptsize}
\begin{minipage}[h]{16.8cm}
\vskip0.1cm\hskip0.0cm
\footnotesize
\tablefoot{ Col (1): object designation in the IRAS Faint source catalogue (FSC). Col (2): other name. Col (3) and (4): right ascension (hours, minutes and seconds) and declination (degrees, arcminutes and arcseconds) from the IRAS FSC. Col (5): redshift of the IRAS sources from the NASA Extragalactic Database (NED). Col (6): luminosity distances assuming a $\Lambda$DCM cosmology with H$_0$ = 70 km/s/Mpc, $\Omega_M$ = 0.3 and $\Omega_\Lambda$ = 0.7, using the Edward L. Wright Cosmology calculator. Col (7): scale. Col (8): infrared luminosity (L$_{IR}$= L(8-1000) $\mu$m), in units of solar bolometric luminosity, calculated using the fluxes in the four IRAS bands as given in \cite{sanders03} when available. Otherwise the standard prescription in \cite{SM96} with the values in the IRAS Point and Faint source catalogues was used. Col (9): Morphology class. For further information see table 1 in \cite{RZ10}.}
\tablebib{(1) \cite{V95}; (2) \cite{C03}. }
\end{minipage}
\end{table*}
\subsection{Observations}
The observations were carried out in service mode using the Integral Field Unit of VIMOS (\citealt{Lfevre03}), at the Very Large Telescope (VLT), covering the spectral range $(5250-7400)$ \AA\ with the high resolution mode `HR-orange' (grating GG435) and a mean spectral resolution of 3470. The field of view (FoV) in this configuration is 27$^{\prime\prime}$ $\times$ 27$^{\prime\prime}$, with a spaxel scale of 0.67$^{\prime\prime}$ per fiber (i.e., 1600 spectra are obtained simultaneously from 40 $\times$ 40 fibers array). A square 4 pointing dithering pattern is used, with a relative offset of 2.7$^{\prime\prime}$ (i.e., 4 spaxels). The exposure time per pointing ranges from 720 to 850 seconds, such that, the total integration time per galaxy is between 2880 and 3400 seconds. For further details about the observations see Paper I.
\subsection{Data reduction}
\label{reduc}
The VIMOS data are reduced with a combination of the pipeline {\it Esorex} (version 3.5.1 and 3.6.5) included in the pipeline provided by ESO, and different customized IDL and IRAF scripts.
The basic data reduction is performed using the {\it Esorex} pipeline (bias subtraction, flat field correction, spectra tracing and extraction, correction of fiber and pixel transmission and relative flux calibration). Then four quadrants per pointing are reduced individually and combined in a single data cube associated for each pointing. After that, the four independent dithered pointings are combined together to end up with the final `super-cube', containing 44 $\times$ 44 spaxels for each object (i.e., 1936 spectra). Further details about the data reduction for the whole sample can be found in Paper I and in Paper III.
The wavelength calibration, the instrumental profile and fiber-to-fiber transmission correction are checked for the four galaxies using the [OI]$\lambda$6300.3 \AA\ sky line as in Paper I. The results is focused on the H$\alpha$ and [NII]$\lambda\lambda$6548.1, 6583.4 \AA\ emission lines, so that the sky line here considered is suitable for its proximity to these lines. In table \ref{calibration} the average values for the central wavelength $<\lambda_c>$ and (instrumental) width $<\sigma_{INS}>$, with their standard deviations, are shown for the four galaxies.
\begin{table}[h]
\centering
\caption{Results of the calibration check using the [OI]$\lambda$6300.3 \AA\ sky line. }
\label{calibration}
\begin{small}
\begin{tabular}{ l c c c }
\hline\hline\noalign{\smallskip}
IRAS Galaxy & $<\lambda_c> \pm \Delta\lambda_c$ [\AA] & $<\sigma_{INS}>\pm std$ [\AA] \\
\hline\hline\noalign{\smallskip}
F11255-4120 & 6300.42 $\pm$ 0.15 & 0.78 $\pm$ 0.09 \\
\hline\noalign{\smallskip}
F10567-4310 & 6300.17 $\pm$ 0.13 & 0.77 $\pm$ 0.09 \\
\hline\noalign{\smallskip}
F04315-0840 & 6300.24 $\pm$ 0.10 & 0.78 $\pm$ 0.08 \\
\hline\noalign{\smallskip}
F21453-3511 & 6300.27 $\pm$ 0.11 & 0.77 $\pm$ 0.06 \\
\hline
\end{tabular}
\end{small}
\begin{minipage}[h]{9cm}
\vskip0.2cm\hskip0.0cm
\footnotesize
\tablefoot{Typical values of the central wavelength and width distribution of the [OI]$\lambda$6300.3 \AA\ sky line registered in the data cubes along with their standard deviations (errors). }
\end{minipage}
\end{table}
\subsection{Line fitting and map construction}
As mentioned in the previous paragraph, we are interested in the H$\alpha$ - [NII] emission line complex analysis. The observed emission profiles of the individual spectra are fitted to a Gaussian function using an IDL routine (i.e., MPFITEXPR, implemented by C. B. Markwardt). This algorithm finds the best set of model parameters which match the data and it is able to fix the line intensity ratios and the wavelength differences according to the atomic parameters when adjusting multiple emission lines (i.e., the H$\alpha$ - [NII] complex). The MPFIT routine provides errors for the output parameters and gives an estimate of the goodness of the fit.
For each emission line we end up with the following information: central wavelength ($\lambda_c$), intrinsic width ($\sigma_{line})$ (i.e., $\sigma_{line} = \sqrt{\sigma_{obs}^2 - \sigma_{INS}^2}$) and flux intensity along with their respective errors. To obtain the radial velocity uncertainty estimates, wavelength calibration and fitting errors were combined in quadrature, giving a global wavelength error of $ \Delta\lambda_{tot} = \sqrt{\Delta\lambda_{MPFIT}^2 + \Delta\lambda_{c}^2}$. For high S/N spectra the wavelength calibration error can be considered the main source of uncertainty since the fitting errors are typically substantially smaller (and vice versa for low S/N spectra). For the fitting errors, we derive values generally smaller than 0.2 \AA\ while the wavelength calibration errors are of the order of $\sim$ 0.12 \AA\, as shown in Tab. \ref{calibration}. An average value of the $\sigma_{INS}$ of (85 $\pm$ 9) km/s is derived at the corresponding wavelength, along with a mean FWHM of (1.82 $\pm$ 0.19) \AA\ {}.
We start fitting all the lines to single Gaussian profiles since the main component usually extends over the entire galaxy. Two-Gaussian profiles (i.e., main and broad components) are needed for the inner regions of the four objects. Using a collection of procedures written in IDL code (i.e, {\bf jmaplot}, \citealt{MA04}), we generate flux intensity, velocity field and velocity dispersion maps, respectively, for the main and broad components (Figures \ref{class0_1} - \ref{class2}) along with a VIMOS continuum image. When HST images are available (i.e, NICMOS F160W (H-band) and ACS F814W (I band)) they are also shown in the panels.
\section {Kinematic properties of individual objects}
In this section we describe the global kinematic behavior as inferred from our IFS maps (i.e, flux intensity, velocity field and velocity dispersion). We will also infer some kinematic parameters (e.g., $\sigma_c$, v$_c/\sigma_c$, v$_{shear}/\Sigma$ and dynamical mass M$_{dyn}$) useful to characterize these systems.
The ratio of the maximum circular rotation velocity v$_c$ (i.e., the half of the observed peak-to-peak velocity) and the central velocity dispersion $\sigma_c$ measures the nature of the gravitational support of a system in equilibrium. A v$_c$/$\sigma_c$ $\geq$ 1 is the signature of a rotation-dominated system while a lower value (i.e., v$_c$/$\sigma_c <$ 1) means that the object is dispersion-dominated, as in the case of elliptical galaxies, where the galaxy is sustained against gravitational collapse by the pressure originated by random motions of the stars. When analyzing the v$_c/\sigma_c$ parameter it is important to realize that the presence of flows, superwinds or other merger-induced processes are factors that may lead to large v$_c$ and/or $\sigma_c$ values. Therefore we compute the circular velocity v$_c$ and the central velocity dispersion $\sigma_c$ using the component of one-Gaussian model fit (i.e., 1c) and the main (i.e., the one defining the systemic behaviour) component of the two-Gaussian model fit (i.e., 2c).
Since class 0 objects show the features of an {\it ideal rotating disk} \footnote{ We refer to {\it ideal rotating disk} as a thin disk with gas clouds kinematically characterized by: a velocity field that peaks at the galaxy major axis and goes to zero along the minor axis of each orbit, while the velocity dispersion will be constant long each orbit and will decrease between orbits of increasing major axes (see description in Sec. 4.1)} we correct their kinematic values for their respective inclination; class 2 objects show the velocity fields distorted such that we derive their kinematic values with and without correcting for the inclination.
The ratio between the velocity shear v$_{shear}$ and the global velocity dispersion in the whole galaxy $\Sigma$ has been derived too, as done in \cite{Gon10}. The v$_{shear}$ (not corrected for the inclination of the object) has been computed as the median of the 5-percentile at each end of the velocity distribution, for the v$_{max}$ and v$_{min}$ and then defined as v$_{shear}$ = $\frac{1}{2} (v_{max} - v_{min}$).
We have also used the kinematic information to derive the dynamical mass $M_{dyn}$. This mass takes into account the whole gravitational field and so dark matter, gas and stellar components are included. Spiral galaxies can be modeled using a flat component for the disk and a spheroidal component for the central concentration (i.e., a de Vaucouleurs profile and a massive halo). Assuming that the central regions are virialized, the observed velocity dispersion of the ionized gas $\sigma_c$ can provide a good estimate for the disk's dynamical mass ($M_{dyn}$, e.g., \citealt{Co05}). We use the following relation:
\begin{equation}
\hspace{1cm}M_{dyn} = m \hspace{1mm} 10^6 \hspace{1mm}R_{hm}\hspace{1mm}\sigma_c^2\hspace{1mm} M_{\odot}
\end{equation}
where $R_{hm}$ is the half-mass radius in $kpc$ and $\sigma_c$ is the central velocity dispersion in $km/s$. The $R_{hm}$ parameter is not an observable itself and therefore it cannot be measured directly, so it can be inferred from measuring the radius that encloses half luminosity (i.e., $R_{eff}$). The R$_{eff}$ were obtained from existing 2MASS near-infrared imaging. The extinction in the near-IR is a factor of 5-10 smaller than that in the optical and a proper way to minimize these effects is to measure the half-light radius in the $H$ or $K$ bands as done here. The parameter $m$ depends on the mass distribution: its value ranges from 1.4, for a King stellar mass distribution with a tidal-to-core ratio of 50, which is a good representation of elliptical galaxies (\citealt{T02}), to 1.75 for a polytropic sphere with a density index covering a range of values (\citealt{Spitzer87}), to 2.1 for a de Vaucouleurs mass distribution (\citealt{Combes95}). As in \cite{Co05} we will assume $m =1.75$. All these results are shown in the table \ref{pixel}; their inclinations are drawn from the NED site and are consistent with our own estimates from the H$\alpha$ distribution.
\subsection {\bf IRAS F11255-4120 (ESO 319-G022) }
This is a class 0 object, appearing to be a single isolated galaxy with a relatively symmetric morphology according to its DSS image. It is a barred spiral with a ring extended up to $\sim$ 4 kpc from the nuclear region, clearly detected in our H$\alpha$ image. The orientation of the bar is different in the VIMOS continuum image (P.A. $\sim$ 110$^{\circ}$) and in the H$\alpha$ image (P.A. $\sim$ 150$^{\circ}$).
For the main component, as shown in the top panel of Fig. \ref{class0_1}, the velocity dispersion ($\sigma$) map has a (almost) centrally peaked structure with values of 120 km/s (high values also in the bar structure) and a lower mean value ($\sim$ 20 km/s) is found in the ring. The location of this peak does not agree with those of the H$\alpha$ flux intensity and continuum maps with an offset of about 0.6 kpc (i.e., $\sim$ 2 arcsec). The kinematic center of the velocity field well agrees with the H$\alpha$ flux peak. The projection of the rotation axis (minor kinematic axis) seems to be shifted by about 30$^{\circ}$ with respect to the orientation of the bar in the H$\alpha$ flux intensity map and in agreement with the bar of the continuum. The observed velocity amplitude for the main component at a distance of 4 kpc from the H$_\alpha$ peak is of the order of (300 $\pm$ 8) km/s.
A small region of about $\sim$ 0.67 kpc x 0.89 kpc in the nuclear region shows traces of a second component, indicating the presence of a non-rotational mechanisms (e.g., outflow / wind). This component is blueshifted by 80 km/s with respect to the main one and shows a broad velocity dispersion $\sigma$ in the range (160 - 220) km/s and a velocity amplitude of the order of $(60 \pm 7)$ km/s.
The v$_c/\sigma_c$ and v$_{shear}/\Sigma$ have values of 2.3 and 2.7 respectively, supporting the idea that this galaxy is rotation-dominated: the kinematic properties of this galaxy seem to be consistent with its morphology (i.e., a rotating disk).
\begin{figure}
\includegraphics[width=0.48\textwidth]{images_11255.ps}
\includegraphics[width=0.48\textwidth]{spectra_11255.ps}
\caption{\small {\bf Top panel:} H$\alpha$ maps for the class 0 object IRAS F11255-4120. VIMOS continuum image; HST continuum images are not available for this galaxy. {\it Middle row:} the flux intensity, velocity dispersion $\sigma$ (km/s) and velocity field $v$ maps (km/s) for the main component. {\it On the bottom:} respective maps for the broad component. Note the different FoV between the main and the broad components. The latter has been zoomed in since it covers a small area. The flux intensity maps are represented in logarithmic scale and in arbitrary flux units. All the images are centered using the H$\alpha$ flux intensity peak and the iso-countors of the H$\alpha$ flux are overplotted. {\bf Bottom panel:} H$\alpha$-[NII] observed spectra of IRAS F11255-4120 for selected inner regions (indicated by the coordinates in the top label using the same reference system as in the top panel) where the main and broad component coexist. The red curve shows the total H$\alpha$-[NII] components as obtained from multi-components Gaussian fits. The green and blue curves represent respectively the main and broad components.}
\label{class0_1}
\end{figure}
\subsection {\bf{IRAS F10567-4310 (ESO 264-G057)} }
From the DSS image IRAS F10567-4310 is morphologically classified as class 0 object (see Paper III). The H$\alpha$ flux map inferred from our IFU data (top panel of Fig. \ref{class0}) shows a ring with a patchy distribution and its peak is in good positional agreement with the center of the of galaxy as inferred from the continuum.
The main component shows a centrally peaked velocity dispersion map with typical values of 60 - 70 km/s. Its general morphology resembles that of the H$\alpha$ flux map (i.e., regions with high H$\alpha$ surface brightness tend to have large velocity dispersion, with a mean value of 40 km/s). This component shows a disk-like regular velocity field with a clear rotation pattern, with the kinematic center in a good positional agreement with the H$\alpha$ flux peak. The observed velocity amplitude is of $(300 \pm 6)$ km/s computed at a distance of 4.7 kpc from the kinematic center.
A secondary kinematic distinct component is identified in the nuclear region over an extension of $\sim$ 1.2 kpc x 0.9 kpc; it shows a velocity amplitude of (128 $\pm$ 9) km/s and it is blue-shifted by $\sim$ 70 km/s with respect to the main one; its velocity dispersion is in the range (90 - 180) km/s.
The v$_c/\sigma_c$ parameter clearly classifies this object as rotation-dominated with v$_c/\sigma_c$ $\sim$ 4.6 and v$_{shear}/\Sigma \sim$ 3. Kinematics of this class 0 object is also consistent with its morphology.
\begin{figure}
\begin{center}
\includegraphics[width=0.48\textwidth]{images_10567.ps}
\includegraphics[width=0.48\textwidth]{spectra_10567.ps}
\caption{\small {\bf Top panel:} H$\alpha$ maps for the class 0 object IRAS F10567-4310. VIMOS continuum image; HST continuum images are not available for this galaxy. {\it Middle row:} the flux intensity, velocity dispersion $\sigma$ (km/s) and velocity field $v$ maps (km/s) for the main component. {\it On the bottom:} respective maps for the broad component. Note the different FoV between the main and the broad components. The latter has been zoomed in since it covers a small area. The flux intensity maps are represented in logarithmic scale and in arbitrary flux units. All the images are centered using the H$_{\alpha}$ flux intensity peak and the iso-countors of the H$_{\alpha}$ flux are overplotted. {\bf Bottom panel:} H$_\alpha$-[NII] observed spectra of {IRAS F10567-4310} for selected inner regions (indicated by the coordinates in the top label using the same reference system as in the top panel) where the main and broad component coexist. The red curve shows the total H$\alpha$-[NII] components as obtained from multi-components Gaussian fits. The green and blue curves represent respectively the main and broad components.}
\label{class0}
\end{center}
\end{figure}
\subsection {\bf IRAS F04315-0840 (NGC 1614)}
This is a class 2 object (Fig. \ref{class2_2}) with a relatively asymmetric morphology: the DSS image shows a tidal tail extending for 13 kpc from the nuclear region. This is a well studied, post-coalescence late merger, with a bright, spiral structure at scale of few kpc (1 - 3 kpc). The spiral structure in H$\alpha$ flux map shows a different orientation with respect to the continuum.
\begin{figure}[!h]
\begin{center}
\includegraphics[width=0.48\textwidth]{images_04315.ps}
\includegraphics[width=0.48\textwidth]{spectra_04315.ps}
\caption{\small {\bf Top panel:} H$\alpha$ maps for the class 2 object IRAS F04315-0840. VIMOS continuum image and HST continuum images (i.e., H and I bands). {\it Middle row:} flux intensity, velocity dispersion $\sigma$ (km/s) and velocity field $v$ maps (km/s) for the main component. {\it On the bottom:} respective maps for the broad component. Note the different FoV between the main and the broad components. The latter has been zoomed in since it covers a small area. The flux intensity maps are represented in logarithmic scale and in arbitrary flux units. All the images are centered using the H$\alpha$ peak and the iso-countors of the H$\alpha$ flux are overplotted. {\bf Bottom panel:} H$\alpha$-[NII] observed spectra of {IRAS F04315-0840} for selected inner regions (indicated in the top label using the same reference system as in the top panel) where the main and broad component coexist. The red curve shows the total H$\alpha$-[NII] components as obtained from multi-components Gaussian fits. The green and blue curves represent respectively the main and broad components.}
\label{class2_2}
\end{center}
\end{figure}
The velocity dispersion map of the main component has an irregular structure. Its peak (i.e., 220 km/s) is found at 2.4 kpc from the nucleus (i.e., H$\alpha$ flux peak) in the northern arm. The velocity field of the main component is somewhat distorted and chaotic with an amplitude of $(325 \pm 5)$ km/s. The projection of the rotation axis in the outer part (connecting the N-E to S-W part) is not aligned with that of the inner part (aligned as N-S direction).
A broad component is found in the inner region and covers a relatively large area of $\sim$ 2.4 kpc x 2.7 kpc. This component is blue-shifted by 50 km/s with respect to the main component. Its velocity dispersion is in the range (70 - 400) km/s with a velocity amplitude of (760 $\pm$ 13) km/s. The fact that the projection of the kinematic axis of this broad component has a shift of almost 90$^{\circ}$ with respect to the main component and the blue-shifted region of the velocity field shows the largest velocity dispersion (i.e., $\sim$ 400 km/s) supports the hypothesis of a dusty outflow, where the receding components (which are behind the disk) are obscured, making the whole profile relatively narrow with respect to the approaching component.
We derive a $v_{c}/\sigma_c$ $>$ 1 even when the inclination correction is not included (i.e., $v_{c}/\sigma_c$ $\approx$ 1.4 - 2.3): it shows the dominance of an intrinsic rotation over random motions. Therefore, this parameter would classify this object as rotation dominated.
\subsection {\bf IRAS F21453-3511 (NGC 7130)}
{IRAS F21453-3511} is a peculiar class 2 object, with traces of spiral and asymmetric morphology from the HST and DSS images. The ionized gas, as traced by the H$\alpha$ emission, is concentrated in the nuclear region and mostly in the northern spiral arm.
The asymmetric velocity dispersion map of the main component shows higher values in the northern arm and its central part (i.e., two local maxima can be revealed) with values of 80 km/s. The velocity field of the same component shows an asymmetric behavior where three main regions can be identified: the SW part having a mean value of 4900 km/s, the NW one with 4850 km/s and the E part with 4780 km/s. The NW part could have different inclination than the rest of the observed regions, so explaining this peculiar velocity structure. The kinematic center can be well identified with the H$_\alpha$ flux peak although the rotational axis it is not well defined. The photometric major axis in the NICMOS image seems to be orientated in the N-S direction and the ACS image seems to reveal a ring of knots with a major axis orientated along P.A. $\sim$ 135$^{\circ}$.
A small region of about 1.5 kpc x 1.3 kpc in the nuclear part shows a second component. The blue-shifted region of its velocity field shows the largest velocity dispersion (i.e., $\sigma$ $\sim$ 400 km/s, see Fig. \ref{class2}). This feature suggests that an outflow is present in the inner part of the galaxy. The amplitude of the velocity dispersion map is about $(285 \pm 20)$ km/s while the velocity field amplitude is of $(270 \pm 16)$ km/s. This component is blue-shifted by 150 km/s with respect to the main component.
The derived $v_{c}/\sigma_c$ and v$_{shear}/\Sigma$ parameters are respectively 3 and 2: in this object the random motions do not seem to dominate although in its velocity field and velocity dispersion maps there are some anomalies. With such a parameter we would classify this source as $rotation$ dominated.
\begin{figure}
\begin{center}
\includegraphics[width=0.48\textwidth]{images_21453.ps}
\includegraphics[width=0.48\textwidth]{spectra_21453.ps}
\caption{\small{\bf Top panel:} H$\alpha$ maps for the class 2 object {IRAS F21453-3511}. VIMOS continuum image and HST continuum images (i.e., H and I bands). {\it Middle row:} flux intensity, velocity dispersion $\sigma$ (km/s) and velocity field $v$ maps (km/s) for the main component. {\it On the bottom:} respective maps for the broad component. Note the different FoV between the main and the broad components. The latter has been zoomed in since it covers a small area. The flux intensity maps are represented in logarithmic scale and in arbitrary flux units. All the images are centered using the H$\alpha$ peak and the iso-countors of the H$\alpha$ flux are overplotted. {\bf Bottom panel:} H$\alpha$-[NII] observed spectra of {IRAS F21453-3511} for a selected inner region (indicated by the coordinates in the top label using the same reference systems as in the top figure) where main and broad component coexist. The red curve show the total H$\alpha$-[NII] components as obtained from multi-components Gaussian fits. The green and blue curves represent respectively the main and broad components.}
\label{class2}
\end{center}
\end{figure}
\subsection{ \bf Summary of the global kinematics properties}
From the kinematic maps and the results derived so far (see table \ref{pixel}) we can draw some general conclusions. Class 0 objects are characterized by point-antisymmetric velocity fields, showing the {\it spider diagram} typical of a ideal rotating disk; their velocity dispersion maps are centrally peaked with a radial decay at larger radii. On the other hand, class 2 objects show velocity fields which do not follow this pattern. Despite they are to first approximation point-antisymmetric, they also show clear disturbances and asymmetries. These could be explained by a bent disk and/or other tidal motions, consequence of the past merger. Their $\sigma$ maps are more asymmetric with off-nuclear regions of peculiar high velocity dispersion values (e.g., {IRAS F04315-0840}), and showing asymmetrical structures in the nuclear regions (e.g., IRAS F21453-3511).
All of our sources show signs of a broad and blue-shifted components (i.e., with velocity shift of $\Delta$v = (50 - 150) km/s with respect to the main component) in the inner part (nucleus) of the galaxy. They show mean velocity dispersions ranging between (140 - 260) km/s, which are 2 - 4 times higher than those of the main components. These components cover a small area in $disk$ objects (i.e., $\sim$ 1 kpc$^2$) while larger areas (i.e., 2 - 6 kpc$^2$) are involved for the post-coalescence $mergers$, showing more complex and distorted spectra. Although this component is more prominent for the class 2 objects, the fact that it is blueshifted for the four sources and have high associated velocities suggests that an outflow can exist in their inner regions. Furthermore, the velocity field patterns of this component (i.e., kinematic axes perpendicular to those of the main component) support the dusty outflow hypothesis.
From our kinematic results we see that the sole use of 1D parameters $v_c/\sigma_c$ and / or v$_{shear}/\Sigma$ does not seem to give us a unique classification for our sources. Indeed, according to them, our objects seem to be {\it rotation dominated,} even when the inclination correction is not included. Their v$_c/\sigma_c$ values are between those typical of local spiral galaxies (i.e., v$_c /\sigma_c \sim$ 5 - 15, \citealt{Epi10}) and those obtained for Lyman Break Analogs at z $\sim$ 0.2 (with v$_c /\sigma_c$ $\sim$ 0.4 - 1.8, \citealt{Gon10}). The derived global velocity dispersions $\Sigma$ generally are between (40 - 60) km/s, quite comparable to those obtained in \cite{Gon10} for LBAs, with a typical median value of $\sim$ 67 km/s, but much higher than those observed in other lower luminosity local star-forming galaxies (i.e., typical $\sigma\sim$ 5 - 15 km/s, e.g., \citealt{Dib06}). For a comparison we also derived the v$_{shear}$ (not corrected for the inclination of the galaxy) as explained before. Our values ranges between 50 (i.e., IRAS F04315-0840) and 130 km/s. This parameter reveals higher degree of rotation if compared with the values obtained for LBAs (i.e., v$_{shear} <$ 70 km/s) where in many cases a significant velocity gradient has not been observed, and no actual rotation can be identified. Then, we derive v$_{shear}/\Sigma$ showing values typical of rotation dominated objects (i.e., $\sim 2 -3$, \citealt{Epi10}) with the exception of the galaxy {IRAS F04315-0840} (see Tab. \ref{pixel}).
The derived dynamical masses classify our LIRGs as moderate mass systems, characterized by a mean value of (1.8 $\pm$ 0.4) $\cdot $ 10$^{10}$ M$_\odot$. Our mass estimates agree with those obtained for LIRGs in \cite{H06} and \cite{Vais08}, which are in the range (10$^{10}$ - 10$^{11}$) M$_\odot$.
The post-coalescence mergers do not show extreme kinematic asymmetries in their maps, making them good examples to test the potential of $kinemetry$ for characterizing moderate asymmetries.
\section {Kinemetry analysis}
The goal of this section is to investigate the potential of the $kinemetry$ method, developed by \citet{K06} (hereafter K06), when analyzing the kinematic maps of (U)LIRGs. In particular we want to investigate how powerful this methodology is for studying the kinematic asymmetries. First we will simply apply the $kinemetry$ method to the sample, drawing preliminary conclusions on the kinematics of these objects, then we will apply the same criteria as those proposed by \citet{S08} and also explore the potential of a new criterion to study kinematic asymmetries. Therefore, we expect to find out lower asymmetries for our two post-coalescence merger systems with respect to those considered by S08, which are more kinematically disturbed due to recent merger activity. In our post-coalescence systems, the inner parts are expected to be relaxed into an almost virialized disk with a large rotational component while the outer parts should still retain asymmetries associated to the merger events. Finally we analyze the resolution / redshift effects on these results.
\subsection{The method}
The $kinemetry$ method, developed by \cite{K06}, comprises a decomposition of the moment maps into Fourier components using ellipses. For clarity, we will briefly describe again the main steps presented in K06 for a better understanding of this analysis. The Fourier analysis is the most straightforward approach to characterize any periodic phenomenon: the periodicity of a kinematic moment can easily be seen by expressing the moment in polar coordinates: K (x, y) $\rightarrow$ K (r, $\psi$). The map K(r, $\psi$) can be expanded as follows to a finite number (N+1) of harmonic terms (frequencies):
\begin{equation}
K(r, \psi) = A_{0}(r) + \sum _{n=1}^N A_{n}(r) \hspace{1mm}sin(n \cdot \psi) + B_{n}(r)\hspace{1mm} cos(n \cdot\psi) ,
\end{equation}
where $\psi$ is the azimuthal angle in the plane of the galaxy (measured from the major axis) and $r$ is the radius of a generic the ellipse. The harmonic series can be presented in a more compact way
\begin{equation}
K(r, \psi) = A_{0}(r) + \sum _{n=1}^N k_{n}(r) \cdot cos[n(\psi-\phi_{n}(r))] ,
\label{kin}
\end{equation}
where the amplitude and the phase coefficients ($k_{n}$, $\phi_{n}$) are easily calculated from the $A_{n}$, $B_{n}$ coefficients: $k_{n} = \sqrt{A_{n}^{2} + B_{n}^{2}}$ \hspace{1mm} and $\phi_{n} = arctan \left(\frac{A_{n}}{B_{n}}\right)$. Thus, for an \textit{ideal rotating disk} one would expect the velocity profile to be dominated by the $B_1$ term while the velocity dispersion profile dominated by the $A_0$ term. In the case of an {\it odd} moment ($\mu_{odd}$), the sampling ellipse parameters are determined by requiring that the profile along the ellipse is well described by $\mu_{odd}(\psi, r) \approx B_{1}(r) \cdot cos(\psi)$ since the velocity field peaks at the galaxy major axis ($\psi$ = 0) and goes to zero along the minor axis ($\psi$ = $\pi$/2). So, the power in the $B_{1,v}$ term therefore represents the {\it circular} velocity in each ring $r$, while power in other coefficients represent deviations from circular motion. On the other hand, the zeroth-order term, \textbf{\textit{$A_{0}$}} gives the {\it systemic velocity} of each ring. The velocity dispersion field is an {\it even} moment ($\mu_{even}$) of the velocity distribution, so that its kinematic analysis is identical to traditional surface photometry. In an ideal rotating disk the velocity dispersion has to be constant along each ring of the ellipse ($\mu_{even}(\psi, r) \approx A_0(r)$) and decreases when the semi-major axis length increases. In this context, the $A_{0}$ term represents the velocity dispersion profile. Higher order terms ($A_{n}$, $B_{n}$) will identify deviation from symmetry (\citealt{K06}).
There are two effects which limit the reliability with which coefficients in the expansion can be determined: (i) the absolute number of points sampled along the ellipse, and (ii) the regularity with which these points sample the ellipse as a function of angle, $\theta$. Ellipse parameters (i.e., position angle $\Gamma$, centre and flattening $q =b/a$) can be determined by minimizing a small number of harmonic terms; in the code some of them can be constrained (e.g., fix the center allows the position angle $\Gamma$ and flattening $q$ to freely change or remain fixed). The choice of a correct center is important in the analysis in order to avoid artificial overestimations of the asymmetries.
\subsection { {\it Kinemetry} of LIRGs sample}
In our analysis, the position of the galaxy's center is considered at the peak of the H$\alpha$ flux intensity map. In the two class 0 objects, the H$\alpha$ peak is in good positional agreement with the center of symmetry of the velocity field (i.e. kinematic center). For the class 2 objects, the kinematic center of the velocity field is not so well defined, but still the H$\alpha$ peak is in a reasonable symmetric position as well. The position angle $\Gamma$ is left free to vary. The flattening $q$ has also been left to vary, but within a physically meaningful range (i.e. 0.2-1). This allows us to consider general cases, such as tilted/wrapped disks. In Sec. 4.3.1. (i.e., Fig. \ref{Tot_plots}) we discuss further the effects of the different choices regarding these input parameters.
On figures \ref{map_rec_11255} - \ref{map_rec_21453} we present for the four galaxies the original kinematic maps (i.e., velocity field and velocity dispersion) along with their reconstructed maps (i.e., obtained using all the coefficients of the harmonic expansion up to the fifth order corrections). The residual maps (i.e., original - model) are also shown with a typical RMS of 10 km/s. Therefore, in all four cases, the fits are good and the reconstructed maps recover the properties of the original data with great detail. W e also present the behavior of the kinematic parameters defined in the previous section (i.e., position angle, $\Gamma$, and the flattening, $q$) along with some of the $kinemetry$ coefficients (k$_1$ and k$_5$ normalized to the k$_1$).
As for the ellipse parameters, in general we find a good optimization for $\Gamma$ at the different radii for the four objects. As for $q$, there are several cases which reach the boundary of the range of acceptable values. In general this happens in the innermost regions where the rotation component, and therefore the amplitude of the sinusoidal velocity profile along the ellipse, are small. Therefore, the irregularities in the velocity field are relatively more important when compared with the (rotational) amplitude of the velocity profile. Since the problem of finding the best ellipse (i.e., best $q$ and $\Gamma$ combination) can be quite degenerate with many local minima, a low $q$ (high ellipticity) facilitates the minimization of $\chi^2$ since the latter is sensitive to a change in $\Gamma$. On the contrary, if $q$ is high (circular ellipse), the fit ($\chi^2$) is insensitive to a change in $\Gamma$ making more difficult the $\chi^2$ minimization process. As a result, $q$ tends to have low values in the inner regions, reaching in some cases the minimum acceptable value. In addition the $q$ parameter is less constrained at low radii as a consequence of the relatively fewer data points involved and the seeing smearing, showing larger errors\footnote{Note that $kinemetry$ provides the errors on the ellipse parameters (i.e., $q$, $\Gamma$), which are determined by the Levenberg-Marquardt least-squares minimization (MPFIT) fit with the formal 1$\sigma$ uncertainties computed from the covariance matrix. Similarly, the harmonic terms have their formal 1$\sigma$ errors estimated from the diagonal elements of the corresponding covariance matrix obtained with a linear least-square fit.}. This does not mean, however, that the harmonic expansion to describe the kinematics is not well constrained. Indeed, the reconstructed velocity maps are in excellent agreement with the real velocity field for the inner regions (i.e., low $q$ values), even for the innermost region (see figures \ref{map_rec_11255} - \ref{map_rec_21453}). For large r, the amplitude of the velocity profile along the ellipse is larger, and more data point contribute to constrain the ellipse, being less sensitive to local kinematic irregularities and, therefore,providing in general a better optimization of the parameters. At large radii we also find good agreement between the fit and the data.
It follows some comments on the $kinemetry$ results for each galaxy of the sample.
\begin{itemize}
\item {\bf IRAS F11255-4120}
The position angle $\Gamma$ is quite stable ($\sim$ 200$^\circ$ $\pm$ 15) over most the sampled radius. In the inner part (i.e. r $< 5^{\prime\prime}$), $q$ has values of 0.2-0.4 and it has some instabilities at r $\sim$ 7$^{\prime\prime}$, likely due to the inner bar structure. However the expansion recovers very well the data in all this region. k$_1$ is mainly dominated by rotation (i.e, B$_1$) and increases radially up to a value of 140 km/s. The k$_5$/k$_1$ term, which measures the small scale kinematic asymmetries, has low values up to a maximum of 0.1, pointing some minor peculiarities for r $<$ 7$^{\prime\prime}$, likely due to the bar structure.These asymmetries become smaller in the outer part. The error bars are not visible being smaller than the black points in the plot.
\begin{figure}[!h]
\begin{center}
\includegraphics[width=0.48\textwidth]{kin_11255_map.ps}
\includegraphics[width=0.48\textwidth]{k_11255.ps}
\caption{\small {\bf Upper panel:} Maps of H$\alpha$ Gaussian fit velocities (top left), H$\alpha$ Gaussian fit dispersion (bottom left) and their respective reconstructed (middle) and residual (data-model) maps (on the right) for {IRAS F11255-4120}. {\bf Lower panel:} Radial profiles of the kinematic properties, obtained using $kinemetry$ program. The position angle $\Gamma$ and the flattening $q$ of the best fitting ellipses as well as the first and the fifth order Fourier terms (respectively, k$_1$ and k$_5$) are plotted as a function of the radius.}
\label{map_rec_11255}
\end{center}
\end{figure}
\vspace{5mm}
\item{\bf IRAS F10567-4310}
The position angle $\Gamma$ is quite stable over the sampled radii, with values between 210 and 250 degrees. The $q$ parameter reaches the value of 0.2 in the inner region (i.e., r $< 4^{\prime\prime}$), where the reconstructed map shows a very good agreement with the data. k$_1$ is mainly dominated by rotation (i.e, B$_1$) and increases radially up to a value of 140 km/s, revealing a possible minor anomaly at r $\sim$ 5$^{\prime\prime}$. The k$_5$/k$_1$ term has very low values, with a maximum of 0.05 over the sampled radii, meaning that this object is close to an ideal rotating disk structure.
\begin{figure}[!h]
\begin{center}
\includegraphics[width=0.48\textwidth]{kin_10567_map.ps}
\includegraphics[width=0.48\textwidth]{k_10567.ps}
\caption{\small {\bf Upper panel:} Maps of H$\alpha$ Gaussian fit velocities (top left), H$\alpha$ Gaussian fit dispersion (bottom left) and their respective reconstructed (middle) and residual (data-model) maps (on the right) for {IRAS F10567-4310}. {\bf Lower panel:} Radial profiles of the kinematic properties, obtained using $kinemetry$ program. The position angle $\Gamma$ and the flattening $q$ of the best fitting ellipses as well as the first and the fifth order Fourier terms (respectively, k$_1$ and k$_5$) are plotted as a function of the radius.}
\label{map_rec_10567}
\end{center}
\end{figure}
\vspace{5mm}
\item{\bf IRAS F04315-0840}
The position angle $\Gamma$ typically spans from 350$^\circ$ up to 380$^\circ$\footnote{The position angle is measured from the North (0$^\circ$=360$^\circ$) anti-clockwise. In order to make the two values more easily comparable we choose to add 360$^\circ$ in one case.}, except for (r $\sim$ 3$^{\prime\prime}$) which drops to around 320$^\circ$ as a consequence of one of the two distinct redshifted peaks present in the velocity field map. For most of the radii $q$ is close to 1(circle-ellipses). k$_1$ increases radially up to a value of 120 km/s ($r< 5^{\prime\prime}$) and then decreases to reach a value of 100 km/s at r $\sim$ 15$^{\prime\prime}$. The k$_5$/k$_1$ term is between 0 and 0.3, with an constant increasing behaviour. This illustrates that departures from rotation are mainly found in the outer parts.
\begin{figure}[!h]
\begin{center}
\includegraphics[width=0.48\textwidth]{kin_04315_map.ps}
\includegraphics[width=0.48\textwidth]{k_04315.ps}
\caption{\small {\bf Upper panel:} Maps of H$\alpha$ Gaussian fit velocities (top left), H$\alpha$ Gaussian fit dispersion (bottom left) and their respective reconstructed (middle) and residual (data-model) maps (on the right) for {IRAS F04315-0840}. {\bf Lower panel:} Radial profiles of the kinematic properties, obtained using $kinemetry$ program. The position angle $\Gamma$ and the flattening $q$ of the best fitting ellipses as well as the first and the fifth order Fourier terms (respectively, k$_1$ and k$_5$) are plotted as a function of the radius.}
\label{map_rec_04315}
\end{center}
\end{figure}
\vspace{5mm}
\item{\bf IRAS F21453-3511}
The position angle $\Gamma$ almost keeps constant to $\sim$ 240$^\circ$ for r $>$~6$^{\prime\prime}$ while it shows lower and more unstable values for r $<$~6$^{\prime\prime}$. The $q$ reaches 0.2 in the region inside a radius of r $<$~6$^{\prime\prime}$, mainly motivated by the relatively small rotation component in this region. However the agreement between the kinematic fit and data is very good for these radii. At r =7$^{\prime\prime}$ $q$ reaches a maximum ($\sim$ 1) to decrease constantly to a value of 0.4. k$_1$ does not reveal any strong rotational component ranging between 10 up to 40 km/s. It increases in the inner region for $r<4^{\prime\prime}$ and then remains constant. The k$_5$/k$_1$ term ranges between 0 and 0.2, showing a quite stable trend. For radii larger than 10$^{\prime\prime}$ it shows an opposite behavior with respect to the k$_1$ term. In this object the deviations from pure rotation are not as high as those of IRAS F04315-0840, but they are higher than those for the class 0 objects.
\begin{figure}[!h]
\begin{center}
\includegraphics[width=0.48\textwidth]{kin_21453_map.ps}
\includegraphics[width=0.48\textwidth]{k_21453.ps}
\caption{\small {\bf Upper panel:} Maps of H$\alpha$ Gaussian fit velocities (top left), H$\alpha$ Gaussian fit dispersion (bottom left) and their respective reconstructed (middle) and residual (data-model) maps (on the right) for {IRAS F21453-3511}. {\bf Lower panel:} Radial profiles of the kinematic properties, obtained using $kinemetry$ program. The position angle $\Gamma$ and the flattening $q$ of the best fitting ellipses as well as the first and the fifth order Fourier terms (respectively, k$_1$ and k$_5$) are plotted as a function of the radius.}
\label{map_rec_21453}
\end{center}
\end{figure}
\end{itemize}
In general we have found that for class 0 sources the higher order deviations (from pure rotation) are small (i.e., k$_5$/k$_1<$ 0.1) while for class 2 objects the deviations are higher, mainly in the outer regions (k$_5$/k$_1$ $\leq$ 0.4). For all of them the rotation curve (i.e., k$_1$ parameter) seem to characterize these objects as rotating. As discussed before, these galaxies do not show extreme asymmetries in their kinematic maps and this is confirmed by the $kinemetry$ results obtained so far.
In the following sections we will study several kinematic criteria with the aim of better classifying these systems.
\subsection {\textbf{Sample of local LIRG systems in} the [$\sigma_a$ - v$_a$] plane }
In order to reveal the presence of rotational/non-rotational motions within the dynamics of the gas in each galaxy we will first consider the same criteria as the one proposed by S08. It is worth mentioning that they compare two main classes of systems: those which have suffered a recent major merger event (i.e., \textit{mergers}) and those without sings of interacting or merger activity (i.e., \textit{disks}). For further details see S08. They define the asymmetries in the velocity and velocity dispersion fields as:
\begin{equation}
\hspace{1cm} v_{asym} = \left\langle \frac{ k_{avg, v} } {B_{1, v} } \right\rangle_r \hspace{1cm} {\sigma_{asym} =\left\langle \frac{k_{avg, \sigma}}{B_{1, v}}\right\rangle _r},
\end{equation}
\vspace{5mm}
where $k_{avg, v} =(k_{2, v} + k_{3, v} + k_{4, v} + k_{5, v})/4$ and $k_{avg, \sigma} =(k_{1, \sigma} + k_{2, \sigma} + k_{3, \sigma} + k_{4, \sigma} + k_{5, \sigma})/5$. For an ideal rotating disk, we expect the velocity profile to be perfectly antisymmetric where the $B_1$ term would dominate the Fourier expansion, while the velocity dispersion map is expected to be perfectly symmetric and therefore all terms except $A_0$ would vanish.
In Fig. \ref{Shap_10} we show the results for our galaxies in the [$\sigma_a - v_a$] plane. As expected, class 0 objects have lower values of the asymmetries than class 2 objects. Indeed, looking at their velocity and velocity dispersion maps, the class 0 maps (Figs. \ref{map_rec_11255} - \ref{map_rec_10567}) resemble to those of an ideal rotating disk (i.e., `spider diagram' structure for the velocity field and centrally peaked velocity dispersion map) while class 2 objects present more distorted velocity fields and irregular dispersion maps (Figs. \ref{map_rec_04315} - \ref{map_rec_21453}). Therefore, taking into account that the objects in the present sample were classified as class 0 or class 2 on the basis of pure morphological arguments, we can conclude that their morphological and kinematics classification in the [$\sigma_a$ - v$_a$] plane are consistent.
In order to analyze the robustness of the results in Fig. \ref{Shap_10}, we will analyze in the following sections their dependence on the input parameters considered as well as in the error in the radial velocities and velocity dispersion measurements.
Note that $kinemetry$ uses the full 2D kinematic information of the velocity field and velocity dispersion map, allowing us a good characterization of the asymmetries, something which is more efficient than using 1D parameters such as v$_c/\sigma_c$ and v$_{shear}/\Sigma$ as seen in section 3.
\begin{figure}
\begin{center}
\includegraphics[width=0.48\textwidth]{Shapiro_4gal_10.ps}
\caption{\small Asymmetry measure of the velocity v$_{asym}$ and velocity dispersion $\sigma_{asym}$ fields for the four galaxies analyzed here. Symbol types distinguish among different systems: the square represents {IRAS F10567-4310}, the triangle is IRAS F11255-4120, the circle is {IRAS F04315-0840} and diamond represents {IRAS F21453-3511}. Blue and green symbols stand for different morphological types, respectively class 0 and class 2 objects.}
\label{Shap_10}
\end{center}
\end{figure}
\subsubsection{Dependence on the input parameters}
\label{constrains}
As pointed out by K06, a good choice of input parameters is important in order to avoid an artificial overestimation of the asymmetries. To perform a $kinemetry$ analysis we require to set the dynamical center, as well as to specify different levels of constrains of the input parameters. For instance, the kinematic position angle $\Gamma$ and the flattening $q$ of the ellipses can be fixed or left free to vary for the fitting at different radii. The COVER parameter, which controls the radius at which the process stops by setting the fraction of the ellipse that has to be covered by data\footnote{For instance, COVER = 0.7 means that if less than 70\% of the points along an ellipse are not covered by data the program stops. This value makes sure that kinematic coefficients are robust. Sometimes it is necessary to relax this condition especially when reconstructing maps. See further details in \cite{K06}.}, has to be assigned too.
In Fig. \ref{Tot_plots} we present the results in the [$\sigma_a$ - v$_a$] plane for our galaxies for different sets of input parameters. In each panel one parameter at a time is changed: respectively, the COVER (top-left), the position angle $\Gamma$ (or PA, top-right), the flattening $q$ (bottom-left) and the CENTER (bottom-right) of the ellipses. We can find that the results are stable to a reasonable choice of values. Indeed, looking at the COVER panel, the four galaxies give similar results up to cover=0.5; {IRAS F11255-4120} deviates significantly for COVER=0.3. {IRAS F21453-3511} seems to be sensitive to the galaxy center but it maintains within the region of high-asymmetries in all the (extreme) cases considered. As shown, the choice of free or fixed position angle $\Gamma$ or flattening $q$ do not affect so much the final results. In particular the computed asymmetries are quite insensitive to the choice for $q$, especially for the class 0 galaxies. In general, the results obtained with a fixed $\Gamma$ / $q$ are somewhat higher due to the fact that there is less degree of freedom such that the asymmetries cannot be well accounted by the fitting. Obviously the proper choice of free / fixed parameters depends on the level of S/N (i.e., if the deviations are associated to a true feature or noise).
According to these results we have selected the following set of input parameters for the remainder analysis (i.e., COVER = 0.7, $\Gamma$ is completely free to vary and $q$ free to vary in the range [$0.2-1$], center = H$\alpha$ flux peak).
\begin{figure}
\begin{center}
\includegraphics[width=0.48\textwidth]{SUPER-4.ps}
\caption{\small Results for the asymmetry measures when different sets of input parameters are considered with 10 harmonic terms analysis. Symbol types and colors distinguish among the different systems as explain in the legend. In each panel one parameter at a time is changed: respectively, COVER, $\Gamma$ (or PA), $q$ and the CENTER of the ellipses. {\it On the top-left:} Different results for the COVER parameter (i.e., 0.3, 0.5, 0.7, 0.9) where the size of the symbol is proportional to the COVER value (the biggest symbol corresponds to the highest cover value and vice versa). {\it On the top-right:} The results obtained considering a {\it position angle} $\Gamma$ free to vary (filled symbols) or fixed to its mean value (empty symbols) are shown. {\it On the bottom-left:} The results shown are obtained when considering $q$ free to vary and constant with the radius to its mean value. {\it On the bottom-right:} Results achieved choosing 5 different centers for each galaxy: filled symbols represent results obtained from the `standard' analysis (H$\alpha$ flux peak), empty symbols the results from shifting their center of 1 pixel (horizontally {\it and} vertically, with respect to the H$\alpha$ peak pixel corresponding to a shift of 0.95$^{\prime\prime}$). }
\label{Tot_plots}
\end{center}
\end{figure}
\subsubsection{Monte Carlo simulations}
In order to analyze the dependence of the $kinemetry$ results to the uncertainties in the radial velocities and velocity dispersion values, we measure the probability distribution function (PDFs) of the asymmetries in these systems using Monte Carlo (MC) simulations, as done in S08, since the $kinemetry$ method does not lend itself to a straightforward error propagation.
Therefore, for each template, we create 150 different realizations of the moment maps (i.e., velocity field and velocity dispersion) based on their corresponding error maps. These error maps correspond to the measurement errors of the velocity moments, as derived when fitting the kinematics from the data cube along with wavelength calibration errors, as explained in section 2.4. For each moment map we perturb the observed data points by randomizing them, using Gaussian noise parameterized by the measured 1$\sigma$ errors. The new maps created are then used to rerun \textit{kinemetry} and apply the analysis described before. In figure \ref{MC_1} the results are shown. They show that the results in the [$\sigma_a$ - v$_a$] plane are stable to the velocity errors, with relatively well defined regions for disks and merger galaxies.
\begin{figure}[h]
\begin{center}
\includegraphics[width=0.45\textwidth]{3_sigma_shap_08.ps}
\includegraphics[width=0.4\textwidth, angle=90]{histo_S08.ps}
\caption{\small {\bf On the top:} Asymmetry measures of the velocity v$_{asym}$ and velocity dispersion $\sigma_{asym}$ fields as derived from the Monte Carlo realizations for the four objects. For each source 150 MC simulations are run but here only the $\pm$3$\sigma$ results are shown. The solid red line indicates the division between class 0 (disks) and class 2 (post-coalescence mergers) at $<K_{tot}>$ = 0.135. {\bf On the bottom:} The probability distribution function (PDFs) as derived from MC realizations. The empirical delineation where $K_{lim}=0.135$ can cleanly separate the two classes (i.e., $K_{lim}$ area). The symbols are the same as the ones used before.}
\label{MC_1}
\end{center}
\end{figure}
\subsection{Total kinematic asymmetry of disk, on-going and post-coalescence merger systems }
As done in S08 we compute the total kinematic asymmetry K$_{tot}$ border in order to separate class 0 (disks) and class 2 (post-coalescence merger) objects. For the four objects we obtained a mean value of $<K_{tot}> = 0.135$: this is a significant lower value than that obtained in S08 (i.e., $<K_{tot}> = 0.5$). The difference could arise from the fact that our \textit{mergers} are in a post-coalescence phase showing relaxation in the innermost regions with a large rotation component, while in S08 they are mostly pre-coalescence merger systems dominated by dispersion. Also, the limit defined in S08 comes out after excluding {\it IRAS 12112+0305}, a pre-coalescence merger pair (i.e., \citealt{GM09}) that looks like a \textit{disk} at high redshift. If the classification of this object is considered, the total kinematic asymmetry border derived for the whole S08 sample would have been considerably lower (i.e., $<K_{tot}> \sim$ 0.3, see Figure 5 in S08) and therefore the discrepancy with our finding reduced.
In any case, the fact that the border $disk/merger$ for the S08 sample is so dependent on the classification of a single object (i.e., IRAS 12112+0305) illustrates its relatively large associated uncertainties. A reduction of the total asymmetry border K$_{tot}$, as our results (and S08, after reclassifying {\it IRAS 12112+0305}) suggests, would imply that the relative frequency of $disks$ and $mergers$ in a given sample changes, increasing the fraction of $mergers$. Changing the relative frequencies of $disk/merger$ have obvious implications when interpreting the data in terms of the different evolutionary scenario mentioned in the introduction. On the other hand our sample is admittedly too small and formed by objects with relatively homogeneous properties (i.e., LIRGs classified as disks and post-coalescence mergers) for a robust determination of general use. Therefore, further efforts to constrain its value as well as to understand how it depends on several instrumental and observational factors are required.
\subsection {A new criterion to distinguish kinematic asymmetries between post-coalescence mergers from disks}
In order to better assess the presence of asymmetries in the kinematic maps we will explore a new kinematic criterion.
As described in \cite{kron07}, when considering recent (i.e., $\leq$ 100 Myr after the first encounter) or ongoing major mergers of equal mass galaxies (i.e., Milky Way type), the inner regions\footnote{If a galaxy covers a FoV of 30 kpc $\times$ 30 kpc, the regions more affected by chaotic motions are those at galactocentric distances smaller than 10 kpc.} of the galaxy are usually affected by more chaotic motions, as revealed by a quite irregular rotational curve and higher order deviations (i.e., k$_5$/k$_1$) at small radii. As the major merger evolves, the inner regions rapidly relax into a rotating disk, while the outer parts are still out of equilibrium. This implies that the velocity field in a post-coalescence system could be dominated by rotation in the inner regions with large kinematic asymmetries in the outer parts, as it is actually observed in our systems. Provided that the outer regions retain better the memory of a merger event, we propose a criterion which enhances the relative importance of the asymmetries at larger radii.
Indeed, instead of simply averaging the asymmetries over all radii (as in S08), these are weighted according to the number of data points used in their determination. Since the number of data points of the outer ellipses is larger than for the inner ones, the asymmetries found in the outer ellipses contribute more significantly to the average for obtaining v$_{asym}$ and $\sigma_{asym}$. As the number of data points is in first approximation proportional to the circumference of the ellipse, for practical reasons we used this to weight the asymmetries found for the different ellipses.
The circumferences of the ellipses are computed using the truncated `infinite sum' formula, that is a function of the ellipticity (i.e., $e = \sqrt{1-q^2}$) and the semi major axis of the ellipse (r):
\begin{equation}
\hspace{5mm}C(e, r) \approx 2 \pi r \left[1- \left(\frac{1}{2}\right)^2 e^2 - \left(\frac{1\cdot 3}{2 \cdot 4}\right)^2 \cdot \frac{e^4}{3} \right]
\label{peri}
\end{equation}
The final formula to compute the weighted velocity and velocity dispersion asymmetries are respectively:
\begin{equation}
\hspace{5mm}v_{asym} = \sum_{n=1}^N \left (\frac{ k_{avg, n}^v}{B_{1, n}^v } \cdot C_n \right )\cdot \frac{1}{\sum_{n=1}^N C_n}
\label{v_weighted}
\end{equation}
\begin{equation}
\hspace{5mm}\sigma_{asym} = \sum_{n=1}^N \left ( \frac{ k_{avg, n}^\sigma}{B_{1, n}^v } \cdot C_n \right )\cdot \frac{1}{\sum_{n=1}^N C_n}
\label{s_weighted}
\end{equation}
\vspace{5mm}
where {\it N} is the total number of radii considered, $C_n$ the value of the circumference for a given ellipse, the different k$_n$ (k$_n^ v$ and k$_n^ \sigma$) are the deviations concerning respectively the velocity field and velocity dispersion maps, while $B_{1}^ v$ is the rotational curves. We will refer to this approach as the {\it `weighted'} method and its associated plane W-[$\sigma_a$ - v$_a$]). In Fig. \ref{MC_w} the results in the W-[$\sigma_a$ - v$_a$] plane for our four galaxies are shown where MC have been performed similarly as above (Sect. 4.3.2).
The results follow the same general trend but the two classes are separated somewhat better than in the unweighted case (Sec. 4.3). Indeed, for class 0 objects the {\it weighted} velocity asymmetries are lower while, for class 2 objects are somewhat higher than in the [$\sigma_a$ - v$_a$] plane. Therefore, it enhances the fact that post-coalescence mergers have larger deviations at larger radii with respect to pure rotational motions while disks have still lower deviations than those obtained using [$\sigma_a$ - v$_a$]. In this case, the total kinematic asymmetry border, which distinguishes the two disks and the two post-coalescence mergers, is characterized by a mean value of 0.146.
\begin{figure}[h]
\begin{center}
\includegraphics[width=0.45\textwidth]{3_sigma_shap_weigh.ps}
\includegraphics[width=0.4\textwidth, angle=90]{istogram_NEW_weight.ps}
\caption{\small {\bf On the top:} Weighted asymmetry measures of the velocity v$_{asym}^{(w)}$ and velocity dispersion $\sigma_{asym}^{(w)}$ fields as derived from the Monte Carlo realizations for the four objects. For each source 150 MC simulations are run but only the $\pm$3$\sigma$ results are shown. The solid red line indicates the empirical division between disks and post-coalescence mergers at $<K_{tot}>$ = 0.146. {\bf On the bottom:} The probability distribution function (PDFs) as derived from MC realizations. The empirical delineation K$_{lim}$ = 0.146 can well separate the two classes along with a large range of other values (i.e., dashed $K_{lim}$ area).}
\label{MC_w}
\end{center}
\end{figure}
\subsection {Angular resolution / Redshift dependence}
The angular resolution effects on the distortions of the velocity fields produced by mergers have been discussed by \citet{kron07} on the basis of simulated velocity fields as a function of redshift (i.e., $0<z<1$). They found that for large (Milky Way type) galaxies the distortions are still visible at intermediate redshifts but partially smeared out, while for small galaxies even strong distortions are not visible in the velocity field at z $\approx$ 0.5.
\cite{Gon10}, simulating LBAs at redshift $z\sim$ 2, found that, in general, galaxies at high redshift present smaller values of K$_{asym}$, i.e., they appear more disky than they actually are. The percentage of galaxies classified as mergers drop from $\sim$ 70 \% to $\sim$ 38 \% from $z= 0$ to $z=3$ according to their simulations.
In order to investigate the resolution effects on our results we simulate to `observe' these systems at $z=3$ with a typical pixel scale of 0.1$^{\prime\prime}$ (the same pixel scale of the IFU NIRSpec/JWST). At this redshift the current FoV of our images is about 1$^{\prime\prime}$ x 1$^{\prime\prime}$ with a typical scale of 7.83 kpc/arcsec, assuming a $\Lambda$DCM cosmology with H$_0$ = 70 km/s/Mpc, $\Omega_M$ = 0.3 and $\Omega_\Lambda$ = 0.7. The simulated maps are shown in Fig. \ref{simulate_maps_highz}. We apply $kinemetry$ using these maps and, following the same procedures as before, obtain the results shown in the [$\sigma_a$ - v$_a$] (Fig. \ref{altoz}) and W-[$\sigma_a$ - v$_a$] (Fig. \ref{ancora}) planes. The [$\sigma_a$ - v$_a$] plane shows the expected trend, where both classes appear more symmetric when observed at high redshift. In this case a lower value of the total kinematic asymmetry K$_{asym}$ border is derived (red dashed line), as expected (i.e., K$_{asym}$ = 0.096). Thus, shifting the sample from $z=0$ to $z=3$, the frontier between {\it disks / post-coalescence mergers} in the [$\sigma_a$ - v$_a$] plane changes from 0.135 to 0.096.
\begin{figure}
\begin{center}
\includegraphics[width=0.48\textwidth]{new_11255_map.ps}\\
\vspace{2mm}
\includegraphics[width=0.48\textwidth]{new_10567_map.ps}\\
\vspace{2mm}
\includegraphics[width=0.48\textwidth]{new_04315_map.ps}\\
\vspace{2mm}
\includegraphics[width=0.48\textwidth]{new_21453_map.ps}\\
\caption{H$\alpha$ maps for the four objects as observed at $z=3$ with a spatial sampling of 0.1$^{\prime\prime}$. The flux intensity, velocity dispersion $\sigma$ (km/s) and velocity fields $v$ maps (km/s) for the main component are shown. The flux intensity maps are represented in logarithmic scale and in arbitrary flux units. All the images are centered using the H$\alpha$ peak and the iso-contours of the H$\alpha$ flux are over-plotted.}
\label{simulate_maps_highz}\end{center}
\end{figure}
The W-[$\sigma_a$ - v$_a$] plane is less sensitive to resolution effects after redshifting our sample at $z = 3$ (see Fig. \ref{ancora}). This is due to the fact that the larger / outer regions are the ones less affected by resolution effects. As with this criterion the associated asymmetries weight more than those present at inner radii, the computed asymmetries (i.e., v$_{asym}$, $\sigma_{asym}$) are less affected by resolution. Therefore, the total kinematic asymmetry distinguishing the two classes changes only from 0.146 to 0.130 between $z= 0$ and $z=3$.
\begin{figure}
\begin{center}
\includegraphics[width=0.4\textwidth]{highz.ps}
\caption{\small The comparison between $local$ and $z=3$ results in the [$\sigma_a$ - v$_a$] plane. Empty symbols represent low-z results while filled ones are for $z=3$. Symbols have the same meaning as in Fig. \ref{Shap_10}. The red dashed line is the {\it high-z} frontier with a value of 0.096 as explained in the text while the dotted one is for the $local$ analysis (i.e., 0.135).}
\label{altoz}
\end{center}
\end{figure}
Summarizing, resolution effects tend to smooth kinematic deviations making objects to appear more disky than they actually are. This effects are more significant when analyzing the kinematic asymmetries in the (unweighted) [$\sigma_a$~-v$_a$] plane than in the W-[$\sigma_a$ - v$_a$] one. In particular, when compared our local seeing limited observations with simulated data at $z=3$ (and 0.1$^{\prime\prime}$/spaxel), the total kinematic asymmetry border value is reduced by a 30\% from $z=0$ to $z=3$ in the [$\sigma_a$ - v$_a$] plane, while it is only shifted by 11\% in the W-[$\sigma_a$ - v$_a$] plane.
\begin{figure}
\begin{center}
\includegraphics[width=0.4\textwidth]{3_sigma_NEW_weigh.ps}
\caption{\small Comparison between $local$ and $z=3$ results in the W-[$\sigma_a$ - v$_a$] plane. Empty symbols represent low-z results while filled ones are for z=3. The red dashed line represents the {\it high-z} frontier, while the dotted one is for the local case. }
\label{ancora}
\end{center}
\end{figure}
Comparing our results with those obtained in S08 when considering local spirals and toy-disk models as $observed$ at high redshift, we notice that the low value of our K$_{tot}$ ($\sim$ 0.1) classifies those galaxies as $mergers$ and, on the other hand, the frontier defined in S08 classifies our post-coalescence mergers and IRAS F12112+0305 as $disks$. This illustrates that the definition of $merger$ is a crucial point to define the asymmetry frontier. Indeed, a lower value of the fraction merger/disk (i.e., a higher value of the K$_{tot}$ parameter) can be derived if only pre-coalescence on-going mergers are considered as `true' mergers. The {\it weighted} criterion proposed here should be applied to larger and more diverse samples in order to understand the uncertainties associated to this type of classifications.
\section {Conclusions}
This paper presents the results from spatially resolved kinematics of four local (z $\sim$ 0.016) luminous infrared galaxies (i.e., LIRGs) observed with the VLT/VIMOS IFU as part of a larger project to characterize the properties of (U)LIRGs on the basis of optical and infrared Integral Field Spectroscopy. The four galaxies are at a similar distance ($\sim$ 70 Mpc) and, on the basis of its morphology, two of them have been classified as {\it isolated disks} and two as post-coalescence $merger$ objects (see Paper III). The velocity field and velocity dispersion maps are analyzed with the aim of studying in detail their kinematics. The $kinemetry$ method (developed by Krajnovi{\'c} and coworkers) is used to characterize the kinematic asymmetries of these galaxies and several criteria are discussed to distinguishing their status. We can draw the following conclusions from this study:
\begin{itemize}
\item
The general kinematic properties of the four LIRGs are consistent with their morphological classification: {\it isolated disks} reveal quite regular velocity fields and centrally peaked velocity dispersion maps consistent with a single rotating disk interpretation while the remaining two show departures from this behavior. In particular we found that post-coalescence {\it mergers} show more irregular velocity fields and velocity dispersion maps showing off-nuclear dispersion peaks (up to 220 km/s at 2.4 kpc from the H$_\alpha$ peak for {IRAS F04315-0840}) or nuclear asymmetric structures;
\item
We have found double peaked emission line profiles in the inner regions of the four galaxies. The secondary broad components (i.e., $\sigma\sim $ 70 - 450 km/s) are in all cases blue-shifted ($\Delta v \sim $ 50 - 150 km/s) and, taking into account the large velocities involved, they can be well explained by the presence of an $outflow$ in a dusty environment. The pattern of the 2D kinematic maps of the secondary broad component for the two post-coalescence $mergers$ (with kinematic axes perpendicular to those of the main component) further supports this interpretation. In the particular case of {IRAS F04315-0840} the broad component is found over a quite extended area ($\sim$ 2.4 kpc x 2.7 kpc);
\item
The v$_c/\sigma_c$ parameter classifies our sources as {\it rotation dominated}. Similar results are obtained when using the quantity v$_{shear}/\Sigma$. This shows that our post-coalescence mergers have a large rotation component and the sole use of these parameters does not allow us to discriminate their kinematic differences with respect to disks;
\item
When the full 2D kinematics information provided by the spatially resolved velocity field and velocity dispersion maps is considered, the kinematic asymmetries are well characterized with $kinemetry$, making the morphological and kinematic classification consistent for the four objects. Disks have lower kinematic asymmetries than those derived for post-coalescence mergers;
\item
We have explored a new criterion to characterize the kinematic asymmetries using {\it kinemetry}. In particular, we introduce a new weighting method which gives weight to the kinematics of the outer regions when computing the total asymmetries v$_{asym}$ and $\sigma_{asym}$. This is motivated by the fact that post-coalescence mergers show relatively small kinematic asymmetries in the inner parts as a consequence of the rapid relaxation into a rotating disk, with the outer parts being still out of equilibrium (i.e., larger asymmetries). The fact that the `frontier' between \textit{disks} and \textit{post-coalescence} systems only changes by 11\% when considering the local and high-z cases suggests that this new criterion is less dependent on angular resolution effects. Thus, the W-[$\sigma_a$ - v$_a$] plane differentiate in a more robust way $disks$ and {\it post-coalescence mergers}.
\item
Classifying $disk/merger$ systems using $kinemetry$ is difficult, and it obviously depends on the definition of $merger$. The `asymmetry frontier' strongly depends on the `type' of mergers considered: if only pre-coalescence on-going mergers are considered as `true' mergers a lower value of the fraction $merger/disk$ systems can be derived. The weighted criterion proposed here helps to characterize in a more robust way post-coalescence asymmetries.
\item
Using previous criteria to classify disks/mergers, our two post-coalescence systems would have been classified as \textit{disks}. This suggests that the ratio of mergers to disks systems at high-z may have been underestimated. Larger and more diverse samples are required to confirm this conclusion.
\end{itemize}
\newpage
\begin{landscape}
\begin{table
\vspace{3cm}
\caption{Kinematic parameters for the sample.}
\vspace{1cm}
\label{pixel}
\begin{scriptsize}
\begin{tabular}{l cccccccccc }
\hline\hline\noalign{\smallskip}
Galaxy ID & Type of fitting $^{a}$ &$i$ $^{b}$ &{}{ $v_c$ $^{c}$ } & { $\sigma_c$ $^{d}$} & { v$_c/ \sigma_c$ } &{} { $v_{shear}$ $^{e}$} & { $\Sigma$ $^{f}$} & { v$_{shear}/ \Sigma$ } & { $R_{eff}$ } & { $M_{dyn}$ } \\
{\smallskip}
{ IRAS code} & {} &{ $degree$} & { $(km/s)$} & { $(km/s)$} & { } & { $(km/s)$} & { $(km/s)$} & { } & { $(kpc)$} & { $(M_\odot)$} \\
\hline\noalign{\smallskip}
{ {\bf F11255-4120} } & {1c }& { 53$^\circ$ $\pm$ 20$^\circ$} & { $187 \pm 72$} & { $104 \pm 3$} & {1.8 $\pm$ 0.7} & {} & {} & {} &{ } & { \bf} \\
{\smallskip}
{ {\bf F11255-4120} } & {2c }& { 53$^\circ$ $\pm$ 20$^\circ$} & { $188 \pm 76$} & { $83 \pm 2$} & { 2.3 $\pm$ 1.0} & {125} & {47} & {2.7} &{ $2.40 \pm 0.57$} & { \bf{$(2.9 \pm 0.8) \cdot 10^{10}$} } \\
\hline\hline\noalign{\smallskip}
{ {\bf F10567-4310}} &{1c}& { 36$^\circ$ $\pm$ 4$^\circ$} & { $253 \pm 32$} & { $56\pm 2$} & { 4.5 $\pm$ 0.8} & {} & {} & {} & { } &{ \bf } \\
{\smallskip}
{ {\bf F10567-4310}} & {2c} & { 36$^\circ$ $\pm$ 4$^\circ$} & { $255 \pm 30$} & { $ 55.0 \pm 1.4$} & { 4.6 $\pm$ 0.7} & {120} & {41} & {3} & { $3.20 \pm 0.40$} &{ \bf {$ (1.7 \pm 0.3) \cdot 10^{10} $} } \\
\hline\hline\noalign{\smallskip}
{ {\bf F04315-0840}} & {1c}& { n.c.} & { $157 \pm 3$} & { $110 \pm 5$} & {1.4 $\pm$ 0.1} &{} & {} & {} & {} & {}\\
{\smallskip}
{ {\bf F04315-0840}} & {2c} & { n.c.} & { $162 \pm 5$} & { $69 \pm$ 3} & {2.3 $\pm$ 0.2} & {51} & {51} & {$\sim$ 1} & { $0.94 \pm 0.14$ } & { $(7.9 \pm 1.9) \cdot 10^9 $ } \\
\hline\noalign{\smallskip}
{ {\bf F04315-0840}} & {1c}& { 29$^\circ\pm3^\circ$} & { $324 \pm 40$} & { $110 \pm 5$} & {2.9 $\pm$ 0.5} &{} & {} & {}& {} & {} \\
{\smallskip}
{ {\bf F04315-0840}} & {2c}& { 29$^\circ\pm3^\circ$} & { $335 \pm 45$} & { $69 \pm 3$} & {4.8 $\pm$ 0.9} &{51} & {51} & {$\sim$ 1} &{ $0.94 \pm 0.14$ } & { $(7.9 \pm 1.9) \cdot 10^9 $ }\\
\hline\hline\noalign{\smallskip}
{ {\bf F21453-3511}} & {1c} & { n.c.} & { $90 \pm 1$} & { $69 \pm 3$} & {1.30 $\pm$ 0.07} & {} & {} & {} & { } & { \bf } \\
{\smallskip}
{ {\bf F21453-3511}} &{2c} & { n.c.} & { $78\pm 2$} & { $61 \pm 1$} & {1.28 $\pm$ 0.05} & {132} & {65} & {2} &{ $ 2.97 \pm 0.82$} & { $(1.9 \pm 0.6) \cdot 10^{10}$ } \\
\hline\noalign{\smallskip}
{ {\bf F21453-3511}} &{1c} & { 24$^\circ$ $\pm$ 2$^\circ$} & { $222\pm 21$} & { $69 \pm 3$} & {3.2 $\pm$ 0.5} &{} & {} & {} \\
{\smallskip}
{ {\bf F21453-3511}} & {2c} &{ 24$^\circ$ $\pm$ 2$^\circ$} & { $192 \pm 21$} & { $61 \pm 1$} & {3.2 $\pm$ 0.4} &{132} & {65} & {2}&{ $ 2.97 \pm 0.82$} & { $(1.9 \pm 0.6) \cdot 10^{10}$ } \\
\hline\hline\noalign{\smallskip}
\end{tabular}
\vskip0.2cm\hskip0.0cm
\end{scriptsize}
\begin{minipage}[h]{21cm}
\tablefoot{ $^{a}$ 1 component fitting (1c) and 2 component fitting (2c, referees to the main (systemic) component of the two-Gaussian fit) is consider for the four galaxies. $^{b}$ Inclination of the galaxy; {\bf n.c.} is when no inclination correction is applied. $^{c}$ Circular velocity derived as the half of the observed peak-to-peak velocity from the H$\alpha$ kinematic. Corrected and no corrected values for the inclination are shown. $^{d}$ Central velocity dispersion as derived from the $\sigma_{H\alpha}$ maps. $^{e}$ Velocity shear not corrected for the inclination of the galaxy. $^{f}$ Global velocity dispersion in the whole galaxy.\\ }
\end{minipage}
\end{table}
\end{landscape}
\begin{acknowledgements}
We acknowledge the anonymous referee for useful comments and suggestions, that helped us to improve the quality of the paper. We also would like to thank Davor Krajnovi{\'c} for his help and valuable comments on his Kinemetry software. This work was funded in part by the Marie Curie Initial Training Network ELIXIR of the European Commission under contract PITN-GA-2008-214227. This work has been supported by the Spanish Ministry of Science and Innovation (MICINN) under grant ESP2007-65475-C02-01. Based on observations carried out at the European Southern observatory, Paranal (Chile), Programs 076.B-0479(A), 078.B-0072(A) and 081.B-0108(A). This research made use of the NASA/IPAC Extragalactic Database (NED), which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautic and Space Administration.
\end{acknowledgements}
\bibliographystyle{aa}
|
{
"timestamp": "2012-03-12T01:00:09",
"yymm": "1203",
"arxiv_id": "1203.1930",
"language": "en",
"url": "https://arxiv.org/abs/1203.1930"
}
|
\section{Introduction}
Recently the ATLAS~\cite{ATLAS-CONF-2011-163} and CMS~\cite{HIG-11-032} collaborations
reported event excesses, which may imply the Higgs boson with mass of about $125$\,GeV.
While it is difficult to explain the Higgs mass in
the minimal supersymmetric standard model (MSSM) as long as the
sparticle masses are around 1\,TeV~\cite{Okada:1990vk},
such a Higgs mass can be explained if the sparticles are heavier than multi-TeV~\cite{Okada:1990gg,Giudice:2011cg,Ibe:2011aa,Feng:2011aa}.
One of such scenarios is the anomaly-mediated SUSY breaking (AMSB) model~\cite{Giudice:1998xp},
where the sfermions and the gravitino are $\mathcal O(100$--$1000)$\,TeV and the gaugino masses are
$\mathcal O(100$--$1000)$\,GeV, given by the AMSB relation.
Phenomenological aspects of this scenario have been discussed in Refs.~\cite{Ibe:2011aa,Moroi:2011ab,Ibe:2012hu},
and it was shown that it is compatible with thermal leptogenesis~\cite{Fukugita:1986hr,Buchmuller:2003gz},
which requires the reheating temperature as high as $T_{\rm R}\gtrsim 10^9$\,GeV~\cite{Giudice:2003jh}.\footnote{
In this letter $T_{\rm R}$ is defined as $T_{\rm R}\equiv (10/\pi^2g_*)^{1/4}\sqrt{\Gamma_\phi M_P}$
where $\Gamma_\phi$ is the inflaton decay rate.
Note that this definition of $T_{\rm R}$ is smaller than that of Ref.~\cite{Giudice:2003jh}
by a factor $\sqrt{3}$.
}
Another attractive scenario is that all sparticles are $\mathcal O(10)$\,TeV in the gravity-mediated SUSY breaking.
The scenario alleviates the SUSY flavor/CP problems because of the heavy SUSY particles,
while it explains the 125\,GeV Higgs boson for $\tan\beta \gtrsim 5$~\cite{Giudice:2011cg}
and the present dark matter abundance
(see Ref.~\cite{Feng:2011aa} for realization in the focus-point region~\cite{Feng:1999mn}).
However, the scenario suffers from the cosmological Polonyi problem~\cite{Coughlan:1983ci,Banks:1993en},
since there must be a singlet SUSY breaking field, called the Polonyi field, in order to generate
sizable gaugino masses.
Although the Polonyi may decay before the big-bang nucleosynthesis (BBN) begins
for the Polonyi mass $m_z \gtrsim \mathcal O(10)$\,TeV, it releases a huge amount of entropy
because it dominates the Universe before the decay.
Thus the leptogenesis scenario does not work in this setup.
An interesting solution to the Polonyi problem was proposed long ago by
Linde~\cite{Linde:1996cx}. It was pointed out that, if the Polonyi field has a large
Hubble-induced mass, it follows a time-dependent potential minimum
adiabatically and the resultant amplitude of coherent oscillations is exponentially
suppressed. Recently, two of the present authors (FT and TTY) noticed that there
is an upper bound on the reheating temperature
for the adiabatic solution to work~\cite{Takahashi:2011as} and also showed that such a large Hubble mass may be a consequence of the strong dynamics at the Planck scale~\cite{Takahashi:2010uw}
or the fundamental cut-off scale one order of magnitude lower than the
Planck scale~\cite{Takahashi:2011as}. More important, the present authors found that there are generally additional
contributions to the Polonyi abundance which depends on the inflation energy scale, and
we showed that the Polonyi problem is still solved or greatly relaxed in high-scale inflation
models~\cite{Nakayama:2011wqa,Nakayama:2011zy}.
In this solution, we do not need any additional mechanism to dilute the Polonyi abundance.
Therefore, it may revive the conventional Polonyi model as a realistic SUSY breaking model,
which is fully compatible with the current experiments and observations, including the 125\,GeV Higgs boson.
In this letter we study the adiabatic solution in detail, considering various production processes of the
Polonyi field as well as the thermal and non-thermal gravitino production. In particular, we focus on a minimal model
in which only a certain coupling of the inflaton to the Polonyi field is enhanced. We also consider explicit inflation models
to see if there is an allowed parameter space where the Polonyi and gravitino problems are solved.
\section{The Polonyi model for gravity-mediation}
First we briefly review the cosmological Polonyi problem in the gravity mediation.
Let us denote the Polonyi field by $z$, which makes a dominant contribution to the SUSY breaking.
Its $F$-term is given by
$F_z = \sqrt{3}m_{3/2}M_P$ where $m_{3/2}$ is the gravitino mass and $M_P$ is the reduced Planck scale.
It generally couples to the MSSM superfields as
\begin{equation}
\mathcal L = \int d^4\theta \left( -c_Q^2 \frac{|z|^2|Q|^2}{M_P^2} \right )
+ \left( \int d^2\theta c_g \frac{z}{4M_P}W_aW^a+ {\rm h.c.} \right) ,
\label{KW}
\end{equation}
where $Q$ and $W^a$ collectively denote the matter and gauge superfields, respectively,
and $c_Q$ and $c_g$ are constants of order unity.
Here and hereafter, $c_Q$ and $c_g$ are taken to be real and positive, for simplicity.
These couplings give masses of order $m_{3/2}$ to the SUSY particles, as
\begin{equation}
m_{\tilde Q}^2 = (c_Q^2+1)m_{3/2}^2,~~~m_{\tilde g} = \frac{\sqrt{3}c_g}{2}m_{3/2}.
\end{equation}
Note that $z$ must be a singlet field in order to give a sizable mass to the gauginos.
The following term in the K\"ahler potential yields the sizable $\mu$ and $B$ terms~\cite{Giudice:1988yz},
\begin{equation}
K = \frac{c_h}{M_P}z^\dagger H_u H_d + {\rm h.c.},
\label{mu}
\end{equation}
as $\mu = \sqrt{3}c_h m_{3/2}$ and $B = m_{3/2}$.
Thus the framework naturally solves the $\mu/B\mu$ problem.
If one takes the gravitino mass to be as large as 10\,TeV, the SUSY flavor and CP problems
are greatly relaxed and the cosmological gravitino problem is also ameliorated.
It also explains the 125\,GeV Higgs boson without tuning the
$A$-parameter~\cite{Okada:1990gg,Giudice:2011cg,Ibe:2011aa,Feng:2011aa}.
Therefore the $\mathcal O(10)$\,TeV SUSY is plausible from these phenomenological point of view.
However, the model suffers from the cosmological Polonyi problem, which inevitably arises in the gravity-mediation scenario.
The Polonyi abundance is estimated as
\begin{equation}
\frac{\rho_z}{s} \simeq \frac{1}{8}T_{\rm R}\left( \frac{z_i}{M_P} \right)^2,
\label{rhoz}
\end{equation}
where $z_i$ is the initial amplitude, which is in general of the order of $M_P$. The reheating
temperature $T_{\rm R}$ is defined by
\begin{equation}
T_{\rm R}\;\equiv\; \lrfp{10}{\pi^2g_*}{1/4}\sqrt{\Gamma_{\rm tot} M_P},
\end{equation}
where $\Gamma_{\rm tot}$ is the inflaton decay rate, and $g_*$ counts the relativistic degrees
of freedom at the reheating. Here we have assumed that the potential for $z$ can be well
approximated by a quadratic term for $|z| \lesssim z_i$, and that the $z$ starts to oscillate before
the reheating.
The Polonyi abundance (\ref{rhoz}) is so large that the $z$
dominates the energy density of the Universe soon after the reheating, and causes
cosmological problems.
The Polonyi decays into gauge bosons through the interaction (\ref{KW}) with the decay rate given by
\begin{equation}
\Gamma(z \to gg) \simeq \frac{3 c_g^2}{32\pi} \frac{m_z^3}{M_P^2},
\end{equation}
where $m_z$ is the Polonyi mass at the zero temperature.
The decay into gauginos is suppressed by $(m_{\tilde g}/m_z)^2$ or $(m_{3/2}/m_z)^2$,
where $m_{\tilde g}$ denotes the gaugino mass, and as we will see later,
the Polonyi mass is considered to be slightly enhanced compared to $m_{\tilde g}$ or $m_{3/2}$.
We parametrize it as $m_z = c_z m_{3/2}$ with $c_z \gtrsim 1$.
The interaction (\ref{mu}) induces the decay of the Polonyi into the Higgs boson pair~\cite{Endo:2006ix},
\begin{equation}
\Gamma(z \to HH) \simeq \frac{c_h^2}{8\pi} \frac{m_z^3}{M_P^2},
\end{equation}
while the decay into a higgsino pair is suppressed by a factor of $(m_{3/2}/m_z)^2$.
The Polonyi also decays into a pair of gravitinos if kinematically allowed~\cite{Endo:2006zj}. The decay rate is given by
\begin{equation}
\Gamma(z \to \psi_{3/2}\psi_{3/2}) \simeq \frac{1}{96\pi} \frac{m_z^5}{m_{3/2}^2M_P^2}.
\end{equation}
For example, if the decay into gauge bosons is the dominant decay mode, the lifetime of the Polonyi is given by
\begin{equation}
\tau_z \simeq 1.3\times 10^{-1}c_g^{-2}\left( \frac{100\,{\rm TeV}}{m_z} \right)^3 \,{\rm sec}.
\end{equation}
If the decay into the gravitino pair is dominant, the lifetime is given by
\begin{equation}
\tau_z \simeq 1.2\times 10^{-2}\left( \frac{100\,{\rm TeV}}{m_z} \right)^5
\left( \frac{m_{3/2}}{10\,{\rm TeV}} \right)^2 \,{\rm sec}.
\end{equation}
The lifetime must be (much) shorter than 1\,sec in order not to spoil the success of BBN~\cite{Kawasaki:2004qu}.
Even if it decays before BBN, it dilutes the pre-existing baryon asymmetry of the Universe.
The dilution factor is roughly given by $\sim T_d / T_{\rm R}$, where $T_d$ is the Polonyi decay temperature.
The dilution factor is so large that thermal and non-thermal leptogenesis scenarios do not work.
Therefore some involved mechanisms to create the baryon asymmetry is required if
the Polonyi problem is solved by increasing the Polonyi mass.
In the next section we will consider another attractive solution to the Polonyi problem in which there is no
late-time entropy production.
\begin{figure}
\begin{center}
\includegraphics[width=0.6\linewidth]{polonyi1.eps}
\includegraphics[width=0.6\linewidth]{polonyi2.eps}
\caption{
Constraints on the $H_{\rm inf}$ and $T_{\rm R}$ plane from thermally produced gravitinos and
the Polonyi coherent oscillation for $m_{3/2}=$10\,TeV (upper panel)
and $50$\,TeV (lower panel) with $c_g=0.1$. We set $c_h = 1$.
The dotted line shows the upper bound from the gravitino thermal production.
The gray band shows the upper bound on the reheating temperature from the Polonyi
coherent oscillation and thermal production.
The upper edge of the band corresponds to $c_z =50, c_X = 100, \Delta z = 0.1M_P/c_X$
and lower one to $c_z =5, c_X = 50, \Delta z = M_P/c_X$.
In the lower panel, the constraint comes from the LSP overproduction, hence
all the constraints disappear if the R-parity is broken by a small amount.
}
\label{fig:1}
\end{center}
\end{figure}
\section{Solution to the Polonyi problem and implications}
Now we revisit the Polonyi model in light of the recent developments in the
suppression mechanism for the moduli abundance~\cite{Nakayama:2011wqa}.
Let us introduce the inflaton fields $X$ and $\phi$, which have $R$-charges of $+2$ and $0$, respectively.
The inflaton superpotential has the form
\begin{equation}
W = X f(\phi),
\end{equation}
where $f(\phi)$ is some function of $\phi$.
The $F$-term of $X$ dominates the potential energy during inflation.
Many known inflation models in supergravity fall into this category.
The Polonyi field in general couples to the inflaton fields as
\begin{equation}
K = - c_X^2\frac{|X|^2|z-z_X|^2}{M_P^2} - c_\phi^2\frac{|\phi|^2|z-z_\phi|^2}{M_P^2},
\end{equation}
where $c_X$ and $c_\phi$ are taken to be real and positive.
The adiabatic suppression mechanism works if $c_X \gg 1$~\cite{Takahashi:2010uw}.
However, the inflaton dynamics just after the inflation induces a non-negligible amount
of the coherent oscillations of the Polonyi field, which
is estimated as~\cite{Nakayama:2011wqa}
\begin{equation}
\frac{\rho_z}{s} \simeq \frac{1}{8}T_{\rm R}\left( \frac{\Delta z}{M_P} \right)^2
\left( \frac{c_\phi^4 m_z}{c_X^3 H_{\rm inf}} \right),
\label{CO}
\end{equation}
where $\Delta z = |z_X-z_\phi|$ and $H_{\rm inf}$ is the Hubble scale at the end of inflation.
This expression is valid for $c_\phi \gtrsim 1$.
For $c_\phi \ll 1$, there remains a contribution like (\ref{CO}) with $c_\phi$ replaced by $\mathcal O(1)$.
This is much smaller than the naive estimate (\ref{rhoz}) if $H_{\rm inf}\gg m_z$,
which is satisfied for the most known inflation models.
From this expression, we can see that the Polonyi abundance is suppressed
for $c_X/c_\phi \gg 1$ and large inflation scale.
Hereafter we take $c_\phi=1$ for simplicity.
Now let us see how the present model is constrained from cosmological arguments.
First, gravitinos are effectively produced at the reheating, and its abundance is proportional
to the reheating temperature.
If the gravitino is heavier than the lightest SUSY particle (LSP), it is unstable and decays
emitting energetic particles. Such late gravitino decay changes the Helium-4 abundance~\cite{Kawasaki:2004qu},
and produces LSPs non-thermally.
The Polonyi causes similar effects: the Polonyi decay may alter the standard BBN results and yield too many LSPs.
If the Polonyi decays into the gravitino, the subsequent gravitino decay also causes similar effects.
Notice that the Polonyi abundance is given by the sum of the coherent oscillation (\ref{CO})
and thermal production, the latter of which is comparable to the abundance of the (transverse components of) gravitino
if $c_g \sim 1$.
Fig.~\ref{fig:1} shows constraints on the $H_{\rm inf}$ and $T_{\rm R}$ plane from thermally produced gravitinos and
the Polonyi coherent oscillations and thermal production for $m_{3/2}=$10\,TeV and $c_g=0.1$ (upper panel)
and $50$\,TeV and $c_g=0.05$ (lower panel).
The choice of relatively small $c_g$ is motivated by the existence of the focus-point region,
and the Polonyi mainly decays into gravitinos in this case.
The dotted line shows the upper bound on $T_{\rm R}$ from the thermal production of gravitinos.
The gray band shows the upper bound on $T_{\rm R}$ from the Polonyi
coherent oscillation and thermal production, and the width of the band represents uncertainty of the Polonyi abundance
and couplings.
The upper edge of the band corresponds to $c_z =30, c_X = 100, \Delta z = 0.1M_P/c_X$, while
the lower one to $c_z =5, c_X = 50, \Delta z = M_P/c_X$,
where $m_z = c_z m_{3/2}$.
The Polonyi mass is varied because it is strongly coupled with the inflaton $X$ $(c_X \gg 1)$,
and the Polonyi self interaction of the form $K \sim -c_z^2|z|^4/M_P^2$ with $c_z \gg 1$ is expected
in the K\"ahler potential.
With the present parameter choice, the Bino is the LSP of mass $360$\,GeV (upper panel) and $900$\,GeV (lower panel).
The thermal relic abundance of the Bino LSP is not taken into account in Fig.~\ref{fig:1}, because it strongly depends on
the mass spectrum. For instance,
if it has a sizable mixing with higgsino or wino, the thermal relic abundance can be smaller than the present DM abundance.
(In the latter case, we need to relax the GUT relation on the gaugino mass.) We note that,
in the lower panel, the constraint comes from the LSP overproduction from the gravitino/Polonyi decay, hence
all the constraints disappear if the R-parity is broken by a small amount.
It is seen that the reheating temperature of $T_{\rm R}\simeq 10^9$\,GeV is allowed
for $H_{\rm inf} \gtrsim 10^9$--$10^{12}$\,GeV.
It is important that we do not need any additional late-time entropy production for solving the Polonyi problem.
Thus the conventional Polonyi model for the gravity-mediation for relatively heavy SUSY scale
of $\mathcal O(10)$\,TeV can be compatible with leptogenesis scenario
once we assume the Polonyi coupling to the inflaton is enhanced.
\section{Inflation model}
Now let us see if the above solution works in some known inflation models in supergravity.
In particular, we will show that there are consistent parameter regions where
thermal~\cite{Fukugita:1986hr} or non-thermal~\cite{Lazarides:1991wu,Kumekawa:1994gx,Asaka:1999yd} leptogenesis scenario works,
avoiding the Polonyi and gravitino problems.
\subsection{Hybrid inflation }
First, let us consider the SUSY hybrid inflation model~\cite{Copeland:1994vg,Dvali:1994ms,Linde:1997sj}.
The superpotential is given by
\begin{equation}
W = \kappa X (\phi\bar\phi - M^2) + W_0,
\end{equation}
where $W_0=m_{3/2}M_P^2$.
The waterfall fields, $\phi$ and $\bar\phi$, can be identified with the Higgs fields which break
U(1)$_{\rm B-L}$ gauge symmetry.
This model, including the constant term $W_0$, was analyzed in detail in Refs.~\cite{Buchmuller:2000zm,Senoguz:2004vu,Nakayama:2010xf}.
We assume that the inflaton dominantly decays into the right-handed neutrinos $N_i$ $(i=1,2,3)$ through the interaction
\begin{equation}
W = \frac{1}{2}y_{i} \phi N_i N_i.
\label{yNN}
\end{equation}
The inflaton decay rate into the right-handed (s)neutrino pair is given by
\begin{equation}
\Gamma (\phi \to N_1N_1,\tilde N_1\tilde N_1) \simeq \frac{1}{64\pi}y_{1}^2m_\phi,
\label{Gamma}
\end{equation}
where we have taken into account a mixing between $X$ and $\phi$ (and $\bar \phi$) due to the
constant term~\cite{Kawasaki:2006gs}.
Here we consider only the decay into the lightest right-handed neutrino.
On the other hand, the inflaton decays into a pair of gravitinos through the interaction in the
K\"ahler potential~\cite{Kawasaki:2006gs,Dine:2006ii,Endo:2006tf},
\begin{equation}
K = \frac{1}{2M_P^2}(c_{\phi zz}^2 |\phi|^2 +c_{\bar\phi zz}^2 |\bar\phi|^2) zz + {\rm h.c.}
\end{equation}
The decay rate into the gravitino pair is given by~\cite{Endo:2006tf}
\begin{equation}
\Gamma_{\rm grav}\equiv\Gamma(\phi \to \psi_{3/2}\psi_{3/2}) = \frac{1}{32\pi} \lrfp{c_{\phi zz}^2 + c_{\bar \phi zz}^2}{2}{2}
\left( \frac{\langle\phi\rangle}{M_P} \right)^2\frac{m_\phi^3}{M_P^2},
\end{equation}
where the mixing between $X$ and $\phi$ (and $\bar \phi$) is taken into account~\cite{Kawasaki:2006gs}.
Notice that the same interaction induces the inflaton decay into the Polonyi pair $(\phi\to zz)$ with the same decay rate.
Since each Polonyi field mainly decays into a pair of the gravitino,
the gravitino abundance produced non-thermally by the inflaton decay is given by
\begin{equation}
Y_{3/2} = \frac{3}{2}\frac{T_{\rm R}}{m_\phi} \frac{3\Gamma_{\rm grav}}{\Gamma_{\rm tot}},
\label{Ygrav}
\end{equation}
where the total decay rate is approximately given by $\Gamma_{\rm tot} \approx \Gamma (\phi \to NN)$.
This imposes severe constraints on the parameter space.
We have scanned parameters $(\kappa, M)$, which are rewritten in terms of
$H_{\rm inf}$ and $T_{\rm R}$ through the relation $H_{\rm inf}=\kappa M^2 /\sqrt{3}M_P$
and $T_{\rm R} = (10/\pi^2 g_*)^{1/4}\sqrt{\Gamma_{\rm tot}M_P}$.
We have also fixed $m_N=0.02m_\phi$: the non-thermal leptogenesis works for
$T_{\rm R}\gtrsim 10^8$\,GeV in this case.
Fig.~\ref{fig:hybrid} shows constraints in the $H_{\rm inf}$ and $T_{\rm R}$ plane for the hybrid inflation model
with $m_{3/2}=$10\,TeV (upper panel) and $50$\,TeV (lower panel).
The red dashed line shows the lower bound on $T_{\rm R}$ from the cosmic string.
The blue band shows the lower bound on $T_{\rm R}$
from the non-thermal gravitinos for $c_{\phi zz}=1$ (upper edge) and $0.1$ (lower edge).
The meanings of the gray band and the black dotted line are same as Fig.~\ref{fig:1} :
they set upper bounds on $T_{\rm R}$ from the Polonyi and thermal gravitino.
The density perturbation with a correct magnitude is generated on the solid line labels by ``WMAP normalization".
It is seen that there is a consistent parameter regions
around $H_{\rm inf}\sim 5\times 10^9$\,GeV and $T_{\rm R}\sim 10^9$\,GeV
where the Polonyi problem is solved within the framework of SUSY hybrid inflation model.
Note that the constraints from the Polonyi and gravitinos disappear if the R-parity is broken slightly for $m_{3/2} \gtrsim 30$\,TeV,
as already explained.
\begin{figure}
\begin{center}
\includegraphics[width=0.6\linewidth]{hybrid1.eps}
\includegraphics[width=.6\linewidth]{hybrid2.eps}
\caption{
Constraints in the $H_{\rm inf}$ and $T_{\rm R}$ plane for the hybrid inflation model
with $m_{3/2}=$10\,TeV (upper panel) and $50$\,TeV (lower panel).
The red dashed line shows the lower bound on $T_{\rm R}$ from the cosmic string.
The blue band shows the lower bound on $T_{\rm R}$
from the non-thermal gravitinos for $c_{\phi zz}=1$ (upper edge) and $0.1$ (lower edge).
The meanings of gray band and the black dotted line are same as Fig.~\ref{fig:1} :
they set upper bounds on $T_{\rm R}$ from the Polonyi and thermal gravitino.
The density perturbation with a correct magnitude is generated on the solid line
denoted by ``WMAP normalization".
}
\label{fig:hybrid}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=0.6\linewidth]{smooth1.eps}
\includegraphics[width=0.6\linewidth]{smooth2.eps}
\caption{
Same as Fig.~\ref{fig:hybrid}, but for the smooth-hybrid inflation model.
The blue band shows the lower bound on $T_{\rm R}$
from the non-thermal gravitinos for $c_{\phi zz}=1$ (upper edge) and $0.05$ (lower edge).
}
\label{fig:smooth}
\end{center}
\end{figure}
\subsection{Smooth hybrid inflation }
Let us consider the smooth-hybrid inflation model~\cite{Lazarides:1995vr} where
the inflaton superpotential is given by
\begin{equation}
W = X\left( \mu^2 - \frac{(\phi\bar\phi)^m}{M^{2m-2}} \right) + W_0, \label{smooth}
\end{equation}
where $m \geq 2$ is an integer.
The model has a discrete symmetry $Z_m$ under which $\phi \bar\phi$ has a charge $+1$ and $X$ has a zero charge.
This model has an advantage that it does not suffer from problematic topological defects formation
since the $\phi$ and $\bar\phi$ have nonzero VEVs during inflation and topological defects are inflated away.
Hereafter we consider the case of $m=2$ for simplicity.
Results do not much affected by this choice.
The gravitino abundance is similarly estimated by Eq.~(\ref{Ygrav}).
The inflaton can decay into ordinary particles through non-renormalizable interactions, for example,
\begin{equation}
K=\frac{|\phi|^2 H_u H_d}{M_c^2}+{\rm h.c.},
\end{equation}
with cutoff parameter $M_c$.
The decay rate into the Higgs boson and higgsino pair is given by
\begin{equation}
\Gamma(\phi \to HH) = \frac{1}{16\pi}
\left( \frac{\langle\phi\rangle}{M_c} \right)^2\frac{m_\phi^3}{M_c^2},
\end{equation}
where the mixing between $X$ and $\phi$ (and $\bar \phi$) is taken into account.
If the right-handed neutrino mass is not much smaller than the inflaton mass,
the decay rate into them through the operator $K=|\phi|^2|N|^2/M_c^2$ is comparable to the above expression.
We have scanned the parameters $(\mu, M)$, in the range $\mu < M$
so that the effective theory (\ref{smooth}) below the scale $M$ remains valid,
which are rewritten in terms of $H_{\rm inf}$ and $T_{\rm R}$ through the relation $H_{\rm inf}=\mu^2 /\sqrt{3}M_P$
and $T_{\rm R} = (10/\pi^2 g_*)^{1/4}\sqrt{\Gamma_{\rm tot}M_P}$.
We have fixed $M_c=6\times 10^{17}$\,GeV.
Fig.~\ref{fig:smooth} shows constraints on the $H_{\rm inf}$ and $T_{\rm R}$ plane for the hybrid inflation model
with $m_{3/2}=$10\,TeV (upper panel) and $50$\,TeV (lower panel).
The blue band shows the lower bound on $T_{\rm R}$
from the non-thermal gravitinos for $c_{\phi zz}=1$ (upper edge) and $0.05$ (lower edge).
The meanings of the gray band and the black dotted line are same as Fig.~\ref{fig:1} :
they set upper bounds on $T_{\rm R}$ from the Polonyi and thermal gravitino.
The WMAP normalization for the density perturbation is satisfied on the solid line.
The scalar spectral index $n_s$ also fits well with the WMAP result: $n_s \sim 0.968 \pm 0.012$~\cite{Komatsu:2010fb}.
It is seen that there is a consistent parameter regions
around $H_{\rm inf}\sim 10^{10-11}$\,GeV and $T_{\rm R}\sim 10^{8-9}$\,GeV
where the Polonyi problem and the gravitino problem are solved within the framework of smooth-hybrid inflation model.
Note again that the constraints from the Polonyi and gravitinos disappear if R-parity is broken slightly for $m_{3/2} \gtrsim 30$\,TeV,
as already explained.
\section{Conclusion}
We have revisited the Polonyi model for gravity mediation with a relatively high-scale SUSY breaking
of $\mathcal O(10)$\,TeV.
The Higgs boson mass of around 125\,GeV indicated by the recent LHC data is naturally explained
in this framework, while constraints from flavor/CP violating processes are alleviated.
The model, however, is plagued with the notorious cosmological Polonyi problem.
We have shown that the Polonyi problem is solved once we assume the relatively enhanced coupling
of the Polonyi to the inflaton. We have also considered explicit inflation models (hybrid and smooth hybrid
inflation), and shown that there is a parameter space where thermal and/or non-thermal leptogenesis scenarios work successfully,
avoiding the Polonyi and gravitino problems.
Thus, our solution revives the conventional Polonyi model as a realistic SUSY breaking model.
\begin{acknowledgments}
This work was supported by the Grant-in-Aid for Scientific Research on
Innovative Areas (No. 21111006) [KN and FT], Scientific Research (A)
(No. 22244030 [KN and FT], 21244033 [FT], 22244021 [TTY]), and JSPS Grant-in-Aid for
Young Scientists (B) (No. 21740160) [FT]. This work was also
supported by World Premier International Center Initiative (WPI
Program), MEXT, Japan.
\end{acknowledgments}
|
{
"timestamp": "2012-03-12T01:01:37",
"yymm": "1203",
"arxiv_id": "1203.2085",
"language": "en",
"url": "https://arxiv.org/abs/1203.2085"
}
|
\section{Introduction}
Behavioral realism has been one of the promising directions in the development of on-screen conversational agents and robots capable of natural language dialogue (see \cite{RichSidner2009} for an overview). For example, interactions with a robot receptionist that evoke user's social response are associated with better engagement and lower rate of breakdowns during information-seeking dialogues~\cite{Simmons2011}. A necessary step in designing such interactions is to identify behaviors with a potential to evoke a desired user response.
Data sources that can be used to harvest behavior candidates include ethnographic and controlled studies. Ethnographic studies provide an opportunity for collection of naturalistic conversational data, but often face the issues of unclear sample population and coarse granularity of captured data~\cite{BeebeCummings1996}. On the other hand, collecting high resolution data in a controlled setting may hamper spontaneity and naturalness of the interaction. In general, data collection methodology can influence both the sociopragmatic choices, namely, what speech act to say, and their pragmalinguistic realization, namely, how to say it (see~\cite{BeebeCummings1996} for a discussion).
These methodological difficulties, combined with the challenges of annotating multimodal data, result in the lack of annotated corpora of naturalistic interactions for many scenarios that are currently relevant for human-robot interaction research. The corpus of role plays between a visitor and a receptionist in a realistic environment that we present in this paper attempts to help fill this gap.
In the next section, we describe related work on corpora of service encounters. After that, we introduce our data collection methodology and the annotation scheme we use. We conclude with the discussion of possible uses of the corpus.
\section{Corpora of service encounters}
Audio corpora of human service encounters have been used for analysis of linguistic and paralinguistic features, such as timing and prosody. For example, Vienna-Oxford International Corpus of English~(VOICE) \cite{voice} includes service encounters between speakers of English as a lingua franca. Audio recordings of Syrian shopping interactions were collected and analyzed by Traverso~\cite{Traverso2001}. Service encounters gathered in public offices and shops of Catalonia were examined with respect to how bilinguals negotiate code (language) of their interaction. Audio recordings have been used to analyze politeness strategies in shopping interactions (see, for example, \cite{Kong2010}).
The importance of gaze (see~\cite{Montague2011} for an overview) and smile (see, for example, \cite{Kim2009}) in defining the outcome of the service interactions suggest the need for capturing and studying nonverbal behaviors in videos.
For instance, customers reported higher satisfaction when they interacted face-to-face with a bank teller who responded with contingent smile, rather than constant neutral or constant smiling expression~\cite{Kim2009}. The same data showed that amused and polite smiles differ with respect to their temporal properties~\cite{Hoque2011}. Analysis of verbal and nonverbal expressions in the videos of interethnic encounters of Korean retailers with Korean and African-American customers showed that these language communities had different perception of function of socially minimal and socially expanded encounters~\cite{Bailey1997}.
Receptionist interactions, a subtype of service encounters, were analyzed with respect to their verbal content via role plays in~\cite{Chee2010}. Hewitt et al.~\cite{Hewitt2009} conducted discourse analysis of dialogues involving hospital receptionists. The openly accessible CUBE-G corpus of nonverbal behaviors from role plays of German and Japanese participants covers scenarios that may be relevant for service encounters, including first meeting, negotiation and status difference~\cite{Rehm2009}. The original Map Task~\cite{Anderson1991} and followup projects collect direction-giving dialogues that may be relevant to some receptionist encounters.
We were not able to find any nonverbal corpora of human receptionist interactions. With respect to availability, among all the corpora mentioned above only VOICE, CUBE-G and Map Task related corpora are freely accessible. Hence, our corpus may be the first annotated corpus of nonverbal behaviors in receptionist interactions, and the first nonverbal corpus (excluding the original video and audio data) of service encounters freely available online~\cite{ReceptionistCorpus2012}.
\section{Data collection}
\subsection{Participants}
We recruited via emails and posters in Education City, in Doha, Qatar and via announcements posted on bulletin boards across CMU campus in Pittsburgh, USA. The recruitment materials specified that we were looking for native speakers of American English or Arabic. Majority of the participants (17 of 22) were university students, staff, or faculty. The participants filled demographic surveys and evaluated themselves on ten-item personality inventory (TIPI)~\cite{Gosling2003} and 20-item positive and negative affect scale (PANAS)~\cite{Watson1988}. The distribution of participants is shown in Table~\ref{table:participants}.
\begin{table}[htb]
\begin{center}
\begin{tabular}{|l|l|l|l|}
\hline
\multirow{4}{*}{Doha} & \multirow{2}{*}{Arabic} & Females & 2\\
& & Males & 6 \\ \cline{2-4}
& \multirow{2}{*}{American English} & Females & 2\\
& & Males & 3 \\ \hline
\multirow{4}{*}{Pittsburgh} & \multirow{2}{*}{Arabic} & Females & 1 \\
& & Males & 1\\ \cline{2-4}
& \multirow{2}{*}{American English} & Females & 5\\
& & Males & 1 \\ \hline
\end{tabular}
\end{center}
\caption{Distribution of participants between Doha and Pittsburgh experiment sites}
\label{table:participants}
\end{table}
People apply different criteria when they report their native language and mother tongue~\cite{Laitin2000}. To control for this, we asked the participants to list the countries they lived in for more than a year, and their age at the time of moving in and out of the country. All but 3 participants (who were all in the American English condition in Doha) spent the majority of their lives in the country where their native language is a primary spoken language. A female participant in Doha changed her reported native language from American English to Tulu, after asking the experimenter a clarification question. Her data remains in the corpus although she is not included in the Table~\ref{table:participants}.
Mean age of participants in Doha was 25 years ($SD=7.8$). In Pittsburgh, average age was 28.7 years ($SD=12.7$). Native speakers of Arabic were on average 23.2 years old ($SD=4.2$), while average age of native speakers of American English was 30.9 years ($SD=12.5$).
\subsection{Procedure}
After filling out the questionnaires, one of the participants was asked to play the role of a receptionist while another was asked to imagine themselves as a first-time visitor looking for a particular location inside the building. The location was picked by the experimenter from the following list: library, restroom, cafeteria, student recreation room, a professor A's office, etc. Visitors were asked to seek help of the receptionist for directions using English and then to proceed towards their destination.
Most of the participant pairs were not familiar with each other. The fact of familiarity, when clear, is noted in the annotations. Similarly, the annotations include information on whether the participant has a thorough (works or studies inside the building) or passing (works or studies in a nearby building) familiarity with the experiment site.
In both sites, the receptionist would occupy the actual receptionist area in the lobby of the building. In Doha, on-duty security guards were present in the vicinity of the reception desk.
Each pair of participants would have 2-3 interactions with one of the subjects as a receptionist, and then they would switch roles and have 2 or 3 more interactions, depending on allotted time. After that, the participants were debriefed on their experiences. Overall, more than 60 interactions were recorded.
The interactions were recorded with 2 or 3 consumer-level high definition cameras. Visitor and receptionist were each dedicated a camera capturing their torso, arms and face that was positioned about 45 degrees off their default line of sight (namely, the line of sight that is perpendicular to the front edge of the rectangular reception desk). Most of the interactions would have a third camera capturing the side view of the scene. All cameras were in plain view. In addition to the audio captured by the cameras, an audio recorder (iPod) was placed on the receptionist desk.
\section{Annotation scheme}
The main goal of our corpus is to analyze occurrences and timing of verbal and nonverbal behaviors. Consequently, we have chosen to annotate the data at the level of granularity that minimizes the coding effort while at the same time allowing to capture timing and major features of communicative events. For example, instead of annotating each of preparation, hold, stroke, and retraction phases of a hand gesture~\cite{Kita1998} we annotate an interval between beginnings of the stroke and retraction phases. Similarly, facial expression are annotated as intervals approximately from the beginning of rise to the beginning of decay \cite{Hoque2011} phases, with some error inherent to manual annotation. The annotation scheme, developed in the process of annotating the corpus, is summarized in Table~\ref{table:annotation}.
\begin{table}[htb]
\begin{center}
\begin{tabular}{|l|p{2in}|}
\hline
Modality & Values \\ \hline\hline
Speech & Transcribed utterances, including non-words \\ \hline
Eye gaze & Pointing (self-initiated), pointing (following interlocutor), focus (interlocutor, guard, desktop, down, up, left, right, front, back, scattered, destination)\\ \hline
Face & smile (open or closed mouth)\\ \hline
Head & nod, half nod, double nod, multiple nod, upward nod, multiple upward nod, micro nod, shake \\ \hline
Hand & Pointing (left or right hand), finger only \\ \hline
Torso & Sitting, standing, focus (left, right, front, back, destination, interlocutor, desk) \\ \hline
\end{tabular}
\end{center}
\caption{Annotation scheme}
\label{table:annotation}
\end{table}
Coding nonverbal expressions, as well as transcribing ambiguous speech involves a degree of subjectivity. For example, the exact point of gaze fixation within the recipient's face is hard to identify even by the recipient himself~\cite{Cranach1973}. In fact, a typical direct eye contact consists of a sequence of fixations on different points on the face~\cite{Cook1977}. Since it is unclear whether the exact fixation pattern has any influence on social communication, in this study we do not distinguish between different fixation points within the general face area (neither does the video fidelity allow that). We plan to validate the annotations by employing a second annotator.
The annotation is done using the multi-track video annotation tool Advene~\cite{Aubert2005}.
\section{Discussion}
While the small number of individual participants makes this corpus unsuitable for cross-subject analysis, the multiple trials may be accounted for by mixed-effects models~\cite{PinheiroBates2000}. More appropriately, the corpus should be used for qualitative analysis and formation of hypothesis for further studies. For example, compare the gaze behaviors of a native Arabic-speaking female S4 (Subject 4) playing a receptionist responding to native Arabic-speaking male S1 playing a visitor (Fig.~\ref{fig:v1r4i3}) versus the dialogue with the subjects' roles reversed (Fig.~\ref{fig:v4r1i1}). Notice that both subjects gazed at their interlocutor more in the visitor role. This appears to be a trend that can be explained in part by the receptionist looking towards the destination during the direction-giving speech, while the visitor may continue looking at the receptionist.
Now, compare a receptionist gaze of S4 (Fig.~\ref{fig:v1r4i3}) with one of S12 (Fig.~\ref{fig:v11r12i3}), who is a female native speaker of American English. Notice the short glances that punctuate fragments of the directions sequence spoken by S12. These glances appear to precede visitor's backchannels and therefore may play a role in connection events~\cite{Sidner2010}. Receptionist S4, on the contrary, did not glance at the visitor until the very end of the directions sequence. These different gaze behaviors may reflect individual styles, genders and cultures of receptionist-visitor pairs, or levels of comfort and expertise, among other possibilities. Further, more controlled, studies may address these hypothesis.
\begin{figure*}[tb]
\centering
\includegraphics[width=6in]{./timeline_v1r4i3_header.pdf}
\caption{Interaction between S1 as a visitor and S4 as a receptionist. Wide vertical stripes represent intervals of speech. Narrow vertical stripes represent (from left to right): intervals of visitor's and receptionist's gaze towards the direction pointed by the receptionist, and visitor's and receptionist's gaze towards each other. Color coding of these modalities is specified by the icons in the upper part of the plots.}
\label{fig:v1r4i3}
\end{figure*}
\begin{figure*}[tb]
\centering
\includegraphics[width=6in]{./timeline_v4r1i1_header.pdf}
\caption{Interaction between S4 as a visitor and S1 as a receptionist.
Wide vertical stripes represent intervals of speech. Narrow vertical stripes represent (from left to right): intervals of visitor's and receptionist's gaze towards the direction pointed by the receptionist, and visitor's and receptionist's gaze towards each other.
Color coding of these modalities is specified by the icons in the upper part of the plots.}
\label{fig:v4r1i1}
\end{figure*}
\begin{figure*}[tb]
\centering
\includegraphics[width=6in]{./timeline_v11r12i3_header.pdf}
\caption{Interaction between S11 as a visitor and S12 as a receptionist. The visitor's eye gaze for this particular dialogue is partially inferred from his head gaze.
Wide vertical stripes represent intervals of speech. Narrow vertical stripes represent (from left to right): intervals of visitor's and receptionist's gaze towards the direction pointed by the receptionist, and visitor's and receptionist's gaze towards each other.
Color coding of these modalities is specified by the icons in the upper part of the plots.}
\label{fig:v11r12i3}
\end{figure*}
\section{Acknowledgments}
This publication was made possible by the support of an NPRP grant from the Qatar National Research Fund. The authors would like to express their gratitude to Michael Agar, Mark Barker, Justine Cassell, Anwar El-Shamy, Ismet Hajdarovic, Alicia Holland, Carol Miller, Dudley Reynolds, Michele de la Reza, Candace Sidner, Mark Stehlik, Mark C. Thompson, security and receptionist staff of CMU Qatar, and the study participants.
\bibliographystyle{abbrv}
|
{
"timestamp": "2012-03-13T01:01:45",
"yymm": "1203",
"arxiv_id": "1203.2299",
"language": "en",
"url": "https://arxiv.org/abs/1203.2299"
}
|
\section{Derivation of Eq.\eqref{eq:objfunc_blktrans}}
\label{sec:app_blktrans}
With the constraint of Eq.\eqref{eq:constraint_g}, we have
\begin{align}
& p(Z_{t+1}^j|Z_t^i) \nonumber \\
= & \sum \limits_{r, q} p(Q_{t+1}^r, Z_{t+1}^j | Q_t^q, Z_t^i) p(Q_t^q | Z_t^i) \nonumber \\
= & p(Q_{t+1}^{g(j)}, Z_{t+1}^j | Q_t^{g(i)}, Z_t^i) \nonumber \\
\label{eq:tran_cnvrt}
= & p(Q_{t+1}^{g(j)} | Q_t^{g(i)}) p(Z_{t+1}^j | Q_{t+1}^{g(j)}, Z_t^i) \\
= & p(Q_{t+1}^{g(j)} | Q_t^{g(i)}) \frac{p(Z_{t+1}^j, Q_{t+1}^{g(j)} | Z_t^i)}{p(Q_{t+1}^{g(j)} | Z_t^i)}\nonumber \\
= & p(Q_{t+1}^{g(j)} | Q_t^{g(i)}) \frac{p(Q_{t+1}^{g(j)} | Z_t^i, Z_{t+1}^j) p(Z_{t+1}^j | Z_t^i)}
{\sum_{j'} p(Q_{t+1}^{g(j)} | Z_t^i, Z_{t+1}^{j'}) p(Z_{t+1}^{j'} | Z_t^i)} \nonumber \\
= & p(Q_{t+1}^{g(j)} | Q_t^{g(i)}) \frac{p(Z_{t+1}^j | Z_t^i)}
{\sum_{j' \in \mathcal{G}(g(j))} p(Z_{t+1}^{j'} | Z_t^i)} \nonumber
\end{align}
Equivalently,
\begin{equation}
\label{eq:cond_QZ2Z}
p(Q_{t+1}^{g(j)} | Q_t^{g(i)}) = \sum_{j' \in \mathcal{G}(g(j))} p(Z_{t+1}^{j'} | Z_t^i), \;\;\; \forall i, j
\end{equation}
Eq.\eqref{eq:tran_cnvrt} shows that we can eliminate $Q$ from the substructure transition model,
which results in the simplified objective function in Eq.\eqref{eq:objfunc_blktrans}.
Eq.\eqref{eq:cond_QZ2Z} leads to the equality constraint in Eq.\eqref{eq:objfunc_blktrans}.
\section{Derivation of Eq.\eqref{eq:theta_hat}}
\label{sec:app_thetahat}
From the KKT conditions, we have:
\begin{eqnarray}
& & \xi_{ij} - (\lambda_{i, g(j)} - \gamma_i + \mu_{ij})\hat{\theta}_{ij} = 0 \nonumber \\
& \Rightarrow & \xi_{ij} - (\lambda_{i, g(j)} - \gamma_i)\hat{\theta}_{ij} = 0 \nonumber \\
& \Rightarrow & \hat{\theta}_{ij} \propto \xi_{ij}, \;\;\; \forall i, r, j \in \mathcal{G}(r) \nonumber \\
& \Rightarrow & \hat{\theta}_{ij} = \hat{\phi}_{g(i), g(j)} \frac{\xi_{ij}}{\sum_{j' \in \mathcal{G}(g(j))} \xi_{ij'}} \nonumber
\end{eqnarray}
\begin{eqnarray}
& & -\alpha_{qr} + \Sigma_{i \in \mathcal{G}(q)} \lambda_{ir}\hat{\phi}_{qr} = 0 \nonumber \\
& \Rightarrow & -\alpha_{qr} + \Sigma_{i \in \mathcal{G}(q), j \in \mathcal{G}(r)} \lambda_{ir} \hat{\theta}_{ij} = 0 \nonumber \\
& \Rightarrow & -\alpha_{qr} + \Sigma_{i \in \mathcal{G}(q), j \in \mathcal{G}(r)}
\left( \xi_{ij} + \gamma_i \hat{\theta}_{ij} \right) = 0 \nonumber \\
& \Rightarrow & \Sigma_{i \in \mathcal{G}(q), j \in \mathcal{G}(r)} \xi_{ij} -\alpha_{qr} +
\Sigma_{i \in \mathcal{G}(q)} \gamma_i \hat{\phi}_{qr} = 0 \nonumber \\
& \Rightarrow & \hat{\phi}_{qr} \propto \Sigma_{i \in \mathcal{G}(q), j \in \mathcal{G}(r)} \xi_{ij} -\alpha_{qr} \nonumber
\end{eqnarray}
Note that $\hat{\phi}_{qr} \geq 0, \sum_{r}\hat{\phi}_{qr}=1$, and we obtain the second equation in
Eq.\eqref{eq:theta_hat}.
\section{Derivation of Eq.\eqref{eq:rbpf_sdz} and \eqref{eq:int_norm_logit}}
\label{sec:app_rbpfsdz}
Denote $\mathbf{X}_t=(S_t, D_t, Z_t, X_t)$, and we have:
\begin{align}
& p(\mathbf{X}_t | \mathbf{y}_{1:t}) \nonumber \\
= & \int \frac{p(\mathbf{y}_t|\mathbf{X}_t) p(\mathbf{X}_t|\mathbf{X}_{t-1}) p(\mathbf{X}_{t-1} | \mathbf{y}_{1:t-1})}
{p(\mathbf{y}_t|\mathbf{y}_{1:t-1})} d \mathbf{X}_{t-1} \nonumber \\
\propto & \int \sum_n w^{(n)}_{t-1} \delta_{S_{t-1}}(s^{(n)}_{t-1}) \delta_{D_{t-1}}(d^{(n)}_{t-1})
\delta_{Z_{t-1}}(z^{(n)}_{t-1}) \nonumber \\
& \;\; \times \chi_{t-1}^{(n)}(X_{t-1}) p(\mathbf{y}_t|\mathbf{X}_t) p(\mathbf{X}_t|\mathbf{X}_{t-1})
d \mathbf{X}_{t-1} \nonumber \\
\propto & \sum_n w^{(n)}_{t-1} p(\mathbf{y}_t|S_t, Z_t, X_t) p(S_t|D_t, s^{(n)}_{t-1}) \nonumber \\
& \;\; \times p(Z_t | S_t, D_t, z^{(n)}_{t-1}) \int p(D_t | d^{(n)}_{t-1}, s^{(n)}_{t-1}, z^{(n)}_{t-1}, \mathbf{x}_{t-1}) \nonumber \\
& \;\; \times p(X_t|\mathbf{x}_{t-1}, S_t, Z_t) \chi_{t-1}^{(n)}(\mathbf{x}_{t-1}) d \mathbf{x}_{t-1} \nonumber
\end{align}
Taking integral with respect to $X_t$, we get:
\begin{align}
& p(S_t, D_t, Z_t | \mathbf{y}_{1:t}) \nonumber \\
\propto & \sum_n w^{(n)}_{t-1} p(S_t| D_t, s^{(n)}_{t-1}) p(Z_t | S_t, D_t, z^{(n)}_{t-1}) \nonumber \\
& \;\; \times \int \int p(\mathbf{y}_t|S_t, Z_t, \mathbf{x}_t) p(\mathbf{x}_t|\mathbf{x}_{t-1}, S_t, Z_t) d \mathbf{x}_t \nonumber \\
& \;\; \times p(D_t|d^{(n)}_{t-1}, s^{(n)}_{t-1}, z^{(n)}_{t-1}, \mathbf{x}_{t-1}) \chi_{t-1}^{(n)}(\mathbf{x}_{t-1})
d \mathbf{x}_{t-1} \nonumber
\end{align}
Eq.\eqref{eq:rbpf_sdz} and \eqref{eq:int_norm_logit} can be obtained by
replacing the inner integral with $p(\mathbf{y}_t|\mathbf{x}_{t-1}, S_t, Z_t)$.
\section{Evaluation of Eq.\eqref{eq:int_norm_logit}}
\label{sec:app_logit}
From Eq.\eqref{eq:lds_y} and Eq.\eqref{eq:lds_x}, we have:
\begin{align}
p(\mathbf{y}_t | S^i_t, Z^j_t, \mathbf{x}_t) & = \mathcal{N}(\mathbf{y}_t; \mathbf{B}^{ij} \mathbf{x}_t, \mathbf{R}^{ij}) \nonumber \\
p(\mathbf{x}_t|\mathbf{x}_{t-1}, S^i_t, Z^j_t) & = \mathcal{N}(\mathbf{x}_t; \mathbf{A}^{ij} \mathbf{x}_{t-1}, \mathbf{Q}^{ij}) \nonumber
\end{align}
which leads to:
\begin{align}
& p(\mathbf{y}_t | \mathbf{x}_{t-1}, S^i_t, Z^j_t) \nonumber \\
= & \mathcal{N}(\mathbf{y}_t; \mathbf{B}^{ij} \mathbf{A}^{ij} \mathbf{x}_{t-1},
\mathbf{B}^{ij} \mathbf{Q}^{ij} {\mathbf{B}^{ij}}^T + \mathbf{R}^{ij}) \nonumber \\
= & \mathcal{N}(\mathbf{y}_t; \bm{\mu}_Y, \mathbf{\Sigma}_Y)
=\mathcal{N}(\mathbf{y}_t; \mathbf{A}\mathbf{x}_{t-1}, \mathbf{\Sigma}_Y) \nonumber
\end{align}
where $\bm{\mu}_Y \triangleq \mathbf{B}^{ij} \mathbf{A}^{ij} \mathbf{x}_{t-1}$,
$\mathbf{\Sigma}_Y \triangleq \mathbf{B}^{ij} \mathbf{Q}^{ij} {\mathbf{B}^{ij}}^T + \mathbf{R}^{ij}$,
and $\mathbf{A} \triangleq \mathbf{B}^{ij} \mathbf{A}^{ij}$.
We also define, $\bm{\mu}_X \triangleq \hat{\mathbf{x}}^{(n)}_{t-1}$,
$\mathbf{\Sigma}_X \triangleq \mathbf{P}^{(n)}_{t-1}$, and have:
\begin{equation}
\chi_{t-1}^{(n)}(\mathbf{x}_{t-1}) = \mathcal{N}(\mathbf{x}_{t-1}; \bm{\mu}_X, \mathbf{\Sigma}_X) \nonumber
\end{equation}
The above two Gaussian distributions, \emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot the first two terms in the integral of Eq.\eqref{eq:int_norm_logit},
can be combined as a single Gaussian of $\mathbf{x}_{t-1}$.
Omit all the subscriptions, and the product of exponential terms is:
\begin{align}
& (\mathbf{x}-\bm{\mu}_X)^T \mathbf{\Sigma}^{-1}_X (\mathbf{x}-\bm{\mu}_X)
+ (\mathbf{y}-\mathbf{A}\mathbf{x})^T \mathbf{\Sigma}^{-1}_Y (\mathbf{y}-\mathbf{A}\mathbf{x}) \nonumber \\
= & \mathbf{x}^T\mathbf{\Sigma}^{-1}_X\mathbf{x} + \mathbf{x}^T\mathbf{A}^T\mathbf{\Sigma}^{-1}_Y\mathbf{A}\mathbf{x}
- 2\mathbf{x}^T\mathbf{\Sigma}^{-1}_X\bm{\mu}_X \nonumber \\
&- 2\mathbf{x}^T\mathbf{A}^T\mathbf{\Sigma}^{-1}_Y\mathbf{y} + c_1 \nonumber \\
= & (\mathbf{x}-\bm{\mu})^T \mathbf{\Sigma}^{-1} (\mathbf{x}-\bm{\mu}) + c_2 \nonumber
\end{align}
where
\begin{align}
\mathbf{\Sigma}^{-1} & = \mathbf{\Sigma}^{-1}_X + \mathbf{A}^T\mathbf{\Sigma}^{-1}_Y\mathbf{A} \nonumber \\
\bm{\mu} & = \mathbf{\Sigma}(\mathbf{\Sigma}^{-1}_X\bm{\mu}_X + \mathbf{A}^T\mathbf{\Sigma}^{-1}_Y\mathbf{y}) \nonumber \\
c_1 & = \bm{\mu}^T_X\mathbf{\Sigma}^{-1}_X\bm{\mu}_X + \mathbf{y}^T\mathbf{\Sigma}^{-1}_Y\mathbf{y} \nonumber \\
c_2 & = -\bm{\mu}^T\mathbf{\Sigma}^{-1}\bm{\mu} + c_1 \nonumber
\end{align}
Therefore, the product of two Gaussian is:
\begin{align}
& \mathcal{N}(\mathbf{x}; \bm{\mu}_X, \mathbf{\Sigma}_X)
\times \mathcal{N}(\mathbf{y};\bm{\mu}_Y, \mathbf{\Sigma}_Y) \nonumber \\
= & \frac{1}{\sqrt{(2\pi)^{d_Y} \det(\mathbf{\Sigma}_Y) }} \sqrt{\frac{\det(\mathbf{\Sigma})}{\det(\mathbf{\Sigma}_X)}}
\exp\left\{-\frac{1}{2}c_2\right\} \nonumber \\
& \times \frac{1}{\sqrt{(2\pi)^{d_X} \det(\mathbf{\Sigma}) }}
\exp\left\{-\frac{1}{2} (\mathbf{x}-\bm{\mu})^T \mathbf{\Sigma}^{-1} (\mathbf{x}-\bm{\mu}) \right\} \nonumber \\
= & c_3 \cdot \mathcal{N}(\mathbf{x}; \bm{\mu}, \mathbf{\Sigma}) \nonumber
\end{align}
where
\begin{equation}
c_3 = \frac{e^{-c_2/2}}{\sqrt{(2\pi)^{d_Y}}} \sqrt{\frac{\det(\mathbf{\Sigma})}{\det(\mathbf{\Sigma}_X)\det(\mathbf{\Sigma}_Y)}}
\nonumber
\end{equation}
The third term in the integral of Eq.\eqref{eq:int_norm_logit}, defined in Eq.\eqref{eq:pD_trans_disc},
can be re-written as:
\begin{align}
& p(D_t^{d+1}|D_{t-1}^d, S_{t-1}^k, Z^l_{t-1}, X^{\mathbf{x}}_{t-1}) \nonumber \\
= & \frac{1}{1+e^{\nu_k (d-\beta_k) + \bm{\omega}_{kl}^T \mathbf{x}}} \nonumber \\
= & \frac{1}{1+e^{-\left[\beta + \bm{\omega}^T \mathbf{x} \right]}} \nonumber \\
= & \mathcal{F}\left(\vec{\bm{\omega}}^T \mathbf{x} + \frac{\beta}{||\bm{\omega}||}; \frac{1}{||\bm{\omega}||} \right) \nonumber
\end{align}
where $\beta=-\nu_k (d-\beta_k)$, $\bm{\omega}=-\bm{\omega}_{kl} = ||\bm{\omega}||\cdot\vec{\bm{\omega}}$,
and $\mathcal{F}(x;\alpha)=\frac{1}{1+e^{-x/\alpha}}$ is logistic (or Fermi) function.
The probability for $p(D_t^{1}|\cdot)$ can be obtained accordingly.
To convert the integral in Eq.\eqref{eq:int_norm_logit} into a single variable integral,
we further introduce a linear transformation:
\begin{equation}
\mathbf{v} = \mathbf{W}^T \mathbf{x} \nonumber
\end{equation}
where $\mathbf{W}^T\mathbf{W}=\mathbf{I}$ is orthonormal, and $\mathbf{W}(:, 1)=\bm{\vec{\omega}}$.
For Gaussian variable, we have:
\begin{align}
& (\mathbf{x}-\bm{\mu})^T \mathbf{\Sigma}^{-1} (\mathbf{x}-\bm{\mu}) \nonumber \\
= & (\mathbf{W}\mathbf{v}-\bm{\mu})^T \mathbf{\Sigma}^{-1} (\mathbf{W}\mathbf{v}-\bm{\mu}) \nonumber \\
= & (\mathbf{v}-\mathbf{W}^T\bm{\mu})^T \mathbf{W}^T\mathbf{\Sigma}^{-1}\mathbf{W} (\mathbf{v}-\mathbf{W}^T\bm{\mu}) \nonumber
\end{align}
Therefore,
\begin{equation}
\mathcal{N}(\mathbf{x}; \bm{\mu}, \mathbf{\Sigma}) =
\mathcal{N}(\mathbf{v}; \mathbf{W}^T\bm{\mu}, \mathbf{W}^T\mathbf{\Sigma}\mathbf{W}) \nonumber
\end{equation}
Now we are ready to evaluate Eq.\eqref{eq:int_norm_logit} as:
\begin{align}
& c_3 \int \mathcal{N}(\mathbf{x}; \bm{\mu}, \mathbf{\Sigma})
\mathcal{F}\left( \bm{\vec{\omega}}^T \mathbf{x} + \beta/||\bm{\omega}||; 1/||\bm{\omega}|| \right)
d\mathbf{x} \nonumber \\
= & \frac{c_3}{|\det(\mathbf{W}^T)|} \int \mathcal{N}(\mathbf{v}; \mathbf{W}^T\bm{\mu}, \mathbf{W}^T\mathbf{\Sigma}\mathbf{W})
\nonumber \\
& \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \times \mathcal{F}\left( v_1 + \beta/||\bm{\omega}||; 1/||\bm{\omega}|| \right)
d\mathbf{v} \nonumber \\
= & c_3 \int \mathcal{N}(v_1; \bm{\vec{\omega}}^T\bm{\mu}, \bm{\vec{\omega}}^T\mathbf{\Sigma}\bm{\vec{\omega}})
\mathcal{F}\left( v_1 + \beta/||\bm{\omega}||; 1/||\bm{\omega}|| \right) d{v_1} \nonumber \\
= & c_3 \int \mathcal{N}(v; 0, \bm{\vec{\omega}}^T\mathbf{\Sigma}\bm{\vec{\omega}})
\mathcal{F}\left( v + \beta/||\bm{\omega}|| + \bm{\vec{\omega}}^T\bm{\mu}; 1/||\bm{\omega}|| \right) d{v} \nonumber \\
\approx & c_3 \cdot \mathcal{F}\left( \frac{1}{||\bm{\omega}||}(\beta + \bm{\omega}^T\bm{\mu});
\sqrt{1+\frac{\pi}{8} \bm{\omega}^T\mathbf{\Sigma}\bm{\omega}} \right) \nonumber
\end{align}
where the approximation follows from \cite{maragakis08}.
\section{Conclusion and Future Work}
\label{sec:con}
In this paper, we introduce an improved SSM with two added layers
modeling the substructure transition dynamics and duration
distribution for human action.
The first layer encodes the sparse and global temporal transition
structure of action primitives and also maintains action variations.
The second layer injects discriminative information into a logistic duration model
and discovers action boundaries more adaptively.
We design a Rao-Blackwellised particle filter for efficient inference.
Our comprehensive experimental results validate the effectiveness of both two
layers of our model in continuous action recognition.
As future work we plan to apply our model to actions in less constrained
scenarios and use more advanced low-level descriptors to deal with
unreliable observations.
\section{Experimental Results}
\label{sec:exp}
Our model is tested on four datasets for continuous action recognition.
In all the experiments, we have used parameters $N_Q=3$, $N_Z=5$, $N_P=200$.
First STM is trained independently for each action
using the segmented sequences in training set; then DBM is learned
from the inferred terminal stage of each sequence. The overall learning procedure
follows EM paradigm where the beginning and terminating stages are initially
set as the first and last $15\%$ of each sequence, and
the initial action primitives are obtained from K-means clustering.
In testing, after the online inference using particle filter, we further
adjust each action boundary using an off-line inference within a local neighborhood
of length $40$ centered at the initial boundary; in this way, the locally ``full'' posterior
in Sec. \ref{sec:rbpf} is considered.
We evaluate the recognition performance by per-frame accuracy.
Contribution from each model component (STM and DBM) is analyzed separately.
\subsection{Public Dataset} \label{subsec_pubset}
The first public dataset used is the IXMAS
dataset~\cite{Weinland:Free}. The dataset contains 11 actions, each
performed 3 times by 10 actors. The videos are acquired
using 5 synchronized cameras from different angles, and the actors
freely changed their orientation in acquisition.
We calculate dense optical flow in the silhouette area
of each subject, from which Locality-constrained
Linear Coding features (LLC)\footnote{implementation from author's
website}~\cite{Jinjun:LLC} are extracted as the observation
in each frame.
We have used 32 codewords and $4 \times 4$, $2 \times 2$ and $1 \times 1$
spatial pyramid~\cite{Lazebnik:Beyond}.
Table~\ref{tab_exp_ixmas} reports the continuous action recognition
results, in comparison with SLDS\footnote{implementation based on
BNT from http://code.google.com/p/bnt/}~\cite{Oh:Learning},
CRF\footnotemark[1]~\cite{Lafferty:Conditional} and
LDCRF\footnotemark[1]~\cite{Morency:Latent}.
Our proposed model (and each of its components) achieves a recognition accuracy
higher than all the other methods by more than $10\%$.
\begin{table}[th]
\caption{Continuous action recognition for IXMAS dataset}\label{tab_exp_ixmas}
\centering \small
\begin{tabular}{|c|c|c||c|c|c|}
\hline
SLDS & CRF & LDCRF & STM & DBM & STM+DBM\\
\hline
53.6\% & 60.6\% & 57.8\% & 70.2\% & 74.5\% & \textbf{76.5}\%\\
\hline
\end{tabular}
\end{table}
The second public dataset used is the CMU MoCap
dataset~\footnote{http://mocap.cs.cmu.edu/}. For comparison purpose,
we report the results from the complete subset of subject 86. The
subset has 14 sequences with 122 actions in 8 category.
Quaternion feature is derived from the raw MoCap data as our observation
for inference. Table~\ref{tab_exp_cmu} lists the continuous action
recognition results, in comparison with the same set of benchmark
techniques as in the first experiment, as well
as~\cite{Ozay:Sequential, Raptis:Spike}. Similarly, results from
this experiment demonstrated the superior performance of our method.
It is interesting to note that, in Table~\ref{tab_exp_cmu}, the
frame-level accuracy by using DBM alone is a little higher than its combination
with STM. This is because there's only one subject in this
experiment and no significant variation in substructure is presented
in each action type, so temporal duration plays a more important
role in recognition. Nevertheless, the result attained by STM+DBM
is superior than all benchmark methods.
\subsection{In-house dataset} \label{subsec_house}
\begin{figure}[t]
\center \small
\addtolength{\tabcolsep}{-7pt}
\begin{tabular}{cccccccc}
\epsfig{file=stack_1_rgb.eps, width=0.8in, height=0.5in} &
\epsfig{file=stack_2_rgb.eps, width=0.8in, height=0.5in} &
\epsfig{file=stack_3_rgb.eps, width=0.8in, height=0.5in} &
\epsfig{file=stack_4_rgb.eps, width=0.8in, height=0.5in} &
\epsfig{file=stack_5_rgb.eps, width=0.8in, height=0.5in} &
\epsfig{file=stack_6_rgb.eps, width=0.8in, height=0.5in} &
\epsfig{file=stack_7_rgb.eps, width=0.8in, height=0.5in} &
\epsfig{file=stack_8_rgb.eps, width=0.8in, height=0.5in} \\
\epsfig{file=stack_1_depth.eps, width=0.8in, height=0.5in} &
\epsfig{file=stack_2_depth.eps, width=0.8in, height=0.5in} &
\epsfig{file=stack_3_depth.eps, width=0.8in, height=0.5in} &
\epsfig{file=stack_4_depth.eps, width=0.8in, height=0.5in} &
\epsfig{file=stack_5_depth.eps, width=0.8in, height=0.5in} &
\epsfig{file=stack_6_depth.eps, width=0.8in, height=0.5in} &
\epsfig{file=stack_7_depth.eps, width=0.8in, height=0.5in} &
\epsfig{file=stack_8_depth.eps, width=0.8in, height=0.5in} \\
\end{tabular}
\caption{Example frames from the ``stacking'' dataset. \emph{top-row: RGB images, bottom-row: aligned depth images.}}
\label{fig_stacking}
\end{figure}
\begin{figure}[t]
\center \small
\addtolength{\tabcolsep}{-7pt}
\begin{tabular}{cccccccc}
\epsfig{file=assmble_1_rgb.eps, width=0.8in, height=0.5in} &
\epsfig{file=assmble_2_rgb.eps, width=0.8in, height=0.5in} &
\epsfig{file=assmble_3_rgb.eps, width=0.8in, height=0.5in} &
\epsfig{file=assmble_4_rgb.eps, width=0.8in, height=0.5in} &
\epsfig{file=assmble_5_rgb.eps, width=0.8in, height=0.5in} &
\epsfig{file=assmble_6_rgb.eps, width=0.8in, height=0.5in} &
\epsfig{file=assmble_7_rgb.eps, width=0.8in, height=0.5in} &
\epsfig{file=assmble_8_rgb.eps, width=0.8in, height=0.5in} \\
\epsfig{file=assmble_1_depth.eps, width=0.8in, height=0.5in} &
\epsfig{file=assmble_2_depth.eps, width=0.8in, height=0.5in} &
\epsfig{file=assmble_3_depth.eps, width=0.8in, height=0.5in} &
\epsfig{file=assmble_4_depth.eps, width=0.8in, height=0.5in} &
\epsfig{file=assmble_5_depth.eps, width=0.8in, height=0.5in} &
\epsfig{file=assmble_6_depth.eps, width=0.8in, height=0.5in} &
\epsfig{file=assmble_7_depth.eps, width=0.8in, height=0.5in} &
\epsfig{file=assmble_8_depth.eps, width=0.8in, height=0.5in} \\
\end{tabular}
\caption{Example frames from the ``assembling'' dataset. \emph{top-row: RGB images, bottom-row: aligned depth images.}}
\label{fig_assembling}
\end{figure}
\begin{table}[t]
\caption{Continuous action recognition for CMU MoCap dataset}\label{tab_exp_cmu}
\centering \small
\begin{tabular}{|c|c|c|c|c||c|c|c|}
\hline
SLDS & CRF & LDCRF & \cite{Ozay:Sequential}& \cite{Raptis:Spike} & STM & DBM & STM+DBM \\
\hline
80.0\% & 77.2\% & 82.5\% & 72.3\% & 90.9\% & 81.0\% & \textbf{93.3}\% & 92.1\% \\
\hline
\end{tabular}
\vspace{-0.1in}
\end{table}
In addition to the above two public datasets, two in-house datasets
were also captured. The actions in these two sets feature
stronger hierarchical substructure.
The first dataset contains videos of stacking/unstacking three colored boxes,
which involves actions of ``move-arm'', ``pick-up'' and ``put-down''.
13 sequences with 567 actions were recorded in both RGB and depth videos
with one Microsoft Kinect sensor~\footnote{http://www.xbox.com/kinect}
(Fig.~\ref{fig_stacking}).
Then object tracking and 3-D reconstruction were performed to obtain the 3D trajectories of
two hands and three boxes. In this way an observation sequence in
$\mathbb{R}^{15}$ is generated. In the experiments, leave-one-out
cross-validation was performed on the 13 sequences. The continuous
recognition results are listed in Table~\ref{tab_exp_stacking}.
It is noticed that, among the four benchmark techniques, the
performance of SLDS and CRF are comparable, while LDCRF achieved the
best performance. This is reasonable because during the stacking
process, each box can be moved/stacked at any place on the desk,
which leads to large spatial variations that cannot be well modeled
by a Bayesian Network of only two layers. LDCRF applied a third
layer to capture such ``latent dynamics'', and hence achieved best
accuracy. For our proposed models, the STM alone brings LDS to a
comparable accuracy to LDCRF because it also models the substructure
transition pattern. By further incorporating duration information,
our model outperforms all other existing approaches.
The second in-house dataset is more complicated than the first one.
It involves five actions, ``move-arm'', ``pick-up'', ``put-down'',
``plug-in'' and ``plug-out'', in a printer part assembling task
(Fig.~\ref{fig_assembling}). The 3D trajectories of two hands and
two printer parts were extracted using the same Kinect sensor
system. 8 sequences were recorded and tested with leave-one-out
cross-validation. As can be seen from
Table~\ref{tab_exp_assembling}, our proposed model with both STM and
DBM outperforms other benchmark approaches by a large margin.
\begin{figure}[th]
\centering \small
\begin{tabular}{c}
\epsfig{file=stacking.eps, height=30mm} \\
(a) Set I: Stacking \\
\epsfig{file=assembling.eps, height=30mm} \\
(b) Set II: Assembling \\
\end{tabular}
\caption{Continuous recognition for in-house datasets}
\label{fig_ixmas_seq}
\end{figure}
\begin{table}[th]
\caption{Continuous action recognition for Set I: Stacking}\label{tab_exp_stacking}
\centering \small
\begin{tabular}{|c|c|c||c|c|c|}
\hline
SLDS & CRF & LDCRF & STM & DBM & STM+DBM\\
\hline
64.4\% & 79.6\% & 90.3\% &88.5\% & 81.3\% & \textbf{94.4}\% \\
\hline
\end{tabular}
\vspace{-0.1in}
\end{table}
\begin{table}[th]
\caption{Continuous action recognition for Set II: Assembling}\label{tab_exp_assembling}
\centering \small
\begin{tabular}{|c|c|c||c|c|c|}
\hline
SLDS & CRF & LDCRF & STM & DBM & STM+DBM\\
\hline
68.2\% & 77.7\% & 88.5\% & 88.7\% & 69.0\% & \textbf{92.9}\%\\
\hline
\end{tabular}
\vspace{-0.1in}
\end{table}
\subsection{Discussion} \label{subsec_discuss}
To provide more insightful comparison between the proposed algorithm
and other benchmark algorithms, we show two examples of continuous
action recognition results from the in-house datasets in
Fig.~\ref{fig_ixmas_seq}. The result given by SLDS contains short
and frequent switchings between incorrect action types. This is
caused by the false matching of motion patterns to an incorrect
action model. dSLDS~\cite{Sang:Parameterized} and LDCRF eliminate
the short transitions by considering additional context information;
however, their performances degrade severely around noisy or
ambiguous action periods (\emph{e.g}\onedot} \def\Eg{\emph{E.g}\onedot the beginning of the sequence in
Fig.~\ref{fig_ixmas_seq}.(b)) due to false duration prior or
overdependence on discriminative classifier. Our proposed STM+DBM
approach does not suffer from any of these problems, because STM
helps to identify all action classes disregarding their variations,
and DBM further helps to improve the precision of boundaries with
both generative and discriminative duration knowledge. Another
interesting finding shown in the last rows of (a) and (b) is that
the substructure node $Z$ can be interpreted by concrete physical
meanings. For all the actions in these experiments, we find
different object involved in an action corresponds to a different
value of $Z$, which dominates the infer values $\hat{Z}_{1:T}$ in
that action. Therefore, in addition to estimating action class, we
can also find the object associated with the action by majority voting
based on $\hat{Z}_{1:T}$. In our experiments, all the inferred object
associations agree with ground truth.
\section{Introduction} \label{sec_intro}
Understanding continuous human activities from videos, \emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot
simultaneous segmentation and classification of actions, is a
fundamental yet challenging problem in computer vision. Many
existing works approach the problem using bottom-up
methods~\cite{Satkin:Modeling}, where segmentation is performed as
preprocessing to partition videos into coherent constituent parts,
and action recognition is then applied as an isolated classification step.
Although a rich literature exists for segmentation of time series,
such as change point detection~\cite{Harchaoui:Kernel}, periodicity
of cyclic events modeling~\cite{Cutler:Robust} and frame
clustering~\cite{Zhou:Aligned}, the methods tend to detect local
boundaries and lack the ability to incorporate global dynamics
of temporal events, which leads to under or over segmentation that
severely affects the recognition performance, especially for complex
actions with diversified local motion statistics~\cite{Hoai:Joint}.
\begin{figure}[t]
\center
\subfigure[]{\epsfig{file=LDS.eps, width=25mm,bbllx=52,bblly=52,bburx=224,bbury=218,clip=}}
\hfil
\subfigure[]{\epsfig{file=overall_struct.eps, width=40mm, bbllx=70,bblly=50,bburx=350,bbury=330,clip=}}
\caption{(a) Tradition SLDS model for continuous action recognition, where each action is
represented by an LDS; (b) the structure of our proposed model, in which each action is represented
by an SLDS with substructure transition, and the inter action transition is by controlled by discriminative boundary model.}
\label{fig:overall_struct}
\vspace{-0.15in}
\end{figure}
The limitation of the bottom-up approaches has been addressed by
performing concurrent top-down recognition using variants of Dynamic
Bayesian Network (DBN), where the dynamics of temporal events are
modeled as transitions in a latent~\cite{McCallum:Maximum,
Lafferty:Conditional} or partially observed state
space~\cite{Hoffken:Switching, Oh:Learning}. The technique has been
successfully used in speech recognition and natural language
processing, while the performance of existing DBN based approaches
for action recognition~\cite{Oh:Learning, Fox:Nonparametric,
Sminchisescu:Conditional, Sung:Human, Kjellstrom:Simultaneous,
Ning:Conditional} tends to be relatively lower~\cite{Hoai:Joint},
mostly due to the difficulty in interpreting the physical meaning of
latent states. Thus, it becomes difficult to impose additional prior
knowledge with clear physical meaning into an existing graphical
structure to further improve its performance.
To tackle the problem, in this paper, we show how two additional
sources of information with clear physical interpretations can be
considered in a general graphical structure for state-space model (SSM)
in Figure~\ref{fig:overall_struct}. Compared to a standard Switching
Linear Dynamic System (SLDS)~\cite{Oh:Learning} model in
Figure~\ref{fig:overall_struct}.(a), where $X$, $Y$ and $S$ are
respectively the hidden state, observation and label, the proposed model
in Figure~\ref{fig:overall_struct}.(b) is augmented with
two additional nodes, $Z$ and $D$, to describe the
substructure transition and duration statistics of actions:
\textbf{Substructure transition} Rather than a uniform motion type,
a real-world human action is usually characterized by a set of inhomogeneous units
with some instinct structure, which we call \emph{substructure}.
Action substructure arises from two factors: (1) the
hierarchical nature of human activity, where one action can be
temporally decomposed into a series of primitives with
spatial-temporal constraints; (2) the large variance of
action dynamics due to differences in kinematical property of subjects,
feedback from environment, or interaction with objects.
For the first factor,
Hoai~\emph{et al}\onedot~\cite{Hoai:Joint} used multi-class Support Vector Machine
(SVM) with Dynamic Programming to recognize coherent motion
constituent parts in an action; Liu \emph{et al}\onedot~\cite{Liu:Recognizing}
applied latent-SVM for temporal evolving of ``attributes'' in actions;
Sung \emph{et al}\onedot~\cite{Sung:Human} introduced a two-layer
Maximum Entropy Markov Models to recognize the correspondence
between sub-activities and human skeletal features.
For the second factor, considerations have been paid to
the substructure variance caused by subject-object interaction
using Connected Hierarchic Conditional Random Field (CRF)~\cite{Kjellstrom:Simultaneous},
and the substructure variance caused by pose using Latent Pose CRF~\cite{Ning:Conditional}.
In more general cases, Morency \emph{et al}\onedot presented the Latent Dynamic CRF
(LDCRF) algorithm by adding a ``latent-dynamic'' layer into CRF for
hidden substructure transition~\cite{Morency:Latent}. The
limitation of CRF as a discriminative method is that,
one single pseudo-likelihood score is estimated for
an entire sequence which is incapable to interpret the probability
of each individual frame.
To solve the problem, we instead design a generative model as
in Figure.\ref{fig:overall_struct}.(b), with extra hidden
node $Z$ gating the transition amongst a set of dynamic systems,
and the posterior for every action can be inferred strictly
under Bayesian framework for each frame.
The dimension of state space increases geometrically with an
extra hidden node, so we introduce effective transition prior constraints
in Section~\ref{sec:stm} to avoid over-fitting on a limited amount of training data.
\textbf{Duration model} The duration statistics of actions is
important in determining the boundary where one action
transits to another in continuous recognition tasks.
Duration model has been
widely adopted in Hidden Markov Model (HMM) based methods, such as
the explicit duration HMM~\cite{Ferguson:Variable} or more generally
the Hidden Semi Markov Model (HSMM)~\cite{Yu:Hidden}. Incorporating
duration model into SSM is more challenging than HMM because SSM has
continuous state space, and exact inference in SSM is usually
intractable~\cite{Lerner:inferencein}. Some works reported in this
line include Cemgil \emph{et al}\onedot\cite{Cemgil:generative} for music
transcription and Chib and Dueker~\cite{Chib:Non} for economics. Oh
\emph{et al}\onedot\cite{Sang:Parameterized} imposed the duration constraint at
the top level of SLDS and achieved improved performance for honeybee
behavior analysis~\cite{Oh:Learning}. In general, naive integration
of duration model into SSM is not effective, because duration
patterns vary significantly across visual data and limited training
samples may bias the model with incorrect duration patterns.
To address this problem, in Figure~\ref{fig:overall_struct}.(b) we
correlate duration node $D$ with the continuous hidden
state node $X$ and the substructure transition node $Z$ via
logistic regression as explained in Section~\ref{sec:abm}.
In this way, the proposed duration model
becomes more discriminative than conventional generative models,
and the data-driven boundary locating process can accommodate
more variation in duration length.
In summary, the major contribution of the paper is to incorporate
two additional models into a general SSM, namely the Substructure
Transition Model (STM) and the Discriminative Boundary Model (DBM).
We also design a Rao-Blackwellised particle filter for
efficient inference of proposed model in Section~\ref{sec:rbpf}.
Experiments in Section~\ref{sec:exp} demonstrate the
superior performance of our proposed system over several existing
state-of-the-arts in continuous action recognition.
Conclusion is drawn in Section~\ref{sec:con}.
\section{Substructure Transition Model}
\label{sec:stm}
Linear Dynamic Systems (LDS) is the most commonly used SSM to
describe visual features of human motions. LDS is modeled by linear
Gaussian distributions:
\begin{equation}
\label{eq:lds_y}
p(Y_t=\mathbf{y}_t|X_t=\mathbf{x}_t) = \mathcal{N}(\mathbf{y}_t; \mathbf{B} \mathbf{x}_t, \mathbf{R})
\end{equation}
\begin{equation}
\label{eq:lds_x}
p(X_{t+1}=\mathbf{x}_{t+1}|X_t=\mathbf{x}_t) = \mathcal{N}(\mathbf{x}_{t+1}; \mathbf{A} \mathbf{x}_t, \mathbf{Q})
\end{equation}
where $Y_t$ is the observation at time $t$, $X_t$ is a latent state,
$\mathcal{N}(\mathbf{x}; \bm{\mu}, \mathbf{\Sigma})$ is
multivariate normal distribution of $\mathbf{x}$ with mean
$\bm{\mu}$ and covariance $\mathbf{\Sigma}$.
To consider multiple actions, SLDS~\cite{Oh:Learning} is formulated as a mixture of LDS's with
the switching among them controlled by action class $S_t$.
However, each LDS can only model an action with homogenous motion, ignoring
the complex substructure within the action.
We introduce a discrete hidden variable $Z_t \in \{1,...,N_Z\}$ to explicitly represent such information,
and the \emph{substructured} SSM can be stated as:
\begin{equation}
p(Y_t=\mathbf{y}_t|X_t=\mathbf{x}_t, S_t^i, Z_t^j) = \mathcal{N}(\mathbf{y}_t; \mathbf{B}^{ij} \mathbf{x}_t, \mathbf{R}^{ij})
\end{equation}
\begin{equation}
p(X_{t+1}=\mathbf{x}_{t+1}|X_t=\mathbf{x}_t, S_{t+1}^i, Z_{t+1}^j) =
\mathcal{N}(\mathbf{x}_{t+1}; \mathbf{A}^{ij} \mathbf{x}_t, \mathbf{Q}^{ij})
\end{equation}
where $\mathbf{A}^{ij}$, $\mathbf{B}^{ij}$, $\mathbf{Q}^{ij}$, and
$\mathbf{R}^{ij}$ are the LDS parameters for the $j^{th}$ action primitive in the
substructure of $i^{th}$ action class. $\{Z_t\}$ is modeled as a
Markov chain and the transition probability is specified by
multinomial distribution:
\begin{equation}
\label{eq:trans_Z}
p(Z_{t+1}^j|Z_t^i, S_{t+1}^k)=\theta_{ijk}
\end{equation}
In the following, the term STM may refer to either the transition matrix
in Eq. \eqref{eq:trans_Z} or the overall substructured SSM depending on its context.
Some examples of STM are given in Fig.~\ref{fig:stm_eg}, which are to be
explained in detail in the remainder of this section.
\begin{figure}[t]
\center
\subfigure[]{\epsfig{file=STM2.eps, width=45mm,bbllx=161,bblly=254,bburx=462,bbury=546,clip}}
\hfil
\subfigure[]{\epsfig{file=STM1.eps, width=45mm,bbllx=161,bblly=254,bburx=462,bbury=546,clip}}
\hfil
\subfigure[]{\epsfig{file=STM3.eps, width=45mm,bbllx=161,bblly=254,bburx=462,bbury=546,clip}}
\caption{STM trained for action ``move-arm'' in stacking dataset using
left-to-right (a), sparse (b), and block-wise sparse (c) constraints, with $N_Z=5$ and $N_Q=3$.
STM in (c) captures global ordering and local details better than the other two.}
\label{fig:stm_eg}
\vspace{-0.1in}
\end{figure}
\subsection{Sparsity Constrained STM}
We use simplified notation $\mathbf{\Theta} = \{\theta_{ij}\}$ for
the STM within a single action.
In general, $\mathbf{\Theta}$ can be any matrix as long as each row of it is
a probability vector, which allows the substructure of action primitives
to be organized arbitrarily.
For most real-world human actions, however, there is a strong
temporal ordering associated with the primitive units. Such order
relationship can be vital to accurate action recognition,
since a different temporal ordering can define a totally different
action even if the composing primitive units are the same.
Moreover, if prior information of substructure is incorporated in model estimation,
the learning process can become more robust to noise and outliers.
There have been some attempts to characterize the order relationship of primitive units by
restricting the structure of transition matrix $\mathbf{\Theta}$.
For example, left-to-right HMM~\cite{Bakis:Continuous} is proposed to model sequential or cyclic ordering of primitive units,
and the corresponding $\mathbf{\Theta}$ has non-zero values only on or directly above diagonal,
as illustrated in Fig.~\ref{fig:stm_eg} (a).
Sequential ordering is a very strong assumption; to describe actions with more flexible temporal patterns,
people have resorted to switching HMM~\cite{Hoffken:Switching} or factorial HMM~\cite{Ghahramani:Factorial},
which model action variations with multiple sequential orderings.
All the works above assume the order relationship between action units are given \textit{a priori};
\emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot, the number of non-zero entries in $\mathbf{\Theta}$ is small and their locations are all known.
In many cases, however, it is difficult to specify such information exactly,
and making a wrong assumption can bias the estimation of action model.
A more practical approach is to impose a sparse transition constraint while
leaving the discovery of exact order relationship to training phase.
Along this direction, negative Dirichlet distribution has been
proposed in \cite{Bicego:Sparseness} as a prior for each row $\bm{\theta}_i$
in $\mathbf{\Theta}$:
\begin{equation}
p(\bm{\theta}_i) \propto \prod \limits_j \theta_{ij}^{-\alpha}
\end{equation}
where $\alpha$ is a pseudo count penalty. The MAP estimation of
parameter is
\begin{equation}
\hat{\theta}_{ij} = \frac{\max(\xi_{ij}-\alpha, 0)}{\sum_t \max(\xi_{it}-\alpha, 0)}
\end{equation}
where $\xi_{ij}$ is the sufficient statistics of $(Z_t^i,
Z_{t+1}^j)$. When the number of transitions from $z^i$ to $z^j$ in
training data is less than $\alpha$, the probability $\theta_{ij}$
is set to zero. The sparsity enforced in this way often leads to
local transition patterns which might be actually caused by noise or
incomplete data, as shown in Fig.~\ref{fig:stm_eg} (b).
Also, the penalty term $\alpha$ introduces bias to
the proportion of non-zero transition probabilities, \emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot
$\frac{\hat{\theta}_{ij}}{\hat{\theta}_{ik}} \neq
\frac{\xi_{ij}}{\xi_{ik}}$. This bias can be severe especially when
$\xi_{ij}$ is small.
\subsection{Block-wise Sparse STM}
\label{subsec:bsst}
As we have seen, the sequential order assumption about the transition between action units is too strong, while
the sparse prior on transition probability is biased and cannot globally regularize the STM.
Here we propose a block-wise sparse STM which can achieve tradeoff between model sparsity and
flexibility.
The idea is to divide an action into several stages and each stage comprises of a subset of action primitives.
The transition between stages is encouraged to be sequential but sparse,
such that the global action structure can be modeled. At the same time,
the action primitives within each stage can propagate freely from one to another
so that variation in action styles and parameters is also preserved.
Our stage-wise transition model is also favorable in regard of continuous action segmentation, since
the starting and terminating stages can be explicitly modeled to enhance discrimination
on action boundaries.
Formally, define discrete variable $Q_t \in \{1, ..., N_Q\}$ as the
current stage index of action, and assume a surjective mapping
$g(\cdot)$ is given which assigns each action primitive $Z_t$ to its
corresponding stage $Q_t$:
\begin{equation}
\label{eq:constraint_g}
\left\{
\begin{array}{ll}
p(Q_t^q, Z_t^i) >0\,, & \textrm{if $g(i)=q$}\\
p(Q_t^q, Z_t^i) =0\,. & \textrm{otherwise}
\end{array}
\vspace{-1mm}
\right.
\end{equation}
The choice of $g(\cdot)$ depends on the nature of action. Intuitively, we can assign more
action primitives to a stage with diversified motion patterns and less action primitives
to a stage with restricted pattern.
The joint dynamic transition distribution of $Q_t$ and $Z_t$ is defined as:
\begin{equation}
\label{eq:trans_QZ}
p(Q_{t+1}, Z_{t+1} | Q_{t}, Z_{t})
= p(Q_{t+1} | Q_{t}) p(Z_{t+1} | Q_{t+1}, Z_{t})
\end{equation}
The second term of Eq. \eqref{eq:trans_QZ} specifies the transition
between action primitives, which we want to keep as flexible as
possible to model diversified local action patterns. The first term
captures the global structure between different action stages, and
therefore we impose an \emph{ordered} negative Dirichlet
distribution as its hyper-prior:
\begin{equation}
\label{eq:prior_phi}
p(\mathbf{\Phi}) \propto \prod \limits_{q \neq r, q+1 \neq r} \phi_{qr}^{-\alpha}
\end{equation}
where $\mathbf{\Phi}=\{ \phi_{qr} \}$ is the stage transition
probability matrix, $\phi_{qr} = p(Q_{t+1}^r | Q_{t}^q)$, and
$\alpha$ is a constant for pseudo count penalty. The ordered
negative Dirichlet prior encodes both sequential order information
and sparsity constraint. It promotes statistically a global
transition path $Q^1 \rightarrow Q^2 \rightarrow ... \rightarrow
Q^{N_Q}$ which can be learned from training data rather than
heuristically defined as in left-to-right HMM
\cite{Bakis:Continuous}. An example of the resulting STM
is shown in Fig.~\ref{fig:stm_eg} (c).
Note that no in-coming{\slash}out-going
transition is encouraged for $Q^1${\slash}$Q^{N_Q}$, which stands
for starting{\slash}terminating stage. The identification of these
two special stages is helpful for segmenting continuous actions, as
will be discussed in Sec. \ref{sec:ddm}.
\subsection{Learning STM}
The MAP model estimation requires maximizing the product of
likelihood (\ref{eq:trans_QZ}) and prior (\ref{eq:prior_phi}) under
the constraint of (\ref{eq:constraint_g}).
There are two interdependent nodes, $Q$ and $Z$, involved in the optimization,
which make the problem complicated.
Fortunately, as shown in Appendix \ref{sec:app_blktrans}, Eq. \eqref{eq:trans_QZ}
can be replaced with the transition distribution of single variable $Z$ and
a constraint exists for the relationship between $\mathbf{\Theta}$ and $\mathbf{\Phi}$.
Therefore, the node $Q$ (and the associated parameter $\mathbf{\Phi}$) serves only for
conceptual purpose and can be eliminated in our model construction.
The MAP estimation can be converted to the following constrained
optimization problem:
\begin{eqnarray}
\label{eq:objfunc_blktrans}
& \max \limits_{\mathbf{\Theta}} & \mathcal{L}(\mathbf{\Theta}) = \sum_{i, j} \xi_{ij} \log \theta_{ij}
- \sum_{\substack{q \neq r \\ q+1 \neq r }} \alpha \log \phi_{qr} , \\
& {\rm s.t.} & \phi_{qr} = \Sigma_{j \in \mathcal{G}(r)} \theta_{ij}, \;\;\; i \in \mathcal{G}(q), \; \forall q, r \nonumber \\
& & \Sigma_{j} \theta_{ij} = 1, \;\; \forall i \;\;\; \;\;\; \;\;\;
\theta_{ij}\geq 0, \;\; \forall i, j \nonumber
\end{eqnarray}
where $\xi_{ij}$ is the sufficient statistics of $(Z_t^i, Z_{t+1}^j)$,
$\mathcal{G}(q) \triangleq \{ i | g(i)=q \}$,
and $\{\phi_{qr}\}$ are just auxiliary variables.
The KKT (Karush-Kuhn-Tucker) conditions for optimal solution $\mathbf{\hat{\Theta}}$ are:
\begin{eqnarray}
\frac{\xi_{ij}}{\hat{\theta}_{ij}} - \lambda_{i, g(j)} + \gamma_i - \mu_{ij} & = & 0, \;\; \forall i, j \nonumber \\
-\frac{\alpha_{qr}}{\hat{\phi}_{qr}} + \sum_{i \in \mathcal{G}(q)} \lambda_{ir} & = & 0, \;\; \forall q, r \nonumber \\
\mu_{ij} \geq 0, \;\; \mu_{ij}\hat{\theta}_{ij} & = & 0, \;\; \forall i, j \nonumber
\end{eqnarray}
where $\lambda_{ir}$, $\gamma_i$, and $\mu_{ij}$ are constant multipliers;
$\alpha_{qr}$ is equal to $\alpha$ if $q \neq r$ or $q+1 \neq r$, and $0$ otherwise.
Solving the equation set as in Appendix \ref{sec:app_thetahat} gives
the MAP parameter estimation:
\begin{eqnarray}
\label{eq:theta_hat}
\hat{\theta}_{ij} &=& \hat{\phi}_{g(i), g(j)} \frac{\xi_{ij}}{\sum_{j' \in \mathcal{G}(g(j))} \xi_{ij'}} \\
\hat{\phi}_{qr} &=&
\frac{\max(\sum_{i \in \mathcal{G}(q), j \in \mathcal{G}(r)} \xi_{ij} - \alpha_{qr}, 0)}
{\sum_{r'} \max(\sum_{i \in \mathcal{G}(q), j \in \mathcal{G}(r')} \xi_{ij} - \alpha_{qr'}, 0)}
\nonumber
\end{eqnarray}
As we can see, the resultant transition
matrix $\hat{\mathbf{\Theta}}$ is a block-wise sparse matrix, which
can characterize both the global structure and local detail of
action dynamics. Also, within each block (stage), there is no bias
in $\hat{\theta}_{ij}$.
\section{Discriminative Boundary Model}
\label{sec:abm}
\subsection{Logistic Duration Model}
\label{sec:ldm}
It is straightforward to use a Markov chain to model the transition
of action $S_t$ where $p(S^j_{t+1}|S^i_t)=a_{ij}$. The duration
information of the $i^{th}$ action is naively incorporated into its
self-transition probability $a_{ii}$, which leads to an
action duration model with exponential distribution:
\begin{equation}
p(dur_i=\tau) = a_{ii}^{\tau-1} (1-a_{ii}), \;\; \tau=1, 2, 3 ... \nonumber
\end{equation}
Unfortunately, only a limited number of real-life events have an
exponentially diminishing duration. Inaccurate duration modeling can
severely affect our ability to segment consecutive actions and
identify their boundaries.
Non-exponential duration distribution can be implemented with
duration-dependent transition matrix, such as the one used in
HSMM~\cite{Yu:Hidden}. Fitting a transition matrix for each epoch within
the maximum length of duration is often impossible given a limited
number of training sequences, even when parameter hyperprior
such as hierarchical Dirichlet distribution~\cite{Wang:Event}
is used to restrict model freedom.
Parametric duration distributions such as
gamma~\cite{Levinson:Continuously}
and Gaussian~\cite{Yoshimura:Duration} provide a more compact way
to represent duration and show good performance in signal synthesis.
However, they are less useful in inference because the corresponding
transition probability is not easy to evaluate.
Here a new logistic duration model is proposed to address the above
limitations. We introduce a variable $D_t$ to represent the length of
time current action has been lasting. $\{D_t\}$ is a counting
process starting from $1$, and the beginning of a new action
is triggered whenever it is reset to $1$:
\begin{equation}
\label{eq:pS_trans} p(S_{t+1}^j | S_t^i, D^d_{t+1}) =
\left\{
\begin{array}{ll}
\delta(j-i), & \textrm{if $d>1$}\\
a_{ij}, & \textrm{if $d=1$}
\end{array}
\right.
\vspace{-1mm}
\end{equation}
where $a_{ij}$ is the probability of transiting from previous action
$i$ to new action $j$. Notice that the same type of action can be
repeated if we have $a_{ii}>0$.
Instead of modeling action duration distribution directly, we model
the transition distribution of $D_t$ as a logistic function of
its previous value:
\begin{equation}
\label{eq:pD_trans}
p(D_{t+1}^c|S_{t}^i, D_{t}^d) = \frac{e^{\nu_i(d-\beta_i)}\delta(c-1)+\delta(c-d-1)}{1+e^{\nu_i(d-\beta_i)}}
\end{equation}
\begin{equation}
\label{eq:pD_init}
p(D_1^c) = \delta(c-1)
\end{equation}
where $\nu_i$ and $\beta_i$ are positive logistic regression
weights. Eq. \eqref{eq:pD_trans} immediately leads to the duration
distribution for action class $i$:
\begin{equation}
\label{eq:pDur_logistic}
p(dur_i=\tau)=\prod_{d=1}^{\tau} \frac{1}{1+e^{\nu_i(d-\beta_i)}} \times e^{\nu_i(\tau-\beta_i)}
\end{equation}
Fig.~\ref{fig:logistic_duration_curve} (a) shows how the resetting probability
of $D_{t+1}$ changes as a function of $D_{t}$ with different
parameter sets, and the corresponding duration distributions
are plotted in (b). The increasing probability of
transiting to a new action leads to a peaked duration distribution,
with center and width controlled by $\beta_i$ and $\nu_i$, respectively.
Our logistic duration model
can be easily extended to represent multiple-mode durations if
double logistic function~\cite{Lipovetsky:Double} is used.
\begin{figure}[t]
\centerline{
\subfigure[]{\includegraphics[width=60mm]{pD1.eps}}
\hfil
\subfigure[]{\includegraphics[width=60mm]{pDur.eps}} }
\caption{(a) Resetting probability $p(D_{t+1}=1|D_t, S_t)$ and (b)
duration distribution for logistic duration model.
Plotted with different color/line style for different $\nu$/$\beta$.}
\label{fig:logistic_duration_curve}
\vspace{-0.1in}
\end{figure}
\subsection{Discriminative Boundary Model}
\label{sec:ddm}
The logistic duration model can be integrated with STM
by stacking the two layers of nodes ($D$-$S$ and $Z$-$X$-$Y$) together.
The resultant generative model, however, is unable to utilize
contextual information for accurate action boundary segmentation.
Discriminative graphic models, such as MEMM~\cite{McCallum:Maximum}
and CRF~\cite{Lafferty:Conditional}, are generally more powerful in
such classification problem except that they ignore data likelihood
or suffer from label bias problem.
To integrate discriminating power into our action boundary model
and at the same time keep the generative nature of the action model
itself, we construct DBM by further augmenting the duration distribution
(which triggers action boundary) with the contextual information
from latent states $X$ and $Z$:
\begin{equation}
\label{eq:pD_trans_disc}
p(D_{t+1}^1|S_{t}^i, D_{t}^d, X^{\mathbf{x}}_t, Z^j_t) =
\frac{e^{\nu_i (d-\beta_i) + \bm{\omega}_{ij}^T \mathbf{x}}}
{1+e^{\nu_i (d-\beta_i) + \bm{\omega}_{ij}^T \mathbf{x}}}
\end{equation}
where $\nu_i$, $\beta_{i}$ have the same meaning as in
Eq. \eqref{eq:pD_trans}, and $\bm{\omega}_{ij}$ are the additional
logistic regression coefficients. When $\bm{\omega}_{ij}^T
\mathbf{x}=0$, no information can be learned from $X_t$ and $Z_t$,
and the DBM reduces to a generative one as
Eq. \eqref{eq:pD_trans}. A similar logistic function has been
employed in augmented SLDS \cite{Barber:Expectation}, where the main
motivation is to distinguish between transitions to different states
based on latent variable. Our DBM is specifically
designed for locating the boundary between contiguous actions. It
relies on both real valued and categorical inputs.
As constrained by the STM in Subsection~\ref{subsec:bsst},
each action is only likely to terminate in stage $N_Q$.
Therefore, $D_{t+1}$ can be reset to $1$
only when the current action in this terminating stage, and we can
modify Eq. \eqref{eq:pD_trans_disc} as: \vspace{-1mm}
\begin{equation}
\addtolength{\tabcolsep}{-3pt}
\label{eq:pD_trans_disc_term}
p(D_{t+1}^1|S_{t}^i, D_{t}^d, X^{\mathbf{x}}_t, Z^j_t) =
\left\{
\begin{tabular}{ll}
\textrm{Eq. \eqref{eq:pD_trans_disc}}, & \textrm{$g(j)=N_Q$}\\
0, & \textrm{otherwise}
\end{tabular}
\right.
\end{equation}
In this way, the number of parameters is greatly reduced and the label
unbalance problem is also ameliorated.
Now, the construction of our action model for continuous recognition
has been completed, with the overall structure shown in Figure~\ref{fig:overall_struct} (b).
\subsection{Learning DBM}
To learn the parameters $\nu$, $\beta$ and $\bm{\omega}$, we use
coordinate descent method to iterate between $\{\nu, \beta\}$ and
$\bm{\omega}$.
For $\nu$ and $\beta$, given a set of $N$ training sequences with class
labels $\{\mathbf{S}^{(n)}\}_{n=1...N}$, we can easily obtain the values
for all duration nodes $\{\mathbf{D}^{(n)}\}_{n=1...N}$
according to Eq. \eqref{eq:pS_trans}-\eqref{eq:pD_init}.
Then fitting the parameters $\nu$ and $\beta$ is equivalent
to performing logistic regression with input-output pairs
$(D^{(n)}_t, \delta(S^{(n)}_{t+1}-S^{(n)}_t))$.
The action transition probability $\{a_{ij}\}$ can be obtained trivially.
To estimate $\bm{\omega}_{ij}$, let $\{ T^{(n)} \}_{n=1...N}$ be our
training set, where each data sample
$T^{(n)}$ is a realization of all the nodes involved in
Eq. \eqref{eq:pD_trans_disc} at a particular time instance $t^{(n)}$
and $S_{t^{(n)}}=i$. Since $X_{t^{(n)}}$ and $Z_{t^{(n)}}$ are
hidden variables, their posterior $p(Z^j_{t^{(n)}}|\cdot)=p_Z^{(n)}$
and $p(X_{t^{(n)}}^{\mathbf{x}}|Z^j_{t^{(n)}},
\cdot)=\mathcal{N}(\mathbf{x}; \bm{\mu}^{(n)},
\mathbf{\Sigma}^{(n)})$ are first inferred from single action STM,
where the posterior of $X_{t^{(n)}}$ is approximated by a
Gaussian. The estimation of $\bm{\omega}_{ij}$ is obtained by
maximizing the expected log likelihood:
\begin{align}
\label{eq:max_omega}
& \max_{\bm{\omega}_{ij}} \sum_{n}
\mathrm{E}_{p(X^{\mathbf{x}}_{t^{(n)}}, Z^j_{t^{(n)}}|\cdot)} \left[ \log l^{(n)}(\mathbf{x}, \bm{\omega}_{ij}) \right] \\
= & \max_{\bm{\omega}_{ij}} \sum_{n}
p_Z^{(n)} \int_{\mathbf{x}} \log l^{(n)}(\mathbf{x}, \bm{\omega}_{ij}) \mathcal{N}(\mathbf{x}; \bm{\mu}^{(n)}, \mathbf{\Sigma}^{(n)})
d \mathbf{x} \nonumber
\end{align}
where
\begin{equation}
l^{(n)}(\mathbf{x}, \bm{\omega}) =
\frac{e^{(c^{(n)} + \bm{\omega}^T \mathbf{x}){b^{(n)}}}}
{1+e^{c^{(n)} + \bm{\omega}^T \mathbf{x}}}
\end{equation}
and $b^{(n)}=p(D_{t^{(n)}+1}=1)$, $c^{(n)}=\nu_i
(D_{t^{(n)}}-\beta_i)$.
The integral in Eq. \eqref{eq:max_omega} cannot be solved
analytically. Instead, we use unscented transform \cite{Julier:new}
to approximate the integral with the average over a set of sigma
points of $\mathcal{N}(\mathbf{x}; \bm{\mu}^{(n)},
\mathbf{\Sigma}^{(n)})$:
\begin{equation}
\mathbf{x}^{(n)}_k =
\left\{
\begin{array}{ll}
\bm{\mu}^{(n)}, & k=0 \\
\bm{\mu}^{(n)}+(\sqrt{M \mathbf{\Sigma}^{(n)}})_k, & k=1,...,M \\
\bm{\mu}^{(n)}-(\sqrt{M \mathbf{\Sigma}^{(n)}})_{k-M}, & k=M+1,...,2M \\
\end{array}
\right.
\nonumber
\end{equation}
where $M$ is the dimension of $\mathbf{x}$,
$(\sqrt{\mathbf{\Sigma}})_k$ is the $k^{th}$ column of the matrix
square root of $\mathbf{\Sigma}$. Therefore, Eq. \eqref{eq:max_omega}
converts to a weighted logistic regression problem with features
$\{\mathbf{x}^{(n)}_k\}$, labels $\{b^{(n)}\}$ and weights
$\{p_Z^{(n)}/(2M+1)\}$.
\section{Inference with Rao-Blackwellised Particle Filter}
\label{sec:rbpf}
In testing, given an observation sequence $\mathbf{y}_{1:T}$, we
want to find the MAP action labels $\hat{S}_{1:T}$ and the boundaries defined
by $\hat{D}_{1:T}$; we are also interested in the style of actions which
can be revealed from $\hat{Z}_{1:T}$. To obtain these MAP estimates,
we are required to find the posterior $p( S_{1:T}, D_{1:T},
Z_{1:T}|\mathbf{y}_{1:T})$, which is a non-trivial job given the
complicated hierarchy and nonlinearity of our model. We propose to
use particle filtering~\cite{Arulampalam02} for online inference due
to its capability in non-linear scenario. Moreover, the latent
variable $X_t$ can be marginalized by
Rao-Blackwellisation~\cite{Doucet00}. In this way, the computation
of particle filtering is significantly reduced since Monte Carlo
sampling is only conducted in the joint space of $(S_t, D_t, Z_t)$,
which has a much lower dimension and a highly compact support (note
the sparse transition probability between these variables).
Formally, we decompose the posterior distribution of all the hidden nodes at time $t$ as
\begin{equation}
p(S_t, D_t, Z_t, X_t | \mathbf{y}_{1:t})
= p(S_t, D_t, Z_t | \mathbf{y}_{1:t}) p(X_t | S_t, D_t, Z_t, \mathbf{y}_{1:t})
\end{equation}
where the second term can be evaluated analytically because $X_t$ depends on other variables
through linear and Gaussian relations.
In Rao-Blackwellised particle filter~\cite{Khan04}, a set of $N_P$ samples
$\{(s^{(n)}_t, d^{(n)}_t, z^{(n)}_t)\}_{n=1}^{N_P}$ and the associated weights $\{w^{(n)}_t\}_{n=1}^{N_P}$
are used to approximate the intractable first term, while the second term is represented by
$\{\chi_t^{(n)}\}_{n=1}^{N_P}$, which are analytical distributions of $X_t$ conditioned on
corresponding samples:
\begin{equation}
\chi_t^{(n)}(X_t) \triangleq p(X_t | s^{(n)}_t, d^{(n)}_t, z^{(n)}_t, \mathbf{y}_{1:t})
\end{equation}
In our model, $\chi_t^{(n)}(X_t) = \mathcal{N}(X_t; \hat{\mathbf{x}}^{(n)}_t, \mathbf{P}^{(n)}_t)$
is a Gaussian distribution. Thus, the posterior can be represented as
\begin{equation}
p(S_t, D_t, Z_t, X_t | \mathbf{y}_{1:t})
\approx \sum \limits_{n=1}^{N_P} w^{(n)}_t \delta_{S_t}(s^{(n)}_t) \delta_{D_t}(d^{(n)}_t)
\delta_{Z_t}(z^{(n)}_t) \chi_t^{(n)}(X_t)
\end{equation}
where the approximation error approaches to zero as $N_P$ increases to infinite.
Given the samples $\{s^{(n)}_{t-1}, d^{(n)}_{t-1}, z^{(n)}_{t-1}, \chi_{t-1}^{(n)}\}$
and weights $\{w^{(n)}_{t-1}\}$ at time $t-1$, the posterior of $(S_t, D_t, Z_t)$ at time $t$ is:
\begin{equation}
\label{eq:rbpf_sdz}
p(S_t, D_t, Z_t | \mathbf{y}_{1:t}) \propto
\sum_n w^{(n)}_{t-1} p(S_t| D_t, s^{(n)}_{t-1})
\times p(Z_t | S_t, D_t, z^{(n)}_{t-1}) \mathcal{L}^{(n)}_{t}(S_t, D_t,Z_t) \nonumber
\end{equation}
where
\begin{equation}
\label{eq:int_norm_logit}
\mathcal{L}^{(n)}_{t}(S_t, D_t, Z_t) = \int p(\mathbf{y}_t | \mathbf{x}_{t-1}, S_t, Z_t)
\chi_{t-1}^{(n)}(\mathbf{x}_{t-1})
\times p(D_t|s^{(n)}_{t-1}, d^{(n)}_{t-1}, z^{(n)}_{t-1}, \mathbf{x}_{t-1}) d\mathbf{x}_{t-1}
\end{equation}
The detailed derivation is shown in Appendix \ref{sec:app_rbpfsdz}.
It is hard to draw sample from Eq. \eqref{eq:int_norm_logit}. Instead, we draw new samples
$(s^{(n)}_{t}, d^{(n)}_{t}, z^{(n)}_{t})$ from a proposal density defined as:
\begin{equation}
q(S_t, D_t, Z_t | \cdot) = p(S_t| D_t, s^{(n)}_{t-1}) p(Z_t | S_t, D_t, z^{(n)}_{t-1})
\times p(D_t | s^{(n)}_{t-1}, d^{(n)}_{t-1}, z^{(n)}_{t-1}, \hat{\mathbf{x}}_{t-1}^{(n)} )
\end{equation}
The new sample weights are then updated as follows:
\begin{equation}
w^{(n)}_t \propto w^{(n)}_{t-1}
\frac{\mathcal{L}^{(n)}_{t}(s^{(n)}_t, d^{(n)}_t, z^{(n)}_t)}
{p(d^{(n)}_t | s^{(n)}_{t-1}, d^{(n)}_{t-1}, z^{(n)}_{t-1}, \hat{\mathbf{x}}_{t-1}^{(n)} )}
\end{equation}
$\mathcal{L}^{(n)}_{t}(\cdot)$ is essentially the integral of a
Gaussian function with a logistic function. Although not solvable
analytically, it can be well approximated by a re-parameterized
logistic function according to \cite{maragakis08}.
Details on how to evaluate $\mathcal{L}^{(n)}_{t}(\cdot)$
can be found in Appendix \ref{sec:app_logit}.
Once we get $s^{(n)}_{t}$ and $z^{(n)}_{t}$,
$\chi_t^{(n)}$ is simply updated by Kalman filter.
Re-sampling and normalization procedures are applied after all the
samples are updated as in~\cite{Doucet00}.
|
{
"timestamp": "2012-03-12T01:00:38",
"yymm": "1203",
"arxiv_id": "1203.1985",
"language": "en",
"url": "https://arxiv.org/abs/1203.1985"
}
|
\section{Introduction}
Invariants have been used in several cases to provide insight into the behavior of rational difference equations, see [1]-[4], [6]-[10], [12]-[14], [16]-[22], [24], [25], and [38]-[40]. In this article, we introduce six new algebraic invariants for six second order rational difference equations. Two of the six second order rational difference equations are linear fractional rational difference equations. We use these invariants to perform a reduction of order in each case. This reduction of order allows us to find forbidden sets in each case. In each case, the second order rational difference equation is transformed, via the invariant, to a family of first order linear fractional rational difference equations.
Since both the forbidden set, and the closed form solution are known for first order linear fractional rational difference equations, we use this information to find the forbidden set and closed form solution for the equations given in Section 4. For the reader's convenience, the forbidden set and closed form solution for first order linear fractional rational difference equations are presented in Section 3. In all six cases, we give a closed form solution for all initial conditions which are not in the forbidden set. In all six cases, the initial conditions and parameters are assumed to be arbitrary complex numbers.
In Section 2, we give some background information on some well known examples of invariants and forbidden sets for rational difference equations.
\section{Background on invariants and forbidden sets for rational difference equations}
When building an understanding of invariants for rational difference equations it is helpful to use the following equations as prototypes:
\begin{equation}
x_{n+1}=\frac{\alpha + x_n}{x_{n-1}},\quad n=0,1,2,\dots,
\end{equation}
\begin{equation}
x_{n+1}=\frac{\alpha + x_n +x_{n-1} }{x_{n-2}},\quad n=0,1,2,\dots,
\end{equation}
with $\alpha>0$ and positive initial conditions.
Equation (1) is known by the cognomen Lyness' equation. Equation (2) is known by the cognomen Todd's equation and has also been referred to as the third order Lyness' recurrence, see for example \cite{gsm3}.
Equations (1) and (2) are discussed in the following references: [2], [4], [6]-[8], [12], [13], [15]-[26], and [38]-[40]. In \cite{klr} it is first shown for Equations (1) and (2) that we have an algebraic invariant in each case, particularly for Equation (1) we have:
$$\left(\alpha + x_{n-1}+ x_{n}\right)\left(1+\frac{1}{x_{n-1}}\right)\left(1+\frac{1}{x_{n}}\right)= constant.$$
For Equation (2) we have the algebraic invariant,
$$\left(\alpha + x_{n-2} + x_{n-1} + x_{n}\right)\left(1+\frac{1}{x_{n}}\right)\left(1+\frac{1}{x_{n-1}}\right)\left(1+\frac{1}{x_{n-2}}\right)= constant.$$\par
Now, in the case of Equation (1), it has been shown in the unpublished paper of Zeeman \cite{z} that the map induced by Equation (1) is conjugated to a rotation.
Moreover, in \cite{gsm3}, it has been shown that the three dimensional case given by Equation (2) is also conjugated to a rotation. It has been shown, see \cite{z} and \cite{gsm3}, that in both Equations (1) and (2), the phase space of the associated dynamical system is foliated by invariant curves.
These invariant curves, which comprise the leaves of the foliation, are algebraic curves which degenerate into isolated points in some places. \par
The Lyness invariants can be generalized for the following $k^{th}$ order rational difference equation, sometimes called the $k^{th}$ order Lyness equation:
\begin{equation}
x_{n+1}=\frac{\alpha + \sum^{k-1}_{i=0}x_{n-i} }{x_{n-k}},\quad n=0,1,2,\dots.
\end{equation}
with $\alpha\geq 0$ and positive initial conditions. With this generalization we obtain the following algebraic invariant:
$$\left(\prod^{k}_{i=0}\left( \frac{1}{x_{n-i}} +1 \right)\right)\left( \alpha + \sum^{k}_{i=0}x_{n-i} \right)= constant.$$
Significant work has been done on this equation, see for example \cite{br2} and \cite{gsm5}. Further invariants are given in \cite{gki} for $k$ sufficiently large.
Recently geometric objects have been used effectively to obtain information about rational difference equations with invariants. A good example of this is the use of the Lie symmetry of the associated map by A. Cima, A. Gasull, and V. Ma\~{n}osa in \cite{gsm3} and \cite{gsm4}.
Another technique, which makes use of algebraic geometry, can be found in \cite{ag}.\par
The forbidden set of a rational difference equation is the set of initial conditions which eventually map to a singularity. Finding such sets has become a topic of recent interest in the literature, see for example [5], [11], [27]-[37]. Few techniques for determining forbidden sets are known. One such technique is the use of a semiconjugate factorization, see [29], [30], [32], and [34]-[37] for more on this topic.
In Section 4, we will find invariants for the given second order difference equations which allow us to perform a reduction of order in each case. In each case, the second order rational difference equation is transformed, via the invariant, to a family of first order linear fractional rational difference equations.
Since the forbidden set and closed form solution is known for first order linear fractional rational difference equations, we use this information to find the forbidden set and closed form solution for the equations given in Section 4. For the reader's convenience, the forbidden set and closed form solution for first order linear fractional rational difference equations are presented in Section 3.
\section{The Riccati Difference Equation}
In this section, we briefly summarize the known results on the Riccati difference equation, see \cite{gks}, \cite{riccati} and \cite{kulenovicladas}. The branch cut for the complex square root will be taken to be the negative real numbers for the remainder of the article.
\begin{thm}
Consider the rational difference equation
$$x_{n+1}=\frac{\alpha + \beta x_{n}}{A+ Bx_{n}},\quad n=0,1,\dots,$$
where the parameters $\alpha , \beta , A, B$ and the initial condition $x_{0}$ are complex numbers.
There are seven possibilities.
\begin{enumerate}
\item Suppose $A=B=0$, then $\mathfrak{F}=\mathbb{C}$.
\item Suppose $B=0$, and $A\neq 0$, then $x_{n+1}=\frac{\alpha + \beta x_{n}}{A}$. So $\mathfrak{F}=\emptyset$ and $x_{n}=\frac{\beta^{n}x_{0}}{A^{n}}+\sum^{n}_{i=1}\frac{\alpha \beta^{n-1}}{A^{n}}$ for all $n\geq 1$.
\item Suppose $B\neq 0$ and $\alpha B - \beta A = 0$, then the forbidden set, $\mathfrak{F}= \{\frac{-A}{B}\}$ and $x_{n}=\frac{\beta}{B}$ for all $n\geq 1$ whenever $x_{0}\not\in \{\frac{-A}{B}\}$.
\item Suppose $B\neq 0$, $\alpha B - \beta A \neq 0$, and $\beta +A =0$, then the forbidden set, $\mathfrak{F}= \{\frac{-A}{B}\}$. Furthermore, $x_{2n+1}=\frac{\alpha + \beta x_{0}}{A+ B x_{0}}$ for all $n\geq 0$, and $x_{2n}=x_{0}$ for all $n\geq 0$, whenever $x_{0}\not\in \{\frac{-A}{B}\}$.
\item Suppose $B\neq 0$, $\alpha B - \beta A \neq 0$, $\beta +A \neq 0$, and $\frac{\beta A - \alpha B}{(\beta + A)^{2}}\in \mathbb{C}\setminus [\frac{1}{4},\infty)$, and let $$w_{-}=\frac{1-\sqrt{1-4\left(\frac{\beta A - \alpha B}{(\beta + A)^{2}}\right)}}{2}\quad and \quad w_{+}=\frac{1+\sqrt{1-4\left(\frac{\beta A - \alpha B}{(\beta + A)^{2}}\right)}}{2},$$
then the forbidden set, $\mathfrak{F}= \left\{ \frac{\beta +A}{B}\left(\frac{w^{n-1}_{+}-w^{n-1}_{-}}{w^{n}_{+}-w^{n}_{-}}\right)w_{+}w_{-} - \frac{A}{B}|n\in\mathbb{N}\right\}$. Furthermore
$$x_{n}=\frac{\beta +A}{B}\left(\frac{(\frac{Bx_{0}+A}{\beta + A}-w_{-})w^{n+1}_{+}-(w_{+}-\frac{Bx_{0}+A}{\beta + A})w^{n+1}_{-}}{(\frac{Bx_{0}+A}{\beta + A}-w_{-})w^{n}_{+}-(w_{+}-\frac{Bx_{0}+A}{\beta + A})w^{n}_{-}}\right) - \frac{A}{B},\quad for\; all\; n\in\mathbb{N},$$
whenever $x_{0}\not\in \left\{ \frac{\beta +A}{B}\left(\frac{w^{n-1}_{+}-w^{n-1}_{-}}{w^{n}_{+}-w^{n}_{-}}\right)w_{+}w_{-} - \frac{A}{B}|n\in\mathbb{N}\right\}$.
\item Suppose $B\neq 0$, $\alpha B - \beta A \neq 0$, $\beta +A \neq 0$, and $\frac{\beta A - \alpha B}{(\beta + A)^{2}}=\frac{1}{4}$,
then the forbidden set, $\mathfrak{F}= \left\{ \frac{\beta +A}{B}\left(\frac{n-1}{2n}\right) - \frac{A}{B}|n\in\mathbb{N}\right\}$. Furthermore
$$x_{n}=\frac{\beta +A}{B}\left(\frac{1+(\frac{2Bx_{0}+2A}{\beta + A}-1)(n+1)}{2+2(\frac{2Bx_{0}+2A}{\beta + A}-1)n}\right) - \frac{A}{B},\quad for\; all\; n\in\mathbb{N},$$
whenever $x_{0}\not\in \left\{ \frac{\beta +A}{B}\left(\frac{n-1}{2n}\right) - \frac{A}{B}|n\in\mathbb{N}\right\}$.
\item Suppose $B\neq 0$, $\alpha B - \beta A \neq 0$, $\beta +A \neq 0$, and $R=\frac{\beta A - \alpha B}{(\beta + A)^{2}}\in (\frac{1}{4},\infty)$,
let $\phi = \arccos{\left(\frac{1}{2}\sqrt{\frac{1}{R}}\right)}$,then the forbidden set, $\mathfrak{F}= \left\{ \frac{\beta +A}{2B}\left(1-\sqrt{4R-1}\cot{\left(n\phi\right)}\right) - \frac{A}{B}|n\in\mathbb{N}\right\}$. Furthermore, for all $n\in\mathbb{N}$,
$$x_{n}=\frac{\beta +A}{B}\left(\sqrt{R}\right)\left(\frac{\sqrt{4R-1}\cos{\left((n+1)\phi\right)}-(\frac{2Bx_{0}+2A}{\beta + A}-1)\sin{\left((n+1)\phi\right)}}{\sqrt{4R-1}\cos{\left(n\phi\right)}-(\frac{2Bx_{0}+2A}{\beta + A}-1)\sin{\left(n\phi\right)}}\right) - \frac{A}{B},$$
whenever $x_{0}\not\in \left\{ \frac{\beta +A}{2B}\left(1-\sqrt{4R-1}\cot{n\phi}\right) - \frac{A}{B}|n\in\mathbb{N}\right\}$.
\end{enumerate}
\end{thm}
\section{Using invariants to find forbidden sets}
Here we present six rational difference equations, each of order 2, which possess algebraic invariants. The invariants allow for a reduction of order in each case so that the dynamics of the equation can be described by either a family of Riccati equations, or a family of linear equations.
The examples here have nice algebraic properties which are conducive to this type of approach. A remaining question of importance which is left to further work is whether this approach can be adapted to yield forbidden sets and explicit closed form solutions for other rational equations. \par
In the remainder of the article we make the notational convention of representing the set of inital conditions as a set in $\mathbb{C}^{2}$ with coordinates $(z_{0},z_{-1})$ this will be important for the remainder of the article, especially as it is needed to accurately describe the forbidden sets. Moreover, to accommodate the large formulae necessary to give a complete description of the forbidden sets, the forbidden sets are included in figures 1 and 2.
\begin{thm}
Consider the rational difference equation,
\begin{equation}
z_{n+1}=\frac{z_{n}}{1+B z_{n-1}-B z_{n}},\quad n=0,1,\dots,
\end{equation}
with $B\in \mathbb{C}\setminus \{0\}$ and with initial conditions $z_{0},z_{-1}\in \mathbb{C}$. Then the forbidden set, $\mathfrak{F}=S_{1}$, where $S_{1}$ is given in Figure 1. Also given $(z_{0},z_{-1})\notin \mathfrak{F}$ there are four possibilities:
\begin{enumerate}[i.]
\item $z_{0}=0$, in which case $z_{n}=0$ for all $n\geq 0$.
\item $z_{0}\neq 0$ and $z_{-1}=\frac{-1}{B}$, in which case $z_{2n+1}=\frac{-1}{B}$ for all $n\geq 0$ and $z_{2n}=\frac{-1}{2B+B^{2}z_{2n-2}}$ for all $n\geq 1$. Thus
$$z_{2n}=\frac{n+2+nBz_{0}+Bz_{0}}{nB+B+nB^{2}z_{0}}-\frac{2}{B},\quad n\geq 0.$$
\item $z_{0}=\frac{-1}{B}$, in which case $z_{2n}=\frac{-1}{B}$ for all $n\geq 0$ and $z_{2n+1}=\frac{-1}{2B+B^{2}z_{2n-1}}$ for all $n\geq 0$. Thus
$$z_{2n+1}=\frac{n+3+nBz_{-1}+2Bz_{-1}}{nB+2B+nB^{2}z_{-1}+B^{2}z_{-1}}-\frac{2}{B},\quad n\geq -1.$$
\item $z_{0}\neq 0, \frac{-1}{B}$ and $z_{-1}\neq \frac{-1}{B}$, in which case $z_{n+1}=\frac{1+Bz_{n}}{C-B-B^{2}z_{n}}$ for all $n\geq 0$, where $C=\left(\frac{1}{z_{0}} + B \right)\left(1+ B z_{-1} \right)$. This implies the following:
\begin{enumerate}[a.]
\item If $\frac{B}{C}\in \mathbb{C}\setminus \left[\frac{1}{4},\infty\right)$, then call $C-B-B^{2}z_{0}-C\lambda_{1}=M_{1}$ and $C\lambda_{2}+B+B^{2}z_{0}-C=M_{2}$, and $$z_{n}=\frac{-C}{B^{2}}\left(\frac{M_{2}\lambda^{n+1}_{1}+M_{1}\lambda^{n+1}_{2}}{M_{2}\lambda^{n}_{1}+M_{1}\lambda^{n}_{2}}\right)+\frac{C}{B^{2}}-\frac{1}{B},\quad n\geq 0.$$
Where $$\lambda_{1}=\frac{1-\sqrt{1-\frac{4B}{C}}}{2},\quad and \quad \lambda_{2}=\frac{1+\sqrt{1-\frac{4B}{C}}}{2}.$$
\item If $C=4B$, then
$$z_{n}=\frac{4+(n+1)\left(2-2Bz_{0} \right)}{nB^{2}z_{0}-2B-nB}+\frac{3}{B},\quad n\geq 0.$$
\item If $\frac{C}{B}\in (0,4)$, then call $\arccos\left(\sqrt{\frac{C}{4B}}\right)=\rho$, and for $n\geq 0$ we have,
$$z_{n}=\frac{-\sqrt{\frac{C}{B}}}{B}\left(\frac{\left(\sqrt{\frac{4B}{C}-1}\right)\cos\left((n+1)\rho\right)+(2w_{0}-1)\sin\left((n+1)\rho\right)}{\left(\sqrt{\frac{4B}{C}-1}\right)\cos\left(n\rho\right)+(2w_{0}-1)\sin\left(n\rho\right)}\right)+\frac{C-B}{B^{2}}.$$
Where $$w_{0}=\frac{-B^{2}z_{0}+C-B}{C}.$$
\end{enumerate}\end{enumerate}
\end{thm}
\begin{proof}
Let us begin by finding the forbidden set for our Equation (4). Let $\mathfrak{F}_{1}$ be the forbidden set with $z_{0}=0$. Let $\mathfrak{F}_{2}$ be the forbidden set with $z_{0}\neq 0$ and $z_{-1}=\frac{-1}{B}$. Let $\mathfrak{F}_{3}$ be the forbidden set with $z_{0}=\frac{-1}{B}$. Let $\mathfrak{F}_{4}$ be the forbidden set with $z_{0}\neq 0,\frac{-1}{B}$ and $z_{-1}\neq\frac{-1}{B}$.
Then the forbidden set $\mathfrak{F}=\bigcup^{4}_{i=1}\mathfrak{F}_{i}$. So we must find all $\mathfrak{F}_{i}$ with $1\leq i\leq 4$.\par
Let us begin with $\mathfrak{F}_{1}$. We have assumed $z_{0}=0$. Let us further assume that $z_{-1}\neq \frac{-1}{B}$, then $z_{1}$ is well defined and equal to $0$. Whenever $(z_{n},z_{n-1})=(0,0)$, then $z_{n+1}$ is well defined and equal to $0$. Thus, by induction, $z_{n}$ is well defined and equal to $0$ for all $n\in\mathbb{N}$.
Thus, if $z_{-1}\neq \frac{-1}{B}$, then $(0,z_{-1})\not\in \mathfrak{F}_{1}$. On the other hand, assume $z_{0}=0$ and $z_{-1}=\frac{-1}{B}$, then $z_{1}$ is not well defined and so $(0,\frac{-1}{B})\in \mathfrak{F}_{1}$. Thus, $\mathfrak{F}_{1}=\{(0,\frac{-1}{B})\}$.\par
Now, let us find $\mathfrak{F}_{2}$. In this case we have assumed $z_{0}\neq 0$ and $z_{-1}=\frac{-1}{B}$. Assume $z_{2n}$ is well defined for $0\leq n\leq N$, then we may make an induction argument with $z_{-1}$ as the base case.
Assume that $z_{2k+1}=\frac{-1}{B}$ for $k<N$, then, by our earlier assumption, $z_{2k+2}$ is well defined and
$$z_{2k+2}= \frac{-1}{2B+B^{2}z_{2k}}\neq 0.$$ So,
$$z_{2k+3}=\frac{z_{2k+2}}{1+B z_{2k+1}-B z_{2k+2}}= \frac{z_{2k+2}}{-B z_{2k+2}}= \frac{-1}{B}.$$
By this induction argument, $z_{2n+1}=\frac{-1}{B}$ and assuming $z_{2N+2}$ is well defined,
$$z_{2n+2}= \frac{-1}{2B+B^{2}z_{2n}},$$
for $0\leq n\leq N$. Call the forbidden set of the following first order difference equation $\hat{\mathfrak{F}}$,
$$x_{n+1}= \frac{-1}{2B+B^{2}x_{n}}, \quad n=0,1,2,\dots.$$
Now, suppose $z_{0}\neq 0$, $z_{-1}=\frac{-1}{B}$, and $z_{0}\not\in \hat{\mathfrak{F}}$, and assume that $z_{2n}$ is well defined for $n\leq N$, then $z_{2N+1}=\frac{-1}{B}$ and
$$1+Bz_{2N}-Bz_{2N+1}=2 + Bz_{2N}=\frac{2B + B^{2}z_{2N}}{B}\neq 0,$$
since $z_{0}\not\in \hat{\mathfrak{F}}$. Thus, $z_{2n}$ is well defined for $n\leq N+1$. By induction, $z_{n}$ is well defined for all $n\in\mathbb{N}$. Thus, $(z_{0},\frac{-1}{B})\not\in \mathfrak{F}_{2}$.\par
Now, suppose $z_{0}\neq 0$, $z_{-1}=\frac{-1}{B}$, and $z_{0}\in \hat{\mathfrak{F}}$. Further assume for the sake of contradiction that $(z_{0},\frac{-1}{B})\not\in \mathfrak{F}_{2}$. Then, since $(z_{0},\frac{-1}{B})\not\in \mathfrak{F}_{2}$, $z_{n}$ is well defined for all $n\in\mathbb{N}$, but also
$2B+B^{2}z_{2N}=0$ for some $N\in\mathbb{N}$, since $z_{0}\in \hat{\mathfrak{F}}$. But then
$$1+Bz_{2N}-Bz_{2N+1}=2 + Bz_{2N}=\frac{2B + B^{2}z_{2N}}{B}= 0.$$ This is a contradiction. Thus, $(z_{0},\frac{-1}{B})\in \mathfrak{F}_{2}$. So $\mathfrak{F}_{2}=\left(\hat{\mathfrak{F}}\setminus \{0\}\right)\times \{\frac{-1}{B}\}$.\par
Next, let us find $\mathfrak{F}_{3}$. In this case we have assumed $z_{0}=\frac{-1}{B}$. Assume $z_{2n-1}$ is well defined for $1\leq n\leq N$, then we may make an induction argument with $z_{0}$ as the base case.
Assume that $z_{2k}=\frac{-1}{B}$ for $k<N$, then, by our earlier assumption, $z_{2k+1}$ is well defined and
$$z_{2k+1}= \frac{-1}{2B+B^{2}z_{2k}}\neq 0.$$ So,
$$z_{2k+2}=\frac{z_{2k+1}}{1+B z_{2k}-B z_{2k+1}}= \frac{z_{2k+1}}{-B z_{2k+1}}= \frac{-1}{B}.$$
By this induction argument, $z_{2n}=\frac{-1}{B}$ and assuming $z_{2N+1}$ is well defined,
$$z_{2n+1}= \frac{-1}{2B+B^{2}z_{2n}},$$
for $0\leq n\leq N$. Call the forbidden set of the following first order difference equation $\hat{\mathfrak{F}}$,
$$x_{n+1}= \frac{-1}{2B+B^{2}x_{n}}, \quad n=0,1,2,\dots.$$
Now, suppose $z_{0}=\frac{-1}{B}$ and $z_{-1}\not\in \hat{\mathfrak{F}}$, and assume that $z_{2n-1}$ is well defined for $n\leq N$, then $z_{2N}=\frac{-1}{B}$ and
$$1+Bz_{2N-1}-Bz_{2N}=2 + Bz_{2N-1}=\frac{2B + B^{2}z_{2N-1}}{B}\neq 0,$$
since $z_{-1}\not\in \hat{\mathfrak{F}}$. Thus, $z_{2n-1}$ is well defined for $n\leq N+1$. By induction, $z_{n}$ is well defined for all $n\in\mathbb{N}$. Thus, $(\frac{-1}{B},z_{-1})\not\in \mathfrak{F}_{3}$.\par
Now, suppose $z_{0}=\frac{-1}{B}$ and $z_{-1}\in \hat{\mathfrak{F}}$. Further assume for the sake of contradiction that $(\frac{-1}{B},z_{-1})\not\in \mathfrak{F}_{3}$. Then, since $(\frac{-1}{B},z_{-1})\not\in \mathfrak{F}_{3}$, $z_{n}$ is well defined for all $n\in\mathbb{N}$, but also
$2B+B^{2}z_{2N-1}=0$ for some $N\in\mathbb{N}$, since $z_{-1}\in \hat{\mathfrak{F}}$. But then
$$1+Bz_{2N-1}-Bz_{2N}=2 + Bz_{2N-1}=\frac{2B + B^{2}z_{2N-1}}{B}= 0.$$ This is a contradiction. Thus, $(\frac{-1}{B},z_{-1})\in \mathfrak{F}_{3}$. So, $\mathfrak{F}_{3}= \{\frac{-1}{B}\} \times \hat{\mathfrak{F}}$.\par
Finally, let us find $\mathfrak{F}_{4}$. In this case we have assumed $z_{0}\neq 0,\frac{-1}{B}$ and $z_{-1}\neq \frac{-1}{B}$. Assume $z_{n}$ is well defined for $n\leq N$, then a simple induction argument shows $z_{n}\neq 0$ for $1\leq n\leq N$. Using this we get that for $n<N$,
$$\left(\frac{1}{z_{n}} + B \right)\left(1+ B z_{n-1} \right)= \left(\frac{1+B z_{n-1}}{z_{n}}\right) \left(1+ B z_{n} \right)=$$
$$ \left(\frac{1+B z_{n-1}-Bz_{n}+Bz_{n}}{z_{n}}\right) \left(1+ B z_{n} \right)=$$
$$\left(\frac{1+B z_{n-1}-Bz_{n}}{z_{n}} + B\right) \left(1+ B z_{n} \right)=\left(\frac{1}{z_{n+1}} + B \right)\left(1+ B z_{n} \right).$$
So,
$$\left(\frac{1}{z_{n}} + B \right)\left(1+ B z_{n-1} \right)=constant,$$
for $0\leq n\leq N$. Thus, $z_{n}\neq \frac{-1}{B}$ for $n\leq N$ and assuming $z_{N+1}$ is well defined,
$$z_{n+1}= \frac{1+Bz_{n}}{C-B-B^{2}z_{n}},$$
for $0\leq n\leq N$. Where $C=\left(\frac{1}{z_{0}} + B \right)\left(1+ B z_{-1} \right)$.
Call the forbidden set of the following first order difference equation $\mathfrak{F}_{C}$,
$$x_{n+1}= \frac{1+Bx_{n}}{C-B-B^{2}x_{n}}, \quad n=0,1,2,\dots.$$
Note that the set $\mathfrak{F}_{C}$ changes depending on the value of $C$.
Now, suppose $z_{0}\neq 0,\frac{-1}{B}$, $z_{-1}\neq \frac{-1}{B}$, $C=\left(\frac{1}{z_{0}} + B \right)\left(1+ B z_{-1} \right)$, and $z_{0}\not\in \mathfrak{F}_{C}$, and assume that $z_{n}$ is well defined for $n\leq N$. Recall that we have shown that this implies $z_{n}\neq \frac{-1}{B}$ for $n\leq N$. Then
$$1+Bz_{N-1}-Bz_{N}=\frac{C}{B+\frac{1}{z_{N}}}-Bz_{N}=\frac{C-B - B^{2}z_{N}}{B+\frac{1}{z_{N}}}\neq 0,$$
since $z_{0}\not\in \mathfrak{F}_{C}$. Thus, $z_{n}$ is well defined for $n\leq N+1$. By induction, $z_{n}$ is well defined for all $n\in\mathbb{N}$. Thus, $(z_{0},z_{-1})\not\in \mathfrak{F}_{4}$.\par
Now, suppose $z_{0}\neq 0,\frac{-1}{B}$, $z_{-1}\neq \frac{-1}{B}$, $C=\left(\frac{1}{z_{0}} + B \right)\left(1+ B z_{-1} \right)$, and $z_{0}\in \mathfrak{F}_{C}$. Further assume for the sake of contradiction that $(z_{0},z_{-1})\not\in \mathfrak{F}_{4}$. Then, since $(z_{0},z_{-1})\not\in \mathfrak{F}_{4}$, $z_{n}$ is well defined for all $n\in\mathbb{N}$, but also
$C-B - B^{2}z_{N}=0$ for some $N\in\mathbb{N}$, since $z_{0}\in \mathfrak{F}_{C}$. But then
$$1+Bz_{N-1}-Bz_{N}=\frac{C}{B+\frac{1}{z_{N}}}-Bz_{N}=\frac{C-B - B^{2}z_{N}}{B+\frac{1}{z_{N}}}= 0.$$ This is a contradiction. Thus, $(z_{0},z_{-1})\in \mathfrak{F}_{4}$. So,
$$\mathfrak{F}_{4}=\bigcup_{b\neq \frac{-1}{B}} \bigcup_{C\neq 0} \left(\left(\left(\mathfrak{F}_{C}\setminus\left\{0,\frac{-1}{B}\right\}\right)\times \{b\}\right)\cap \left\{(a,b)|C=\left(\frac{1}{a} + B \right)\left(1+ B b \right)\right\}\right).$$
Reducing the above expression, we get
$$\mathfrak{F}_{4}= \bigcup_{C\neq 0} \left\{ \left(a,\frac{Ca-Ba-1}{B^{2}a+B}\right)| a\in\mathfrak{F}_{C}\setminus\left\{0,\frac{-1}{B}\right\}\right\}.$$
From the above characterization of $\mathfrak{F}_{1},\dots ,\mathfrak{F}_{4} $ and from the facts about the forbidden sets of the Riccati difference equation in Section 3, we get $\mathfrak{F}=S_{1}$, where $S_{1}$ is given in Figure 1. Now, let us describe the behavior when $(z_{0},z_{-1})\notin \mathfrak{F}$. Our analysis of this case will be broken into four subcases as shown in the statement of Theorem 2.
Let us first consider case (i). In this case $z_{0}=0$ and also, since $(z_{0},z_{-1})\notin \mathfrak{F}$, there will never be division by zero in our solution.
It is immediately clear from these two facts and from a basic induction argument that $z_{n}=0$ for all $n\geq 0$.
Now, let us consider case (ii). In this case $z_{0}\neq 0$ and $z_{-1}=\frac{-1}{B}$. Also, since $(z_{0},z_{-1})\notin \mathfrak{F}$, there will never be division by zero in our solution.
This allows us to prove by induction that $z_{n}\neq 0$ for all $n\geq 0$. The induction argument for this piece is straightforward and is omitted. Now, we will show by induction that $z_{2n+1}=\frac{-1}{B}$ for all $n\geq 0$.
Since $z_{0}\neq 0$, we have
$$z_{1}= \frac{z_{0}}{1+B z_{-1}-B z_{0}}= \frac{z_{0}}{-B z_{0}}= \frac{-1}{B}.$$
This provides the base case for our induction argument. Now, suppose $z_{2n-1}=\frac{-1}{B}$, then since $z_{n}\neq 0$ for all $n\geq 0$, we have
$$z_{2n+1}=\frac{z_{2n}}{1+B z_{2n-1}-B z_{2n}}= \frac{z_{2n}}{-B z_{2n}}= \frac{-1}{B}.$$
Thus, we have shown that $z_{2n+1}=\frac{-1}{B}$ for all $n\geq 0$. This fact and Equation (4) immediately yields,
$$z_{2n}=\frac{z_{2n-1}}{1+B z_{2n-2}-B z_{2n-1}}= \frac{-1}{2B+B^{2}z_{2n-2}}, \quad n=1,2,\dots.$$
Thus, the even terms are defined recursively by the above equation. Notice that since we have only rewritten the recursive Equation (4) we cannot have division by zero in this equation with our choice of initial conditions. In other words, $z_{2n}\neq \frac{-2}{B}$ for any $n\geq 0$.
Notice that this is a Riccati equation after a change of variables. Since we already know the closed form solution for any Riccati equation, we may obtain a closed form solution for $z_{2n}$, and thus a closed form solution for $z_{n}$ in this case. We use the known results for Riccati equations restated in Section 3 to obtain the closed form solutions in the statement of the theorem.\par
Now, let us consider case (iii). In this case, $z_{0}=\frac{-1}{B}$. Also, since $(z_{0},z_{-1})\notin \mathfrak{F}$, there will never be division by zero in our solution.
This allows us to prove by induction that $z_{n}\neq 0$ for all $n\geq 0$. The induction argument for this piece is straightforward and is omitted. Now, we will show by induction that $z_{2n}=\frac{-1}{B}$ for all $n\geq 0$.
Since $z_{1}\neq 0$, we have
$$z_{2}= \frac{z_{1}}{1+B z_{0}-B z_{1}}= \frac{z_{1}}{-B z_{1}}= \frac{-1}{B}.$$
This provides the base case for our induction argument. Now, suppose $z_{2n-2}=\frac{-1}{B}$, then since $z_{n}\neq 0$ for all $n\geq 0$, we have
$$z_{2n}=\frac{z_{2n-1}}{1+B z_{2n-2}-B z_{2n-1}}= \frac{z_{2n-1}}{-B z_{2n-1}}= \frac{-1}{B}.$$
Thus, we have shown that $z_{2n}=\frac{-1}{B}$ for all $n\geq 0$. This fact and Equation (4) immediately yields,
$$z_{2n+1}=\frac{z_{2n}}{1+B z_{2n-1}-B z_{2n}}= \frac{-1}{2B+B^{2}z_{2n-1}}, \quad n=0,1,2,\dots.$$
Thus, the odd terms are defined recursively by the above equation. Notice that since we have only rewritten the recursive Equation (4) we cannot have division by zero in this equation with our choice of initial conditions. In other words, $z_{2n-1}\neq \frac{-2}{B}$ for any $n\geq 0$.
Notice that this is a Riccati equation after a change of variables. Since we already know the closed form solution for any Riccati equation, we may obtain a closed form solution for $z_{2n+1}$, and thus a closed form solution for $z_{n}$ in this case. We use the known results for Riccati equations restated in Section 3 to obtain the closed form solutions in the statement of the theorem.\par
Let us finally consider case (iv). In this case $z_{0}\neq 0, \frac{-1}{B}$ and $z_{-1}\neq \frac{-1}{B}$. Also, since $(z_{0},z_{-1})\notin \mathfrak{F}$, there will never be division by zero in our solution.
This allows us to prove by induction that $z_{n}\neq 0$ for all $n\geq 0$. The induction argument for this piece is straightforward and is omitted. Since there is never division by zero in our solution, and since $z_{n}\neq 0$ for all $n\geq 0$, the following algebraic computation is well defined.
$$\left(\frac{1}{z_{n}} + B \right)\left(1+ B z_{n-1} \right)= \left(\frac{1+B z_{n-1}}{z_{n}}\right) \left(1+ B z_{n} \right)=$$ $$\left(\frac{1+B z_{n-1}-Bz_{n}+Bz_{n}}{z_{n}}\right) \left(1+ B z_{n} \right)=$$ $$\left(\frac{1+B z_{n-1}-Bz_{n}}{z_{n}} + B\right) \left(1+ B z_{n} \right)=\left(\frac{1}{z_{n+1}} + B \right)\left(1+ B z_{n} \right).$$
Thus, we have the following algebraic invariant:
$$\left(\frac{1}{z_{n}} + B \right)\left(1+ B z_{n-1} \right)=constant.$$
For our fixed but arbitrary initial conditions with $z_{0}\neq 0, \frac{-1}{B}$ and $z_{-1}\neq \frac{-1}{B}$ let us denote $C=\left(\frac{1}{z_{0}} + B \right)\left(1+ B z_{-1} \right)$. Since $z_{0}\neq 0$, $C$ is well defined, and since $z_{0},z_{-1}\neq \frac{-1}{B}$, $C\neq 0$.
Since $C\neq 0$, this forces $z_{n}\neq \frac{-1}{B}$ for all $n\geq 0$. Now, we claim that since $(z_{0},z_{-1})\notin \mathfrak{F}$, $z_{n}\neq \frac{C-B}{B^{2}}$ for all $n\geq 0$. In the case where $C=B$ it follows from the fact that $z_{n}\neq 0$ for all $n\geq 0$. In the remaining case, suppose there were such an $N$, then:
$$\left(\frac{B^{2}}{C-B} + B \right)\left(1+ B z_{N-1} \right)=C.$$
This implies that $z_{N}=\frac{C-B}{B^{2}}$ and $z_{N-1}= \frac{C-2B}{B^{2}}$, but then $1+Bz_{N-1}-Bz_{N}= 0$. However, this contradicts the fact that $(z_{0},z_{-1})\notin \mathfrak{F}$. Thus, we have that $z_{n}\neq \frac{C-B}{B^{2}}$ for all $n\geq 0$.
Algebraic manipulations of our invariant yield the following:
$$z_{n}=\frac{1+Bz_{n-1}}{C-B-B^{2}z_{n-1}},\quad n\geq 1.$$
Since $z_{n}\neq \frac{C-B}{B^{2}}$ for all $n\geq 0$, this equation is well-defined for all $n\geq 1$. Thus the dynamics of $\{z_{n}\}^{\infty}_{n=-1}$ are given by a Riccati equation in this case. Since we already know the closed form solution for any Riccati equation, we may obtain a closed form solution for $\{z_{n}\}^{\infty}_{n=-1}$ in this case. We use the known results for Riccati equations restated in Section 3 to obtain the closed form solutions in the statement of the theorem.
\end{proof}
\begin{thm}
Consider the rational difference equation,
\begin{equation}
z_{n+1}=\frac{z_{n-1}}{1+B z_{n}-B z_{n-1}},\quad n=0,1,\dots,
\end{equation}
with $B\in \mathbb{C}\setminus \{0\}$ and with initial conditions $z_{0},z_{-1}\in \mathbb{C}$. Then the forbidden set, $\mathfrak{F}=S_{2}$, where $S_{2}$ is given in Figure 1.
Also, given $(z_{0},z_{-1})\notin \mathfrak{F}$, there are four possibilities:
\begin{enumerate}[i.]
\item $z_{0}=0$, in which case $z_{2n}=0$ for all $n\geq 0$ and $z_{2n+1}=\frac{z_{2n-1}}{1-Bz_{2n-1}}$ for all $n\geq 0$. This implies
$$z_{2n+1}=\frac{1}{-B}\left(\frac{1-(n+2)Bz_{-1}}{1-(n+1)Bz_{-1}}\right)+\frac{1}{B},\quad n\geq -1.$$
\item $z_{-1}=0$, in which case $z_{2n+1}=0$ for all $n\geq 0$ and $z_{2n}=\frac{z_{2n-2}}{1-Bz_{2n-2}}$ for all $n\geq 1$. This implies
$$z_{2n}=\frac{1}{-B}\left(\frac{1-(n+1)Bz_{0}}{1-nBz_{0}}\right)+\frac{1}{B},\quad n\geq 0.$$
\item $z_{0}=\frac{-1}{B}$, in which case $z_{n}=\frac{-1}{B}$ for all $n\geq 0$.
\item $z_{0}\neq 0, \frac{-1}{B}$ and $z_{-1}\neq 0$, in which case $z_{n+1}=\frac{1}{Cz_{n}-B}$ for all $n\geq 0$, where $C=\left(\frac{1}{z_{0}} + B \right)\left(\frac{1}{z_{-1}} \right)$. This implies the following:
\begin{enumerate}[a.]
\item If $\frac{-C}{B^{2}}\in \mathbb{C}\setminus \left[\frac{1}{4},\infty\right)$, then $$z_{n}=\frac{-B}{C}\left(\frac{(B\lambda_{2}+Cz_{0}-B)\lambda^{n+1}_{1}+(B-Cz_{0}-B\lambda_{1})\lambda^{n+1}_{2}}{(B\lambda_{2}+Cz_{0}-B)\lambda^{n}_{1}+(B-Cz_{0}-B\lambda_{1})\lambda^{n}_{2}}\right)+\frac{B}{C},\quad n\geq 0.$$
Where $$\lambda_{1}=\frac{1-\sqrt{1+\frac{4C}{B^{2}}}}{2},\quad and \quad \lambda_{2}=\frac{1+\sqrt{1+\frac{4C}{B^{2}}}}{2}.$$
\item If $\frac{-C}{B^{2}}=\frac{1}{4}$, then
$$z_{n}=\frac{-B}{C}\left(\frac{-B+(n+1)\left(2Cz_{0}-B \right)}{-2B+4nCz_{0}-2nB}\right)+\frac{B}{C},\quad n\geq 0.$$
\item If $\frac{-C}{B^{2}}\in \left(\frac{1}{4},\infty\right)$, then call $B\sqrt{\frac{-4C}{B^{2}}-1}=D$ and call $\arccos\left(\sqrt{\frac{-B^{2}}{4C}}\right)=\rho$ and for $n\geq 0$, we have
$$z_{n}=\sqrt{\frac{-B^{2}}{C}}\left(\frac{D\cos\left((n+1)\rho\right)+(B-2Cz_{0})\sin\left((n+1)\rho\right)}{BD\cos\left(n\rho\right)+(B^{2}-2CBz_{0})\sin\left(n\rho\right)}\right)+\frac{B}{C}.$$
\end{enumerate}
\end{enumerate}
\end{thm}
\begin{proof}
We begin by finding the forbidden set for our Equation (5). Let $\mathfrak{F}_{1}$ be the forbidden set with $z_{0}=0$. Let $\mathfrak{F}_{2}$ be the forbidden set with $z_{-1}=0$. Let $\mathfrak{F}_{3}$ be the forbidden set with $z_{0}=\frac{-1}{B}$. Let $\mathfrak{F}_{4}$ be the forbidden set with $z_{0}\neq 0,\frac{-1}{B}$ and $z_{-1}\neq 0$.
Then the forbidden set $\mathfrak{F}=\bigcup^{4}_{i=1}\mathfrak{F}_{i}$. So we must find all $\mathfrak{F}_{i}$ with $1\leq i\leq 4$.\par
Let us begin with $\mathfrak{F}_{1}$. We have assumed $z_{0}=0$. Assume $z_{n}$ is well defined for $0\leq n\leq 2N$, then by a simple induction argument $z_{2n}=0$ for $0\leq n\leq N$ and so assuming $z_{2N+1}$ is well defined,
$$z_{2n+1}= \frac{z_{2n-1}}{1-Bz_{2n-1}},$$
for $0\leq n\leq N$. Call the forbidden set of the following first order difference equation $\hat{\mathfrak{F}}$,
$$x_{n+1}= \frac{x_{n}}{1-Bx_{n}}, \quad n=0,1,2,\dots.$$
Now, suppose $z_{0}=0$ and $z_{-1}\not\in \hat{\mathfrak{F}}$, and assume that $z_{n}$ is well defined for $n\leq 2N$, then $z_{2N}=0$ and
$$1+Bz_{2N}-Bz_{2N-1}=1 - Bz_{2N-1}\neq 0,$$
since $z_{-1}\not\in \hat{\mathfrak{F}}$. Thus, $z_{n}$ is well defined for $n\leq 2N+1$ and
$$z_{2N+1}= \frac{z_{2N-1}}{1-Bz_{2N-1}}.$$
So,
$$1+Bz_{2N+1}-Bz_{2N}=1 + Bz_{2N+1}= \frac{1-Bz_{2N-1} + Bz_{2N-1}}{1-Bz_{2N-1}}= \frac{1}{1-Bz_{2N-1}}\neq 0,$$
since $1 - Bz_{2N-1}\neq 0$. Thus $z_{n}$ is well defined for $n\leq 2N+2$. By induction $z_{n}$ is well defined for all $n\in\mathbb{N}$. Thus $(0,z_{-1})\not\in \mathfrak{F}_{1}$.\par
Now suppose $z_{0}=0$ and $z_{-1}\in \hat{\mathfrak{F}}$. Further assume for the sake of contradiction that $(0,z_{-1})\not\in \mathfrak{F}_{1}$. Then, since $(0,z_{-1})\not\in \mathfrak{F}_{1}$, $z_{n}$ is well defined for all $n\in\mathbb{N}$, but also
$1-Bz_{2N-1}=0$ for some $N\in\mathbb{N}$ since $z_{-1}\in \hat{\mathfrak{F}}$. But then
$$1+Bz_{2N}-Bz_{2N-1}=1 - Bz_{2N-1}= 0.$$ This is a contradiction. Thus $(0,z_{-1})\in \mathfrak{F}_{1}$. So $\mathfrak{F}_{1}=\{0\}\times \hat{\mathfrak{F}} $.\par
Next, let us find $\mathfrak{F}_{2}$. In this case, we have assumed $z_{-1}=0$. Assume $z_{n}$ is well defined for $0\leq n\leq 2N+1$, then by a simple induction argment $z_{2n+1}=0$ for $0\leq n\leq N$ and so assuming $z_{2N+2}$ is well defined,
$$z_{2n+2}= \frac{z_{2n}}{1-Bz_{2n}},$$
for $0\leq n\leq N$. Call the forbidden set of the following first order difference equation $\hat{\mathfrak{F}}$,
$$x_{n+1}= \frac{x_{n}}{1-Bx_{n}}, \quad n=0,1,2,\dots.$$
Now, suppose $z_{-1}=0$ and $z_{0}\not\in \hat{\mathfrak{F}}$ and assume that $z_{n}$ is well defined for $n\leq 2N+1$, then $z_{2N+1}=0$ and
$$1+Bz_{2N+1}-Bz_{2N}=1 - Bz_{2N}\neq 0,$$
since $z_{0}\not\in \hat{\mathfrak{F}}$. Thus, $z_{n}$ is well defined for $n\leq 2N+2$ and
$$z_{2N+2}= \frac{z_{2N}}{1-Bz_{2N}}.$$
So,
$$1+Bz_{2N+2}-Bz_{2N+1}=1 + Bz_{2N+2}= \frac{1-Bz_{2N} + Bz_{2N}}{1-Bz_{2N}}= \frac{1}{1-Bz_{2N}}\neq 0,$$
since $1 - Bz_{2N}\neq 0$. Thus, $z_{n}$ is well defined for $n\leq 2N+3$. By induction $z_{n}$ is well defined for all $n\in\mathbb{N}$. Thus, $(z_{0},0)\not\in \mathfrak{F}_{2}$.\par
Now, suppose $z_{-1}=0$ and $z_{0}\in \hat{\mathfrak{F}}$. Further assume for the sake of contradiction that $(z_{0},0)\not\in \mathfrak{F}_{2}$. Then, since $(z_{0},0)\not\in \mathfrak{F}_{2}$, $z_{n}$ is well defined for all $n\in\mathbb{N}$, but also
$1-Bz_{2N}=0$ for some $N\in\mathbb{N}$, since $z_{0}\in \hat{\mathfrak{F}}$. But then
$$1+Bz_{2N+1}-Bz_{2N}=1 - Bz_{2N}= 0.$$ This is a contradiction. Thus, $(z_{0},0)\in \mathfrak{F}_{2}$. So $\mathfrak{F}_{2}= \hat{\mathfrak{F}}\times \{0\} $.\par
Now, let us find $\mathfrak{F}_{3}$. We have assumed $z_{0}=\frac{-1}{B}$. Let us further assume that $z_{-1}\neq 0$, then $z_{1}$ is well defined and equal to $\frac{-1}{B}$. Whenever $(z_{n},z_{n-1})=(\frac{-1}{B},\frac{-1}{B})$ then $z_{n+1}$ is well defined and equal to $\frac{-1}{B}$. Thus, by induction, $z_{n}$ is well defined and equal to $\frac{-1}{B}$ for all $n\in\mathbb{N}$.
Thus, if $z_{-1}\neq 0$, then $(\frac{-1}{B},z_{-1})\not\in \mathfrak{F}_{3}$. On the other hand, assume $z_{0}=\frac{-1}{B}$ and $z_{-1}=0$. Then $z_{1}$ is not well defined, and so $(\frac{-1}{B},0)\in \mathfrak{F}_{3}$. Thus, $\mathfrak{F}_{3}=\{(\frac{-1}{B},0)\}$.\par
Finally, let us find $\mathfrak{F}_{4}$. In this case we have assumed $z_{0}\neq 0,\frac{-1}{B}$ and $z_{-1}\neq 0$. Assume $z_{n}$ is well defined for $n\leq N$, then a simple induction argument shows $z_{n}\neq 0$ for $n\leq N$, and so for $n < N$,
$$\left(\frac{1}{z_{n+1}} + B \right)\left(\frac{1}{z_{n}} \right)= \left(\frac{1+ B z_{n} - B z_{n-1}}{z_{n-1}} + B \right) \left(\frac{1}{ z_{n}} \right)=$$
$$ \left(\frac{1+ B z_{n}}{z_{n-1}} \right) \left(\frac{1}{ z_{n}} \right)=\left(\frac{1+ B z_{n}}{z_{n}} \right) \left(\frac{1}{ z_{n-1}} \right)=\left(\frac{1}{z_{n}} + B \right)\left(\frac{1}{z_{n-1}} \right).$$
So,
$$\left(\frac{1}{z_{n}} + B \right)\left(\frac{1}{z_{n-1}} \right)=constant,$$
for $0\leq n\leq N$. Thus, $z_{n}\neq \frac{-1}{B}$ for $n\leq N$ and assuming $z_{N+1}$ is well defined,
$$z_{n+1}= \frac{1}{Cz_{n}-B},$$
for $0\leq n\leq N$. Where $C=\left(\frac{1}{z_{0}} + B \right)\left(\frac{1}{z_{-1}} \right)$.
Call the forbidden set of the following first order difference equation $\mathfrak{F}_{C}$,
$$x_{n+1}= \frac{1}{Cx_{n}-B}, \quad n=0,1,2,\dots.$$
Note that the set $\mathfrak{F}_{C}$ changes depending on the value of $C$.
Now, suppose $z_{0}\neq 0,\frac{-1}{B}$, $z_{-1}\neq 0$, $C=\left(\frac{1}{z_{0}} + B \right)\left(\frac{1}{z_{-1}} \right)$, and $z_{0}\not\in \mathfrak{F}_{C}$, and assume that $z_{n}$ is well defined for $n\leq N$. Then, notice that $$z_{N-1}=\frac{1+Bz_{N}}{Cz_{N}}.$$
Thus $$1+Bz_{N}-Bz_{N-1}=1+Bz_{N}-\left(\frac{B+B^{2}z_{N}}{Cz_{N}}\right)=\frac{Cz_{N}+BCz^{2}_{N}-B-B^{2}z_{N}}{Cz_{N}}$$$$=\frac{(Cz_{N}-B)(Bz_{N}+1)}{Cz_{N}}\neq 0,$$
since $z_{0}\not\in \mathfrak{F}_{C}$ and $z_{n}\neq 0,\frac{-1}{B}$ for all $n\leq N$. Thus, $z_{n}$ is well defined for $n\leq N+1$. By induction, $z_{n}$ is well defined for all $n\in\mathbb{N}$. Thus $(z_{0},z_{-1})\not\in \mathfrak{F}_{4}$.\newline
Now, suppose $z_{0}\neq 0,\frac{-1}{B}$, $z_{-1}\neq 0$, $C=\left(\frac{1}{z_{0}} + B \right)\left(\frac{1}{z_{-1}} \right)$, and $z_{0}\in \mathfrak{F}_{C}$. Further assume for the sake of contradiction that $(z_{0},z_{-1})\not\in \mathfrak{F}_{4}$. Then, since $(z_{0},z_{-1})\not\in \mathfrak{F}_{4}$, $z_{n}$ is well defined for all $n\in\mathbb{N}$, but also
$Cz_{N}-B=0$ for some $N\in\mathbb{N}$ since $z_{0}\in \mathfrak{F}_{C}$. But then
$$1+Bz_{N}-Bz_{N-1}=1+Bz_{N}-\left(\frac{B+B^{2}z_{N}}{Cz_{N}}\right)=\frac{(Cz_{N}-B)(Bz_{N}+1)}{Cz_{N}}= 0.$$ This is a contradiction. Thus, $(z_{0},z_{-1})\in \mathfrak{F}_{4}$.
So, $$\mathfrak{F}_{4}=\bigcup_{b\neq 0} \bigcup_{C\neq 0} \left(\left(\left(\mathfrak{F}_{C}\setminus\left\{0,\frac{-1}{B}\right\}\right)\times \{b\}\right)\cap \left\{(a,b)|C=\left(\frac{1}{a} + B \right)\left(\frac{1}{b} \right)\right\}\right).$$
Reducing the above expression, we get
$$\mathfrak{F}_{4}= \bigcup_{C\neq 0} \left\{ \left(a,\frac{Ba+1}{Ca}\right)| a\in\mathfrak{F}_{C}\setminus\left\{0,\frac{-1}{B}\right\}\right\}.$$
From the above characterization of $\mathfrak{F}_{1},\dots ,\mathfrak{F}_{4} $ and from the facts about the forbidden sets of the Riccati difference equation in Section 3, we get $\mathfrak{F}=S_{2}$, where $S_{2}$ is given in Figure 1.
Now, we will describe the behavior when $(z_{0},z_{-1})\notin \mathfrak{F}$. Our analysis of this case will be broken into four subcases as shown in the statement of Theorem 3.
Let us first consider case (i). In this case, $z_{0}=0$ and also since $(z_{0},z_{-1})\notin \mathfrak{F}$ there will never be division by zero in our solution.
It is immediately clear from these two facts and from a basic induction argument that $z_{2n}=0$ for all $n\geq 0$. This fact and Equation (5) immediately yields,
$$z_{2n+1}=\frac{z_{2n-1}}{1+Bz_{2n}-Bz_{2n-1}}=\frac{z_{2n-1}}{1-Bz_{2n-1}} , \quad n=0,1,2,\dots.$$
Thus, the odd terms are defined recursively by the above equation. Notice that since we have only rewritten the recursive Equation (5), we cannot have division by zero in this equation with our choice of initial conditions. In other words, $z_{2n+1}\neq \frac{1}{B}$ for any $n\geq 0$.
Notice that this is a Riccati equation after a change of variables. Since we already know the closed form solution for any Riccati equation, we may obtain a closed form solution for $z_{2n+1}$, and thus a closed form solution for $z_{n}$ in this case. We use the known results for Riccati equations restated in Section 3 to obtain the closed form solutions in the statement of the theorem.\par
Now, let us consider case (ii). In this case $z_{-1}=0$ and also since $(z_{0},z_{-1})\notin \mathfrak{F}$ there will never be division by zero in our solution.
It is immediately clear from these two facts and from a basic induction argument that $z_{2n+1}=0$ for all $n\geq 0$. This fact and Equation (5) immediately yields,
$$z_{2n}=\frac{z_{2n-2}}{1+Bz_{2n-1}-Bz_{2n-2}}=\frac{z_{2n-2}}{1-Bz_{2n-2}} , \quad n=1,2,\dots.$$
Thus, the even terms are defined recursively by the above equation. Notice that since we have only rewritten the recursive Equation (5), we cannot have division by zero in this equation with our choice of initial conditions. In other words, $z_{2n}\neq \frac{1}{B}$ for any $n\geq 0$.
Notice that this is a Riccati equation after a change of variables. Since we already know the closed form solution for any Riccati equation, we may obtain a closed form solution for $z_{2n}$, and thus a closed form solution for $z_{n}$ in this case. We use the known results for Riccati equations restated in Section 3 to obtain the closed form solutions in the statement of the theorem.\par
Now, let us consider case (iii). In this case, $z_{0}=\frac{-1}{B}$. Also, since $(z_{0},z_{-1})\notin \mathfrak{F}$, there will never be division by zero in our solution.
This allows us to prove by induction that $z_{n} = \frac{-1}{B}$ for all $n\geq 0$. The case $z_{0}= \frac{-1}{B}$ provides the base case. Assume that $z_{n}= \frac{-1}{B}$, since there is never division by zero from Equation (5) we get,
$$ z_{n+1}=\frac{z_{n-1}}{1+B z_{n}-B z_{n-1}}= \frac{z_{n-1}}{-Bz_{n-1}}= \frac{-1}{B}.$$
Thus we have shown that $z_{n} = \frac{-1}{B}$ for all $n\geq 0$.\par
Let us finally consider case (iv). In this case, $z_{0}\neq 0, \frac{-1}{B}$ and $z_{-1}\neq 0$. Also, since $(z_{0},z_{-1})\notin \mathfrak{F}$, there will never be division by zero in our solution.
This allows us to prove by induction that $z_{n}\neq 0$ for all $n\geq 0$. The induction argument for this piece is straightforward and is omitted. Since there is never division by zero in our solution and since $z_{n}\neq 0$ for all $n\geq 0$, the following algebraic computation is well defined.
$$\left(\frac{1}{z_{n+1}} + B \right)\left(\frac{1}{z_{n}} \right)= \left(\frac{1+ B z_{n} - B z_{n-1}}{z_{n-1}} + B \right) \left(\frac{1}{ z_{n}} \right)= \left(\frac{1+ B z_{n}}{z_{n-1}} \right) \left(\frac{1}{ z_{n}} \right)=$$
$$\left(\frac{1+ B z_{n}}{z_{n}} \right) \left(\frac{1}{ z_{n-1}} \right)=\left(\frac{1}{z_{n}} + B \right)\left(\frac{1}{z_{n-1}} \right).$$
Thus, we have the following algebraic invariant:
$$\left(\frac{1}{z_{n}} + B \right)\left(\frac{1}{z_{n-1}} \right)=constant.$$
For our fixed but arbitrary initial conditions with $z_{0}\neq 0, \frac{-1}{B}$ and $z_{-1}\neq 0$, let us denote $C=\left(\frac{1}{z_{0}} + B \right)\left(\frac{1}{ z_{-1}} \right)$. Since $z_{0}\neq 0$, $C$ is well defined and since $z_{0}\neq \frac{-1}{B}$, $C\neq 0$.
Since $C\neq 0$, this forces $z_{n}\neq \frac{-1}{B}$ for all $n\geq 0$. Now, we claim that since $(z_{0},z_{-1})\notin \mathfrak{F}$, $z_{n}\neq \frac{B}{C}$ for all $n\geq 0$. In the case where $C=-B^{2}$, it follows from the fact that $z_{n}\neq \frac{-1}{B}$ for all $n\geq 0$. In the remaining case, suppose there were such an $N$, then:
$$\left(\frac{C}{B} + B \right)\left(\frac{1}{z_{N-1}} \right)=C.$$
This implies that $z_{N}=\frac{B}{C}$ and $z_{N-1}= \frac{B^{2}+C}{CB}$, but then $1+Bz_{N}-Bz_{N-1}= 0$. However, this contradicts the fact that $(z_{0},z_{-1})\notin \mathfrak{F}$. Thus, we have that $z_{n}\neq \frac{B}{C}$ for all $n\geq 0$.
Algebraic manipulations of our invariant yield the following:
$$z_{n}=\frac{1}{Cz_{n-1}- B},\quad n\geq 1.$$
Since $z_{n}\neq \frac{B}{C}$ for all $n\geq 0$, this equation is well-defined for all $n\geq 1$. Thus, the dynamics of $\{z_{n}\}^{\infty}_{n=-1}$ are given by a Riccati equation in this case. Since we already know the closed form solution for any Riccati equation, we may obtain a closed form solution for $\{z_{n}\}^{\infty}_{n=-1}$ in this case. We use the known results for Riccati equations restated in Section 3 to obtain the closed form solutions in the statement of the theorem.
\end{proof}
\begin{thm}
Consider the rational difference equation,
\begin{equation}
z_{n+1}=\frac{z^{2}_{n}+Bz_{n} - Bz_{n-1}}{z_{n-1}},\quad n=0,1,\dots,
\end{equation}
with $B\in \mathbb{C}$ and with initial conditions $z_{0},z_{-1}\in \mathbb{C}$. Then the forbidden set, $\mathfrak{F}=S_{3}$, where $S_{3}$ is given in Figure 2.
Also, given $(z_{0},z_{-1})\notin \mathfrak{F}$, $z_{n+1}=Cz_{n}-B$ for all $n\geq 0$, where $C=\frac{z_{0} + B }{z_{-1}}$. This implies $$z_{n}=C^{n}z_{0}-\sum^{n}_{k=1}C^{k-1}B,\quad n\geq 1.$$
\end{thm}
\begin{proof}
Let us first consider the case where $(z_{0},z_{-1})\notin \mathfrak{F}$. In this case clearly $z_{n}\neq 0$ for $n\geq -1$ or else we would have division by zero.
Since $z_{n}\neq 0$ for all $n\geq -1$ the following algebraic computation is well defined:
$$\frac{z_{n+1}+B}{z_{n}}=\frac{\frac{z^{2}_{n}+Bz_{n} - Bz_{n-1}}{z_{n-1}}+B}{z_{n}}= \frac{z^{2}_{n}+Bz_{n}}{z_{n}z_{n-1}}= \frac{z_{n}+B}{z_{n-1}}.$$
Thus, we have the following algebraic invariant:
$$\frac{z_{n}+B}{z_{n-1}}=constant.$$
For our fixed but arbitrary initial conditions, let us denote $C=\frac{z_{0} + B }{z_{-1}}$. Since $z_{-1}\neq 0$, $C$ is well defined.
Algebraic manipulations of our invariant yield the following:
$$z_{n}=Cz_{n-1}-B,\quad n\geq 1.$$
Thus, the dynamics of $\{z_{n}\}^{\infty}_{n=-1}$ are given by a linear equation in this case. Since we already know the closed form solution for any linear equation, we may obtain a closed form solution for $\{z_{n}\}^{\infty}_{n=-1}$ in this case.
The resulting closed form solution is $$z_{n}=C^{n}z_{0}-\sum^{n}_{k=1}C^{k-1}B,\quad n\geq 1.$$
Now, we must find the forbidden set for our Equation (6). Let $\mathfrak{F}$ be the forbidden set.
Suppose $z_{0},z_{-1}\neq 0$, $C=\frac{z_{0} + B }{z_{-1}}\neq 1$, and $z_{0}\not\in \left\{\frac{B-BC^{n}}{C^{n}-C^{n+1}}|n\in\mathbb{N}\right\}$, and assume that $z_{n}$ is well defined for $n\leq N$.
Then $$z_{N-1}\neq 0,$$
since $z_{0}\not\in \{\frac{B-BC^{n}}{C^{n}-C^{n+1}}|n\in\mathbb{N}\}$. Thus, $z_{n}$ is well defined for $n\leq N+1$. By induction, $z_{n}$ is well defined for all $n\in\mathbb{N}$. Thus, $(z_{0},z_{-1})\not\in \mathfrak{F}$.\par
Now, suppose $z_{0},z_{-1}\neq 0$, $C=\frac{z_{0} + B }{z_{-1}}\neq 1$, and $z_{0}\in \left\{\frac{B-BC^{n}}{C^{n}-C^{n+1}}|n\in\mathbb{N}\right\}$. Further assume for the sake of contradiction that $(z_{0},z_{-1})\not\in \mathfrak{F}$. Then since $(z_{0},z_{-1})\not\in \mathfrak{F}$, $z_{n}$ is well defined for all $n\in\mathbb{N}$, but also
$z_{N}=0$ for some $N\in\mathbb{N}$, since $z_{0}\in \left\{\frac{B-BC^{n}}{C^{n}-C^{n+1}}|n\in\mathbb{N}\right\}$. This is a contradiction. Thus, $(z_{0},z_{-1})\in \mathfrak{F}$. \par
Suppose $z_{0},z_{-1}\neq 0$, $C=\frac{z_{0} + B }{z_{-1}}= 1$, and $z_{0}\not\in \left\{nB|n\in\mathbb{N}\right\}$, and assume that $z_{n}$ is well defined for $n\leq N$.
Then $$z_{N-1}\neq 0,$$
since $z_{0}\not\in \{nB|n\in\mathbb{N}\}$. Thus, $z_{n}$ is well defined for $n\leq N+1$. By induction, $z_{n}$ is well defined for all $n\in\mathbb{N}$. Thus, $(z_{0},z_{-1})\not\in \mathfrak{F}$.\par
Now, suppose $z_{0},z_{-1}\neq 0$, $C=\frac{z_{0} + B }{z_{-1}}=1$, and $z_{0}\in \left\{nB|n\in\mathbb{N}\right\}$. Further assume for the sake of contradiction that $(z_{0},z_{-1})\not\in \mathfrak{F}$. Then, since $(z_{0},z_{-1})\not\in \mathfrak{F}$, $z_{n}$ is well defined for all $n\in\mathbb{N}$, but also
$z_{N}=0$ for some $N\in\mathbb{N}$, since $z_{0}\in \left\{nB|n\in\mathbb{N}\right\}$. This is a contradiction. Thus, $(z_{0},z_{-1})\in \mathfrak{F}$. \par
Finally, suppose either $z_{0}=0$ or $z_{-1}=0$, then $(z_{0},z_{-1})\in \mathfrak{F}$. Thus, $\mathfrak{F}=S_{3}$, where $S_{3}$ is given in Figure 2.
\end{proof}
\begin{thm}
Consider the rational difference equation,
\begin{equation}
z_{n+1}=\frac{z^{2}_{n} + B z_{n}}{ z_{n-1} + B},\quad n=0,1,\dots,
\end{equation}
with $B\in \mathbb{C}\setminus \{0\}$ and with initial conditions $z_{0},z_{-1}\in \mathbb{C}$. Then the forbidden set, $\mathfrak{F}=S_{4}$, where $S_{4}$ is given in Figure 2.
Also, given $(z_{0},z_{-1})\notin \mathfrak{F}$ there are two possibilities:
\begin{enumerate}[i.]
\item $z_{0}=0$, in which case $z_{n}=0$ for all $n\geq 0$;
\item $z_{0}\neq 0$, in which case $z_{n+1}=\frac{z_{n}+B}{C}$ for all $n\geq 0$, where $C=\frac{z_{-1}+B}{z_{0}}$. This implies $$z_{n}=\frac{z_{0}}{C^{n}}+\sum^{n}_{k=1}\frac{B}{C^{k}},\quad n\geq 1.$$
\end{enumerate}
\end{thm}
\begin{proof}
Let us first consider the case where $(z_{0},z_{-1})\notin \mathfrak{F}$. Our analysis of this case will be broken into two subcases as shown in the statement of Theorem 5.
Let us first consider case (i). In this case, $z_{0}=0$ and also since $(z_{0},z_{-1})\notin \mathfrak{F}$, there will never be division by zero in our solution.
It is immediately clear from these two facts and from a basic induction argument that $z_{n}=0$ for all $n\geq 0$.
Now, let us consider case (ii). In this case $z_{0}\neq 0$. Also, since $(z_{0},z_{-1})\notin \mathfrak{F}$, there will never be division by zero in our solution.
Thus, $z_{n}\neq -B$ for all $n\geq -1$ or else we would have division by zero.
This allows us to prove by induction that $z_{n}\neq 0$ for all $n\geq 0$. The induction argument for this piece is straightforward and is omitted. Since there is never division by zero in our solution, and since $z_{n}\neq 0$ for all $n\geq 0$, the following algebraic computation is well defined.
$$\frac{z_{n}+B}{z_{n+1}}= \frac{(z_{n}+B)(z_{n-1}+B)}{z^{2}_{n}+Bz_{n}}= \frac{z_{n-1}+B}{z_{n}}.$$
Thus, we have the following algebraic invariant:
$$\frac{z_{n-1}+B}{z_{n}}=constant.$$
For our fixed but arbitrary initial conditions with $z_{0}\neq 0$, let us denote $C=\frac{z_{-1}+B}{z_{0}}$. Since $z_{0}\neq 0$, $C$ is well defined and since $z_{-1}\neq -B$, $C\neq 0$.
Algebraic manipulations of our invariant yield the following:
$$z_{n}=\frac{z_{n-1}+B}{C} ,\quad n\geq 1.$$
Thus, the dynamics of $\{z_{n}\}^{\infty}_{n=-1}$ are given by a linear equation in this case. Since we already know the closed form solution for any linear equation, we may obtain a closed form solution for $\{z_{n}\}^{\infty}_{n=-1}$ in this case.
The resulting closed form solution is $$z_{n}=\frac{z_{0}}{C^{n}}+\sum^{n}_{k=1}\frac{B}{C^{k}},\quad n\geq 1.$$
Now, we must find the forbidden set for our Equation (7). Let $\mathfrak{F}$ be the forbidden set.
Suppose $z_{0},z_{-1}\neq -B$, $C=\frac{z_{-1} + B }{z_{0}}\neq 1$, and $z_{0}\not\in \left\{\frac{B-BC^{n+1}}{C-1}|n\in\mathbb{N}\right\}$, and assume that $z_{n}$ is well defined for $n\leq N$.
Then $$z_{N-1}\neq -B,$$
since $z_{0}\not\in \left\{\frac{B-BC^{n+1}}{C-1}|n\in\mathbb{N}\right\}$. Thus, $z_{n}$ is well defined for $n\leq N+1$. By induction, $z_{n}$ is well defined for all $n\in\mathbb{N}$. Thus, $(z_{0},z_{-1})\not\in \mathfrak{F}$.\par
Now, suppose $z_{0},z_{-1}\neq -B$, $C=\frac{z_{-1} + B }{z_{0}}\neq 1$, and $z_{0}\in \left\{\frac{B-BC^{n+1}}{C-1}|n\in\mathbb{N}\right\}$. Further assume for the sake of contradiction that $(z_{0},z_{-1})\not\in \mathfrak{F}$. Then, since $(z_{0},z_{-1})\not\in \mathfrak{F}$, $z_{n}$ is well defined for all $n\in\mathbb{N}$, but also
$z_{N}=-B$ for some $N\in\mathbb{N}$, since $z_{0}\in \left\{\frac{B-BC^{n+1}}{C-1}|n\in\mathbb{N}\right\}$. This is a contradiction. Thus, $(z_{0},z_{-1})\in \mathfrak{F}$.\par
Suppose $z_{0},z_{-1}\neq -B$, $C=\frac{z_{-1} + B }{z_{0}}= 1$, and $z_{0}\not\in \left\{-nB-B|n\in\mathbb{N}\right\}$, and assume that $z_{n}$ is well defined for $n\leq N$.
Then $$z_{N-1}\neq -B,$$
since $z_{0}\not\in \{-nB-B|n\in\mathbb{N}\}$. Thus, $z_{n}$ is well defined for $n\leq N+1$. By induction, $z_{n}$ is well defined for all $n\in\mathbb{N}$. Thus, $(z_{0},z_{-1})\not\in \mathfrak{F}$.\par
Now, suppose $z_{0},z_{-1}\neq -B$, $C=\frac{z_{-1} + B }{z_{0}}=1$, and $z_{0}\in \left\{-nB-B|n\in\mathbb{N}\right\}$. Further assume for the sake of contradiction that $(z_{0},z_{-1})\not\in \mathfrak{F}$. Then, since $(z_{0},z_{-1})\not\in \mathfrak{F}$, $z_{n}$ is well defined for all $n\in\mathbb{N}$, but also
$z_{N}=-B$ for some $N\in\mathbb{N}$, since $z_{0}\in \left\{-nB-B|n\in\mathbb{N}\right\}$. This is a contradiction. Thus, $(z_{0},z_{-1})\in \mathfrak{F}$.\par
Finally, suppose either $z_{0}=-B$ or $z_{-1}=-B$, then $(z_{0},z_{-1})\in \mathfrak{F}$. Thus,
$\mathfrak{F}=S_{4}$, where $S_{4}$ is given in Figure 2.
\end{proof}
\begin{thm}
Consider the rational difference equation,
\begin{equation}
z_{n+1}=\frac{z_{n}z_{n-1}+Bz_{n}}{B + z_{n}},\quad n=0,1,\dots,
\end{equation}
with $B\in \mathbb{C}\setminus \{0\}$ and with initial conditions $z_{0},z_{-1}\in \mathbb{C}$. Then the forbidden set, $\mathfrak{F}=S_{5}$, where $S_{5}$ is given in Figure 2. Also, given $(z_{0},z_{-1})\notin \mathfrak{F}$, $z_{n+1}=\frac{C}{z_{n}+B}$ for all $n\geq 0$, where $C=\left(z_{-1} + B \right)\left(z_{0}\right)$. This implies the following:
\begin{enumerate}[a.]
\item If $C=0$, then $z_{n}=0$ for all $n\geq 0$.
\item If $\frac{-C}{B^{2}}\in \mathbb{C}\setminus \left[\frac{1}{4},\infty\right)$, then $$z_{n}=B\left(\frac{(B\lambda_{2}-z_{0}-B)\lambda^{n+1}_{1}+(z_{0}+B-B\lambda_{1})\lambda^{n+1}_{2}}{(B\lambda_{2}-z_{0}-B)\lambda^{n}_{1}+(z_{0}+B-B\lambda_{1})\lambda^{n}_{2}}\right)-B,\quad n\geq 0.$$
Where $$\lambda_{1}=\frac{1-\sqrt{1+\frac{4C}{B^{2}}}}{2},\quad and \quad \lambda_{2}=\frac{1+\sqrt{1+\frac{4C}{B^{2}}}}{2}.$$
\item If $\frac{-C}{B^{2}}=\frac{1}{4}$, then
$$z_{n}=B\left(\frac{B+(n+1)\left(2z_{0}+B \right)}{2B+4nz_{0}+2nB}\right)-B,\quad n\geq 0.$$
\item If $\frac{-C}{B^{2}}\in \left(\frac{1}{4},\infty\right)$, then call $B\sqrt{\frac{-4C}{B^{2}}-1}=D$ and call $\arccos\left(\sqrt{\frac{B^{2}}{-4C}}\right)=\rho$, and for $n\geq 0$, we have
$$z_{n}=B\sqrt{\frac{-C}{B^{2}}}\left(\frac{D\cos\left((n+1)\rho\right)+(B+2z_{0})\sin\left((n+1)\rho\right)}{D\cos\left(n\rho\right)+(B+2z_{0})\sin\left(n\rho\right)} \right) -B.$$
\end{enumerate}\end{thm}
\begin{proof}
Let us first consider the case where $(z_{0},z_{-1})\notin \mathfrak{F}$. In this case clearly $z_{n}\neq -B$ for $n\geq 0$, or else we would have division by zero.
Since $z_{n}\neq -B$ for all $n\geq 0$, the following algebraic computation is well defined:
$$z_{n+1}(z_{n}+B)=\left(\frac{z_{n}z_{n-1}+Bz_{n}}{B + z_{n}}\right)(z_{n}+B)= z_{n}(z_{n-1}+B).$$
Thus we have the following algebraic invariant:
$$z_{n}(z_{n-1}+B)=constant.$$
For our fixed but arbitrary initial conditions, let us denote $C=z_{0}(z_{-1}+B)$.
Algebraic manipulations of our invariant yield the following:
$$z_{n}=\frac{C}{z_{n-1}+ B},\quad n\geq 1.$$
Since $z_{n}\neq -B$ for all $n\geq 0$, this equation is well-defined for all $n\geq 1$. Thus the dynamics of $\{z_{n}\}^{\infty}_{n=-1}$ are given by a Riccati equation in this case. Since we already know the closed form solution for any Riccati equation, we may obtain a closed form solution for $\{z_{n}\}^{\infty}_{n=-1}$ in this case. We use the known results for Riccati equations restated in Section 3 to obtain the closed form solutions in the statement of the theorem.\par
Now, we must find the forbidden set for our Equation (8).
Let $\mathfrak{F}$ be the forbidden set. Assume $z_{n}$ is well defined for $n\leq N$, then $z_{n}\neq -B$ for $1\leq n\leq N-1$. Using this we get that for $n<N$,
$$z_{n+1}(z_{n}+B)=\left(\frac{z_{n}z_{n-1}+Bz_{n}}{B + z_{n}}\right)(z_{n}+B)= z_{n}(z_{n-1}+B).$$
So,
$$z_{n}(z_{n-1}+B)=constant.$$
for $0\leq n\leq N$. Thus, assuming $z_{N+1}$ is well defined,
$$z_{n+1}= \frac{C}{z_{n}+B},$$
for $0\leq n\leq N$. Where $C=z_{0}(z_{-1}+B)$.
Call the forbidden set of the following first order difference equation $\mathfrak{F}_{C}$,
$$x_{n+1}= \frac{C}{x_{n}+B}, \quad n=0,1,2,\dots.$$
Note that the set $\mathfrak{F}_{C}$ changes depending on the value of $C$.
Now, suppose $C=z_{0}(z_{-1}+B)$ and $z_{0}\not\in \mathfrak{F}_{C}$, and assume that $z_{n}$ is well defined for $n\leq N$. Recall that we have shown that this implies $z_{n}\neq -B$ for $n < N$. Then
$$B+z_{n}\neq 0,$$
since $z_{0}\not\in \mathfrak{F}_{C}$. Thus, $z_{n}$ is well defined for $n\leq N+1$. By induction, $z_{n}$ is well defined for all $n\in\mathbb{N}$. Thus, $(z_{0},z_{-1})\not\in \mathfrak{F}$.\par
Now, suppose $C=z_{0}(z_{-1}+B)$ and $z_{0}\in \mathfrak{F}_{C}$. Further assume for the sake of contradiction that $(z_{0},z_{-1})\not\in \mathfrak{F}$. Then, since $(z_{0},z_{-1})\not\in \mathfrak{F}$, $z_{n}$ is well defined for all $n\in\mathbb{N}$, but also
$B+z_{N}=0$ for some $N\in\mathbb{N}$, since $z_{0}\in \mathfrak{F}_{C}$. This is a contradiction. Thus $(z_{0},z_{-1})\in \mathfrak{F}$. So,
$$\mathfrak{F}=\bigcup_{b\in\mathbb{C}} \bigcup_{C\in\mathbb{C}} \left(\left(\mathfrak{F}_{C}\times \{b\}\right)\cap \left\{(a,b)|C=a(b+B)\right\}\right).$$
Notice that $\left(\{0\}\times \mathbb{C}\right)\cap \mathfrak{F}=\emptyset$ since if $z_{0}=0$ then a simple induction argument tells us that $z_{n}=0$ for all $n\in\mathbb{N}$, so there will never be division by zero in such a case.
So we may reduce the above expression as follows,
$$\mathfrak{F}= \bigcup_{C\in\mathbb{C}} \left\{ \left(a,\frac{C-Ba}{a}\right)| a\in\mathfrak{F}_{C}\setminus\left\{0\right\}\right\}.$$
From the above reduction, and from the facts about the forbidden sets of the Riccati difference equation in Section 3, we get $\mathfrak{F}=S_{5}$, where $S_{5}$ is given in Figure 2.
\end{proof}
\begin{thm}
Consider the rational difference equation,
\begin{equation}
z_{n+1}=\frac{z_{n}z_{n-1}+Bz_{n-1} - Bz_{n}}{z_{n}},\quad n=0,1,\dots,
\end{equation}
with $B\in \mathbb{C}\setminus\{0\}$ and with initial conditions $z_{0},z_{-1}\in \mathbb{C}$. Then the forbidden set, $\mathfrak{F}=S_{6}$, where $S_{6}$ is given in Figure 2. Also, given $(z_{0},z_{-1})\notin \mathfrak{F}$, $z_{n+1}=\frac{C - B z_{n}}{z_{n}}$ for all $n\geq 0$,
where $C=z_{-1}(z_{0}+B)$. This implies the following:
\begin{enumerate}[a.]
\item If $C=0$, then $z_{n}=-B$ for all $n\geq 0$.
\item If $\frac{-C}{B^{2}}\in \mathbb{C}\setminus \left[\frac{1}{4},\infty\right)$, then $$z_{n}=-B\left(\frac{(B\lambda_{2}+z_{0})\lambda^{n+1}_{1}-(z_{0}+B\lambda_{1})\lambda^{n+1}_{2}}{(B\lambda_{2}+z_{0})\lambda^{n}_{1}-(z_{0}+B\lambda_{1})\lambda^{n}_{2}}\right),\quad n\geq 0.$$
Where $$\lambda_{1}=\frac{1-\sqrt{1+\frac{4C}{B^{2}}}}{2},\quad and \quad \lambda_{2}=\frac{1+\sqrt{1+\frac{4C}{B^{2}}}}{2}.$$
\item If $\frac{-C}{B^{2}}=\frac{1}{4}$, then
$$z_{n}=-B\left(\frac{-B+(n+1)\left(2z_{0}+B \right)}{-2B+4nz_{0}+2nB}\right),\quad n\geq 0.$$
\item If $\frac{-C}{B^{2}}\in \left(\frac{1}{4},\infty\right)$, then call $B\sqrt{\frac{-4C}{B^{2}}-1}=D$ and $\arccos\left(\sqrt{\frac{B^{2}}{-4C}}\right)=\rho$, and for $n\geq 0$, we get
$$z_{n}=-B\sqrt{\frac{-C}{B^{2}}}\left(\frac{D\cos\left((n+1)\rho\right)+(-2z_{0}-B)\sin\left((n+1)\rho\right)}{D\cos\left(n\rho\right)+(-2z_{0}-B)\sin\left(n\rho\right)}\right).$$
\end{enumerate}
\end{thm}
\begin{proof}
Let us first consider the case where $(z_{0},z_{-1})\notin \mathfrak{F}$. In this case clearly $z_{n}\neq 0$ for $n\geq 0$, or else we would have division by zero.
Since $z_{n}\neq 0$ for all $n\geq 0$, the following algebraic computation is well defined:
$$z_{n}(z_{n+1}+B)=\left(\frac{z_{n}z_{n-1}+Bz_{n-1} - Bz_{n}}{z_{n}}+B\right)(z_{n})= \left(\frac{z_{n}z_{n-1}+Bz_{n-1}}{z_{n}}\right)(z_{n})= z_{n-1}(z_{n}+B).$$
Thus, we have the following algebraic invariant:
$$z_{n-1}(z_{n}+B)=constant.$$
For our fixed but arbitrary initial conditions, let us denote $C=z_{-1}(z_{0}+B)$.
Algebraic manipulations of our invariant yield the following:
$$z_{n}=\frac{C - B z_{n-1}}{z_{n-1}},\quad n\geq 1.$$
Since $z_{n}\neq 0$ for all $n\geq 0$, this equation is well-defined for all $n\geq 1$. Thus, the dynamics of $\{z_{n}\}^{\infty}_{n=-1}$ are given by a Riccati equation in this case. Since we already know the closed form solution for any Riccati equation, we may obtain a closed form solution for $\{z_{n}\}^{\infty}_{n=-1}$ in this case. We use the known results for Riccati equations restated in Section 3 to obtain the closed form solutions in the statement of the theorem.
Now, we must find the forbidden set for our Equation (9).
Let $\mathfrak{F}$ be the forbidden set. Assume $z_{n}$ is well defined for $n\leq N$, then $z_{n}\neq 0$ for $1\leq n\leq N-1$. Using this we get that for $n<N$
$$z_{n}(z_{n+1}+B)=\left(\frac{z_{n}z_{n-1}+Bz_{n-1} - Bz_{n}}{z_{n}}+B\right)(z_{n})=$$$$\left(\frac{z_{n}z_{n-1}+Bz_{n-1}}{z_{n}}\right)(z_{n})= z_{n-1}(z_{n}+B).$$
So,
$$z_{n}(z_{n-1}+B)=constant.$$
for $0\leq n\leq N$. Thus, assuming $z_{N+1}$ is well defined,
$$z_{n+1}= \frac{C-Bz_{n}}{z_{n}},$$
for $0\leq n\leq N$, where $C=z_{-1}(z_{0}+B)$.
Call the forbidden set of the following first order difference equation $\mathfrak{F}_{C}$,
$$x_{n+1}= \frac{C-Bx_{n}}{x_{n}}, \quad n=0,1,2,\dots.$$
Note that the set $\mathfrak{F}_{C}$ changes depending on the value of $C$.
Now, suppose $C=z_{-1}(z_{0}+B)$ and $z_{0}\not\in \mathfrak{F}_{C}$, and assume that $z_{n}$ is well defined for $n\leq N$. Recall that we have shown that this implies $z_{n}\neq 0$ for $n < N$. Then
$$z_{N}\neq 0,$$
since $z_{0}\not\in \mathfrak{F}_{C}$. Thus $z_{n}$ is well defined for $n\leq N+1$. By induction, $z_{n}$ is well defined for all $n\in\mathbb{N}$. Thus, $(z_{0},z_{-1})\not\in \mathfrak{F}$.\newline
Now, suppose $C=z_{-1}(z_{0}+B)$ and $z_{0}\in \mathfrak{F}_{C}$. Further assume for the sake of contradiction that $(z_{0},z_{-1})\not\in \mathfrak{F}$. Then, since $(z_{0},z_{-1})\not\in \mathfrak{F}$, $z_{n}$ is well defined for all $n\in\mathbb{N}$, but also
$z_{N}=0$ for some $N\in\mathbb{N}$, since $z_{0}\in \mathfrak{F}_{C}$. This is a contradiction. Thus, $(z_{0},z_{-1})\in \mathfrak{F}$. So
$$\mathfrak{F}=\bigcup_{z_{-1}\in\mathbb{C}} \bigcup_{C\in\mathbb{C}} \left(\left(\mathfrak{F}_{C}\times \{z_{-1}\}\right)\cap \left\{(z_{0},z_{-1})|C=z_{-1}(z_{0}+B)\right\}\right).$$
Notice that $\left(\{-B\}\times \mathbb{C}\right)\cap \mathfrak{F}=\emptyset$, since if $z_{0}=-B$, then a simple induction argument tells us that $z_{n}=-B$ for all $n\in\mathbb{N}$, so there will never be division by zero in such a case.
So, we may reduce the above expression as follows,
$$\mathfrak{F}= \bigcup_{C\in\mathbb{C}} \left\{ \left(a,\frac{C}{a+B}\right)| a\in\mathfrak{F}_{C}\setminus\left\{-B\right\}\right\}.$$
From the above reduction, and from the facts about the forbidden sets of the Riccati difference equation in Section 3, we get $\mathfrak{F}=S_{6}$, where $S_{6}$ is given in Figure 2.
\end{proof}
\section{Conclusion}
We have introduced some new invariants for rational difference equations, including some invariants for certain cases of the second order rational difference equation of the form,
$$x_{n+1}=\frac{\alpha + \beta x_{n} + \gamma x_{n-1}}{A + B x_{n} + C x_{n-1}},\quad n=0,1,2,\dots.$$
This second order linear fractional rational difference equation has been studied extensively in the case of nonnegative parameters and nonnegative initial conditions.
See \cite{kulenovicladas} for more information about this case. However, the invariants we have found only apply in a region of the parameters where at least one of the parameters cannot be a nonnegative real number.
For this reason, these particular examples were overlooked in \cite{kulenovicladas} as well as in the subsequent literature.\par
What is particularly interesting about the presented cases is that we are able to use invariants we find to obtain both the forbidden set, and a closed form solution for our rational difference equations through reduction of order.
\vfill
\begin{center}
\slide{
$$S_{1}=\left\{\left(0,\frac{-1}{B}\right)\right\}\bigcup \left\{\left(\frac{-1}{B}\left(\frac{n+1}{n}\right),\frac{-1}{B}\right)|n\in\mathbb{N}\right\}\bigcup \left\{\left(\frac{-1}{B},\frac{-1}{B}\left(\frac{n+1}{n}\right)\right)|n\in\mathbb{N}\right\}\bigcup $$
$$\bigcup_{D\in\mathbb{C}\setminus [0,4]}\left\{\left(a,\frac{DBa-Ba-1}{B^{2}a+B}\right)\left|a\in\left\{\frac{-2}{B}\left(\frac{\left(1+\sqrt{1-\frac{4}{D}}\right)^{n-1}-\left(1-\sqrt{1-\frac{4}{D}}\right)^{n-1}}{\left(1+\sqrt{1-\frac{4}{D}}\right)^{n}-\left(1-\sqrt{1-\frac{4}{D}}\right)^{n}}\right)+\frac{D-1}{B}\right|n\in\mathbb{N}\right\}\setminus\left\{0,\frac{-1}{B}\right\}\right\}$$
$$\bigcup_{D\in (0,4)} \left\{\left(a,\frac{DBa-Ba-1}{B^{2}a+B}\right)\left|a\in\left\{\frac{-D}{B}\left(1-\sqrt{\frac{4}{D}-1}\cot\left(n\cdot \arccos\left(\frac{\sqrt{D}}{2}\right)\right)\right)+\frac{D-1}{B}\right|n\in\mathbb{N}\right\}\setminus\left\{0,\frac{-1}{B}\right\}\right\}$$
$$\bigcup \left\{\left(a,\frac{3Ba-1}{B^{2}a+B}\right)\left|a\in \left\{\frac{-2}{B}\left(\frac{n-1}{n}\right)+\frac{3}{B}\right|n\in\mathbb{N}\right\}\setminus\left\{0,\frac{-1}{B}\right\}\right\}.$$
$$S_{2}=\left\{\left(\frac{-1}{B},0\right)\right\}\bigcup \left\{\left(0,\frac{1}{Bn}\right)|n\in\mathbb{N}\right\}\bigcup \left\{\left(\frac{1}{Bn},0\right)|n\in\mathbb{N}\right\}\bigcup $$
$$\bigcup_{D\in\mathbb{C}\setminus \left((-\infty,\frac{-1}{4}]\cup\{0\}\right)}\left\{\left(a,\frac{Ba+1}{DB^{2}a}\right)\left|a\in\left\{\frac{2}{B}\left(\frac{\left(1+\sqrt{1+4D}\right)^{n-1}-\left(1-\sqrt{1+4D}\right)^{n-1}}{\left(1+\sqrt{1+4D}\right)^{n}-\left(1-\sqrt{1+4D}\right)^{n}}\right)+\frac{1}{DB}\right|n\in\mathbb{N}\right\}\setminus\left\{0,\frac{-1}{B}\right\}\right\}$$
$$\bigcup_{D\in (-\infty,\frac{-1}{4})} \left\{\left(a,\frac{Ba+1}{DB^{2}a}\right)\left|a\in\left\{\frac{-1}{2DB}\left(-1-\sqrt{-4D-1}\cot\left(n\cdot \arccos\left(\frac{1}{2\sqrt{-D}}\right)\right)\right)\right|n\in\mathbb{N}\right\}\setminus\left\{0,\frac{-1}{B}\right\}\right\}$$
$$\bigcup \left\{ \left(a,\frac{-4Ba-4}{B^{2}a}\right)\left| a\in\left\{\frac{-2n-2}{Bn}\right|n\in\mathbb{N}\right\}\setminus\left\{0,\frac{-1}{B}\right\}\right\}.$$
\hspace{9cm} Figure 1.
}
\slide{
$$S_{3}=\bigcup_{C\neq 1}\left\{\left(\frac{B-BC^{n}}{C^{n}-C^{n+1}},\frac{B-BC^{n+1}}{C^{n+1}-C^{n+2}}\right)|n\in\mathbb{N}\right\}\bigcup\left(\{0\}\times \mathbb{C}\right)\bigcup\left(\mathbb{C}\times \{0\}\right)\bigcup\left\{\left(nB,nB+B\right)|n\in\mathbb{N}\right\}.$$
$$S_{4}=\bigcup_{C\neq 1}\left\{\left(\frac{B-BC^{n+1}}{C-1},\frac{B-BC^{n+2}}{C-1}\right)|n\in\mathbb{N}\right\}\bigcup\left(\{-B\}\times \mathbb{C}\right)\bigcup\left(\mathbb{C}\times \{-B\}\right)\bigcup\left\{\left(-nB-B,-nB-2B\right)|n\in\mathbb{N}\right\}.$$
$$S_{5}= \{-B,-B\}\bigcup \left\{ \left(a,\frac{B^{2}}{-4a}-B\right)\left| a\in\left\{\frac{-Bn-B}{2n}\right|n\in\mathbb{N}\right\}\right\}\bigcup$$
$$\bigcup_{D\in\mathbb{C}\setminus \left((-\infty,\frac{-1}{4}]\cup\{0\}\right)}\left\{\left(a,\frac{DB^{2}}{a}-B\right)\left|a\in\left\{-2DB\left(\frac{\left(1+\sqrt{1+4D}\right)^{n-1}-\left(1-\sqrt{1+4D}\right)^{n-1}}{\left(1+\sqrt{1+4D}\right)^{n}-\left(1-\sqrt{1+4D}\right)^{n}}\right)-B\right|n\in\mathbb{N}\right\}\setminus\left\{0\right\}\right\}$$
$$\bigcup_{D\in (-\infty,\frac{-1}{4})} \left\{\left(a,\frac{DB^{2}}{a}-B\right)\left|a\in\left\{\frac{B}{2}\left(-1-\sqrt{-4D-1}\cot\left(n\cdot \arccos\left(\frac{1}{2\sqrt{-D}}\right)\right)\right)\right|n\in\mathbb{N}\right\}\setminus\left\{0\right\}\right\}.$$
$$S_{6}= \{0,0\}\bigcup \left\{ \left(a,\frac{-B^{2}}{4a+4B}\right)\left| a\in\left\{\frac{-Bn+B}{2n}\right|n\in\mathbb{N}\right\}\right\}\bigcup$$
$$\bigcup_{D\in\mathbb{C}\setminus \left((-\infty,\frac{-1}{4}]\cup\{0\}\right)}\left\{\left(a,\frac{DB^{2}}{a+B}\right)\left|a\in\left\{2DB\left(\frac{\left(1+\sqrt{1+4D}\right)^{n-1}-\left(1-\sqrt{1+4D}\right)^{n-1}}{\left(1+\sqrt{1+4D}\right)^{n}-\left(1-\sqrt{1+4D}\right)^{n}}\right)\right|n\in\mathbb{N}\right\}\setminus\left\{-B\right\}\right\}$$
$$\bigcup_{D\in (-\infty,\frac{-1}{4})} \left\{\left(a,\frac{DB^{2}}{a+B}\right)\left|a\in\left\{\frac{-B}{2}\left(1-\sqrt{-4D-1}\cot\left(n\cdot \arccos\left(\frac{1}{2\sqrt{-D}}\right)\right)\right)\right|n\in\mathbb{N}\right\}\setminus\left\{-B\right\}\right\}.$$
\hspace{9cm} Figure 2.
}
\end{center}
\par\vspace{0.2 cm}
|
{
"timestamp": "2012-05-29T02:01:10",
"yymm": "1203",
"arxiv_id": "1203.2170",
"language": "en",
"url": "https://arxiv.org/abs/1203.2170"
}
|
\section{Introduction} \seclab{intro}
A \emph{reflection framework} is a planar structure made of \emph{fixed-length bars} connected by
\emph{universal joints} with full rotational freedom. Additionally, the bars and joints are symmetric with
respect to a reflection through a fixed axis. The allowed motions preserve the \emph{length} and
\emph{connectivity} of the bars and \emph{symmetry} with respect to some reflection.
This model is very similar to that of \emph{cone frameworks} that we introduced in \cite{MT11}; the
difference is that the symmetry group $\mathbb{Z}/2\mathbb{Z}$ acts on the plane by reflection instead of rotation
through angle $\pi$.
When all the allowed motions are Euclidean isometries, a reflection framework is \emph{rigid} and otherwise it is
\emph{flexible}. In this paper, we give a \emph{combinatorial} characterization of minimally rigid, generic
reflection frameworks.
\subsection{The algebraic setup and combinatorial model}
Formally a reflection framework is given by a triple $(\tilde{G},\varphi,\tilde{\bm{\ell}})$,
where $\tilde{G}$ is a finite graph, $\varphi$ is a $\mathbb{Z}/2\mathbb{Z}$-action on $\tilde{G}$ that is
free on the vertices and edges, and $\tilde{\bm{\ell}} = (\ell_{ij})_{ij\in E(\tilde{G})}$
is a vector of non-negative \emph{edge lengths} assigned to the edges of $\tilde{G}$. A
\emph{realization} $\tilde{G}(\vec p,\Phi)$ is an assignment of points $\vec p = (\vec p_i)_{i\in V(\tilde{G})}$
and a representation of $\mathbb{Z}/2\mathbb{Z}$ by a reflection $\Phi\in \operatorname{Euc}(2)$ such that:
\begin{eqnarray}\eqlab{lengths-1}
||\vec p_j - \vec p_i||^2 = \ell_{ij}^2 & \qquad \text{for all edges $ij\in E(\tilde{G})$} \\
\eqlab{lengths-2}
\vec p_{\varphi(\gamma)\cdot i} = \Phi(\gamma)\cdot\vec p_i & \qquad
\text{for all $\gamma\in \mathbb{Z}/2\mathbb{Z}$ and $i\in V(\tilde{G})$}
\end{eqnarray}
The set of all realizations is defined to be the \emph{realization space}
$\mathcal{R}(\tilde{G},\varphi,\bm{\ell})$ and its quotient by the Euclidean isometries
$\mathcal{C}(\tilde{G},\varphi,\bm{\ell}) = \mathcal{R}(\tilde{G},\varphi,\bm{\ell})/\operatorname{Euc}(2)$
to be the configuration space. A realization is \emph{rigid} if it is isolated in the
configuration space and otherwise \emph{flexible}.
As the combinatorial model for reflection frameworks it will be more convenient
to use colored graphs. A \emph{colored graph} $(G,\bm{\gamma})$ is a finite,
directed%
\footnote{For the group $\mathbb{Z}/2\mathbb{Z}$, the orientation of the edges do not play a role, but we give the standard
definition for consistency.} %
graph $G$, with an assignment $\bm{\gamma} = (\gamma_{ij})_{ij\in E(G)}$
of an element of a group $\Gamma$ to each edge. In this paper $\Gamma$ is always
$\mathbb{Z}/2\mathbb{Z}$. There is a standard dictionary
\cite[Section 9]{MT11} associating $(\tilde{G},\varphi)$ with a colored
graph $(G,\bm{\gamma})$: $G$ is the quotient of $\tilde{G}$ by $\Gamma$, and the
colors encode the covering map via a natural map $\rho : \pi_1(G,b) \to \Gamma$.
In this setting, the choice of base vertex does not matter, and indeed, we may
define $\rho : \HH_1(G, \mathbb{Z})\to \mathbb{Z}/2\mathbb{Z}$ and obtain the same theory.
\subsection{Main Theorem}
We can now state the main result of this paper.
\begin{theorem}[\reflectionlaman]\theolab{reflection-laman}
A generic reflection framework is minimally rigid if and only if its associated colored
graph is reflection-Laman.
\end{theorem}
The \emph{reflection-Laman graphs}
appearing in the statement are defined in \secref{matroid}. Genericity has
its standard meaning from algebraic geometry: the set of non-generic reflection
frameworks is a measure-zero algebraic set, and a small \emph{geometric} perturbation
of a non-generic reflection framework yields a generic one.
\subsection{Infinitesimal rigidity and direction networks}
As in all known proofs of ``Maxwell-Laman-type'' theorems such as \theoref{reflection-laman},
we give a combinatorial characterization of a linearization of the problem known as
\emph{infinitesimal rigidity}. To do this, we use a \emph{direction network} method
(cf. \cite{W88,ST10,MT10,MT11}). A \emph{reflection direction network} $(\tilde{G},\varphi,\vec d)$ is a
symmetric graph, along with an assignment of a \emph{direction} $\vec d_{ij}$ to each edge. The
\emph{realization space} of a direction network is the set of solutions $\tilde{G}(\vec p)$ to the system of
equations:
\begin{eqnarray}
\eqlab{dn-realization1}
\iprod{\vec p_j - \vec p_i}{\vec d^{\perp}_{ij}} = 0 & \qquad \text{for all edges $ij\in E(\tilde{G})$} \\
\eqlab{dn-realization2}
\vec p_{\varphi(\gamma)\cdot i} = \Phi(\gamma)\cdot\vec p_i & \qquad
\text{for all $\gamma\in \mathbb{Z}/2\mathbb{Z}$ and $i\in V(\tilde{G})$}
\end{eqnarray}
where the $\mathbb{Z}/2\mathbb{Z}$-action $\Phi$ on the plane is by reflection through the $y$-axis.
A reflection direction network is
determined by assigning a direction to each edge of the colored quotient graph $(G,\bm{\gamma})$
of $(\tilde{G},\varphi)$ (cf. \cite[Lemma 17.2]{MT11}).
Since all the direction networks in this paper are
reflection direction networks, we will refer to them simply as ``direction networks''
to keep the terminology manageable. A realization of a direction network is
\emph{faithful} if none of the edges of its graph have coincident endpoints and
\emph{collapsed} if all the endpoints are coincident.
A basic fact in the theory of finite planar frameworks \cite{W88,ST10,DMR07}
is that, if a direction network has faithful realizations, the dimension of the realization
space is equal to that of the space of infinitesimal motions of a generic framework with the same
underlying graph. In \cite{MT10,MT11}, we adapted this idea to the symmetric case when all the
symmetries act by rotations and translations.
As discussed in \cite[Section 1.8]{MT11}, this so-called ``parallel redrawing trick''%
\footnote{This terminology comes from the engineering community, in which the basic idea has been folklore for
quite some time.} %
described above does \emph{not} apply verbatim to reflection frameworks.
Thus, we rely on the somewhat technical (cf. \cite[Theorem B]{MT10},
\cite[Theorem 2]{MT11}) \theoref{direction-network}, which we state after giving an important
definition.
Let $(\tilde{G},\varphi,\vec d)$ be a direction network and define $(\tilde{G},\varphi,\vec d^{\perp})$
to be the direction network with $(\vec d^\perp)_{ij} = (\vec d_{ij})^\perp$. These two direction
networks form a \emph{special pair} if:
\begin{itemize}
\item $(\tilde{G},\varphi,\vec d)$ has a faithful realization.
\item $(\tilde{G},\varphi,\vec d^\perp)$ has only collapsed realizations.
\end{itemize}
\begin{theorem}[\linkeddirectionnetworks]\theolab{direction-network}
Let $(G,\bm{\gamma})$ be a colored graph with $n$ vertices, $2n-1$ edges, and lift $(\tilde{G},\varphi)$.
Then there are directions $\vec d$ such that the direction networks $(\tilde{G},\varphi,\vec d)$
and $(\tilde{G},\varphi,\vec d^\perp)$ are a special pair if and only if $(G,\bm{\gamma})$ is reflection-Laman.
\end{theorem}
Briefly, we will use \theoref{direction-network} as follows: the faithful realization of
$(\tilde{G},\varphi,\vec d)$ gives a symmetric immersion of the graph $\tilde{G}$ that can be interpreted
as a framework, and the fact that $(\tilde{G},\varphi,\vec d^\perp)$ has only collapsed realizations
will imply that the only symmetric infinitesimal motions of this framework correspond to translation
parallel to the reflection axis.
\subsection{Notations and terminology} In this paper, all graphs $G=(V,E)$ may be multi-graphs. Typically,
the number of vertices, edges, and connected components are denoted by $n$, $m$, and $c$, respectively.
The notation for a colored graph is $(G,\bm{\gamma})$, and a symmetric graph with a free $\mathbb{Z}/2\mathbb{Z}$-action
is denoted by $(\tilde{G},\varphi)$. If $(\tilde{G},\varphi)$ is the lift of $(G,\bm{\gamma})$, we
denote the fiber over a vertex $i\in V(G)$ by $\tilde{i}_\gamma$, with $\gamma\in \mathbb{Z}/2\mathbb{Z}$, and the fiber over a directed
edge $ij$ with color $\gamma_{ij}$ by $\tilde{i}_\gamma \tilde{j}_{\gamma+\gamma_{ij}}$.
We also use \emph{$(k,\ell)$-sparse graphs} \cite{LS08} and their generalizations. For a graph $G$, a
\emph{$(k,\ell)$-basis} is a maximal $(k,\ell)$-sparse subgraph; a \emph{$(k,\ell)$-circuit} is an edge-wise
minimal subgraph that is not $(k,\ell)$-sparse; and a \emph{$(k,\ell)$-component} is a maximal subgraph
that has a spanning $(k,\ell)$-graph.
Points in $\mathbb{R}^2$ are denoted by $\vec p_i = (x_i,y_i)$, indexed sets of points by $\vec p = (\vec p_i)$,
and direction vectors by $\vec d$ and $\vec v$. Realizations
of a reflection direction network $(\tilde{G},\varphi,\vec d)$ are written as $\tilde{G}(\vec p)$, as are realizations
of abstract reflection frameworks. Context will always make clear the type of realization under consideration.
\subsection{Acknowledgements}
LT is supported by the European Research Council under the European Union's Seventh Framework
Programme (FP7/2007-2013) / ERC grant agreement no 247029-SDModels. JM is supported by NSF
CDI-I grant DMR 0835586.
\section{Reflection-Laman graphs} \seclab{matroid}
In this short section we introduce the combinatorial families of sparse colored graphs we use.
\subsection{The map $\rho$}
Let $(G,\bm{\gamma})$ be a $\mathbb{Z}/2\mathbb{Z}$-colored graph. Since all the colored graphs in this paper have $\mathbb{Z}/2\mathbb{Z}$ colors,
from now on we make this assumption and write simply ``colored graph''. We recall two key definitions
from \cite{MT11}.
The map $\rho : \HH_1(G, \mathbb{Z})\to \mathbb{Z}/2\mathbb{Z}$ is defined on cycles by adding up the colors on the edges. (The directions of the
edges don't matter for $\mathbb{Z}/2\mathbb{Z}$ colors. Similarly, neither does the traversal order.) As the notation suggests,
$\rho$ extends to a homomorphism from $\HH_1(G, \mathbb{Z})$ to $\mathbb{Z}/2\mathbb{Z}$, and it is well-defined even if $G$ is not connected.
\subsection{Reflection-Laman graphs}
Let $(G,\bm{\gamma})$ be a colored graph with $n$ vertices and $m$ edges. We define $(G,\bm{\gamma})$ to be a
\emph{reflection-Laman graph} if: the number of edges $m=2n-1$, and for all subgraphs $G'$, spanning $n'$
vertices, $m'$ edges, $c'$ connected components with non-trivial $\rho$-image and $c'_0$ connected components
with trivial $\rho$-image
\begin{equation}\eqlab{cone-laman}
m'\le 2n' - c' - 3c'_0
\end{equation}
This definition is equivalent to that of \emph{cone-Laman graphs} in \cite[Section 15.4]{MT11}.
The underlying graph $G$ of a reflection-Laman graph is a $(2,1)$-graph.
\subsection{Ross graphs and circuits}
Another family we need is that of \emph{Ross graphs}
(see \cite{BHMT11} for an explanation of the terminology). These are colored graphs with $n$ vertices,
$m = 2n - 2$ edges, satisfying the sparsity counts
\begin{equation}\eqlab{ross}
m'\le 2n' - 2c' - 3c'_0
\end{equation}
using the same notations as in \eqref{cone-laman}. In particular, Ross graphs $(G,\bm{\gamma})$
have as their underlying graph, a $(2,2)$-graph $G$, and are thus connected \cite{LS08}.
A \emph{Ross-circuit}%
\footnote{The matroid of Ross graphs has more circuits, but these are the ones
we are interested in here. See \secref{reflection-22}.} %
is a colored graph that becomes a Ross graph after removing \emph{any} edge. The underlying
graph $G$ of a Ross-circuit $(G,\bm{\gamma})$ is a $(2,2)$-circuit, and these are also known
to be connected \cite{LS08}, so, in particular, a Ross-circuit has $c'_0=0$, and
thus satisfies \eqref{cone-laman} on the whole graph. Since \eqref{cone-laman} is
always at least \eqref{ross}, we see that every Ross-circuit is reflection-Laman.
Because reflection-Laman graphs are $(2,1)$-graphs and subgraphs that are $(2,2)$-sparse
are, in addition, Ross-sparse, we get the following structural result.
\begin{prop}[\xyzzy][{\cite[Proposition 5.1]{MT12},\cite[Lemma 11]{BHMT11}}]\proplab{ross-circuit-decomp}
Let $(G,\bm{\gamma})$ be a reflection-Laman graph. Then each $(2,2)$-component of $G$ contains at most one
Ross-circuit, and in particular, the Ross-circuits in $(G,\bm{\gamma})$ are vertex disjoint.
\end{prop}
\subsection{Reflection-$(2,2)$ graphs}\seclab{reflection-22}
The next family of graphs we work with is new. A colored graph $(G,\bm{\gamma})$ is defined to be
a \emph{reflection-$(2,2)$} graph, if it has $n$ vertices, $m=2n-1$ edges, and satisfies the
sparsity counts
\begin{equation} \eqlab{ref22a}
m' \le 2n' - c' - 2c'_0
\end{equation}
using the same notations as in \eqref{cone-laman}.
The relationship between Ross graphs and reflection-$(2,2)$ graphs we will need is:
\begin{prop} \proplab{ross-adding}
Let $(G,\bm{\gamma})$ be a Ross-graph. Then for either
\begin{itemize}
\item an edge $ij$ with any color where $i \neq j$
\item or a self-loop $\ell$ at any vertex $i$ colored by $1$
\end{itemize}
the graph $(G+ij,\bm{\gamma})$ or $(G+\ell,\bm{\gamma})$ is reflection-$(2,2)$.
\end{prop}
\begin{proof}
Adding $ij$ with any color to a Ross $(G,\bm{\gamma})$ creates either a Ross-circuit, for which
$c'_0=0$ or a Laman-circuit with trivial $\rho$-image. Both of these types of
graph meet this count, and so the whole of $(G+ij,\bm{\gamma})$ does as well.
\end{proof}
It is easy to see that every reflection-Laman graph is a reflection-$(2,2)$ graph. The
converse is not true.
\begin{prop}\proplab{reflection-laman-vs-reflection-22}
A colored graph $(G,\bm{\gamma})$ is a reflection-Laman graph if and only if it is a
reflection-$(2,2)$ graph and no subgraph with trivial $\rho$-image is a $(2,2)$-block.\hfill$\qed$
\end{prop}
Let $(G,\bm{\gamma})$ be a reflection-Laman graph, and let $G_1,G_2,\ldots,G_t$ be the Ross-circuits
in $(G,\bm{\gamma})$. Define the \emph{reduced graph} $(G^*,\bm{\gamma})$ of $(G,\bm{\gamma})$ to be the
colored graph obtained by contracting each $G_i$, which is not already a single vertex with a self-loop (this is necessarily
colored $1$), into a new vertex $v_i$, removing any self-loops
created in the process, and then adding a new self-loop with color $1$ to each of the $v_i$.
By \propref{ross-circuit-decomp} the reduced graph is well-defined.
\begin{prop}\proplab{reduced-graph}
Let $(G,\bm{\gamma})$ be a reflection-Laman graph. Then its reduced graph is a reflection-$(2,2)$ graph.
\end{prop}
\begin{proof}
Let $(G,\bm{\gamma})$ be a reflection-Laman graph with $t$ Ross-circuits with vertex sets $V_1,\ldots,V_t$.
By \propref{ross-circuit-decomp}, the $V_i$ are all
disjoint. Now select a Ross-basis $(G',\bm{\gamma})$ of $(G,\bm{\gamma})$. The graph $G'$
is also a $(2,2)$-basis of $G$, with $2n-1 - t$ edges, and each of the $V_i$ spans a $(2,2)$-block
in $G'$. The $(k,\ell)$-sparse graph
Structure Theorem \cite[Theorem 5]{LS08} implies that contracting each of the $V_i$ into a new vertex $v_i$
and discarding any self-loops created, yields a $(2,2)$-sparse graph $G^+$ on $n^+$ vertices and $2n^+ - 1 - t$
edges. It is then easy to check that adding a self-loop colored $1$ at each of the $v_i$ produces
a colored graph satisfying the reflection-$(2,2)$ counts \eqref{ref22a} with exactly
$2n^+ -1$ edges. Since this is the reduced graph, we are done.
\end{proof}
\subsection{Decomposition characterizations}
A \emph{map-graph} is a graph with exactly one cycle per connected component. A \emph{reflection-$(1,1)$} graph
is defined to be a colored graph $(G,\bm{\gamma})$ where $G$, taken as an undirected graph, is a map-graph and
the $\rho$-image of each connected component is non-trivial.
\begin{lemma}\lemlab{reflection-22-decomp}
Let $(G,\bm{\gamma})$ be a colored graph. Then $(G,\bm{\gamma})$ is a reflection-$(2,2)$ graph if and only
if it is the union of a spanning tree and a reflection-$(1,1)$ graph.
\end{lemma}
\begin{proof}
By \cite[Lemma 15.1]{MT11}, reflection-$(1,1)$ graphs are equivalent to graphs satisfying
\begin{equation} \eqlab{ref11a}
m' \le n' - c'_0
\end{equation}
for every subgraph $G'$. Thus, \eqref{ref22a} is
\begin{equation}\eqlab{ref22redux}
m' \le (n' - c'_0) + (n' - c' - c'_0)
\end{equation}
The second term in \eqref{ref22redux} is well-known to be the rank function of the graphic matroid, and
the Lemma follows from the Edmonds-Rota construction \cite{ER66} and the Matroid Union Theorem.
\end{proof}
In the next section, it will be convenient to use this slight refinement of \lemref{reflection-22-decomp}.
\begin{prop}\proplab{reflection-22-nice-decomp}
Let $(G,\bm{\gamma})$ be a reflection-$(2,2)$ graph. Then there is a coloring $\bm{\gamma}'$ of the edges of
$G$ such that:
\begin{itemize}
\item The $\rho$-image of every subgraph in $(G,\bm{\gamma}')$ is the same as in $(G,\bm{\gamma})$.
\item There is a decomposition of $(G,\bm{\gamma}')$ as in \lemref{reflection-22-decomp}
in which the spanning tree has all edges colored by the identity.
\end{itemize}
\end{prop}
\begin{proof}
It is shown in \cite[Lemma 2.2]{MT10} that $\rho$ is determined by its image on a
homology basis of $G$. Thus, we may start with an arbitrary decomposition of $(G,\bm{\gamma})$
into a spanning tree $T$ and a reflection-$(1,1)$ graph $X$, as provided by
\lemref{reflection-22-decomp}, and define $\bm{\gamma}'$ by coloring the edges of $T$ with the identity
and the edges of $X$ with the $\rho$-image of their fundamental cycle in $T$ in $(G,\bm{\gamma})$.
\end{proof}
\propref{reflection-22-nice-decomp} has the following re-interpretation in terms of the symmetric
lift $(\tilde{G},\varphi)$:
\begin{prop}\proplab{reflection-laman-decomp-lift}
Let $(G,\bm{\gamma})$ be a reflection-$(2,2)$ graph. Then for a decomposition, as provided
by \propref{reflection-22-nice-decomp}, into a spanning tree $T$ and a reflection-$(1,1)$
graph $X$:
\begin{itemize}
\item Every edge $ij\in T$ lifts to the two edges $i_0j_0$ and $i_1j_1$.
(In other words, the vertex representatives in the lift all lie in a single connected
component of the lift of $T$.)
\item Each connected component of $X$ lifts to a connected graph.
\end{itemize}
\end{prop}
\section{Special pairs of reflection direction networks} \seclab{direction-network}
We recall, from the introduction, that for reflection direction networks, $\mathbb{Z}/2\mathbb{Z}$ acts on the plane
by reflection through the $y$-axis, and in the rest of this section $\Phi(\gamma)$ refers to this action.
\subsection{The colored realization system}
The system of equations \eqref{dn-realization1}--\eqref{dn-realization2}
defining the realization space of a reflection direction network
$(\tilde{G},\varphi,\vec d)$ is linear, and
as such has a well-defined dimension. Let $(G,\bm{\gamma})$ be the
colored quotient graph of $(\tilde{G},\varphi)$.
To be realizable at all, the directions on the edges in the fiber over $ij\in E(G)$
need to be reflections of each other. Thus, we see that the realization system
is canonically identified with the solutions to the system:
\begin{eqnarray}\eqlab{colored-system}
\iprod{\Phi(\gamma_{ij})\cdot\vec p_j - \vec p_i}{\vec d_{ij}} = 0 & \qquad
\text{for all edges $ij\in E(G)$}
\end{eqnarray}
From now on, we will implicitly switch between the two formalisms when it is
convenient.
\subsection{Genericity}
Let $(G,\bm{\gamma})$ be a colored graph with $m$ edges. A statement about direction networks
$(\tilde{G},\varphi,\vec d)$ is \emph{generic} if it holds on the complement of a
proper algebraic subset of the possible direction assignments, which is canonically identified
with $\mathbb{R}^{2m}$. Some facts about generic statements that
we use frequently are:
\begin{itemize}
\item Almost all direction assignments are generic.
\item If a set of directions is generic, then so are all sufficiently
small perturbations of it.
\item If two properties are generic, then their intersection is as well.
\item The maximum rank of \eqref{colored-system} is a generic property.
\end{itemize}
\subsection{Direction networks on Ross graphs}
We first characterize the colored graphs for which generic direction networks have
strongly faithful realizations. A realization is \emph{strongly faithful} if no
two vertices lie on top of each other. This is a stronger condition than simply being
faithful which only requires that edges not be collapsed.
\begin{prop}\proplab{ross-realizations}
A generic direction network $(\tilde{G},\varphi,\vec d)$
has a unique, up to translation and scaling, strongly faithful realization
if and only if its associated colored graph is a Ross graph.
\end{prop}
To prove \propref{ross-realizations} we expand upon the method from \cite[Section 20.2]{MT11},
and use the following proposition.
\begin{prop}\proplab{reflection-22-collapse}
Let $(G,\bm{\gamma})$ be a reflection-$(2,2)$ graph. Then a generic direction network on the
symmetric lift $(\tilde{G},\varphi)$ of $(G,\bm{\gamma})$ has only collapsed realizations.
\end{prop}
Since the proof of \propref{reflection-22-collapse} requires a detailed construction,
we first show how it implies \propref{ross-realizations}.
\subsection{Proof that \propref{reflection-22-collapse} implies \propref{ross-realizations}}
Let $(G,\bm{\gamma})$ be a Ross graph, and assign directions $\vec d$ to the edges of $G$ such that,
for any extension $(G+ij,\bm{\gamma})$ of $(G,\bm{\gamma})$ to a reflection-$(2,2)$ graph as
in \propref{ross-adding}, $\vec d$ can be extended to a set of directions that is generic
in the sense of \propref{reflection-22-collapse}. This is possible because there are a finite
number of such extensions.
For this choice of $\vec d$, the realization space of the direction network $(\tilde{G},\varphi,\vec d)$
is $2$-dimensional. Since solutions to \eqref{colored-system} may be scaled or translated in the vertical
direction, all solutions to $(\tilde{G},\varphi,\vec d)$ are related by scaling and translation. It
then follows that a pair of vertices in the fibers over $i$ and $j$ are either distinct from each other
in all non-zero solutions to \eqref{colored-system} or always coincide. In the latter case, adding the
edge $ij$ with any direction does not change the dimension of the solution space, no matter
what direction we assign to it. It then follows that the solution spaces of generic
direction networks on $(\tilde{G},\varphi,\vec d)$ and $(\widetilde{G+ij},\varphi,\vec d)$ have
the same dimension, which is a contradiction by \propref{reflection-22-collapse}.
\hfill$\qed$
\subsection{Proof of \propref{reflection-22-collapse}}
It is sufficient to construct a specific set of directions with this property. The
rest of the proof gives such a construction and verifies that all the solutions
are collapsed. Let $(G,\bm{\gamma})$ be a reflection-$(2,2)$ graph.
\paragraph{Combinatorial decomposition}
We apply \propref{reflection-22-nice-decomp} to decompose $(G,\bm{\gamma})$ into a
spanning tree $T$ with all colors the identity and a reflection-$(1,1)$ graph $X$. For
now, we further assume that $X$ is connected.
\paragraph{Assigning directions}
Let $\vec v$ be a direction vector that is not
horizontal or vertical. For each edge $ij\in T$, set $\vec d_{ij} = \vec v$.
Assign all the edges of $X$ the vertical
direction. Denote by $\vec d$ this assignment of directions.
\begin{figure}[htbp]
\centering
\includegraphics[width=.3\textwidth]{ref-22-collapse}
\caption{Schematic of the proof of \propref{reflection-22-collapse}: the $y$-axis
is shown as a dashed line. The directions on the edges of the lift of the tree
$T$ force all the vertices to be on one of the two lines meeting at the $y$-axis,
and the directions on the reflection-$(1,1)$ graph $X$ force all the vertices to
be on the $y$-axis.}
\label{fig:ref-22-collapse}
\end{figure}
\paragraph{All realizations are collapsed}
We now show that the only realizations of $(\tilde{G},\varphi,\vec d)$ have
all vertices on top of each other. By \propref{reflection-laman-decomp-lift}
$T$ lifts to two copies of itself, in $\tilde{G}$. It then follows from
the connectivity of $T$ and the construction of $\vec d$ that, in any
realization, there is a line $L$ with direction $\vec v$ such that every vertex
of $\tilde{G}$ must lie on $L$ or its reflection. Since the vertical direction
is preserved by reflection, the connectivity of the lift of $X$, again from
\propref{reflection-laman-decomp-lift}, implies that every vertex of $\tilde{G}$
lies on a single vertical line, which must be the $y$-axis by reflective symmetry.
Thus, in any realization of $(\tilde{G},\varphi,\vec d)$ all the vertices lie
at the intersection of $L$, the reflection of $L$
through the $y$-axis and the $y$-axis itself. This is a single point, as
desired. \figref{ref-22-collapse} shows a schematic of this argument.
\paragraph{$X$ does not need to be connected}
Finally, we can remove the assumption that $X$ was connected by repeating the argument
for each connected component of $X$ separately.
\hfill$\qed$
\subsection{Special pairs for Ross-circuits}
The full \theoref{direction-network} will reduce to the case of a Ross-circuit.
\begin{prop}\proplab{ross-circuit-pairs}
Let $(G,\bm{\gamma})$ be a Ross circuit with lift $(\tilde{G},\varphi)$. Then there
is an edge $i'j'$ such that, for a generic direction network $(\tilde{G'},\varphi,\vec d')$ with colored graph
$(G-i'j',\bm{\gamma})$:
\begin{itemize}
\item The solution space of $(\tilde{G'},\varphi,\vec d')$ induces a well-defined direction
$\vec d_{ij}$ between $i$ and $j$, yielding an assignment of directions $\vec d$ to the edges
of $G$.
\item The direction networks $(\tilde{G},\varphi,\vec d)$
and $(\tilde{G},\varphi,(\vec d)^\perp)$ are a special pair.
\end{itemize}
\end{prop}
Before giving the proof, we describe the idea. We are after sets of directions that
lead to faithful realizations of Ross-circuits. By \propref{reflection-22-collapse},
these directions must be non-generic. A natural way to obtain such a set of directions
is to discard an edge $ij$ from the colored quotient graph,
apply \propref{ross-realizations} to obtain a generic set of directions $\vec d'$ with a
strongly faithful realization $\tilde{G}'(\vec p)$, and then simply set the directions on the
edges in the fiber over $ij$ to be the difference vectors between the points.
\propref{ross-realizations} tells us that this procedure induces a well-defined
direction for the edge $ij$, allowing us to extend $\vec d'$ to $\vec d$ in a controlled way.
However, it does \emph{not} tell us that rank
of $(\tilde{G},\varphi,\vec d)$ will rise when the directions are turned
by angle $\pi/2$, and this seems hard to do directly. Instead,
we construct a set of directions $\vec d$ so that $(\tilde{G},\varphi,\vec d)$ is rank deficient
and has faithful realizations, and $(\tilde{G},\varphi,\vec d^\perp)$ is generic.
Then we make a perturbation argument to show the existence of a special pair.
The construction we use is, essentially, the one used in the proof of \propref{reflection-22-collapse}
but turned through angle $\pi/2$. The key geometric insight is that horizontal edge directions
are preserved by the reflection, so the ``gadget'' of a line and its reflection crossing on the $y$-axis, as in
\figref{ref-22-collapse}, degenerates to just a single line.
\subsection{Proof of \propref{ross-circuit-pairs}}
Let $(G,\bm{\gamma})$ be a Ross-circuit; recall that this implies that $(G,\bm{\gamma})$ is a
reflection-Laman graph.
\paragraph{Combinatorial decomposition}
We decompose $(G,\bm{\gamma})$ into a spanning tree $T$ and a
reflection-$(1,1)$ graph $X$ as in \propref{reflection-laman-decomp-lift}. In particular,
we again have all edges in $T$ colored by the identity. For now,
we \emph{assume that $X$ is connected}, and we fix $i'j'$ to be an edge that is on the cycle in $X$
with $\gamma_{i'j'}\neq 0$; such an edge must exist by the hypothesis that $X$ is
reflection-$(1,1)$. Let $G' = G\setminus i'j'$. Furthermore, let $T_0$ and $T_1$
be the two connected components of the lift of $T$. For a vertex $i \in G$, we denote
the lift in $T_0$ by $i_0$ and the lift in $T_1$ by $i_1$. We similarly denote the
lifts of $i'$ and $j'$ by $i_0', i_1'$ and $j_0', j_1'$.
\paragraph{Assigning directions}
The assignment of directions is as follows: to the edges of $T$, we assign a direction
$\vec v$ that is neither vertical nor horizontal. To the edges of $X$
we assign the horizontal direction. Define the resulting direction network to be
$(\tilde{G},\varphi,\vec d)$, and the direction network induced on the lift of $G'$ to be
$(\tilde{G'},\varphi,\vec d)$.
\paragraph{The realization space of $(\tilde{G},\varphi,\vec d)$}
\figref{ross-circuit-special-pair} contains a schematic picture of the arguments that
follow.
\begin{lemma}\lemlab{RC-proof-1}
The realization space of $(\tilde{G},\varphi,\vec d)$ is $2$-dimensional
and parameterized by exactly one representative in the fiber over the
vertex $i$ selected above.
\end{lemma}
\begin{proof}
In a manner similar to
the proof of \propref{reflection-22-collapse}, the directions on the edges of $T$ force every
vertex to lie either on a line $L$ in the direction $\vec v$ or its reflection. Since the lift
of $X$ is connected, we further conclude that all the vertices lie on a single horizontal line.
Thus, all the points $\vec p_{j_0}$ are at the intersection of the same horizontal line and $L$
or its reflection. These determine the locations of the $\vec p_{j_1}$, so the
realization space is parameterized by the location of $\vec p_{i'_0}$.
\end{proof}
Inspecting the argument more closely, we find that:
\begin{lemma}
In any realization $\tilde{G}(\vec p)$ of $(\tilde{G},\varphi,\vec d)$,
all the $\vec p_{j_0}$ are equal and all the $\vec p_{j_1}$ are equal.
\end{lemma}
\begin{proof}
Because the colors on the edges of $T$ are all zero, it lifts to two copies of itself,
one of which spans the vertex set $\{\tilde{j_0} : j\in V(G)\}$ and one which spans
$\{\tilde{j_1} : j\in V(G)\}$. It follows that in a realization, we have all the
$\vec p_{j_0}$ on $L$ and the $\vec p_{j_1}$ on the
reflection of $L$.
\end{proof}
In particular, because the color $\gamma_{i'j'}$ on the edge $i'j'$ is $1$, we obtain the following.
\begin{lemma}\lemlab{RC-proof-5}
The realization space of $(\tilde{G},\varphi,\vec d)$ contains points
where the fiber over the edge $i'j'$ is not collapsed.
\end{lemma}
\begin{figure}[htbp]
\centering
\includegraphics[width=.3\textwidth]{ross-circuit-1}
\caption{Schematic of the proof of \propref{ross-circuit-pairs}: the $y$-axis
is shown as a dashed line. The directions on the edges of the lift of the tree
$T$ force all the vertices to be on one of the two lines meeting at the $y$-axis.
The horizontal directions on the connected reflection-$(1,1)$ graph $X$ force the
point $\vec p_{j_0}$ to be at the intersection marked by the black dot and
$\vec p_{j_1}$ to be at the intersection marked by the gray one.}
\label{fig:ross-circuit-special-pair}
\end{figure}
\paragraph{The realization space of $(\tilde{G}',\varphi,\vec d)$}
The conclusion of \lemref{RC-proof-1} implies that the realization
system for $(\tilde{G},\varphi,\vec d)$ is rank deficient by one.
Next we show that removing the edge $i'j'$ results in a
direction network that has full rank on the colored graph $(G',\bm{\gamma})$.
\begin{lemma}\lemlab{RC-proof-2}
The realization space of $(\tilde{G},\varphi,\vec d)$ is canonically identified
with that of $(\tilde{G}',\varphi,\vec d)$.
\end{lemma}
\begin{proof}
In the proof of \lemref{RC-proof-1}, that $X$ lifts to a connected subgraph of $\tilde{G}$
was not essential. Because a horizontal line is preserved by the reflection,
realizations will take on the same structure provided that $X$ lifts to a subgraph
with two connected components. Removing $i'j'$
from $X$ leaves a graph $X'$ with this property since $X'$ is a tree.
It follows that the equation corresponding to the edge $i'j'$ in \eqref{colored-system}
was dependent.
\end{proof}
\paragraph{The realization space of $(\tilde{G},\varphi,\vec d^\perp)$}
Next, we consider what happens when we turn all the directions by $\pi/2$.
\begin{lemma}\lemlab{RC-proof-3}
The realization space of $(\tilde{G},\varphi,\vec d^\perp)$ has only collapsed solutions.
\end{lemma}
\begin{proof}
This is exactly the construction used to prove \propref{reflection-22-collapse}.
\end{proof}
\paragraph{Perturbing $(\tilde{G},\varphi,\vec d)$}
To summarize what we have shown so far:
\begin{itemize}
\item[(a)] $(\tilde{G},\varphi,\vec d)$ has a $2$-dimensional realization space parameterized
by $\vec p_{i'_0}$ and identified with that of a full-rank direction network on the Ross graph
$(G',\bm{\gamma})$.
\item[(b)] There are points $\tilde{G}(\vec p)$ in this realization space where
$\vec p_{i'_0}\neq \vec p_{j'_1}$.
\item[(c)] $(\tilde{G},\varphi,\vec d)$ has a $1$-dimensional realization space containing only collapsed
solutions.
\end{itemize}
What we have not shown is that the realization space of $(\tilde{G},\varphi,\vec d)$
has \emph{faithful} realizations, since the ones we constructed all have many
coincident vertices. \propref{ross-realizations} will imply the rest of the
theorem, provided that the above properties hold for any small perturbation of
$\vec d$, since some small perturbation of \emph{any} assignment of directions
to the edges of $(G',\bm{\gamma})$ has only faithful realizations.
\begin{lemma}\lemlab{RC-proof-4}
Let $\vec{ \hat d'}$ be a perturbation of the directions $\vec d'$ on the edges of $G'$. If $\vec{ \hat d'}$
is sufficiently close to $\vec d'$ ,
then there are realizations of the direction network
$(\tilde{G}',\varphi,\vec{ \hat d'})$ such that $\vec p_{i'_0}\neq \vec p_{j'_1}$.
\end{lemma}
\begin{proof}
The realization space is parameterized by $\vec p_{i'_0}$, and so $\vec p_{j'_1}$ varies
continuously with the directions on the edges and $\vec p_{i'_0}$. Since there are realizations of $(\tilde{G}', \varphi, \vec d)$
with $\vec p_{i_0} \neq \vec p_{j_1}$, the Lemma follows.
\end{proof}
\lemref{RC-proof-4} implies that any sufficiently small perturbation of the directions assigned to the
edges of $G'$ gives a direction network that induces a well-defined direction on the edge $i'j'$ which
is itself a small perturbation of $\vec d_{i'j'}$. Since the ranks of $(\tilde{G'},\varphi,\vec d')$
and $(\tilde{G},\varphi,\vec d^\perp)$ are stable under small perturbations, this implies that
we can perturb $\vec d'$ to a $\vec{\hat d'}$ that is generic in the sense of \propref{ross-realizations},
while preserving faithful realizability of $(\tilde{G},\varphi,\hat{\vec d})$ and full rank of the
realization system for $(\tilde{G},\varphi,\hat{\vec d}^\perp)$. The Proposition is proved for when $X$ is
connected.
\paragraph{$X$ need not be connected}
The proof is then complete once we remove the additional assumption that $X$ was
connected. Let $X$ have connected
components $X_1, X_2,\ldots,X_c$. For each of the $X_i$, we can identify an edge
$(i'j')_k$ with the same properties as $i'j'$ above.
Assign directions to the tree $T$ as above.
For $X_1$, we assign directions exactly as above. For each of the $X_k$ with $k\ge 2$,
we assign the edges of $X_k\setminus (i'j')_k$ the horizontal direction and $(i'j')_k$
a direction that is a small perturbation of horizontal.
With this assignment $\vec d$ we see that
for any realization of $(\tilde{G},\varphi,\vec d)$, each of the $X_k$, for $k\ge 2$
is realized as completely collapsed to a single point
at the intersection of the line $L$ and the $y$-axis. Moreover,
in the direction network on $\vec d^\perp$, the directions on these $X_i$ are a small
perturbation of the ones used on $X$ in the proof of \propref{reflection-22-collapse}.
From this is follows that, in any realization $(\tilde{G},\varphi,\vec d^\perp)$,
is completely collapsed and hence full rank.
We now see that this new set of directions has properties (a), (b), and (c) above
required for the perturbation argument. Since that argument makes no reference
to the decomposition, it applies verbatim to the case where $X$ is disconnected.
\hfill$\qed$
\subsection{Proof of \theoref{direction-network}}
The easier direction to check is necessity.
\paragraph{The Maxwell-direction}
If $(G,\bm{\gamma})$ is not reflection-Laman, then it contains either a
Laman-circuit with trivial $\rho$-image, or a violation of $(2,1)$-sparsity. If there is a
Laman-circuit with trivial $\rho$-image, the Parallel Redrawing Theorem \cite[Theorem 4.1.4]{W96}
in the form \cite[Theorem 3]{ST10} implies that this subgraph has no faithful realizations
for
$(G,\varphi,\vec d)$ only if it does in $(G,\varphi,\vec d^\perp)$ if rank-deficient.
A violation of $(2,1)$-sparsity implies that the realization system \eqref{colored-system}
of $(\tilde{G},\varphi,\vec d^\perp)$ has a dependency, since the realization space is always at
least $1$-dimensional.
\paragraph{The Laman direction}
Now let $(G,\bm{\gamma})$ be a reflection-Laman graph and let $(G',\bm{\gamma})$
be a Ross-basis of $(G,\bm{\gamma})$. For any edge $ij \notin G'$, adding it to
$G'$ induces a Ross-circuit which contains some edge $i'j'$ having the property
specified in \propref{ross-circuit-pairs}. Note that $G' - ij +i'j'$ is again a Ross-basis.
We therefore can assume (after edge-swapping in this manner) for all $ij \notin G'$ that
$ij$ has the property from \propref{ross-circuit-pairs} in the Ross-circuit it induces.
We assign directions $\vec d'$ to the edges of $G'$ such that:
\begin{itemize}
\item The directions on each of the intersections of the Ross-circuits with $G'$ are generic in the sense
of \propref{ross-circuit-pairs}.
\item The directions on the edges of $G'$ that remain in the reduced graph $(G^*,\bm{\gamma})$
are perpendicular to an assignment of directions on $G^*$ that is
generic in the sense of \propref{reflection-22-collapse}.
\item The directions on the edges of $G'$ are generic in the sense of \propref{ross-realizations}.
\end{itemize}
This is possible because the set of disallowed directions is the union of a finite number of
proper algebraic subsets in the space of direction assignments. Extend to directions $\vec d$ on $G$
by assigning directions to the remaining edges as specified by \propref{ross-circuit-pairs}. By construction,
we know that:
\begin{lemma}\lemlab{laman-1}
The direction network $(\tilde{G},\varphi,\vec d)$ has faithful realizations.
\end{lemma}
\begin{proof}
The realization space is identified with that of $(\tilde{G'},\varphi,\vec d')$, and $\vec d'$
is chosen so that \propref{ross-realizations} applies.
\end{proof}
\begin{lemma}\lemlab{laman-2}
In any realization of $(\tilde{G},\varphi,\vec d^{\perp})$, the Ross-circuits are realized with all their
vertices coincident and on the $y$-axis.
\end{lemma}
\begin{proof}
This follows from how we chose $\vec d$ and \propref{ross-circuit-pairs}.
\end{proof}
As a consequence of \lemref{laman-2}, and the fact that we picked $\vec d$ so that $\vec d^\perp$
extends to a generic assignment of directions $(\vec d^*)^\perp$ on the reduced graph $(G^*,\bm{\gamma})$
we have:
\begin{lemma}
The realization space of $(\tilde{G},\varphi,\vec d^\perp)$ is identified with that of
$(\tilde{G^*},\varphi,\vec (d^*)^\perp)$ which, furthermore, contains only collapsed solutions.
\end{lemma}
Observe that a direction network for a single self-loop (colored $1$) with a generic direction only has solutions where
vertices are collapsed and on the $y$-axis. Consequently, replacing a Ross-circuit with a single vertex
and a self-loop yields isomorphic realization spaces. Since the reduced graph is reflection-$(2,2)$ by \propref{reduced-graph} and the directions
assigned to its edges were chosen generically for \propref{reflection-22-collapse},
that $(\tilde{G},\varphi,\vec d^\perp)$ has only collapsed solutions follows.
Thus, we have exhibited a special pair, completing the proof.
\hfill$\qed$
\paragraph{Remark} It can be seen that the realization space of a direction network as supplied by
\theoref{direction-network} has at least one degree of freedom for each edge that is not in a Ross basis. Thus,
the statement cannot be improved to, e.g., a unique realization up to translation and scale.
\section{Infinitesimal rigidity of reflection frameworks} \seclab{reflection-laman-proof}
Let $(\tilde{G},\varphi,\bm{\ell})$ be a reflection framework and let $(G,\bm{\gamma})$
be the quotient graph. The configuration space, which is the set of
solutions to the quadratic system \eqref{lengths-1}--\eqref{lengths-2} is canonically
identified with the solutions to:
\begin{eqnarray}\eqlab{colored-lengths}
||\Phi(\gamma_{ij})\cdot \vec p_j - \vec p_i||^2 = \ell^2_{ij} & \qquad
\text{for all edges $ij\in E(G)$}
\end{eqnarray}
where $\Phi$ acts on the plane by reflection through the $y$-axis. (That ``pinning down''
$\Phi$ does not affect the theory is straightforward from the definition of the configuration
space: it simply removes rotation and translation in the $x$-direction from the
set of trivial motions.)
\subsection{Infinitesimal rigidity}
Computing the formal differential of \eqref{colored-lengths}, we obtain the system
\begin{eqnarray}\eqlab{colored-inf}
\iprod{\Phi(\gamma_{ij})\cdot \vec p_j - \vec p_i}{\vec v_j - \vec v_i} = 0 & \qquad
\text{for all edges $ij\in E(G)$}
\end{eqnarray}
where the unknowns are the \emph{velocity vectors} $\vec v_i$. A standard kind of result (cf. \cite{AR78})
is the following.
\begin{prop}\proplab{ar-direction}
Let $\tilde{G}(\vec p,\Phi)$ be a realization of an abstract framework $(\tilde{G},\varphi,\bm{\ell})$.
If the corank of the system \eqref{colored-inf} is one, then $\tilde{G}(\vec p)$ is rigid.
\end{prop}
Thus, we define a realization to be \emph{infinitesimally rigid} if the system \eqref{colored-inf}
has maximal rank, and \emph{minimally infinitesimally rigid} if it is infinitesimally rigid but ceases to
be so after removing any edge from the colored quotient graph.
By definition, infinitesimal rigidity is defined by a polynomial condition in the coordinates of
the points $\vec p_i$, so it is a generic property associated with the colored graph $(G,\bm{\gamma})$.
\subsection{Relation to direction networks}
Here is the core of the direction network method for reflection frameworks:
we can understand the rank of \eqref{colored-inf} in terms of a direction network.
\begin{prop}\proplab{rigidity-vs-directions}
Let $\tilde{G}(\vec p,\Phi)$ be a realization of a reflection framework with $\Phi$
acting by reflection through the $y$-axis. Define the direction $\vec d_{ij}$ to be
$\vec \Phi(\gamma_{ij})\cdot \vec p_j - \vec p_i$. Then the rank of \eqref{colored-inf}
is equal to that of \eqref{colored-system} for the direction network
$(G,\bm{\gamma},\vec d^{\perp})$.
\end{prop}
\begin{proof}
Exchange the roles of $\vec v_i$ and $\vec p_i$ in \eqref{colored-inf}.
\end{proof}
\subsection{Proof of \theoref{reflection-laman}}
The, more difficult, ``Laman direction'' of the Main Theorem follows immediately from
\theoref{direction-network} and \propref{rigidity-vs-directions}:
given a reflection-Laman graph \theoref{direction-network} produces a realization with
no coincident endpoints and a certificate that \eqref{colored-inf} has corank one.
\hfill$\qed$
\subsection{Remarks}
The statement of \propref{rigidity-vs-directions} is \emph{exactly the same} as the analogous statement for
orientation-preserving cases of this theory. What is different is that, for reflection frameworks,
the rank of $(G,\bm{\gamma},\vec d^{\perp})$ is \emph{not}, the same
as that of $(G,\bm{\gamma},\vec d)$. By \propref{reflection-22-collapse}, the set of directions arising as
the difference vectors from point sets are \emph{always non-generic} on reflection-Laman graphs, so
we are forced to introduce the notion of a special pair as in \secref{direction-network}.
\bibliographystyle{plainnat}
|
{
"timestamp": "2012-03-13T01:01:27",
"yymm": "1203",
"arxiv_id": "1203.2276",
"language": "en",
"url": "https://arxiv.org/abs/1203.2276"
}
|
\section*{Introduction}
Let us start by introducing the context of this work, that can
summarized by the following diagram.
\begin{equation*}
\xymatrix{
\mathbf{Sym} \ar@{ ->}[r] & \operatorname{Dend} \ar@{ ->}[r] & \mathbf{FQSym} \\
\mathbf{Sym} \cap \operatorname{Lie} \ar@{ ->}[r]\ar[u] & \operatorname{Dend} \cap \operatorname{Lie} \ar@{ ->}[r]\ar@{ ->}[u] & \operatorname{Lie}\ar@{ ->}[u] \\
& \operatorname{PreLie} \ar@{ ->}[u]^{\phi}_{\simeq ?} &
}
\end{equation*}
At the top left corner, $\mathbf{Sym}$ is the graded Hopf algebra of
non-commutative symmetric functions \cite{ncsf}, which has a basis indexed by
compositions of integers. At the top right corner, $\mathbf{FQSym}$ is the
graded Hopf algebra of free quasi-symmetric functions, also known as
the Malvenuto-Reutenauer algebra \cite{malvenuto}, which has a basis
indexed by permutations. These two Hopf algebras can be considered as
non-commutative analogues of the classical Hopf algebra of symmetric
functions. They have been studied a lot, and have proved to be useful
in algebraic combinatorics, see for example \cite{thibon_lectures,ncsf6}.
At the middle of the top line, $\operatorname{Dend}$ is the free Dendriform algebra
on one generator. This is also a graded Hopf algebra, also known as
the Loday-Ronco Hopf algebra \cite{loday_pbt}, and has a basis indexed
by planar binary trees. The horizontal morphisms of the first line are
inclusions of Hopf algebras, and can be described using appropriate
equivalence relations on permutations, see for instance
\cite{loday_pbt}.
On the second line, the subspace $\operatorname{Lie}$ of $\mathbf{FQSym}$ has two equivalent
descriptions. First, one can map $\mathbf{FQSym}$ into a space of rational
moulds, as described in \cite{moulds}. Then $\operatorname{Lie}$ is the subspace of
alternal elements, in the terminology of the mould calculus of Ecalle
\cite{ecalle_ari,ecalle_tale}. One can also identify $\mathbf{FQSym}$ with the
direct sum of all group rings of symmetric groups, and therefore to the
associative operad. Then $\operatorname{Lie}$ is the space of Lie elements, or the
image of the $\operatorname{Lie}$ operad in the associative operad.
On the left of second line is the intersection of the subspaces
$\mathbf{Sym}$ and $\operatorname{Lie}$ of $\mathbf{FQSym}$. It is known to be exactly the subspace
of primitive elements in the Hopf algebra $\mathbf{Sym}$, by results of
\cite{ncsf}.
The intersection at the middle of the second line is quite
interesting. Starting from the usual injective morphism from the
Pre-Lie operad to the Dendriform operad, one gets an injective
morphism $\phi$ from the free Pre-Lie algebra on one generator,
denoted here by $\operatorname{PreLie}$, to $\operatorname{Dend}$. It was proved in \cite{moulds}
that its image is contained in the intersection $\operatorname{Dend} \cap \operatorname{Lie}$.
It is conjectured that $\phi$ is an isomorphism from $\operatorname{PreLie}$ to
$\operatorname{Dend} \cap \operatorname{Lie}$. This has been checked for small degrees. If this
isomorphism holds, it would have interesting consequences for the
theory of Lie idempotents, that we will now present.
Recall that a Lie idempotent is an element $\theta$ in the group ring
$\mathbb{Q}[\mathfrak{S}_n]$ of the symmetric group, such that $\theta$ is
idempotent, and such that the product by $\theta$ is a projector onto
the subspace of Lie elements. The set of Lie idempotents is an affine
subspace of the group ring $\mathbb{Q}[\mathfrak{S}_n]$. There are many known
examples of Lie idempotents, and most of them belong to a sub-algebra
of $\mathbb{Q}[\mathfrak{S}_n]$, the Solomon descent algebra.
There is a natural way to identify $\mathbb{Q}[\mathfrak{S}_n]$ with the graded
component of degree $n$ of $\mathbf{FQSym}$. By this isomorphism, Solomon
descent algebra is identified with the graded component of degree $n$
of $\mathbf{Sym}$. Moreover, the subspace of primitive elements of $\mathbf{Sym}$
corresponds to the intersection of Solomon descent algebras with the
vector space spanned by Lie idempotents \cite{ncsf}.
From all this, one can deduce that any Lie idempotent in the descent
algebra gives an element in the intersection $\mathbf{Sym} \cap \operatorname{Lie}$ and
therefore also in $\operatorname{Dend} \cap \operatorname{Lie}$. If $\phi$ is an isomorphism, this
element will come from an element of $\operatorname{PreLie}$. Conversely, given an
element of $\operatorname{PreLie}$, if one can check that its image by $\phi$
belongs to $\mathbf{Sym}$, then it will belong to $\mathbf{Sym} \cap \operatorname{Lie}$ and
will define, up to multiplication by a scalar, a Lie idempotent in the
descent algebra.
Given any specific Lie idempotent in the descent algebra, one can
therefore ask for a description of its pre-image by $\phi$. This has
been obtained in \cite{qidempotent} for a one parameter familly of Lie
idempotents. The starting point of this article was to do the same for
a specific familly of Lie idempotents, that has just been recently
introduced. Let us now present them briefly.
Inspired by previous works by Ecalle and Menous
\cite{menous_bm,ecalle_menous} on the Alien calculus, Menous, Novelli
and Thibon have defined in \cite{menoth} a sequence of Lie idempotents
$\sD_n$ in the descent algebra of the symmetric group $\mathfrak{S}_n$. The
coefficients of $\sD_n$ in the basis of ribbon Schur functions are given by
homogeneous polynomials in two variables $a$ and $b$, more precisely
products of powers of $a$ and $b$ and Narayana polynomials in $a$ and
$b$. By homogeneity, one can let $a=1$ in the coefficients of $\sD_n$
without losing any information. We will therefore work with
polynomials in $b$ only.
By computing, for small $n$, the elements $\xD_n$ in $\operatorname{PreLie}$ whose
image by $\phi$ is $\sD_n$, one observes that their coefficients are
positive polynomials in $b$ and seem to factorise according to
subtrees, with factors being also positive polynomials in $b$.
The first result of the present article is a combinatorial description
of the coefficients of $\xD_n$ and their factors, in terms of flows on
rooted trees.
To achieve this, one works inside groups of operadic series,
associated with the Pre-Lie and Dendriform operads. All the
idempotents $\sD_n$ are gathered into one series $\sD$ in the group
$\mathsf{G}_{\operatorname{Dend}}$ associated with the Dendriform operad. Their pre-images
$\xD_n$ by $\phi$ are similarly grouped in a series $\xD$ in the group
$\mathsf{G}_{\operatorname{PreLie}}$ associated with the Pre-Lie operad.
We proceed in the following order. First, we introduce the
combinatorial notion of flow on a rooted tree, and describe its
properties. Next, we obtain, from combinatorial arguments, various
functional equations satisfied by several series in the Pre-Lie group,
whose coefficients count different kinds of flows. We then go on to
introduce some series in the Dendriform group, and to show, by
algebraic means, that they satisfy another set of functional
equations. By comparing the functional equations in the Pre-Lie and
Dendriform cases, one can then recognize among the dendriform series
the images by $\phi$ of some of the Pre-Lie series.
On the way, one uses many auxiliary series, and some of them have
interesting properties. In particular, one does not only recover the
Lie idempotents $\sD_n$ of \cite{menoth}, but also gets a new familly
$\sF_n$ of Lie idempotents, related to closed connected
flows. Moreover, two other conjectural famillies $\sZ_n$ and
$\sF_{n,t}$ of Lie idempotents are proposed, for which we have not
been able to obtain a full proof. In the case of $\sF_{n,t}$, one is
missing a combinatorial proof of the existence of a Pre-Lie series
$\xF_t$ and so we do not know if the dendriform series $\sF_{t}$ is a
Lie element or not. In the case of $\sZ_n$, one only has a conjectural
description of the coefficients of the dendriform series $\sZ$, and so
we do not know if it belong to $\mathbf{Sym}$.
\medskip
We gather in an appendix some technical tools that are necessary to
turn combinatorial bijections into equalities of series in groups
associated with operads. The notions of rooted-operad and
rooted-monoid that are introduced here may be of independent interest.
\section{Rooted trees and the $\operatorname{PreLie}$ operad}
\subsection{Notations for rooted trees}
\label{section_pl}
A \textbf{rooted tree} is a finite connected and simply connected graph,
together with a distinguished vertex called the root.
Rooted trees will be considered implicitly as directed graphs by
orienting every edge towards the root.
The \textbf{valency} $v_s$ of a vertex $s$ in a rooted tree is the
number of incoming edges.
The \textbf{height} of a vertex $s$ in a rooted tree is defined as
follows: the height of the root is $0$, and the height of the source
of every edge is $1$ more than the height of its end.
Rooted trees of maximal height at most $1$ are called \textbf{corollas}. Rooted
trees of maximal valency at most $1$ are called \textbf{linear trees}.
A rooted tree $T$ will sometimes be considered as a partially ordered
set whose Hasse diagram is given by the orientation towards the root, with
the root as the unique minimal element.
A \textbf{leaf} in a rooted tree $T$ is a vertex of valency $0$. A leaf can
also be defined as a maximal vertex.
Rooted trees will be drawn with their root at the bottom and leaves at the top.
If $T_1,\dots,T_k$ are rooted trees, we will denote
$B_+(T_1,\dots,T_k)$ the rooted tree obtained by grafting together
$T_1,\dots,T_k$ on a new common root.
Let $\arb{0}$ be the rooted tree with one vertex.
Let $\mathtt{Lnr}_\ell$ be the linear rooted tree with $\ell$ vertices,
defined by induction:
\begin{equation*}
\mathtt{Lnr}_1=\arb{0}\quad\text{and}\quad \mathtt{Lnr}_{\ell+1}= B_+(\mathtt{Lnr}_\ell).
\end{equation*}
Let $\mathtt{Crl}_n$ be the corolla with $n+1$ vertices, defined by
\begin{equation*}
B_+(\arb{0},\dots,\arb{0}),
\end{equation*}
with $n$ copies of $\arb{0}$.
Let $\mathtt{Frk}_{i,n-i}$ be the fork with $n$ vertices, with stem of size
$i$, defined by induction:
\begin{equation*}
\mathtt{Frk}_{1,\ell}=\mathtt{Crl}_{\ell}\quad\text{and}\quad \mathtt{Frk}_{k+1,\ell}=B_+(\mathtt{Frk}_{k,\ell}).
\end{equation*}
Examples of linear trees, corollas and forks are depicted in figure
\ref{fig:fork}.
The number of vertices of a rooted tree $T$ will be denoted by $\#T$.
\begin{figure}
\centering
\includegraphics[height=3cm]{fourcorolle.pdf}
\caption{Linear tree $\mathtt{Lnr}_4$, corolla $\mathtt{Crl}_3$ and fork $\mathtt{Frk}_{3,4}$.}
\label{fig:fork}
\end{figure}
\subsection{The group of rooted trees}
For more details on the general construction of the group of series
associated $\mathsf{G}_\mathcal{P}$ with an operad $\mathcal{P}$, the reader may consult the
appendix \ref{appA}, \cite{chaplive2} and \cite[App. A]{qidempotent}.
We will work in the group of series $\mathsf{G}_{\operatorname{PreLie}}$ associated with
the Pre-Lie operad. This group is contained in the free Pre-Lie
algebra on one generator, denoted here by $\operatorname{PreLie}$.
The Pre-Lie operad has a basis indexed by labelled rooted
trees \cite{chaplive1}. It follows that the Pre-Lie algebra on one
generator has a basis index by (unlabelled) rooted trees.
For a series $\xD$ in the group of rooted trees, we will use $\xD_T$
to denote the coefficient of the rooted tree $T$ in $\xD$, in the
following sense:
\begin{equation}
\xD = \sum_{T} \frac{\xD_T}{\operatorname{aut}(T)} T,
\end{equation}
where $\operatorname{aut}(T)$ is the cardinal of the automorphism group of $T$.
The homogeneous component of $\xD$ of degree $n$ will be denoted by
$\xD_n$.
We will use the following special notation for the sum of all corollas:
\begin{equation}
\label{defi_crls}
\textsc{Crls}=\sum_{n \geq 0} \frac{\mathtt{Crl}_n}{n!}.
\end{equation}
Let $\textsc{H}_k$ be the element
\begin{equation}
\textsc{H}_k = \sum_{T} k^{\#T-1} \frac{T}{\operatorname{aut}_T}
\end{equation}
of the group $\mathsf{G}_{\operatorname{PreLie}}$. Its coefficients are polynomials in the
variable $k$.
\begin{lemma}
\label{inversion_somme_tous}
One has
\begin{equation}
\textsc{H}_{k} \circ \textsc{H}_{\ell} = \textsc{H}_{k+\ell},
\end{equation}
where $k$ and $\ell$ are formal variables. In particular, when $k$
is a positive integer, $\textsc{H}_k$ is the $k^{th}$ power of $\textsc{H}_1$ for the
group law of $\mathsf{G}_{\operatorname{PreLie}}$. The inverse of $\textsc{H}_k$ is $\textsc{H}_{-k}$.
\end{lemma}
\begin{proof}
It is enough to prove this identity for $k$ and $\ell$ positive
integers, by polynomiality.
Let $\mathsf{K}$ and $\mathsf{L}$ be finite sets of cardinality $k$
and $\ell$. Elements of this sets are considered as colors.
One applies proposition \ref{circ_prop} for the rooted-operad
$\operatorname{PreLie}$, with $A$ the species of rooted trees with edges colored
by elements of $\mathsf{K}$, $B$ the species of rooted trees with
edges colored by elements of $\mathsf{L}$ and $C$ the species of
rooted trees with edges colored by elements of $\mathsf{K}\sqcup
\mathsf{L}$. The series $s_A$, $s_B$ and $s_C$ are clearly just
$\textsc{H}_{k}$, $\textsc{H}_{\ell}$ and $\textsc{H}_{k+\ell}$.
The necessary bijection (\textbf{hypothesis} $H_\sharp(A,B,C)$) is
obtained as follows. Pick any rooted tree $T$ with edges colored by
$\mathsf{K}\sqcup \mathsf{L}$. One considers the connected
components in $T$ with respect to the edges with color in
$\mathsf{L}$. Each connected component is a rooted tree. Collapsing
every connected component to a point, one obtains a rooted tree
$\tau$ with edges colored by $\mathsf{K}$. To recover the original
rooted tree $T$, one has to know how to glue back the connected
components into $\tau$. The different ways to do that are exactly
counted by a constant of structure of the global composition map of
the Pre-Lie operad.
\end{proof}
The suspension $\Sigma$ is defined by
\begin{equation}
\Sigma \big{(} \sum_{n \geq 1}a_n \big{)} = \sum_{n \geq 1} (-1)^{n-1} a_n,
\end{equation}
where $a_n$ is homogeneous of degree $n$.
\section{Combinatorics of flows}
\subsection{Definition}
\begin{figure}
\centering
\includegraphics[height=4cm]{flot_ferme.pdf}\includegraphics[height=4cm]{flot_ouvert.pdf}
\caption{Two flows of size $4$, on the same rooted tree with $14$
vertices. Only the left one is closed.}
\label{fig:flow14}
\end{figure}
Let $T$ be a rooted tree. We will call a \textbf{flow} on $T$ of size
$k$ the data of
\begin{itemize}
\item $k$ distinct vertices of $T$ (\textbf{outputs}),
\item vertices of $T$ (\textbf{inputs}), distinct from outputs,
and that can be taken with multiplicities,
\end{itemize}
that has to satisfy the condition that we will introduce next.
Given inputs and $k$ outputs as above, one can define a \textbf{rate}
in $\mathbb{Z}$ on every edge of $T$ as follows.
\begin{itemize}
\item If the vertex $v$ is neither an input nor an output, the sum of
incoming rates in $v$ is equal to the outgoing rate of $v$.
\item If the vertex $v$ is an input with multiplicity $\ell$, the
outgoing rate of $v$ is the sum of incoming rates in $v$ plus
$\ell$.
\item If the vertex $v$ is an output, the
outgoing rate of $v$ is the sum of incoming rates in $v$ minus $1$.
\end{itemize}
The main requirement is that \textit{all rates are in $\mathbb{N}$}.
Note that, by convention, the incoming rate in leaves is $0$, but the
outgoing rate at the root (\textbf{exit rate}) can be an arbitrary
positive integer.
If the exit rate is $0$, the flow is \textbf{closed}.
This definition is illustrated in Figure \ref{fig:flow14}, where
outputs are depicted by red squares {\color{red}\rule{2mm}{2mm}} and
inputs by green circles {\color{green} $\bullet$} with their
multiplicity. The rates, between $0$ and $3$, are drawn with
increasing width.
\begin{lemma}
A closed flow of size $k$ can also be described as
\begin{itemize}
\item $k$ distinct vertices of $T$ (outputs),
\item $k$ vertices of $T$ (inputs), distinct from outputs, and that
can be taken with multiplicities,
\end{itemize}
such that there exists $k$ decreasing paths from one input to an
output that make a one-to-one matching of inputs with outputs.
\end{lemma}
\begin{proof}
Let us see why the data of a closed flow is equivalent to the
existence of $k$ paths with the required properties.
Given $k$ decreasing paths matching inputs with outputs, one can
find the rate of an edge by counting how many paths go through this
edge. This rate function on edges does satisfy all the desired
properties, and defines a closed flow.
Conversely, given a rate function on edges defining a closed flow,
one can find paths, by induction on the size $k$. Let us pick an
output and choose an increasing path of edges of strictly positive
rate, until one reaches an input. This defines a path from the
reached input to the chosen output. Removing this input and this
output and subtracting $1$ to the rate function for every edge of
this path, one find another admissible rate function with $k$
decreased by $1$. Then by induction, one gets $k$ paths with the
expected matching property.
\end{proof}
Let $\mathbb{F}(T)$ be the set of flows on $T$ and $\mathbb{F}(T,k,i)$ be the finite
set of flows of size $k\in \mathbb{N}$ with exit rate $i\in\mathbb{N}$.
\subsection{Properties of closed flows}
Let us give some simple properties of the definition of closed flows.
For every rooted tree $T$, there is exactly one closed flow of size
$0$, which is the empty flow, with no input vertex and no output
vertex, where every edge has rate $0$.
For a rooted tree $T$, closed flows of size $1$ are in bijection with
pairs of distinct comparable vertices of $T$. The number of closed
flows of size $1$ is therefore the sum of the heights of the vertices
of $T$.
\begin{lemma}
\label{lemma_maxi}
For a rooted tree $T$, the maximal size of a flow on $T$ is the
number of non-leaf vertices of $T$. There always exist a closed flow
having this exact size.
\end{lemma}
\begin{proof}
Indeed, any output must be a non-leaf vertex, because it has to be
smaller than an input vertex. Conversely, one can find a closed flow
of this size by putting an output on every non-leaf vertex and
inputs on leaves as follows. Going upwards in the tree, one can
choose at each output where the incoming flow should come from,
until one reaches leaves.
\end{proof}
\subsection{Small flows}
Let us say that a flow $\psi \in \mathbb{F}(T)$ is \textbf{small} if the root
is neither an output nor an input.
If the flow is closed, it is equivalent to require that the rate of
every edge incoming in the root of $T$ is $0$.
Denote by $\mathbb{F}^{s}(T)$ the set of small flows on $T$.
\begin{lemma}
\label{relation_produit_D_E}
If $T=B_+(T_1,\dots,T_k)$, there is a bijection
\begin{equation}
\mathbb{F}^{s}(T) \simeq \prod_{i=1}^{k} \mathbb{F}(T_i),
\end{equation}
where the factors are given by restriction of the flow to subtrees.
\end{lemma}
\subsection{Inductive description of flows}
Let $\xE_{T,t}$ be the generating function of flows on $T$ with respect to size and exit rate:
\begin{equation}
\xE_{T,t} = \sum_{k,i \geq 0} \sum_{\psi \in \mathbb{F}(T,k,i)} b^{k} t^i,
\end{equation}
and let $\xD_{T,t}$ be the similar generating function of small flows on $T$:
\begin{equation}
\xD_{T,t} = \sum_{k,i \geq 0} \sum_{\psi \in \mathbb{F}^{s}(T,k,i)} b^{k} t^i.
\end{equation}
Recall that Lemma \ref{lemma_maxi} says in particular that the size of
a flow on $T$ is bounded by the number of non-leaf vertices of
$T$. Therefore the generating functions $\xE_{T,t}$ and $\xD_{T,t}$ are
polynomials in $b$ with coefficients that are formal power series in
$t$. We will see later that they are in fact polynomials in $b$ with
coefficients that are rational functions in $t$.
By Lemma \ref{lemma_maxi}, the degree of $\xE_T$ as a polynomial in
$b$ is exactly the number of non-leaf vertices of $T$. The constant
term of $\xE_T$ with respect to $b$ is ${1}/{(1-t)}^{\# T}$, because a
flow without outputs is just the choice of how many inputs there are
at every vertex.
For example, when $T$ is the fork $\mathtt{Frk}_{2,2}$, one gets
\begin{equation*}
\xE_{T,t}=\frac{ 1+ 5 b + 3 b^{2} - t (9 b + 8 b^{2} )+ t^2 ( 5 b + 7 b^{2} ) - t^3 (b + 2 b^{2} ) }{(1-t)^4}.
\end{equation*}
We will see later how to compute this by induction.
We will use the general convention that the value at $t=0$ of a series
denoted by a symbol with index $t$ will be denoted by the same symbol
without index $t$. For instance, let $\xE_{T}$ and $\xD_{T}$ be the
value at $t=0$ of $\xE_{T,t}$ and $\xD_{T,t}$.
\begin{lemma}
\label{lemme_d_exp_e}
One has $\xD_{B_+(T_1,\dots,T_k),t}=\prod_{i=1}^k\xE_{T_i,t}$.
\end{lemma}
\begin{proof}
This follows from the bijection of Lemma \ref{relation_produit_D_E},
and its simple behaviour with respect to size and exit rate.
\end{proof}
We will now proceed to give an inductive description of the series
$\xE_{T,t}$ and $\xD_{T,t}$.
Let $T$ be a tree and $v \to u$ be an edge of $T$, with $u$ closer to
the root. Let $T\curvearrowleft_v w$ be the tree obtained by adding a new vertex
$w$ on top of $v$. Let $T\curvearrowleft_u w$ be the tree obtained by adding a new
vertex $w$ on top of $u$. Let $S$ and $T_1,\dots,T_k$ be the trees
obtained from $T$ by removing the edges incoming in $v$. Here $S$ is
the bottom tree (containing the root of $T$) and $T_1,\dots,T_k$ are
the top trees. This is illustrated in figure \ref{fig:pullup}.
\begin{figure}
\centering
\includegraphics[height=4cm]{propriete_serie_DE.pdf}
\caption{From left to right: $T\curvearrowleft_v w$, $T\curvearrowleft_u w$ and $S$ under $T_1,\dots,T_k$.}
\label{fig:pullup}
\end{figure}
\begin{theorem}
\label{main}
With the previous notations, one has the following equalities:
\begin{equation}
\label{pullE}
\xE_{T\curvearrowleft_v w,t} = \xE_{T\curvearrowleft_u w,t} + b\, \xE_{S,t} \prod_{i=1}^{k} \xE_{T_i},
\end{equation}
and
\begin{equation}
\label{pullD}
\xD_{T\curvearrowleft_v w,t} = \xD_{T\curvearrowleft_u w,t} + b \,\xD_{S,t} \prod_{i=1}^{k} \xE_{T_i}.
\end{equation}
\end{theorem}
\begin{proof}
Let us prove the first equation.
Let us consider a flow on the tree $T\curvearrowleft_v w$. Let $\alpha$ be the
rate of $w \rightarrow v$ and $\beta$ be the rate of $v \rightarrow
u$. One can distinguish two cases.
Either $\alpha=\beta+1$, in which case $v$ is an output, and all
other edges incoming in $v$ have rate $0$. This kind of flow can be
described in a bijective way using closed flows on the trees
$T_1,\dots,T_k$ and one flow on the tree $S$. This gives the
rightmost term.
Otherwise $\alpha \leq \beta$. One can then define a flow on $T\curvearrowleft_u
w$ as follows. One moves down the end of the edge $w \to v$ which
becomes an edge $w \to u$ and keep the rate $\alpha$. The rate of
the edge $u-v$ is set to $\beta-\alpha$ and remains positive. This
clearly defines a bijection, and one gets the leftmost term.
Requiring in addition that the root is empty, the same proof
gives the second identity.
\end{proof}
The simplest case of this induction is when $v$ is a leaf in $T$, in
which case the rightmost term has just the factor associated with $S$.
This theorem can be used to compute $\xE_{T,t}$ from smaller cases, by
choosing a leaf $w$ of height at least $2$. This is always possible,
unless $T$ is a corolla.
There is a nice commuting property to this induction. Indeed, one can
use it in several different ways to compute $\xE_{T,t}$, by choosing
different leaves. This happens first for trees with $5$ vertices.
One has the following consequence:
\begin{corollary}
\label{plus_large}
Let $T$ be a rooted tree. Then one has
\begin{equation}
\xE_{B_+(\includegraphics[height=2.5mm]{a0.pdf},T_1,\dots,T_k),t} = \frac{1}{1-t}\left(\xE_{B_+(T_1,\dots,T_k),t} + b\, \prod_{i=1}^k \xE_{T_i}\right).
\end{equation}
\end{corollary}
\begin{proof}
This follows from equation \eqref{pullD}. Indeed, one has
\begin{equation*}
\xD_{B_+(B_+(\includegraphics[height=2.5mm]{a0.pdf},T_1,\dots,T_k)),t}=\xD_{B_+(\includegraphics[height=2.5mm]{a0.pdf},B_+(T_1,\dots,T_k)),t}+b \xD_{B_+(\includegraphics[height=2.5mm]{a0.pdf}),t} \prod_{i=1}^k \xE_{T_i}.
\end{equation*}
One can then use Lemma \ref{lemme_d_exp_e}.
\end{proof}
Corollary \ref{plus_large} can be used to compute the coefficients
$\xE_{\mathtt{Crl}_n,t}$ for corollas, by induction on $n$.
\begin{remark}
When $t=0$, Theorem \ref{main} implies that the coefficients of
$\xE_{T}$ (as a polynomial in $b$) grow when a leaf is pulled up, as
the rightmost term of \eqref{pullE} has positive coefficients.
\end{remark}
\subsection{Properties of $\xE_T$}
\begin{lemma}
\label{rational}
For every rooted tree $T$, the series $\xE_{T,t}$ is a polynomial in
$b$ of degree the number of non-leaf vertices of $T$, with
coefficients that are rational functions in $t$, with poles only at
$t=1$. The common denominator of $\xE_{T,t}$ is $(1-t)^{\# T}$.
\end{lemma}
\begin{proof}
The polynomial behaviour with respect to $b$ follows from the upper
bound on the number of outputs, given by the number of non-leaf
vertices, see Lemma \ref{lemma_maxi}. There always exists at least
one flow with outputs at every non-leaf vertex, for example by
placing sufficiently many inputs in every leaf. Therefore the degree
of the polynomial is the number of non-leaf vertices.
Let us now show that the coefficients of this polynomial in $b$ are
rational functions in $t$ with poles only at $t=1$ and of order at
most the size of $T$. This is true for the rooted tree $\arb{0}$, as
$\xE_{\includegraphics[height=2.5mm]{a0.pdf},t}=1/(1-t)$. By corollary \ref{plus_large}, this is true
for all corollas, by induction. One can then use induction on the
sum of heights of the vertices and on the number of vertices. Let
$T$ be a tree which is not a corolla, and let $w$ be a leaf of
maximal height in $T$. Take $v$ to be the vertex under $w$ and $u$
the vertex under $v$. Then one can apply Theorem \ref{main} to prove
the induction step.
It remains to show that the order of the pole at $1$ of $\xE_{T,t}$
is exactly the size of $T$. This follows from the obvious fact that
the constant term with respect to $b$ is exactly $1/(1-t)^{\# T}$.
\end{proof}
The same kind of properties holds for $\xD_{T,t}$, thanks to Lemma
\ref{lemme_d_exp_e} and the obvious initial conditions
$\xD_{{\mathtt{Crl}_n},t}=1/(1-t)^{n}$.
\subsection{Connected flows}
Let us say that two vertices $u,v$ of $T$ are connected by the flow
$\psi$ on $T$ if every edge of the unique path from $u$ to $v$ does
have a strictly positive rate in $\psi$.
One can then define connected components with respect to the flow
$\psi$, namely sets of vertices connected by the flow $\psi$. Each
connected component with respect to a flow is a rooted tree.
A flow is called \textbf{connected} if it has exactly one connected component.
Let $\mathbb{F}^{c}(T)$ be the set of connected flows on $T$.
\begin{lemma}
\label{connexe_un}
If a rooted tree $T$ admits a closed connected flow, its root has
valency at most $1$.
\end{lemma}
\begin{proof}
The statement holds for the tree with one vertex. One can therefore
assume that the tree $T$ is not the trivial tree $\arb{0}$. By connectedness,
every edge incident to the root contributes at least $1$ to the
total rate entering the root. By closure, the root is then
necessarily an output, and it can only accept a rate of
$1$. Therefore there is exactly one incident edge to the root.
\end{proof}
We will consider now the question of what rooted trees admit a
closed connected flow.
\subsection{Trees with a closed connected flow}
We will now give a description of the rooted trees that admit a closed
connected flow, using a function defined by Jean-Claude Arditti
\cite{arditti,arditti_cori} in relation to rooted trees with
Hamiltonian comparability graphs. One can note that these references
also use some kind of flows on rooted trees. To avoid possible
confusion, we will call this function the valor, which is not the
original terminology.
Let $T$ be a rooted tree. The \textbf{valor} $\mathbb{V}(f)$ of a leaf $f$
is $1$. The valor $\mathbb{V}(v)$ of a vertex $v$ is
\begin{equation}
\max(1,-1+\sum_{s\to v} \mathbb{V}(s)).
\end{equation}
\begin{lemma}
The valor of the root of $T$ is the minimal value of the exit rate
among all connected flows on $T$ with non-zero exit rate.
\end{lemma}
\begin{proof}
By induction on the size of the tree $T$. This is true for the tree
$\arb{0}$, which has minimal non-zero exit rate $1$. Let
$T=B_+(T_1,\dots,T_k)$. Then the minimal exit rate of a connected
flow on $T$ is the sum of the minimal non-zero exit rates of
$T_1,\dots,T_k$, minus $1$ corresponding to an output at the root of
$T$. If this is at least $1$, this is the minimum non-zero exit
rate. If this is zero, the minimum non-zero exit rate is $1$, and
can be obtained by adding $1$ to the rate along the path from the
root to any chosen leaf.
This proves that the minimal non-zero exit rate satisfies the same
recursion as the valor.
\end{proof}
\begin{proposition}
A rooted tree $B_+(T)$ admits a closed connected flow if and only if
the root of $T$ has valor $1$.
\end{proposition}
\begin{proof}
Using Lemma \ref{connexe_un}, the rooted tree $B_+(T)$ admits a
closed connected flow if and only if the rooted tree $T$ admits a
connected flow with exit rate $1$. By the previous lemma, this is
equivalent to say that the valor of the root of $T$ is $1$.
\end{proof}
\subsection{Inductive description of connected flows}
Let us introduce a generating function for connected flows:
\begin{equation}
\xE^c_{T,t} = \sum_{k,i \geq 0} \sum_{\psi \in \mathbb{F}^{c}(T,k,i)} b^{k} t^i.
\end{equation}
By Lemma \ref{connexe_un}, a rooted tree (different from $\arb{0}$)
which admits a closed connected flow can be written $B_+(T)$. Let us
denote by $\xF_T$ the generating series of connected flows on $T$ with
exit rate $1$.
We will now obtain an inductive description of the coefficients $\xF_T$.
Let us consider the situation depicted in figure \ref{fig:pullupF},
with the same notations as for Theorem \ref{main}. The tree $S$ is
obtained from $T$ by removing everything above $v$. The tree $S'$ is
obtained as the subtree of $T\curvearrowleft_v w$ with root $v$.
\begin{figure}
\centering
\includegraphics[height=4cm]{propriete_serie_F.pdf}
\caption{From left to right: $T\curvearrowleft_v w$, $T\curvearrowleft_u w$ and $S$ under $S'$.}
\label{fig:pullupF}
\end{figure}
\begin{theorem}
\label{mainF}
With the previous notations, one has the following equalities:
\begin{equation}
\label{pullF}
\xF_{T\curvearrowleft_v w} = \xF_{T\curvearrowleft_u w} + \xF_{S}\xF_{S'}.
\end{equation}
\end{theorem}
\begin{proof}
Let us consider a connected flow on $T \curvearrowleft_v w$ with exit rate $1$.
Let $\alpha\geq 1$ be the rate of the edge $w \to v$ and $\beta\geq
1$ be the rate of the edge $v \to u$.
If $\beta \geq \alpha+1$, then one can define a connected flow on $T
\curvearrowleft_u w$ with exit rate $1$ as follows. One replaces the edge $w\to
v$ by an edge $w\to u$ with rate $\alpha$, and assign the rate
$\beta-\alpha \geq 1$ to the edge $v \to u$. This is clearly a
bijection, and gives the leftmost term.
Otherwise, one has $\beta \leq \alpha$. One can then define a
connected flow on $S'$ with exit rate $1$ and a connected flow on
$S$ with exit rate $1$, as follows. On the bottom tree $S$, the
vertex $v$ becomes an input with exit rate $\beta$, and all rates
are unchanged. On the top tree $S'$, the vertex $v$ has the same
content as the vertex $v$ of $T\curvearrowleft_v w$, either input or output. One
assigns to the edge $w \to v$ the rate $\alpha-\beta+1\geq 1$. One
can check that the exit rate of this connected flow on $S'$ is
$1$. This construction is clearly a bijection, and one obtains the
rightmost term.
\end{proof}
For example, one can compute using this theorem that
$\xF_{\mathtt{Frk}_{2,2}}$ is $2b(1+b)$.
\begin{corollary}
For every rooted tree $T$ with $n$ vertices, the coefficient of
$b^k$ and the coefficient of $b^{n-1-k}$ in $\xF_T$ are equal.
\end{corollary}
\begin{proof}
This is certainly true for small corollas by inspection, and
$\xF_{\mathtt{Crl}_n}$ vanishes if $n\geq 3$. Then one can proceed by
induction on the size and the total height, using \eqref{pullF}.
\end{proof}
One may wonder whether this unexpected symmetry has a combinatorial
description.
\medskip
It appears that it may be possible to introduce a parameter $t$ in the
inductive definition \eqref{pullF}.
\begin{conjecture}
\label{conjecture_F}
We keep the same notations as for Theorem \ref{mainF}. There exists
rational functions $\xF_{T,t}$, such that
\begin{equation}
\xF_{T\curvearrowleft_v w,t} = \xF_{T\curvearrowleft_u w,t} + (1-t) \xF_{S,t}\xF_{S',t},
\end{equation}
and such that
\begin{equation}
\xF_{\mathtt{Lnr}(n),t}=\xE_{\mathtt{Lnr}(n),t} \quad \text{for}\quad n\geq 1,
\end{equation}
and
\begin{equation}
\xF_{\mathtt{Crl}_n}=b(-t)^{n-2}/(1-t)^{n-1} \quad \text{for}\quad n\geq 2.
\end{equation}
\end{conjecture}
It is easy to prove that this defines uniquely the fractions
$\xF_{T,t}$, if they exist.
For example, one gets that
\begin{equation*}
\xF_{\mathtt{Frk}_{2,2},t} = \frac{b}{(1-t)^2}+ \frac{b(1+2b)}{1-t}.
\end{equation*}
Looking at the first fractions $\xF_{T,t}$, one observes that they do
not have positive coefficients as formal power series in $t$ and $b$,
for example for the rooted tree $B_+(\mathtt{Crl}_2,\arb{0},\arb{0})$. They
can therefore not be given a combinatorial description similar to the
one for $\xF_T$ in terms of connected flows with exit rate $1$.
\subsection{Flows on linear trees and Dyck paths}
\label{dyck}
Let us consider the case of the linear trees. We first show that
closed flows on linear trees are in bijection with very classical
objects, namely Dyck paths.
Recall that a \textbf{Dyck path} of length $2n$ is a plane lattice
path from $(0,0)$ to $(n,n)$ using steps $(0,1)$ (up) and $(1,0)$
(right) and keeping above the diagonal line $y=x$. A Dyck path of
length at least $2$ is called \textbf{indecomposable} if it only
touches the diagonal line at its extremities. Every Dyck path can by
uniquely written as the concatenation of indecomposable Dyck
paths. Every indecomposable Dyck path can be uniquely written
$(0,1)D(1,0)$ where $D$ is a Dyck path. A \textbf{peak} in a Dyck path
is a factor $(0,1)(1,0)$. We say that two letters $(0,1)$ and $(1,0)$
appearing in this order in a Dyck path are \textbf{matched} if the
factor between them is a Dyck path.
\begin{figure}
\centering
\includegraphics[height=3cm]{bijection_dyck.pdf}
\caption{Bijection between Dyck paths and closed flows on linear
trees. In this example, the flow has two connected components.}
\label{fig:bijection}
\end{figure}
\begin{proposition}
There exists a bijection $\rho$ between closed flows on $\mathtt{Lnr}_n$
and Dyck paths of length $2n$ through which
\begin{itemize}
\item connected components correspond to indecomposable factors,
\item outputs correspond to matched pairs of steps that do not form a
peak.
\end{itemize}
\end{proposition}
\begin{proof}
The bijection is defined by induction on $n$. If $n=1$, there is
only one closed flow on $\arb{0}$, which has no output, and only one
Dyck path, which is $(0,1)(1,0)$.
Assume now that $n$ is at least $2$, and the bijection $\rho$ is
defined for smaller $n$.
Any closed flow can be written as a list of connected components,
starting from the component containing the root. Its image by $\rho$
is defined as the concatenation of the images by $\rho$ of the
connected components.
If there are at least $2$ connected components, this defines $\rho$
by induction.
If not, the closed flow is connected. Then the root is an
output. One can remove $1$ to the rate of every edge and remove the
root. This defines a closed flow on the linear tree with one vertex
less. Its image by $\rho$ is taken to be $(0,1)D(1,0)$, where $D$ is
the image by $\rho$ of the smaller flow, defined by induction.
This decomposition is obviously mapped to the similar classical
decomposition of Dyck paths, using sub-Dyck paths and down-moving of
indecomposable paths. The inverse bijection is immediate.
The statement on outputs follows easily by inspection of the
bijection.
\end{proof}
The bijection is illustrated in figure \ref{fig:bijection}.
Let $\operatorname{\mathbf{ca}}_{n,t}$ be the generating series $\xE_{\mathtt{Lnr}_n,t}$ and let
$\operatorname{\mathbf{ca}}_{n}$ be the polynomial $\xE_{\mathtt{Lnr}_n}$.
The first few values of $\operatorname{\mathbf{ca}}_{n,t}$ are
\begin{align*}
\operatorname{\mathbf{ca}}_{1,t}& =\frac{1}{1 - t},\quad
\operatorname{\mathbf{ca}}_{2,t}= \frac{1 + b - t b}{(1-t)^2}, \\
\operatorname{\mathbf{ca}}_{3,t}&= \frac{1 + 3 b + b^{2} - t (4 b + 2 b^{2}) + t^{2} (b + b^{2})}{(1-t)^3}
\end{align*}
From the bijection above, it follows that $\operatorname{\mathbf{ca}}_n$ counts Dyck paths
according to the number of peaks. These polynomials are classical in
combinatorics, and known as the Narayana polynomials, see for example
\cite{kostov_et_al}. We will call $\operatorname{\mathbf{ca}}_{n,t}$ a $t$-Narayana fraction.
Let us introduce ordinary generating series
\begin{equation}
E = \sum_{n \geq 1} \operatorname{\mathbf{ca}}_n x^n \quad \text{and} \quad E_t = \sum_{n \geq 1} \operatorname{\mathbf{ca}}_{n,t} x^n,
\end{equation}
and let $E^c$ be the similar series for closed connected flows on linear
trees.
The analogous series for small flows are just $x(1 + E)$ and $x(1
+ E_t)$, because a small flow on $\mathtt{Lnr}_{n+1}$ can be described by a
flow on $\mathtt{Lnr}_n$.
From the combinatorial decomposition used in the bijection with Dyck
paths, one deduces that
\begin{equation}
\label{usual_eq}
E=E^c/(1-E^c)\quad\text{and}\quad E^c=x(1+b E).
\end{equation}
By decomposing a flow according to whether the root is an output or
not, one obtains the equation
\begin{equation*}
E_t=x/(1-t)(1+E_t)+bx/t(E_t-E).
\end{equation*}
This is a special case of the global equation for flows
\eqref{master_eq_E}, that we will prove later.
It follows from all this that $E$ and $E^c$ are algebraic over
$\mathbb{Q}(x)$ and that $E_t$ is algebraic over $\mathbb{Q}(x,t)$.
\subsection{Conjectural formula for closed flows on forks}
Recall from \S \ref{section_pl} that $\mathtt{Frk}_{i,n-i}$ is the fork
with $n$ vertices, with stem of size $i$.
\begin{conjecture}
The number of closed flows of size $k$ on the fork $\mathtt{Frk}_{i,n-i}$ is given by
\begin{equation}
\# \mathbb{F}(\mathtt{Frk}_{i,n-i},k,0)=\binom{i}{k}\binom{n}{k}-\binom{i+1}{k+1}\binom{n-1}{k-1}.
\end{equation}
\end{conjecture}
For $i=n-1$ or $i=n$, corresponding to linear trees, this formula
gives the Narayana numbers, which is the correct result for the linear
trees (see section \ref{dyck}). One can easily check that this also
gives the correct answer for $i=1$, namely for corollas.
\subsection{Zeroes of flow polynomials}
After inspection of some examples, one is tempted to ask the following
question.
\begin{question}
Let $T$ be a rooted tree. Are the zeroes of $\xF_T$ real and
negative ? Are the zeroes of $\xE_T$ real and negative ?
\end{question}
It is known, for the Narayana polynomials, that all roots are real,
simple and negative, see for example \cite{kostov_et_al}. Therefore
the question has a positive answer for linear trees. One can also
check easily that this is true for corollas.
\section{Series of flows}
\subsection{Global equations for flows}
Let us introduce now two series
\begin{equation}
\xE_t=\sum_T \xE_{T,t} \frac{T}{\operatorname{aut}(T)} \quad\text{and}\quad \xD_t=\sum_T \xD_{T,t} \frac{T}{\operatorname{aut}(T)},
\end{equation}
in the group $\mathsf{G}_{\operatorname{PreLie}}$ associated with the Pre-Lie operad.
Let $\xE$ (resp. $\xD$) be the value at $t=0$ of $\xE_t$
(resp. $\xD_t$).
\begin{theorem}
\label{th_d_corolle_e}
The following identity holds:
\begin{equation}
\label{eq_d_corolle_e}
\xD_t= \textsc{Crls} \diamond (\arb{0},\xE_t).
\end{equation}
\end{theorem}
\begin{proof}
This is essentially a restatement of Lemma
\ref{relation_produit_D_E}, using the notation defined in
\eqref{defi_crls} and the results of the appendix \ref{appA}.
Namely, one applies Prop. \ref{diam_prop} of the appendix, with $A$
the species of corollas, $B$ the species made only of the rooted
tree on one vertex, $C$ the species of flows on rooted trees and $D$
the species of small flows on rooted trees.
\end{proof}
\begin{theorem}
\label{master_th_E}
One has
\begin{equation}
\label{master_eq_E}
\xE_t = \frac{1}{1-t}\xD_t + \frac{b}{t} \left( \xD_t - \xD \right).
\end{equation}
\end{theorem}
\begin{proof}
Consider a rooted tree $T=B_+(T_1,\dots,T_k)$ endowed with a flow.
Either the root is an input vertex with multiplicity $\ell$ for some
$\ell \geq 0$. This can be described using a small flow and the
integer $\ell$. One obtains
\begin{equation*}
\frac{1}{1-t} \xD_t.
\end{equation*}
The other possibility is that the root is an output vertex. Removing
the output, one gets a small flow with the condition that the exit
rate is not zero. This gives the term
\begin{equation*}
\frac{b}{t} \left(\xD_t - \xD\right).
\end{equation*}
\end{proof}
\subsection{Global equations for connected flows}
Let $\xE^c_t$ be the global series of connected flows:
\begin{equation}
\xE^c_t=\sum_T \xE^c_{T,t} \frac{T}{\operatorname{aut}(T)},
\end{equation}
and let $\xE^c$ be its value at $t=0$.
\begin{theorem}
The series $\xE^c_t$ satisfies the following equation
\begin{equation}
\label{global_eq_connected}
\xE^{c}_t = \frac{1}{1-t} \textsc{Crls} \diamond (\arb{0},\xE^c_t-\xE^c) + \frac{b}{t} \left( \textsc{Crls} \diamond (\arb{0}, \xE_t^c-\xE^c) - \arb{0} \right).
\end{equation}
\end{theorem}
\begin{proof}
This is similar to the proof of Theorems \ref{th_d_corolle_e}
and \ref{master_th_E}. One has to distinguish according to the status
of the root.
If the root is an input (possibly empty), the restriction to every
subtree is an arbitrary connected flow with non-zero outgoing rate
at the root. We obtain the first term of the right-hand side.
If the root is an output, there must be at least one subtree, and
the restriction to every subtree is an arbitrary connected flow with
non-zero outgoing rate at the root. This gives the second term of the
right-hand-side.
\end{proof}
The series $\xE_t$ of flows can be recovered from the series $\xE^c_t$
of connected flows.
\begin{theorem}
There holds
\begin{equation}
\label{rela_ect_e}
\xE_t = \left( \sum_T \frac{T}{\operatorname{aut}(T)} \right) \diamond( \xE^c_t, \xE^c).
\end{equation}
\end{theorem}
\begin{proof}
This follows from Prop. \ref{diam_prop} applied to the following four
species: $A$ is the species of rooted trees, $B$ the species of
connected flows, $C$ the species of closed connected flows and $D$
the species of flows.
The necessary bijection (\textbf{hypothesis} $H_\natural(A,B,C,D)$)
is rather clear. Indeed, given any flow, one can define connected
flows on its connected components, closed if not containing the
root. One can also make a rooted tree $\tau$ with vertices the
connected components. To be able to recover the flow, one has to
know how to glue back components into the tree $\tau$. This is given
by a constant of structure of the global composition of the Pre-Lie
operad.
\end{proof}
In words, this theorem says that the series $\xE_t$ of flows is
obtained from the series of all trees, by insertion of $\xE^c_t$ in
the root and insertion of $\xE^c$ in all other vertices.
When $t=0$, this reduces to the factorisation of series
\begin{equation}
\label{flow_is_tree_of_connected}
\xE = \left( \sum_T \frac{T}{\operatorname{aut}(T)} \right) \circ \xE^c,
\end{equation}
in the group $\mathsf{G}_{\operatorname{PreLie}}$, which means that a closed flow is made
by gluing closed connected flows along a rooted tree.
\medskip
Because rooted trees that support closed connected flows have
root-valency at most $1$ by Lemma \ref{connexe_un}, one can write
\begin{equation}
\label{from_EC_to_F}
\xE^c= \arb{0}+ b\, \arb{0} \curvearrowleft \xF,
\end{equation}
for some series $\xF$. We will use this series later on.
\subsection{Quotient series $\xE \circ \xD^{-1}$}
A \textbf{saturated flow} is a closed connected flow where every
non-leaf vertex is an output.
Let $\xE_T^s$ be the generating series for saturated flows on
$T$. Note that this is a monomial in the variable $b$, of degree the
number of non-leaf vertices of $T$.
\begin{lemma}
\label{same_support}
Let $T$ be a rooted tree that admits a closed connected flow. Then
$T$ admits a saturated flow.
\end{lemma}
\begin{proof}
Pick a closed connected flow on $T$. The proof is by induction on
the number of non-leaf vertices which are not outputs. If the chosen
flow is saturated, there is nothing to do. Otherwise, let $v$ be a
non-leaf vertex which is not an output.
If $v$ is not an input, one can put an output in $v$, choose a path
from $v$ to some leaf $w$ of the subtree at $v$, and add $1$ to the
rate on every edge of this path and $1$ input on $w$.
If $v$ is an input, one can first move this input to a leaf, by
choosing a path from $v$ to a leaf $w$ of the subtree at $v$, and
adding $1$ to the rate on every edge of this path. Then one gets
back to the previous case.
\end{proof}
Therefore rooted trees that admit closed connected flows are exactly
the same as rooted trees that admit saturated flows.
Let now $\xY$ be the quotient series $\xE \circ \xD^{-1}$ in the group
$\mathsf{G}_{\operatorname{PreLie}}$. One observes a surprising property.
\begin{conjecture}
The coefficient $\xY_T$ of a rooted tree $T$ in $\xY$ is the monomial
\begin{equation}
(-1)^{L(T)-1} \xE_T^s,
\end{equation}
where $L(T)$ is the number of leaves of $T$.
\end{conjecture}
If this is true, then by Lemma \ref{same_support}, the support of
$\xY$ is the same as the support of $\xE^c$, and one can write
\begin{equation}
\label{Z_here}
\xY= \arb{0}+ b\, \arb{0} \curvearrowleft \xZ,
\end{equation}
for some series $\xZ$. We will consider this series again later.
\section{Planar binary trees, dendriform operad and $\mathbf{Sym}$}
\subsection{Notations for planar binary trees}
A \textbf{planar binary tree} on $n$ vertices is either the tree $1=|$ with no
inner vertex or a pair of two planar binary trees. Planar binary trees
will be drawn with their root at the bottom and leaves at the top,
aligned on a horizontal line. Examples are depicted in figure
\ref{fig:expl_bt}.
There is a natural involution on the set of planar binary trees, given
by left-right reversal, as shown in figure \ref{fig:expl_bt}.
The \textbf{canopy} of a planar binary tree is a sequence of letters
$\boxed{\color{red}+}$ and $\boxed{\color{blue}-}$ of length $n-1$. There is a letter for each leaf but
the leftmost and rightmost one. The letter is $\boxed{\color{blue}-}$ is the leaf is the
left son of its parent vertex, and $\boxed{\color{red}+}$ is the leaf is the right son
of its parent vertex.
For example, the canopy of the planar binary tree at the left of of figure
\ref{fig:expl_bt} is $\boxed{\color{blue}-} \boxed{\color{red}+} \boxed{\color{red}+} \boxed{\color{blue}-} \boxed{\color{red}+} $.
We will also use the following variants: the full canopy is obtained
from the canopy by adding $\boxed{\color{blue}-}$ at the beginning and $\boxed{\color{red}+}$ at the end,
the left-completed canopy by adding $\boxed{\color{blue}-}$ at the beginning, and the
right-completed canopy by adding $\boxed{\color{red}+}$ at the end.
\begin{figure}
\centering
\includegraphics[height=2cm]{expl_arbre.pdf}\hspace{1cm}\includegraphics[height=2cm]{reverse_arbre.pdf}
\caption{A planar binary tree with $6$ inner vertices, and its image by reversal.}
\label{fig:expl_bt}
\end{figure}
\subsection{The Dendriform operad}
The Dendriform operad, introduced by Loday, is a non-symmetric operad
with a basis indexed by planar binary trees. The free dendriform
algebra is just the direct sum of all components of the Dendriform
operad. We refer the reader to \cite{loday_lnm} for more information
on the dendriform algebras.
On the free dendriform algebra $\operatorname{Dend}$, there are two dendriform
products $\prec$ and $\succ$, that satisfy the $3$ dendriform
axioms. In particular, their sum defines an associative product
\begin{equation}
x * y = x\succ y + x \prec y,
\end{equation}
which is the product used in the Hopf algebra structure of $\operatorname{Dend}$.
We will use the following notation:
\begin{equation}
\label{def_vee}
x \vee_y z = x \succ y \prec z.
\end{equation}
By one of the dendriform axioms, no parentheses are needed in this
expression. When $y$ is the planar binary tree $\arb{1}$, the
operation $ x \vee z$ can be described as the gluing of $x$ and $z$ on
a common vertex.
From the dendriform axioms, one can deduce the following relations :
\begin{equation}
\label{vee_star}
(x \vee y) \prec z = x \vee (y*z) \quad \text{and} \quad x \succ (y \vee z) = (x*y) \vee z.
\end{equation}
Oen can extend (in a unique way) the notation $x \vee_y z$ to the
cases where $x$ or $z$ are the unit tree $1$, with the same
properties.
Let $\phi$ be the operad morphism from the $\operatorname{PreLie}$ operad to the
$\operatorname{Dend}$ operad defined by its value on the labelled generator:
\begin{equation}
\label{defi_phi}
\phi(x \curvearrowleft y) =y \succ x - x \prec y.
\end{equation}
Therefore, the map $\phi$ sends $\arb{10}$ to $\arb{12}-\arb{21}$. One
can show that the morphism $\phi$ is injective by using that it
factorises through the Brace operad.
From now on, the expression ``dendriform image'' will mean the image
by $\phi$.
We will work in the group $\mathsf{G}_{\operatorname{Dend}}$ associated with the dendriform
operad. This is an open subset in the free dendriform algebra on one
generator $\operatorname{Dend}$.
\begin{lemma}
\label{racine_map}
Let $T$ be a labeled rooted tree, with $i$ the label of the root.
The dendriform image of $T$ is a linear combination of labelled planar binary
trees whose root is labeled by $i$.
\end{lemma}
\begin{proof}
One has to show that $\phi$ is a morphism of rooted-operads, in the
language of the appendix \ref{appA}. This is clear on the generators
by \eqref{defi_phi}, hence one can apply Lemma \ref{check_on_gen}.
\end{proof}
\begin{lemma}
\label{diamant_vee}
Let $x,y,z,t$ in $\operatorname{Dend}$. Then
\begin{equation}
(x \vee y) \diamond (z,t) = (x \circ t) \vee_z (y \circ t).
\end{equation}
\end{lemma}
\begin{proof}
This is an easy consequence of the definition \eqref{def_vee} of
$\vee$ and of the general definition of the operations $\diamond$
and $\circ$ in appendix \ref{appA}.
\end{proof}
\begin{lemma}
\label{double_inversion}
Let $x,y,z,t, u$ in $\operatorname{Dend}$. Let $ v=(y \vee_z t).$ Then
\begin{equation*}
x \vee_{v} u = (x * y) \vee_z (t * u).
\end{equation*}
\end{lemma}
\begin{proof}
This is a simple computation in the dendriform operad, starting from
the definition \eqref{def_vee}.
\end{proof}
The suspension $\Sigma$ is defined by
\begin{equation}
\Sigma \big{(} \sum_{n \geq 1}a_n \big{)} = \sum_{n \geq 1} (-1)^{n-1} a_n,
\end{equation}
where $a_n$ is homogeneous of degree $n$.
We will also use the \textbf{bar involution}, which is the composition of
suspension and reversal, that are two commuting involutions.
\subsection{The subalgebra $\mathbf{Sym}$ of $\operatorname{Dend}$}
Let $\mathbf{Sym}$ be the algebra of non-commutative symmetric
functions. This is the free associative algebra generated by one
generator in every positive degree. We will use the basis of ribbon
Schur functions, indexed by compositions of $n$ in degree $n$. For
more information, the reader may consult
\cite{ncsf, thibon_lectures, ncsf6}.
Compositions of $n$ will be identified with strings of $n-1$ symbols
$\boxed{\color{red}+}$ and $\boxed{\color{blue}-}$, by the convention that a $\boxed{\color{blue}-}$ symbol means ``cut
here'' and a $\boxed{\color{red}+}$ symbols means ``do not cut here''. For example,
\begin{equation}
1|4|1|2 \longleftrightarrow \boxed{\color{blue}-} \boxed{\color{red}+} \boxed{\color{red}+} \boxed{\color{red}+} \boxed{\color{blue}-} \boxed{\color{blue}-} \boxed{\color{red}+}.
\end{equation}
The product in the basis of ribbon Schur functions is given by the rule
\begin{equation}
\epsilon * \delta = \epsilon\boxed{\color{red}+}\delta + \epsilon\boxed{\color{blue}-}\delta.
\end{equation}
The inclusion from $\mathbf{Sym}$ to $\operatorname{Dend}$ is defined on the basis of Schur
function by sending a sequence of elements of $\{\boxed{\color{red}+},\boxed{\color{blue}-}\}$ to the sum
of all planar binary having this sequence as canopy. This is a
morphism of algebras.
One will need the following lemma.
\begin{lemma}
\label{coeff_id}
Let $\theta$ be a Lie idempotent in the descent algebra of $\mathfrak{S}_n$,
seen as an element of $\mathbf{Sym}$. Then the coefficient of the ribbon
Schur function with index $\boxed{\color{red}+}^{n-1}$ in $\theta$ is $1/n$.
\end{lemma}
\begin{proof}
By \cite[Prop. 2.4]{schocker}, the coefficient of $\operatorname{Id}$ in the
expansion of any Lie idempotent in the usual basis of the symmetric
group ring $\mathbb{Q}[\mathfrak{S}_n]$ is $1/n$. By the inclusion of $\mathbf{Sym}$ in
$\mathbf{FQSym}$, a ribbon Schur function is mapped to the sum of all
permutations with a fixed descent set, depending on its index. For
the ribbon Schur function with index $\boxed{\color{red}+}^{n-1}$, the image is just
the permutation $\operatorname{Id}$. For a Lie idempotent in the descent algebra,
the coefficient of the ribbon Schur function with index $\boxed{\color{red}+}^{n-1}$
is therefore $1/n$.
\end{proof}
\subsection{Known series in the dendriform group}
Let us recall some elements of $\mathsf{G}_{\operatorname{Dend}}$ and their properties.
Let $\sR$ be the positive sum of all right combs,
\begin{equation*}
\sR = \includegraphics[height=3mm]{a1.pdf} + \arb{21}+ \arb{321} + \dots
\end{equation*}
This is the unique solution of the equation
\begin{equation}
\label{carac_R}
\sR = \includegraphics[height=3mm]{a1.pdf} + \includegraphics[height=3mm]{a1.pdf} \prec \sR.
\end{equation}
Let $\sL$ be the alternative sum of all left combs,
\begin{equation*}
\sL = - \includegraphics[height=3mm]{a1.pdf} + \arb{12} - \arb{123} + \dots
\end{equation*}
This is the unique solution of the equation
\begin{equation}
\label{carac_L}
\sL=- \includegraphics[height=3mm]{a1.pdf} - \sL \succ \includegraphics[height=3mm]{a1.pdf}.
\end{equation}
The bar involution maps $\sR$ to $-\sL$.
\begin{lemma}
\label{L_inverse_R}
The following inversion relation holds:
\begin{equation}
(1+\sL) * (1+ \sR)=1.
\end{equation}
\end{lemma}
\begin{proof}
Both $1+\sL$ and $1+\sR$ belong to the subalgebra $\mathbf{Sym}$. Indeed
$\sL$ and $\sR$ are the same as
\begin{equation*}
\sum_{k \geq 0} (-1)^{k+1} \boxed{\color{red}+}^k \quad \text{and}\quad \sum_{k \geq 0} \boxed{\color{blue}-}^k.
\end{equation*}
With the product rule $A * B = A \boxed{\color{red}+} B+A \boxed{\color{blue}-} B$ of $\mathbf{Sym}$, this
identity is easily proved there. Another proof can be found in
\cite[Prop. 5.1]{qidempotent}.
\end{proof}
\begin{proposition}
\label{image_des_corolles}
The dendriform image of
\begin{equation}
\textsc{Crls} = \sum_{n\geq 0} \frac{\mathtt{Crl}_n}{n!} \quad \text{is} \quad (1+\sR) \vee (1+\sL).
\end{equation}
\end{proposition}
\begin{proof}
This was proved in \cite{ronco}.
\end{proof}
We will now recall and extend some results of \cite{qidempotent}.
Beware that this article uses slightly different notations.
Recall from section \ref{section_pl} that $\mathtt{Lnr}_\ell$ is the linear
rooted tree with $\ell$ vertices.
\begin{lemma}
\label{lemme_auxi_1}
The dendriform image of $\sum_T \frac{T}{\operatorname{aut}(T)}$ is given by
\begin{equation}
(1-\Sigma \sL) * \phi\left(\sum_{\ell \geq 1} \mathtt{Lnr}_\ell \right) * (1-\Sigma \sR).
\end{equation}
\end{lemma}
\begin{proof}
This follows from \cite[Prop. 5.6]{qidempotent} (at $q=\infty$) and \cite[Prop. 6.4]{qidempotent}. One also uses Lemma \ref{L_inverse_R}.
\end{proof}
\begin{lemma}
\label{lemme_auxi_2}
One has
\begin{equation}
(1-\Sigma \sL) * \phi\left(\sum_{\ell \geq 1} \mathtt{Lnr}_\ell \right)=
\sum_{n \geq 1} n L_n,
\end{equation}
where $L_n$ is the left comb with $n$ vertices.
\end{lemma}
\begin{proof}
This is essentially \cite[Prop. 5.3]{qidempotent}.
\end{proof}
\begin{lemma}
\label{lemme_auxi_3}
There holds
\begin{equation}
\left(\sum_{n \geq 1} n L_n \right)* (1-\Sigma \sR) = (1-\Sigma\sL)\vee (1-\Sigma\sR).
\end{equation}
\end{lemma}
\begin{proof}
This is a simple computation in the dendriform algebra, or even in
the sub-algebra $\mathbf{Sym}$, with easy cancellations.
\end{proof}
\begin{proposition}
\label{image_de_tout}
The dendriform image of
\begin{equation}
\textsc{H}_1 = \sum_T \frac{T}{\operatorname{aut}(T)} \quad \text{is} \quad (1- \Sigma \sL) \vee (1- \Sigma \sR).
\end{equation}
\end{proposition}
\begin{proof}
This follows from the Lemmas \ref{lemme_auxi_1},
\ref{lemme_auxi_2} and \ref{lemme_auxi_3}.
\end{proof}
\begin{lemma}
\label{inverse_cool}
In $\mathsf{G}_{\operatorname{Dend}}$, the inverse of $(1- \Sigma \sL) \vee (1- \Sigma
\sR)$ is $(1+ \sL) \vee (1+ \sR)$.
\end{lemma}
\begin{proof}
By lemma \ref{inversion_somme_tous}, the inverse of $\textsc{H}_1$ is
$\textsc{H}_{-1}$, which is the suspension of $\textsc{H}_1$.
The result then follows from proposition \ref{image_de_tout}, by
functoriality of the group construction.
\end{proof}
\begin{lemma}
\label{nice_compo}
One has
\begin{equation}
\Sigma \sR \circ ((1+\sL) \vee (1+\sR)) = -\sL.
\end{equation}
\end{lemma}
\begin{proof}
By equation \eqref{carac_L}, it is enough to prove that
\begin{equation*}
\Sigma\sR \circ \left((1+\sL)\vee (1+\sR)\right) = \includegraphics[height=3mm]{a1.pdf} - (\Sigma\sR \circ \left((1+\sL)\vee (1+\sR)\right)) \succ \includegraphics[height=3mm]{a1.pdf}.
\end{equation*}
By composition with the inverse of $(1+\sL)\vee (1+\sR)$ given by
lemma \ref{inverse_cool}, this is equivalent to
\begin{equation*}
\Sigma\sR = (1-\Sigma \sL)\vee (1-\Sigma \sR) - \Sigma\sR \succ \left((1-\Sigma \sL)\vee (1-\Sigma \sR) \right).
\end{equation*}
By suspension, this is the same as
\begin{equation*}
\sR = (1+ \sL)\vee (1+ \sR) + \sR \succ \left((1+ \sL)\vee (1+ \sR) \right).
\end{equation*}
By Lemma \ref{L_inverse_R} and \eqref{vee_star}, this is equivalent to
\begin{equation*}
\sR = 1 \vee (1 + \sR),
\end{equation*}
which is just the equation \eqref{carac_R}.
\end{proof}
\section{Series in $\mathbf{Sym}$}
Let $\dP_t$ and $\dN_t$ be series in variables $\boxed{\color{red}+},\boxed{\color{blue}-}$ defined by
\begin{equation}
\dP_t=\sum_{k \geq 1} \operatorname{\mathbf{ca}}_{k,t} \boxed{\color{red}+}^k\quad \text{and}\quad \dN_t=\sum_{k \geq 1} (-1)^k \operatorname{\mathbf{ca}}_{k,t} \boxed{\color{blue}-}^k,
\end{equation}
where $ca_{k,t}$ are the $t$-Narayana fractions defined in \S \ref{dyck}, and let $\dP$ (resp. $\dN$) be $\dP_{t=0}$ (resp. $\dN_{t=0}$).
These series can be considered as ordinary generating series for flows
on linear trees, see section \ref{dyck}.
\begin{lemma}
\label{equation_pt_nt}
One has
\begin{equation}
\label{defi_pt}
\dP_t = \frac{1}{1-t} (\emptyset+\dP_t) \boxed{\color{red}+} +\frac{b}{t} (\dP_t - \dP) \boxed{\color{red}+}
\end{equation}
and
\begin{equation}
\dN_t = \frac{-1}{1-t} \boxed{\color{blue}-} (\emptyset+\dN_t) -\frac{b}{t} \boxed{\color{blue}-} (\dN_t - \dN).
\end{equation}
\end{lemma}
\begin{proof}
This follows from the fact that the coefficients $\operatorname{\mathbf{ca}}_{k,t}$ count
flows on linear rooted trees. One has to decompose according to
whether the root is an output or not, as already done in the proof
of Theorem \ref{main}.
\end{proof}
One will also need connected variants of $\dP$ and $\dN$, defined by
\begin{equation}
\label{defi_pc_nc}
\dP^c= (\emptyset+b\dP)\boxed{\color{red}+} \quad \text{and}\quad \dN^c= -\boxed{\color{blue}-}(\emptyset+b\dN).
\end{equation}
By \eqref{usual_eq}, these series are generating series for connected
closed flows on linear trees. Let us now consider the similar series
$\dP^c_t$ for arbitrary connected flows on linear trees.
Every flow on a linear tree can be decomposed as a list of connected
components, all but one are closed. One therefore has
\begin{equation}
\label{decoupe_flot_lineaire}
\dP_t=\dP^c_{t}+\dP_t \dP^c.
\end{equation}
A connected flow on a linear tree is either closed, or one can remove one layer of rate on every edge, and obtain any linear flow. This implies that
\begin{equation}
\label{flot_lineaire_connexe}
\dP^c_t = \dP^c + t \dP_t.
\end{equation}
\begin{proposition}
The series $\dP^c$ and $\dN^c$ satisfy
\begin{equation}
\label{prop_pc_nc}
(1-t) \dP_t=\dP^c+ \dP^c \dP_t \quad \text{and} \quad (1-t) \dN_t=\dN^c+ \dN_t \dN^c.
\end{equation}
\end{proposition}
\begin{proof}
It is enough to consider the case of $\dP$, by symmetry under the
exchange of $\boxed{\color{red}+}$ and $-\boxed{\color{blue}-}$. The equation follows directly from
\eqref{decoupe_flot_lineaire} and \eqref{flot_lineaire_connexe}.
\end{proof}
Let us now define three series involving both variables $\boxed{\color{red}+}$ and
$\boxed{\color{blue}-}$.
The series $\dT$ is defined by
\begin{equation}
\label{defi_dt}
\dT=\sum_{k \geq 0} b^k (\dP \dN)^k,
\end{equation}
and does not depend on the variable $t$.
The series $\dU_t$ is then defined by
\begin{equation}
\label{def_u}
\dU_t=(\emptyset+b \dN)\dT\dP_t.
\end{equation}
The first few terms of $\dU_t$ are
\begin{equation*}
\operatorname{\mathbf{ca}}_{1,t} \boxed{\color{red}+} + \operatorname{\mathbf{ca}}_{2,t} \boxed{\color{red}+}\pp -b\operatorname{\mathbf{ca}}_{1,t}\boxed{\color{blue}-}\boxed{\color{red}+} + \dots
\end{equation*}
The series $\dV_t$ is similarly defined by
\begin{equation}
\label{def_v}
\dV_t=\dN_t \dT (\emptyset+b \dP) .
\end{equation}
Its first few terms are
\begin{equation*}
- \operatorname{\mathbf{ca}}_{1,t} \boxed{\color{blue}-} + \operatorname{\mathbf{ca}}_{2,t} \boxed{\color{blue}-}\mm - b\operatorname{\mathbf{ca}}_{1,t}\boxed{\color{blue}-}\boxed{\color{red}+} + \dots
\end{equation*}
Let $\dU$ (resp. $\dV$) be $\dU_{t=0}$ (resp. $\dV_{t=0}$).
\begin{lemma}
One has
\begin{equation}
\label{droite_u_plus}
\dU_t = \frac{1}{1-t} ((\emptyset+b\dN)\dT+\dU_t) \boxed{\color{red}+} +\frac{b}{t} (\dU_t - \dU) \boxed{\color{red}+}
\end{equation}
and
\begin{equation}
\dV_t = \frac{-1}{1-t} \boxed{\color{blue}-} (\dT(\emptyset+b\dP)+\dV_t) -\frac{b}{t} \boxed{\color{blue}-} (\dV_t - \dV).
\end{equation}
\end{lemma}
\begin{proof}
This follows from Lemma \ref{equation_pt_nt} and the definition of
$\dU_t$ and $\dV_t$.
\end{proof}
One will need an involution (called the \textbf{bar involution}) on
the space of non-commutative formal power series in two variables
$\boxed{\color{blue}-},\boxed{\color{red}+}$. It is the unique anti-morphism of algebra defined on
generators by $\overline{\boxed{\color{red}+}}=-\boxed{\color{blue}-}$ and $\overline{\boxed{\color{blue}-}}=-\boxed{\color{red}+}$.
Under the bar involution, $\dN_t$ and $\dP_t$ are exchanged, $\dT$ is
fixed and $\dU_t$ and $\dV_t$ are exchanged.
\section{Series in the dendriform group}
Let $\sU_t$ be the
unique dendriform series whose right-completed canopy is given by $\dU_t$ :
\begin{equation}
\sU_t= \operatorname{\mathbf{ca}}_{1,t} \includegraphics[height=3mm]{a1.pdf} + \operatorname{\mathbf{ca}}_{2,t} \arb{12} -b\operatorname{\mathbf{ca}}_{1,t}\arb{21} + \dots
\end{equation}
and let $\sV_t$ be the unique dendriform series whose left-completed canopy is given
by $\dV_t$ :
\begin{equation}
\sV_t= - \operatorname{\mathbf{ca}}_{1,t} \includegraphics[height=3mm]{a1.pdf} + \operatorname{\mathbf{ca}}_{2,t} \arb{21} - b\operatorname{\mathbf{ca}}_{1,t}\arb{12} + \dots.
\end{equation}
It follows from this definition that, under the bar involution on
dendriform series, one has $\overline{\sU}_t=-\sV_t$.
\begin{lemma}
\label{lemmeVU}
Let $u,v$ be two indeterminates. One has
\begin{equation}
(1+\sV_v) * (1+ \sU_u) = 1 + (v-u) \dN_v T \dP_u,
\end{equation}
where $\dN_v T \dP_u$ has to be interpreted as the sum over planar
binary trees with the given full canopy.
\end{lemma}
\begin{proof}
This is in fact a computation inside series in $\boxed{\color{red}+}$ and $\boxed{\color{blue}-}$, by
the correspondence between a monomial in $\boxed{\color{red}+}$ and $\boxed{\color{blue}-}$ and the sum
of all planar binary trees having this monomial as their full
canopy.
Let us compute $(1+\sV_v) * (1+ \sU_u) -1$. One finds
\begin{multline*}
\dN_v\dT(\emptyset+b\dP)\boxed{\color{red}+} + \boxed{\color{blue}-} (\emptyset + b\dN) \dT \dP_u + \dN_v\dT(\emptyset+b\dP)\boxed{\color{red}+}(\emptyset + b\dN) \dT \dP_u \\+\dN_v\dT(\emptyset+b\dP) \boxed{\color{blue}-} (\emptyset + b\dN) \dT \dP_u .
\end{multline*}
Using the definition \eqref{defi_pc_nc} of $\dN^c$ and $\dP^c$, one gets
\begin{equation*}
\dN_v\dT\dP^c -\dN^c \dT \dP_u + \dN_v\dT\dP ^c (\emptyset + b\dN) \dT \dP_u -\dN_v\dT(\emptyset+b\dP)\dN^c \dT \dP_u .
\end{equation*}
Expanding the products, one obtains
\begin{multline*}
\dN_v\dT\dP^c -\dN^c \dT \dP_u + \dN_v\dT\dP ^c \dT \dP_u+ b \dN_v\dT\dP ^c \dN \dT \dP_u \\ -\dN_v\dT\dN^c \dT \dP_u-b \dN_v\dT \dP\dN^c \dT \dP_u .
\end{multline*}
One can then use the fact that $\dT=\emptyset+b \dT \dP \dN=\emptyset+b \dP \dN \dT$ to split the third and fifth terms, getting
\begin{multline*}
\dN_v\dT\dP^c -\dN^c \dT \dP_u + \dN_v\dT \dP ^c \dP_u+ b \dN_v\dT \dP ^c \dP \dN \dT \dP_u+ b \dN_v\dT\dP ^c \dN\dT \dP_u \\-\dN_v\dN^c \dT \dP_u -b \dN_v\dT \dP \dN \dN^c \dT \dP_u-b \dN_v\dT \dP\dN^c \dT \dP_u .
\end{multline*}
Now gathering terms by pairs and using four times the equation
\eqref{prop_pc_nc}, one gets, after some
cancellations,
\begin{equation*}
(1-u) \dN_v \dT \dP_u - (1-v) \dN_v \dT \dP_u,
\end{equation*}
which is the expected result.
\end{proof}
\subsection{Flows in the dendriform group}
Let us now consider two series $\sD_t$ and $\sE_t$. Our aim will be to
show that they are the respective dendriform images of the series
$\xD_t$ and $\xE_t$.
The series $\sD_t$ is defined by
\begin{equation}
\label{defi_dt}
\sD_t= (1+\sU_t)\vee(1+\sV_t),
\end{equation}
and its first few terms are
\begin{equation*}
\includegraphics[height=3mm]{a1.pdf} + \operatorname{\mathbf{ca}}_{1,t} \arb{12} - \operatorname{\mathbf{ca}}_{1,t} \arb{21} +\dots
\end{equation*}
The series $\sD$ is the value of $\sD_t$ at $t=0$.
The series $\sE_t$ is then defined by
\begin{equation}
\label{defi_et}
\sE_t =\frac{1}{1-t} \sD_t + \frac{b}{t} (\sD_t-\sD).
\end{equation}
Its first few terms are
\begin{equation*}
\operatorname{\mathbf{ca}}_{1,t} \includegraphics[height=3mm]{a1.pdf} + \operatorname{\mathbf{ca}}_{2,t} \arb{12} - \operatorname{\mathbf{ca}}_{2,t} \arb{21} + \dots
\end{equation*}
The series $\sE$ is the value of $\sE_t$ at $t=0$.
From these definitions, it results that both $\sE_t$ and $\sD_t$ are
fixed under the bar involution of $\operatorname{Dend}$.
\begin{proposition}
\label{uv_from_et}
One has the following relations
\begin{equation}
\sU_t = \sR \circ \sE_t \quad \text{and}\quad \sV_t = \sL \circ \sE_t.
\end{equation}
\end{proposition}
\begin{proof}
By symmetry under the bar involution, it is enough to prove the
first equation. By the characteristic property \eqref{carac_R} of
right combs, one just has to show that
\begin{equation*}
\sU_t = \sE_t + \sE_t \prec \sU_t.
\end{equation*}
Let us compute the right hand side using \eqref{defi_et}. One finds
\begin{equation*}
\frac{1}{1-t} \sD_t + \frac{b}{t} (\sD_t-\sD) + \frac{1}{1-t} \sD_t \prec \sU_t + \frac{b}{t} (\sD_t\prec \sU_t-\sD\prec \sU_t).
\end{equation*}
Using then \eqref{defi_dt}, one gets
\begin{multline*}
\frac{1}{1-t} (1+\sU_t)\vee(1+\sV_t) + \frac{b}{t}
((1+\sU_t)\vee(1+\sV_t) - (1+\sU)\vee(1+\sV) )+ \\ \frac{1}{1-t}
((1+\sU_t)\vee(1+\sV_t)) \prec \sU_t + \frac{b}{t} ((
(1+\sU_t)\vee(1+\sV_t))\prec \sU_t-( (1+\sU)\vee(1+\sV))\prec
\sU_t),
\end{multline*}
which can be rewritten by \eqref{vee_star} as
\begin{multline*}
\frac{1}{1-t} (1+\sU_t)\vee((1+\sV_t)*(1+\sU_t)) + \frac{b}{t}
((1+\sU_t)\vee((1+\sV_t)*(1+\sU_t)) \\ - (1+\sU)\vee ((1+\sV)*(1+\sU_t) ).
\end{multline*}
Using Lemma \ref{lemmeVU}, one can replace $(1+\sV_t)*(1+\sU_t)$
by $1$. One obtains
\begin{equation*}
\frac{1}{1-t} (1+\sU_t)\vee 1 +\frac{b}{t}(1+\sU_t)\vee 1 -\frac{b}{t} (1+\sU)\vee ((1+\sV) * (1+\sU_t)).
\end{equation*}
Using Lemma \ref{lemmeVU} again, one finds
\begin{equation*}
\frac{1}{1-t} (1+\sU_t)\vee 1+\frac{b}{t}(1+\sU_t)\vee 1 - \frac{b}{t} (1+\sU)\vee (1-t(\dN T \dP_t)).
\end{equation*}
Expanding that, one gets
\begin{equation}
\label{pas_final}
\frac{1}{1-t} 1\vee 1 +\frac{1}{1-t} \sU_t\vee 1+\frac{b}{t} \sU_t\vee 1 - \frac{b}{t} \sU \vee 1 + b 1 \vee (\dN T \dP_t)+ b \sU \vee (\dN T \dP_t).
\end{equation}
We therefore have to show that this expression is simply $\sU_t$.
To prove that, let us decompose $\sU_t$ according to the position of
the root in the trees. There are four ways to place the root in the
full canopy:
\begin{itemize}
\item the tree is $\includegraphics[height=3mm]{a1.pdf}$, the root can be put between $\boxed{\color{blue}-}$ and $\boxed{\color{red}+}$,
\item the full canopy ends by $\boxed{\color{red}+}\pp$, the root can be put between them,
\item the full canopy starts by $\boxed{\color{blue}-}\mm$, the root can be put
between them,
\item the root can be put after any $\boxed{\color{red}+}$ followed by $\boxed{\color{blue}-}$ in the
full canopy.
\end{itemize}
Using equation \eqref{droite_u_plus}, let us describe the first two cases. One gets
\begin{itemize}
\item $\frac{1}{1-t} 1 \vee 1$ for the tree $\includegraphics[height=3mm]{a1.pdf}$,
\item $\frac{1}{1-t} \sU_t\vee 1+\frac{b}{t} \sU_t\vee 1 - \frac{b}{t} \sU \vee 1 $ for the root between $\boxed{\color{red}+}$ and $\boxed{\color{red}+}$.
\end{itemize}
Using equation \eqref{def_u}, let us describe the last two cases. One gets
\begin{itemize}
\item $b 1 \vee (\dN T \dP_t)$ for the root between $\boxed{\color{blue}-}$ and $\boxed{\color{blue}-}$,
\item $b \sU \vee (\dN T \dP_t)$ for the root between $\boxed{\color{red}+}$ and $\boxed{\color{blue}-}$.
\end{itemize}
Note that the last case is slightly more subtle, as the cut takes
places inside the $\dT$ factor and one has to use the expression
\eqref{defi_dt} for the series $\dT$.
It follows that \eqref{pas_final} is exactly the expansion of
$\sU_t$ according to the possible positions of the root in the
canopy.
\end{proof}
\begin{corollary}
One has
\begin{equation}
\sD_t= ((1+\sR) \circ \sE_t) \vee ((1+\sL)\circ \sE_t).
\end{equation}
\end{corollary}
\begin{proof}
This follows from \eqref{defi_dt} and Proposition \ref{uv_from_et}.
\end{proof}
This is readily reformulated by lemma \ref{diamant_vee} using the
$\diamond$ operation as
\begin{equation}
\label{d_corolle_e_dend}
\sD_t= ((1+\sR) \vee (1+\sL)) \diamond (\arb{1},\sE_t).
\end{equation}
\begin{theorem}
The dendriform images of $\xE_t$ and $\xD_t$ are $\sE_t$ and $\sD_t$.
\end{theorem}
\begin{proof}
The series $\xE_t$ and $\xD_t$ are characterized by the equations
\eqref{master_eq_E} and \eqref{eq_d_corolle_e}. By Proposition
\ref{image_des_corolles} and results of appendix \ref{appA}, the
dendriform image of \eqref{eq_d_corolle_e} is exactly
\eqref{d_corolle_e_dend}. The dendriform image of
\eqref{master_eq_E} is exactly \eqref{defi_et}. Therefore $\sE_t$
and $\sD_t$ satisfy equations that characterize the dendriform
images of $\xE_t$ and $\xD_t$, and the statement follows.
\end{proof}
\subsection{Explicit product formulas for coefficients}
The equations \eqref{def_u} and \eqref{def_v} provide an explicit
description of the coefficients of the series $\sU_t$ and $\sV_t$.
More precisely, the coefficient of a planar binary tree $\tau$ in the
series $\sU_t$ can be found as follows. One considers the
right-completed canopy of $\tau$ (including the rightmost leaf but not
the leftmost leaf). It admits a unique coarsest decomposition into
blocks of the shape $\boxed{\color{red}+}^k$ and $\boxed{\color{blue}-}^\ell$ for $k,\ell \geq 1$. Every
block of length $\ell$ in this decomposition contributes a Narayana
factor $\operatorname{\mathbf{ca}}_\ell$, but the rightmost block contributes instead a
$t$-Narayana factor $\operatorname{\mathbf{ca}}_{\ell,t}$. There is an additional factor of
$b$ to the power the number of $\boxed{\color{blue}-}$ blocks and $(-1)$ to the power
the number of $\boxed{\color{blue}-}$.
For example, the coefficient of the leftmost planar binary tree of
figure \ref{fig:expl_bt}, whose right-completed canopy is
$\boxed{\color{blue}-}\boxed{\color{red}+}\pp\boxed{\color{blue}-}\boxed{\color{red}+}\pp$, is
\begin{equation*}
b^2 \operatorname{\mathbf{ca}}_1 \operatorname{\mathbf{ca}}_2 \operatorname{\mathbf{ca}}_1 \operatorname{\mathbf{ca}}_{2,t}.
\end{equation*}
There is a similar description for $\sV_t$. One considers the
left-completed canopy of $\tau$ (including the leftmost leaf but not
the rightmost leaf) and decompose it into maximal blocks of $\boxed{\color{red}+}$ and
$\boxed{\color{blue}-}$. Every such block of length $\ell$ contributes a Narayana factor
$\operatorname{\mathbf{ca}}_\ell$, but the leftmost block contributes instead a $t$-Narayana
factor $\operatorname{\mathbf{ca}}_{\ell,t}$. There is an additional factor of $b$ to the
power the number of $\boxed{\color{red}+}$ blocks and $(-1)$ to the power the number of
$\boxed{\color{blue}-}$.
\medskip
One can also interpret the definition \eqref{defi_dt} as giving the
explicit coefficients of the series $\sD_t$.
More precisely, the coefficient of a planar binary tree $\tau$ in the
series $\sD_t$ can be found as follows. Consider the canopy of $\tau$,
and cut it into two parts according to the position of the root of
tree. Decompose both parts into maximal blocks of $\boxed{\color{red}+}$ and $\boxed{\color{blue}-}$. Every such
block of length $\ell$ contributes a Narayana factor $\operatorname{\mathbf{ca}}_\ell$, but
the two blocks that are closest to the root contributes instead a
$t$-Narayana factor $\operatorname{\mathbf{ca}}_{\ell,t}$. There is an additional factor of
$b$ to the power the number of $\boxed{\color{blue}-}$ blocks in the left part plus the
number of $\boxed{\color{red}+}$ blocks in the right part, and $(-1)$ to the power the
number of $\boxed{\color{blue}-}$.
For example, the coefficient of the leftmost planar binary tree of
figure \ref{fig:expl_bt}, whose canopy is cut into
$\boxed{\color{blue}-}\boxed{\color{red}+}\pp$ and $\boxed{\color{blue}-}\boxed{\color{red}+}$, is
\begin{equation*}
b^2 \operatorname{\mathbf{ca}}_1 \operatorname{\mathbf{ca}}_{2,t} \operatorname{\mathbf{ca}}_{1,t} \operatorname{\mathbf{ca}}_{1}.
\end{equation*}
Letting $t=0$ in this description, one observes that the coefficient
of a tree in $\sD$ depends only on its canopy. This is obvious for the
factors associated with blocks and for the sign. As for the power of
$b$, it can be described as the number of $\boxed{\color{blue}-}$ blocks in the canopy,
excluding the last (rightmost) block.
It follows that $\sD$ is in the descent algebra. Moreover, this
description of $\sD_n$ is exactly the value at $a=1$ of the
description given in \cite[Th. 10.1]{menoth} and we therefore recover
this theorem. Let us give its statement here.
\begin{corollary}
The homogeneous components $\sD_n$ are Lie idempotents and satisfy
\begin{equation}
\sD_n \cdot \sD_n = n \operatorname{\mathbf{ca}}_{n-1} \sD_n,
\end{equation}
in the symmetric group ring of $\mathfrak{S}_n$.
\end{corollary}
To determine the precise constant of proportionality, one uses Lemma
\ref{coeff_id} and the fact that the coefficient of the ribbon Schur
function $\boxed{\color{red}+}^{n-1}$ is $\operatorname{\mathbf{ca}}_{n-1}$.
\medskip
One can now use \eqref{defi_et} to give an explicit description of the
coefficients of the series $\sE_t$.
As $\sD_t$ and $\sD$ have all but two of their factors in common, all
these factors are also in $\sE_t$. The remaining factor is
\begin{equation}
\frac{1}{1-t}\operatorname{\mathbf{ca}}_{k,t} \operatorname{\mathbf{ca}}_{\ell,t}+\frac{b}{t}\left(\operatorname{\mathbf{ca}}_{k,t}
\operatorname{\mathbf{ca}}_{\ell,t}-\operatorname{\mathbf{ca}}_k \operatorname{\mathbf{ca}}_{\ell} \right)
\end{equation}
which is the fraction counting flows on the rooted trees
$B_+(\mathtt{Lnr}_k,\mathtt{Lnr}_{\ell})$.
Therefore, the coefficient of a planar binary tree $\tau$ in the
series $\sE_t$ can be found as follows. Consider the canopy of $\tau$,
and cut it into two parts according to the position of the root of
tree. Decompose both parts into maximal blocks of $\boxed{\color{red}+}$ and $\boxed{\color{blue}-}$,
excluding the two central blocks. Every such block of length $\ell$
contributes a Narayana factor $\operatorname{\mathbf{ca}}_\ell$. The two central blocks
together are of the shape $\boxed{\color{red}+}^k \boxed{\color{blue}-}^\ell$. We associate with this the
coefficient of the rooted tree $B_+(\mathtt{Lnr}_k,\mathtt{Lnr}_{\ell})$ in the
series $\xE_t$.
There is an additional factor of $b$ to the power the number of $\boxed{\color{blue}-}$
blocks in the left part plus the number of $\boxed{\color{red}+}$ blocks in the right
part, and $(-1)$ to the power the number of $\boxed{\color{blue}-}$.
\subsection{Connected flows in the dendriform group}
Let us now introduce the dendriform image $\sE^c_t$ of the series
$\xE^c_t$ of connected flows. As $\xE^c_t$ is related to $\xE_t$ by
equation \eqref{rela_ect_e}, one gets, by using Proposition
\ref{image_de_tout} and results of appendix \ref{appA}, that $\sE^c_t$
is defined by
\begin{equation}
\label{defi_ect_dend}
\sE_t = \left( (1-\Sigma \sL) \vee (1-\Sigma \sR)\right) \diamond ( \sE^c_t , \sE^c ).
\end{equation}
Letting $t=0$, one gets
\begin{equation}
\label{defi_ec}
\sE = \left( (1-\Sigma \sL) \vee (1-\Sigma \sR) \right) \circ \sE^c.
\end{equation}
\begin{proposition}
\label{formule_Ect}
The series $\sE^c_t$ admits the following expression
\begin{multline}
\label{f_ect}
\frac{1}{1-t}\includegraphics[height=3mm]{a1.pdf} + \left(\frac{t}{1-t}+b\right) 1 \vee \dN_t\dT\dP - \left(\frac{t}{1-t}+b\right) \dN\dT\dP_t \vee 1 \\ - t \left(\frac{t}{1-t}+b\right) \dN\dT\dP_t \vee \dN_t\dT\dP.
\end{multline}
\end{proposition}
\begin{proof}
Using lemma \ref{inverse_cool}, one can invert the relation \eqref{defi_ec} between $\sE$ and $\sE^c$ as
\begin{equation}
\label{ec_from_e}
((1+\sL)\vee (1+\sR)) \circ \sE = \sE^c.
\end{equation}
On the other hand, using lemma \ref{diamant_vee}, lemma
\ref{double_inversion} and lemma \ref{L_inverse_R}, one can invert
the relation \eqref{defi_ect_dend} between $\sE_t$ and $\sE^c_t$ to
obtain
\begin{equation*}
\sE_t^c = (1-\Sigma \sR \circ \sE^c) \vee_{\sE_t} (1-\Sigma \sL\circ \sE^c).
\end{equation*}
But by \eqref{ec_from_e} and Lemma \ref{nice_compo}, one deduces that
\begin{equation*}
\Sigma\sR\circ \sE^c = \Sigma\sR \circ ((1+\sL)\vee (1+\sR)) \circ \sE = -\sL \circ \sE,
\end{equation*}
and by symmetry that
\begin{equation*}
\Sigma\sL\circ \sE^c = \sL \circ ((1+\sL)\vee (1+\sR)) \circ \sE = -\sR \circ \sE.
\end{equation*}
One therefore gets using prop. \ref{uv_from_et} that
\begin{equation*}
\sE_t^c = (1 + \sL \circ \sE) \vee_{\sE_t} (1+ \sR\circ \sE)= (1+\sV) \vee_{\sE_t} (1+\sU).
\end{equation*}
By the equation \eqref{defi_et}, one finds
\begin{equation*}
\frac{1}{1-t}(1+\sV) \vee_{\sD_t} (1+\sU) +\frac{b}{t} ((1+\sV) \vee_{\sD_t} (1+\sU) - (1+\sV) \vee_{\sD} (1+\sU) ).
\end{equation*}
Using then the definition \eqref{defi_dt} of $\sD_t$, one gets
\begin{multline*}
\left(\frac{1}{1-t}+\frac{b}{t}\right)(1+\sV)*(1+\sU_t) \vee (1+\sV_t)*(1+\sU)-\frac{b}{t}(1+\sV)*(1+\sU) \vee (1+\sV)*(1+\sU) .
\end{multline*}
By Lemma \ref{lemmeVU}, one gets
\begin{equation*}
\left(\frac{1}{1-t} +\frac{b}{t}\right) (1-t \dN\dT\dP_t ) \vee (1+t \dN_t\dT\dP )-\frac{b}{t} 1 \vee 1.
\end{equation*}
This gives the expected result, after simplification.
\end{proof}
Recall from \eqref{from_EC_to_F} that one can write
\begin{equation}
\sE^c = \includegraphics[height=3mm]{a1.pdf} + b \sF/\includegraphics[height=3mm]{a1.pdf} - b \includegraphics[height=3mm]{a1.pdf} \backslash \sF,
\end{equation}
where $\sF$ is the dendriform image of the series $\xF$ introduced in
\eqref{from_EC_to_F}.
\begin{corollary}
The series $\sE^c$ admits the following expression
\begin{equation}
\includegraphics[height=3mm]{a1.pdf} + b 1 \vee \dN\dT\dP - b \dN\dT\dP \vee 1.
\end{equation}
The series $\sF$ is given by
\begin{equation}
\label{formule_f_dend}
- \dN\dT\dP,
\end{equation}
and belongs to the descent algebra.
\end{corollary}
One can use \eqref{formule_f_dend} to give an explicit
description of the coefficients of $\sF$.
More precisely, let $\tau$ be a planar binary tree. Then consider the
full canopy of $\tau$, and its coarsest decomposition into blocks of
the shape $\boxed{\color{blue}-}^k$ or $\boxed{\color{red}+}^\ell$ for $k,\ell \geq 1$. To each such
block of size $\ell$, one associate a factor $\operatorname{\mathbf{ca}}_{\ell}$.
Then the coefficient of $\tau$ is the product of these factors, times
a power of $b$ given by the number of $\boxed{\color{blue}-}$ blocks minus $1$ and times
$(-1)$ to the power the number of $\boxed{\color{blue}-}$ minus $1$.
For example, the coefficient of the leftmost planar binary tree of
figure \ref{fig:expl_bt}, whose full canopy is
$\boxed{\color{blue}-}\mm\boxed{\color{red}+}\pp\boxed{\color{blue}-}\boxed{\color{red}+}\pp$, is
\begin{equation*}
b \operatorname{\mathbf{ca}}_2 \operatorname{\mathbf{ca}}_2 \operatorname{\mathbf{ca}}_1 \operatorname{\mathbf{ca}}_2.
\end{equation*}
Using Prop. \ref{formule_Ect}, one can also give a description of the
coefficients of the series $\sE^c_t$ of connected flows.
Let us consider a planar binary tree $\tau$ with at least $2$ inner
vertices. One considers the full canopy of $\tau$, and cut it into two
parts by using the position of the root. One distinguish three cases:
the root can either be placed between $\boxed{\color{blue}-}$ and $\boxed{\color{blue}-}$ at the left of
of the full canopy, or between $\boxed{\color{red}+}$ and $\boxed{\color{red}+}$ at the right of the
full canopy, or between $\boxed{\color{red}+}$ and $\boxed{\color{blue}-}$ inside the full canopy. In
each case, one can translate the corresponding term in \eqref{f_ect}
into a description of the factors of the coefficient of $\tau$ in
$\sE^c_t$.
\medskip
The series $\sF$ provides new Lie idempotents.
\begin{proposition}
\label{lie_id_F}
The homogeneous component $\sF_n$ of the series $\sF$ satisfies
\begin{equation}
\sF_n \cdot \sF_n = n \operatorname{\mathbf{ca}}_n \sF_n,
\end{equation}
in the symmetric group ring of $\mathfrak{S}_n$.
\end{proposition}
The constant $n\operatorname{\mathbf{ca}}_n$ is determined by Lemma \ref{coeff_id}, using
that the coefficient of the ribbon Schur function $\boxed{\color{red}+}^{n-1}$
(corresponding to the full canopy $\boxed{\color{blue}-} \boxed{\color{red}+}^n$) is $\operatorname{\mathbf{ca}}_n$.
Let us consider now the series $\sF_t=- (1-t) \dN_t \dT \dP_t$, which gives
back $\sF$ when $t=0$.
Assuming that Conjecture \ref{conjecture_F} holds, one can introduce a
global series $\xF_t$ and propose the following conjecture.
\begin{conjecture}
The series $\sF_t$ is the dendriform image of the series $\xF_t$.
\end{conjecture}
If this is true, then the series $\sF_t$ provides new Lie idempotents.
\begin{conjecture}
\label{lie_id_FT}
The homogeneous component $\sF_{n,t}$ of the series $\sF_t$ satisfies
\begin{equation}
\sF_{n,t} \cdot \sF_{n,t} = n \operatorname{\mathbf{ca}}_{n,t} \sF_{n,t},
\end{equation}
in the symmetric group ring of $\mathfrak{S}_n$.
\end{conjecture}
The constant $n\operatorname{\mathbf{ca}}_{n,t}$ in this conjecture is given by Lemma
\ref{coeff_id}, using that the coefficient of the ribbon Schur
function $\boxed{\color{red}+}^{n-1}$ (corresponding to the full canopy $\boxed{\color{blue}-} \boxed{\color{red}+}^n$) is
$\operatorname{\mathbf{ca}}_{n,t}$.
Conjecture \ref{lie_id_FT} has been checked up to $\mathfrak{S}_6$ included.
\subsection{Description of $\sZ$}
Let $\sZ$ be the dendriform image of $\xZ$, introduced in
\eqref{Z_here}. We propose here a conjectural description of the
coefficients of $\sZ$.
For positive integers $ p$ and $ q$, let us define polynomials
\begin{equation*}
z_{p,q}=\sum_{k\geq 0} \binom{p}{k}\binom{q}{k} b^{p+q+1-k}.
\end{equation*}
If $p=q$, this polynomial is essentially a Narayana polynomial of type
$B$.
Let now $\tau$ be a planar binary tree of size $n$. Consider the full
canopy of $\tau$ and decompose it into blocks of the shape
$\boxed{\color{blue}-}^{p}\boxed{\color{red}+}^{q}$ with $p,q\geq 1$. To each such block
$\boxed{\color{blue}-}^{p}\boxed{\color{red}+}^{q}$, one associates a factor $(-1)^{p-1} z_{p-1,q-1}$.
The coefficient of $\tau$ in the series $\xZ$ seems to be the product
of these factors associated with blocks, divided by $b$. The total
degree with respect to $b$ is $n$ minus the number of blocks.
If this description holds, the coefficient of $\tau$ would depend only
on its canopy. This would imply the following result.
\begin{conjecture}
The homogeneous component $\sZ_n$ of the series $\sZ$ is in the
descent algebra and satisfies
\begin{equation}
\sZ_n \cdot \sZ_n = n b^{n-1} \sZ_n,
\end{equation}
in the symmetric group ring of $\mathfrak{S}_n$.
\end{conjecture}
Note that one uses Lemma \ref{coeff_id} to get this precise statement.
\begin{question}
Are the zeroes of the polynomials $z_{p,q}$ real ?
\end{question}
It is known that the generalized Narayana numbers associated with
finite Coxeter groups have only real roots, see \cite[\S
5.2]{reiner_welker}.
\begin{remark}
The polynomials $z_{p,p+1}$ and $z_{p,p}$, as well as the
polynomials $\xF$ for forks seem to appear in the article
\cite{lassalle}, which deals with symmetric functions. The
relationship with the present work is not clear to us.
\end{remark}
|
{
"timestamp": "2012-03-09T02:02:43",
"yymm": "1203",
"arxiv_id": "1203.1780",
"language": "en",
"url": "https://arxiv.org/abs/1203.1780"
}
|
\section{Introduction}
We consider the well-posedness for the Cauchy problem of the Kawahara equation which is one of the fifth order KdV type equations.
\begin{align} \label{KT_1}
\begin{cases}
& \p_t u+ \alpha \p_{x}^5 u+\beta \p_x^3 u + \gamma \p_x ( u^2 )=0, \hspace{0.3cm} (t,x) \in [0,T] \times \mathbb{T}, \\
& u(0, x)=u_0(x), \hspace{0.3cm} x \in \mathbb{T},
\end{cases}
\end{align}
where $\alpha, \beta, \gamma \in \mathbb{R}$ with $\alpha, \gamma \neq 0$ and
$\mathbb{T}:= \mathbb{R}/ 2 \pi \mathbb{Z}$.
Here the unknown function $u$ is assumed to
be real valued or complex valued in the case we deal with
the local well-posedness (LWP for short) and
to be real valued when we consider the global well-posedness (GWP for short).
By the renormalization of $u$, we may assume $\alpha=-1$, $\gamma=1$ and $\beta=-1,0$ or
$1$. We put $v=u-a$ where $a$ is the integral mean value of initial data defined as $a:= \int_{\mathbb{T}} u_0 (x) dx$.
If $u$ solves (\ref{KT_1}), then $v$ satisfies the following equation.
\begin{align*}
\begin{cases}
& \p_t v- \p_{x}^5 v+\beta \p_x^3 v + 2a \p_x v+ \p_x ( v^2 )=0, \hspace{0.3cm} (t,x) \in [0,T] \times \mathbb{T}, \\
& v(0, x)=v_0(x), \hspace{0.3cm} x \in \mathbb{T}.
\end{cases}
\end{align*}
Note that the Fourier coefficient $\mathcal{F}_x (v) (0)$ of zero mode vanishes.
It suffices to consider the well-posedness for (\ref{KT_1}) under the mean-zero assumption
$\int_{\mathbb{T}} u_0 (x) dx=0$ because the linear first order term is harmless.
This observation was used by Bourgain \cite{Bo}. Without the mean-zero assumption,
the data-to-solution map fails to be $C^2$ in $H^s(\mathbb{T})$ for any $s \in \mathbb{R}$.
So this assumption is crucial for some of analysis that follows.
From the above argument, we only consider the case $\dot{\mathbb{Z}} := \mathbb{Z} \setminus \{0\}$.
The Kawahara equation models the capillary waves on a shallow layer and
the magneto-sound propagation in plasma (see e.g. \cite{Ka}).
This equation has solitary waves with $\beta=1$ and many conserved quantities.
Our aim is to prove the well-posedness for (\ref{KT_1}) with low regularity data given in the Sobolev space $\dot{H}^s(\mathbb{T})$.
Here $\dot{H}^s(\mathbb{T})$ is defined by the norm,
\begin{align*}
\| u \|_{\dot{H}^s(\mathbb{T})} :=
\| \langle k \rangle^s \mathcal{F}_x u \|_{l_k^2( \dot{\mathbb{Z}} )},
\end{align*}
where $\langle \cdot \rangle:=(1+|\cdot |^2)^{1/2}$ .
We first use the Fourier restriction norm method to prove LWP for (\ref{KT_1}).
This method was introduced by Bourgain \cite{Bo}. Next,
we extend local solutions to global-in-time ones by the I-method which
was exploited by Colliander, Keel, Staffilani, Takaoka and Tao \cite{CoKe}, \cite{I02}.
The Kawahara equation with the periodic boundary does not have the Kato smoothing effect
unlike the case of $\mathbb{R}$, though a weak version of the Strichartz estimate still holds in the periodic setting.
It is possible to make a close investigation into the resonance of nonlinear interactions under the periodic
boundary conditions.
The local well-posedness for the periodic KdV equation has been extensively studied.
Bourgain \cite{Bo} proved LWP in $\dot{H}^s$ for $s \geq 0$.
Kenig, Ponce and Vega \cite{KPV96} refined Bourgain's argument to show LWP in $\dot{H}^s$ for $s>-1/2$.
Moreover, Colliander, Keel, Staffilani, Takaoka and Tao \cite{CoKeSt} obtained LWP in the critical case $s=-1/2$.
On the other hand, Christ, Colliander and Tao \cite{CCT} showed that the data-to-solution map fails to be uniformly
continuous for $-2< s <-1/2$.
We now recall the local well-posedness results for the Kawahara equation.
Hirayama \cite{Hi} proved LWP in $\dot{H}^s$ for $s \geq -1$ in the periodic case,
which was an adaptation of the argument to Kenig, Ponce and Vega \cite{KPV96}.
Moreover, there are many studies in the case of $\mathbb{R}$.
Chen and Guo \cite{CG} proved for $s \geq -7/4$, using some modified Bourgain space $\bar{F}^s$ introduced in \cite{Gu}.
Following an idea of Bejenaru and Tao \cite{BT} and Kishimoto and Tsugawa \cite{KT},
we improved the previous results to $s \geq -2$ in \cite{TK_K}. This result is optimal in such a sense that
the data-to-solution map fails to be continuous when $s<-2$.
Earlier results can be found in \cite{CLMW}, \cite{CDT} and \cite{WCD}.
The main difficulty in obtaining LWP for the periodic equation is to recover no derivatives by the smoothing effects.
So we need to make a more complex modification of the Bourgain space.
Then we find a suitable modification of function spaces to obtain the following theorem.
\begin{thm} \label{LWP_TK}
Let $s \geq -3/2$. Then (\ref{KT_1}) is locally well-posed in $\dot{H}^{s}(\mathbb{T})$.
\end{thm}
On the other hand, we obtain the ill-posedness result in the following sense.
\begin{thm} \label{ill_TK}
Let $s<-3/2$. Then, there is no $T>0$ such that the flow map,
$\dot{H}^s(\mathbb{T}) \ni u_0 \mapsto u(t) \in \dot{H}^s(\mathbb{T})$, can be $C^3$ for
any $t \in (0,T]$
\end{thm}
These theorems imply that the critical regularity is $s=-3/2$.
Moreover, the local solutions
obtained in Theorem~\ref{LWP_TK} are shown to exist on an arbitrary time by the I-method.
Colliander, Keel, Staffilani, Takaoka and Tao \cite{CoKeSt} proved GWP for the periodic case KdV equation when $s>-1/2$,
which was improved to $s \geq -1/2$ in \cite{CKSTT03}.
We now describe the global well-posedness results for the Kawahara equation
in the non-periodic case.
Note that it is difficult to apply the I-method to the Kawahara equation because this equation has less symmetries than
the KdV equation. Chen and Guo \cite{CG} overcame this issue and used the similar argument to \cite{CoKeSt}
to show GWP for $s \geq -7./4$. Recently, we have refined their argument and established GWP for
$s \geq -38/21$ in \cite{TK_G}.
We apply the argument presented for the non-periodic case to the periodic setting so that the following is established.
\begin{thm} \label{GWP_TK}
Let $s \geq -1$. Then (\ref{KT_1}) is globally well-posed in $\dot{H}^s (\mathbb{T})$.
\end{thm}
This result is optimal as long as we use the standard Bourgain space.
\vspace{0.3em}
We now use the scaling argument. For $\lambda \geq 1$,
\begin{align*}
u_{\lambda}(t,x):= \lambda^{-4} u(\lambda^{-5}t, \lambda^{-1} x),
\hspace{0.3cm} u_{0,\lambda}(x):=\lambda^{-4} u_0(\lambda^{-1} x)
\end{align*}
If $u$ solves (\ref{KT_1}), $u_{\lambda}$ satisfies
the following rescaled Cauchy problem;
\begin{align} \label{KT_2}
\begin{cases}
& \p_t u_{\lambda} -\p_x^5 u_{\lambda}
+ \ \lambda^{-2} \beta \p_x^3 u_{\lambda} + \p_x(u_{\lambda}^2 )=0,
\hspace{0.3cm} (t,x) \in [0, \lambda^5 T] \times \mathbb{T}_{\lambda}, \\
& u_{\lambda}(0,x)=u_{0,\lambda} (x),
\hspace{0.3cm} x \in \mathbb{T}_{\lambda},
\end{cases}
\end{align}
where $\mathbb{T}_{\lambda}:= \mathbb{R} / 2 \pi \lambda \mathbb{Z}$. $\widehat{\varphi}$ denotes
the Fourier transform on $\mathbb{T}_{\lambda}$ of $\varphi$ as follows;
\begin{align*}
\widehat{\varphi}(k) := \frac{1}{ \sqrt{2 \pi}} \int_0^{2 \pi \lambda}
e^{-ikx} \varphi(x) dx, \hspace{0.3cm} k \in \dot{\mathbb{Z}}_{\lambda}
:= \frac{1}{\lambda} \dot{\mathbb{Z}}.
\end{align*}
Here the space $\dot{H}^{s} (\mathbb{T}_{\lambda})$ is equipped with the norm
\begin{align*}
\| \varphi \|_{\dot{H}^s(\mathbb{T}_{\lambda})}:=
\| \langle k \rangle^{s} \widehat{\varphi} \|_{l_k^2 (\dot{\mathbb{Z}}_{\lambda})},
\end{align*}
where
\begin{align*}
\| f \|_{l_k^p(\dot{\mathbb{Z}}_{\lambda})}:= \Bigl( \frac{1}{\lambda}
\sum_{k \in \dot{\mathbb{Z}}_{\lambda}} |f (k)|^{p} \Bigr)^{1/p},
\end{align*}
for $1 \leq p \leq \infty$.
A direct calculation shows that
\begin{align} \label{in_sm}
\| u_{0,\lambda} \|_{\dot{H}^s (\mathbb{T}_{\lambda})}
\leq \lambda^{-7/2-s} \| u_0 \|_{ \dot{H}^s (\mathbb{T})}
~~ \text{for} ~~ s<0.
\end{align}
Therefore we can assume smallness of initial data. So it suffices to solve (\ref{KT_2}) for sufficiently small data.
We first summarize the local well-posedness theory.
The main idea is how to define the function space to construct solutions.
When $s$ is small, especially negative, the Bourgain space plays an important role.
The Bourgain space $X^{s,b}(\mathbb{R} \times \mathbb{T}_{\lambda})$ for $2 \pi \lambda$-periodic
is defined by the norm
\begin{align*}
\| u \|_{X^{s,b} (\mathbb{R} \times \mathbb{T_{\lambda}})}
:= \Bigl\| \langle k \rangle^s \langle \tau-p_{\lambda}(k) \rangle^b \widehat{u}
\Bigr\|_{l_k^2(\dot{\mathbb{Z}}_{\lambda} ; L_{\tau}^2 (\mathbb{R}) )},
\end{align*}
where $p_{\lambda}(k):= k^{5}+\beta \lambda^{-2} k^3$.
Remark that
the Bourgain space depends on the linear part of our target equation.
One of the key estimates is the bilinear estimate in $X^{s,b}$ as follows:
\begin{align} \label{BE_T1}
\| \Lambda^{-1} \p_x (uv) \|_{X^{s,b}} \leq C
\| u \|_{X^{s,b}} \| v \|_{X^{s,b}},
\end{align}
where $\Lambda^{b}$ is the Fourier multiplier defined as
$\Lambda^{b}:=
\mathcal{F}_{\tau,k}^{-1} \langle \tau-p_{\lambda}(k) \rangle^{b} \mathcal{F}_{t,x}
$ for $b \in \mathbb{R}$.
From the bilinear estimate and some linear estimates, the standard argument of the Fourier restriction norm method works to obtain LWP.
Hirayama \cite{Hi} showed (\ref{BE_T1}) for $s \geq -1$. On the other hand, he proved that this estimate fails for any $b \in \mathbb{R}$ when $s<-1$.
So it is difficult to construct the local solutions by the iteration argument when $s<-1$.
To avoid this difficulty, we modify the Bourgain space $X^{s,b}$ to control strong nonlinear interactions and
establish the bilinear estimate at the critical regularity $s=-3/2$.
An idea of a modification of $X^{s,b}$ was developed by Bejenaru and Tao \cite{BT}.
They considered the quadratic Schr\"{o}dinger equation with the nonlinearity $u^2$ and
obtained LWP in the critical case $H^{-1}(\mathbb{R})$.
Note that there is no general framework for modifying $X^{s,b}$.
This is one of the most difficult points in our study.
Compared to the non-periodic case, less derivatives can be recovered by the smoothing effects in the
periodic setting.
So nonlinear interactions which we can ignore in the non-periodic case take effect.
Therefore we need to make a more complex modification of $X^{s,b}$ to control three types nonlinear interactions.
We now mention how to modify $X^{s,b}$.
From the counterexamples of (\ref{BE_T1}) in the case $s<-1$, we find the regions in which strong nonlinear interactions appear.
In these domains,
we make a suitable modification of $X^{s,b}$ as follows;
\begin{align*}
\| u \|_{Z^s} & := \| P_{D_1} u \|_{X^{s,3/4}}+ \| P_{D_2} u \|_{X^{-3s-1,s+1}} \\
&~ + \| P_{D_3} u \|_{X^{-s/2-1,s/2+1}} + \| u \|_{Y^s}, \hspace{0.3cm} \text{for}
\hspace{0.3cm} -3/2 \leq s \leq -1,
\end{align*}
where $P_{\Omega}$ is the Fourier projection onto a set
$\Omega \subset \mathbb{R} \times \dot{\mathbb{Z}}_{\lambda}$
and
\begin{align*}
D_1 &:= \bigl\{ (\tau, k) \in \mathbb{R} \times \dot{\mathbb{Z}}_{\lambda}~;~
|\tau -p_{\lambda} (k) | \leq |k|^4/10 ~~\text{and}~~ |k| \geq 1 \bigr\}, \\
D_2 &:= \bigl\{ (\tau, k) \in \mathbb{R} \times \dot{\mathbb{Z}}_{\lambda}~;~ |k|^4/10
\leq |\tau -p_{\lambda} (k) | \leq |k|^5/10 ~~\text{and}~~ |k| \geq 1 \bigr\}, \\
D_3 &:= \bigl\{ (\tau, k) \in \mathbb{R} \times \dot{\mathbb{Z}}_{\lambda}~;~
|\tau -p_{\lambda} (k) | \geq |k|^5/10 ~~\text{and}~~ \frac{1}{\lambda} \leq |k| \leq 1 \bigr\}.
\end{align*}
Here $\| u \|_{Y^s} := \| \langle k \rangle^{s} \widehat{u} \|_{l_k^2 L_{\tau}^1}$
and $Y^s$ is continuously embedded into $C(\mathbb{R} ; \dot{H}^s (\mathbb{T}_{\lambda}))$.
Using the function space above,
we obtain the following bilinear estimate which is one of the main estimates in the present paper.
\begin{prop} \label{prop_BE_2}
Let $-3/2 \leq s <-1$. Then, the following estimate holds.
\begin{align} \label{BE_T2}
\| \Lambda^{-1} \p_x (uv) \|_{Z^s} \leq C \| u \|_{Z^s} \| v \|_{Z^s},
\end{align}
where a positive constant $C$ is independent of $\lambda$.
\end{prop}
\vspace{0.5em}
Next, we extend the local solution obtained above globally in time.
In the case $s$ is negative, we have no conservation laws. To avoid this difficulty,
we apply the I-method exploited by Colliander, Keel, Staffilani, Takaoka and Tao \cite{CoKe}, \cite{I02}.
The main idea is to use a modified energy defined for less regular functions, which is not conserved.
If we control the growth of the modified energy in time, this enables us to
iterate the local theory to continue the solution to any time $T$.
We now mention the definition of the modified energy $E_I^{(2)} (u)$.
The operator $I: H^s \rightarrow L^2$ is the Fourier multiplier defined as
$I= \mathcal{F}_{\xi}^{-1} m (\xi) \mathcal{F}_x$.
Here $m$ is a smooth and monotone function satisfying
\begin{align*}
m(\xi):=
\begin{cases}
1 ~~& \text{for} ~~ |\xi| \leq N \\
|\xi|^{s} N^{-s} ~~ & \text{for} ~~|\xi| \geq 2N,
\end{cases}
\end{align*}
for $s<0$ and $N \gg1$. The modified energy $E_I^{(2)} (u)$ is defined as
$E_{I}^{(2)} (u)(t) := \| I u (t) \|_{L^2}^2 $. In the I-method, the key estimate is
the almost conservation law which implies the increment of the modified energy
is sufficiently small for a short time interval and large $N$.
Following the argument of \cite{CoKe}, we obtain the almost conservation law and show GWP for
$s > -21/26$. However, the growth of the modified energy $E_I^{(2)} (u)$ in time cannot be controlled for
$-1 \leq s \leq -21/26$.
Then we add some correction terms to the original modified energy
$E_I^{(2)} (u)$ to construct a new modified energy in order to remove some oscillations in this functional.
This idea was developed by Colliander, Keel, Staffilani, Takaoka and Tao \cite{CoKeSt}. They \cite{CoKeSt} proved
GWP of the KdV equation for $s>-3/4$ in the case of $\mathbb{R}$ and for $s>-1/2$ in the periodic
case. Chen and Guo \cite{CG} establish the sharp upper bound of some multiplier to show GWP of the Kawahara equation
for $s \geq -7/4$ in the case of $\mathbb{R}$.
Following the argument of \cite{CG}, we obtain the almost conservation law for
the modified energy $E_I^{(4)} (u)$ by adding two suitable correction terms to
the original functional when $s \geq -1$.
On the other hand, the difference between the almost conserved quantities $E_I^{(4)} (u)$ and
the first modified energy $E_I^{(2)} (u)$ can be controlled by $E_I^{(2)}(u)$ when the time is fixed.
This estimate and the almost conservation law imply that
the well-posedness on any time interval. Remark that
we do not expect to recover any derivatives by the bilinear Strichartz estimate in the periodic case
(see Lemma~\ref{lem_L_4} in section 2). This is the reason why it is hard so that
the I-method is applicable when $s<-1$.
\vspace{0.5em}
We use the following notations in this paper.
$A \lesssim B$ means $A \leq C B$ for some positive constant $C$
and $A \sim B$ when both $A \lesssim B$ and $B \lesssim A$.
$c+$ means $c+\varepsilon$, while $c-$ means $c- \varepsilon$ where
$\varepsilon>0$ is enough small.
For a normed space $\mathcal{X}$ and a set $\Omega$,
$\| \cdot \|_{\mathcal{X}(\Omega) }$ denotes $\| f \|_{\mathcal{X} (\Omega)} := \| \chi_{\Omega} f \|_{\mathcal{X}}$
where $\chi_{\Omega}$ is the characteristic function of $\Omega$.
The rest of this paper is planned as follows. In Section 2, we give some preliminary lemmas.
In Section 3, we prove the bilinear estimate (\ref{BE_T2}) and
give the proof of LWP in Section 4.
In Section 5, we show GWP by the I-method, following \cite{CG} and \cite{CoKeSt}
In Section 6, we give the proof of Theorem~\ref{ill_TK} which is based on Bourgain's work \cite{Bo97}.
\vspace{1em}
\noindent
\textbf{Acknowledgment.} The author would like to appreciate his adviser Professor Yoshio Tsutsumi for many helpful conversation and encouragement and
thank Professor Kotaro Tsugawa and Professor Nobu Kishimoto for helpful comments.
\section{Preliminaries}
In this section, we prepare the bilinear Strichartz estimate to show the main estimates. When we use the variables $(\tau,k)$, $(\tau_1,k_1)$ and $(\tau_2, k_2)$, we always assume the relation
\begin{align*}
(\tau, k)=(\tau_1, k_1) + (\tau_2, k_2).
\end{align*}
The bilinear estimate (\ref{BE_T2}) can be established by the H\"{o}lder inequality, the Young inequality and the following estimate.
\begin{lem} \label{lem_L_4}
If $b, b' \in \mathbb{R}$ satisfy $b+b' \geq 29/40$ and $b, b' > 9/40$, then we have
\begin{align} \label{es_L_4_1}
\bigl\| P_{\{ |k| \geq 1 \}} ( uv ) \bigr\|_{L_{t,x}^2} & \lesssim \| u \|_{X^{0,b}} \| v \|_{X^{0,b'}}, \\
\label{es_L_4_2}
\bigl\| u ( P_{\{ |k| \geq 1 \}} v) \bigr\|_{X^{0,-b'}} & \lesssim \| u \|_{X^{0,b}} \| v \|_{L_{t,x}^2}.
\end{align}
\end{lem}
\begin{proof}
For a dyadic number $M \geq 1$, $u_M$ denotes that the support of $\widehat{u}$ is restricted to
the dyadic block $\{ \langle \tau-p_{\lambda}(k) \rangle \sim M \}$. We use the triangle inequality
and the Plancherel theorem to have
\begin{align*}
\bigl\| P_{\{ |k| \geq 1 \}} (uv) \bigr\|_{L_{t,x}^2} & \lesssim
\sum_{M_1,M_2 \geq 1} \bigl\| P_{\{ |k| \geq 1 \}} (u_{M_1} v_{M_2} ) \bigr\|_{L_{t,x}^2} \\
& \sim \sum_{M_1, M_2 \geq 1} \Bigl\| \frac{1}{ \lambda} \sum_{k_1 \in \mathbb{Z}_{\lambda}}
\int_{\mathbb{R}} \widehat{u}_{M_1}(\tau_1, k_1) \widehat{v}_{M_2} (\tau_2 ,k_2)
d\tau_1 \Bigr\|_{l_k^2 L_{\tau}^2 (|k| \geq 1)} .
\end{align*}
Using the Schwarz inequality twice, the above is bounded by
\begin{align} \label{es_dyb}
\sum_{M_1, M_2 \geq 1} \sup_{(\tau,k) \in \mathbb{R} \times \mathbb{Z}_{\lambda} }
\Bigl( \frac{1}{\lambda}
\sum_{k_1 \in \mathbb{Z}_{\lambda}} \int_{\mathbb{R}} \chi_E(\tau,k,\tau_1,k_1)
d\tau_1 \Bigr)^{1/2}
\| u_{M_1} \|_{L_{t,x}^2} \| v_{M_2} \|_{L_{t,x}^2},
\end{align}
where
\begin{align*}
E:=\bigl\{ (\tau,k,\tau_1, k_1) \in ( \mathbb{R} \times \mathbb{Z}_{\lambda} )^2
~;~ |\tau_1-p_{\lambda} (k_1) | \sim M_1, ~ |\tau_2-p_{\lambda} (k_2)| \sim M_2
, ~|k| \geq 1 \bigr\}.
\end{align*}
We now show the following estimate when
$M_1 \geq M_2$.
\begin{align} \label{es_dy_T1}
\sup_{(\tau,k) \in \mathbb{R} \times \mathbb{Z}_{\lambda}} \frac{1}{\lambda}
\sum_{k_1 \in \mathbb{Z}_{\lambda}} \int_{\mathbb{R}} \chi_E(\tau,k,\tau_1,k_1)
d\tau_1 \lesssim M_1^{9/20} M_2.
\end{align}
The identity,
\begin{align*}
& \bigl( \tau- \frac{k^5}{16}- \beta \lambda^{-2} \frac{k^3}{4} \bigr)-
(\tau_1-p_{\lambda}(k_1)) -(\tau_2-p_{\lambda}(k_2)) \\
& \hspace{1.2cm} = \frac{5}{16} k (k_1-k_2)^2
\bigl\{ (k_1-k_2)^2+ 2 k^2 + \frac{12}{5} \beta \lambda^{-2} \bigr\},
\end{align*}
implies
\begin{align*}
& (k_1-k_2)^2= \\
&~~ \Bigl\{ \frac{ L_0+O(\max \{M_1, M_2 \}) }{|k|} +(k^2 +\frac{6}{5} \beta \lambda^{-2})^2 \Bigr\}^{1/2}-(k^2 + \frac{6}{5} \beta \lambda^{-2}),
\end{align*}
where $ \displaystyle L_0:= \frac{16}{5} \bigl| \tau- \frac{k^5}{16}-\beta \lambda^{-2} \frac{k^3}{4} \bigr|$.
Now $(\tau,k)$ is fixed. Then the variation of $k_1$ is bounded by
\begin{align} \label{es_leng1}
& \lambda \frac{\max \{ M_1, M_2\}}{|k|}
\Bigl\{ \frac{L_0+ O(\max\{ M_1, M_2 \} )}{|k|}+
(k^2+ \frac{6}{5} \beta \lambda^{-2} )^2 \Bigr\}^{-1/2} \nonumber \\
\times & \Bigl[ \Bigl\{\frac{L_0+ O(\max \{M_1,M_2 \}) }{|k|} +(k^2+ \frac{6}{5} \beta \lambda^{-2} )^{2} \Bigr\}^{1/2} -(k^2+ \frac{6}{5} \beta \lambda^{-2} ) \Bigr]^{-1/2}.
\end{align}
Note that for $|k| \geq 1$
\begin{align} \label{es_leng2}
& \Bigl[ \Bigl\{\frac{L_0+ O(\max \{M_1,M_2 \}) }{|k|} +(k^2+ \frac{6}{5} \beta \lambda^{-2} )^{2} \Bigr\}^{1/2} -(k^2+ \frac{6}{5} \beta \lambda^{-2} ) \Bigr]^{-1/2}
\nonumber \\
& \hspace{1cm} \gtrsim |k|^{-3/2} O(\max \{M_1, M_2 \})^{1/4}.
\end{align}
We apply (\ref{es_leng2}) and the Young inequality to (\ref{es_leng1}) so that the variation of $k_1$ is at most
\begin{align*}
\lambda |k|^{3/2-5/2p} \max\{M_1, M_2 \}^{3/4-1/2p} \hspace{0.3cm} \text{for} \hspace{0.3em}
1<p<\infty.
\end{align*}
The above is equal to $\lambda \max\{M_1,M_2 \}^{9/20}$ with $p=5/3$.
If we also fix $k_1$, $\tau_1$ is restricted to the interval of measure
$O(\max\{ M_1,M_2 \})$. Therefore we obtain (\ref{es_dy_T1}) when
$M_1 \geq M_2$.
Substituting (\ref{es_dy_T1}) into (\ref{es_dyb}), we have
\begin{align*}
\| P_{\{ |k| \geq 1 \}} uv \|_{L_{t,x}^2} \lesssim &
\sum_{M_1, M_2 \geq 1} M_1^{9/40} M_2^{1/2} \| u_{M_1} \|_{L_{t,x}^2}
\| v_{M_2} \|_{L_{t,x}^2} \\
= & \sum_{N, M_2 \geq 1} M_2^{29/40} N^{9/40} \| u_{N M_2} \|_{L_{t,x}^2}
\| v_{M_2} \|_{L_{t,x}^2} \\
\lesssim &
\sum_{N \geq 1} \sum_{M_2 \geq 1} N^{9/40-b} M_2^{29/40-(b+b')}
(N M_2)^{b} \| u_{N M_2} \|_{L_{t,x}^2} M_2^{b'} \| v_{M_2} \|_{L_{t,x}^2}.
\end{align*}
Applying the Schwarz inequality in $M_2$ and summing over $N$,
we obtain the desired estimate.
On the other hand, we immediately obtain (\ref{es_L_4_2}) from the duality argument.
\end{proof}
We put a one parameter semigroup $U_{\lambda}(t)$ as follow:
\begin{align*}
U_{\lambda}(t):= \mathcal{F}_{ k }^{-1} \exp(i p_{\lambda}(k) t ) \mathcal{F}_{x}.
\end{align*}
For any time interval $I$, we define the restricted space $Z^s (I)$ by the norm
\begin{align*}
\| u \|_{Z^s(I)}:= \inf \bigl\{ \| v \|_{Z^s} ~;~
u(t)=v(t) ~~\text{on} ~~t \in I \bigr\}.
\end{align*}
From the definition, $Z^s ([0,T])$ has the property as follows;
\begin{align*}
X^{s,3/4} ([0,T]) \hookrightarrow
Z^s ([0,T]) \hookrightarrow C([0,T]; \dot{H}^s (\mathbb{T}_{\lambda})) .
\end{align*}
The above property implies the following linear estimates.
\begin{prop} \label{prop_linear1}
Let $s \in \mathbb{R}$, $T>0$ and $\lambda \geq 1$. Then, we have
\begin{align*}
\| U_{\lambda}(t) u_0 \|_{Z^s([0,T]) } \lesssim \| u_0 \|_{\dot{H}^s(\mathbb{T}) } .
\end{align*}
\end{prop}
\begin{prop} \label{prop_linear2}
Let $s \in \mathbb{R}$, $T>0$ and $\lambda \geq 1$. If the bilinear estimate
(\ref{BE_T2}) holds, then we have
\begin{align*}
\bigl\| \int_0^t U_{\lambda}(t-t') F(t') dt' \bigr\|_{Z^s ([0,T]) }
\lesssim \| u \|_{Z^s ([0,T])}
\| v \|_{Z^s([0,T])}.
\end{align*}
\end{prop}
For the proofs of these propositions, see \cite{BT}.
\section{Proof of the bilinear estimate}
\noindent
In this section, we give a proof of the bilinear estimate (\ref{BE_T2}). For simplicity, we introduce the Fourier multiplier
$J^{\sigma}:= \mathcal{F}_{k}^{-1} \langle k \rangle^{\sigma} \mathcal{F}_x$ for $\sigma \in \mathbb{R}$.
Proposition~\ref{prop_BE_2} can be established by H\"{o}lder's and Young's inequalities and Lemma~\ref{lem_L_4}.
\begin{proof}[Proof of Proposition~\ref{prop_BE_2}]
We prove the following two estimates to obtain (\ref{BE_T2}).
\begin{align}
\label{BE_X}
\| \Lambda^{-1} \p_x (uv) \|_{X_{w}^s} & \lesssim
\| u \|_{Z^s} \| v \|_{Z^s}, \\
\label{BE_Y}
\| \Lambda^{-1} \p_{x} (uv) \|_{Y^s} & \lesssim
\| u \|_{Z^s} \| v \|_{Z^s},
\end{align}
where $\| \cdot \|_{X_w^s}$ is the norm
removing $\| \cdot \|_{Y^s} $ from $\| \cdot \|_{Z^s}$.
Firstly, we divide $(\mathbb{R} \times \dot{\mathbb{Z}}_{\lambda})^2$ into
six parts as follows;
\begin{align*}
\Omega_0 & :=\bigl\{ (\tau,k,\tau_1,k_1) \in (\mathbb{R} \times \dot{\mathbb{Z}}_{\lambda})^2~;~
|k|, |k_1| \lesssim 1 \bigr\}, \\
\Omega_1 & := \bigl\{ (\tau,k,\tau_1,k_1) \in (\mathbb{R} \times \dot{\mathbb{Z}}_{\lambda})^2 \setminus \Omega_0~;~
|k_1| \sim |k-k_1| \gg |k| \geq 1 \bigr\}, \\
\Omega_2 & := \bigl\{ (\tau,k,\tau_1,k_1) \in (\mathbb{R} \times \dot{\mathbb{Z}}_{\lambda})^2 \setminus \Omega_0~;~
|k_1| \sim |k-k_1| \gg |k|~\text{and}~ 1 \geq |k| \geq 1/\lambda \bigr\}, \\
\Omega_3 & := \bigl\{ (\tau,k,\tau_1,k_1) \in (\mathbb{R} \times \dot{\mathbb{Z}}_{\lambda})^2 \setminus \Omega_0~;~
|k| \sim |k-k_1| \gg |k_1| \geq 1 \bigr\}, \\
\Omega_4& := \bigl\{ (\tau,k,\tau_1,k_1) \in (\mathbb{R} \times \dot{\mathbb{Z}}_{\lambda})^2 \setminus \Omega_0~;~
|k| \sim |k-k_1| \gg |k_1 | ~\text{and}~1 \geq |k_1| \geq 1/\lambda \bigr\}, \\
\Omega_5 & := \bigl\{ (\tau,k,\tau_1,k_1) \in (\mathbb{R} \times \dot{\mathbb{Z}}_{\lambda})^2 \setminus \Omega_0~;~
|k| \sim |k_1| \sim |k-k_1| \geq 1 \bigr\}.
\end{align*}
Recall that $Z^s$ has the following properties;
\begin{align*}
\| u \|_{X^{s,1/4}} \lesssim \| u \|_{Z^{s}} \lesssim \| u \|_{X^{s,3/4}}
~~ \text{and} ~~\| u \|_{X^{s,1/2}(D_1 \cup D_2)} \lesssim \| u \|_{Z^{s}(D_1 \cup D_2) }.
\end{align*}
\underline{\bf Estimate in $\Omega_0$}
From the property of $Z^s$, we only estimate the norm $X^{s,3/4}$ of $\Lambda^{-1} \p_x (uv)$.
From $|k|, |k_1|, |k-k_1| \lesssim 1$, we use the H\"{o}lder inequality and the Young inequality to have
\begin{align*}
\| |k| \langle \tau-p_{\lambda} (k) \rangle^{-1/4} \widehat{u}* \widehat{v} \|_{l_k^2 L_{\tau}^2}
\lesssim & \| |k| \|_{l_k^2} \| \widehat{u}* \widehat{v} \|_{l_k^{\infty} L_{\tau}^2} \\
\lesssim & \| \widehat{u} \|_{l_k^2 L_{\tau}^2} \| \widehat{v} \|_{l_k^2 L_{\tau}^1},
\end{align*}
which is an appropriate bound.
Here we put
$L_{\max}= \max \{ |\tau-p_{\lambda}(k)|, |\tau_1-p_{\lambda}(k_1)|, | (\tau-\tau_1) -p_{\lambda}(k-k_1)|
\}$. In the remainder case, we often use the algebraic relation as follows;
\begin{align} \label{alg}
L_{\max} \geq &
\frac{1}{3} \Bigl| (\tau-p_{\lambda}(k))- (\tau_1-p_{\lambda}(k_1)) -\bigl\{ (\tau-\tau_1)-p_{\lambda}(k-k_1) \bigr\} \Bigr| \nonumber \\
\geq &
\frac{5}{6} \Bigl|k k_1 (k-k_1) \bigl\{ k^2+k_1^2+(k-k_1)^2+ \frac{6}{5} \beta
\lambda^{-2} \bigr\} \Bigr| .
\end{align}
\vspace{0.3em}
\noindent
(I) We prove the estimate for $\Omega_1$. We first decompose $\Omega_1$ into three parts as follows;
\begin{align*}
\Omega_{11}& := \bigl\{ (\tau,k,\tau_1,k_1) \in \Omega_1~; ~ |\tau-p_{\lambda}(k)|=L_{\max} \bigr\}, \\
\Omega_{12}& := \bigl\{ (\tau,k, \tau_1,k_1) \in \Omega_1~; ~ |\tau_1-p_{\lambda}(k_1)|=L_{\max} \bigr\}, \\
\Omega_{13}& := \bigl\{ (\tau,k, \tau_1,k_1) \in \Omega_1~; ~ |(\tau-\tau_1)-p_{\lambda}(k-k_1)|=L_{\max} \bigr\}.
\end{align*}
The case $\Omega_{13}$ is identical to the case $\Omega_{12}$. So we omit this case.
Note that $L_{\max} \gtrsim |k k_1^4|$ in $\Omega_1$ from (\ref{alg}).
\vspace{0.3em}
\noindent
(Ia) In $\Omega_{11}$,
$\widehat{u}* \widehat{v}$ is supported on $D_3$ from the definition. We use Lemma~\ref{lem_L_4} with $b'=1/4$ and
$b=1/2$ to obtain
\begin{align*}
\| \langle k \rangle^{-s/2} \langle \tau-p_{\lambda} (k) \rangle^{s/2} \widehat{u}* \widehat{v} \|_{l_k^2 L_{\tau}^2}
\lesssim \| J^s u J^s v \|_{L_{t,x}^2} \lesssim \| u \|_{X^{s,1/2}} \| v \|_{X^{s,1/4}},
\end{align*}
which implies the desired estimate except the case $\widehat{u}$ and $\widehat{v}$ are restricted to $D_3$.
Next we consider the case both $\widehat{u}$ and $\widehat{v}$ are supported on $D_3$. We use H\"{o}lder's inequality
and Young's inequality to have
\begin{align*}
& \| \langle k \rangle^{-s/2} \langle \tau-p_{\lambda}(k) \rangle^{s/2} \widehat{u} * \widehat{v} \|_{l_k^{2} L_{\tau}^2}
\lesssim \| \widehat{u} * (\langle k \rangle^{2s} \widehat{v} ) \|_{l_k^2 L_{\tau}^2} \\
& \hspace{0.3cm} \lesssim \| (\langle k \rangle^{-s/2-1} \langle \tau-p_{\lambda}(k) \rangle^{s/2+1} \widehat{u})*
(\langle k \rangle^{-4} \widehat{v}) \|_{l_k^{2} L_{\tau}^2}
\lesssim \| u \|_{X^{-s/2-1,s/2+1}} \| \langle k \rangle^{-4} \widehat{v} \|_{l_k^1 L_{\tau}^1},
\end{align*}
which shows the required estimate since $\| \langle k \rangle^{-4} \widehat{v} \|_{l_k^1 L_{\tau}^1}
\lesssim \| v \|_{Y^s}$ by the Schwarz inequality.
Moreover we estimate the norm $Y^s$ of $\Lambda^{-1} \p_x(uv)$. Following
$|\tau-p_{\lambda}(k)| \gtrsim |k k_1^4|$, we use the H\"{o}lder inequality and the Young inequality to obtain
\begin{align*}
\| \langle k \rangle^{s+1} \langle \tau-p_{\lambda}(k) \rangle^{-1} \widehat{u}*\widehat{v} \|_{l_k^2 L_{\tau}^1} \lesssim
\| (\langle k \rangle^{-2} \widehat{u}) *( \langle k \rangle^{-2} \widehat{v} ) \|_{l_k^{\infty} L_{\tau}^1} \lesssim \| u \|_{Y^s} \| v \|_{Y^s}.
\end{align*}
\vspace{0.3em}
\noindent
(Ib) We show the estimate for $\Omega_{12}$. Consider three subregions
\begin{align*}
\Omega_{12a}& := \bigl\{ (\tau,k, \tau_1,k_1) \in \Omega_{12}~; ~ |\tau_1-p_{\lambda}(k_1)| \sim |k k_1^4| \bigr\}, \\
\Omega_{12b}& := \bigl\{ (\tau,k, \tau_1,k_1) \in \Omega_{12}~; ~ |k_1|^5 \gtrsim |\tau_1-p_{\lambda}(k_1)| \gg |k k_1^4| \bigr\}, \\
\Omega_{12c}& := \bigl\{ (\tau,k, \tau_1,k_1) \in \Omega_{12}~; ~ |\tau_1-p_{\lambda}(k_1)| \gtrsim |k_1|^5 \bigr\},
\end{align*}
In $\Omega_{12a}$, $\widehat{u}$ is restricted to $D_2$. Then we use Lemma~\ref{lem_L_4} with $b'=1/4$ and $b=1/2$ to obtain
\begin{align*}
\| \langle k \rangle^s \langle \tau-p_{\lambda}(k) \rangle^{-1/4} \widehat{u}*\widehat{v} \|_{l_k^2 L_{\tau}^2}
\lesssim & \| (J^{-3s-1} \Lambda^{s+1} u) (J^{-s-3} v )\|_{X^{0,-1/4}} \\
\lesssim & \| u \|_{X^{-3s-1,s+1}} \| v \|_{X^{s,1/2}}.
\end{align*}
In $\Omega_{12b}$ and $\Omega_{12c}$, either $|\tau_1-p_{\lambda}(k_1)| \sim |\tau-p_{\lambda}(k) |$ or
$|\tau_1-p_{\lambda}(k_1)| \sim |(\tau-\tau_1)- p_{\lambda}(k-k_1)| $ happens.
The former case is almost identical to the case (Ia).
So we only consider the latter case.
We prove the estimate for $\Omega_{12b}$.
In this case, we may assume that both $\widehat{u}$ and $\widehat{v}$ are supported on
$D_2$ and $|\tau-p_{\lambda} (k)| \lesssim |k_1|^5$.
From $\langle k_1 \rangle^{3s+1} \langle \tau_1-p_{\lambda} (k_1) \rangle^{-s-1} \lesssim
\langle k_1 \rangle^{-2s-4}$, we use the H\"{o}lder inequality and
the Young inequality to have
\begin{align*}
& \| \langle k \rangle^{s+1} \langle \tau-p_{\lambda}(k ) \rangle^{-1/4}
\widehat{u}* \widehat{v} \|_{l_k^2 L_{\tau}^2}
\lesssim \| \langle k \rangle^{s+3/2} \langle \tau-p_{\lambda}(k)
\rangle^{1/4} \widehat{u}* \widehat{v} \|_{l_k^{\infty} L_{\tau}^{\infty}} \\
& \hspace{0.3cm} \lesssim
\| (\langle k \rangle^{s+11/4} \widehat{u})* \widehat{v} \|_{l_k^{\infty} L_{\tau}^{\infty} }
\lesssim \| \langle k \rangle^{-3s-21/4} \|_{l_k^{\infty}} \| u \|_{X^{-3s-1,s+1}} \| v \|_{X^{-3s-1,s+1}},
\end{align*}
which is an appropriate bound.
\vspace{0.3em}
From the similar argument to above, we obtain the desired estimate for $\Omega_{12c}$.
\vspace{0.5em}
\noindent
\underline{\bf Estimate for $\Omega_2$}\\
\noindent
(II) We divide $\Omega_{2}$ into three parts as follows;
\begin{align*}
\Omega_{21} & := \bigl\{ (\tau,k,\tau_1,k_1) \in \Omega_2~;~
L_{\max}= |\tau-p_{\lambda}(k)| \bigr\}, \\
\Omega_{22} & := \bigl\{ (\tau,k,\tau_1,k_1) \in \Omega_2~;~
L_{\max}= |\tau_1-p_{\lambda}(k_1)| \bigr\}, \\
\Omega_{23} & := \bigl\{ (\tau,k,\tau_1,k_1) \in \Omega_2~;~
L_{\max}= |(\tau-\tau_1)-p_{\lambda}(k-k_1)| \bigr\}.
\end{align*}
We omit the estimate for $\Omega_{23}$ because this case is identical to
$\Omega_{22}$. Note that $\widehat{u}* \widehat{v}$ is supported on
$D_3$ in $\Omega_2$.
When $|k| \lesssim |k_1|^{-4}$, H\"{o}lder's and Young's inequalities show
\begin{align*}
\| |k| \langle \tau-p_{\lambda}(k) \rangle^{-1/4} \widehat{u} * \widehat{v} \|_{l_k^2 L_{\tau}^2}
& \lesssim \| (\langle k\rangle^{-3} \widehat{u})* (\langle k \rangle^{-3} \widehat{v}) \|_{l_k^{\infty}L_{\tau}^2} \\
& \lesssim \| u \|_{X^{-3,0}} \| v \|_{Y^{-3}},
\end{align*}
which implies the desired estimate. So we only deal with the case
$|k_1|^{-4} \lesssim |k| \leq 1$.
\vspace{0.3em}
\noindent
(IIa) We prove the estimate for $\Omega_{21}$. We use the H\"{o}lder inequality and the Young inequality to obtain
\begin{align*}
& \| |k| \langle \tau-p_{\lambda}(k) \rangle^{s/2} \widehat{u}* \widehat{v} \|_{l_k^2 L_{\tau}^2}
\lesssim \| |k|^{1+s/2} (\langle k \rangle^s \widehat{u}) * (\langle k \rangle^{s} \widehat{v}) \|_{l_k^{2} L_{\tau}^2} \\
& \hspace{0.8cm} \lesssim \| (\langle k \rangle^{s} \widehat{u}) * (\langle k \rangle^s \widehat{v} ) \|_{l_k^{\infty} L_{\tau}^2} \lesssim
\| u \|_{X^{s,0}} \| v \|_{Y^s}.
\end{align*}
Next we estimate the $Y^s$ norm of $\Lambda^{-1} \p_x (uv)$. Combining H\"{o}lder's and Young's inequalities, we have
\begin{align*}
\| |k| \langle \tau-p_{\lambda}(k) \rangle^{-1} \widehat{u} * \widehat{v}
\|_{l_k^2 L_{\tau}^1}
\lesssim & \| (\langle k \rangle^{-2} \widehat{u}) * (\langle k \rangle^{-2} \widehat{v}) \|_{l_{k}^{\infty} L_{\tau}^1} \\
\lesssim & \| \langle k \rangle^{-2} \widehat{u} \|_{l_k^2 L_{\tau}^1}
\| \langle k \rangle^{-2} \widehat{v} \|_{l_k^2 L_{\tau}^1},
\end{align*}
which is an appropriate bound.
\noindent
(IIb) We consider the estimate for $\Omega_{22}$. Following $\| u \|_{Y^s} \lesssim \| u \|_{X^{s,1/2+}}$,
it suffices to show
\begin{align} \label{BE_3}
\| |k| \langle \tau-p_{\lambda}(k) \rangle^{-1/2+} \widehat{u}*
\widehat{v} \|_{l_k^2 L_{\tau}^2} \lesssim \| u \|_{Z^s} \| v \|_{Z^s}
\end{align}
in $\Omega_{22}$. We consider three subregions as follows;
\begin{align*}
\Omega_{22a} & := \bigl\{ (\tau,k,\tau_1,k_1) \in \Omega_{22}~;~
|k k_1|^4 \lesssim |\tau_1-p_{\lambda}(k_1)| \lesssim |k_1|^4 \bigr\}, \\
\Omega_{22b} & := \bigl\{ (\tau,k,\tau_1,k_1) \in \Omega_{22}~;~
|k_1^4| \lesssim |\tau_1-p_{\lambda} (k_1)| \lesssim |k_1|^5 \bigr\}, \\
\Omega_{22c} & := \bigl\{ (\tau,k,\tau_1,k_1) \in \Omega_{22}~;~
|k_1|^5 \lesssim |\tau_1-p_{\lambda}(k_1)| \bigr\}.
\end{align*}
In $\Omega_{22a}$, $\widehat{u}$ is restricted to $D_1$. We use the H\"{o}lder inequality and the Young inequality to obtain
\begin{align*}
& \| |k| \langle \tau-p_{\lambda}(k) \rangle^{-1/2+} \widehat{u}* \widehat{v} \|_{l_k^2 L_{\tau}^2}
\lesssim \| |k|^{1/4}
(\langle k \rangle^s \langle \tau-p_{\lambda}(k) \rangle^{3/4} \widehat{u} )
* (\langle k \rangle^{-s-3} \widehat{v}) \|_{l_k^2 L_{\tau}^2} \\
& \hspace{0.3cm} \lesssim \| (\langle k \rangle^{s}
\langle \tau-p_{\lambda}(k) \rangle^{3/4} \widehat{u}) * (\langle k \rangle^s
\widehat{v}) \|_{l_k^{\infty} L_{\tau}^2 }
\lesssim \| u \|_{X^{s,3/4} } \| v \|_{Y^s}.
\end{align*}
We consider the estimate for $\Omega_{22b}$ and $\Omega_{22c}$. From the estimate for $\Omega_{12}$,
we immediately obtain (\ref{BE_3}) in the case
$|\tau-p_{\lambda} (k)| \sim |\tau_1-p_{\lambda} (k_1) |$. So we only deal with the case
$|\tau_1-p_{\lambda}(k_1)| \sim |(\tau-\tau_1) -p_{\lambda}(k-k_1)|$.
In $\Omega_{22b}$, both $\widehat{u}$ and $\widehat{v}$ are restricted to $D_2$.
We use Lemma~\ref{lem_L_4} with $b'=1/2-$ and $b=s/2+1$ to have
\begin{align*}
\| |k| \langle \tau-p_{\lambda}(k) \rangle^{-1/2+} \widehat{u}* \widehat{v} \|_{l_k^2 L_{\tau}^2}
\lesssim & \| (J^{-3s-1} \Lambda^{s+1} u) (J^{-2s-4} v ) \|_{X^{0, -1/2+}} \\
& \lesssim \| u \|_{X^{-3s-1,s+1}} \| v \|_{X^{-2s-4,s/2+1}},
\end{align*}
which shows the desired estimate since $\| v \|_{X^{-2s-4,s/2+1}} \lesssim
\| v \|_{X^{-3s-1,s+1}}$ in $D_{2}$ for $s \geq -3/2$.
The case $\Omega_{22c}$ is almost identical to the above case.
\vspace{0.5em}
\noindent
\underline{\bf Estimate for $\Omega_{3}$} \\
\noindent
(III) From the algebraic relation (\ref{alg}), $L_{\max} \gtrsim |k_1 k^4|$.
We decompose $\Omega_3$ into three parts as follows;
\begin{align*}
\Omega_{31} & := \bigl\{ (\tau,k,\tau_1,k_1) \in \Omega_3~;~
L_{\max}= |\tau_1-p_{\lambda}(k_1)| \bigr\}, \\
\Omega_{32} & := \bigl\{ (\tau,k,\tau_1,k_1) \in \Omega_3~;~
L_{\max}= |\tau-p_{\lambda}(k)| \bigr\}, \\
\Omega_{33} & := \bigl\{ (\tau,k,\tau_1,k_1) \in \Omega_3~;~
L_{\max}= |\tau_2-p_{\lambda}(k_2) | \bigr\}.
\end{align*}
\noindent
(IIIa) Firstly, we consider the case $\widehat{u} * \widehat{v}$ is supported on $D_3$. In this case,
either $|\tau-p_{\lambda}(k)| \sim |\tau_1-p_{\lambda} (k_1) | \gtrsim |k|^5$ or
$|\tau-p_{\lambda} (k)| \sim |\tau_2 -p_{\lambda} (k_2)| \gtrsim |k|^5$ holds.
In the former case, $\widehat{u}$ are supported on $D_3$. We use the Young inequality to obtain
\begin{align*}
\| \langle k \rangle^{-s/2} \langle \tau-p_{\lambda}(k) \rangle^{s/2} \widehat{u} * \widehat{v} \|_{l_k^2 L_{\tau}^2}
& \lesssim \| (J^{-s/2-1} \Lambda^{s/2+1} u) (J^{-4} v) \|_{L_{t,x}^2} \\
& \lesssim \| u \|_{X^{-s/2-1,s/2+1}} \| \langle k \rangle^{-4} v \|_{l_k^{1} L_{\tau}^1},
\end{align*}
which is bounded by $\| u \|_{X^{-s/2-1,s/2+1}} \| v \|_{Y^s}$ from the Schwarz inequality.
The latter case is almost identical to the above case.
Secondly, we deal with the case $\widehat{v}$ is supported on $D_3$.
From (\ref{alg}), $|\tau_2 -p_{\lambda} (k_2)| \sim |\tau-p_{\lambda} (k)| \gtrsim |k|^5$ or
$|\tau_2-p_{\lambda} (k_2) | \sim |\tau_1-p_{\lambda} (k_1)| \gtrsim |k|^5$ holds.
In the former case, we have already proven (\ref{BE_T2}). So we consider the latter case.
We may assume that $\widehat{u}$ is restricted to $D_3$
and $|\tau-p_{\lambda} (k)| \lesssim |k|^5$. We use the H\"{o}lder inequality and the Young inequality to have
\begin{align*}
& \| \langle k \rangle^{s+1} \langle \tau-p_{\lambda}(k) \rangle^{-1/4} \widehat{u} * \widehat{v} \|_{l_k^2 L_{\tau}^2}
\lesssim \| \langle k \rangle^{s+3/2} \langle \tau-p_{\lambda} (k) \rangle^{1/4} \widehat{u}*
\widehat{v} \|_{l_k^{\infty} L_{\tau}^{\infty}} \\
& ~~ \lesssim \| \langle k \rangle^{-3s-21/4} \|_{l_k^{\infty}}
\| u \|_{X^{-s/2-1,s/2+1}} \| v \|_{X^{-s/2-1,s/2+1}},
\end{align*}
which shows the required estimate. Therefore we only deal with the case
both $\widehat{u}* \widehat{v}$ and $\widehat{v}$ are supported on $D_1 \cup D_2$.
\vspace{0.3em}
\noindent
(IIIb) We estimate (\ref{BE_T2}) for $\Omega_{31}$. In $\Omega_{31}$, $\widehat{u}$ is supported on $D_3$ from
$L_{\max} \gtrsim |k_1 k^4|$.
We use Lemma~\ref{lem_L_4} with $b'=1/4$ and $b=1/2$ to obtain
\begin{align*}
\| \langle k \rangle^{s+1} \langle \tau-p_{\lambda}(k) \rangle^{-1/4} \widehat{u} * \widehat{v} \|_{l_k^2 L_{\tau}^2}
& \lesssim \| (J^{-s/2-1} \Lambda^{s/2+1} u) (J^{-s-3} v) \|_{X^{0,-1/4}} \\
& \lesssim \| u \|_{X^{-s/2-1,s/2+1}} \| v \|_{X^{s,1/2}},
\end{align*}
which is an appropriate bound.
\vspace{0.3em}
\noindent
(IIIc)
We consider the estimate for $\Omega_{32}$. From (\ref{alg}),
$\widehat{u}* \widehat{v}$ is supported on $D_2$.
We use Lemma~\ref{lem_L_4} with $b=1/4$ and $b'=1/2$ to have
\begin{align*}
\| \langle k \rangle^{-3s} \langle \tau-p_{\lambda}(k) \rangle^{s}
\widehat{u} * \widehat{v} \|_{l_k^2 L_{\tau}^2}
& \lesssim
\| (J^s u) (J^s v) \|_{L_{t,x}^2} \\
& \lesssim \| u \|_{X^{s,1/4}} \| v \|_{X^{s,1/2}},
\end{align*}
which is an appropriate bound.
Next, we estimate the $Y^s$ norm of $\Lambda^{-1} \p_{x} (uv)$.
The Young inequality shows
\begin{align*}
\| \langle k \rangle^{s+1} \langle \tau-p_{\lambda}(k) \rangle^{-1} \widehat{u} * \widehat{v} \|_{l_k^2 L_{\tau}^1}
& \lesssim \| (\langle k \rangle^{-4} \widehat{u})* (\langle k \rangle^{s} \widehat{v}) \|_{l_k^2 L_{\tau}^1} \\
& \lesssim \| \langle k \rangle^{-4} \widehat{u} \|_{l_k^1 L_{\tau}^2 } \| v \|_{Y^s},
\end{align*}
which implies the desired estimate from the Schwarz inequality.
\vspace{0.3em}
\noindent
(IIId) We consider the estimate for $\Omega_{33}$.
From (\ref{alg}), we may assume that $\widehat{v}$ is supported on $D_2$ and
$|\tau-p_{\lambda} (k)| \lesssim |k|^5$.
In the case $\widehat{u}$ is supported on $D_1 \cup D_2$,
we use Lemma~\ref{lem_L_4} with $b'=1/4$ and $b=1/2$ to obtain
\begin{align*}
\| \langle k \rangle^{s+1} \langle \tau-p_{\lambda} (k) \rangle^{-1/4}
\widehat{u} * \widehat{v} \|_{l_k^2 L_{\tau}^2}
& \lesssim \| (J^{-s-3} u) (J^{-3s-1} \Lambda^{s+1} v ) \|_{X^{0,-1/4}} \\
& \lesssim \| u \|_{X^{s,1/2}} \| v \|_{X^{-3s-1,s+1}}.
\end{align*}
On the other hand, we consider the case $\widehat{u}$ is supported on $D_3$.
Then we use Lemma~\ref{lem_L_4} with $b'=-s/2-1/4$ and $b=s/2+1$ to have
\begin{align*}
& \| \langle k \rangle^{s+1} \langle \tau-p_{\lambda} (k) \rangle^{-1/4}
\widehat{u} * \widehat{v} \|_{l_k^2 L_{\tau}^2}
\lesssim \| \langle k \rangle^{-3s/2-3/2}
\langle \tau-p_{\lambda} (k) \rangle^{s/2+1/4}
\widehat{u} * \widehat{v} \|_{l_k^2 L_{\tau}^2} \\
& \lesssim
\| (J^{-7s/2-11/2} u) (J^{-3s-1 } \Lambda^{s+1} v) \|_{X^{0,s/2+1/2}}
\lesssim \| u \|_{X^{-7s/2-11/2, s/2+1}} \| v \|_{X^{-3s-1,s+1}},
\end{align*}
which shows the desired estimate since $-7s/2 -11/2 \leq -s/2-1$ for $s \geq -3/2$.
\vspace{0.5em}
\noindent
\underline{\bf Estimate for $\Omega_{4}$}\\
\noindent
(VI) In $\Omega_4$, $\widehat{u}$ is restricted to $D_3$.
We divide $\Omega_4$ into three parts as follows;
\begin{align*}
\Omega_{41}:= & \bigl\{ (\tau,k, \tau_1, k_1) \in \Omega_{4}~; ~
L_{\max}= |\tau_1-p_{\lambda} (k_1)| \bigr\}, \\
\Omega_{42}:= & \bigl\{ (\tau,k, \tau_1, k_1) \in \Omega_{4}~; ~
L_{\max}= |\tau-p_{\lambda} (k)| \bigr\}, \\
\Omega_{43}:= & \bigl\{ (\tau,k, \tau_1, k_1) \in \Omega_{4}~; ~
L_{\max}= |\tau_2-p_{\lambda} (k_2)| \bigr\}.
\end{align*}
When $|k_1| \lesssim |k|^{-4}$, we easily obtain the desired estimate combining H\"{o}lder's and Young's inequalities.
So we only deal with the case $|k|^{-4} \lesssim |k_1| \leq 1$.
\vspace{0.3em}
\noindent
(VIa) In $\Omega_{41}$, we use the H\"{o}lder inequality and the Young inequality to obtain
\begin{align*}
& \| \langle k \rangle^{s+1} \langle \tau-p_{\lambda}(k) \rangle^{-1/4} \widehat{u}* \widehat{v} \|_{l_k^2 L_{\tau}^2}
\lesssim \| (|k|^{-1/4} \langle \tau-p_{\lambda}(k) \rangle^{1/4} \widehat{u} ) * ( \langle k \rangle^{s} \widehat{v} ) \|_{l_k^2 L_{\tau}^2} \\
&~~\lesssim \| |k|^{-1/4} \langle \tau-p_{\lambda}(k) \rangle^{1/4} \widehat{u} \|_{l_k^{1} L_{\tau}^2} \| v \|_{Y^s}
\lesssim \| u \|_{X^{0,1/4}} \| v \|_{Y^s}.
\end{align*}
\noindent
(VIb) In $\Omega_{42}$, we use Young's inequality to have
\begin{align*}
\| \langle k \rangle^{s+1} \langle \tau-p_{\lambda}(k) \rangle^{-1/4} \widehat{u} * \widehat{v} \|_{l_k^2 L_{\tau}^2}
& \lesssim \| (|k|^{-1/4} \widehat{u})* (\langle k \rangle^{s} \widehat{v}) \|_{l_{k}^2 L_{\tau}^2} \\
& \lesssim \| |k|^{-1/4} \widehat{u} \|_{l_k^1 L_{\tau}^2} \| v \|_{Y^{s}},
\end{align*}
which is an appropriate bound from Schwarz's inequality.
\noindent
(VIc) From $\langle k_2 \rangle^{-s} \langle \tau_2-p_{\lambda}(k_2) \rangle^{-1/4} \lesssim |k_1|^{-1/4} \langle k_2 \rangle^{-s-1}$ in $\Omega_{43}$,
we use the H\"{o}lder inequality and the Young inequality to have
\begin{align*}
& \| \langle k \rangle^{s+1} \langle \tau-p_{\lambda}(k) \rangle^{-1/4} \widehat{u} * \widehat{v} \|_{l_k^2 L_{\tau}^2}
\lesssim \| ( |k|^{-1/4} \widehat{u} ) * (\langle k \rangle^s \langle \tau-p_{\lambda}(k) \rangle^{1/4} \widehat{v}) \|_{l_k^{2} L_{\tau}^2} \\
& \hspace{0.3cm} \lesssim \| |k|^{-1/4} \widehat{u} \|_{l_k^1 L_{\tau}^1}
\| v \|_{X^{s,1/4}} \lesssim \| u \|_{Y^s} \| v \|_{X^{s,1/4}}.
\end{align*}
\vspace{0.5em}
\noindent
\underline{\bf Estimate for $\Omega_{5}$}\\
We decompose $\Omega_{5}$ into three parts as follows;
\begin{align*}
\Omega_{51}:= & \bigl\{ (\tau,k ,\tau_1, k_1) \in \Omega_{5} ~: ~
L_{\max}= |\tau-p_{\lambda}(k)| \bigr\}, \\
\Omega_{52}:= & \bigl\{ (\tau,k ,\tau_1, k_1) \in \Omega_{5} ~: ~
L_{\max}= |\tau_1-p_{\lambda}(k_1)| \bigr\}, \\
\Omega_{53}:= & \bigl\{ (\tau,k ,\tau_1, k_1) \in \Omega_{5} ~: ~
L_{\max}= |\tau_2-p_{\lambda}(k_2)| \bigr\}.
\end{align*}
\vspace{0.3em}
\noindent
(Va) In $\Omega_{51}$, $\widehat{u} * \widehat{v}$ is supported on $D_3$. We divide this region into
\begin{align*}
\Omega_{51a}:= & \bigl\{ (\tau,k ,\tau_1, k_1) \in \Omega_{51} ~: ~
|\tau-p_{\lambda}(k)| \sim |k^5| \bigr\}, \\
\Omega_{51b}:= & \Omega_{51} \setminus \Omega_{51a}.
\end{align*}
In $\Omega_{51a}$, both $\widehat{u}$ and $\widehat{v}$ are supported on $D_1 \cup D_2$ from (\ref{alg}).
We use Lemma~\ref{lem_L_4} with $b'=1/4$ and $b=1/2$ to have
\begin{align*}
\| \langle k \rangle^{-s/2} \langle \tau-p_{\lambda}(k) \rangle^{s/2}
\widehat{u}* \widehat{v} \|_{l_k^2 L_{\tau}^2}
& \lesssim \|(J^s u) (J^s v ) \|_{L_{t,x}^2} \\
& \lesssim \| u \|_{X^{s,1/2}} \| v \|_{X^{s,1/4}}.
\end{align*}
In $\Omega_{51b}$, either $|\tau-p_{\lambda} (k)| \sim |\tau_1-p_{\lambda} (k_1)| $ or $|\tau-p_{\lambda} (k)| \sim |\tau_2-p_{\lambda} (k_2)|$ holds.
Following the similar argument to the case $\Omega_{11}$, we obtain the desired estimate in $\Omega_{51b}$.
\vspace{0.5em}
\noindent
(Vb) We consider the estimate for $\Omega_{52}$. From (\ref{alg}), $\widehat{u}$ is supported on $D_3$.
We divide $\Omega_{52}$ into
\begin{align*}
\Omega_{52a}:= & \bigl\{ (\tau,k ,\tau_1, k_1) \in \Omega_{52} ~: ~
|\tau_1-p_{\lambda}(k_1)| \sim |k_1^5| \bigr\}, \\
\Omega_{52b}:= & \Omega_{52} \setminus \Omega_{52a}.
\end{align*}
In $\Omega_{52a}$, $\widehat{u} * \widehat{v}$ and $\widehat{v}$ are supported on $D_1 \cup D_2$ under this assumption.
Then we use Lemma~\ref{lem_L_4} with $b'=1/4$ and $b=1/2$ to have
\begin{align*}
\| \langle k \rangle^{s+1} \langle \tau-p_{\lambda}(k) \rangle^{-1/4}
\widehat{u}* \widehat{v} \|_{l_k^2 L_{\tau}^2}
& \lesssim \|(J^{-s/2-1} \Lambda^{s/2+1} u) (J^{-s-3} v ) \|_{X^{0,-1/4}} \\
& \lesssim \| u \|_{X^{-s/2-1,s/2+1}} \| v \|_{X^{s,1/2}},
\end{align*}
which is an appropriate bound.
In $\Omega_{52b}$, either $|\tau_1-p_{\lambda} (k_1)| \sim |\tau-p_{\lambda} (k)|$ or $|\tau_1 -p_{\lambda}(k_1) | \sim |\tau_2-p_{\lambda}(k_2) |$ holds.
These cases are almost identical to the case (IIIa).
In the same manner as above, we obtain the desired estimate in $\Omega_{53}$
by symmetry.
\end{proof}
\section{Proof of the local well-posedness}
In this section, we give the proof of Theorem~\ref{LWP_TK} by the
iteration method.
Here we put $U(t):=\mathcal{F}_{k}^{-1} \exp(i p(k) t) \mathcal{F}_x$ and $p(k):=k^5 +\beta k^3$.
We obtain the local well-posedness result in the following sense.
\begin{prop} \label{prop_well}
Let $-3/2 \leq s \leq -1$ and $r>1$. For any $u_0 \in B_r(\dot{H}^{s})$, there exist $T=T(r)>0$ and a unique solution
$u \in Z^{s}([0,T])$ satisfying the following integral form for (\ref{KT_1});
\begin{align} \label{integral-1}
u(t)= U(t) u_0 -\int_{0}^t U(t-s) \p_x (u (s))^2 ds.
\end{align}
Moreover the data-to-solution map,
$B_r(\dot{H}^{s}) \ni u_0 \mapsto u \in Z^{s} ([0,T])$, is locally Lipschitz continuous.
\end{prop}
\begin{proof}
We first prove the existence of the solution by the fixed point argument. Here $\lambda$ is a
sufficiently large number determined later.
For any $u_0 \in B_r (\dot{H}^s)$, from (\ref{in_sm}),
$\| u_{0,\lambda} \|_{\dot{H}^s} \leq \lambda^{-2} r $ when $-3/2 \leq s <0$.
Therefore we prove, for any $u_{0,\lambda} \in B_{\lambda^{-2} r}(\dot{H}^s)$, there exists
$u_{\lambda} \in Z^s([0,1])$ satisfying
\begin{align} \label{integral-2}
M[u_{\lambda} ] (t) =u_{\lambda} (t), \hspace{0.3em}
M[u_{\lambda}] (t)=U_{\lambda} (t) u_{0,\lambda}-\int_0^t
U_{\lambda} (t-s) \p_x (u_{\lambda}(s))^2 ds.
\end{align}
Following Propositions~\ref{prop_BE_2} and \ref{prop_linear2},
we obtain the bilinear estimate as follows;
\begin{align} \label{BE_time}
\Bigl\| \int_0^t U_{\lambda} (t-s) \p_x (u_{\lambda} (s) v_{\lambda} (s)) ds \Bigr\|_{Z^s([0,1])}
\leq C_1 \| u_{\lambda} \|_{Z^s([0,1])} \| v_{\lambda} \|_{Z^s([0,1])},
\end{align}
for some constant $C_1>0$.
From Proposition~\ref{prop_linear1} and (\ref{BE_time}), we have
\begin{align*}
\| M [u_{\lambda}] \|_{Z^s ([0,1])} \leq C_1 \bigl( \| u_{0,\lambda} \|_{\dot{H}^s}+
\| u_{\lambda} \|_{Z^s([0,1])}^2 \bigr).
\end{align*}
Here we choose $\lambda^2 \geq 8 C_1^2 r$ so that
$M$ is a map from $B_{2C_1 \lambda^{-2} r} (Z^s ([0,1]))$ to itself.
In the same manner as above, we obtain
\begin{align*}
\| M [u_{\lambda}] -M [v_{\lambda}] \|_{Z^s ([0,1])} & \leq C_1
\| u_{\lambda} + v_{\lambda} \|_{Z^s ([0,1])}
\| u_{\lambda}-v_{\lambda} \|_{Z^s ([0,1])} \\
& \leq 4 \lambda^{-2} C_1^2 r \| u_{\lambda} -v_{\lambda} \|_{Z^s ([0,1])}
\leq \frac{1}{2} \| u_{\lambda} -v_{\lambda} \|_{Z^s ([0,1])},
\end{align*}
which implies that $M$ is a contraction map on $B_{2 C_1 \lambda^{-2} r} (Z^s([0,1]))$.
From the fixed point argument,
we construct the solution to (\ref{integral-2}) on $[0,1]$. Here we put
$u(t,x):= \lambda^{4} u_{\lambda} (\lambda^5 t, \lambda x)$. Then $u$ solves (\ref{integral-1})
on $[0,T]$ where the lifetime $T$ satisfies $T \sim \lambda^{-5} \sim r^{-5/2}$. Moreover, following the standard
argument, we show that the data-to-solution map is locally Lipschitz continuous.
Moreover uniqueness can be extended to the whole $Z^s([0,T])$.
This proof is based on Muramatu and Taoka's work \cite{MT}.
For the details, see \cite{TK_K}.
\end{proof}
\section{Proof of the global well-posedness}
In this section, we extend the local solution obtained above globally in time by the I-method.
When $s \geq -1$,
Hirayama \cite{Hi} obtained LWP for (\ref{KT_1}) in the function space $W^s([0,T])$ equipped with the norm
\begin{align*}
\| u \|_{W^s([0,T])}:= \| u \|_{X^{s,1/2}([0,T])}+ \| u \|_{Y^{s}([0,T])}.
\end{align*}
These local-in-time solutions are shown to exist on an arbitrary time interval for $0> s \geq -1$.
Note that $s=-1$ is optimal in such sense that the bilinear estimate in the standard Bourgain space
fails for $s<-1$.
The proof is an adaptation of the argument presented for the periodic KdV equation in
\cite{CoKeSt}. Remark that we encounter difficulty such that the Kawahara equation
has less symmetries than the KdV equation.
Before modified energies are introduced, we prepare some notations.
A $l$ multiplier is a function $M; \mathbb{R}^l \rightarrow \mathbb{C}$.
We say a $l$ multiplier $M$ is symmetric if
$M(k_1, k_2, \cdots, k_l)=M(k_{\sigma(1)} , k_{\sigma(2) }, \cdots, k_{\sigma(l)} )$ for
all $\sigma \in S_l$.
The symmetrization of a $l$ multiplier $M$ is defined by
\begin{align*}
[M]_{sym} (k_1, k_2, \cdots, k_l) := \frac{1}{ l! } \sum_{\sigma \in S_l}
M (k_{\sigma(1)}, k_{\sigma(2)}, \cdots, k_{\sigma(l)} ).
\end{align*}
We define a $l$-linear functional associated to the function $M$ acting on $l$ functions $u_1,u_2, \cdots, u_l$,
\begin{align*}
\Lambda_l (M; u_1, u_2, \cdots, u_l):= \int_{k_1+k_2+\cdots +k_l =0 } M(k_1,k_2, \cdots, k_l)
\prod_{i=1}^{l} \widehat{u}_i(k_i).
\end{align*}
$ \Lambda_l (M; u, \cdots,u )$ is simply written as $\Lambda_l (M)$.
We recall the original modified energy $E_I^{(2)} (u) (t) = \| I u (t) \|_{L^2}^2$.
We use this functional to obtain GWP for $-21/26< s <0$ but not
$-1 \leq s \leq -21/26$. Then we construct new modified energies by adding some correction terms to
$E_I^{(2)}(u)$, following the argument to \cite{CoKeSt}.
Using $u$ is real valued and $m$ is even, we use the Plancherel theorem to have
\begin{align*}
E_I^{(2)} (u) (t)= \Lambda_2 ( m(k_1) m(k_2)) (t).
\end{align*}
Here $a_l$, $b_l$ denote $a_l =i \sum_{j=1}^{l} k_j^5$ and $b_l= i \sum_{j=1}^{l} k_j^{3}$.
We compute the time derivative of the modified energy $E_I^{(2)} (u)$ to have
\begin{align*}
\frac{d}{dt} E_{I}^{(2)} (u) (t)= & \Lambda_{2} ((a_2+ \lambda^{-2} \beta b_2) m(\xi_1) m(\xi_2) ) (t) \\
- & 2i \Lambda_3 ([ ( k_2+ k_3) m(k_1) m(k_2+ k_3) ]_{sym}) (t) .
\end{align*}
Here the first term vanishes because $a_2=0$ and $b_2=0$. Therefore the time derivative of $E_I^{(2)}(u)$ has the cubic form as follows;
\begin{align*}
\frac{d}{ dt} E_I^{(2)} (u) (t)= \Lambda_3 (M_3) (t), \hspace{0.3cm}
M_3(k_1, k_2,k_3) = -2i [ m(k_1) m(k_{23}) k_{23}]_{sym},
\end{align*}
where $k_{ij}= k_i+ k_j$ for $i \neq j$.
We add a correction term $\Lambda_3( \sigma_3)$ to the modified energy $E_I^{(2)} (u)$ to construct a new modified energy $E_I^{(3)} (u)$.
Namely,
\begin{align*}
E_I^{(3)} (u) (t) =E_I^{(2)} (u)(t)+\Lambda_3 (\sigma_3) (t),
\end{align*}
where the symmetric function $\sigma_3$ is determined later. Similarly, the time derivative of $E_I^{(3)} (u)$ is expressed by
\begin{align*}
\frac{d}{dt} E_I^{(3)} (u) (t) = & \Lambda_3( M_3 ) (t) +\Lambda_3 ( (a_3+ \lambda^{-2} \beta b_3 ) \sigma_3) (t) \\
- & 3i \Lambda_4 ([ \sigma_3 (k_1, k_2, k_{34}) k_{34} ]_{sym} ) (t).
\end{align*}
Here we choose $\sigma_3= - M_3 /( a_3+ \lambda^{-2} \beta b_3 )$ to cancel the cubic terms.
Then,
\begin{align*}
\frac{d}{ d t} E_I^{(3)} (u)(t)=\Lambda_4 (M_4) (t), \hspace{0.3cm}
M_4 (k_1,k_2,k_3,k_4)
:=- 3i \Lambda_4 ([ \sigma_3 (k_1, k_2, k_{34}) k_{34} ]_{sym} ).
\end{align*}
In the same manner, we define the third modified energy as follows;
\begin{align*}
E_I^{(4)} (u) (t):= E_I^{(3)} (u)(t)+ \Lambda_4(\sigma_4) (t), \hspace{0.3em}
\sigma_4:=- M_4/(a_4+\lambda^{-2} \beta b_4).
\end{align*}
Then we have
\begin{align*}
\frac{d}{ d t} E_I^{(4)} (u) (t):= & \Lambda_5 (M_5) (t), \\
M_5 (k_1, k_2, k_3,k_4, k_5):= & -4 i [ \sigma_4 (k_1,k_2,k_3, k_{45}) k_{45} ]_{sym}.
\end{align*}
Chen and Guo \cite{CG} obtained the upper bound of $M_4$ as follows.
\begin{lem} \label{lem_M_4}
Let $|k_1| \geq |k_2| \geq |k_3| \geq |k_4|$. Then we have
\begin{align} \label{M_4}
|M_4 (k_1, k_2, k_3, k_4)| \lesssim \frac{ |a_4+ \beta \lambda^{-2} b_4| m( k_4^{*}) }
{ (N+|k_1|)^2 ( N+|k_2|)^2 (N+|k_3|)^3 (N+|k_4|) }
\end{align}
where $k_4^{*}:=\min \{ |k_l|, |k_{ij}| \} $.
\end{lem}
To establish this upper bound for the Kawahara equation is difficult
because this equation has less symmetries than the KdV equation.
Combining the bilinear Strichartz estimate (\ref{es_L_4_1}) and this upper bound (\ref{M_4}),
we establish the following almost conservation law which
controls the increment of the modified energy $E_I^{(4)} (u)$ in time.
\begin{prop} \label{prop_ACL}
Let $ 0>s \geq -1$ and $N \gg 1$. Then there exists $C_1>0$ such that
\begin{align} \label{ACL1}
\bigl| E_I^{(4)}(u)(t)-E_I^{(4)}(u)(t_0) \bigr| \leq C_1 N^{5s}
\| I u (t_0) \|_{W^{0} ([t_0-1,t_0+1])}^5,
\end{align}
for any $t_0 \in \mathbb{R}$ and $t \in [t_0-1,t_0+1]$.
\end{prop}
\begin{proof}
We may assume $t_0=0$ and $ \widehat{u} $ is non-negative. Since
\begin{align*}
| E_I^{(4)} (u) (t) -E_I^{(4)} (u) (0) | \lesssim \int_{-1}^{1} \Lambda(M_5 ) (t) dt,
\end{align*}
for any $t \in [-1,1]$, it suffices to show that
\begin{align} \label{ACL2}
\int_{-1}^1 \Lambda_5 \Bigl(\frac{M_5 (k_1,k_2,k_3,k_4,k_5)}{ m(k_1) m(k_2) m(k_3) m(k_4) m(k_5)} \Bigr)
(t) dt \lesssim N^{5s} \| u \|_{W^0([-1,1])}^5.
\end{align}
We suppose that $|k_1| \geq |k_2| \geq |k_3| \geq |k_4| \geq |k_5|$ without loss of generality.
$M_5$ vanishes when $|k_i| \ll N$ for any $i=1,2,3,4,5$. So we can assume
$|k_1| \sim |k_2| \gtrsim N$. From the definition of $M_5$, we have
\begin{align*}
|M_5 (k_1,k_2,k_3,k_4,k_5)| \lesssim |\sigma_4(k_3,k_4,k_5,k_{12}) k_{12}|.
\end{align*}
From $k_3+k_4+k_5+k_{12}=0$, we only consider two cases as follows;
\begin{align*}
D_1:=& \bigl\{ (\vec{\tau}, \vec{k}) \in \mathbb{R}^5 \times \dot{\mathbb{Z}}_{\lambda}^5 ~; ~
|k_3| \sim |k_{12}| \gtrsim |k_4| \geq |k_5| \text{ and } |k_3 | \sim |k_{12}| \gtrsim N \bigr\}, \\
D_2:=& \bigl\{ (\vec{\tau}, \vec{k}) \in \mathbb{R}^5 \times \dot{\mathbb{Z}}_{\lambda}^5 ~; ~
|k_3| \sim |k_4| \gg \max \{ |k_{12}|, k_5| \} \text{ and } |k_3 | \sim |k_4| \gtrsim N \bigr\}.
\end{align*}
where $\vec{\tau}:=(\tau_1,\tau_2, \cdots ,\tau_5)$ and
$\vec{k}:= (k_1, k_2 , \cdots, k_5)$.
\vspace{0.3em}
\noindent
(I) Firstly, we prove (\ref{ACL2}) in $D_1$.
From (\ref{M_4}), we easily obtain the upper bound of $M_5$ as follows;
\begin{align*}
| M_5(k_1,k_2, k_3, k_4, k_5)| \lesssim \frac{|k_{12}| }{ (N+|k_3|)^4 (N+|k_4|)^3 (N+|k_5|)^1}.
\end{align*}
From $|k_{12}| \sim |k_3| \gtrsim N$, we substitute this estimate into (\ref{ACL2}) and use the dyadic decompositions
to have
\begin{align*}
(\text{L. H. S. of (\ref{ACL2}} ) ) & \lesssim
N^{5s} \int_{-1}^1 \Lambda_5 \bigl( |k_{12}| \langle k_1 \rangle^{-s}
\langle k_2 \rangle^{-s} \langle k_3 \rangle^{-s-4} \langle k_4 \rangle^{-s-3} \langle k_5 \rangle^{-s-1} \bigr) (t) dt \\
& \lesssim N^{5s} \sum_{N_1} \sum_{N_2 \sim N_1} \sum_{N_3 \leq N_2} \sum_{N_4 \leq N_3} \sum_{N_5 \leq N_4} \\
& \hspace{0.3cm} \times N_1^{-s} N_2^{-s} N_3^{-s-4} \langle N_4 \rangle^{-s-3} \langle N_5 \rangle^{-s-1}
\prod_{i=1}^5 \| u_{N_i} \|_{L_x^1 L_{t \in [-1,1]}^1 }
\end{align*}
where $u_{N_i} := P_{\{ |k_i| \sim N_i \} } u$ for dyadic numbers $N_i$ with $i=1,2,3,4,5$.
From the Schwarz inequality, (\ref{ACL2}) is reduced to two estimates as follows;
\begin{align} \label{BE_G}
N_1^{-s} N_2^{-s} \| |\p_x| u_{N_1} u_{N_2} \|_{X^{s,-1/2} } & \lesssim N_1^{-s-1} N_2^{-s-1} \| u_{N_1} \|_{W^0}
\| u_{N_2} \|_{W^0}, \\
\label{TR_G}
N_3^{-s-4} \langle N_4 \rangle^{-s-3} \langle N_5 \rangle^{-s-1} \| \prod_{i=3}^5 u_{N_i} \|_{X^{-s,1/2}} &
\lesssim N_3^{-2s-2} \langle N_4 \rangle^{-s-2} \prod_{i=3}^5 \| u_{N_i} \|_{W^0}.
\end{align}
If these estimates hold, the left hand side of (\ref{ACL2}) is bounded by
\begin{align*}
& N^{5s} \sum_{N_1} \sum_{N_2 \sim N_1} \sum_{N_3 \leq N_2} \sum_{N_4 \leq N_3} \sum_{N_5 \leq N_4}
N_1^{-s-1} N_2^{-s-1} N_3^{-2s-2} \langle N_4 \rangle^{-s-2} \\
& \hspace{0.8cm} \times
\| u_{N_1} \|_{W^{0} ([-1,1])} \| u_{N_2} \|_{W^{0}([-1,1]) } \| u \|_{W^{0} ([-1,1]) }^3 \\
& \hspace{0.5cm} \lesssim
N^{5s} \sum_{N_1} \sum_{N_2 \sim N_1}
N_1^{-4s-4} \| u_{N_1} \|_{W^{0} ([-1,1])} \| u_{N_2} \|_{W^0 ([-1,1])} \| u \|_{W^0 ([-1,1])}^3,
\end{align*}
which shows the desired estimate for $-1 \leq s <0$.
The bilinear estimate (\ref{BE_G}) has been already proven by Hirayama \cite{Hi}.
So we only prove the trilinear estimate (\ref{TR_G}). From the Plancherel theorem,
we have the identity,
\begin{align*}
\| u_{N_3} u_{N_4} u_{N_5} \|_{X^{-s,1/2}} = \bigl\| \langle k \rangle^{-s} \langle \tau-p_{\lambda} (k) \rangle^{1/2}
\prod_{i=3}^5 \widehat{u}_{N_i} (\tau_i, k_i) \bigr\|_{l_k^2 L_{\tau}^2 },
\end{align*}
where $k=k_3+k_4+k_5$ and $\tau=\tau_3+ \tau_4 +\tau_5$. From the definition, $|k| \sim |k_{12}| \sim N_3$ in this case.
\vspace{0.3em}
\noindent
(Ia) We first consider the case
$\langle \tau-p_{\lambda} (k) \rangle \lesssim \langle \tau_i-p_{\lambda} (k_i) \rangle$ for
some $i=3,4,5$. By symmetry, we may assume
$\langle \tau-p_{\lambda} (k) \rangle \lesssim \langle \tau_3 -p_{\lambda} (k_3) \rangle$.
It suffices show that
\begin{align} \label{red_1}
& N_3^{-2s-4} \langle N_4 \rangle^{-s-3} \langle N_5 \rangle^{-s-1} \| u_{N_3} u_{N_4} u_{N_5} \|_{L_{t,x}^2} \nonumber \\
& \hspace{0.5cm} \lesssim N_{3}^{-2s-4} \langle N_4 \rangle^{-s-2} \langle N_5 \rangle^{-s-1} \| u_{N_3} \|_{X^{0,0} } \| u_{N_4} \|_{Y^0} \| u_{N_5} \|_{Y^0}.
\end{align}
H\"{o}lder's and Young's inequalities imply
\begin{align*}
\| \prod_{i=3}^{5} \widehat{u}_{N_i} (\tau_i, k_i) \|_{l_k^2 L_{\tau}^2}
\lesssim & \| \widehat{u}_{N_3} \|_{l_k^2 L_{\tau}^2 } \| \widehat{u}_{N_4} \|_{l_k^1 L_{\tau}^1}
\| \widehat{u}_{N_5} \|_{l_k^1 L_{\tau}^1} \\
\lesssim &
N_4^{1/2} N_5^{1/2}
\| u_{N_3} \|_{X^{0,0}} \| u_{N_4} \|_{Y^0} \| u_{N_5} \|_{Y^0}.
\end{align*}
We insert this into the left hand side of (\ref{red_1}) to obtain the required estimate.
\vspace{0.3em}
\noindent
(Ib) Next, we consider the case
$ \langle \tau-p_{\lambda} (k) \rangle \gg \langle \tau_i-p_{\lambda}(k_i) \rangle $
for all $i=3,4,5$.
In this case, we use the algebraic relation to have
\begin{align} \label{res_es}
|\tau-p_{\lambda} (k)| \sim |-p_{\lambda} (k) +
p_{\lambda}(k_1)+p_{\lambda} (k_2) +p_{\lambda} (k_3) | \lesssim |k_3|^4 |k_4|
\end{align}
We use (\ref{res_es}) and the H\"{o}lder inequality to obtain
\begin{align*}
& N_3^{-2s-4} \langle N_4 \rangle^{-s-3} \langle N_5 \rangle^{-s-1}
\| u_{N_3} u_{N_4} u_{N_5} \|_{X^{0,1/2}} \\
& \hspace{0.8cm} \lesssim
N_3^{-2s-2} \langle N_4 \rangle^{-s-5/2} \langle N_5 \rangle^{-s-1}
\| u_{N_3} u_{N_4} u_{N_5} \|_{L_{t,x}^2} \\
& \hspace{0.8cm} \lesssim N_3^{-2s-2} \langle N_4 \rangle^{-s-5/2} \langle N_5 \rangle^{-s-1}
\| u_{N_3} u_{N_4} \|_{L_{t,x}^2} \| u_{N_5} \|_{L_{t,x}^{\infty}} .
\end{align*}
When $|k_{34}| \geq 1$, from (\ref{es_L_4_1}) and the Sobolev inequality, the right hand side is bounded by
\begin{align*}
N_3^{-2s-2} \langle N_4 \rangle^{-s-5/2} \langle N_5 \rangle^{-s-1/2} \| u_{N_3} \|_{X^{0,3/8} } \| u_{N_4} \|_{X^{0,3/8}} \| u_{N_5} \|_{Y^0},
\end{align*}
which shows the required estimate.
On the other hand, we deal with the case $|k_{34}| \leq 1 $. Combining the H\"{o}lder inequality and the Young inequality, we have
\begin{align*}
\| u_{N_3} u_{N_4} \|_{L_{t,x}^2} \| u_{N_5} \|_{L_{t,x}^{\infty}} & \lesssim
N_5^{1/2} \| \widehat{u}_{N_3} * \widehat{u}_{N_4} \|_{l_k^{\infty} L_{\tau}^2} \| u_{N_5} \|_{Y^0} \\
& \lesssim
N_5^{1/2} \| u_{N_3} \|_{L_{t,x}^2} \| u_{N_4} \|_{Y^0} \| u_{N_5} \|_{Y^0}.
\end{align*}
From this, we immediately obtain the desired estimate.
\vspace{0.5em}
\noindent
(II) Secondly, we prove (\ref{ACL2}) in $D_2$.
In this case, we have the upper bound of $M_5$ as follows;
\begin{align*}
| M_5(k_1,k_2, k_3, k_4, k_5)| \lesssim \frac{|k_{12}| }{ (N+|k_3|)^2 (N+|k_4|)^2 (N+|k_{12}|)^2 (N+|k_5|)^2 }.
\end{align*}
In the same manner as above, (\ref{ACL2}) is reduced to (\ref{BE_G}) and
\begin{align} \label{TR_G2}
N_3^{-s-2} N_4^{-s-2} \langle N_5 \rangle^{-s-2} \bigl\| \prod_{i=3}^5 \widehat{u}_{N_i} (\tau_i,k_i)
\bigr\|_{X^{-s-2,1/2} }
\lesssim N_3^{-2s-2} \langle N_5 \rangle^{-s-3/2} \prod_{i=3}^5 \| u_{N_i} \|_{W^0}.
\end{align}
We now show the trilinear estimate (\ref{TR_G2}).
\vspace{0.3em}
\noindent
(IIa) We first consider the case $|\tau-p_{\lambda} (k)| \lesssim |\tau_3-p_{\lambda} (k_3) | $.
We use H\"{o}lder's inequality and Young's inequality to have
\begin{align*}
& \| \langle k \rangle^{-s-2} \prod_{i=3}^5 \widehat{u}_{N_i} \|_{l_k^2 L_{\tau}^2} \lesssim
\| \prod_{i=3}^5 \widehat{u}_{N_i} \|_{l_k^{\infty} L_{\tau}^2} \\
& \hspace{0.3cm} \lesssim \| \widehat{u}_{N_3} \|_{l_k^2 L_{\tau}^2}
\| \widehat{u}_{N_4} \|_{l_k^{2} L_{\tau}^1} \| \widehat{u}_{N_5} \|_{l_k^1 L_{\tau}^1}
\lesssim N_5^{1/2} \| u_{N_3} \|_{L_{t,x}^2 } \| u_{N_4} \|_{Y^0} \| u_{N_5} \|_{Y^0},
\end{align*}
which implies the desired estimate.
\vspace{0.3em}
\noindent
(IIb) Next, we consider the case $|\tau-p_{\lambda} (k)| \gg |\tau_i-p_{\lambda} (k_i)| $ for all $i=3,4,5$.
In this case, the algebraic relation implies
\begin{align*}
|\tau-p_{\lambda} (k) | \lesssim \max \{ |k|, |k_5| \} |k_3|^4.
\end{align*}
We use the H\"{o}lder inequality and the Young inequality to obtain
\begin{align*}
\text{(L. H. S. of (\ref{TR_G2}))} & \lesssim
N_3^{-2s-2} \langle N_5 \rangle^{-s-3/2} \| \prod_{i=3}^5 \widehat{u}_{N_i} \|_{l_k^{\infty} L_{\tau}^2} \\
& \lesssim N_{3}^{-2s-2} \langle N_{5} \rangle^{-s-3/2} \| u_{N_3} u_{N_4} \|_{L_{t,x}^2} \| u_{N_5} \|_{Y^0},
\end{align*}
which is an appropriate bound from the above argument.
\end{proof}
Next, we estimate the difference between the almost conserved quantity $E_I^{(4)} (u)$ and the first modified energy $E_{I}^{(2)} (u)$ when
the time is fixed. We call this estimate the fixed time difference.
\begin{prop} \label{prop_FTD}
Let $0> s \geq -1$ and $N \gg 1$. Then there exists $C_2>0$ such that
\begin{align} \label{FTD}
| E_I^{(4)} (u)(t_0) -E_I^{(2)} (u) (t_0) | \leq C_2 (\| I u(t_0) \|_{L_x^2}^3+ \| I u(t_0) \|_{L_x^2}^4 ),
\end{align}
for any $t_0 \in \mathbb{R}$.
\end{prop}
\begin{proof}
From the definition of the modified energies, it suffices to show that
\begin{align*}
| \Lambda_3 (\sigma_3) (t_0) | \lesssim \| I u(t_0) \|_{L^2}^3, \hspace{0.3cm}
|\Lambda_4 (\sigma_4) (t_0) | \lesssim \| I u(t_0) \|_{L^2}^4.
\end{align*}
These estimates are reduced to the following estimates.
\begin{align} \label{FTD_C}
\Bigl| \Lambda_3 \Bigl( \frac{M_3 (k_1,k_2,k_3) }{ (a_3+ \beta \lambda^{-2} b_3) m(k_1) m(k_2) m(k_3) } (t_0) \Bigr)
\Bigr| & \lesssim \| u (t_0) \|_{L^2}^3, \\
\label{FTD_Q}
\Bigl| \Lambda_4 \Bigl( \frac{M_4 (k_1,k_2,k_3,k_4) }{ (a_4+ \beta \lambda^{-2} b_4) m(k_1) m(k_2) m(k_3) m(k_4) } (t_0) \Bigr)
\Bigr| & \lesssim \| u (t_0) \|_{L^2}^4.
\end{align}
Firstly, we prove (\ref{FTD_C}) when $|k_1| \geq |k_2| \geq |k_3|$. Following the mean value theorem, we easily obtain
the upper bound of $M_3$ as follows;
\begin{align*}
| M_3 (k_1, k_2,k_3)| \lesssim |k_3| m(k_3)^2.
\end{align*}
If $|k_i| \ll N$ for all $i=1,2,3$, then $M_3$ vanishes. So we only consider the case $|k_1| \sim |k_2| \gtrsim N$.
The algebraic relation shows
\begin{align*}
|a_3+ \beta \lambda^{-2} b_3| \sim |k_1|^4 |k_3|
\end{align*}
Following these, we use the H\"{o}lder inequality and the Sobolev inequality to have
\begin{align*}
& (\text{L. H. S. of (\ref{FTD_C})}) \lesssim N^{2s} \int |\langle \p_x \rangle^{-2-s} u(t_0) |^2 | I u(t_0) | dx \\
& \hspace{0.3cm} \lesssim N^{2s} \| \langle k \rangle^{-2-s} u(t_0) \|_{L^4}^2 \| I u(t_0) \|_{L^2}
\lesssim N^{2s} \| u (t_0) \|_{L^2}^2 \| m \widehat{u} (t_0) \|_{L_{\xi}^2},
\end{align*}
which is bounded by $N^{2s} \| u (t_0) \|_{L^2}^3$ from the definition of $m$.
Secondly, we prove (\ref{FTD_Q}) when $|k_1| \geq |k_2| \geq |k_3| \geq |k_4|$.
From (\ref{M_4}) and Sobolev's inequality, the left hand side of (\ref{FTD_Q}) is bounded by
\begin{align*}
\Bigl| \Lambda_4 \Bigl( \frac{1}{ \prod_{i=1}^4 (N+|k_i|)^{2} m(k_i) } \Bigr) (t_0) \Bigr|
& \lesssim N^{4s} \int | \langle \p_x \rangle^{-s-2} u (t_0) |^4 dx \\
& \lesssim N^{4s} \| \langle \p_x \rangle^{-s-2} u(t_0) \|_{L^4}^4 \lesssim N^{4s} \| u (t_0) \|_{L^2}^4.
\end{align*}
\end{proof}
Propositions~\ref{prop_ACL} and \ref{prop_FTD} imply that we can find a constant $C_3>0$ such that
\begin{align} \label{es_ACL3}
\sup_{-N^{-5s} \leq t \leq N^{-5s}} \| I u (t) \|_{L^2} \leq C_3 \| I u(0) \|_{L^2}
\end{align}
For the details of the proof, see \cite{CoKeSt}.
A direct calculation shows that
\begin{align} \label{es_ACL4}
\| I u_{\lambda} (0, \cdot) \|_{L^2} \leq C_0 \lambda^{-s-7/2} N^{-s} \| u_0 \|_{ \dot{H}^s}
\end{align}
for some constant $C_0>0$.
Here we take $\lambda \geq 1$ satisfying the following condition.
\begin{align*}
\lambda^{-s-7/2} N^{-s} =\varepsilon_0 \ll 1.
\end{align*}
Then we combine (\ref{es_ACL3}) and (\ref{es_ACL4}) to have
\begin{align*}
\sup_{-T \leq t \leq T} \| u(t) \|_{\dot{H}^s} \leq & \lambda^{7/2} \sup_{-\lambda^5 T \leq t \leq \lambda^5 T}
\| I u_{\lambda} (t) \|_{L^2} \\
\leq & C_3 \lambda^{7/2} \| I u_{\lambda} (0) \|_{L^2}
\leq \varepsilon_0 C_1 C_3 \lambda^{-s} N^{-s} \| u_0 \|_{\dot{H}^s},
\end{align*}
when $\lambda^{5} T \leq N^{-5s}$.
Therefore we have the following upper bound of the growth order of $\dot{H}^s$,
\begin{align*}
\sup_{-T \leq t \leq T} \| u(t) \|_{\dot{H}^s} \leq C T^{7 /5 (2 s+5)} \| u_0 \|_{\dot{H}^s},
\end{align*}
for $-1 \leq s <0$.
\section{Proof of the ill-posedness}
In this section, we give the proof Theorem~\ref{ill_TK} which is based on \cite{Bo97}.
From the argument to \cite{Ho}, it suffices to show that
we seek for the initial data such that, for $|t|$ bounded,
\begin{align} \label{es_ill}
\| A_3 (u_0) (t) \|_{\dot{H}^s} \lesssim \| u_0 \|_{ \dot{H}^s}^3,
\end{align}
fails when $s<-3/2$. Here $A_3(u_0)$ is the cubic term of the Taylor expansion of the flow map as follows;
\begin{align} \label{def_cub}
A_3(u_0)(t) =2 \int_0^{t} U(t-s) \p_x(u_1(s) A_2 (u_0) (s)) ds,
\end{align}
where $u_1(t)=U(t) u_0$ and
\begin{align*}
A_2(u_0) (t)= \int_0^t U(t-s) \p_x(u_1(s)^2 ) ds,
\end{align*}
which is the quadratic term of the Taylor expansion of the flow map.
We put a sequence of initial data $\{ \phi_N \}_{N=1}^{\infty} \in H^{\infty}$ as follows;
\begin{align} \label{def_ini}
\widehat{\phi}_N(k)= N^{-s} (\chi_N (k) + \chi_{-N} (k) ).
\end{align}
Clearly $\| \phi \|_{\dot{H}^s} \sim 1$.
A simple computation shows that
\begin{align*}
\widehat{A}_2 (u_0) (t)= \sum_{k_1 \neq 0 ,k \neq k_1}
k~\frac{ e^{i p(k) t} -e^{ i p(k_1)t+i p(k-k_1) t} }{ q_0(k_1,k-k_1) } \widehat{u}_0 (k_1) \widehat{u}_0(k-k_1),
\end{align*}
where
\begin{align*}
q_0(k_1,k-k_1) := \frac{5}{2} k k_1 (k-k_1) \bigl\{ k^2+ k_1^2+(k-k_1)^2+\frac{6}{5} \beta \bigr\}.
\end{align*}
Substituting this into (\ref{def_cub}), we use the Fourier inversion formula to have
\begin{align} \label{cub_2}
A_3(u_0)(t)=& 2 \sum_{k_1 \neq 0} \sum_{k_2 \neq 0} \sum_{k_3 \neq 0}
e^{i (k_1+k_2+k_3) x+ i p(k_1+k_2+k_3) t} \Bigl( -\frac{1-e^{-i q_1 t}}{q_1} +\frac{1-e^{-i q_2 t} }{q_2}
\Bigr) \nonumber \\
& \hspace{0.5em} \times \frac{ (k_1+k_2+k_3) (k_2+ k_3) }{q_0(k_2,k_3) }
\widehat{u}_0 (k_1) \widehat{u}_0 (k_2 ) \widehat{u}_0 (k_3),
\end{align}
where
\begin{align*}
q_1:= & \frac{5}{2} (k_1+k_2) (k_1+k_3) (k_2+k_3) \bigl\{ (k_1+k_2)^2+ (k_1+k_3)^2+(k_2+k_3)^2+\frac{6}{5} \beta \bigr\},\\
q_2:= & \frac{5}{2} k_1 (k_2+k_3) (k_1+k_2+k_3) \bigl\{ k_1^2+ (k_2+k_3)^2+(k_1+k_2+k_3)^2+\frac{6}{5} \beta \bigr\}.
\end{align*}
Note that $q_2$ does not vanish but $q_1 $ vanishes when $k_1=-N$ and $k_2=k_3=N$.
In this case, inserting (\ref{def_ini}) into (\ref{cub_2}), we obtain
\begin{align*}
|\widehat{A}_3 (\phi_N) (t)| \gtrsim C_1 |t| N^{-3s-4} |k| \chi_{N}(k) -C_2 N^{-3s-8} \chi_{N}(k)
\end{align*}
for some constants $C_1>0$ and $C_2 \geq 0$. So there exists $C_3>0$ such that
\begin{align*}
\| A_3 (\phi_N) (t) \|_{\dot{H}^s} \gtrsim C_3 N^{-2s-3}
\end{align*}
for $|t|$ bounded. From $\| \phi_N \|_{\dot{H}^s} \sim 1$, (\ref{es_ill}) fails for $s <-3/2$.
|
{
"timestamp": "2012-03-13T01:01:26",
"yymm": "1203",
"arxiv_id": "1203.2275",
"language": "en",
"url": "https://arxiv.org/abs/1203.2275"
}
|
\section{Introduction} The most extraordinary phenomenon of
wave transport in random media is Anderson
localization\cite{Anderson1958}, which is shared by various systems,
such as, electronic, photonic and accoustic
systems\cite{ShengSoukoulisBook}. In Anderson localized system the
ensemble average of logarithmic transmission $\left<lnT\right>$, not
transmission $\left<T\right>$, is additive with system length
$L$\cite{Abrahams1979,Anderson1980}. Naturally, the localization
length can be defined as $\xi=-2L/\left<lnT\right>$. In strongly
localized systems($L\gg\xi$), the transmission is generally small,
but also shows large fluctuations\cite{Anderson1980}. Those extreme
large $T$ values will dominate the transmission of localized
systems. In the past years, the study of the physical origins of
those large $T$ values yielded abundant results for understanding
the transport phenomena in random systems and the Anderson
localization\cite{Azbel1983,Lee1985RMP,Pendry,Tartakovskii,Bliokh2004,
Bertolotti2005PRL,Sebbah2006PRL,Bertolotti2006PRE,Bliokh2006PRL,Ghulinyan2007PRA,
Ghulinyan2007PRL,Bliokh2008PRL,Bliokh2008RMP,Vanneste2009PRA,
LiWei2009,Zhang2009PRB,Wang2008Nano,Wang2011PRB,Wang2010PRB,ShengPingPercolation,Chen2011NJP}.
Previously, it was pointed out by M. A. Azbel et al.\cite{Azbel1983} that
the resonant transport through those localized states lying near the system center
can contribute very large $T$ values, being of order unity.
Although such a mechanism can generate high resonant transmission peak
\cite{Azbel1983,Lee1985RMP}, its contribution to
the transmission is vanishingly small in strongly localized system because
the resonant peak is exponentially sharp ($\sim e^{-L/\xi}$)\cite{Azbel1983,Lee1985RMP,Bliokh2004}.
Laterly, Pendry\cite{Pendry} and Tartakovskii\cite{Tartakovskii} independently
predicted a kind of quasi-extended state,
called the \emph{necklace state(NS)}. The NS is formed through the
coupling between nearly-degenerated localized states which are
evenly distributed in the system. Despite the spatial overlaps
between localized states are small, \emph{because of the degenerate
coupling}, the NS contributes very wide transmission ``mini-bands'',
which can dominate the transmission of strongly localized systems.
The NSs have been demonstrated experimentally in photonic
systems\cite{Bertolotti2005PRL,Sebbah2006PRL}. Their fundamental
characters and statistical consequences to the transmission are
widely studied\cite{Bertolotti2006PRE,Ghulinyan2007PRA,
Ghulinyan2007PRL,Bliokh2008PRL,Bliokh2008RMP}. Recently, it is
demonstrated that the NSs have evident contribution to the
short-time transport of wave
package\cite{LiWei2009,Zhang2009PRB,Wang2008Nano,Wang2011PRB} and
the dynamics of fluctuations of localized waves\cite{Wang2010PRB}.
More profoundly, the number of NS can increase dramatically as approaching
the Anderson transition point, which strongly supports a modes-coupling induced
quantum percolation scenario for the Anderson localization-delocalization
transition\cite{ShengPingPercolation,Chen2011NJP}.
Even though NSs can contribute very large transmission, their
formation has rigorous conditions, i.e., the nearly-degenerated
localized states are required to be \emph{evenly distributed} inside
the system\cite{Pendry,Ghulinyan2007PRA,Bliokh2008PRL,Bliokh2008RMP}. In
general the localized states with similar frequency are not such
well distributed in a specific random configuration. In such cases,
\emph{ideal NSs}\cite{Ghulinyan2007PRA,Ghulinyan2007PRL} crossing
the whole configuration are not formed.
But eventually, those nearly-degenerated states which are also
\emph{spatially close to each other} couple together, forming short chains of
coupled-localized-states embedded inside the configuration. Since
those coupled-localized-states have similar properties as the NS, we
call them \emph{short necklace state}(SNS)\cite{JiangArXiv}. The SNS
also manifests as high transmission plateau which has significant
transmission contribution. In transmission spectra the SNS is hardly
to be distinguished from isolated localized states since the valley
between coupled peaks seems very low. But it can be clearly
identified from the logarithmic transmission spectra, for instance,
see the coupled-peaks in Fig.1. Because of the strong coupling
between the neighboring localized states inside the SNS, on top of
the plateau there are sharp peaks and valleys, which correspond to
un-smooth changes of the transmission phase between the
coupled-peaks\cite{Ghulinyan2007PRA}.
In contrast to the NSs, the SNSs have special
properties\cite{JiangArXiv} that (1) the SNSs exist widely in every
random configuration while the NS only occurs once in millions of
random configurations with large length($L\gg\xi$); (2) the
plateau-width of the SNSs only depends on the coupling strength
between the neighboring localized states and is insensitive to the
system length. With these properties, SNSs are superior
\emph{qualitatively} compared with the NSs.
However, how to \emph{quantitatively} characterize statistical
properties and scaling behaviors of SNSs are still unexplored topic.
Even more, some essential quantities of SNS study, such as the
``half-width" of SNS plateaus in $lnT$ spectra, need to be defined
since there is no obvious physical quantity in previous studies
can directly describe SNS.
In this paper, we study the statistical manifestation of the SNS
using physical quantities which are defined in the
\emph{logarithmic} transmission spectra.
To characterize SNS, we find out two different approaches, whose
results can be compared with each other. The first approach is based
on the correlation functions $C_{lnT}$ (defined in the $lnT$
spectra) and $C_t$ (defined in the transmission coefficient $t$
spectra). From $C_{lnT}$, we show that there is a typical frequency
correlation range of logarithmic transmission, which is explained as
the typical SNS plateau-width. From this frequency range and
compared with the results of $C_t$, we can find the most-probable
order of SNS. The second approach is from the direct measurement of
peak/plateau-width in $lnT$ spectra. We find the statistical
distribution of peak/plateau-width is quite abnormal and
can be fitted very well by the summation of two Gaussian distribution
functions, where the primary Gaussian center gives the typical width
of resonant peaks of localized states while the secondary one gives
exactly the typical width of the SNS plateaus. Excellent agreements
are found between two approaches. The dependencies of SNS's
properties on the scaling parameter $L/\xi$ are also studied. We
find the plateau-width of SNS almost does not depend on $L/\xi$
while the peak-width of localized states decays exponentially with
$L/\xi$, indicating the SNSs have more significant transmission
contribution in longer systems.
The essential quantities defined in this work also provide new ways
for further quantitative study on statistical properties of the
transmission of Anderson localized systems.
The rest of this paper is organized as follows. In section II, we
introduce our model and basic properties of the $lnT$ spectra. In
section III, we present our numerical results and theoretical
analysis of three correlation functions: the correlation of
transmission $C_T$, the correlation of logarithmic transmission
$C_{lnT}$ and the field correlation $C_t$. We show that the
half-width of $C_{lnT}$ gives the typical plateau-width of SNS. The
most-probable order of SNS can be obtained by comparing $C_{lnT}$
and $C_t$. In section IV, we study statistical distributions of
the peak/plateau-width (defined in the $lnT$ spectra). The
probability distributions are calculated from very large numbers
of samples and can be fitted very well by the summation of
two Gaussian distributions, where the first
Gaussian center gives the typical width of resonant peaks of
localized states and the second one gives the typical
width of the SNS plateau. In section V, we discuss the system length
dependency of the SNS. Finally, a summary of this work is given in
section VI.
\begin{figure}
\includegraphics[width=1.0 \columnwidth ]{Figure1.eps}
\caption{(Color online) Typical logarithmic transmission spectra (a)
and corresponding transmission spectra (b) of $L=10\xi$ systems. Red
bold line: $\xi=34.13 a$ ($W=0.6$); black thin line: $\xi=215.6 a$
($W=0.24$). The SNS marked in (a) is enlarged in (b) for clear
observation.}
\end{figure}
\section{ The Model } We study 1D random stacks composed of
binary-dielectric layers(called A and B) with thicknesses
$d_A=d_B=500 nm$ and with refractive indices $n_A=1.0$ and
$n_B=3.0*(1+W\gamma)$, where $\gamma$ is a random number uniformly
distributed in $[-0.5,0.5]$ and $W$ gives the randomness strength.
Such a periodic on average model is an optical counterpart of the
Anderson model of electronic systems and is widely used for
localization
studies\cite{Bertolotti2006PRE,Ghulinyan2007PRA,Ghulinyan2007PRL,LiWei2009,Chen2011NJP}.
The transmission coefficients of optical waves are calculated by
standard transfer matrix method\cite{Chen2011NJP}. Without
randomness, $W=0$, the system exhibits a pass band in frequency
range $(89.13, 124.1 THz)$. Our study will focus on the range
$(100.0, 103.2 THz)$, where the localization length is almost a
constant for a certain randomness $W$($0.2 \le W \le
0.6$)\cite{Chen2011NJP}. The localization length is calculated by
$\xi=-2L/\left<lnT\right>$, where $\left<lnT\right>$ is averaged
from the $10^6$ configurations satisfying $L\gg\xi$ ($L\approx
500\xi$).
Fig.1(a) shows two typical logarithmic transmission spectra with the
same $L/\xi=10$ but with different localization lengths,
$\xi=215.6a$(black thin) and $\xi=34.14a$(red bold). We can see they
have similar average height, being approximately $-2L/\xi=-20$.
After averaged over a large number of configurations, the $lnT$
spectrum becomes a flat line exactly falls on the mean value
$\left<lnT\right>=-20$. This is a natural result according to the
single parameter scaling theorem that $lnT$ follows the Gaussian
distribution with mean value $-2L/\xi$.
An interesting phenomenon in the $lnT$ spectra is that there are
some clear plateau-structures, such as the one marked by arrows in
Fig.1(a). From the wave intensity distributions one can find those
plateaus are formed by the SNSs(i.e., from the degenerate coupling
between some spatially close localized states) embedded inside
the configuration\cite{JiangArXiv}.
Since the peaks of localized states in the spectra are extremely sharp ($\sim
e^{-L/\xi}$), the fluctuation of $lnT$ is dominated by those
plateaus. More importantly, these SNS may be essential for
understanding the Anderson transition phenomenon\cite{JiangArXiv}.
After observed a large number of $lnT$ spectra for fixed $\xi$ and
$L/\xi$, we find the plateaus appear with some intuitionistic
regularities, (1) most plateaus have similar frequency width
for a configuration; (2) the number of peaks on each plateau are
similar. For example, for most plateaus of the $\xi=34.14a$ system
shown in Fig.1, the plateau width is about $0.3\sim0.4 THz$ and
the number of peaks are about $3\sim4$. In the following we will
focus on the statistical properties of these SNSs and try to
characterize them quantitatively.
To quantitatively study the SNS, we need to find proper physical
quantities. Fig.1 shows that the plateaus of SNSs can
hardly be distinguished from localized states in a $T$ spectrum (see
Fig.1(b)), but can be clearly identified by the plateaus in the
$lnT$ spectra (Fig.1(a)). So we try to define physical quantities
in the $lnT$ spectra for the SNS study.
In the following, we will define
and study the correlation function of $lnT$ spectra in our first
approach. Then, we will define the $lnT$ peak/plateau
width--$\Gamma$ and study the statistical distribution of $\Gamma$
in our second approach.
\section{ Correlation functions } According to the standard definition of mathematics,
we define the correlation functions of $T$ and $lnT$ in frequency
domain as:
\begin{eqnarray}
C_T(\Delta\omega)=
\frac{Cov(T_{\omega},T_{\omega+\Delta\omega})}{\sigma(T_{\omega})\sigma(T_{\omega+\Delta\omega})}
\end{eqnarray}
\begin{eqnarray}
C_{lnT}(\Delta\omega)=
\frac{Cov(lnT_{\omega},lnT_{\omega+\Delta\omega})}{\sigma(lnT_{\omega})\sigma(lnT_{\omega+\Delta\omega})}
\end{eqnarray}
where $T$ is the transmission coefficient, $\sigma$ is the standard
deviation $\sigma(x)=\sqrt{\left<(x-\left<x\right>)^2\right>}$ and
$Cov$ is the covariance
$Cov(x,y)=\left<x-\left<x\right>\right>\left<y-\left<y\right>\right>$.
Physically $C_T$ implicates the frequency correlation of transmitted
wave intensity. Its Fourier transformation corresponds to the
dynamical response at the outgoing interface, which have been
intensively studied in weakly localized
systems\cite{GenackCorrelation}. If two frequencies are on the same
transmission peak, $C_T$ is close to unity. Otherwise $C_T$ will
close to zero. Hence the half-width of $C_T$ is usually used to
characterize the linewidth of localized
states\cite{GenackCorrelation}. Similarly, on the $lnT$ spectra,
since $lnT$ of two frequencies on the same plateau are much larger
than the mean value $\left<lnT\right>$ and contribute a
significantly large $C_{lnT}$. We expect $C_{lnT}$ is close to unity
when $\Delta\omega$ is smaller than a typical plateau-width and
decays to zero when $\Delta\omega$ is larger than a typical
plateau-width.
Meanwhile, we also study the field correlation function,
\begin{eqnarray}
C_t
(\Delta\omega)=\frac {<t_{\omega} {t_{\omega+\Delta\omega}}^*
+{t_{\omega}}^*t_{\omega+\Delta\omega}>}
{<T_{\omega}>+<T_{\omega+\Delta\omega}> }
\end{eqnarray}
where $t$ is complex transmission coefficient and $T=tt^*$. The
complex amplitude of incident wave is chosen to be $E_{in}(x=0)=1$
so that at the outgoing interface $t=E(x=L)=|E|e^{i\phi}$, where $E$
is the complex electronic field and $\phi$ is its phase. Similar to
$C_T$, the physical implication of $C_t$ is simply the frequency
correlation of transmitted field, i.e., the field correlation($C_T$
is the intensity correlation). The definition of $C_t$ is the same
as that in ref.\cite{Pendry}, which could be negative valued,
depending on the averaged phase difference $\Delta\phi$ between
$\omega$ and $\omega+\Delta\omega$.
Generally, when $\Delta\omega$ crosses a resonant peak, the phase of
$t$ jumps $\pi$\cite{Pendry,Chen2011NJP}.
Hence the phase difference between $t(\omega)$ and $t(\omega+\Delta\omega)$ is
approximately $\Delta\phi=\phi(\omega)-\phi(\omega+\Delta\omega)
\approx \pi$. Then
$t_{\omega}t_{\omega+\Delta\omega}^*+t_{\omega}^*t_{\omega+\Delta\omega}=
|E_{\omega}||E_{\omega+\Delta\omega}|\cdot 2cos(\Delta\phi)$ becomes
negative and so that $C_t$ is negative. When $\Delta\omega$ gets to
the value which typically contains two peaks (plus one average distance
between two peaks), $C_t$ will be positive
since $\Delta\phi\approx 2\pi$, and so forth. Hence when increasing
$\Delta\omega$, we expect $C_t(\Delta\omega)$ to oscillate between
positive and negative. Each time $C_t(\Delta\omega)$ changes its
sign, the average number of resonant peaks in $\Delta\omega$
increases one. So $C_t(\Delta\omega)$ provides a way to check the
average resonant peak number in a certain frequency range.
\begin{figure}
\includegraphics[width=1.0 \columnwidth]{Figure2.eps}
\caption{(Color online) (a): Correlation functions of $L=10\xi$
systems calculated from $2\times 10^7$ configurations. Red:
$\xi=34.13 a$; Black: $\xi=215.6 a$. For each color, the correlation
functions from the top down are respectively $C_{lnT}$, $C_T$ and
$C_t$. (b): Same as (a) but with $\Delta\omega$ shown in logarithmic scale.
(c): Detailed view of the oscillations of $C_t$($\xi=34.13 a$). The
half-width of $C_{lnT}$, $\Omega_S$, gives the typical plateau-width
of SNS. The first minimum of $C_t$, $\Omega_1$, gives the typical
width of a resonant peak.}
\end{figure}
In our calculations we set the original frequency point $\omega=100
THz$ without lose of generality. The correlation functions
calculated from a very large number ($2\times10^7$) of
configurations for different $\xi$ are shown in Fig.2(a), where the
ratio $L/\xi$ is fixed at $10$. The solid, dashed and dotted curves
respectively represents $C_T$, $C_t$ and $C_{lnT}$. The red curves
marked by crosses correspond to the smaller $\xi$ system and the
black curves marked by solid dots correspond to the larger $\xi$
system. In Fig.2(b) the transverse axis is shown in logarithmic
scale.
Let us first discuss $C_T$. Fig.2(a) shows $C_T$ is a very singular
function at the origin $\Delta\omega\rightarrow 0$. The linewidth
$\Delta\omega$ of localized states, corresponding to
$C_T(\Delta\omega)=0.5$, is larger in the smaller $\xi$ systems.
This is in consistent with the observation in Fig.1 that the
linewidth of resonant peaks in smaller $\xi$ system is larger.
Detailed data gives that the halfwidth of the larger $\xi$ system
(black solid curve) is about $0.382GHz$ and the smaller $\xi$ system
(red solid curve) is $2.483GHz$. This is also quantitatively in
consistent with the observation from the $lnT$ spectra.
$C_{lnT}$ is a much smoother function than $C_T$ in the
$\Delta\omega\rightarrow 0$ limit. Take the $\xi=34.13 a$ system
(red curves) for example. $C_{T}$ rapidly falls to zero near
$ln(\Delta\omega) \approx 24$. $C_{lnT}$ exhibits a plateau at the
origin, then drops to $0.5$ at $ln(\Delta\omega) \approx 26.6$,
i.e.,
its half-width is about $0.341THz$, being much larger than that of
$C_T$. Such contrast reflects the different geometry properties
between the $T$ and $lnT$ spectrum. On the $T$ spectrum, $C_T$ falls
to zero typically when two frequencies are not on the same peak.
Since the peaks of localized states are exponentially sharp, $C_T$
decreases rapidly as increasing $\Delta\omega$. However, on the
$lnT$ spectrum, there exist many plateaus formed by SNSs, as shown
in Fig.1(a). Those plateaus are usually higher than the average
transmission background $\left<lnT\right>$. When two frequencies are
on the same plateau but not the same peak, it will still contribute
a significantly large value to $C_{lnT}$. Hence \emph{the halfwidth
of $C_{lnT}$, which is denoted as $\Omega_S$ in this paper,
approximately gives the typical plateau-width of SNS in $lnT$
spectrum}.
With the help of $C_t$, we can find more detailed information of
SNS, such as the average SNS order, which is the average number of
coupled localized states in one SNS. Similar to $C_T$, $C_t$ also
shows sharp singularity at the origin and falls quickly as
increasing $\Delta\omega$. Interestingly, $C_t$ shows some
oscillations around $C_t=0$ in the region where $C_T$ falls to
nearly zero, see Fig.2(c). As discussed earlier in this section,
those oscillations can be understood from the $\pi$-phase jumps of
resonant peaks of the localized states. The different order
minimal/maximal points of the $C_t$ correspond to the frequency
ranges which can accommodate certain number of resonant peaks. We
denote the $\Delta\omega$ at the first minimum of $C_t$ as
$\Omega_1$, which is the the typical frequency width of a resonant peak.
We also denote $n$th order minimal/maximal points
$\Omega_n$ as the average frequency range which can accommodate
$n$ resonant peaks. So $C_t$ is the ruler of the number of resonant
peaks in a frequency range. With $\Omega_n$ in our mind, we can see
that the mean width of SNS plateaus, obtained by $C_{lnT}$, can
accommodate about $3\sim4$ resonant peaks, as indicated by the green
arrows in the Fig.2(c). Actually, this average number of resonant
peaks in a SNS
agrees very well with our direct observation of many spectra.
\begin{figure}[t]
\includegraphics[width=1.0 \columnwidth ]{Figure3.eps}%
\caption{(Color online) Probability distribution of the
peak/plateau-width $\Gamma$(see the text for definition) measured
from logarithmic transmission spectrum. The distribution function
can be fitted very well by the summation(red solid curve) of two
Gaussian functions(dashed curves). The primary(left, blue) Gaussian
center corresponds to the typical width of resonant peaks of localized states.
The secondary(right, green) Gaussian center corresponds to the typical
plateau-width of SNS.}
\end{figure}
\section{ Peak/plateau-width statistics } To make a test and verify of
the characters of SNS obtained from $C_{lnT}$ and $C_t$, we next try
our second approach, \emph{i.e.} direct measure of the
peaks/plateaus-width in a large number of $lnT$ spectra.
Since the ensemble average of $lnT$ spectra--$\left<lnT\right>$ is
well defined, it is natural to \emph{define the peaks/plateaus-width
$\Gamma$ as the frequency interval where $lnT$ is always higher than
$\left<lnT\right>$}. From direct observation of the $lnT$ spectra,
one can find that $\left<lnT\right>$ crosses lots of sharp peaks of
localized states and fewer SNS plateaus. More precise statistical
results should be obtained from a large number of realizations.
We find that the statistical distribution of $\Gamma$ is extremely
skewed. The reason is that in Anderson localized systems the
peak-width of localized states is exponentially small and the width
of coupled-peaks also scales exponentially with the system
length\cite{Azbel1983,Pendry,Chen2011NJP}. This is very similar to
the probability distribution of the dimensionless conductance $g$,
which is extremely skewed (nearly log-normal). Early
study\cite{Anderson1980} on Anderson localization has shown that it
is better to study the additive quantity--$lng$, which is nearly
Gaussian distributed. Similarly, instead of $\Gamma$, we will study the
$ln\Gamma$ distribution, which is likely Gaussian distributed.
\begin{figure}[t]
\includegraphics[width=1.0 \columnwidth ]{Figure4.eps}%
\caption{(Color online) (a): $C_t$ (lower four blue curves) and
$C_{lnT}$ (upper four red curves) for different $L/\xi$ systems. For
each set of four curves, $L/\xi = 6,10,14,18$ from left to right.
The inset gives detailed view at the halfwidth of $C_{lnT}$. (b):
The typical peak-width of localized states $\Omega_1$ (marked by
blue arrows in (a)) as a function of $L/\xi$.
(c): The typical plateau-width of SNS $\Omega_S$ (marked by red
arrows in (a)) as a function of $L/\xi$. $\Omega_1$ decays
exponentially as increasing $L/\xi$ while $\Omega_S$ is insensitive
to $L/\xi$.}
\end{figure}
We have measured $\Gamma$ in $10^5$ spectra with high frequency
precision that can distinguish each resonant peak in the $L=10\xi$
systems\cite{note3}. The probability distributions of $ln\Gamma$ for
two different $\xi$ systems are shown in Fig.3.
Take the $\xi=34.13a$ system as the example. (All the following
discussions also apply to the $\xi=215.6a$ system.) The probability
distribution is roughly Gaussian-like, where a clear maximum is
found at $ln\Gamma \approx 24.3$($\Gamma\approx0.035THz$). This is
exactly the $ln(\Omega_1)$ at first minimum of $C_t$ shown in Fig.2.
As shown before, this value corresponds to the typical frequency
range occupied by one single resonant peak.
Such a result is coincident with our direct observation of the $lnT$
spectra that the most probable peaks/plateaus crossed by
$\left<lnT\right>$ are those single peaks. Suppose the spectrum
contains only sharp peaks but none plateaus. $ln(\Delta\omega)$
should be nearly Gaussian distributed with a single maximum at the
mean value. However, the real probability distribution of $ln\Gamma$
shows a strange shoulder at $ln\Gamma \approx 26.6$($\Gamma\approx
0.341THz$). Comparing with Fig.2 we find it exactly corresponds to
the halfwidth of $C_{lnT}$, $\Omega_S$, which is just the typical
width of SNS plateau, referring to the physical meaning of
$C_{lnT}$. Actually, the measured probability distribution can be
fitted very well by the summation of two Gaussian distributions, as
shown by the curves in Fig.3. The primary(left) Gaussian center gives
the width of single resonant peaks while the secondary(right) one
gives exactly the typical width of the SNS plateau. Such a strange
probability distribution implies substantial SNS plateaus with
similar frequency widths contribute significantly to the probability
distribution function of $ln\Gamma$, resulting the characteristic
shoulder of SNS.
It is first time to see directly from the statistical distribution
that the SNS is clearly distinguished from other localized states.
From the distribution, we can see that the logarithmic plateau-width
of SNS is Gaussian distributed and its mean value agrees excellently
with the results of correlation functions in third section, as
expected. So, both $C_{lnT}$ and the probability distribution of
$ln\Gamma$ provides proper physical values for characterizing the
SNS.
\section{ System length dependencies } Both the correlation
functions and the probability distribution of $ln\Gamma$ suggest
that the SNS favors a specific frequency width and a specific
number(order) of localized states. But those studies are done with
certain $L/\xi$. Next, to drive this conclusion further, we will
calculate the correlation functions
in the systems with different $L/\xi$. Fig.4(a) shows $C_t$ (the
lower four blue curves) and $C_{lnT}$ (the upper four red curves)
functions for several $L/\xi$ values, where $\xi$ is fixed at $34.13
a$ ($W=0.6$) and $L/\xi=6,10,14,18$ from left to right. It clearly
shows the peak width $\Omega_1$ of resonant peaks of
localized states (marked by blue arrows) decreases
dramatically as increasing $L$ while the decrease of plateau-width
$\Omega_S$(halfwidth of $C_{lnT}$ marked by red arrows) is much
smaller. $\Omega_1$ and $\Omega_S$ versus $L/\xi$ are shown in Fig.4(b) and
(c). We can see $\Omega_1$ decays exponentially with $L$ ($\sim
e^{-L/\xi})$, as expected from traditional understanding\cite{Azbel1983,Lee1985RMP,Bliokh2004}.
However, the decrease of $\Omega_S$ is almost negligible and obviously much
slower than $\Omega_1$. Actually, the plateau-width of SNS is
determined by the intrinsic coupling strength(repulsing distance)
between localized states, which almost does not depend on
$L$\cite{JiangArXiv}.
Such a distinction naturally results a picture that, as increasing
$L$ more and more, the peaks of localized states become so sharp
that they are almost undetectable, but the SNS plateaus are still
there. In our numerical statistics, the number(order) of localized
states increases nearly linearly with $L/\xi$. In
Ref.\cite{JiangArXiv} it is further argued that those SNSs (called
intrinsic short necklace states) have main contribution to the
fluctuation of transmission and also affect the value of
localization length.
\section{ Summary }
In summary, we have defined the basic quantities for SNS study and
investigate the statistical properties of SNS in the strongly
localized systems. We find two approaches to quantitatively study
SNS properties. The first approach is based on the correlation
functions and the second one is based on direct measurements of the
peak/plateau-width in logarithmic transmission spectra. In the first
approach, we defined the correlation function $C_{lnT}$ in $lnT$
spectra and show that the typical width of SNS plateaus can be
characterized by the half-width of $C_{lnT}$. And, with the help of
$C_{t}$ (correlation function of transmission coefficient $t$), the
most-probable order of SNS can be obtained. In the second approach,
we defined the peak/plateau-width $\Gamma$ in $lnT$ spectra and
studied the probability distribution of $ln{\Gamma}$ directly measured
from a large number of spectra.
The probability distribution of $ln{\Gamma}$ shows a novel shoulder
and it can be fitted very well by the summation of two Gaussian
distributions. We showed that the first one is from the distribution
of localized state peaks and the second one is from the plateaus of
SNS. The center of the second distribution gives the average
frequency width of SNS plateaus, which agrees very well with the
value obtained from correlation function. As increasing the system
length $L$, the plateau-width of SNS decays very slowly, compared
with the exponentially-decayed peak-width of localized states.
Finally, we note that the methods we used in our paper is not
limited to the transport properties of localized system. They may be
used for studying other statistical quantities which have similar
properties with the transmission of localized systems.
This work is supported by the NSFC (Grant Nos. 11004212, 11174309,
and 60938004), and the STCSM (Grant Nos. 11ZR1443800 and
11JC1414500).
|
{
"timestamp": "2012-03-09T02:01:38",
"yymm": "1203",
"arxiv_id": "1203.1713",
"language": "en",
"url": "https://arxiv.org/abs/1203.1713"
}
|
\section{Introduction}
In recent years the significance of sensor systems/networks is increased
(e.g., \cite{aky02},
\cite{cul04},
\cite{kul11},
\cite{qi01},
\cite{soh07},
\cite{zad07}).
In general,
it may be reasonable to consider a simplified 3-layer architecture of
a
sensor system
(Fig. 1):
(i) sensors and sensor local networks (sensor subsystem layer),
(ii) communication network (transportation layer), and
(iii) management subsystem (
control layer: information analysis and integration/fusion,
decision making and control)
(e.g.,
\cite{kul11},
\cite{qi01},
\cite{zad07}).
In the article, a hierarchical modular design of
configuration for wireless sensor element is examined.
The problem corresponds to layer 1
of the three-layer sensor system structure above.
A real world numerical example is targeted
to a fire alarm wireless sensor element.
Note various approaches have been applied for the design of system configurations
\cite{lev09}:
(1) the shortest path problem \cite{art91};
(2) evolutionary approaches
(e.g.,
\cite{rod05});
(3) multi-agent approaches (e.g.,
\cite{campbell03});
(4) approaches based on fuzzy sets
(e.g., \cite{smirnov04});
(5)
composite constraint satisfaction problems
(e.g.,
\cite{sabin98}, \cite{stefik95});
\begin{center}
\begin{picture}(70,71)
\put(04,00){\makebox(0,0)[bl]{Fig. 1. Sensor system (three layers)
\cite{levfim10}}}
\put(03,46){\line(1,0){64}} \put(03,69){\line(1,0){64}}
\put(03,46){\line(0,1){23}} \put(67,46){\line(0,1){23}}
\put(03.5,46.5){\line(1,0){63}} \put(03.5,68.5){\line(1,0){63}}
\put(03.5,46.5){\line(0,1){22}} \put(66.5,46.5){\line(0,1){22}}
\put(6.5,64){\makebox(0,0)[bl]{Layer 3: Management }}
\put(6.5,60){\makebox(0,0)[bl]{(monitoring processes,
information}}
\put(6.5,56){\makebox(0,0)[bl]{analysis and processing/fusion,}}
\put(6.5,52){\makebox(0,0)[bl]{information collection and store,}}
\put(6.5,48){\makebox(0,0)[bl]{generation of control decisions)}}
\put(10,41){\vector(1,1){5}}
\put(22.5,41){\vector(0,1){5}}
\put(35,41){\vector(0,1){5}}
\put(47.5,41){\vector(0,1){5}}
\put(60,41){\vector(-1,1){5}}
\put(35,31){\oval(70,20)}
\put(5,36){\makebox(0,0)[bl]{Layer 2: Transportation
(information}}
\put(5,32){\makebox(0,0)[bl]{transmission and processing,
routing,}}
\put(5,28){\makebox(0,0)[bl]{scheduling, network
covering/spanning, }}
\put(5,24){\makebox(0,0)[bl]{preliminary data aggregation)}}
\put(05,16){\vector(0,1){5}} \put(15,16){\vector(0,1){5}}
\put(25,16){\vector(0,1){5}} \put(35,16){\vector(0,1){5}}
\put(45,16){\vector(0,1){5}} \put(55,16){\vector(0,1){5}}
\put(65,16){\vector(0,1){5}}
\put(00,06){\line(1,0){70}} \put(00,16){\line(1,0){70}}
\put(00,06){\line(0,1){10}} \put(70,06){\line(0,1){10}}
\put(4,11.5){\makebox(0,0)[bl]{Layer 1: Sensors (including
preliminary}}
\put(4,07.5){\makebox(0,0)[bl]{data processing and transmission)}}
\end{picture}
\end{center}
(6) ontology-based approaches
(e.g.,
\cite{ciuksys07});
(7) multicriteria multiple choice problem
(e.g.,
\cite{poladian06});
(8) hierarchical multicriteria morphological design (HMMD) approach
(\cite{lev98},
\cite{lev06},
\cite{lev09});
(9) AI techniques
(e.g.,
\cite{mcd82},
\cite{pira05},
\cite{sugu08},
\cite{wielinga97});
and
(10) design grammars approaches
(e.g., multidisciplinary grammar approach that includes
production rules and optimization, graph grammar approach)
(e.g.,
\cite{schmidt00}).
A survey of combinatorial optimization approaches to system configuration design
is presented in
\cite{lev09}.
In this article,
a generalized composite design framework
for modular systems
is used (Fig. 2):
selection of design alternatives (DAs) for system
components/parts,
combinatorial synthesis (composition) of the composite
solutions, and
aggregation of the obtained solutions to get a resultant
aggregated solution.
\begin{center}
\begin{picture}(72,79)
\put(02,00){\makebox(0,0)[bl]{Fig. 2.
Selection, composition, aggregation
}}
\put(00,64){\line(1,0){20}} \put(00,78){\line(1,0){20}}
\put(00,64){\line(0,1){14}} \put(20,64){\line(0,1){14}}
\put(01,74){\makebox(0,0)[bl]{Set of DAs}}
\put(01,70){\makebox(0,0)[bl]{for system}}
\put(04,66){\makebox(0,0)[bl]{part \(1\)}}
\put(10,64){\vector(0,-1){4}}
\put(21,71){\makebox(0,0)[bl]{{\bf ...}}}
\put(47,71){\makebox(0,0)[bl]{{\bf ...}}}
\put(26,64){\line(1,0){20}} \put(26,78){\line(1,0){20}}
\put(26,64){\line(0,1){14}} \put(46,64){\line(0,1){14}}
\put(27,74){\makebox(0,0)[bl]{Set of DAs}}
\put(27,70){\makebox(0,0)[bl]{for system }}
\put(30,66){\makebox(0,0)[bl]{part \(i\)}}
\put(36,64){\vector(0,-1){4}}
\put(52,64){\line(1,0){20}} \put(52,78){\line(1,0){20}}
\put(52,64){\line(0,1){14}} \put(72,64){\line(0,1){14}}
\put(53,74){\makebox(0,0)[bl]{Set of DAs}}
\put(53,70){\makebox(0,0)[bl]{for system }}
\put(56,66){\makebox(0,0)[bl]{part \(m\)}}
\put(62,64){\vector(0,-1){4}}
\put(00,50){\line(1,0){20}} \put(00,60){\line(1,0){20}}
\put(00,50){\line(0,1){10}} \put(20,50){\line(0,1){10}}
\put(0.4,50){\line(0,1){10}} \put(19.6,50){\line(0,1){10}}
\put(03,56){\makebox(0,0)[bl]{Ranking}}
\put(03,52){\makebox(0,0)[bl]{of DAs }}
\put(10,50){\vector(1,-1){5}}
\put(21,55){\makebox(0,0)[bl]{{\bf ...}}}
\put(47,55){\makebox(0,0)[bl]{{\bf ...}}}
\put(26,50){\line(1,0){20}} \put(26,60){\line(1,0){20}}
\put(26,50){\line(0,1){10}} \put(46,50){\line(0,1){10}}
\put(26.4,50){\line(0,1){10}} \put(45.6,50){\line(0,1){10}}
\put(29,56){\makebox(0,0)[bl]{Ranking}}
\put(29,52){\makebox(0,0)[bl]{of DAs }}
\put(36,50){\vector(0,-1){5}}
\put(52,50){\line(1,0){20}} \put(52,60){\line(1,0){20}}
\put(52,50){\line(0,1){10}} \put(72,50){\line(0,1){10}}
\put(52.4,50){\line(0,1){10}} \put(71.6,50){\line(0,1){10}}
\put(55,56){\makebox(0,0)[bl]{Ranking}}
\put(55,52){\makebox(0,0)[bl]{of DAs }}
\put(62,50){\vector(-1,-1){5}}
\put(10,38){\line(1,0){52}} \put(10,45){\line(1,0){52}}
\put(10,38){\line(0,1){07}} \put(62,38){\line(0,1){07}}
\put(10.5,38.5){\line(1,0){51}} \put(10.5,44.5){\line(1,0){51}}
\put(10.5,38.5){\line(0,1){06}} \put(61.5,38.5){\line(0,1){06}}
\put(12,40){\makebox(0,0)[bl]{Composition/synthesis process}}
\put(36,38){\vector(0,-1){04}}
\put(36,31){\oval(58,6)} \put(36,31){\oval(57,5)}
\put(08.6,29){\makebox(0,0)[bl]{Designed composite solution(s)
\(\{S\}\)}}
\put(27,28){\vector(0,-1){04}}
\put(34,25.7){\makebox(0,0)[bl]{{\bf ...}}}
\put(45,28){\vector(0,-1){04}}
\put(10,18){\line(1,0){52}} \put(10,24){\line(1,0){52}}
\put(10,18){\line(0,1){06}} \put(62,18){\line(0,1){06}}
\put(10.5,18.5){\line(1,0){51}} \put(10.5,23.5){\line(1,0){51}}
\put(10.5,18.5){\line(0,1){05}} \put(61.5,18.5){\line(0,1){05}}
\put(20,19.2){\makebox(0,0)[bl]{Aggregation process}}
\put(36,18){\vector(0,-1){04}}
\put(36,09.5){\oval(44,9)} \put(36,9.5){\oval(43,8)}
\put(20,9.5){\makebox(0,0)[bl]{Resultant aggregated }}
\put(23,6){\makebox(0,0)[bl]{solution(s) \(S^{agg}\)}}
\end{picture}
\end{center}
The approach is based on three optimization problems:
{\bf I.} Multicriteria ranking
(outranking technique as a modification of ELECTRE method
is used
\cite{roy96}).
{\bf II.} Morphological synthesis based on morphological clique problem
(as Hierarchical Morphological Multicriteria Design - HMMD)
(\cite{lev98},
\cite{lev06},
\cite{lev09}, \cite{lev12}).
{\bf III.} Aggregation of the obtained composite solutions into
the
resultant
aggregated solution(s)
(aggregation strategies are used,
e.g., design of system ``kernel'' and its extension)
\cite{lev11agg}
(here knapsack-like problems are sued).
The illustrative numerical design example involves
hierarchical structure of sensor
(and-or tree model),
design alternatives (DAs) for system parts/components,
Bottom-Up solving process.
Estimates of DAs and their compatibilities are based on expert judgment.
A preliminary material of the paper was published as conference paper
\cite{levfim10}.
\section{Structure of Sensor and Estimates}
The following simplified illustrative hierarchical structure of an alarm wireless sensor
element is examined
(Fig. 3):
\begin{center}
\begin{picture}(74,96)
\put(03,0){\makebox(0,0)[bl]{Fig. 3. Structure of wireless sensor
element}}
\put(1,17){\makebox(0,8)[bl]{\(R_{1}(3)\)}}
\put(1,13){\makebox(0,8)[bl]{\(R_{2}(3)\)}}
\put(1,9){\makebox(0,8)[bl]{\(R_{3}(1)\)}}
\put(1,5){\makebox(0,8)[bl]{\(R_{4}(1)\)}}
\put(11,17){\makebox(0,8)[bl]{\(P_{1}(3)\)}}
\put(11,13){\makebox(0,8)[bl]{\(P_{2}(1)\)}}
\put(11,9){\makebox(0,8)[bl]{\(P_{3}(2)\)}}
\put(21,17){\makebox(0,8)[bl]{\(D_{1}(2)\)}}
\put(21,13){\makebox(0,8)[bl]{\(D_{2}(1)\)}}
\put(21,9){\makebox(0,8)[bl]{\(D_{3}(3)\)}}
\put(31,17){\makebox(0,8)[bl]{\(Q_{1}(3)\)}}
\put(31,13){\makebox(0,8)[bl]{\(Q_{2}(3)\)}}
\put(31,9){\makebox(0,8)[bl]{\(Q_{3}(2)\)}}
\put(31,5){\makebox(0,8)[bl]{\(Q_{4}(1)\)}}
\put(4,27){\line(1,0){30}}
\put(04,27){\line(0,-1){04}} \put(14,27){\line(0,-1){04}}
\put(24,27){\line(0,-1){04}} \put(34,27){\line(0,-1){04}}
\put(04,22){\circle{2}} \put(14,22){\circle{2}}
\put(24,22){\circle{2}} \put(34,22){\circle{2}}
\put(06,23){\makebox(0,8)[bl]{\(R\) }}
\put(16,23){\makebox(0,8)[bl]{\(P\) }}
\put(26,23){\makebox(0,8)[bl]{\(D\) }}
\put(36,23){\makebox(0,8)[bl]{\(Q\) }}
\put(04,27){\line(0,1){20}}
\put(04,37){\circle*{1.7}}
\put(06,37){\makebox(0,8)[bl]{\(M = R \star P \star D \star Q\)}}
\put(05,33){\makebox(0,8)[bl]{\(M_{1} = R_{3} \star P_{3} \star
D_{2} \star Q_{4}\) }}
\put(05,29){\makebox(0,8)[bl]{\(M_{2} = R_{4} \star P_{3} \star
D_{2} \star Q_{4}\) }}
\put(44,31){\makebox(0,8)[bl]{\(U_{1}(1)\)}}
\put(44,27){\makebox(0,8)[bl]{\(U_{2}(2)\)}}
\put(53,31){\makebox(0,8)[bl]{\(Z_{1}(1)\)}}
\put(53,27){\makebox(0,8)[bl]{\(Z_{2}(1)\)}}
\put(53,23){\makebox(0,8)[bl]{\(Z_{3}(2)\)}}
\put(48,38){\makebox(0,8)[bl]{\(U\) }}
\put(57,38){\makebox(0,8)[bl]{\(Z\) }}
\put(46,42){\line(0,-1){04}} \put(55,42){\line(0,-1){04}}
\put(46,37){\circle{2}} \put(55,37){\circle{2}}
\put(04,42){\line(1,0){51}}
\put(4,69){\circle*{2.5}}
\put(4,69){\line(0,-1){22}}
\put(6,69){\makebox(0,8)[bl]{\(H= M \star U \star Z\) }}
\put(6,65){\makebox(0,8)[bl]{\(H_{1} = M_{1} \star U_{1} \star
Z_{1}\) }}
\put(6,61){\makebox(0,8)[bl]{\(H_{2} = M_{2} \star U_{1} \star
Z_{1}\) }}
\put(6,57){\makebox(0,8)[bl]{\(H_{3} = M_{1} \star U_{1} \star
Z_{2}\) }}
\put(6,53){\makebox(0,8)[bl]{\(H_{4} = M_{2} \star U_{1} \star
Z_{2}\) }}
\put(48,52){\makebox(0,8)[bl]{\(Y_{1}(3)\)}}
\put(48,48){\makebox(0,8)[bl]{\(Y_{2}(1)\)}}
\put(48,44){\makebox(0,8)[bl]{\(Y_{3}(2)\)}}
\put(57,52){\makebox(0,8)[bl]{\(O_{1}(1)\)}}
\put(57,48){\makebox(0,8)[bl]{\(O_{2}(2)\)}}
\put(52,59){\makebox(0,8)[bl]{\(Y\) }}
\put(61,59){\makebox(0,8)[bl]{\(O\) }}
\put(50,63){\line(0,-1){04}} \put(59,63){\line(0,-1){04}}
\put(50,58){\circle{2}} \put(59,58){\circle{2}}
\put(50,63){\line(1,0){9}}
\put(50,59){\line(0,1){10}} \put(50,69){\circle*{2.5}}
\put(52,72){\makebox(0,8)[bl]{\(W=Y\star O\) }}
\put(52,68){\makebox(0,8)[bl]{\(W_{1}=Y_{3}\star O_{1}\) }}
\put(52,64){\makebox(0,8)[bl]{\(W_{2}=Y_{2}\star O_{2}\) }}
\put(04,69){\line(0,1){05}} \put(50,69){\line(0,1){05}}
\put(04,74){\line(1,0){46}}
\put(04,74){\line(0,1){18}}
\put(04,92){\circle*{3}}
\put(7,92){\makebox(0,8)[bl]{\(S = H \star W\) }}
\put(6,88){\makebox(0,8)[bl]{\(S_{1} = H_{1} \star W_{1}\),
\(S_{2} = H_{2} \star W_{1}\), }}
\put(6,84){\makebox(0,8)[bl]{\(S_{3} = H_{3} \star W_{1}\),
\(S_{4} = H_{4} \star W_{1}\), }}
\put(6,80){\makebox(0,8)[bl]{\(S_{5} = H_{1} \star W_{2}\),
\(S_{6} = H_{2} \star W_{2}\), }}
\put(6,76){\makebox(0,8)[bl]{\(S_{7} = H_{3} \star W_{2}\),
\(S_{8} = H_{4} \star W_{2}\)}}
\end{picture}
\end{center}
{\bf 0.} Alarm wireless sensor element ~\(S = H \star W\).
{\bf 1.} Hardware ~ \(H = M \star U \star Z\).
{\it 1.1.} Microelectronic components ~
\(M = R \star P \star D \star Q\).
{\it 1.1.1.} Radio \(R\):~
Chipcon CC2420 Radio \(R_{1}(3)\),
Chipcon CC1000 Radio \(R_{2}(4)\),
Semtech XE1205 Radio \(R_{3}(2)\),
Infineon TDA5250 Radio \(R_{4}(1)\).
{\it 1.1.2.} Microprocessor \(P\):~
Atmel ATmega128 with 10-bit ADC \(P_{1}(3)\),
Atmel AVR AT90S2313 \(P_{2}(1)\),
Texas Instruments MSP430F16 with 12-bit ADC/DAC \(P_{3}(2)\).
{\it 1.1.3.} DAC/ADC \(D\):~
Atmel ATmega128L embedded 10-bit ADC \(D_{1}(2)\),
Texas Instruments MSP430F16 embedded 12-bit ADC/DAC \(D_{2}(1)\),
Analog Devices 14-bit AD679 \(D_{3}(3)\).
{\it 1.1.4.} Memory \(Q\):~
No external memory \(Q_{1}(4)\),
4 Kb EEPROM \(Q_{2}(3)\),
128 Kb Flash \(Q_{3}(2)\),
1 Mb Flash \(Q_{4}(1)\).
{\it 1.2.} Power supply \(U\):~
2800 mAh NiMh Battery \(U_{1}(1)\),
1500 mAh Li-Ion Battery \(U_{2}(2)\).
{\it 1.3.} Sensor \(Z\):~
Edwards 284b-pl Heat Detector \(Z_{1}(1)\),
123 Security Systems Photoelectric 2-Wire Smoke \(Z_{2}(2)\),
Multisensing Fire Detector \(Z_{3}(3)\).
{\bf 2.} Software~ \(W = Y \star O\).
{\it 2.1.} Sensor software \(Y\):~
Zigbee/802.15.4 \& Application \(Y_{1}(3)\),
TinyOS BMAC \& Application \(Y_{2}(1)\),
Ad-Hoc software \& Application~ \(Y_{3}(2)\).
{\it 2.2.} OS \(O\):~
No OS, Simple run-time environment \(O_{1}(1)\),
TinyOS~ \(O_{2}(2)\).
\begin{center}
\begin{picture}(73,114)
\put(04.5,110){\makebox(0,0)[bl]{Table 1. Estimates of DAs upon
criteria}}
\put(00,0){\line(1,0){73}} \put(00,98){\line(1,0){73}}
\put(00,108){\line(1,0){73}}
\put(00,0){\line(0,1){108}} \put(09,0){\line(0,1){108}}
\put(73,0){\line(0,1){108}}
\put(01,94){\makebox(0,0)[bl]{\(R_{1}\)}}
\put(01,90){\makebox(0,0)[bl]{\(R_{2}\)}}
\put(01,86){\makebox(0,0)[bl]{\(R_{3}\)}}
\put(01,82){\makebox(0,0)[bl]{\(R_{4}\)}}
\put(01,78){\makebox(0,0)[bl]{\(P_{1}\)}}
\put(01,74){\makebox(0,0)[bl]{\(P_{2}\)}}
\put(01,70){\makebox(0,0)[bl]{\(P_{3}\)}}
\put(01,66){\makebox(0,0)[bl]{\(D_{1}\)}}
\put(01,62){\makebox(0,0)[bl]{\(D_{2}\)}}
\put(01,58){\makebox(0,0)[bl]{\(D_{3}\)}}
\put(01,54){\makebox(0,0)[bl]{\(Q_{1}\)}}
\put(01,50){\makebox(0,0)[bl]{\(Q_{2}\)}}
\put(01,46){\makebox(0,0)[bl]{\(Q_{3}\)}}
\put(01,42){\makebox(0,0)[bl]{\(Q_{4}\)}}
\put(01,38){\makebox(0,0)[bl]{\(U_{1}\)}}
\put(01,34){\makebox(0,0)[bl]{\(U_{2}\)}}
\put(01,30){\makebox(0,0)[bl]{\(Z_{1}\)}}
\put(01,26){\makebox(0,0)[bl]{\(Z_{2}\)}}
\put(01,22){\makebox(0,0)[bl]{\(Z_{3}\)}}
\put(01,18){\makebox(0,0)[bl]{\(Y_{1}\)}}
\put(01,14){\makebox(0,0)[bl]{\(Y_{2}\)}}
\put(01,10){\makebox(0,0)[bl]{\(Y_{3}\)}}
\put(01,06){\makebox(0,0)[bl]{\(O_{1}\)}}
\put(01,02){\makebox(0,0)[bl]{\(O_{2}\)}}
\put(16,98){\line(0,1){10}} \put(24,98){\line(0,1){10}}
\put(31,98){\line(0,1){10}} \put(38,98){\line(0,1){10}}
\put(44,98){\line(0,1){10}} \put(58,98){\line(0,1){10}}
\put(64,98){\line(0,1){10}}
\put(11,104){\makebox(0,0)[bl]{\(C_{1}\)}}
\put(18,104){\makebox(0,0)[bl]{\(C_{2}\)}}
\put(25,104){\makebox(0,0)[bl]{\(C_{3}\)}}
\put(32,104){\makebox(0,0)[bl]{\(C_{4}\)}}
\put(39,104){\makebox(0,0)[bl]{\(C_{5}\)}}
\put(49,104){\makebox(0,0)[bl]{\(C_{6}\)}}
\put(59,104){\makebox(0,0)[bl]{\(C_{7}\)}}
\put(64.7,104){\makebox(0,0)[bl]{Prio-}}
\put(64.7,100){\makebox(0,0)[bl]{rity}}
\put(11,94){\makebox(0,0)[bl]{\(13\)}}
\put(18,94){\makebox(0,0)[bl]{\(80\)}}
\put(25.5,94){\makebox(0,0)[bl]{\(25\)}}
\put(32,94){\makebox(0,0)[bl]{\(250\)}}
\put(67,94){\makebox(0,0)[bl]{\(3\)}}
\put(11,90){\makebox(0,0)[bl]{\(11\)}}
\put(17,90){\makebox(0,0)[bl]{\(160\)}}
\put(25.5,90){\makebox(0,0)[bl]{\(29\)}}
\put(33,90){\makebox(0,0)[bl]{\(76\)}}
\put(40,90){\makebox(0,0)[bl]{\(\)}}
\put(67,90){\makebox(0,0)[bl]{\(3\)}}
\put(12,86){\makebox(0,0)[bl]{\(6\)}}
\put(17,86){\makebox(0,0)[bl]{\(600\)}}
\put(25.5,86){\makebox(0,0)[bl]{\(25\)}}
\put(33,86){\makebox(0,0)[bl]{\(76\)}}
\put(40,86){\makebox(0,0)[bl]{\(\)}}
\put(67,86){\makebox(0,0)[bl]{\(1\)}}
\put(12,82){\makebox(0,0)[bl]{\(8\)}}
\put(17,82){\makebox(0,0)[bl]{\(200\)}}
\put(25.5,82){\makebox(0,0)[bl]{\(17\)}}
\put(33,82){\makebox(0,0)[bl]{\(64\)}}
\put(67,82){\makebox(0,0)[bl]{\(1\)}}
\put(12,78){\makebox(0,0)[bl]{\(8\)}}
\put(13,78){\makebox(0,0)[bl]{\(\)}}
\put(26.5,78){\makebox(0,0)[bl]{\(8\)}}
\put(33,78){\makebox(0,0)[bl]{\(16\)}}
\put(67,78){\makebox(0,0)[bl]{\(3\)}}
\put(10.5,74){\makebox(0,0)[bl]{\(2.5\)}}
\put(13,74){\makebox(0,0)[bl]{\(\)}}
\put(26.5,74){\makebox(0,0)[bl]{\(5\)}}
\put(33,74){\makebox(0,0)[bl]{\(10\)}}
\put(67,74){\makebox(0,0)[bl]{\(1\)}}
\put(11,70){\makebox(0,0)[bl]{\(11\)}}
\put(13,70){\makebox(0,0)[bl]{\(\)}}
\put(26.5,70){\makebox(0,0)[bl]{\(2\)}}
\put(33,70){\makebox(0,0)[bl]{\(12\)}}
\put(67,70){\makebox(0,0)[bl]{\(2\)}}
\put(12,66){\makebox(0,0)[bl]{\(0\)}}
\put(26.5,66){\makebox(0,0)[bl]{\(2\)}}
\put(32,66){\makebox(0,0)[bl]{\(150\)}}
\put(39.5,66){\makebox(0,0)[bl]{\(10\)}}
\put(67,66){\makebox(0,0)[bl]{\(2\)}}
\put(12,62){\makebox(0,0)[bl]{\(0\)}}
\put(26.5,62){\makebox(0,0)[bl]{\(1\)}}
\put(32,62){\makebox(0,0)[bl]{\(200\)}}
\put(39.5,62){\makebox(0,0)[bl]{\(12\)}}
\put(67,62){\makebox(0,0)[bl]{\(1\)}}
\put(12,58){\makebox(0,0)[bl]{\(4\)}}
\put(26.5,58){\makebox(0,0)[bl]{\(4\)}}
\put(32,58){\makebox(0,0)[bl]{\(250\)}}
\put(39.5,58){\makebox(0,0)[bl]{\(14\)}}
\put(67,58){\makebox(0,0)[bl]{\(3\)}}
\put(12,54){\makebox(0,0)[bl]{\(0\)}}
\put(26.5,54){\makebox(0,0)[bl]{\(0\)}}
\put(33.5,54){\makebox(0,0)[bl]{\(0\)}}
\put(50,54){\makebox(0,0)[bl]{\(0\)}}
\put(67,54){\makebox(0,0)[bl]{\(3\)}}
\put(12,50){\makebox(0,0)[bl]{\(1\)}}
\put(26.5,50){\makebox(0,0)[bl]{\(2\)}}
\put(33.5,50){\makebox(0,0)[bl]{\(3\)}}
\put(47,50){\makebox(0,0)[bl]{\(1024\)}}
\put(67,50){\makebox(0,0)[bl]{\(3\)}}
\put(12,46){\makebox(0,0)[bl]{\(3\)}}
\put(26.5,46){\makebox(0,0)[bl]{\(3\)}}
\put(33.5,46){\makebox(0,0)[bl]{\(2\)}}
\put(45.5,46){\makebox(0,0)[bl]{\(131072\)}}
\put(67,46){\makebox(0,0)[bl]{\(2\)}}
\put(12,42){\makebox(0,0)[bl]{\(3\)}}
\put(26.5,42){\makebox(0,0)[bl]{\(3\)}}
\put(33.5,42){\makebox(0,0)[bl]{\(2\)}}
\put(45,42){\makebox(0,0)[bl]{\(1048576\)}}
\put(67,42){\makebox(0,0)[bl]{\(1\)}}
\put(12,38){\makebox(0,0)[bl]{\(3\)}}
\put(47.5,38){\makebox(0,0)[bl]{\(2800\)}}
\put(67,38){\makebox(0,0)[bl]{\(1\)}}
\put(11,34){\makebox(0,0)[bl]{\(10\)}}
\put(47.5,34){\makebox(0,0)[bl]{\(1500\)}}
\put(67,34){\makebox(0,0)[bl]{\(2\)}}
\put(11,30){\makebox(0,0)[bl]{\(10\)}}
\put(40,30){\makebox(0,0)[bl]{\(2\)}}
\put(67,30){\makebox(0,0)[bl]{\(1\)}}
\put(11,26){\makebox(0,0)[bl]{\(25\)}}
\put(40,26){\makebox(0,0)[bl]{\(5\)}}
\put(67,26){\makebox(0,0)[bl]{\(1\)}}
\put(11,22){\makebox(0,0)[bl]{\(50\)}}
\put(39.5,22){\makebox(0,0)[bl]{\(16\)}}
\put(67,22){\makebox(0,0)[bl]{\(3\)}}
\put(10,18){\makebox(0,0)[bl]{\(100\)}}
\put(47,18){\makebox(0,0)[bl]{\(15000\)}}
\put(60,18){\makebox(0,0)[bl]{\(5\)}}
\put(67,18){\makebox(0,0)[bl]{\(3\)}}
\put(11,14){\makebox(0,0)[bl]{\(50\)}}
\put(47.5,14){\makebox(0,0)[bl]{\(6000\)}}
\put(60,14){\makebox(0,0)[bl]{\(6\)}}
\put(67,14){\makebox(0,0)[bl]{\(1\)}}
\put(10,10){\makebox(0,0)[bl]{\(100\)}}
\put(32,10){\makebox(0,0)[bl]{\(\)}}
\put(47.5,10){\makebox(0,0)[bl]{\(4000\)}}
\put(59,10){\makebox(0,0)[bl]{\(11\)}}
\put(67,10){\makebox(0,0)[bl]{\(2\)}}
\put(12,06){\makebox(0,0)[bl]{\(0\)}}
\put(47.5,06){\makebox(0,0)[bl]{\(2000\)}}
\put(60,06){\makebox(0,0)[bl]{\(4\)}}
\put(67,06){\makebox(0,0)[bl]{\(1\)}}
\put(12,2){\makebox(0,0)[bl]{\(0\)}}
\put(47.5,2){\makebox(0,0)[bl]{\(4500\)}}
\put(60,2){\makebox(0,0)[bl]{\(0\)}}
\put(67,2){\makebox(0,0)[bl]{\(2\)}}
\end{picture}
\end{center}
The following generalized set of criteria for DAs is used
(criteria weights are shown in parentheses, symbol \(-\)
corresponds to the case when minimum value is the best one):
cost \(C_{1}\) (-100),
radius \(C_{2}\) (1),
power consumption \(C_{3}\) (-80),
speed/frequency \(C_{4}\) (1),
fidelity \(C_{5}\) (10),
capacity(memory) \(C_{6}\) (0.5), and
development duration \(C_{7}\) (1000).
Estimates of DAs upon the criteria
are presented in Table 1 (expert judgment).
The resultant priorities of DAs
are pointed out in Fig. 3
(priorities are shown in parentheses).
and in Table 1
(a modification
of outranking technique ELECTRE was used \cite{roy96}).
Table 2 and Table 3 contain estimates of compatibility between DAs.
Mainly, estimates are illustrative ones.
For components of \(M\), \(U\) and \(S\) equal compatibility estimates
(between corresponding local DAs) are considered.
\begin{center}
\begin{picture}(67,54)
\put(16,50){\makebox(0,0)[bl]{Table 2. Compatibility }}
\put(00,0){\line(1,0){67}} \put(00,42){\line(1,0){67}}
\put(00,48){\line(1,0){67}}
\put(00,0){\line(0,1){48}} \put(07,0){\line(0,1){48}}
\put(67,0){\line(0,1){48}}
\put(01,38){\makebox(0,0)[bl]{\(R_{1}\)}}
\put(01,34){\makebox(0,0)[bl]{\(R_{2}\)}}
\put(01,30){\makebox(0,0)[bl]{\(R_{3}\)}}
\put(01,26){\makebox(0,0)[bl]{\(R_{4}\)}}
\put(01,22){\makebox(0,0)[bl]{\(P_{1}\)}}
\put(01,18){\makebox(0,0)[bl]{\(P_{2}\)}}
\put(01,14){\makebox(0,0)[bl]{\(P_{3}\)}}
\put(01,10){\makebox(0,0)[bl]{\(D_{1}\)}}
\put(01,06){\makebox(0,0)[bl]{\(D_{2}\)}}
\put(01,02){\makebox(0,0)[bl]{\(D_{3}\)}}
\put(13,42){\line(0,1){6}} \put(19,42){\line(0,1){6}}
\put(25,42){\line(0,1){6}} \put(31,42){\line(0,1){6}}
\put(37,42){\line(0,1){6}} \put(43,42){\line(0,1){6}}
\put(49,42){\line(0,1){6}} \put(55,42){\line(0,1){6}}
\put(61,42){\line(0,1){6}}
\put(08,44){\makebox(0,0)[bl]{\(P_{1}\)}}
\put(14,44){\makebox(0,0)[bl]{\(P_{2}\)}}
\put(20,44){\makebox(0,0)[bl]{\(P_{3}\)}}
\put(26,44){\makebox(0,0)[bl]{\(D_{1}\)}}
\put(32,44){\makebox(0,0)[bl]{\(D_{2}\)}}
\put(38,44){\makebox(0,0)[bl]{\(D_{3}\)}}
\put(44,44){\makebox(0,0)[bl]{\(Q_{1}\)}}
\put(50,44){\makebox(0,0)[bl]{\(Q_{2}\)}}
\put(56,44){\makebox(0,0)[bl]{\(Q_{3}\)}}
\put(62,44){\makebox(0,0)[bl]{\(Q_{4}\)}}
\put(09,38){\makebox(0,0)[bl]{\(3\)}}
\put(15,38){\makebox(0,0)[bl]{\(3\)}}
\put(21,38){\makebox(0,0)[bl]{\(3\)}}
\put(27,38){\makebox(0,0)[bl]{\(3\)}}
\put(33,38){\makebox(0,0)[bl]{\(3\)}}
\put(39,38){\makebox(0,0)[bl]{\(3\)}}
\put(45,38){\makebox(0,0)[bl]{\(3\)}}
\put(51,38){\makebox(0,0)[bl]{\(3\)}}
\put(57,38){\makebox(0,0)[bl]{\(3\)}}
\put(63,38){\makebox(0,0)[bl]{\(3\)}}
\put(09,34){\makebox(0,0)[bl]{\(3\)}}
\put(15,34){\makebox(0,0)[bl]{\(3\)}}
\put(21,34){\makebox(0,0)[bl]{\(3\)}}
\put(27,34){\makebox(0,0)[bl]{\(3\)}}
\put(33,34){\makebox(0,0)[bl]{\(3\)}}
\put(39,34){\makebox(0,0)[bl]{\(3\)}}
\put(45,34){\makebox(0,0)[bl]{\(3\)}}
\put(51,34){\makebox(0,0)[bl]{\(3\)}}
\put(57,34){\makebox(0,0)[bl]{\(3\)}}
\put(63,34){\makebox(0,0)[bl]{\(3\)}}
\put(09,30){\makebox(0,0)[bl]{\(3\)}}
\put(15,30){\makebox(0,0)[bl]{\(3\)}}
\put(21,30){\makebox(0,0)[bl]{\(3\)}}
\put(27,30){\makebox(0,0)[bl]{\(3\)}}
\put(33,30){\makebox(0,0)[bl]{\(3\)}}
\put(39,30){\makebox(0,0)[bl]{\(3\)}}
\put(45,30){\makebox(0,0)[bl]{\(3\)}}
\put(51,30){\makebox(0,0)[bl]{\(3\)}}
\put(57,30){\makebox(0,0)[bl]{\(3\)}}
\put(63,30){\makebox(0,0)[bl]{\(3\)}}
\put(09,26){\makebox(0,0)[bl]{\(3\)}}
\put(15,26){\makebox(0,0)[bl]{\(3\)}}
\put(21,26){\makebox(0,0)[bl]{\(3\)}}
\put(27,26){\makebox(0,0)[bl]{\(3\)}}
\put(33,26){\makebox(0,0)[bl]{\(3\)}}
\put(39,26){\makebox(0,0)[bl]{\(3\)}}
\put(45,26){\makebox(0,0)[bl]{\(3\)}}
\put(51,26){\makebox(0,0)[bl]{\(3\)}}
\put(57,26){\makebox(0,0)[bl]{\(3\)}}
\put(63,26){\makebox(0,0)[bl]{\(3\)}}
\put(09,22){\makebox(0,0)[bl]{\(\)}}
\put(15,22){\makebox(0,0)[bl]{\(\)}}
\put(21,22){\makebox(0,0)[bl]{\(\)}}
\put(27,22){\makebox(0,0)[bl]{\(3\)}}
\put(33,22){\makebox(0,0)[bl]{\(0\)}}
\put(39,22){\makebox(0,0)[bl]{\(1\)}}
\put(45,22){\makebox(0,0)[bl]{\(3\)}}
\put(51,22){\makebox(0,0)[bl]{\(3\)}}
\put(57,22){\makebox(0,0)[bl]{\(3\)}}
\put(63,22){\makebox(0,0)[bl]{\(3\)}}
\put(09,18){\makebox(0,0)[bl]{\(\)}}
\put(15,18){\makebox(0,0)[bl]{\(\)}}
\put(21,18){\makebox(0,0)[bl]{\(\)}}
\put(27,18){\makebox(0,0)[bl]{\(0\)}}
\put(33,18){\makebox(0,0)[bl]{\(0\)}}
\put(39,18){\makebox(0,0)[bl]{\(1\)}}
\put(45,18){\makebox(0,0)[bl]{\(3\)}}
\put(51,18){\makebox(0,0)[bl]{\(3\)}}
\put(57,18){\makebox(0,0)[bl]{\(3\)}}
\put(63,18){\makebox(0,0)[bl]{\(3\)}}
\put(09,14){\makebox(0,0)[bl]{\(\)}}
\put(15,14){\makebox(0,0)[bl]{\(\)}}
\put(21,14){\makebox(0,0)[bl]{\(\)}}
\put(27,14){\makebox(0,0)[bl]{\(0\)}}
\put(33,14){\makebox(0,0)[bl]{\(3\)}}
\put(39,14){\makebox(0,0)[bl]{\(1\)}}
\put(45,14){\makebox(0,0)[bl]{\(3\)}}
\put(51,14){\makebox(0,0)[bl]{\(3\)}}
\put(57,14){\makebox(0,0)[bl]{\(3\)}}
\put(63,14){\makebox(0,0)[bl]{\(3\)}}
\put(45,10){\makebox(0,0)[bl]{\(3\)}}
\put(51,10){\makebox(0,0)[bl]{\(3\)}}
\put(57,10){\makebox(0,0)[bl]{\(3\)}}
\put(63,10){\makebox(0,0)[bl]{\(3\)}}
\put(45,6){\makebox(0,0)[bl]{\(3\)}}
\put(51,6){\makebox(0,0)[bl]{\(3\)}}
\put(57,6){\makebox(0,0)[bl]{\(3\)}}
\put(63,6){\makebox(0,0)[bl]{\(3\)}}
\put(45,2){\makebox(0,0)[bl]{\(3\)}}
\put(51,2){\makebox(0,0)[bl]{\(3\)}}
\put(57,2){\makebox(0,0)[bl]{\(3\)}}
\put(63,2){\makebox(0,0)[bl]{\(3\)}}
\end{picture}
\end{center}
\begin{center}
\begin{picture}(40,26)
\put(02,22){\makebox(0,0)[bl]{Table 3. Compatibility}}
\put(10,0){\line(1,0){19}} \put(10,14){\line(1,0){19}}
\put(10,20){\line(1,0){19}}
\put(10,0){\line(0,1){20}} \put(17,0){\line(0,1){20}}
\put(29,0){\line(0,1){20}}
\put(11,10){\makebox(0,0)[bl]{\(Y_{1}\)}}
\put(11,06){\makebox(0,0)[bl]{\(Y_{2}\)}}
\put(11,02){\makebox(0,0)[bl]{\(Y_{3}\)}}
\put(23,14){\line(0,1){6}}
\put(18,16){\makebox(0,0)[bl]{\(O_{1}\)}}
\put(24,16){\makebox(0,0)[bl]{\(O_{2}\)}}
\put(19,10){\makebox(0,0)[bl]{\(1\)}}
\put(25,10){\makebox(0,0)[bl]{\(2\)}}
\put(19,6){\makebox(0,0)[bl]{\(0\)}}
\put(25,6){\makebox(0,0)[bl]{\(3\)}}
\put(19,2){\makebox(0,0)[bl]{\(3\)}}
\put(25,2){\makebox(0,0)[bl]{\(2\)}}
\end{picture}
\end{center}
\section{Combinatorial Synthesis}
Second,
Hierarchical Morphological Multicriteria Design (HMMD)
based on morphological clique problem is considered
(e.g., \cite{lev98},
\cite{lev06},
\cite{lev09},
\cite{lev12}).
HMMD generalizes morphological analysis that was suggested by
F. Zwicky
\cite{zwi69}.
Development stages of morphological analysis based design approaches
are presented in \cite{lev12}.
A examined composite
(modular, decomposable, composable) system consists
of components and their interconnection or compatibility (IC).
Basic assumptions of HMMD are the following:
~(a) a tree-like structure of the system;
~(b) a composite estimate for system quality
that integrates components (subsystems, parts) qualities and
qualities of IC (compatibility) across subsystems;
~(c) monotonic criteria for the system and its components;
~(d) quality estimates of system components and IC are evaluated by
coordinated ordinal scales.
The designations are:
~(1) design alternatives (DAs) for
nodes of the model;
~(2) priorities of DAs (\(r=\overline{1,k}\);
\(1\) corresponds to the best level of quality);
~(3) an ordinal compatibility estimate for each pair of DAs
(\(w=\overline{0,l}\); \(l\) corresponds to the best level of quality).
Generally, the basic phases of HMMD are:
{\bf 1.} Design of the tree-like system model.
{\bf 2.} Generation of DAs for leaf nodes of the model.
{\bf 3.} Hierarchical selection and composing of DAs into composite
DAs for the corresponding higher level of the system
hierarchy.
{\bf 4.} Analysis and improvement of composite DAs (solution(s)).
Let \(S\) be a system consisting of \(m\) parts (components):
\(P(1),...,P(i),...,P(m)\).
A set of design alternatives (DAs)
is generated for each system part above.
The problem is:
{\it Find composite design alternative}
~ \(S=S(1)\star ...\star S(i)\star ...\star S(m)\)~
({\it one representative design alternative} \(S(i)\)
{\it for each system component/part} ~\(P(i)\), \(i=\overline{1,m}\))
{\it with non-zero}~ IC {\it estimates between
the representative
design
alternatives.}
A discrete space of the integrated system excellence is based
on the
following vector:
~~\(N(S)=(w(S);n(S))\),
~where \(w(S)\) is the minimum of pairwise compatibility
between DAs which correspond to different system components
(i.e.,
\(~\forall ~P_{j_{1}}\) and \( P_{j_{2}}\),
\(1 \leq j_{1} \neq j_{2} \leq m\))
in \(S\),
~\(n(S)=(n_{1},...,n_{r},...n_{k})\),
~where \(n_{r}\) is the number of DAs of the \(r\)th quality in \(S\)
~(\(\sum^{k}_{r=1} n_{r} = m\)).
As a result,
we search for composite decisions which are nondominated by \(N(S)\)
(i.e., Pareto-efficient solutions).
Fig. 4 depicts the lattice of system quality
(by elements; \(m=3\),\(k=3\)).
\begin{center}
\begin{picture}(60,81)
\put(0,0){\makebox(0,0)[bl] {Fig. 4. Lattice of quality (by
elements)}}
\put(27,74){\makebox(0,0)[bl]{The ideal}}
\put(27,71){\makebox(0,0)[bl]{point}}
\put(10,73){\makebox(0,0)[bl]{\(<3,0,0>\) }}
\put(17,68){\line(0,1){4}}
\put(10,63){\makebox(0,0)[bl]{\(<2,1,0>\)}}
\put(17,56){\line(0,1){6}}
\put(10,51){\makebox(0,0)[bl]{\(<2,0,1>\) }}
\put(17,44){\line(0,1){6}}
\put(10,39){\makebox(0,0)[bl]{\(<1,1,1>\) }}
\put(17,32){\line(0,1){6}}
\put(10,27){\makebox(0,0)[bl]{\(<1,0,2>\) }}
\put(17,20){\line(0,1){6}}
\put(10,15){\makebox(0,0)[bl]{\(<0,1,2>\) }}
\put(17,10){\line(0,1){4}}
\put(10,05){\makebox(0,0)[bl]{\(<0,0,3>\) }}
\put(29,08){\makebox(0,0)[bl]{The worst}}
\put(29,05){\makebox(0,0)[bl]{point}}
\put(19,59){\line(0,1){3}} \put(35,59){\line(-1,0){16}}
\put(35,56){\line(0,1){3}}
\put(30,51){\makebox(0,0)[bl]{\(<1,2,0>\) }}
\put(35,50){\line(0,-1){3}} \put(35,47){\line(-1,0){16}}
\put(19,47){\line(0,-1){3}}
\put(37,44){\line(0,1){6}}
\put(30,39){\makebox(0,0)[bl]{\(<0,3,0>\) }}
\put(19,35){\line(0,1){3}} \put(35,35){\line(-1,0){16}}
\put(35,32){\line(0,1){3}}
\put(37,32){\line(0,1){6}}
\put(30,27){\makebox(0,0)[bl]{\(<0,2,1>\) }}
\put(35,26){\line(0,-1){3}} \put(35,23){\line(-1,0){16}}
\put(19,23){\line(0,-1){3}}
\end{picture}
\end{center}
Now, let us consider combinatorial synthesis for
the subsystems of wireless sensor.
The obtained
Pareto-efficient
composite DAs for subsystems are the following:
(a) \( W_{1}= Y_{3} \star O_{1}\), ~\(N(W_{1})=(3;1,1,0)\);
(b) \( W_{2}= Y_{2} \star O_{2}\), ~\(N(W_{2})=(3;1,1,0)\);
(c) \( M_{1} = R_{3} \star P_{3} \star D_{2} \star Q_{4}\),
~\(N(M_{1})=(3;3,1,0)\).
(d) \( M_{2} = R_{4} \star P_{3} \star D_{2} \star Q_{4}\),
~\(N(M_{1})=(3;3,1,0)\).
Fig. 5 and Fig. 6 illustrate
solutions for \(M_{1}\) and \(M_{2}\).
\begin{center}
\begin{picture}(50,58)
\put(0,0){\makebox(0,0)[bl] {Fig. 5. Concentric presentation}}
\put(0,30){\line(1,0){24}} \put(0,36){\line(1,0){24}}
\put(0,30){\line(0,1){6}}
\put(1,32){\makebox(0,0)[bl]{\(R_{1}\)}}
\put(6,32){\makebox(0,0)[bl]{\(R_{2}\)}}
\put(11,30){\line(0,1){6}} \put(13,30){\line(0,1){6}}
\put(14,32){\makebox(0,0)[bl]{\(R_{3}\)}}
\put(19,32){\makebox(0,0)[bl]{\(R_{4}\)}}
\put(24,30){\line(0,1){6}}
\put(30,30){\line(1,0){18}} \put(30,36){\line(1,0){18}}
\put(30,30){\line(0,1){6}}
\put(31,32){\makebox(0,0)[bl]{\(D_{2}\)}}
\put(36,30){\line(0,1){6}}
\put(37,32){\makebox(0,0)[bl]{\(D_{1}\)}}
\put(42,30){\line(0,1){6}}
\put(43,32){\makebox(0,0)[bl]{\(D_{3}\)}}
\put(48,30){\line(0,1){6}}
\put(24,39){\line(0,1){16}} \put(30,39){\line(0,1){16}}
\put(24,39){\line(1,0){6}}
\put(25,40.5){\makebox(0,0)[bl]{\(P_{2}\)}}
\put(24,44.5){\line(1,0){6}}
\put(25,46){\makebox(0,0)[bl]{\(P_{3}\)}}
\put(24,50){\line(1,0){6}}
\put(25,51){\makebox(0,0)[bl]{\(P_{1}\)}}
\put(24,55){\line(1,0){6}}
\put(24,06){\line(0,1){21}} \put(30,06){\line(0,1){21}}
\put(24,06){\line(1,0){6}}
\put(25,7.5){\makebox(0,0)[bl]{\(Q_{2}\)}}
\put(25,12){\makebox(0,0)[bl]{\(Q_{1}\)}}
\put(24,17){\line(1,0){6}}
\put(25,18){\makebox(0,0)[bl]{\(Q_{3}\)}}
\put(24,22){\line(1,0){6}}
\put(25,23){\makebox(0,0)[bl]{\(Q_{4}\)}}
\put(24,27){\line(1,0){6}}
\put(20,36){\line(0,1){10}} \put(20,46){\line(1,0){4}}
\put(16,36){\line(0,1){12}} \put(16,48){\line(1,0){8}}
\put(20,30){\line(0,-1){04}} \put(20,26){\line(1,0){4}}
\put(16,30){\line(0,-1){6}} \put(16,24){\line(1,0){8}}
\put(33,36){\line(0,1){10}} \put(33,46){\line(-1,0){3}}
\put(33,30){\line(0,-1){04}} \put(33,26){\line(-1,0){3}}
\put(30,24){\line(1,0){20}} \put(50,24){\line(0,1){24}}
\put(30,48){\line(1,0){20}}
\put(24,33){\line(1,0){06}}
\put(14,36){\line(0,1){21}} \put(14,57){\line(1,0){21}}
\put(35,36){\line(0,1){21}}
\end{picture}
\end{center}
\begin{center}
\begin{picture}(52,65)
\put(02,0){\makebox(0,0)[bl]{Fig. 6. Space of system quality}}
\put(0,010){\line(0,1){40}} \put(0,010){\line(3,4){15}}
\put(0,050){\line(3,-4){15}}
\put(20,015){\line(0,1){40}} \put(20,015){\line(3,4){15}}
\put(20,055){\line(3,-4){15}}
\put(40,020){\line(0,1){40}} \put(40,020){\line(3,4){15}}
\put(40,060){\line(3,-4){15}}
\put(40,53){\circle*{2}}
\put(28,47){\makebox(0,0)[bl]{\(N(M_{1}), N(M_{2})\)}}
\put(40,60){\circle*{1}} \put(40,60){\circle{3}}
\put(23,59){\makebox(0,0)[bl]{The ideal}}
\put(27,56){\makebox(0,0)[bl]{point}}
\put(0,6){\makebox(0,0)[bl]{\(w=1\)}}
\put(20,11){\makebox(0,0)[bl]{\(w=2\)}}
\put(40,16){\makebox(0,0)[bl]{\(w=3\)}}
\put(02,12){\makebox(0,0)[bl]{The worst}}
\put(02,09){\makebox(0,0)[bl]{point}}
\put(0,10){\circle*{0.5}} \put(0,10){\circle{1.6}}
\end{picture}
\end{center}
Further, the solutions for \(H\) are:
\(H_{1} = M_{1} \star U_{1} \star Z_{1} =
(R_{3} \star P_{3} \star D_{2} \star Q_{4}) \star
(U_{1} \star Z_{1})\);
\(H_{2} = M_{2} \star U_{1} \star Z_{2} =
(R_{3} \star P_{3} \star D_{2} \star Q_{4}) \star
(U_{1} \star Z_{2})\);
\(H_{3} = M_{1} \star U_{1} \star Z_{1} =
(R_{4} \star P_{3} \star D_{2} \star Q_{4}) \star
(U_{1} \star Z_{1})\);
\(H_{4} = M_{1} \star U_{1} \star Z_{2} =
(R_{4} \star P_{3} \star D_{2} \star Q_{4}) \star
(U_{1} \star Z_{2})\).
Finally, eight resultant solutions are obtained:
\(S_{1} = H_{1} \star W_{1} =
(M_{1} \star U_{1} \star Z_{1}) \star
(Y_{3} \star O_{1})=\)
\(
((R_{3} \star P_{3} \star D_{2} \star Q_{4})
\star
(U_{1} \star Z_{1}))
\star
(Y_{3} \star O_{1})\);
\(S_{2} = H_{2} \star W_{1} =
(M_{2} \star U_{1} \star Z_{1}) \star
(Y_{3} \star O_{1})=\)
\(
((R_{4} \star P_{3} \star D_{2} \star Q_{4})
\star
(U_{1} \star Z_{1}))
\star
(Y_{3} \star O_{1})\);
\(S_{3} = H_{3} \star W_{1} =
(M_{1} \star U_{1} \star Z_{2}) \star
(Y_{3} \star O_{1})=\)
\(
((R_{3} \star P_{3} \star D_{2} \star Q_{4})
\star
(U_{1} \star Z_{1}))
\star
(Y_{3} \star O_{1})\);
\(S_{4} = H_{4} \star W_{1} =
(M_{2} \star U_{1} \star Z_{2}) \star
(Y_{3} \star O_{1})=\)
\(
((R_{4} \star P_{3} \star D_{2} \star Q_{4})
\star
(U_{1} \star Z_{1}))
\star
(Y_{3} \star O_{1})\);
\(S_{5} = H_{1} \star W_{2} =
(M_{1} \star U_{1} \star Z_{1}) \star
(Y_{2} \star O_{2})=\)
\(
((R_{3} \star P_{3} \star D_{2} \star Q_{4})
\star
(U_{1} \star Z_{1}))
\star
(Y_{2} \star O_{2})\);
\(S_{6} = H_{2} \star W_{2} =
(M_{2} \star U_{1} \star Z_{1}) \star
(Y_{2} \star O_{2})=\)
\(
((R_{4} \star P_{3} \star D_{2} \star Q_{4})
\star
(U_{1} \star Z_{1}))
\star
(Y_{2} \star O_{2})\);
\(S_{7} = H_{3} \star W_{2} =
(M_{1} \star U_{1} \star Z_{2}) \star
(Y_{2} \star O_{2})=\)
\(
((R_{3} \star P_{3} \star D_{2} \star Q_{4})
\star
(U_{1} \star Z_{2}))
\star
(Y_{2} \star O_{2})\);
\(S_{8} = H_{4} \star W_{2} =
(M_{2} \star U_{1} \star Z_{2}) \star
(Y_{2} \star O_{2})=\)
\(
((R_{4} \star P_{3} \star D_{2} \star Q_{4})
\star
(U_{1} \star Z_{2}))
\star
(Y_{2} \star O_{2})\).
Note in the example the initial combinatorial set
includes ~\(5184\) (\(4 \times 3 \times 3 \times 4 \times 2 \times 3 \times 3 \times 2\))
possible composite solutions.
\section{Aggregation of Modular Solutions}
Aggregation of composite systems (as modular solutions)
can be considered as follows
\cite{lev11agg}.
Fig. 7 illustrates substructure and superstructure
for three initial solutions
\(S^{1}\), \(S^{2}\), and \(S^{3}\).
\begin{center}
\begin{picture}(62,28)
\put(00,00){\makebox(0,0)[bl]{Fig. 7. Substructure and
superstructure}}
\put(20,09){\oval(36,6)}
\put(06,8){\makebox(0,8)[bl]{\(S^{1}\)}}
\put(40,09){\oval(30,7.6)}
\put(50,8){\makebox(0,8)[bl]{\(S^{3}\)}}
\put(32,13){\oval(08,15)}
\put(30.5,16.5){\makebox(0,8)[bl]{\(S^{2}\)}}
\put(29,13){\oval(58,17)} \put(29,13){\oval(58.7,17.7)}
\put(00,23){\makebox(0,8)[bl]{Superstructure}}
\put(10,22.5){\line(1,-1){5.6}}
\put(38.5,23.4){\makebox(0,8)[bl]{Substructure}}
\put(47.2,22.6){\line(-1,-1){13.3}}
\put(32,09){\oval(6,4)}
\end{picture}
\end{center}
In \cite{lev11agg},
basic aggregation strategies are described, for example:
{\bf 1.} {\it Extension strategy}:
~{\it 1.1.} building a ``kernel'' for initial solutions
(i.e., substructure/subsolution or an extended subsolution),
~{\it 1.2.} generation of a set of additional solution elements,
~{\it 1.3.} selection of additional elements from the generated
set while taking into account their ``profit'' and resource requirements
(i.e., a total ``profit'' and total resource constraint(s))
(here knapsack-like problems are used).
{\bf 2.} {\it Compression strategy}:
~{\it 2.1.} building a supersolution (as a superstructure),
~{\it 2.2.} generation of a set of solution elements from the
supersolution as candidates for deletion,
~{\it 2.3.} selection of the elements-candidates for deletion
while taking into account their ``profit'' and resource requirements
(i.e., a total profit and total resource constraint(s))
(here knapsack-like problems with minimization of objective function are used).
A general aggregation strategy has to be based on searching for a
consensus/median solution
\(S^{M}\) (``generalized'' median)
for the initial solutions \( \overline{S} = \{S^{1},...,S^{n}\}\)
(e.g., \cite{lev11agg}):
\[ S^{M} = \arg ~ min_{X \in \overline{S}} ~
( \sum_{i=1}^{n} ~ \rho (X, S^{i}) ),\]
where
\(\rho (X, Y)\) is a proximity (e.g., distance) between two solutions \(X\) and
\(Y\).
Mainly,
searching for the median for many structures is usually NP-complete
problem.
In our case, product structures correspond to a combination of
tree, set of DAs, their estimates, matrices of compatibility
estimates. As a result, the proximity between the structures
are more complicated and
the
``generalized'' median problem is very complex.
As a result, simplified (approximate) solving strategies is often used, for example
\cite{lev11agg}:
(a) searching for
``set median'' (i.e., one of the initial solutions is selected),
(b) ``extension strategy'' above,
(c) ``compression strategy'' above.
In the example, eight obtained composite solutions are considered:
~\(S_{1}\), \(S_{2}\), \(S_{3}\), \(S_{4}\), \(S_{5}\),
\(S_{6}\), \(S_{7}\), \(S_{8}\).
The substructure of the eight solutions is
presented in Fig. 8. This substructure is examined as
system ``kernel'' for future extension.
The superstructure is presented in Fig. 9.
\begin{center}
\begin{picture}(67,20)
\put(08,00){\makebox(0,0)[bl] {Fig. 8. Substructure
(``kernel'')}}
\put(05,14){\circle*{1.5}} \put(13,14){\circle*{1.5}}
\put(21,14){\circle*{1.5}} \put(29,14){\circle*{1.5}}
\put(37,14){\circle*{1.5}} \put(45,14){\circle*{1.5}}
\put(53,14){\circle*{1.5}} \put(61,14){\circle*{1.5}}
\put(04,16){\makebox(0,0)[bl]{\(R\)}}
\put(12,16){\makebox(0,0)[bl]{\(P\)}}
\put(20,16){\makebox(0,0)[bl]{\(D\)}}
\put(28,15.5){\makebox(0,0)[bl]{\(Q\)}}
\put(35,16){\makebox(0,0)[bl]{\(U\)}}
\put(43,16){\makebox(0,0)[bl]{\(Z\)}}
\put(52,16){\makebox(0,0)[bl]{\(Y\)}}
\put(60,16){\makebox(0,0)[bl]{\(O\)}}
\put(05,10){\oval(07,08)} \put(13,10){\oval(07,08)}
\put(21,10){\oval(07,08)} \put(29,10){\oval(07,08)}
\put(37,10){\oval(07,08)} \put(45,10){\oval(07,08)}
\put(53,10){\oval(07,08)} \put(61,10){\oval(07,08)}
\put(10.8,09){\makebox(0,0)[bl]{\(P_{3}\)}}
\put(18.8,09){\makebox(0,0)[bl]{\(D_{2}\)}}
\put(26.8,09){\makebox(0,0)[bl]{\(Q_{4}\)}}
\put(34.8,09){\makebox(0,0)[bl]{\(U_{1}\)}}
\end{picture}
\end{center}
\begin{center}
\begin{picture}(67,22)
\put(06.5,00){\makebox(0,0)[bl] {Fig. 9. Superstructure of
solutions}}
\put(05,16){\circle*{1.5}} \put(13,16){\circle*{1.5}}
\put(21,16){\circle*{1.5}} \put(29,16){\circle*{1.5}}
\put(37,16){\circle*{1.5}} \put(45,16){\circle*{1.5}}
\put(53,16){\circle*{1.5}} \put(61,16){\circle*{1.5}}
\put(04,18){\makebox(0,0)[bl]{\(R\)}}
\put(12,18){\makebox(0,0)[bl]{\(P\)}}
\put(20,18){\makebox(0,0)[bl]{\(D\)}}
\put(28,17.5){\makebox(0,0)[bl]{\(Q\)}}
\put(35,18){\makebox(0,0)[bl]{\(U\)}}
\put(43,18){\makebox(0,0)[bl]{\(Z\)}}
\put(52,18){\makebox(0,0)[bl]{\(Y\)}}
\put(60,18){\makebox(0,0)[bl]{\(O\)}}
\put(05,10.5){\oval(07,11)} \put(13,10.5){\oval(07,11)}
\put(21,10.5){\oval(07,11)} \put(29,10.5){\oval(07,11)}
\put(37,10.5){\oval(07,11)} \put(45,10.5){\oval(07,11)}
\put(53,10.5){\oval(07,11)} \put(61,10.5){\oval(07,11)}
\put(02.8,11){\makebox(0,0)[bl]{\(R_{3}\)}}
\put(02.8,07){\makebox(0,0)[bl]{\(R_{4}\)}}
\put(10.8,11){\makebox(0,0)[bl]{\(P_{3}\)}}
\put(18.8,11){\makebox(0,0)[bl]{\(D_{2}\)}}
\put(26.8,11){\makebox(0,0)[bl]{\(Q_{4}\)}}
\put(34.8,11){\makebox(0,0)[bl]{\(U_{1}\)}}
\put(42.8,11){\makebox(0,0)[bl]{\(Z_{1}\)}}
\put(42.8,07){\makebox(0,0)[bl]{\(Z_{2}\)}}
\put(50.8,11){\makebox(0,0)[bl]{\(Y_{2}\)}}
\put(50.8,07){\makebox(0,0)[bl]{\(Y_{3}\)}}
\put(58.8,11){\makebox(0,0)[bl]{\(O_{1}\)}}
\put(58.8,07){\makebox(0,0)[bl]{\(O_{2}\)}}
\end{picture}
\end{center}
The extension procedure based on multiple choice problem
is the following.
Table 4 contains design alternatives (DAs)
and their estimates (ordinal scales, expert judgment).
The design alternatives correspond to superstructure (Fig. 9).
\begin{center}
\begin{picture}(52,52)
\put(03,48){\makebox(0,0)[bl] {Table 4. Design alternatives}}
\put(0,0){\line(1,0){52}} \put(0,35){\line(1,0){52}}
\put(0,45){\line(1,0){52}}
\put(0,0){\line(0,1){45}} \put(05,0){\line(0,1){45}}
\put(18,0){\line(0,1){45}} \put(32,0){\line(0,1){45}}
\put(42,0){\line(0,1){45}} \put(52,0){\line(0,1){45}}
\put(2,41){\makebox(0,0)[bl]{\(\kappa\)}}
\put(7.5,40.5){\makebox(0,0)[bl]{DAs}}
\put(19,40.5){\makebox(0,0)[bl]{Binary}}
\put(19,37){\makebox(0,0)[bl]{variable}}
\put(33,41){\makebox(0,0)[bl]{Cost}}
\put(35,36.5){\makebox(0,0)[bl]{\(a_{ij}\)}}
\put(42.6,41){\makebox(0,0)[bl]{Profit}}
\put(45,36.5){\makebox(0,0)[bl]{\(c_{ij}\)}}
\put(1.6,30){\makebox(0,0)[bl]{\(1\)}}
\put(9,29.5){\makebox(0,0)[bl]{\(R_{3}\)}}
\put(23,30){\makebox(0,0)[bl]{\(x_{11}\)}}
\put(1.6,26){\makebox(0,0)[bl]{\(2\)}}
\put(9,25.5){\makebox(0,0)[bl]{\(R_{4}\) }}
\put(23,26){\makebox(0,0)[bl]{\(x_{12}\)}}
\put(1.6,22){\makebox(0,0)[bl]{\(3\)}}
\put(9,21.5){\makebox(0,0)[bl]{\(Z_{1} \)}}
\put(23,022){\makebox(0,0)[bl]{\(x_{21}\)}}
\put(1.6,18){\makebox(0,0)[bl]{\(4\)}}
\put(9,17.5){\makebox(0,0)[bl]{\(Z_{2} \)}}
\put(23,18){\makebox(0,0)[bl]{\(x_{22}\)}}
\put(1.6,14){\makebox(0,0)[bl]{\(5\)}}
\put(9,013.5){\makebox(0,0)[bl]{\(Y_{2}\)}}
\put(23,14){\makebox(0,0)[bl]{\(x_{31}\)}}
\put(1.6,10){\makebox(0,0)[bl]{\(6\)}}
\put(9,9.5){\makebox(0,0)[bl]{\(Y_{3}\) }}
\put(23,10){\makebox(0,0)[bl]{\(x_{32}\)}}
\put(1.6,6){\makebox(0,0)[bl]{\(7\)}}
\put(9,05.5){\makebox(0,0)[bl]{\(O_{1} \)}}
\put(23,06){\makebox(0,0)[bl]{\(x_{41}\)}}
\put(1.6,2){\makebox(0,0)[bl]{\(8\)}}
\put(9,01.5){\makebox(0,0)[bl]{\(O_{2} \)}}
\put(23,02){\makebox(0,0)[bl]{\(x_{42}\)}}
\put(36,30){\makebox(0,0)[bl]{\(2\)}}
\put(46,30){\makebox(0,0)[bl]{\(3\)}}
\put(36,26){\makebox(0,0)[bl]{\(3\)}}
\put(46,26){\makebox(0,0)[bl]{\(4\)}}
\put(36,22){\makebox(0,0)[bl]{\(4\)}}
\put(46,22){\makebox(0,0)[bl]{\(3\)}}
\put(36,18){\makebox(0,0)[bl]{\(6\)}}
\put(46,18){\makebox(0,0)[bl]{\(3\)}}
\put(36,14){\makebox(0,0)[bl]{\(7\)}}
\put(46,14){\makebox(0,0)[bl]{\(3\)}}
\put(36,10){\makebox(0,0)[bl]{\(8\)}}
\put(46,10){\makebox(0,0)[bl]{\(2\)}}
\put(36,06){\makebox(0,0)[bl]{\(1\)}}
\put(46,06){\makebox(0,0)[bl]{\(3\)}}
\put(36,02){\makebox(0,0)[bl]{\(1\)}}
\put(46,02){\makebox(0,0)[bl]{\(2\)}}
\end{picture}
\end{center}
It is assumed design alternatives for different
product components are compatible.
The multiple choice problem is:
\[\max \sum_{i=1}^{4} \sum_{j=1}^{q_{i}} c_{ij} x_{ij}
~~~s.t.~ \sum_{i=1}^{4} \sum_{j=1}^{q_{i}} a_{ij} x_{ij} \leq
b,
\]
\[\sum_{j=1}^{q_{i}} x_{ij} = 1 ~~ \forall i=\overline{1,4},
~~x_{ij} \in \{0,1\}.\]
Clearly, \(q_{1} = 2\), \(q_{2} = 2\), \(q_{3} = 2\), \(q_{4} =
2\).
The resultant aggregated solutions are
(a simple greedy algorithm was used;
the algorithm is based on ordering of elements by
\(c_{i}/a_{i}\)):
(1) \(b^{1}=14\):~
(\(x_{11} = 1\), \(x_{21} = 1\), \(x_{31} = 1\), \(x_{41} = 1\)),
~\(S^{agg}_{b^{1}} =
R_{3}\star P_{3} \star D_{2}\star Q_{4}\star U_{1}\star Z_{1}
\star Y_{2}\star O_{1} \);
(2) \(b^{2}=15\):~
(\(x_{12} = 1\), \(x_{21} = 1\), \(x_{31} = 1\), \(x_{41} = 1\)),
~\(S^{agg}_{b^{2}} =
R_{4}\star P_{3} \star D_{2}\star Q_{4}\star U_{1}\star Z_{1}
\star Y_{2}\star O_{1} \).
\section{Conclusion}
In the article, hierarchical combinatorial approach to configuration of
modular wireless sensor has been described.
The solving framework is based on
hierarchical model of the sensor element,
morphological design method for combinatorial synthesis
(building a special morphological clique), and
aggregation of the obtained modular solutions
(multiple choice problem).
The suggested approach can be used for many modular systems.
In the future it may be prospective to consider
the following research directions:
{\it 1.} taking into account uncertainty;
{\it 2.} analysis of dynamical design problems;
{\it 3.} usage of AI solving techniques'
and
{\it 4.} usage of the described application and solving approach
in engineering/CS education.
The draft material for the article was prepared
within framework of faculty course ``{\it Design of systems: structural
approach}'' at Moscow Inst. of Physics and Technology (State Univ.)
(creator and lecturer: M.Sh. Levin, 2004...2008)
\cite{lev11edu}.
|
{
"timestamp": "2012-03-12T01:01:08",
"yymm": "1203",
"arxiv_id": "1203.2031",
"language": "en",
"url": "https://arxiv.org/abs/1203.2031"
}
|
\section{Introduction}
Bipolar outflows are one of the most studied phenomena
of the star-formation process.
They result from the supersonic acceleration
of gas in two opposite directions by a newly formed star, and were first
identified more than three decades ago with radio observations
(\cite[Snell, Loren, \& Plambeck 1980]{sne80}).
Since their discovery, bipolar outflows have been identified
around protostars of nearly all masses, from below the brown
dwarf limit to the precursors of the ultra-compact HII regions, and in
environments as different as isolated globules and cluster-forming regions.
This ubiquity of the outflow phenomenon suggest that bipolar outflow
formation is a necessary element of the physics of star formation,
likely associated to the need for the gas to lose angular momentum
in its process of forming a highly compact object (see
\cite[Arce et al. 2007]{arc07} for a recent review).
Interest on the chemistry of bipolar outflows goes back in time
as far as the interest on their physics.
Outflow emission originates from ambient gas that has been accelerated
by a supersonic wind, and therefore has been shock-processed.
As a result, the study of outflow chemistry is unavoidably
intertwined to the study of shock chemistry in the interstellar gas.
Early questions on this chemistry were raised as soon as high velocity
molecular gas was observed, and were
related to the survival or shock-production of the
CO molecules seen at velocities of tens of km~s$^{-1}$
in the wings of outflow spectra (\cite[Kwan \& Scoville 1976]{kwa76})
and the up to a hundred km~s$^{-1}$ velocities seen in some water masers
(\cite[Morris 1976]{mor76}).
Prompted by these observations, the first (and still relevant) models
of molecule formation in shocks soon
appeared (\cite[Hollenbach \& McKee 1979]{hol79}).
\begin{figure}[t]
\begin{center}
\includegraphics[width=4.5in]{overabundances.ps}
\caption{Overabundance factors with respect to dense core values
for a number of molecules towards the outflows from L1448-mm
and IRAS 04166+2706. Data from \cite[Tafalla et al. (2010)]{taf10}.
}
\label{overabundances}
\end{center}
\end{figure}
With time, additional chemical processes have been
identified in the gas accelerated by outflows.
Current interest includes the enhancement in the abundance
of high-density gas tracers like SiO and CH$_3$OH, which is seen
toward a number of outflow sources powered both by high and low-mass
young stellar objects (\cite[Bachiller 1996]{bac96}).
These abundance enhancements are believed to
result from the erosion of dust grains via sputtering and grain-grain
collisions, perhaps combined with a number of gas-phase chemical
reactions favored by the temperature increase caused by the
shock (see \cite[van Dishoeck \& Blake 1998]{van98} for a review).
The amount of the
abundance enhancement depends on the species under consideration,
and typically ranges from factors of a few for species like HCO$^+$ to several
orders of magnitude for species like CH$_3$OH and SiO (Fig.\ref{overabundances}).
Not all outflows show the same degree of molecular richness, however,
and this apparent selectivity of outflow chemistry has lead
to the creation of special category of outflows, the so-called ``chemically
active'' outflows (\cite[Bachiller et al. 2001]{bac01}).
\section{Chemically active outflows}
Although easily distinguishable when observed in
species like CH$_3$OH and SiO, the chemically active outflows do not stand
apart from the rest when observed in CO (the standard outflow tracer),
either by their line strength, velocity extent, or spatial collimation.
It is well established
that the chemically active phase corresponds to an early period
in the outflow development, as the driving sources of this family of
outflows tend to be Class 0 sources. The exact physical cause of the
chemical richness is however unclear. One possibility is that the
youngest outflows must encounter denser envelopes along their paths, and
therefore must naturally give rise to stronger shocks. Another possibility
is that early chemical activity is associated with the higher energy
known to characterize the earliest outflow phases
(\cite[Bontemps et al. 1996]{bon96}).
In either way, the lack of chemical richness in the more evolved outflows
indicates that chemical activity is highly transient, and that any signature
of its presence must disappear quickly in the protostellar life
(near the transition between Class 0 and Class I,
\cite[Tafalla et al. 2000]{taf00}).
Rapid depletion of the enhanced species via freeze out
onto cold dust grains is the most likely cause of this effect
(\cite[Bergin et al. 1998]{ber98}).
\begin{figure}
\begin{center}
\resizebox{12cm}{!}{\includegraphics{l1157_co.ps}
\includegraphics{l1157_survey.ps}}
\caption{{\bf Left: } map of CO(2-1) emission from the L1157 outflow.
{\bf Right: } maps of integrated intensity for different molecular
species towards the blue (southern) lobe of the L1157 outflow. Note
the different extension and peak position of each species.
Figures from \cite[Bachiller et al. (2001)]{bac01}.
}
\label{l1157_plot}
\end{center}
\end{figure}
Among the group of chemically active outflows, the one in the
L1157 dark cloud stands
out for the strength of its lines in species like SiO and CH$_3$OH.
As a result, it has been the subject of a very intense observing campaign,
including the study using the Herschel Space Observatory
presented by Lefloch (2011, this volume).
The L1157 outflow was first identified by \cite[Umemoto et al. (1992)]{ume92}
due to its strong CO lines. Its chemical activity
was recognized very early on (\cite[Mikami et al. 1992]{mik92}),
and has been characterized in great detail by
\cite[Bachiller \& P\'erez-Guti\'errez (1997)]{bac97} and
\cite[Bachiller et al. (2001)]{bac01}.
The chemical activity of L1157 is specially prominent
towards several regions often denoted as B1 and B2
(for blue lobe) and R (for red lobe), and are most likely associated
with regions where the ambient cloud is being strongly shocked by the outflow
wind. Interferometric
observations of these regions reveal a complex pattern
of spatial distributions that indicate a very fragmented
small-scale structure (\cite[Benedettini et al. 2007]{ben07}).
As illustration of this chemical richness, we show in
Fig.\ref{l1157_plot} (right panel) a series of maps of the
southern lobe of the L1157 outflow in a number of molecules
that are selectively enhanced in different parts of the outflow
(especially B1 and B2).
\section{Recent progress trancing outflow gas with molecules}
It is not possible to review here with detail
the rapid progress made in the study of molecules in outflows
during the last few years. In this section we present a selection of
studies that have been carried out since the last IAU Astrochemistry
meeting in 2005. We concentrate on low-mass outflow studies and in molecules
that seem specially relevant to our understanding of shock chemistry.
\subsection{H$_2$O}
Probably the most significant progress since the Asilomar meeting
concerns the
study of the H$_2$O molecule, thanks to the coming on line of a series of
satellite observatories: the Submillimeter Wave Astronomy Observatory
(SWAS), Odin, the Spitzer Space Telescope,
and the Herschel Space Observatory.
Since the early discovery of its maser emission in outflows,
water has been considered a sensitive outflow tracer. The maser
emission is so bright that it can be easily detected across the galaxy,
and the abundance of water required to produce the emission
must have been significantly
enhanced by shocks (\cite[Elitzur \& de Jong 1978]{eli78},
\cite[Kaufman \& Neufeld 1996]{kau96}, \cite[Cernicharo et al.1996]{cer96}).
The thermal emission of
water, on the other hand, has taken longer to become a recognized
tracer of outflow activity, especially in low-mass outflows.
Early observations with the Infrared Space Observatory (ISO)
revealed thermal water emission from a number of
bipolar outflows (\cite[Liseau et al. 1996]{lis96},
\cite[Nisini et al. 1999]{nis99},
\cite[Giannini et al. 2001]{gia01}). These ISO spectra, however,
did not resolve spectrally even the broadest outflow profiles,
a fact that was later overcome by the
SWAS and Odin telescopes, which observed the H$_2$O(1$_{10}$--1$_{01}$) transition
with a resolution better than 1~km~s$^{-1}$.
These instruments, however, had a
limited angular resolution, approximately 4 arcminutes
for SWAS and half that size for Odin, that only allowed to
study global averages of the water emission over the whole outflow,
or at least over each of their lobes.
Compilations of this work have been presented by
\cite[Franklin et al. (2008)]{fra08} (SWAS) and
\cite[Bjerkeli et al. (2009)]{bje09} (Odin).
In the past two years, the Herschel Space Observatory (HSO) has
started to provide water data with spatial resolution comparable to
that of the large radio telescopes from the ground and, thanks
to its Heterodyne Instrument for the Far Infrared (HIFI), also with
velocity resolutions below 1~km~s$^{-1}$. Two HSO Guaranteed Time Key
Programmes have a strong component of low-mass outflow studies, the
Chemical Herschel Surveys of Star Forming Regions (CHESS) and Water in
Star-forming regions with Herschel (WISH, see
\cite[van Dishoeck et al. 2011]{van11}), while
Herschel/HIFI Observations of EXtraOrdinary Sources (HEXOS)
covers Orion. As a number of contributions in this volume by members
of these teams detail the first results of these programs,
here we will only present a brief description of the new exciting
outflow science that the HSO is providing, and we refer the reader to
the contributions by Bertrand Lefloch (CHESS), Lars Kristensen (WISH),
and Nathan Crockett (HEXOS) for further details.
\begin{figure}[t]
\begin{center}
\includegraphics[width=4.5in]{toledo_wish.ps}
\caption{{\bf Top: } excitation temperature between the first two excited
levels of o-H$_2$O as a function of peak intensity in the 557~GHz
H$_2$O line, as measured in a sample of 25 outflows within the
WISH program. Note the constant value at a level of about 25~K
(optically thin conditions have been assumed).
{\bf Bottom: } LVG radiative transfer results
for representative physical conditions of the emitting gas
(n(H$_2$) = $5\; 10^6$~cm$^{-3}$ and N(o-H$_2$O) = $10^{14}$~cm$^{-2}$).
As can be seen, the excitation temperatures derived in the top panel
require gas kinetic temperatures in the range 200-1100~K.
From Tafalla et al. (2011), in preparation.
}
\label{wish}
\end{center}
\end{figure}
It is safe to say that we are still exploring the tip of the iceberg
in our research of water emission from outflows. Much work still needs
to be done to understand the excitation conditions of the H$_2$O
emitting gas, and this understanding is critical to extract the
full potential of the water data. Preliminary work on
water excitation in outflows is presented in this meeting by the
poster contributions of Vasta et al. and Santangelo et al., which summarize
the first results of a multi-transition analysis of water towards the
very young outflows in L1157 and L1448. A more limited treatment in
terms of water lines, but more extensive in terms of outflow sources,
comes from the so-called ``water outflow survey'' carried out also as
part of the WISH project, and illustrated in Fig.~\ref{wish}. This outflow
survey has explored about 25 different objects by observing in each of
them two positions using two ortho-water lines, the 1$_{10}$--1$_{01}$
at 557 GHz with HIFI and the 2$_{12}$--1$_{01}$ at 1670 GHz with
PACS. As seen in Fig.~\ref{wish}, the data from
this survey shows a clear trend for the ratio of
these two lines (when convolved to the same angular resolution) to be
approximately constant independent on the intensity of the 557 GHz
line. This ratio, assuming optically thin conditions (tested with an
LVG code), is proportional to the excitation temperature between the
first two excited levels of ortho-water
(T$_{\mathrm{ex}}(2_{12}-1_{10})$), and as Fig.~\ref{wish} (top panel) shows is
approximately constant over the whole sample at a level of
approximately $25 \pm 5$~K. Such a low level of scatter suggests that
the bulk of the water emission we observe in outflows (at least from
the lowest energy levels) arises from a rather narrow range of
physical conditions.
Although a T$_{\mathrm{ex}}(2_{12}-1_{10})$ value of around 25~K may
seem a relatively low temperature, it is important to remember that the large
Einstein A coefficients of water make this molecule hard to become excited
collisionally. To estimate the kinetic temperature of the gas
responsible for this T$_{\mathrm{ex}}$ of 25~K, we need to estimate
the balance between collisional excitation and radiative de-excitation
(both spontaneous and induced). We have done this using the large
velocity gradient (LVG) approximation together with the recently
calculated collision rates by \cite[Faure et al. (2007)]{fau07}
(as provided by the LAMDA data base, \cite[Sch{\"o}ier et al. 2005]{sch05}),
and assuming values for the gas volume density and H$_2$O
column densities in line with those found by multi-transition analysis
like those in the posters by Vasta et al. and Santangelo et al. As
shown in Fig.~\ref{wish} (bottom panel), T$_{\mathrm{ex}}(2_{12}-1_{10})$
values of around 25~K require relatively high gas temperatures, in the
vicinity of 500~K. This result adds to existing evidence that the gas
responsible for the observed water emission from outflows corresponds
to a component warmer than that typically observed with low-J CO
transitions.
\subsection{H$_2$}
H$_2$ emission from outflows has been observed widely since its discovery
in Orion, and reveals the presence of gas heated to temperatures up to a few thousand
Kelvin (\cite[Gautier et al. 1976]{gau76}). Most of the H$_2$ work has so far
concentrated on the v=1--0 S(1) line at 2.12 $\mu$m, which is
easily observable from the ground. A testimony of the widespread nature
of this emission in outflows is its
use to define the recently created category of
Molecular Hydrogen emission-line Objects (MHOs), which are analogous to
the optically-based Herbig-Haro objects but include more deeply embedded
line-emitting regions
thanks to its use of NIR wavelengths (\cite[Davis et al. 2010]{dav10}).
At the time of writing this article, the MHO on-line catalog
({\tt http://www.jach.hawaii.edu/UKIRT/MHCat/}) is growing steadily and
contains more than 1200 entries.
Although some of the pure rotational transitions of H$_2$ are also observable from
the ground (e.g., \cite[Burton et al. 1989]{bur89}), sampling the full H$_2$ spectrum,
and therefore characterizing the population of its energy levels, requires the
use of space-based telescopes. ISO observations in the late 1990's carried out the
first complete observations of the H$_2$ spectrum in Orion,
and revealed a distribution of excitation temperatures that
ranged from about 600~K in the lowest levels to more than 3000~K
in the highest levels (\cite[Rosenthal et al. 2000]{ros00}).
More recently, the Infrared Spectrograph (IRS) on board of the
Spitzer Space Telescope has covered the 5-33 $\mu$m
wavelength range, and therefore has provided access to the
lowest pure rotational lines of H$_2$.
The high sensitivity of this
instrument has allowed the systematic mapping of
rotational emission from low mass outflows.
\cite[Neufeld et al. (2006)]{neu06},
for example, observed
the S(0)-S(7) transitions towards the HH 7-11 and HH 54 outflows, and found that
the ortho-to-para ratio is spatially variable, and presents
values significantly below the expected equilibrium value
of 3 for the observed gas temperature (which ranges between 400 and 1200~K).
Such non-equilibrium ortho-to-para ratios can be understood as being a remnant
of the gas pre-shock conditions, where the gas temperature was significantly lower
($\sim 50$~K).
A more systematic study of the H$_2$ emission from low mass outflows using Spitzer
observations comes from the series of papers by
\cite[Neufeld et al. (2009)]{neu09} and \cite[Nisini et al. (2010)]{nis10},
and the poster by Giannini et al. in this meeting, which concentrate on the
outflows from L1157, L1448, BHR71, NGC 2071, and VLA 1623. Despite their
differences in evolutionary stage and central source luminosity, all these
outflows present a number of similar trends. One of them is the need for
a multiplicity of temperatures to explain the H$_2$ emission, and which these
authors fit with a distribution of mass with temperature that follows a
power law $T^{-b}$ with $b$ in the range 2.3-3.3. In addition, the ortho-to-para
ratio in all objects with the exception of L1448 is below the equilibrium
value of 3, and suggest the presence of an activation energy in the para-to-ortho
conversion. Similar results were obtained by
\cite[Maret et al. (2009)]{mar09} in their
study of the outflows in the NGC 1333 region.
\subsection{Complex Molecules}
There has also been substantial progress in the study of "complex molecules"
in outflows over the last few years.
These organic species are usually defined as containing 6 or
more atoms, and are found in a number of different environments that range from the
Orion hot core to the cold core of TMC-1 (see
\cite[Herbst \& van Dishoeck 2009]{her09} for a recent review).
In outflows, the most widely observed complex molecule is methanol (CH$_3$OH), whose
abundance is often enhanced by several orders of magnitude, most likely due to its
release from dust grain mantles (see Fig.~\ref{overabundances}).
A recent result concerning methanol
comes from the study of its high energy transitions
by \cite[Codella et al. (2010)]{cod10},
who used the Herschel Space Telescope to observe
L1157-B1 as part of the CHESS project. These authors found that
the excitation of the high-J lines (up to J=13) is characterized by a
rotation temperature of 106~K, which is almost one order of magnitude higher
than the rotation temperature of
12~K derived from the low-J lines
(\cite[Bachiller et al. 1995]{bac95}). Given
the excitation conditions of CH$_3$OH, a rotational temperature of more than
100~K implies a gas kinetic temperature of at least 200~K, and such a warm component
represents a link between the low-excitation gas seen in most molecular tracers
and the hotter gas traced by H$_2$ emission. (See also
\cite[Nisini et al. 2007]{nis07} for a similar
result obtained from SiO observations.)
Molecules larger than CH$_3$OH have also been recently
identified in L1157-B1.
\cite[Arce et al. (2008)]{arc08} carried out deep
integrations with the IRAM 30m telescope towards this position
and detected for the first time in an outflow
HCOOCH$_3$ (methyl formate), CH$_3$CN (methyl cyanide),
HCOOH (formic acid), and C$_2$H$_5$OH (ethanol). The detection
of these molecules shows that outflow shocks can lead to a
very rich chemistry, that in low-mass sources was previously restricted
to the hot corino regions in the vicinity of some protostars
like IRAS 16293-2422 (\cite[Cazaux et al. 2003]{caz03}).
Additional detection of complex molecules in the L1157-B1 region comes from a
3mm spectral line survey being carried out
with the Nobeyama 45m telescope, and whose first results
have been presented by
\cite[Sugimura et al. (2011)]{sug11}
. This survey has already produced
new detections of complex molecules in this chemically active outflow:
CH$_3$CHO (acetaldehyde) and the mono-deuterated variety of methanol
CH$_2$DOH. From the first analysis of the
line survey, \cite[Sugimura et al. (2011)]{sug11}
find that the relative abundance of the complex molecules
with respect to methanol is significantly lower in L1157 than in the prototype hot
corino IRAS 16293-2422. These authors suggest that such a difference reveals a
difference in the chemical processing of the dust in the cloud regions sampled by
the outflow shock compared to the vicinity of the protostar sampled in the hot corino
phase. Detection of more complex molecules is possible given the on-going nature of the
survey.
Another recent detection towards the L1157-B1 position is that of HNCO
(isocyanic acid) by
\cite[Rodr{\'{\i}}guez-Fern{\'a}ndez et al. (2010)]{rod10}.
Although not a
complex molecule, HNCO is a well-known organic tracer of warm
environments, like hot cores, the Galactic center, and even
extragalactic sources, and the good correlation of its distribution
with that of methanol in IC 342 had led to the suggestion that this
species is shock-sensitive (\cite[Meier \& Turner 2005]{mei05}).
The detection of
HNCO in L1157-B1 by \cite[Rodr{\'{\i}}guez-Fern{\'a}ndez et al. (2010)]{rod10}
not only
reinforces this suggestion, but illustrates the important role of
relatively simple regions like L1157-B1, which is far enough from the
YSO to be considered a ``pure shock'' region (without stellar heating),
and therefore provides a clean template to recognize shock chemistry
in more complex environments.
\section{Do we understand outflow chemistry?}
A meeting like this one is an appropriate occasion to look critically at the
state of our understanding of outflow chemistry in a more global way than
usually done in research papers. A look of the literature reveals a
generalized consensus in the belief that shock chemistry can explain most (or all)
abundance anomalies observed in the outflow gas (apart from the
EHV component discussed in the following section). This consensus, however, seems
based more on the lack of observational counter-examples and alternative
models than in the existence of a quantitative proof that a single shock chemistry
model can explain simultaneously the
set of abundances observed in an object like L1157-B1.
This incomplete state of affairs mostly results from a
too-focused approach in outflow-chemistry work. From the
observational point of view, and this paper testifies it, most of this work
has been dedicated to a single object, the L1157 outflow, and more exactly, to
the B1 position of this outflow. Although L1157 is sometimes called ``prototypical''
in terms of outflow chemistry,
the lack of objects with comparable chemical richness and complexity
suggest that L1157 is probably extreme,
if not for its abundances, at least for the column densities responsible for its
very bright lines.
\begin{figure}[t]
\begin{center}
\includegraphics[width=2.0in, angle=-90]{compX-LV-h2coch3oh.4.ps}
\caption{Abundance ratio between H$_2$CO and CH$_3$OH in the outflow gas phase
(squares) and in the ices of dust grains (triangles). The red squares
represent values for a
sample of low-mass outflows from Santiago-Garc\'{\i}a et al.
(in preparation), and the blue squares are values towards high-mass
star forming regions from
\cite[Bisschop et al. 2007]{bis07}. The black triangles are ice
values from \cite[Gibb et al.(2004)]{gib04}.
The good match between gas-phase and solid-phase values reinforces the idea
that the abundance enhancement of H$_2$CO and CH$_3$OH in outflows results from
the release of these molecules from the mantles.
}
\label{h2co_ch3oh}
\end{center}
\end{figure}
Clearly, more general studies of outflow chemistry are needed, not only
to understand how typical L1157 is, but to reconstruct the full cycle of outflow chemical
activity represented by other very young outflows.
An important limitation for any statistical approach is the lack of
good samples. This is partly the result of the small number of outflows that
so far have been explored in molecular lines other than CO, but also because of the
need for full maps to identify chemical hot spots in the flows. Despite the limitations,
some work has been done in this direction.
Fig.~\ref{h2co_ch3oh} shows an example of how an statistical
approach can further illuminate the origin of the
observed abundances. This plot represents the CH$_3$OH/H$_2$CO
ratio for the sample of low-mass
outflows studied by Santiago-Garc\'{\i}a et al. (in preparation), and
shows that this gas-phase ratio remains approximately constant over a
relatively large range of individual abundances. The (gas-phase) ratio,
in addition, matches
closely the solid-phase ratio derived from ice mantles
in a different sample of objects, and this good match
supports strongly the interpretation that the
CH$_3$OH and H$_2$CO abundance enhancements in outflows result from the
release of these species from the mantles of grains due to sputtering in a shock.
While observations tend to concentrate on one object at a time, chemical models tend
to solve one
species at a time, and this makes it difficult to asses whether a single set of physical
conditions can explain the variety of observations, even for a single object like L1157.
A more important limitation
of the chemical models is their reliance on plane-parallel shock models.
In a plane parallel shock, there is a single velocity for the shock, and all
the material suffers the same type of acceleration. As a result, the predicted
emerging spectrum is characterized by a single spike at the velocity of
the shock and a rapidly declining tail of pre-shocked material towards low velocities
(e.g., Figs. 8 in both \cite[Gusdorf et al. 2008]{gus08} and
\cite[Flower \& Pineau Des For{\^e}ts 2010]{flo10}).
This shape of the spectrum is the opposite to what it is observed in a typical
outflow wing, which has most of the material moving at low velocities
and a minority of the gas moving at top speeds (see next
section for a discussion of the different nature of the
spiky extremely high velocity regime). As the optical depth of the
emission is proportional to the amount of material per unit velocity,
the compression of shocked gas into a small
range of velocities predicted by the plane parallel model
leads to a substantial overestimate of the optical depth.
By comparing in the figures mentioned above
the width of the spikes with the full velocity extent of the spectrum, we estimate
that the optical depth overestimate resulting from the artificial
limited range of velocities in the emission must range from factors of a few to
more than order of magnitude. Thus, comparing observations
with the predictions from chemical models must be done with care when the
model predicts a non-negligible of optical depth in the emission.
The systematic reliance on plane-parallel models in the analysis of
outflows is more a reflection of our
poor understanding of the internal kinematics in the outflow gas
than a conviction in the model providing the correct geometry.
In fact,
the observation of a wing feature in the emission spectrum of almost
every outflow indicates that a multiple-velocity description is needed,
and that most of the gas in the outflow
moves at intrinsically low velocities.
How to incorporate this description to chemical models is
not yet clear, although a first step may be to parametrize the mix
of velocities
with a power-law description like that used in the multi-temperature
analysis of the
H$_2$ data by \cite[Neufeld et al. (2006)]{neu06} and described above.
Using such a more realistic description of the outflow velocity
field will clearly bring a better agreement between the predicted
and observed spectra. More interestingly, a multiple-velocity
description of the shocked gas will allow exploring the changes in
the gas composition as a function of the velocity that a number of
observations are starting to uncover
(\cite[Codella et al. 2010]{cod10}, \cite[Tafalla et al. 2010]{taf10}).
\section{The EHV gas, a different chemical component?}
\begin{figure}[t]
\begin{center}
\includegraphics[width=3.5in]{i04166_ehv.ps}
\caption{Average blue lobe spectrum and velocity-integrated maps
of the CO(2-1) emission from the IRAS 04166+2706 outflow illustrating the
different spatial distribution and spectral signature of the wing and the
EHV components. Data from Santiago-Garc\'{\i}a et al. (2009).
}
\label{i04166_ehv}
\end{center}
\end{figure}
So far we have referred to the ``outflow emission'' as the one appearing in the spectra
forming a wing where the intensity decreases steadily with velocity.
Although such type of wing emission is the most characteristic signature of outflow gas, a
few very young Class 0 objects show in their spectra an additional outflow
component that looks like a close-to-gaussian secondary peak. This component, often referred to
as the extremely high velocity (EHV) gas given its high speed (tens of km~s$^{-1}$),
was first discovered in the outflow powered by L1448-mm
(\cite[Bachiller et al. 1990]{bac90}), and by now it has
been observed in a (reduced) number of outflows.
One of the outflows where an EHV component has been recently identified is
the one powered by IRAS 04166+2706 (IRAS 04166 hereafter, see
\cite[Tafalla et al. 2004]{taf04}).
In Fig.~\ref{i04166_ehv} we present the CO(2-1) emission from this outflow
as observed by
\cite[Santiago-Garc{\'{\i}}a et al. (2009)]{san09}
using the Plateau de Bure Interferometer. The
figure shows that the EHV emission appears distinct from the
wing emission both in spectral signature and spatial distribution. As can be seen, the
(lower-velocity) wing component forms in the maps
an X-shaped structure centered on the IRAS source and suggestive of
tracing the walls of a pair of evacuated cavities (a fact confirmed by the
association of the northern blue cavity with an IR cometary nebula).
The EHV emission, on the other hand, appears in the maps as a jet-like
feature that runs along the middle of the cavities and consists of a collection
of discrete peaks. As shown by
\cite[Santiago-Garc{\'{\i}}a et al. (2009)]{san09}, the
peaks of EHV emission are located so symmetrically from the IRAS source
that each has a counterpart on the other side
less than $2''$ away from its expected position.
Such a level of symmetry suggests that the EHV peaks
arise from
events that took place near the central source
and that have since propagated outwards
with the flow.
\begin{figure}[t]
\begin{center}
\resizebox{12cm}{!}{\includegraphics[width=4in,clip]{i04166-mom.ps}
\includegraphics{i04166_co_pv_2.ps}}
\caption{{\bf Left:\ } First momentum plot of the CO(2-1) EHV
emission towards the outflow
from IRAS 04166+2706 illustrating the presence of velocity oscillations.
{\bf Right:\ } Position-velocity diagrams of CO(2-1) along the outflow axis.
Note the sawtooth pattern in the EHV regime. From
\cite[Santiago-Garc\'{\i}a et al. (2009)]{san09}.
}
\label{i04166_vel}
\end{center}
\end{figure}
A clue to the origin of the EHV peaks
comes from a pattern of oscillations in their
velocity field.
As illustrated in Fig.~\ref{i04166_vel},
the oscillations form a sawtooth pattern in the
position-velocity diagram indicative of a combination of a close-to-constant
mean velocity together with strong velocity gradients inside each EHV peak.
The sense of these gradients is the same in all EHV peaks, and indicates
that the upstream gas moves faster than
the downstream gas. This sawtooth velocity pattern in the PV diagram
is in striking agreement
with the predictions for a set of internal shocks in a jet
modeled by \cite[Stone \& Norman (1993)]{sto93} to simulate the
pattern of knots seen in optical jets (see their Figure 16).
Thus, the combination of a symmetric and
fragmented structure together with a sawtooth velocity
pattern suggests that the EHV emission in IRAS 04166
arises not from ambient accelerated gas (like the rest of the
outflow), but from the internal shocks in a pulsating jet.
If the EHV gas represents jet material, it must be coming from the protostar
or its near-most vicinity, and we can therefore expect its chemical composition to
differ substantially from that of the wing gas, which consists
of ambient gas that has been shock-accelerated, and therefore has
had a very different thermal history. To test this possible difference
between these two components of the outflow,
\cite[Tafalla et al. (2010)]{taf10} have carried out the first molecular survey
of EHV gas by making deep integrations towards two outflow positions
known to have bright EHV and wing components, one in the L1448 outflow
and the other in the IRAS 04166 outflow. Previous to this survey, only
CO and SiO had been detected in the EHV gas of any outflow,
despite the large number of species detected in the wing component
of outflows like that of L1157 (see above). Thanks to the new deep
integrations, carried out with the IRAM 30m telescope, and some of them lasting
several hours, the number of species observed in the EHV gas has more than
doubled, with clear detections of SO, CH$_3$OH, and H$_2$CO. Possible
detections of HCO$^+$ and CS have also resulted from these deep integrations,
although their status remains unclear because of possible contamination
with emission from the wing outflow component, which extends at a low
level up to velocities comparable to those of the EHV gas.
(See also \cite[Kristensen et al. 2011]{kri11} for
the recent detection of H$_2$O in the EHV gas of L1448.)
Probably the most interesting result from the EHV molecular survey of
\cite[Tafalla et al. (2010)]{taf10} is the evidence for a change in the
carbon-to-oxygen ratio between the wing and the EHV components.
This is illustrated in Fig.~\ref{l1448_hcn_sio} with a superposition of the
SiO(2--0) and HCN(1-0) spectra towards the target position in the
L1448 outflow (the IRAS 04166 target position presents
a similar behavior but weaker lines).
As can be seen, when scaled appropriately, the
SiO(2-1) and HCN(1-0) spectra track each other in the (red) wing
regime over several tens of km~s$^{-1}$, suggesting that the HCN/SiO
abundance ratio remains constant over this range of velocities.
In contrast with this smooth behavior in the wing regime,
the intensity ratio between HCN and SiO drops by more
than one order of magnitude in the EHV regime. This intensity drop
is unlikely to result from a difference in excitation between the
two velocity
regimes, as multi-transition analysis of SiO, SO, and CH$_3$OH
show only small changes in the excitation temperature of these
molecules between the wing and EHV gas. The most likely cause of
the sudden drop in the HCN intensity towards the EHV regime is a
similar, order-of-magnitude drop in the abundance of this species
towards this outflow component. A similar drop in the abundance of
CS is also seen in the data, suggesting
that the HCN drop is not a peculiarity of this molecule but a common
feature of C-bearing species. Indeed, all molecules with clear detection in
the EHV component have so far been O-bearing (CO, SiO, SO, CH$_3$OH,
H$_2$CO, and H$_2$O).
\begin{figure}[t]
\begin{center}
\includegraphics[width=4.5in]{l1448_hcn_sio.ps}
\caption{Comparison between SiO(2-1) and HCN(1-0) spectra towards
the red lobe of the L1448-mm outflow. The SiO line has
been scaled to match the HCN intensity in the wing, in order
to better illustrate the sudden drop of HCN/SiO
abundance in the EHV component. From \cite[Tafalla et al. (2010)]{taf10}.}
\label{l1448_hcn_sio}
\end{center}
\end{figure}
The finding of chemical differences between the EHV and wing outflow components
in both L1448 and IRAS 04166
opens a new tool to explore the still mysterious mechanism of outflow
acceleration. If the EHV gas truly arises from a jet, it must have
originated in the innermost vicinity of the central object, and
therefore must carry information about the physical conditions in the
outflow acceleration region, which is of a few AU or even
a few stellar radii, depending on the model, and therefore
inaccessible to current instrumentation. To fully exploit the
potential of this information, however, a new generation of
chemical models is needed. The chemistry of protostellar
winds was originally studied by
\cite[Glassgold et al. (1991)]{gla91},
but little progress has been done on this issue in the twenty
years past since this pioneering work. It is reassuring to see
that there is a renewed interest in the topic, as illustrated
by the poster contribution in this meeting of Yvart et al.
Clearly more modeling and observational progress is needed
in this new and exciting line of study of molecules in outflows.
Let's hope we can all see its results in the next IAU Astrochemistry
Symposium a few years from now.
|
{
"timestamp": "2012-03-13T01:00:05",
"yymm": "1203",
"arxiv_id": "1203.2181",
"language": "en",
"url": "https://arxiv.org/abs/1203.2181"
}
|
\section{Introduction}\label{sect:introduction}
\noindent
Black holes (BH) in Einstein's theory of general relativity are described by only three parameters: mass, spin and charge.
Astrophysical black holes are even simpler, as charge can be neglected. So, besides their masses, $M_{\rm BH}$, they are completely characterized by their dimensionless spin parameter, $a \equiv c \, J_{BH}/G \, M_{\rm BH}^2$, where
$c$ is the speed of light,
$G$ the gravitation constant,
$J_{\rm BH}$ is the angular momentum of the black hole, and
$0\le a\lesssim 1$ \cite[e.g.][]{Kerr1963,Peters1964,BoyerLindquist1967,Carter1968,Felice1968,Bardeenetal1972}.
\\
Many previous theoretical efforts have focused on the mass growth of SMBHs that reside in galaxy centers and power quasars \cite[e.g.][and references therein]{Lynden-Bell1969,Bardeen1970,ShakuraSunyaev1973,ShakuraSunyaev1976,Blandford1977,Abramowicz1978,Rees1982,WilsonColbert1995,ModerskiSikora1996,GhoshAbramowicz1997,Moderski1998,Silk1998,HNR1998,Livio1999,DiMatteo2005,Dimatteo2008,Bower2006,Croton2006,Li2006,King2006,KingPringle2007,Sijacki2007,Booth2009,Sijacki2009,hopkins2005,Hopkins2011,hopkinsquataert2010,Fabian2010,Dubois2012}.
Thus, we will focus here on the processes taking place in their environments and on the consequent implications for the other property of SMBHs: spin, which is of no less importance.
\\
Indeed, the spin of a black hole affects the efficiency \cite[e.g.][and references therein]{NovikovThorne1973,BambiBarausse2011} of `classical' accretion processes.
In radiatively efficient thin accretion disks the mass-to-energy conversion efficiency, $\epsilon$, equals \cite[][]{NovikovThorne1973} $\epsilon\equiv 1- E/c^2$, where $E$ is the binding energy per unit mass of a particle at the last stable orbit.
The closer the last stable orbit is to the horizon, the higher the mass-to-energy conversion efficiency, which increases from $\epsilon\simeq 5.7\%$, for a non-rotating ($a=0$) hole, to $\epsilon\simeq 42\%$, for its maximally rotating counterpart
(see figure~\ref{fig:epsilona}).
\\
The mass-to-energy conversion directly affects the mass growth rate of black holes \cite[][]{Soltan1982,YuTremaine2002,Shankar2009} according to
\begin{equation}
\epsilon~\frac{\d M}{\d t} = f_{\rm Edd}~\frac{L_{\rm Edd}}{c^2}
\end{equation}
where $ f_{\rm Edd} $ is the Eddington efficiency, and $L_{\rm Edd}$ the Eddington luminosity:
high radiative efficiency, $\epsilon$, implies slow growth, as more mass is radiated away instead of being fed into the black hole.
For instance, for a black hole accreting at a fraction $f_{\rm Edd}$
of the Eddington rate, $L_{\rm Edd}$, assuming constant $\epsilon$, the
mass increases with time as:
\begin{equation} \label{eq:Mt}
M(t)=M(0)\,\exp\!\left\{{f_{\rm Edd} {(1-\epsilon)}\over{\epsilon}}\frac{t}{t_{\rm Edd}}\right\},
\end{equation}
where $M(0)$ is the initial mass, and the Eddington timescale, $t_{\rm Edd}$, is $c^2$ divided by the Eddington luminosity per unit mass,
\begin{equation}
t_{\rm Edd} = \frac{ \sigma_T c}{4 \pi G m_p} \simeq 0.45\,{\rm Gyr},
\end{equation}
with
$\sigma_T$ Thomson cross section, and $m_p$ proton mass.
Moreover, the higher the spin parameter, $a$, the higher $\epsilon$, implying longer timescales to grow the black-hole mass (see also later figure~\ref{fig:efficiency}).
Going from $\epsilon\simeq 5.7\%$ to $\epsilon\simeq 42\%$, the difference in mass amounts to 6 orders of magnitude at $t=t_{\rm Edd}$, for $f_{Edd} = 1$.
The typical spin therefore impacts the overall mass growth of SMBHs.
This is particularly important for the case of SMBHs powering the highest-redshift ($z\gsim 6$) quasars \cite[][]{KingPringle2007}, since these SMBHs must grow in mass by gas accretion in less than a billion year.
\\
\begin{figure}
\centering
\includegraphics[width=0.4\textwidth, height=0.18\textheight]{radiative_efficiency.ps}
\caption[Radiative efficiency]{\small
Radiative efficiency as a function of the spin parameter, $a$ \cite[][]{Bardeenetal1972, NovikovThorne1973}. Positive values of $a$ correspond to SMBHs co-rotating with their accretion disks, while negative values for $a$ indicate counter-rotating SMBHs.
}
\label{fig:epsilona}
\end{figure}
\noindent
Although the initial mass of the SMBH plays a crucial role in this game, all the SMBH formation models proposed to date require significant and fast mass accretion to result in $10^9 \msun$ SMBHs at redshift $\gsim 6$. As an example, under the assumption of a constant spin and radiative efficient accretion, a non-rotating ($a=0$, $\epsilon \simeq 5.7\%$) SMBH accreting at its Eddington limit ($f_{\rm Edd} = 1$) can grow its mass of 6 orders of magnitude in only $ < 0.4\,\rm Gyr$.
A maximally rotating ($a=1$, $\epsilon \simeq 42\%$) SMBH would accrete (at $f_{\rm Edd} = 1$) the same mass in more than a billion year.
Note however that assuming a constant $a=0$ is highly unphysical, since non rotating SMBHs are spun-up by every accretion event with finite angular momentum.
Fortunately, the constraint on the spin parameter are not too stringent, since
(see figure~\ref{fig:epsilona})
the radiative efficiency is a very steep function of $a$ only for $a \gtrsim 0.9$ \citep{NovikovThorne1973, KingPringleHofmann2008}, with typical values varying from a few per cents up to $\epsilon \approx 10\%$ for $a \lesssim 0.7$, and slightly increasing to $\epsilon \approx 15\%$ for $a \approx 0.9$.
As an example, from equation~(\ref{eq:Mt}) one can estimate that a SMBH with $a \approx 0.8$ ($\epsilon \approx 11.5\%$) would grow by 6 orders of magnitude in roughly two times the time needed by a non-rotating SMBH still well within the age of the Universe at $z\approx 6$ (of about $ 1~\,\rm Gyr$).
\\
The magnitude and orientation of SMBH spins also affect the frequency of SMBHs in galaxies, via `gravitational recoil'.
When two black holes coalesce, they may recoil due to anisotropic emission of gravitational waves. If this recoil were sufficiently violent, the newly merged hole would escape from the host galaxy. Recent breakthroughs in numerical relativity have allowed reliable
computations of recoil velocities. Non-spinning black holes are
expected to recoil with velocities below 200 $\rm{km\,s^{-1}}$. The recoil is much larger, up to thousands $\rm{km\,s^{-1}}$, for SMBHs with large spins in non-aligned configurations
\cite[e.g.][]{Campanelli2007, Lousto2012}.
\\
Finally, the value of the spin parameter, $a$, in a Kerr black hole \cite[][]{Kerr1963, BoyerLindquist1967} also determines how much energy is in principle extractable from the black hole itself \cite[][]{Blandford1977, BlandfordPayne1982}, and it may be important \cite[as argued by e.g.][]{MacDonaldThorne1982, ModerskiSikora1996, GhoshAbramowicz1997, McKinney2005} in relation to jet production \cite[but see][]{Livio1999}.
\\
In general, since SMBH growth is supposed to be led by mass accretion \cite[][]{Soltan1982,YuTremaine2002,Shankar2009,Shankar2011arXiv}, it is the way SMBHs accrete gas that will determine their spin.
Spin-up is a natural consequence of prolonged disk-mode accretion:
any hole that has, for instance, doubled its mass
by capturing material with constant angular momentum axis (coherent
accretion) would end up spinning rapidly, close to the maximum allowed value \cite[][]{Thorne1974}.
For example, if a BH accretes at values of order $\sim 10\%$ the Eddington limit (i.e. $f_{\rm Edd} = 0.1$), then it can be spun up to high values in less than 10~Myr (figure~\ref{fig:efficiency}).
The radiative efficiency evolves much more slowly, since as discussed previously, it is a shallow function of the spin until $a\sim 0.9$.
\\
On the other hand, a black hole which had gained its mass from capturing many low-mass objects in randomly-oriented orbits or from small (and short) uncorrelated accretion episodes (chaotic accretion), would keep small spin \citep[$a<0.2$, e.g.][]{King2006,KingPringle2007}.
\\
The main unknown here is therefore the typical angular momentum of the material feeding the SMBH: does the vector maintain a roughly constant preferential direction, or does the direction change chaotically?
\\
Since SMBHs are expected to have grown in mass mostly during active phases as quasars, the evolution of SMBH spins has to be addressed by simulating the typical environmental conditions expected around quasars, where the properties of the accretion flow are established.
When accretion is triggered by galaxy mergers, as expected for quasars \cite[][]{SandersMirabel1996, Downes1998, Davies2003, Davies2004, DownesEckart2007}, the material that feeds a SMBH is expected to assemble into a circum-nuclear disk (CND), as also found by numerical simulations \cite[e.g.][]{Mayer2007}.
These are therefore the most relevant circumstances to be explored.
CNDs are also associated to very high levels of star formation that are expected to influence SMBH spin evolution: local star formation injects energy and momentum into the gas near a SMBH, possibly breaking the coherency of the gas inflow.
A quasar could be fueled by a sequence of well-separated short-lived events with randomly distributed angular momentum vectors, leading to a chaotic gas flow \cite[][]{King2008}.
\\
In this work we will try to understand if a coherent circum-nuclear disk around a SMBH can be affected by the gaseous and stellar processes taking place during infall.
Moreover, we will try to address if these phenomena can break or alter the coherency of the matter flow, independently from larger-scale effects, like mergers or SMBH-SMBH interactions
\cite[as already studied by e.g.][]{Hopkins2011}.
\\
Therefore, we simulate the gaseous and stellar environment around a $4 \times 10^6\,\msun $ SMBH placed in a circum-nuclear disk of $10^8\, \msun$, by using the parallel tree/SPH code Gadget-3, an improved version of the publicly available code Gadget-2 \cite[][]{gadget,Springel2010}, including cooling
\cite[][]{Maio2007,Maio2010,Maio2011}, star formation
\cite[][]{SpringelHernquist2003}, stellar, and radiative feedback
\cite[][]{Petkova2011}.
We make sure that our spatial resolution (from 0.1 to 1~pc) is sufficient to resolve the `sphere of influence' within which the SMBH dominates gravity ($\sim$ 10-20~pc), allowing us to study the dynamics of the inflowing material \cite[e.g.][]{Dotti2010,Hopkins2011}.
This also allows us to model gas cooling, star formation, and stellar feedback accurately.
In our investigation we focus on the effect of local perturbations, neglecting large-scale instabilities that can alter the angular momentum of the nuclear gas \cite[e.g.][]{Hopkins2011}.
\\
\begin{figure}
\centering
\includegraphics[width=0.45\textwidth]{spin_acc_eff_time.ps}
\caption[Radiative efficiency]{\small
Time evolution of the properties of a black hole with initial mass $M(0)$, when accreting from a prograde accretion disk at a rate corresponding to $10\%$ the Eddington ratio.
Top panel: relative mass increase, $M(t)/M(0)$.
Middle panel: spin parameter evolution, $a$, derived according to \cite{Bardeen1970}'s formalism.
Bottom panel: corresponding evolution of the radiative efficiency, $\epsilon$.
}
\label{fig:efficiency}
\end{figure}
The paper is organized as follows: in section~\ref{sect:simulations}, we introduce and describe the simulations we performed; in section~\ref{sect:spin} and \ref{sect:spin results}, we outline the technique used to address spin evolution and show our main results; in section \ref{sect:conclusions}, we comment and conclude.
\section{Simulations}\label{sect:simulations}
\noindent
In the following we will describe the code used, the physical processes included in our treatment, and the set of 3D simulations performed.
\subsection{Implementation}\label{sect:implementation}
\noindent
We run 3D N-body/hydro simulations by using the parallel tree, SPH code Gadget3, an extended version of the publicly available code Gadget2 \cite[][]{Springel2005,Springel2010}. Besides gravity and hydrodynamics, the code includes gas cooling down to low temperatures \cite[][]{Maio2007,Maio2010} and a multi-phase model for star formation \cite[][]{Springel2003,Hernquist2003}, inspired by the works of \cite{KatzGunn1991, Cen1992, CenOstriker1992, Katz1992, Katz_et_al_1992, Katz_et_al_1996}.
Additionally, mechanical feedback \cite[][]{Aguirre_et_al_2001} from supernovae (SN) is taken into account \cite[][]{Springel2003}.
We also implement radiative transfer (RT) prescriptions \cite[][]{Petkova2009,Petkova2011,PetkovaMaio2011} to model the
effects onto the surrounding medium of photo-ionization and photo-heating from stellar radiation and gaseous radiative processes.
\\
The main agents of gas cooling are atomic resonant transitions that can rapidly bring the temperatures down to $\sim 10^4\,\rm K$. In metal-rich environments, atomic fine-structure transitions can lower temperatures further to $\sim$ 10 K \cite[][]{Maio2007,Maio2010b}. We will additionally show that detailed metallicity evolution can slightly influence the resulting star formation and feedback processes, but has no drastic effects on the spin evolution of the central SMBH and helps stabilizing the disk against star-formation-induced chaotic motions.
\\
When the medium cools and gets denser than a given density threshold of $10^4$ cm$^{-3}$, stochastic star formation is assumed, cold gas is gradually converted into stars, and entropy feedback is injected into the surrounding environment, leading to a self-regulated star forming regime. The only free parameter in this model is the maximum star formation timescale, $\tsfr$, which determines the amplitude of the typical star formation timescale, $t_\star$, at any gas density, $\rho$, above the mass-density threshold, $\rho_{\rm th}$,
\begin{equation}
\label{tsfr}
t_\star(\rho) = \tsfr \left( \frac{\rho}{\rho_{\rm th}} \right)^{-1/2}.
\end{equation}
In some astrophysical problems, it is possible to fix $\tsfr$ by matching with observations\footnote{
For example, in galactic disks, $\tsfr=2.1\,\rm Gyr$ allows us to fit the Kennicutt-Schmidt law \cite[][]{Kennicutt1998,Springel2003}.
}.
In strongly dynamically evolving systems, such as intense star-bursts in galactic nuclear regions, however, it is impossible to define a unique characteristic timescale for star formation.
Thus, we will assume two extreme timescales, a very long one ($\tsfr=300\,\rm Myr$), and a very short one ($\tsfr=1\,\rm Myr$), as limiting cases \cite[e.g.][]{Krumholz2011}.
This means that the maximum timescales at the threshold for star formation are $\rm 300\,\rm Myr$ and $1\,\rm Myr$, respectively, while at $\rho > \rho_{th}$, the time over which stars can form, $t_\star(\rho)$, becomes shorter and shorter.
Moreover, we will also explore the implications of intermediate, more realistic values ($\tsfr=6, 10, 20, 50, 100\,\rm Myr$) in terms of star formation efficiency, and the effects of full stellar evolution with consistent metal enrichment calculations.
\\
At each timestep, $\Delta t \ll 1\,\rm Myr$, collisionles star particles are spawned with a probability distribution
\begin{equation}\label{probability_stars}
p_{\star} = \frac{m}{m_\star} \left[ 1 - e^{-(1-\beta)x\Delta t / t_\star} \right],
\end{equation}
where $m$ is the current mass of the gas particle, $m_\star$ is the mass of each star particle, $\beta\simeq 0.1$ is the
fraction\footnote{Here a Salpeter IMF has been assumed over the range [0.1,100] M$_\odot$.
}
of short-lived stars which die instantly as supernov{\ae}, and $x$ is the cold-cloud mass fraction, ranging roughly from $0.85$ up to $\sim 1$ \cite[][]{Kennicutt1998, Springel2003}.
The star mass is given by $m_\star=m_0/N_g$, with $N_g$ an integer number indicating how many stars each gas particle can generate --
usually of the order of unity or a few \cite[][]{Springel2003, Hernquist2003}) -- and $m_0$ the initial gas particle mass, which is decreased (and reduced to the current gas mass $m$) accordingly to the spawned stars.
The resulting star formation rate is simply given by
$\dot M_\star = (1-\beta)xm / t_\star$.
The initial position and velocity of each star particle are
assumed to be the same as the parent gas particle.
We note that relations (\ref{tsfr}) and (\ref{probability_stars}) imply higher probability of forming stars at higher densities.
Once the star forming regime kicks in, supernova energy heats the
ambient medium and cold clouds are evaporated by thermal conduction: this process is parameterized via the evaporation factor, $A_0$, dependent on the environment.
Assuming that thermal instabilities in the gas become active at
$\sim 10^5\,\rm K$ and typical SN temperatures are $T_{SN}\sim 10^8\,\rm K$, the evaporation factor is $A_0=T_{SN}/10^5\rm K\sim 10^3$.
However, different SN and cooling features would give different values, thus, we will assume $A_0 = 10^2$, $10^4$, and $10^6$.
For further details see \cite{Springel2003}.
Similarly, outflows are phenomenologically taken into account according to the following probability distribution:
\begin{equation}
\label{probability_winds}
p_{w} = 1 - e^{-\eta(1-\beta)x\Delta t / t_\star},
\end{equation}
where the wind efficiency, $\eta\sim 2$ \cite[][]{Martin1999,Springel2003},
is the ratio between the mass-loss rate and the star formation rate of each gas particle.
Winds carry an additional kinetic energy that is a fraction, $\chi\simeq 0.25$ \cite[][]{Springel2003}, of the supernova energy.
Therefore, the wind velocity, $v_w$, is $ v_{w} = \sqrt{2\chi\varepsilon_{\rm SN}/\eta \phantom{i}}$, where $\varepsilon_{\rm SN}=4 \times 10^{48}$ erg $\msun^{-1}$ is the supernova energy injected in the medium for each solar mass in stars formed \cite[assuming a Salpeter IMF,][]{Springel2003}.
The corresponding wind velocity has typical magnitudes of a few hundreds km/s in a direction that can be randomly chosen either on the unit sphere (isotropic winds) or along the axis uniquely defined by ${\bf v}~\times~{\bf\nabla}~\phi$, with velocity vector ${\bf v}$ and gravitational potential $\phi$ (axial winds).
\\
The photo-ionization and photo-heating of the gas is followed by a radiative transfer scheme extended to a multi-frequency regime, with four frequency bins in the range 13.6 -- 60~eV, and corresponding to the ionization potential of neutral hydrogen (13.6~eV), neutral helium (24.6~eV), singly ionized helium (54.4~eV), and one arbitrarily higher value (60~eV). The radiative transfer equations are coupled to hydrodynamics and solved self-consistently taking account both gas cooling and radiative heating from stellar sources \cite[for further details see][]{Petkova2009,Petkova2011,PetkovaMaio2011}. Ionizing radiation from stars and gas is diffused throughout the simulation volume, interacting with both hydrogen and helium. The stellar emissivity is assumed to have a black-body spectrum with effective temperature $T_{\rm eff} = 5\times 10^4 \, \rm K$ (usual emission from short-lived massive stars). The gas emissivity due to cooling processes is also a black-body spectrum with effective temperature equal to its own temperature.
\\
A few studies \cite[][]{wada2009,schartmann2009,schartmann2011} attempted analyses of the gas properties around SMBHs. None of them, though, was suited to study how star formation influences the evolution of SMBH spins, which is the main goal of our investigation. Previous works, in fact, made several simplifications on the physical processes considered, for instance neglecting either radiative transfer effects or the fundamental link between stellar evolution and the gaseous environment where stars form. Our simulations, instead, follow star formation and stellar evolution explicitly, so that SN explosions are driven from the hydrodynamical evolution of the gas. The ensuing feedback effects are therefore fully self-consistent with the dynamics and thermodynamics of the circum-nuclear disk. These features allow us to investigate accurately how stellar and SN feedback affect the angular momentum of the gas near SMBHs, and, as a consequence, how star-bursts impact SMBH spins.
\subsection{Initial conditions}\label{sect:ICs}
\noindent
We initialize every simulation with a SMBH of $4\times 10^6 \msun$ at rest, at the center of a purely gaseous CND. The SMBH is treated as an active evolving collisionles particle, and reacts to the gravitational torques exerted by the stellar and gaseous background.
\\
The SMBH and the disk are embedded in a spherical stellar bulge \cite[][]{Magorrian1998} with total mass of $2\times 10^9 \msun$ and scale length $b=10^3\,\rm pc$, always modelled using $10^5$ particles of $2\times 10^4\msun$. The bulge follows a Hernquist mass density profile \cite[][]{Hernquist1990,Tremaine_et_al_1994}, which reproduces the $R^{1/4}$-law well \cite[][]{deVaucouleurs1948} and mimics the potential of the host galaxy.
\\
The disk is assumed to have a mass of $10^8 \msun$ (25 times larger than the central SMBH) and is modelled using $2 \times 10^4$, $2\times 10^5$, $2\times 10^6$, or $2 \times 10^7$ gas particles in our low, standard, high, and very-high resolution runs, such that the corresponding gas particle masses are $5000\,\msun$, $500\,\msun$, $50\,\msun$, and $5\,\msun$.
The CND is initially composed only of gas particles, since we aim at modelling the evolution of SMBHs in a recently formed CND. We note, however, that the CND becomes populated of young stars as the simulation evolves. The absence of an old population of stars in our initial conditions does not affect our results: only young stars, through their powerful winds or their deaths as supernovae, can significantly alter the properties of the gas.
The disk initially follows an exponential surface density profile, and has a scale length of $100\,\rm~pc$. It is pressure supported in the vertical direction, with an aspect ratio $H/R \equiv 0.1$. The disk internal energy is initially fixed according to a polytropic equation of state with adiabatic index $\gamma=5/3$ and cosmic composition for H and He.
\\
We evolve adiabatically our initial conditions for $\approx 100\,\rm Myr$, to ensure that the system reaches a stable configuration without fragmenting nor developing any strong asymmetry and instability, before turning on cooling, star formation, winds, supernova and radiative feedback.
\subsection{Runs}\label{sect:runs}
\noindent
\begin{table*}
\centering
{\footnotesize
\caption[Simulation set-up]{\small
Initial parameters for the different runs: the black-hole mass is always fixed to $4\times 10^6\rm\msun$, the disk gas mass to $10^8\msun$, and the stellar bulge mass to $2\times 10^9\msun$.
}
\begin{tabular}{lccccccc}
\hline
\hline
Runs & Particle number & Particle mass [$\rm \msun$]& Evap. & SF time& Low-T & RT &Softening\\
& gas $\quad$ (bulge) & gas $\quad$ (bulge) & factor & [Myr] &cooling & & length [pc]\\
\hline
EVP1e2-SFRT1 & $2\times 10^5\quad$ ($10^5$) & $500\quad$ ($2\times 10^4$) & $10^2$ & 1 & off &off & 1.0 \\
EVP1e2-SFRT1-LowT$^{*}$ & $2\times 10^5\quad$ ($10^5$) & $500\quad$ ($2\times 10^4$) & $10^2$ & 1 & on &off & 1.0 \\
EVP1e2-SFRT300 & $2\times 10^5\quad$ ($10^5$) & $500\quad$ ($2\times 10^4$) & $10^2$ & 300 & off &off & 1.0 \\
EVP1e2-SFRT300-LowT$^{*}$ & $2\times 10^5\quad$ ($10^5$) & $500\quad$ ($2\times 10^4$) & $10^2$ & 300 & on &off & 1.0 \\
\hline
EVP1e4-SFRT1 & $2\times 10^5\quad$ ($10^5$) & $500\quad$ ($2\times 10^4$) & $10^4$ & 1 & off &off & 1.0 \\
EVP1e4-SFRT1-LowT$^{*}$ & $2\times 10^5\quad$ ($10^5$) & $500\quad$ ($2\times 10^4$) & $10^4$ & 1 & on &off & 1.0 \\
EVP1e4-SFRT300 & $2\times 10^5\quad$ ($10^5$) & $500\quad$ ($2\times 10^4$) & $10^4$ & 300 & off &off & 1.0 \\
EVP1e4-SFRT300-LowT$^{*}$ & $2\times 10^5\quad$ ($10^5$) & $500\quad$ ($2\times 10^4$) & $10^4$ & 300 & on &off & 1.0\\
\hline
EVP1e6-SFRT1 & $2\times 10^5\quad$ ($10^5$) & $500\quad$ ($2\times 10^4$) & $10^6$ & 1 & off &off & 1.0 \\
EVP1e6-SFRT1-LowT$^{*}$ & $2\times 10^5\quad$ ($10^5$) & $500\quad$ ($2\times 10^4$) & $10^6$ & 1 & on &off & 1.0 \\
EVP1e6-SFRT300 & $2\times 10^5\quad$ ($10^5$) & $500\quad$ ($2\times 10^4$) & $10^6$ & 300 & off &off & 1.0 \\
EVP1e6-SFRT300-LowT$^{*}$ & $2\times 10^5\quad$ ($10^5$) & $500\quad$ ($2\times 10^4$) & $10^6$ & 300 & on &off & 1.0 \\
\hline
EVP1e2-SFRT1-IW$^{\diamondsuit}$ & $2\times 10^5\quad$ ($10^5$) & $500\quad$ ($2\times 10^4$) & $10^2$ & 1 & off &off & 1.0 \\
EVP1e2-SFRT300-IW$^{\diamondsuit}$ & $2\times 10^5\quad$ ($10^5$) & $500\quad$ ($2\times 10^4$) & $10^2$ & 300 & off &off & 1.0 \\
\hline
EVP1e2-SFRT1-LR$^\dag$ & $2\times 10^4\quad$ ($10^5$) & $5000\quad$ ($2\times 10^4$) & $10^2$ & 1 & off &off & 1.0 \\
EVP1e2-SFRT1-HR$^\star$ & $2\times 10^6\quad$ ($10^5$) & $50 \qquad$ ($2\times 10^4$) & $10^2$ & 1 & off &off & 1.0 \\
EVP1e2-SFRT1-HR-SS$^\star$ & $2\times 10^6\quad$ ($10^5$) & $50 \qquad$ ($2\times 10^4$) & $10^2$ & 1 & off &off & 0.1 \\
EVP1e2-SFRT1-VHR-SS$^\star$ & $2\times 10^7\quad$ ($10^5$) & $5 \,\,\,\qquad$ ($2\times 10^4$) & $10^2$ & 1 & off &off & 0.1 \\
\hline
EVP1e2-SFRT1-RT & $2\times 10^5\quad$ ($10^5$) & $500\quad$ ($2\times 10^4$) & $10^2$ & 1 & off &on & 1.0 \\
EVP1e2-SFRT300-RT & $2\times 10^5\quad$ ($10^5$) & $500\quad$ ($2\times 10^4$) & $10^2$ & 300 & off &on & 1.0 \\
\hline
EVP1e2-SFRT6 & $2\times 10^5\quad$ ($10^5$) & $500\quad$ ($2\times 10^4$) & $10^2$ & 6 & off &off & 1.0 \\
EVP1e2-SFRT10 & $2\times 10^5\quad$ ($10^5$) & $500\quad$ ($2\times 10^4$) & $10^2$ & 10 & off &off & 1.0 \\
EVP1e2-SFRT20 & $2\times 10^5\quad$ ($10^5$) & $500\quad$ ($2\times 10^4$) & $10^2$ & 20 & off &off & 1.0 \\
EVP1e2-SFRT50 & $2\times 10^5\quad$ ($10^5$) & $500\quad$ ($2\times 10^4$) & $10^2$ & 50 & off &off & 1.0 \\
EVP1e2-SFRT100 & $2\times 10^5\quad$ ($10^5$) & $500\quad$ ($2\times 10^4$) & $10^2$ & 100 & off &off & 1.0 \\
\hline
EVP1e2-SFRT1-SE$^{\ddag}$ & $2\times 10^5\quad$ ($10^5$) & $500\quad$ ($2\times 10^4$) & $10^2$ & 1 & on &off & 1.0 \\
EVP1e2-SFRT300-SE$^{\ddag}$ & $2\times 10^5\quad$ ($10^5$) & $500\quad$ ($2\times 10^4$) & $10^2$ & 300 & on &off & 1.0 \\
\hline
\label{tab:runs}
\end{tabular}}
\begin{flushleft}
\vspace{-0.5cm}
{\small
$\phantom{}^{*}$ The LowT label indicates runs with low-temperature metal cooling for $Z=Z_\odot$.\\
$\phantom{}^{\diamondsuit}$ The IW label indicates runs with isotropic winds. The others are performed with axial winds.\\
$^\dag$ The LR label stands for low resolution.\\
$^\star$ The HR label stands for high resolution; the HR-SS label stands for high resolution and small softening; the VHR-SS label stands for very high resolution and small softening (see text in Sect.~\ref{sect:runs} for more details).\\
$\phantom{}^{\ddag}$ The SE label indicates runs with consistent stellar evolution calculations, and metal-dependent cooling in the temperature range $\sim 10-10^9\,\rm K$, for $Z$ evolving according to the proper stellar lifetimes and yields.
}
\end{flushleft}
\end{table*}
\begin{figure*}
\begin{center}
\includegraphics[width=0.52\textwidth]{pgplot_EVP1e2_SFRT1_VHR_SmallSoftening.ps}
\hspace{-0.9cm}
\includegraphics[width=0.52\textwidth]{pgplot_EVP1e6_SFRT300_LowT.ps}
\end{center}
\caption{\small
Face--on view of the inner part of the CND at $t=10\,\rm Myr$.
The color (logarithmic) scale refers to the gas surface density in units of $\rm 10^6\,M_\odot/pc^2$. Axes are in pc.
{\it Left panel}: EVP1e2-SFRT1-VHR-SS.
{\it Right panel}: EVP1e6-SFRT300-LowT.
}
\label{fig:maps}
\end{figure*}
\noindent
The runs we performed involve the study of several parameters and physical processes and we refer to Table~\ref{tab:runs} for the general outline of the features in the different simulations\footnote{
In the runs, we used 32 SPH neighbours for the density calculations, however, we checked the bahaviours up to 64 neighbours, as well, and found no significant difference in the results (mass profiles, star formation rates, angular momentum, relative angles, accretion rates, etc.).
}.
\\
The first, second, and third blocks refer to the simulations with axial wind feedback and evaporation factor A$_0=10^2$, A$_0=10^4$, and A$_0=10^6$, respectively.
For each case we run four simulations:
two with $\tsfr=1\,\rm Myr$ (one with metallicity $Z=0$, and H and He cooling down to $\sim 10^4\,\rm K$, and one with additional metal low-T cooling down to $\sim 10\,\rm K$ for an assumed constant $Z=Z_\odot$) and two with $\tsfr=300\,\rm Myr$ (one with $Z=0$, H and He cooling down to $\sim 10^4\,\rm K$, and one with additional metal low-T cooling down to $\sim 10\,\rm K$ with $Z=Z_\odot$).
\\
The corresponding complete, more detailed runs including (i) self-consistent metal production from stellar evolution according to the proper yields and lifetimes, (ii) exact abundance calculations, and (iii) metal-dependent cooling in the range $\sim 10-10^9\,\rm K$, will be also discussed in the following sections for the two limiting cases $\tsfr=1\,\rm Myr$ and $\tsfr=300\,\rm Myr$ (both with $A_0 = 10^2$).
\\
The resulting consequences from the cooling down to low temperatures on disk stability and fragmentation are readily seen in the density maps of figure~\ref{fig:maps}.
\\
In order to check the effects of outflows, we also run additional simulations with isotropic winds (IW), for the case with $A_0=10^2$, cooling down to $\sim 10^4\,\rm K$, and either $\tsfr=1\,\rm Myr$ or $\tsfr=300\,\rm Myr$.
\\
We carry out convergence studies over the dependence on mass and spatial resolution by performing four additional runs, with $A_0=10^2$, $\tsfr=1\,\rm Myr$ and no low-T cooling.
The ``standard'' runs have a gravitational softening length of $1$ pc and $2 \times 10^5$ gas particles, with corresponding gas mass of 500 $\msun$.
By comparison, the low-resolution (LR) run has the same gravitational softening length, but ten times less gas particles. The two high-resolution (HR) simulations have 2 million gas particles (50 $\msun$ per particle) and are run with either a gravitational softening of $1$ pc or with a smaller softening of $0.1$ pc. Similarly, the very-high-resolution run (VHR) is run with a gravitational softening of 0.1 pc, but sampling our circum-nuclear disk with 20 million gas particles, corresponding to a gas mass resolution of 5 $\msun$.
\\
To check more realistic values of star formation timescales, we re-run the previous standard cases also with $\tsfr=6, 10, 20, 50, 100~\rm Myr$, besides the two $\tsfr=1~\rm Myr$ and $\tsfr=300~\rm Myr$ reference values.
\\
We then perform two simulations considering radiative transfer effects, as described in section~\ref{sect:implementation}, with standard resolution, assuming $A_0=10^2$ and either $\tsfr=1\,\rm Myr$ or $\tsfr=300\,\rm Myr$.
\\
Finally, we study the role of stellar evolution (SE) with consequent metal spreading by starting the simulations with a pristine gaseous environment ($Z=0$), by polluting it with metal yields \cite[][]{WW1995, vandenHoekGroenewegen1997, Thielemann2003}, from stellar evolution, according to the stellar lifetimes by \cite{PadovaniMatteucci1993}, and by consistently following the full chemical composition of the medium for H, He, O, C, Mg, S, Si, Fe.
For the technical details on this topic and studies on parameter dependencies, we refer the reader to \cite{Tornatore2004, Tornatore2007, Maio2007, Maio2010, Maio2011}.
\subsection{Star formation} \label{sect:SF}
\noindent
In all our runs, independently of the resolution, the softening, or other parameters, the disk experiences a burst of star formation, with a peak of a few solar masses per year, lasting for $\sim 1 - 10\,\rm Myr$ (see figure~\ref{fig:sfr}).
These values are compatible, although on the low side, with the typical values of star formation rates in circum-nuclear disks associated with star-bursts and late stages of galaxy interactions, from a few solar masses per year up to several hundreds \cite[][]{Kennicutt1998, Downes1998, Davies2003, Davies2004, Evans2006, DownesEckart2007}.
After $t\approx 100\,\rm Myr$, the star formation rate decreases by 1--2 orders of magnitude, depending on the particular parameters used, but it always peaks at $3-5\,\rm Myr$ for $\tsfr=1\,\rm Myr$, and the global behaviour is quite independent of $A_0$ (left panel of figure~\ref{fig:sfr}) and of resolution (right panel of figure~\ref{fig:sfr}).
The cases with $\tsfr=300\,\rm Myr$ result in a slightly later peak of star formation at $\sim 5-30\,\rm Myr$
with slightly larger values.
This is due to the longer timescale considered and the resulting higher clumpiness reached in the star forming regions.
However, in all the simulations the central region of the disk is efficiently converted into stars, that become a relevant component after $\sim 10\,\rm Myr$.
\\
In figure~\ref{fig:sfr2}, we also show the star formation rates for different $\tsfr$ assumptions: $\tsfr=$~1, 6, 10, 20, 50, 100, 300~Myr (left panel)\footnote{
We note that a value of $\tsfr=1\rm Myr$ (with gas threshold of $10^4\,\rm cm^{-3}$) would correspond to star formation efficiencies of roughly $\sim 50\%$, while $\tsfr=300\rm Myr$ would correspond to a very low value of $\sim 0.15\%$.
As mentioned in the Introduction, these are quite extreme numbers, compared to expected values of a few per cents, or so.
A more realistic assumption of star formation efficiency of $\sim 1\%$ would require $\tsfr\sim 50\,\rm Myr$.
Even though there are some cases (that we also take into account) for which timescales can be significantly shorter, as in the Milky Way, for example, for which it is expected $\sim 5-6\,\rm Myr$ \cite[][]{Burkert2006}, or the T-Tauri complex for which observed timescales are of the order of $\sim 1-3\,\rm Myr$ \cite[][]{PallaStahler2000, Hartmann2001}.
},
and additional calculations including stellar evolution and metal production (right panel).
We see (left panel of figure~\ref{fig:sfr2}) that when increasing $\tsfr$ -- i.e. when lowering star formation efficiency -- the SFRs during the initial phases (i.e., by $\lesssim 10\,\rm Myr$) get shallower and the bulk of formation of stars gets gradually shifted to later times.
After the first burst, efficiently star forming models with short $\tsfr$ (say around $\lesssim 50 \,\rm Myr$) show a clearly declining trend, due to the lack of remaining cold gas, that has been rapidly consumed during the initial phases. Feedback processes take place immediately, heat the surrounding medium, and inhibit further star formation.
In less efficiently star forming models, with larger $\tsfr$ values, we assist to a slowly increasing trend of the SFRs for the initial tens Myrs. In these scenarios, gas can cool and condense for longer time without forming significant amounts of stars.
At $\gtrsim 20\,\rm Myr$, cooling processes finally lead to a number of strong SFR peaks, that take place when the medium is, in average, colder and more clumpy than in the previous cases.
Thus, only after these episodes feedback effects become efficient in heating the gas and halting further stellar production.
Interestingly, a limiting situation is represented by $\tsfr = 50\,\rm Myr$, for which the SFR is roughly flat over the entire lapse of time simulated, and reflects an almost continuous star formation process during which feedback effects and gas cooling are always well regulated.\\
The inclusion of detailed stellar evolution with self-consistent metal production (right panel of figure~\ref{fig:sfr2}) determines some difference in the time evolution of the SFR due to the increasing metallicities that slightly boost star formation. However, no major effects in the final results are found.
\\
In figure~\ref{fig:sfr3}, we finally plot the evolution of the SFRs in models with different wind shapes (left) and different gas density thresholds (right).
On the left panel, we show results from simulations with $\tsfr=1\,\rm Myr$ and $A_0 = 10^2$, considering:
axial winds with coupling with the ISM after a travel length of 1~pc (black solid line, simply labeled as "Axial winds");
axial winds with coupling with the ISM after a travel length of 10~pc (red dotted line, labeled as "Extended winds");
axial winds with coupling with the ISM after a travel length of 10~pc and density contrast below 0.1 (blue dashed line, labeled as "Extended low-density winds");
isotropic winds (cyan dot-dashed line).
While the first and second case are basically very similar, some slight difference is found for the third case, where axial winds start interacting with the medium only in under-dense regions, and hence, lead to less star formation after the first few Myrs.
In the case of isotropic winds, particles are not collimated in one direction, but can interact with the surrounding gas at $4\pi$, compressing the gas in all directions, hence, a larger SFR is obtained.
Quantitatively, the boost to the SFR due to isotropy is only a factor of a few, though.
On the right panel, SFRs for the same initial setup and axial winds, but with gas density thresholds of $10^2-10^6\,\rm cm^{-3}$ are plotted.
Since star formation is strongly dependent on density, larger thresholds determine higher SFRs and lower thresholds determine lower SFRs.
In the cases considered, differences can reach up to $\sim 2$ orders of magnitude, ranging from $\lesssim 10^{-2}\,\rm M_\odot/yr$ (for a threshold of $10^2\,\rm cm^{-3}$) up to several $ \rm M_\odot/yr $ (for a threshold of $10^6\,\rm cm^{-3}$).
The obvious consequence is that the little amount of stars formed in low-density-threshold scenarios will trigger feedback processes acting essentially on low-density gas, and that could be also effective in randomizing the dynamics of the medium (see more discussion in the parameter study later).
\begin{figure*}
\begin{center}
\includegraphics[width=0.45\textwidth]{sfr_SFRT1_LocalWinds.ps}
\includegraphics[width=0.45\textwidth]{sfr_resolution.ps}
\end{center}
\caption{\small
Star formation rates for simulations evaporation parameters, $A_0$ (left), and resolution (right), as explained in the legends. The cases refer to $\tsfr=1\,\rm Myr$ with cooling down to $\sim 10^4\,\rm K$. The initial drop is due to the denser central regions of the CND that undergo star formation and get evacuated during the first few Myrs.
}
\label{fig:sfr}
\end{figure*}
\begin{figure*}
\begin{center}
\includegraphics[width=0.45\textwidth]{sfr_tscale.ps}
\includegraphics[width=0.45\textwidth]{sfr_zmetals_se.ps}
\end{center}
\caption{\small
{\it Left panel}:
Star formation rates for simulations with different star formation time scales, $\tsfr = $~1, 6, 10, 20, 50, 100, 300~Myr.
{\it Right panel}: Star formation rates for simulations with inclusion of stellar evolution (SE) treatment and metal pollution process compared to the corresponding star formation rates in the runs with no SE treatment, for $\tsfr=1\,\rm Myr$ and $\tsfr=300\,\rm Myr$, with $A_0 = 10^2$.
}
\label{fig:sfr2}
\end{figure*}
\begin{figure*}
\begin{center}
\includegraphics[width=0.45\textwidth]{sfr_winds.ps}
\includegraphics[width=0.45\textwidth]{sfr_nth.ps}
\end{center}
\caption{\small
{\it Left panel}:
Star formation rates for simulations with different wind parameters for the $\tsfr=1\,\rm Myr$ and $A_0 = 10^2$ case:
axial winds with coupling with the ISM after a travel length of 1~pc (black solid line, simply labeled as "Axial winds");
axial winds with coupling with the ISM after a travel length of 10~pc (red dotted line, labeled as "Extended winds");
axial winds with coupling with the ISM after a travel length of 10~pc and density contrast below 0.1 (blue dashed line, labeled as "Extended low-density winds");
isotropic winds (cyan dot-dashed line).
{\it Right panel}:
Star formation rates for simulations with gas density thresholds of
$10^2 - 10^6 \,\rm cm^{-3}$, as indicated by the labels, for the $\tsfr=1\,\rm Myr$ and $A_0 = 10^2$ case with axial winds.
}
\label{fig:sfr3}
\end{figure*}
\section{Spin}\label{sect:spin}
\noindent
We take advantage of the extremely wide variety of our simulations to track the dynamics of the stream of particles inflowing towards the center and show the results for different cases.
\subsection{Method}
\noindent
We estimate the accretion rate and angular momentum of the inflowing material through the smallest surface surrounding the SMBH that can be numerically resolved\footnote{
We note that the sphere of influence (extending out to $\sim 20\,\rm pc$) is largely resolved in all our runs.
}
(e.g., 3~pc and 0.3~pc for simulations with a 1~pc and 0.1~pc resolution, respectively).
Due to the large range of scales involved, the formation of accretion disks around the SMBH could not be resolved down to $\lsim 0.01$ pc, where the gas can be modelled as a standard accretion disk.
Therefore, the relevant properties of the disk and their evolution in time have been simply inferred from the properties of the particles inflowing through the previously mentioned surface.
In particular, we assume that: 1) the accretion rate onto the hole is equal to the accretion rate at the smallest resolved scales; 2) the direction of the orbital angular momentum in the external part of the unresolved accretion disk is parallel to the direction of the total angular momentum (relative to the SMBH) of the infalling gas at the resolution limit.
We notice that all these assumptions are necessary working hypotheses.
Nevertheless, we stress that these information can be gathered only by performing high-resolution numerical simulations that accurately resolve the ``sphere of influence'' of the black hole.
We limit the gas inflow to the Eddington rate for the SMBHs, and use the SMBH accretion histories obtained from our SPH simulations to evolve the SMBH spin vector.
We notice that while the SMBH spin and mass evolve in the semianalytical model we use to postprocess our simulations (described in section~\ref{sect:spin method}), the SMBH mass is constant in our SPH simulations, and the gas flowing toward the SMBH is not removed from the simulations.
As shown in section~\ref{sect:spin results}, we limit our simulations to $t\lsim 20~\rm Myr$, to limit the mis-match between the SMBH mass in the simulations and in the semianalytic model to a factor of a few.
We stress that such a small mass growth would not change substantially the gas dynamics, since the CND is significantly more massive than the SMBH in our initial set-up.
\\
If the orbital angular momentum of the disk around the SMBH is misaligned with respect to the SMBH spin, the coupled action of viscosity and relativistic Lense-Thirring precession warps the disk in its innermost region \cite[][]{bardeenpetterson1975} over a timescale shorter than the viscous/accretion timescale \cite[][]{Scheuer1996, NatarajanPringle1998, Bogdanovic2007}.
We develop an analytical framework to model the response of the accretion disk (in the innermost non-resolved region) to the angular momentum of the mass inflow as described below.
At each timestep we can calculate the evolution of the magnitude and direction of the SMBH spin \cite[][]{Perego2009, Dotti2010}.
\subsection{Bardeen-Petterson effect}\label{sect:spin method}
\noindent
We use the SMBH accretion histories obtained from our SPH simulations to follow the evolution of each SMBH spin vector, ${\mathbf J}_{\rm BH}=(aGM_{\rm BH}^2/c){{\mathbf j}}_{\rm BH}$, where $0 \leq a \lesssim 1$ is the dimensionless spin parameter and ${\mathbf j}_{\rm BH}$ is the spin unit vector. We will use the term maximally rotating for a black hole with $a=0.998$, following \cite{Thorne1974} who showed that accretion driven spin-up is limited to such value.
Magnetic fields connecting material in the disk and the plunging region may further reduce the equilibrium spin. Magneto-hydrodynamic simulations for a series of thick accretion disks suggest an asymptotic equilibrium spin at $a\approx 0.9$ \cite[][]{Gammie2004}.
\\
The scheme we adopt to study spin evolution is based on the model developed in \cite{Perego2009}.
Here we summarize the algorithm.
We assume that inflowing gas forms a geometrically thin/optically thick $\alpha$-disk \cite[][]{ShakuraSunyaev1973, ShakuraSunyaev1976} on sub-parsec scales (not resolved in the simulation), and that the outer disk orientation is defined by the unit vector ${\mathbf l}_{\rm edge}$.
The evolution of the $\alpha$-disk is related to the radial viscosity, $\nu_1$, and the vertical viscosity, $\nu_2$:
$\nu_1$ is the radial shear viscosity, while $\nu_2$ is the vertical shear viscosity associated to the diffusion of vertical warps through the disk.
The two viscosities can be described in terms of two different dimensionless viscosity parameters, $\alpha_1$ and $\alpha_2$, through the relations
\begin{equation}
\nu_{1,2} = \alpha_{1,2}Hc_{\rm s},
\end{equation}
where $H$ is the disk vertical scale height and $c_{\rm s}$ is the sound speed of the gas in the accretion disk.
Both of them depend on $R$, $M_{\rm BH}$, $\dot{M}$ and $\alpha_1$.
Here we assume the \cite{ShakuraSunyaev1973} solutions for the gas pressure dominated zone.
\\
We further assume \cite[][]{LodatoPringle2007}
\begin{equation}
\alpha_2=f_2 / (2 \alpha_1),
\end{equation}
with\footnote{
These values are slightly different from others also found in literature \cite[as e.g. in][]{BateLodatoPringle2010}.
}
\begin{equation}
\alpha_1 = 0.1
\end{equation}
and $f_2 = 0.6$, i.e.
\begin{equation}
\alpha_2 = 3,
\end{equation}
and power law profiles for the two viscosities
\cite[][]{ShakuraSunyaev1973,ShakuraSunyaev1976},
\begin{equation}
\label{norm}
\nu_{1,2} \propto R^{3/4}.
\end{equation}
\noindent
If the orbital angular momentum of the disk around the SMBH is misaligned with respect to the SMBH spin, the coupled action of viscosity and relativistic Lense-Thirring precession warps the disk in its innermost region forcing the fluid to rotate in the equatorial plane of the spinning SMBH \cite[][]{bardeenpetterson1975,Kumar1985}.
The timescale of propagation of the warp is short compared with the viscous/accretion timescale \cite[][]{Scheuer1996, NatarajanPringle1998}, so that the deformed disk reaches an equilibrium profile that can be computed by solving the equation of conservation of angular momentum
\begin{eqnarray}
\label{eqn:angular momentum}
\frac{1}{R}\frac{\partial}{\partial R}(R {\mathbf L} v_{\rm R}) =
\frac{1}{R}\frac{\partial}{\partial R}\left(\nu_1 \Sigma R^3 \frac{d\Omega}{dR}~ {\mathbf l} \right)+ \nonumber \\
+\frac{1}{R}\frac{\partial}{\partial R}\left(\frac{1}{2}\nu_2 R L \frac{\partial {\mathbf l}}{\partial R} \right)
+ \frac{2G}{c^2} \frac{{\mathbf J}_{\rm BH} \times {\mathbf L}} {R^3}
\end{eqnarray}
where $v_R$ is the radial drift velocity, $\Sigma$ is the surface density, $\Omega$ is the Keplerian velocity of the gas in the disk, and $\mathbf{L}$ is the local angular momentum surface density of the disk, defined by its modulus $L$ and the versor ${\mathbf l}$ that defines its direction.\\
The boundary conditions to eq.~(\ref{eqn:angular momentum}) are the direction of ${\mathbf L}$ at the outer edge ${\mathbf l}_{\rm edge}$, the mass accretion rate (that fixes the magnitude of $\Sigma$), and the values of mass and spin of the massive BH.
\\
The values of ${\mathbf l}_{\rm edge}$ and of the mass accretion rate are directly extracted from the simulations:
in particular, they are computed considering those SPH particles nearing the MBH gravitational sphere of influence that are accreted.
\\
The spin value, $\rm J_{BH}$, is treated on post-processing in a Monte Carlo fashion, by randomly assuming several different realizations at the initial time, and evolving each of them.
More exactly, the direction of the SMBH spin changes in response to its gravito-magnetic interaction with the disk on a timescale longer than the time scale of warp propagation \cite[][]{Perego2009}.
This interaction tends to reduce the degree of misalignment between the disk and the SMBH spin, decreasing with time the angle between ${\mathbf J}_{\rm BH}$ and ${\mathbf l}_{\rm edge}$ \cite[][]{Bogdanovic2007}.
So, the SMBH spin evolution is followed by solving for the equation
\begin{equation}
\label{eqn:jbh precession-disk}
\frac{d{\mathbf J}_{\rm BH}}{dt} =\dot{M}\Lambda(R_{\rm ISO}) {\mathbf l}(R_{\rm ISO}) + \frac{4\pi G}{c^2}\int_{\rm disk}\frac{{\mathbf L} \times {\mathbf J}_{\rm BH}}{R^2}dR.
\end{equation}
The first term in eq.~(\ref{eqn:jbh precession-disk}) accounts for the angular momentum deposited onto the SMBH by the accreted particles at the innermost stable orbit (ISO), where $\Lambda(R_{\rm ISO})$
denotes the specific angular momentum at $R_{\rm ISO}$ and $ {\mathbf l}(R_{\rm ISO})$ the unit vector parallel to ${\mathbf J}_{\rm BH}$, describing the warped shape of the disk.
The second term instead accounts for the gravo-magnetic interaction of the SMBH spin with the warped disk. It modifies only the SMBH spin direction (and not its modulus), conserving the total angular momentum of the composite (MBH+disk) system \cite[][]{King2005}.
The integrand peaks at the warp radius ($R_{\rm warp}$), where the disk deformation is the largest\footnote{
The exact definition of $R_{\rm warp}$ is where the vertical viscous timescale $t_{\nu_2}\simeq R^2/\nu_2$ in the disk is comparable to the Lense-Thirring precession timescale. Because $R_{\rm warp}$ and the radius at which the disk is maximally deformed are comparable \cite[][]{Perego2009}, we simplify the notation in the paper using only $R_{\rm warp}$.
}.
The equation incorporates two timescales: the accretion time related to the first right-hand term describing the $e-$folding increase of the spin modulus, and the shorter timescale of SMBH spin alignment
\begin{equation}
\tau_{\rm al} \sim 10^5 a^{5/7} \left(\frac{M_{\rm BH}}{4 \times 10^6 \msun}\right)^{-2/35} f_{\rm Edd}^{-32/35} {\rm yr},
\label{eq:alignts}
\end{equation}
that will ensure a high degree of SMBH-disk gravito-magnetic coupling during SMBH inspiral.
In Eq~(\ref{eq:alignts}), $f_{\rm Edd}$ is the SMBH luminosity in units of $L_{\rm Edd}$.
\\
We apply iteratively eq.~(\ref{eqn:angular momentum}) and (\ref{eqn:jbh precession-disk}), by using inputs from the SPH simulations that give the values of the mass accretion rate, the SMBH
mass and the direction of ${\mathbf l}_{\rm edge}$. The algorithm returns, as output, the spin vector, that is, its magnitude and direction.
At each timestep our code therefore provides the angle between the spin vector of the SMBH and the global angular momentum vector of the circum-nuclear disk.
\subsection{Resulting spin alignment and magnitude} \label{sect:spin results}
\begin{figure*}
\begin{center}
\includegraphics[width=0.49\textwidth]{Dist_EVP1e2-SFRT1_shortskip.ps}
\includegraphics[width=0.49\textwidth]{new_comparison.ps}
\includegraphics[width=0.49\textwidth]{Dist_EVP1e2-SFRT1-IW_complete.ps}
\includegraphics[width=0.49\textwidth]{Dist_EVP1e2-SFRT1-RT_complete.ps}\\
\end{center}
\caption{\small
For each panel, the time evolution of the spin parameter (top), of the spin orientation angle (middle), and of the luminosity in units of the Eddington luminosity, $f_{\rm Edd}$ (bottom), computed at a distance of 3~pc from the central BH, are shown. In the upper and middle panels the solid lines refer to the mean values of the spin magnitude and orientation $\theta$, averaged over 100 realizations. When $\theta$ approaches zero, the accreted material is aligned with the SMBH spin (``coherent'' accretion). When $\theta$ is large, material is flowing with angular momentum misaligned with respect to the spin, causing the spin to decrease. The darker and lighter (red and orange in the online version) shaded areas refer to the regions encompassing 1- and 2-sigma around the means. The different panels refer to the runs with $500\,\rm M_\odot$ mass resolution: EVP1e2-SFRT1 (top left), EVP1e2-SFRT1-LowT (top right), EVP1e2-SFRT1-IW (bottom left), EVP1e2-SFRT1-RT (bottom right). In the top right panel the evolution of $f_{\rm Edd}$ for the EVP1e2-SFRT1 run is shown with a dashed line to facilitate the comparison.
}
\label{fig:spins}
\end{figure*}
\noindent
We will now discuss the results expected for the spin behaviour in the various runs presented before. We will start with general trends and then show also one of the highest resolution runs.
\\
Figure~\ref{fig:spins} shows the time evolution of the spin magnitudes (top panel).
We show here the results of 100 Monte Carlo realizations starting from a flat distribution in both spin magnitudes and initial orientations for four representative cases (the alignment process and the evolution of the spin parameter are almost identical in all other runs).
The time evolution of the relative angle, $\theta$, between the spin of the SMBH and the orbital angular momentum of the circum-nuclear disk is also shown (central panel).
The inner (outer) shaded areas in the upper and central insets correspond to 1-sigma (2-sigma) deviations from the mean values, over 100 random initial spin realizations\footnote{
We checked that changing the number of initial realizations the final results are not affected significantly.
}.
The corresponding luminosity in units of the Eddington luminosity, $f_{\rm Edd}$, is plotted, as well (bottom panel).
All the results discussed in the following, where not specified otherwise, are obtained computing the properties of the accreting material at a distance from the central SMBH of 3~pc, i.e. three softening lengths of the low resolution runs, and for star formation timescale of $\tsfr = 1\,\rm Myr$.
Top-left panels refer to the reference run with axial wind and radiative cooling down to $\sim 10^4\,\rm K$, top-right panels to the corresponding run with low-temperature cooling (below $10^4\,\rm K$), bottom-left panels to the run with isotropic winds and cooling down to $\sim 10^4\,\rm K$, and bottom-right panels to the reference run including RT.
\\
In general, SMBH spins rapidly loose memory of their initial orientation, and accretion torques suffice to align the spins with the angular momentum of the disk orbit on a short timescale ($\simlt 1-2$ Myr).
In all the cases, ${\mathbf J}_{\rm BH}$ tends to align with the angular momentum of the accreting gas, as extracted by the simulations and highlighted by the black lines in the upper and central insets, within a few Myr, regardless the initial orientation and magnitude of the SMBH spin, even if initially completely anti-aligned.
After the first, fast alignment of the spin with the orbital angular momentum of the circum-nuclear disk, the accretion flow is mostly coherent, i.e. the inflow traces the angular momentum of the large-scale circum-nuclear disk that feeds the SMBH, and the gas luminosity near the center is very
close to the Eddington luminosity.
Only in the RT runs (bottom-right panel) we find inclination angles that are slightly larger: this is due to the more realistic treatment of entropy injection by radiative sources, that is powered by radiation heating (missing in the other cases).
This does not change our overall conclusions, because typical
values are still $< ~50^{\circ}$, and only rarely isolated spikes are found.
After the ``fast'' spin alignment, accretion proceeds essentially coherently, i.e. the gas accretes on planes that have an inclination angle with respect to ${\mathbf J}_{\rm BH}$ that is significantly smaller than $90^{\circ}$.
As a consequence, $a$ monotonically increases with time, except for $a$ very close to unity, where it stays roughly constant, or decreases slightly.
Since for $a\approx 0$ the alignment timescale ($\propto a^{5/7}$) is extremely short \cite[][]{Perego2009}, slowly rotating SMBHs start immediately to accrete in a prograde fashion, and the lower boundary of the shaded areas are not expected to decrease with time on timescales of $\sim \rm Myr$.
Once a BH starts accreting from a prograde disk, the spin magnitude also increases over short timescales, close to the maximum allowed one.
\\
The resulting accretion rate, $f_{\rm Edd}$, shown in the lower insets, is more seriously affected by the different physical mechanisms
By comparing the EVP1e2-SFRT1 and the EVP1e2-SFRT1-LowT runs (upper right panel)
it emerges that low-temperature cooling contributes to a slight enhancement of $f_{\rm Edd}$.
More quantitatively, the average increment during the first 15~Myr is roughly $\sim 4$ per cent,
with temporary peaks of a factor of $\sim 2$.
In fact, at $\sim 5\,\rm Myr$, $f_{\rm Edd}$ results to be $\sim 0.3$ for the standard run, and $\sim 0.6$ for the low-temperature case.
This is related to the more efficient gas condensation process that can take place in such conditions.
In this case, gas can loose more pressure support, reach lower temperatures, and form smaller clumps with very high densities on short timescales.
Both formation of dense structures and gas compression from SN shocks reshuffle the local angular momentum of the disk strengthening inflows onto the central black hole \cite[][]{wada2009, kawakatu2009, hobbs2011}.
The overall effect is limited on short timescales, since in the long run (e.g. $\gtrsim 15\,\rm Myr$) the trends tend to converge mainly because of gas consumption, as shown by the decrement in the star formation activity.
\\
When comparing isotropic winds to axial winds (see upper-left and lower-left panels), we notice an initial drop in accretion rate for the isotropic case.
The very first episodes of star formation happen close to the black hole, where the gas density reaches initially its peak. Because of the disk-like geometry of our gas distribution, isotropic winds can interact with close gas particle more efficiently, initially evacuating the central region of the disk, and hence decreasing the accretion rate.
However, as soon as star formation picks up in the outer regions of the disk, the accretion rate onto the black hole raises again, up to values even slightly higher than in the runs with axial winds.
This is because isotropic winds mix the angular momentum of gas more efficiently.
In the RT case (bottom-right panel), heating from radiative feedback keeps the gas hotter, strongly suppressing the formation of dense substructures and star formation.
As a consequence, $f_{\rm Edd}$ is decreased by a factor of a few, with peak values of $f_{\rm Edd} \lesssim 1/2$.
\\
In our highest resolution run (figure~\ref{fig:hires}), we can resolve almost one star at a time. Even in this run we find results similar to the lower resolution runs, with the only difference that we can follow gas inflow down to sub-parsec scales (0.3~pc), deeper in the potential well of the central SMBH, where the accretion rate is, in average, a factor of $\sim 2-3$ larger.
\\
We emphasize that these results apply only to quasars in merger remnants, not to those fed by disk instabilities and cold streams. In those cases, massive clumps may chaoticize accretion \cite[e.g.][]{Bournaud_et_al_2011, Dubois2012}, since they would usually infall from large scales carrying their own angular momentum.
\begin{figure}
\centering
\includegraphics[width=0.49\textwidth]{Dist_EVP1e2-SFRT1-VHR_SS_complete.ps}
\caption{\small
Same as figure~\ref{fig:spins} for the highest resolution run (simulation EVP1e2-SFRT1-VHR-SS).
}
\label{fig:hires}
\end{figure}
\subsection{Parameter study}
\begin{figure*}
\centering
\includegraphics[width=0.32\textwidth]{t50ev1e2.ps}
\includegraphics[width=0.32\textwidth]{t100ev1e2.ps}
\includegraphics[width=0.32\textwidth]{t300ev1e2.ps}
\caption{\small
As in figure~\ref{fig:spins} for runs performed assuming different star formation timescales: $\tsfr = $~50 (EVP1e2-SFRT50, left panel), 100 (EVP1e2-SFRT100, middle panel), and 300~Myr (EVP1e2-SFRT300, right panel).
}
\label{fig:tsfr}
\end{figure*}
\begin{figure*}
\centering
\includegraphics[width=0.49\textwidth]{plot_EVP1e2_SFRT1_LocalWinds_Z2.ps}
\includegraphics[width=0.49\textwidth]{plot_EVP1e2_SFRT300_LocalWinds_Z2.ps}
\caption{\small
As in figure~\ref{fig:spins} for runs assuming star formation timescales of 1 (left) and 300 (right) Myr, both with full stellar evolution and metal enrichment calculations (SE). For a direct comparison with the corresponding $Z=0$ cases, see top-left plot in figure~\ref{fig:spins}, and right plot in figure~\ref{fig:tsfr}, respectively.
}
\label{fig:Z}
\end{figure*}
\noindent
To draw more robust and definitive conclusions, we finish by showing the main findings arising from a parameter study for different cases.
In particular, to check the dependencies on star formation we explore the changes induced by adopting different star formation timescales, gas density thresholds, evaporation factors, self-consistent stellar evolution, and metal spreading.
\\
Similarly to figure~\ref{fig:spins} and \ref{fig:hires}, where we explored the case $\tsfr = 1\,\rm Myr$, in figure~\ref{fig:tsfr} we display the time evolution of the spin parameter, of the spin orientation angle, and of the luminosity in Eddington units, $f_{\rm Edd}$, also for the runs performed with $\tsfr = $~50, 100, 300~Myr (in the cases with $\tsfr = 1-20\,\rm Myr$ no relevant differences are found, hence we will not focus on each of them). The general trends show that, despite the good agreements of the spin parameters (in all the cases ranging between $\sim 0.6$ and 0.9 at $t \sim 10\,\rm Myr$) and of the inclination angles (always quite small), models with larger star formation timescales (i.e. smaller efficiency), as the one with $\tsfr = 300\,\rm Myr$, can nevertheless accrete slightly more mass and sustain $f_{\rm Edd}$ at slightly larger values ($f_{\rm Edd}\sim 0.8-1$ at $\gtrsim 15\,\rm Myr$) than models with more efficient star formation and shorter $\tsfr$.
We also notice that, in models with larger $\tsfr$, the formation of long-lived gas clumps that could stir the medium seems not to be very efficient in increasing the chaotic behaviour of the CND, with $f_{\rm Edd}$ reaching roughly unity after $\gtrsim 10-15~\rm Myr$.
\\
Our conclusions could be affected by the ability of resolving properly the cold, dense, star forming regions. As a check, we analyze the cases carried out by assuming different gas density thresholds for star formation in the range $10^2 - 10^8\,\rm cm^{-3}$ (see supplementary online material).
It emerges that models with low-density thresholds have a more chaotic behaviour and show larger spreads in the results.
Indeed, for the scenarios with a threshold of $\sim 10^2-10^3\,\rm cm^{-3}$, the spin parameter is roughly $a\sim 0.4 - 0.95$ and $\theta$ can smoothly increase above $\sim 30^o$, or so.
The more chaotic behavior is visible mainly at initial times, when first bursts of star formation take place, and the spreads in the data are wider than in the higher-density-threshold cases.
Thus, the accretion rate is easily inhibited (with $f_{\rm Edd}$ dropping below $\sim 0.1$), because even material at relatively low-densities experiences star formation and gets evacuated by the consequent thermal heating and feedback processes.
Larger, more realistic, density thresholds, instead, allow the gas of the CND to cool and actually condense, so that star formation takes place only in very clumpy regions on the disk.
Dense, cold fragments are affected in a minor way by feedback processes, can survive them, and continue accreting on the central BH.
As a consequence, the CND can keep a more stable and ordered shape, and the cases with a threshold $\gtrsim 10^5\,\rm cm^{-3}$ converge rapidly (in $\lesssim 1\,\rm Myr$) to similar trends.
Moreover, they are compatible with the fiducial value of $\sim 10^4\,\rm cm^{-3}$ used throughout this paper.
\\
The effects of changes in the evaporation factor (see supplementary online material)
lead to similar spin evolution, irrespectively from the detailed values used, and to inclination angles always below $\sim 50^o$.
In general, despite the non-very-regular trends, some boosts in $\theta$ are present when there is strong star formation activity (figure~\ref{fig:sfr}) for all the cases, with corresponding decrements in the Eddington fraction, since hot gas gets more pressure supported and provokes smaller inflows towards the central regions. Instead, when star formation rates drop down, the resulting $\theta$ values decrease and the $f_{\rm Edd}$ values get closer to unity.
This variability is a consequence of the interplay between star formation heating and gas cooling that regulate the behaviour of the environment.
Stellar feedback heats the medium, while simultaneous cooling processes cool down the gas, stabilize the disk, and trigger the next star formation episodes.
\\
Complete full stellar evolution and metal spreading, displayed in figure~\ref{fig:Z}, does not imply dramatic changes in the spin results.
The sharper variations (on timescales of $\sim 1 \,\rm Myr$) of the inclination angle and of $f_{\rm Edd}$ are due to the larger radiative rates in presence of metals, and, hence, faster cooling and star formation processes, that respectively boost and inhibit gas inflows.
This is mostly evident for $ \tsfr=1 \,\rm Myr$, where metal pollution takes place earlier than in the case of longer $\tsfr=300 \,\rm Myr$.
In the former case, the effects of short star formation timescales (see also right panel in figure~\ref{fig:sfr2}) are enhanced by the stronger radiative losses of the newly enriched material, that quickly cool the ambient medium and temporarily enhance gas infall.
In the latter case, the star formation rate at the same times is a factor of a few to $\sim 10$ more limited, and cooling from metals is able to efficiently stabilize the disk against the chaotic motions from the rarer stellar feedback.
These conclusions on the stabilizing role of metals are easily cross-checked by a direct comparison with the corresponding $Z=0$ cases, in the top-left plot of figure~\ref{fig:spins}, and in the right plot of figure~\ref{fig:tsfr}, for $\tsfr=1~\rm Myr$ and $\tsfr=300~\rm Myr$, respectively.
\section{Discussion and conclusions}\label{sect:conclusions}
\noindent
We have presented a number of thre-dimensional, N-body, hydrodynamical, chemistry simulations of the behaviour of circum-nuclear discs around super-massive black holes.
We considered the relevant local physical processes that could alter the coherency of gaseous flows onto the central regions, i.e.: cooling, star formation, feedback effects, and radiative transfer from stellar sources.
Our goal was to explore the implications of these several phenomena on the stability of circum-nuclear discs around SMBHs and the consequent effects on the black-hole spin evolution.
Our investigation considers mainly those ``local'' processes that could increase the turbulence of the gas (such as the gas self-gravity, cooling, star-formation, and feedback from young stars), while global processes, such as the precession of the nuclear gas structures in a less symmetric, tri-axial potential, are not taken in account, as we started with an axisymmetric distribution of gas and stars. Global instabilities or asymmetries that could decrease the degree of coherency in the BH fuelling \cite[as discussed e.g. in][]{Hopkins2011} are beyond the aims of this work.
\\
Our study does not include AGN feedback. The energy liberated by an AGN may be by far larger than the typical energies delivered through stellar feedback in the central regions of the galaxy.
The effectiveness of the AGN in altering the CND gas dynamics, however, depends on the jet geometry and on the fraction of energy actually deposited in the nuclear regions.
In general terms, we can make some inferences on the expected impact of AGN feedback from the results of (two-dimensional) dedicated simulations such as those by \cite{Novak2011}.
They find that mechanical feedback (winds) dominates the effect on the local gas dynamics, and that this effect is stronger when the wind is confined in a cone. Nevertheless, even during a high-accretion event a beamed wind does not strongly affect the gas located perpendicularly to the outflow.
These simulations differ from our three-dimensional setup, though. On the one hand, they do not include any dense and rotationally supported gas structure, that would reduce even further the effect of the AGN feedback. On the other hand, they keep the orientation of the conical outflow constant. If the orientation would follow the rearrangement of the SMBH spin, it would in some cases transiently impinge on the CND, modifying more strongly its dynamics and the properties of the accreting flow.
Definitely, more self-consistent investigations are needed to assess the interplay between AGN feedback and CND dynamics, but this first study, focusing on the dynamical perturbations self-generated by the CND, will facilitate distinguishing the effects of AGN feedback in future works.
\\
Summarizing, our analysis suggests that in most of the cases local star formation episodes and feedback effects are able to partially break the general coherency of the gas flow, but do not inject enough energy and momentum into the gas near SMBHs to create a strongly chaotic environment that completely randomizes the orbits of the inflowing particles. As a conclusion, we find that initially maximally rotating SMBHs are slightly spun down, and initially slow-rotating black holes are spun up, leading to upper-intermediate equilibrium values, $a\simeq 0.6-0.9$ (corresponding to radiative efficiencies $\epsilon \simeq 9\%-15\%$).
\\
Considering the whole parameter space, it results that different feedback mechanisms can affect the CND gas dynamics and coherency in different ways.
Mechanical (kinetic and thermal) feedback processes are responsible for causing temporary, moderate, changes in the inclination angle and Eddington fraction.
Metal feedback, instead, introduces sharper variations, but has a generally stabilizing role for the CND gas flows, since it facilitates the formation of cold clumps.
Radiative feedback is responsible for inducing a more chaotic regime with increments of the alignment angle of at least a factor of a few (up to $\sim 50^o$), and for limiting the accretion capabilities of the central BH of a factor $\gtrsim 2$.
\\
We note that our results are based on the angular momentum of the material that is effectively accreted on the SMBH, and not on the global alignment of the nuclear disc with the large scale properties of the host galaxy \cite[][]{Hopkins2011}.
While the nuclear disc can be misaligned with respect to the galaxy disc, the relevant quantity to address spin evolution is the direction of the angular momentum of the material within the nuclear disc that eventually feeds the SMBH, not the alignment or misalignment between nuclear disc and the host.
These results are in line with the suggestion that jets are preferentially perpendicular to dust lanes \cite[][]{Verdoes2005}, which are believed to be the remnants of galaxy mergers -- a low-mass analogue of nuclear discs. Further hints for at least partial alignment are obtained from analyses of active galactic nuclei (AGN) and their hosts in the Sloan survey \cite[][]{Lagos2011}. According to these studies, the material feeding the black hole is expected to accrete in a somewhat coherent fashion.
\\
Observationally, spin magnitudes of SMBHs powering AGNs can be estimated directly by measuring the width of the K$_\alpha$ iron line, at 6.4~keV, through X-ray spectroscopy: recent determinations with
{\it Chandra}\footnote{http://chandra.si.edu/},
{\it XMM-Newton}\footnote{http://xmm.esac.esa.int/},
and {\it Suzaku}\footnote{http://www.isas.jaxa.jp/e/enterp/missions/suzaku/}
have revealed variable, relativistic iron emission lines from the inner disc in many Seyfert galaxies \cite[][]{Brenneman2011}\footnote{ This iron line originates near the black hole, and it is distorted by relativistic effects \cite[][]{Fabian1989,Laor1991}.
In particular, the width of the line is much broader in the rotating case, because the last stable orbit is closer in, and the gravitational redshift is stronger.
}.
It is suggestive that all SMBHs with spin measured directly through X-ray spectroscopy have spin parameters between $\sim$~0.6 and 1 \cite[][]{Brenneman2011}, in broad agreement with our results. Indeed, if after the quasar phase the mass of a SMBH has not changed much through gas accretion or SMBH mergers, the expectation is that today's spins are similar to the final spin gained during the last growth spurt.
\section*{Acknowledgments}
\noindent
We acknowledge V.~Springel for providing access to Gadget-3 code. U.M. acknowledges financial contribution from the Project HPC-Europa2, funded under the European Seventh Framework Programme, Infrastructure, Grant Agreement n. 228398, and kind hospitality at the Italian computing center (CINECA); he also acknowedges funding from the European Commission Seventh Framework Programme (FP7/2007-2013) Grant Agreement n. 267251.
M.P. research is part of the project GLENCO, funded under the European Seventh Framework Programme, Ideas, Grant Agreement n. 259349.
M.V. acknowledges support from NASA, through award ATP NNX10AC84G; from NSF, through award AST 1107675; and the European Seventh Programme FP7/PEOPLE/2012-CIG (PCIG-GA-2011-303609). For the bibliographic research we made use of the tools offered by the NASA Astrophysics Data System (ADS) and by the JSTOR archive.
|
{
"timestamp": "2013-02-25T02:02:29",
"yymm": "1203",
"arxiv_id": "1203.1877",
"language": "en",
"url": "https://arxiv.org/abs/1203.1877"
}
|
\section{Introduction}
The inequalities discovered by C. Hermite and J. Hadamard for convex
functions are very important in the literature (see, e.g.,\cite[p.137
{pecaric}, \cite{dragomir1}). These inequalities state that if
f:I\rightarrow \mathbb{R}$ is a convex function on the interval $I$ of real
numbers and $a,b\in I$ with $a<b$, then
\begin{equation}
f\left( \frac{a+b}{2}\right) \leq \frac{1}{b-a}\int_{a}^{b}f(x)dx\leq \frac
f\left( a\right) +f\left( b\right) }{2}. \label{E1}
\end{equation
The inequality (\ref{E1}) has evoked the interest of many mathematicians.
Especially in the last three decades numerous generalizations, variants and
extensions of this inequality have been obtained, to mention a few, see
\cite{alomari}-\cite{ngoc}) and the references cited therein.
\begin{definition}
The function $f:[a,b]\subset \mathbb{R}\rightarrow \mathbb{R}$, is said to
be convex if the following inequality hold
\begin{equation*}
f(\lambda x+(1-\lambda )y)\leq \lambda f(x)+(1-\lambda )f(y)
\end{equation*
for all $x,y\in \lbrack a,b]$ and $\lambda \in \left[ 0,1\right] .$ We say
that $f$ is concave if $(-f)$ is convex.
\end{definition}
In \cite{pec2}, Pearce et. al. generalized this inequality to $r$-convex
positive function $f$ which defined on an interval $[a,b]$, for all $x,y\in
\lbrack a,b]$ and $t\in \lbrack 0,1]
\begin{equation*}
f\left( tx+(1-t)y\right) \leq \left\{
\begin{array}{ll}
\left( t\left[ f\left( x\right) \right] ^{r}+\left( 1-t\right) \left[
f\left( y\right) \right] ^{r}\right) ^{\frac{1}{r}}, & \text{if }r\neq 0 \\
\left[ f\left( x\right) \right] ^{t}\left[ f\left( y\right) \right] ^{1-t},
& \text{if }r=0
\end{array
\right.
\end{equation*
We have that $0$-convex functions are simply $\log $-convex functions and $1
-convex functions are ordinary convex functions.
Recently, the generalizations of the Hermite-Hadamard's inequality to the
integral power mean of a positive convex function on an interval $[a,b]$,
and to that of a positive $r$-convex function on an interval $[a,b]$ are
obtained by Pearce and Pecaric, and others (see \cite{pec2}-\cite{ngoc}).
A function $f:I\rightarrow \lbrack 0,\infty )$ is said to be log-convex or
multiplicatively convex if $\log t$ is convex, or, equivalently, if for all
x,y\in I$ and $t\in \left[ 0,1\right] $ one has the inequality:
\begin{equation}
f\left( tx+\left( 1-t\right) y\right) \leq \left[ f\left( x\right) \right]
^{t}\left[ f\left( y\right) \right] ^{1-t}. \label{E2}
\end{equation}
We note that if $f$ and $g$ are convex and $g$ is increasing, then $g\circ f$
is convex; moreover, since $f=\exp \left( \log f\right) $, it follows that a
log-convex function is convex, but the converse may not necessarily be true
\cite{pec2}. This follows directly from (\ref{E2}) because, by the
arithmetic-geometric mean inequality, we have
\begin{equation*}
\left[ f\left( x\right) \right] ^{t}\left[ f\left( y\right) \right]
^{1-t}\leq tf\left( x\right) +\left( 1-t\right) f\left( y\right)
\end{equation*
for all $x,y\in I$ and $t\in \left[ 0,1\right] $.
For some results related to this classical results, (see\cite{dragomir1}
\cite{dragomir2},\cite{set1},\cite{set2}$)$ and the references therein.
Dragomir and Mond \cite{dragomir1} proved the following Hermite-Hadamard
type inequalities for the $\log $-convex functions:
\begin{eqnarray}
f\left( \frac{a+b}{2}\right) &\leq &\exp \left[ \frac{1}{b-a
\int\limits_{a}^{b}\ln \left[ f\left( x\right) \right] dx\right] \label{z2}
\\
&\leq &\frac{1}{b-a}\int\limits_{a}^{b}G\left( f\left( x\right) ,f\left(
a+b-x\right) \right) dx \notag \\
&\leq &\frac{1}{b-a}\int\limits_{a}^{b}f\left( x\right) dx \notag \\
&\leq &L\left( f\left( a\right) ,f\left( b\right) \right) \notag \\
&\leq &\frac{f\left( a\right) +f\left( b\right) }{2}, \notag
\end{eqnarray
where $G\left( p,q\right) =\sqrt{pq}$ is the geometric mean and $L\left(
p,q\right) =\frac{p-q}{\ln p-\ln q}$ $\left( p\neq q\right) $ is the
logarithmic mean of the positive real numbers $p,q$ $\left( \text{for }p=q
\text{ we put }L\left( p,q\right) =p\right) $.
Recall also that a function $f:I\rightarrow R$ is called strongly convex
with modulus $c>0,$ i
\begin{equation*}
f\left( tx+\left( 1-t\right) y\right) \leq tf\left( x\right) +\left(
1-t\right) f\left( y\right) -ct(1-t)(x-y)^{2}
\end{equation*
for all $x,y\in I$ and $t\in (0,1).$ Strongly convex functions have been
introduced by Polyak in \cite{polyak}\ and they play an important role in
optimization theory and mathematical economics. Various properties and
applicatins of them can be found in the literature see (\cite{polyak}-\cit
{angu}) and the references cited therein.
In this paper we introduce the notation of strongly logarithmic convex with
respect to $c>0$ and versions of Hermite-Hadamard-type inequalities for
strongly logarithmic convex with respect to $c>0$ are presented. This result
generalizes the Hermite-Hadamard-type inequalities obtained in \cit
{dragomir1}\ for log-convex functions with $c=0.$
\section{Main Results}
We will say that a positive fuction $f:I\rightarrow \left( 0,\infty \right) $
is strongly log-convex with respect to $c>0$ if
\begin{equation*}
f\left( \lambda x+\left( 1-\lambda \right) y\right) \leq \left[ f\left(
x\right) \right] ^{\lambda }\left[ f\left( y\right) \right] ^{1-\lambda
}-c\lambda \left( 1-\lambda \right) \left( x-y\right) ^{2}
\end{equation*
for all $x,y\in I$ and $\lambda \in (0,1).$ In particular, from the above
definition, by the arithmetic-geometric mean inequality, we hav
\begin{eqnarray}
f\left( \lambda x+\left( 1-\lambda \right) y\right) &\leq &\left[ f\left(
x\right) \right] ^{\lambda }\left[ f\left( y\right) \right] ^{1-\lambda
}-c\lambda \left( 1-\lambda \right) \left( x-y\right) ^{2} \label{E} \\
&\leq &\lambda f\left( x\right) +\left( 1-\lambda \right) f\left( y\right)
-c\lambda \left( 1-\lambda \right) \left( x-y\right) ^{2} \notag \\
&\leq &\max \left\{ f\left( x\right) ,f\left( y\right) \right\} -c\lambda
\left( 1-\lambda \right) \left( x-y\right) ^{2} \notag
\end{eqnarray}
\begin{theorem}
If a function $f:I\rightarrow \left( 0,\infty \right) $ be a strongly
log-convex with respect to $c>0$ and Lebesgue integrable on $I$, we hav
\begin{eqnarray}
f\left( \frac{a+b}{2}\right) +\frac{c\left( b-a\right) ^{2}}{12} &\leq
\frac{1}{b-a}\dint\limits_{a}^{b}G\left( f\left( x\right) ,f\left(
a+b-x\right) \right) dx \label{E3} \\
&\leq &\frac{1}{b-a}\dint\limits_{a}^{b}f\left( x\right) dx \notag \\
&\leq &L\left( f\left( a\right) ,f\left( b\right) \right) -\frac{c\left(
b-a\right) ^{2}}{6} \notag \\
&\leq &\frac{f\left( a\right) +f\left( b\right) }{2}-\frac{c\left(
b-a\right) ^{2}}{6} \notag
\end{eqnarray
for all $a,b\in I$ with $a<b.$
\end{theorem}
\begin{proof}
From (\ref{E}), we hav
\begin{eqnarray}
f\left( \lambda x+\left( 1-\lambda \right) y\right) &\leq &\left[ f\left(
x\right) \right] ^{\lambda }\left[ f\left( y\right) \right] ^{1-\lambda
}-c\lambda \left( 1-\lambda \right) \left( x-y\right) ^{2} \label{E4} \\
&\leq &\lambda f\left( x\right) +\left( 1-\lambda \right) f\left( y\right)
-c\lambda \left( 1-\lambda \right) \left( x-y\right) ^{2}. \notag
\end{eqnarray
Since $f$ is a strongly log-convex function on $I$, we have for $x,y\in I$
with $\lambda =\frac{1}{2}
\begin{eqnarray}
f\left( \frac{x+y}{2}\right) &\leq &\sqrt{f\left( x\right) f\left( y\right)
}-\frac{c\left( x-y\right) ^{2}}{4} \label{E5} \\
&\leq &\frac{f\left( x\right) +f\left( y\right) }{2}-\frac{c\left(
x-y\right) ^{2}}{4} \notag
\end{eqnarray
i.e., with $x=ta+\left( 1-t\right) b$, $y=\left( 1-t\right) a+tb$
\begin{eqnarray}
&&f\left( \frac{a+b}{2}\right) \label{E6} \\
&\leq &\sqrt{f\left( ta+\left( 1-t\right) b\right) f\left( \left( 1-t\right)
a+tb\right) }-\frac{c\left( b-a\right) ^{2}\left( 1-2t\right) ^{2}}{4}
\notag \\
&\leq &f\left( ta+\left( 1-t\right) b\right) +f\left( \left( 1-t\right)
a+tb\right) -\frac{c\left( b-a\right) ^{2}\left( 1-2t\right) ^{2}}{4}.
\notag
\end{eqnarray
Integrating the inequality (\ref{E6}) with respect to $t$ over $\left(
0,1\right) $, we obtai
\begin{eqnarray*}
f\left( \frac{a+b}{2}\right) &\leq &\frac{1}{b-a}\dint\limits_{a}^{b}\sqrt
f\left( x\right) f\left( a+b-x\right) }dx-\frac{c\left( b-a\right) ^{2}}{12}
\\
&\leq &\frac{1}{b-a}\dint\limits_{a}^{b}A\left( f\left( x\right) ,f\left(
a+b-x\right) \right) dx-\frac{c\left( b-a\right) ^{2}}{12},
\end{eqnarray*
and so for $\dint\limits_{a}^{b}f\left( x\right)
dx=\dint\limits_{a}^{b}f\left( a+b-x\right) dx,
\begin{eqnarray}
f\left( \frac{a+b}{2}\right) +\frac{c\left( b-a\right) ^{2}}{12} &\leq
\frac{1}{b-a}\dint\limits_{a}^{b}G\left( f\left( x\right) ,f\left(
a+b-x\right) \right) dx \label{E7} \\
&\leq &\frac{1}{b-a}\dint\limits_{a}^{b}f\left( x\right) dx. \notag
\end{eqnarray
Since $f$ is a strongly log-convex function on $I,$ for $x=a$ and $y=b,$ we
write
\begin{eqnarray}
f\left( ta+\left( 1-t\right) b\right) &\leq &\left[ f\left( a\right) \right]
^{t}\left[ f\left( b\right) \right] ^{1-t}-ct\left( 1-y\right) \left(
a-b\right) ^{2} \label{z1} \\
&\leq &tf\left( a\right) +\left( 1-t\right) f\left( b\right) -ct\left(
1-t\right) \left( a-b\right) ^{2}. \notag
\end{eqnarray
Integrating the inequality (\ref{z1}) with respect to $t$ over $\left(
0,1\right) $, we obtain
\begin{eqnarray*}
\frac{1}{b-a}\dint\limits_{a}^{b}f\left( x\right) dx &\leq &f\left( b\right)
\dint\limits_{0}^{1}\left[ \frac{f\left( a\right) }{f\left( b\right) }\right]
^{t}dt-c\left( b-a\right) ^{2}\dint\limits_{0}^{1}t\left( 1-t\right) dt \\
&\leq &f\left( a\right) \dint\limits_{0}^{1}tdt+f\left( b\right)
\dint\limits_{0}^{1}\left( 1-t\right) dt-c\left( b-a\right)
^{2}\dint\limits_{0}^{1}t\left( 1-t\right) dt,
\end{eqnarray*
and s
\begin{equation}
\frac{1}{b-a}\dint\limits_{a}^{b}f\left( x\right) dx\leq L\left( f\left(
a\right) ,f\left( b\right) \right) -\frac{c\left( b-a\right) ^{2}}{6}\leq
\frac{f\left( a\right) +f\left( b\right) }{2}-\frac{c\left( b-a\right) ^{2}}
6}. \label{E8}
\end{equation
Thus, from (\ref{E7}) and (\ref{E8}), we obtain the inequality of (\ref{E3
). This completes to proof.
\end{proof}
\begin{theorem}
Let a function $f:I\rightarrow \lbrack 0,\infty )$ be a strongly log-convex
with respect to $c>0$ and Lebesgue integrable on $I$, then the following
inequality holds
\begin{eqnarray}
&&\frac{1}{b-a}\dint\limits_{a}^{b}f\left( x\right) f\left( a+b-x\right)
dx\leq f\left( a\right) f\left( b\right) +\frac{c^{2}\left( b-a\right) ^{4}}
30} \notag \\
&& \label{E9} \\
&&-\frac{4c\left( b-a\right) ^{2}}{\left[ \ln \left( f\left( b\right)
-f\left( a\right) \right) \right] ^{2}}\left[ A\left( f\left( a\right)
,f\left( b\right) \right) +L\left( f\left( a\right) ,f\left( b\right)
\right) \right] \notag
\end{eqnarray
for all $a,b\in I$ with $a<b.$
\end{theorem}
\begin{proof}
Since $f$ is strongly log-convex with respect to $c>0$, we have that for all
$t\in \left( 0,1\right)
\begin{eqnarray}
f\left( ta+\left( 1-t\right) b\right) &\leq &\left[ f\left( a\right) \right]
^{t}\left[ f\left( b\right) \right] ^{1-t}-ct\left( 1-t\right) \left(
b-a\right) ^{2} \notag \\
&& \label{E10} \\
&\leq &tf\left( a\right) +\left( 1-t\right) f\left( b\right) -ct\left(
1-t\right) \left( b-a\right) ^{2} \notag
\end{eqnarray
and
\begin{eqnarray}
f\left( \left( 1-t\right) a+tb\right) &\leq &\left[ f\left( a\right) \right]
^{1-t}\left[ f\left( b\right) \right] ^{t}-ct\left( 1-t\right) \left(
b-a\right) ^{2} \notag \\
&& \label{E11} \\
&\leq &\left( 1-t\right) f\left( a\right) +tf\left( b\right) -ct\left(
1-t\right) \left( b-a\right) ^{2}. \notag
\end{eqnarray
Multiplying both sides of (\ref{E10}) by (\ref{E11}), it follows tha
\begin{eqnarray}
f\left( ta+\left( 1-t\right) b\right) f\left( \left( 1-t\right) a+tb\right)
&\leq &f\left( a\right) f\left( b\right) +c^{2}\left( b-a\right)
^{4}t^{2}\left( 1-t\right) ^{2} \notag \\
&& \notag \\
&&-c\left( b-a\right) ^{2}t\left( 1-t\right) \left( f\left( b\right) \left[
\frac{f\left( a\right) }{f\left( b\right) }\right] ^{t}+f\left( a\right)
\left[ \frac{f\left( b\right) }{f\left( a\right) }\right] ^{t}\right) .
\label{E12}
\end{eqnarray
Integrating the inequality (\ref{E12}) with respect to $t$ over $\left(
0,1\right) $, we obtai
\begin{eqnarray}
&&\dint\limits_{a}^{b}f\left( ta+\left( 1-t\right) b\right) f\left( \left(
1-t\right) a+tb\right) dt\leq \dint\limits_{0}^{1}f\left( a\right) f\left(
b\right) dt+c^{2}\left( b-a\right) ^{4}\dint\limits_{0}^{1}t^{2}\left(
1-t\right) ^{2}dt \notag \\
&& \label{E13} \\
&&-c\left( b-a\right) ^{2}f\left( b\right) \dint\limits_{0}^{1}t\left(
1-t\right) \left[ \frac{f\left( a\right) }{f\left( b\right) }\right]
^{t}dt-c\left( b-a\right) ^{2}f\left( a\right) \dint\limits_{0}^{1}t\left(
1-t\right) \left[ \frac{f\left( b\right) }{f\left( a\right) }\right] ^{t}dt
\notag \\
&& \notag \\
&=&\dint\limits_{0}^{1}f\left( a\right) f\left( b\right) dt+c^{2}\left(
b-a\right) ^{4}\dint\limits_{0}^{1}t^{2}\left( 1-t\right) ^{2}dt-c\left(
b-a\right) ^{2}f\left( b\right) I_{1}-c\left( b-a\right) ^{2}f\left(
a\right) I_{2}. \notag
\end{eqnarray
Integrating by parts for $I_{1}$ and $I_{2}$ integrals, we obtai
\begin{eqnarray}
&&I_{1}=\dint\limits_{0}^{1}t\left( 1-t\right) \left[ \frac{f\left( a\right)
}{f\left( b\right) }\right] ^{t}dt \notag \\
&& \notag \\
&=&\left. t\left( 1-t\right) \frac{1}{\ln \left[ \frac{f\left( a\right) }
f\left( b\right) }\right] }\left[ \frac{f\left( a\right) }{f\left( b\right)
\right] ^{t}\right\vert _{0}^{1}-\frac{1}{\ln \left[ \frac{f\left( a\right)
}{f\left( b\right) }\right] }\dint\limits_{0}^{1}\left( 1-2t\right) \left[
\frac{f\left( a\right) }{f\left( b\right) }\right] ^{t}dt \notag \\
&& \label{E14} \\
&=&-\frac{1}{\ln \left[ \frac{f\left( a\right) }{f\left( b\right) }\right]
\left[ \left. \left( 1-2t\right) \frac{1}{\ln \left[ \frac{f\left( a\right)
}{f\left( b\right) }\right] }\left[ \frac{f\left( a\right) }{f\left(
b\right) }\right] ^{t}\right\vert _{0}^{1}+\frac{2}{\ln \left[ \frac{f\left(
a\right) }{f\left( b\right) }\right] }\dint\limits_{0}^{1}\left[ \frac
f\left( a\right) }{f\left( b\right) }\right] ^{t}dt\right] \notag \\
&& \notag \\
&=&\frac{1}{f\left( b\right) }\frac{f\left( a\right) +f\left( b\right) }
\left[ \ln \left( f\left( a\right) -f\left( b\right) \right) \right] ^{2}}
\frac{2f\left( a\right) -2f\left( b\right) }{\left[ \ln \left( f\left(
a\right) -f\left( b\right) \right) \right] ^{2}}, \notag
\end{eqnarray
and similarly we get
\begin{eqnarray}
I_{2} &=&\dint\limits_{0}^{1}t\left( 1-t\right) \left[ \frac{f\left(
b\right) }{f\left( a\right) }\right] ^{t}dt \label{E15} \\
&& \notag \\
&=&\frac{1}{f\left( a\right) }\frac{f\left( a\right) +f\left( b\right) }
\left[ \ln \left( f\left( b\right) -f\left( a\right) \right) \right] ^{2}}
\frac{2f\left( b\right) -2f\left( a\right) }{\left[ \ln \left( f\left(
b\right) -f\left( a\right) \right) \right] ^{2}}. \notag
\end{eqnarray
Putting (\ref{E14}) and (\ref{E15}) in (\ref{E13}), and if we change the
variable $x:=ta+\left( 1-t\right) b$, $t\in \left( 0,1\right) $, we get the
required inequality in (\ref{E9}). This proves the theorem.
\end{proof}
|
{
"timestamp": "2012-03-13T01:01:31",
"yymm": "1203",
"arxiv_id": "1203.2281",
"language": "en",
"url": "https://arxiv.org/abs/1203.2281"
}
|
\section{Introduction}
Ultracold quantum gases have provided an exceptionally idealized
playground for emulating condensed matter systems \cite{bloch2008}.
Recently, Lin {\it et al.} have achieved a breakthrough in
quantum simulations of condensed matter systems by realizing an
effective spin-orbit (SO) coupling in $^{87}$Rb atoms
\cite{lin2011}. The SO coupling produces non-abelian gauge fields
\cite{dalibard2011}, plays a key role in spintronics
\cite{zutic2004} such as spin-polarized transport, spin injection,
and spin relaxation, and yields many other interesting phenomena such as
the quantum spin Hall effect and topological insulators
\cite{qi2010}. The experimental realization of synthetic gauge
fields has stimulated tremendous efforts on SO-coupled
quantum gases
\cite{stanescu2008,wu2011,wang2010,ho2011,yip2011,xu2011,
kawakami2011,xqxu2011,hu2012,sinha2011,radic2011,zhou2011,deng2011}.
The SO interaction couples spin and linear momentum,
thereby significantly modifying single-particle spectra.
For the Rashba-type SO coupling \cite{ruseckas2005,campbell2011,sau2011,xu2011b}, the conserved quantity
is $L_z+F_z$, where $L_z$ and $F_z$ are the projected
orbital and spin angular momenta along the $z$-axis, respectively.
In homogeneous systems, an axisymmetric SO coupling will change the parabolic
single-particle spectrum of a spin-$F$ atom into $2F+1$ energy bands with the lowest one featuring a Mexican
hat, causing a circular degeneracy of single-particle
ground states. For trapped systems, such circular dengeneracy is
reduced into double and no degeneracy for half-integer and integer spin systems,
respectively.
In spinor Bose-Einstein condensates (BECs) \cite{ueda2010}, the coupling between spin
and linear momentum will cooperate or compete with spin-dependent or
spin-independent interactions, giving rise to
many exotic ground states with or without harmonic trapping potentials,
such as plane-wave, stripe, triangular, and square lattice phases \cite{wang2010,xu2011,kawakami2011}.
Firstly, the plane-wave phase and the stripe phase are found in
SO-coupled pseudo spin-1/2 or spin-1 BECs \cite{wang2010}.
Later, we found two different lattice phases where each spin component shows
trianguar- or square-lattice density distributions in SO-coupled
spin-2 BECs with cyclic interactions \cite{xu2011}.
As an axisymmetry harmonic trap is turned on, more phases are
found as ground states of SO-coupled spin-1/2 BECs \cite{hu2012,sinha2011}.
Although many phases are found in SO-coupled spinor BECs,
a systematic understanding on them is still elusive as previous results
largely depend on numerical solutions of the coupled Gross-Pitaevskii equations.
In this article, alternatively we present a symmetry classification scheme to
investigate ground states of SO-coupled spinor condensates.
We can then not only understand different lattice phases already found,
but also find two different kagaome-lattice phases and a nematic vortex lattice phase,
both of which emerge spontaneously without lattice potentials.
This paper is organized as follows. Section II describes the model Hamiltonian
used in the present paper. Section III analyzes various phases that spontaneously
emerge in SO-coupled spinor condensates based on symmetry considerations.
The following three sections discuss properties of the three typical phases:
triangular-, square-, and kagome-lattice phases. Section VII discusses a
possibility of a fragmented ground state of a SO-coupled spin-1/2 system
with time-reversal symmetry. Section VIII summarizes the main results of this paper.
\section{Model Hamiltonian}
\label{model}
We consider a SO-coupled spinor BEC
with $N$ atoms, including pseudo spin-1/2, spin-1, and spin-2 condensates in a pancake-shaped
quasi-two-dimensional harmonic potential.
The effective Hamiltonian is given by $\mathcal{H}=\mathcal{H}_0+\mathcal{H}_{\rm int}$, with
\begin{eqnarray}
\mathcal{H}_0=\int d\bm{\rho}\hat{\psi}^{\dag}\left[\frac{\mathbf{p}^2}{2M}+\mathcal{V}_o
+\frac{v}{F}(p_xF_x+p_yF_y)\right]\hat{\psi},
\label{singleparticle}
\end{eqnarray}
where $\hat{\psi}=(\hat{\psi}_{F},\dots,\hat{\psi}_{-F})^T$,
$M$ is the atomic mass, $\bm{\rho}\equiv(x,y)$, $\mathcal{V}_o=M\omega_{\perp}^2\bm{\rho}^2/2$,
$v (>0)$ describes the strength of the SO coupling,
and $F_{x,y}$ are the spin-$F$ matrices.
For the $F=1/2$ case, we have
\begin{eqnarray}
\mathcal{H}_{\rm int}&=&\frac{1}{2}\int d\bm{\rho}\left( g\hat{n}_{1/2}^2+g\hat{n}_{-1/2}^2+2g'\hat{n}_{1/2}\hat{n}_{-1/2}\right)
\nonumber\\
&=&\frac{1}{2}\int d\bm{\rho}\left(\alpha \hat{n}^2+\beta\hat{S}_z^2\right),
\label{spinhalf}
\end{eqnarray}
where $g$ and $g'$ denote the strengths of the intra- and inter-component contact interactions, respectively.
Here, $\hat{n}_{\pm1/2}=\hat{\psi}_{\pm1/2}^{\dag}\hat{\psi}_{\pm1/2}^{\dag}\hat{\psi}_{\pm1/2}\hat{\psi}_{\pm1/2}$,
$\hat{n}=\hat{n}_{1/2}+\hat{n}_{-1/2}$, $\hat{S}_z=\hat{n}_{1/2}-\hat{n}_{-1/2}$, $\alpha=(g+g')/2$,
and $\beta=(g-g')/2$.
For the integer spin cases, we take the same interaction Hamiltonian
from spinor BECs \cite{ueda2010}.
For the $F=1$ case, the interaction part is given by
\begin{eqnarray}
\mathcal{H}_{\rm int}=\frac{1}{2}\int d\bm{\rho}\left(\alpha \hat{\psi}_i^{\dag}\hat{\psi}_j^{\dag}\hat{\psi}_j\hat{\psi}_i
+\beta\hat{\psi}_i^{\dag}\hat{\psi}_k^{\dag}\vec{F}_{ij}\cdot\vec{F}_{kl}\hat{\psi}_l\hat{\psi}_j\right),
\label{spinone}
\end{eqnarray}
where $\alpha$ and $\beta$ give the strengths of density-density and spin-exchange interactions, respectively.
Here, the indices that appear twice are to be summed over $-F, \dots, F$.
For the $F=2$ case, the interaction Hamiltonian is given by
\begin{eqnarray}
\mathcal{H}_{\rm int}&=&\frac{1}{2}\int d\bm{\rho}\left(\alpha \hat{\psi}_i^{\dag}\hat{\psi}_j^{\dag}\hat{\psi}_j\hat{\psi}_i
+\beta\hat{\psi}_i^{\dag}\hat{\psi}_k^{\dag}\vec{F}_{ij}\cdot\vec{F}_{kl}\hat{\psi}_l\hat{\psi}_j\right.
\nonumber\\
&&+\left.\gamma (-1)^{i+j}\hat{\psi}_i^{\dag}\hat{\psi}^{\dag}_{-i} \hat{\psi}_j\hat{\psi}_{-j}\right),
\label{spintwo}
\end{eqnarray}
where $\alpha$, $\beta$ and $\gamma$ give the strengths of the density-density, spin-exchange,
and singlet-pairing interactions, respectively.
Defining the harmonic-oscillator length $a_{\perp}\equiv\sqrt{\hbar/M\omega_{\perp}}$, we introduce
dimensionless parameters $v'=v/\omega_{\perp}a_{\perp}$ and
$(\alpha',\beta',\gamma')=(\alpha,\beta,\gamma)N/\hbar\omega_{\perp}$.
\section{Symmetry analysis}
\label{symmetry}
Under a mean-field approximation, we assume
that all atoms are condensed into a common single-particle state,
and therefore the symmetries of the Hamiltonian are spontaneously
broken. We can classify the ground states according to the remaining symmetries \cite{volovik1985,bruder1986,makela2007,yip2007,kawaguchi2011}.
\begin{figure*}[t]
\centering
\includegraphics[width=4.1in]{fig1.eps}
\caption{(Color online). Schematic diagrams for states preserving the
combined symmetry of spin-space rotation $\mathcal{C}_{nz}$,
gauge transformation, and time reversal.
Figures (a)--(f) correspond to $n=1$, $2$, $3$, $4$, $6$, and $\infty$,
respectively, where arrows denote the direction of spin, solid closed loops indicate
regions with nonzero momentum distributions $|\psi(k)|^2$, and
$n$ points on the circle are denoted as $\mathcal{S}_0$, \dots, $\mathcal{S}_{n-1}$,
respectively.
Characteristic momentum distributions
of the ground state $\psi$ for a trapped system
are shown in the lower right corner of each figure.
Black (yellow) color refers to the region of small (large) amplitude. }
\label{fig1}
\end{figure*}
For spinor BECs \cite{ueda2010} without SO-coupling, we usually focus only on
the spin degrees of freedom because they are decoupled from the orbital degrees of freedom.
As a result, the Hamiltonian is invariant under the U(1) global gauge transformation,
the SO(2) spin rotation along $z$-axis for the $F=1/2$ case or
the SO(3) spin rotation for the integer-spin cases, and the time reversal $\mathcal{T}\equiv e^{-i\pi F_y}\mathcal{K}$,
where $\mathcal{K}$ takes complex conjugation.
In contrast, for SO-coupled spinor BECs, to make the Hamiltonian invariant, we should simultaneously rotate the spin and the space.
Therefore, the Hamiltonian $\mathcal{H}$ described in Sec.\ref{model} is invariant under the global
U(1) gauge transformation, simultaneous SO(2) global spin and space (spin-space) rotations, and time reversal $\mathcal{T}$.
According to the symmetry classification scheme \cite{volovik1985,bruder1986,makela2007,yip2007,kawaguchi2011},
our task is: (1) to find the full symmetry group of the Hamiltonian, which in the present case is $\rm G=U(1)\times SO(2)\times \mathcal{T}$;
(2) to list all subgroups H of G; (3) to find the order parameter that is invariant under H.
If the order parameter can be uniquely determined from $h\psi=\psi$ for $\forall h\in \rm H$,
the state $\psi$ is an inert state, which is always a stationary point of the energy functional.
On the other hand, if the order parameter that is invariant under H is not uniquely determined,
we need to minimize the energy functional within the restricted manifold.
Such a state is called a non-inert state \cite{bruder1986,makela2007,yip2007,kawaguchi2011}.
The high-symmetry states in the SO-coupled system are all non-inert states.
To see this more clearly, we consider the eigenstates of simultaneous spin-space rotation $\mathcal{C}_{nz}$,
where $\mathcal{C}_{nz}\equiv\mathcal{R}_F (2\pi/n)\mathcal{R}_{\rho}(2\pi/n)$
is the generator of a discrete cyclic subgroup of SO(2) with $\mathcal{R}_F=\exp(-iF_z2\pi/n)$
and $\mathcal{R}_{\rho}$ being respectively the $2\pi/n$ spin and space rotation operators about the $z$-axis.
For the integer-spin cases, $\mathcal{C}_{nz}$ has $n$ different eigenvalues $\exp(-i2\mathbb{N}\pi/n)$ with $\mathbb{N}=0,\dots,n-1$,
whereas for the $F=1/2$ case, $\mathcal{C}_{nz}$ has $n$ different eigenvalues $\exp\{-i(2\mathbb{N}+1)\pi/n\}$
because $(\mathcal{C}_{nz})^n=-1$.
If the eigenvalue of $\mathcal{C}_{nz}$ is not $1$,
we can infer that this state is invariant under the combined U(1) gauge transformation
and spin-space rotation $\mathcal{C}_{nz}$, namely,
$\mathcal{O}_{nz}(\mathbb{N})\psi = \psi$
where $\mathcal{O}_{nz}(\mathbb{N})=\exp\{i(2\mathbb{N}+1)\pi/n\}\mathcal{C}_{nz}$ for the $F=1/2$ case,
and $\mathcal{O}_{nz}(\mathbb{N})=\exp(i2\mathbb{N}\pi/n)\mathcal{C}_{nz}$ for the $F=1$ and $2$ cases.
For a spin-$F$ system, we can construct eigenstates of $\mathcal{O}_{nz}(\mathbb{N})$ by using a complete basis set of plane waves as
\begin{eqnarray}
\psi=\sum\limits_{j=0}^{n-1}\left[\mathcal{O}_{nz}(\mathbb{N})\right]^j\sum\limits_{\mathbf{k}\in \bar{\mathcal{S}},\sigma}
D^{n}_{\sigma}(\mathbf{k})e^{i\mathbf{k}\cdot\bm{\rho}}|\sigma\rangle,
\label{rotation}
\end{eqnarray}
where $\mathbf{k}$ is the wave vector,
$\bar{\mathcal{S}}=\{\mathbf{k}| -\pi/n+\bar{\varphi}\le\varphi_k<\pi/n+\bar{\varphi}, \varphi_k\equiv\arg(k_x+ik_y)\}$
with $\bar{\varphi}$ being arbitrary, $|\sigma\rangle$ ($\sigma=-F,-F+1,\dots,F$)
denotes the spin state with $M_F=\sigma$ under the $z$-axis quantization,
and $D_{\sigma}^{n}(\mathbf{k})$ are the expansion coefficients.
Unless we know the detail of $D_{\sigma}^{n}(\mathbf{k})$,
we cannot uniquely determine the state that is invaraint under ${\rm H}=\{\mathbb{E}, \mathcal{O}_{nz}(\mathbb{N}),\dots,\mathcal{O}_{nz}^{n-1}(\mathbb{N})\}$
with $\mathbb{E}$ being the identity operator.
Similar arguments can be applied to other subgroups of G.
Since there are no inert states for spin-orbit coupled spinor BECs,
we cannot find the order parameter by analyzing its symmetry only.
However, such a difficulty can be alleviated if we take the $\it ansatz$ that the state
is a superposition of several degenerate single-particle ground states of the Hamiltonian in Eq. (\ref{singleparticle})
with $\omega_{\perp}=0$:
\begin{eqnarray}
\psi=\sum\limits_{j=0}^{n-1}e^{i\phi_j}\mathcal{PW}\left(k=k_g,\varphi_k=\bar{\varphi}+j\frac{2\pi}{n}\right),
\label{orderparameters}
\end{eqnarray}
where $\mathcal{PW}(k_g,\varphi_{k})=e^{i\mathbf{k}_g\cdot\bm{\rho}}\zeta_{-F}(\varphi_{k})$
are degenerate single-particle ground states
with $k_g=mv/\hbar$, $\varphi_k=\arg(k_x+ik_y)$,
and $\bar{\varphi}$ being arbitrary. For the cases of $F=1/2$, $1$ and $2$,
we have respectively
\begin{eqnarray}
&\zeta_{-1/2}(\varphi_k)=(1,-e^{i\varphi_k})^T/\sqrt{2},&\nonumber\\
&\zeta_{-1}(\varphi_k)=(e^{-i\varphi_k},-\sqrt{2},e^{i\varphi_k})^T/2,&\nonumber\\
&\zeta_{-2}(\varphi_k)=(e^{-2i\varphi_k},-2e^{-i\varphi_k},\sqrt{6},-2e^{i\varphi_k},
e^{2i\varphi_k})^T/4.&
\label{planewaves}
\end{eqnarray}
The corresponding spin expectation value is anti-parallel to the wave vector:
$(\langle F_x\rangle, \langle F_y \rangle)=-F(\cos\varphi_k,\sin\varphi_k)$.
If we take $\phi_j=j2\mathbb{N}\pi/n$, the state $\psi$ in Eq. (\ref{orderparameters})
is invariant under $\mathcal{O}_{nz}(\mathbb{N})$. This state is a special case of
that in Eq. (\ref{rotation}) with the set $\bar{\mathcal{S}}$ shrinks into only one
element and the allowed values of $\mathbf{k}$ are fixed at discrete points
as $D_\sigma^n({\bf k})=\delta(k-k_g)[\zeta_{-F}(\bar{\varphi})]_\sigma$.
Figure 1 illustrates schematic diagrams for the states of Eq. (\ref{orderparameters}),
where $n$ single-particle ground states $\mathcal{PW}(k_g,\varphi_{k})$ are denoted by
points $\mathcal{S}_j$.
\begin{figure*}[t]
\centering
\includegraphics[width=4.1in]{fig2.eps}
\caption{(Color online). Ground states of trapped SO-coupled spin-1 condensates,
with $v'=15$, $\alpha'=0.5$ and $\beta'=0.1$.
(a) Density distributions of three spin components $M_F=1,0,-1$ from left to right.
(b) Spatial variation of the corresponding order parameter visualized by plotting
$\sum_{m}\xi_{m}Y_F^{m}(\theta,\varphi)$,
where $Y_F^{M_F}$ is the spherical harmonics and $\xi_m$ is the $m$-th component
of the spin wave function. }
\label{fig2}
\end{figure*}
We choose the $\it ansatz$ in Eq. (\ref{orderparameters})
based on previous understanding on a 2D system and our numerical results
given in the following sections.
For a 2D homogenous system with $\omega_{\perp}=0$,
as discussed in Refs. \cite{wang2010,xu2011},
plane-wave and stripe phases are found to be the ground states for
the $F=1/2$, $1$ and $2$ condensates \cite{wang2010,xu2011},
whereas triangular and square lattice phases exist
only for spin-2 condensates with cyclic interactions \cite{xu2011}.
Such phases can all be described by Eq. (\ref{orderparameters}).
For a trapped system, translation symmetry is broken, and linear momentum is no
longer conserved. Therefore, in momentum space, there are infinite points forming
regions, where momentum distributions $|\psi(\mathbf{k})|^2$ of the
ground state $\psi$ are nonzero. In Fig. \ref{fig1}, we use solid closed
loops (circles for $n=\infty$ ) to denote such regions.
In the case of strong SO couplings, we can approximate low-lying single-particle eigenstates as \cite{jacob2009,sinha2011}
\begin{eqnarray}
\Psi_{\bar{n},m}(k')\propto k'^{-1/2}e^{-(k'-v')^2/2}H_{\bar{n}}(k'-v')e^{im\varphi_k}\zeta_{-F}(\varphi_k),
\label{eigenstates}
\end{eqnarray}
with eigenenergies
$E_{\bar{n},m}=(m+1/2)^2/2v'^2+\bar{n}+1/2-v'^2/2$, $(m^2+1/4)/2v'^2+\bar{n}+1/2-v'^2/2$,
and $(m^2+3/4)/2v'^2+\bar{n}+1/2-v'^2/2$ for $F=1/2$, $1$ and $2$ respectively.
Here, $\bar{n}=0,1,2,\dots$, $m=0,\pm1,\pm2, \dots$, $k'=ka_{\perp}$, and $H_{\bar{n}}$ are Hermite polynomials.
When the interatomic interactions are weak, the ground states can be constructed only from
the states with $\bar{n}=0$, which are superpositions of single-particle
eigenstates $\zeta_{-F}(\mathbf{k})$ with weight $k'^{-1/2}e^{-(k'-v')^2/2}\simeq v'^{-1/2}e^{-(k'-v')^2/2}$.
In the limit of $\omega_{\perp}\rightarrow0$, $e^{-(k'-v')^2/2}\rightarrow \delta(k'-v')/\sqrt{2\pi}$,
which implies that the ground states will involve momenta with its magnitude close to $k_g$.
Furthermore, by choosing specific weak interactions, axial rotational symmetry in the single-particle
states of Eq. (\ref{eigenstates}) can break into discrete rotation symmetry,
and 2D lattice phases appear, as already found in the $F=1/2$ case \cite{hu2012,sinha2011}.
In the limit of weak trapping potential, strong SO couplings, and weak interatomic interactions,
there are ground states whose momentum distributions split into several small regions
as those illustrated in Fig. \ref{fig1}.
As long as the area of the allowed momentum is small enough,
these ground states can still be well described by Eq. (\ref{orderparameters}).
The validity of the {\it ansatz} has been numerically confirmed \cite{numerical},
because we find several ground states which can be understood by Eq. (\ref{orderparameters}).
In the lower right corner of each figure in Fig. \ref{fig1}, we
illustrate the characteristic momentum distributions of these ground states.
For the case of $n=1$ and $2$, the state in Eq. (\ref{orderparameters})
can describe respectively the plane-wave phase and the stripe phase found in a homogeneous system \cite{wang2010,xu2011}.
For the case of $n=\infty$, the state is invariant under the combined SO(2) spin-space rotations and U(1) gauge transformation,
and has been predicted for the $F=1/2$ case \cite{hu2012,sinha2011},
where single-particle eigenstates of Eq.~(\ref{eigenstates}) will be stabilized in some parameter regions.
For the $F=1$ and $F=2$ cases, similar arguments can be applied, and more single-particle eigenstates
of Eq.~(\ref{eigenstates}) can be stabilized.
In the following, we start from the ansatz of Eq. (\ref{orderparameters}).
and find several high-symmetry states, where a set of $\phi_j$'s
are determined by requiring the wave function of Eq. (\ref{orderparameters}) to preserve a certain symmetry.
Some of them become ground states of a trapped SO-coupled system,
where triangular, square and kagaome lattices are formed
spontaneously without lattice potentials.
These states can be described by Eq. (\ref{orderparameters})
with $n=3$, $4$, and $6$, respectively.
Before discussing lattice phases, we point out that
(1) the spin and space rotation operations commute with time reversal
$[\mathcal{T},\mathcal{C}_{nz}]=0$, and that
(2) for the $F=1/2$ case, the ground-state order parameter $\psi$ always breaks time-reversal symmetry
because $\mathcal{T}^2=-1$, which means $\psi$ will only be invariant under $\mathcal{C}_{nz}$
or $\mathcal{TC}_{nz}$, and possibly combined with the U(1) gauge transformation for each $n$.
For the integer-spin case, the ground-state order parameter will be invariant under one or all of the three operations:
$\mathcal{T}$, $\mathcal{C}_{nz}$ and $\mathcal{TC}_{nz}$,
and possibly combined with the U(1) gauge transformation for each $n$.
\section{Triangular-lattice phase}
A triangular-lattice phase is described by
Eq.~(\ref{orderparameters}) with $n=3$ as
\begin{eqnarray}
\psi&=&e^{i\phi_0}e^{ik_gx}\zeta_{-F}(0)+e^{i\phi_1}e^{ik_g(-x/2+\sqrt{3}y/2)}\zeta_{-F}(2\pi/3)\nonumber\\
&&+e^{i\phi_2}e^{ik_g(-x/2-\sqrt{3}y/2)}\zeta_{-F}(4\pi/3),
\label{triangular}
\end{eqnarray}
where we take $\bar{\varphi}=0$ as an example.
According to the symmetry classification scheme described in Sec.\ref{symmetry},
we can apply specific symmetries on the state of Eq. (\ref{triangular})
to evaluate the value of $\phi_j$.
We find the only possible symmetry is described by the group
${\rm H}=\{\mathbb{E}, \mathcal{O}_{3z}(\mathbb{N}),\mathcal{O}_{3z}^{2}(\mathbb{N})\}$,
resulting in $\phi_j=j2\mathbb{N}\pi/3+\rm const$.
Meanwhile, the states of Eq. (\ref{triangular}) with different values of $\phi_j$
are connected by a global U(1) phase change and a global coordinate translation.
This is because for arbitrary change $\delta\phi_j$ in $\phi_j$,
there are always solutions for $(\delta x,\delta y)$ and $\delta U$ that satisfy
\begin{eqnarray}
&\exp\{i[\delta \phi_0+k_g\delta x+\delta U)\}=1,&\nonumber\\
&\exp\{i[\delta \phi_1+k_g(-\delta x/2+\sqrt{3}\delta y/2)+\delta U)\}=1,&\nonumber\\
& \exp\{i[\delta \phi_2+k_g(-\delta x/2-\sqrt{3}\delta y/2)+\delta U)\}=1,&
\end{eqnarray}
where $(\delta x, \delta y)$ and $\delta U$ describe
the amount of the coordinate translation and the U(1) phase change, respectively.
As a result, there is only one type of triangular-lattice phase.
By choosing a proper center of lattice, such a ground state
becomes invariant under $\mathcal{O}_{3z}$. Note that in a harmonic trap,
the $\mathcal{O}_{3z}$ symmetry axis sometimes deviates from the trap center to
gain the interaction energies.
In Fig.~\ref{fig2}, we illustrate
a triangular-lattice phase which appears in the ground state of SO coupled spin-1 condensates with
$v'=15$, $\alpha'=0.5$ and $\beta'=0.1$,
showing triangular-lattice density distributions for each spin component.
Figure \ref{fig2}(b) shows the spatial variation of the corresponding order parameter.
At three lattice sites, the order paramters
can be written as $\mathcal{P}_1: (0,e^{-i\pi/3},0)^T$, $\mathcal{P}_2: (0,e^{i\pi/3},0)^T$
and $\mathcal{P}_3: (0,e^{i\pi},0)^T$.
To go from $\mathcal{P}_1$ to $\mathcal{P}_2$,
a $\pi$ spin rotation along $\mathcal{P}_1\mathcal{P}_2$ and the $e^{-i\pi/3}$ gauge transformation
are needed. Similar transformations are needed to go from $\mathcal{P}_2$ to $\mathcal{P}_3$
or from $\mathcal{P}_3$ to $\mathcal{P}_1$.
Going along the loop of $\mathcal{P}_1\mathcal{P}_2\mathcal{P}_3\mathcal{P}_1$,
the order parameter undergoes a $\pi$ spin rotation along an axis on the $x$-$y$ plane and
the $-\pi$ gauge transformation,
which, however, does not imply that the mass circulation here is fractional with $-1/2$ winding \cite{fractional}.
To check whether this vortex is fractional or not, we should calculate the
circulation \cite{ueda2010} along the loop of $\mathcal{P}_1\mathcal{P}_2\mathcal{P}_3\mathcal{P}_1$.
Numerically we found this vortex is not fractional.
For the triangular-lattice phase, two (three) spin components of spin-1/2 (spin-1) condensates
tend to occupy different spaces due to the interferences of three plane waves.
For spin-2 condensates, five spin components are divided into
three classes: (1) $M_F=2,-1$; (2) $M_F=1,-2$; (3) $M_F=0$.
In the cases of (1) and (2), two different spin components
show the same wave function, which can be
understood from Eq.~(\ref{triangular}). These properties
are consistent with our previous work \cite{xu2011}.
\begin{figure*}
\centering
\includegraphics[width=4.1in]{fig3.eps}
\caption{(Color online). Ground states of trapped SO-coupled spinor BECs,
with $v'=15$. (a) Density distributions of $M_F=1$ (left),
$0$ (middle), $-1$ (right) of the spin-1 case with $\alpha'=0.5$, $\beta'=-0.1$.
(b) The corresponding order parameter, where F is used to denote ferromagnetic vortex cores.
(c) Density distributions of $M_F=2$ (left), $1$ (middel), and $0$ (right) of the spin-2 case
with $\alpha'=0.5$, $\beta'=0.1$ and $\gamma'=-0.1$. (d) The corresponding order parameter.
The uniaxial and biaxial nematic vortex cores are denoted by UN and BN, respectivley.
Here, black (yellow) color refers to the low (high) density region.}
\label{fig3}
\end{figure*}
\begin{figure*}
\centering
\includegraphics[width=4.1in]{fig4.eps}
\caption{(Color online). Ground states of trapped SO-coupled pseudo spin-1/2 BECs,
with $v'=15$ and (a,b) $\alpha'=0.3$, $\beta'=0.06$,
(c,d) $\alpha'=0.4$, $\beta'=0.08$. Here, black (yellow) color refers to the low (high) density region.
From left to right, we show the spin-up, spin-down and total density distributions.
Figures (a) and (c) are numerically calculated for a trapped system,
while (b) and (d) are obtained by superposition
of six single-particle states $\mathcal{PW}(\varphi_k)$.
In (b) and (d), we use symbols ``-'',``+'', and ``++'' to denote vorticities
$-\hbar$, $\hbar$ and $2\hbar$, respectively. The size of figure (b) [(c)]
is the same as that of (d) [(a)]. }
\label{fig4}
\end{figure*}
\section{Square-lattice phase}
When $n=4$, the state in Eq. (\ref{orderparameters}) can be simplified as
\begin{eqnarray}
\psi&=&e^{i\phi_0}e^{ik_gx}\zeta_{-F}(0)+e^{i\phi_1}e^{ik_gy}\zeta_{-F}(\pi/2)\nonumber\\
&+&e^{i\phi_2}e^{-ik_gx}\zeta_{-F}(\pi)+e^{i\phi_3}e^{-ik_gy}\zeta_{-F}(3\pi/2),
\label{square}
\end{eqnarray}
where we again take $\bar{\varphi}=0$ as an example.
This state can only potentially be invariant under $\mathcal{O}_{nz}$ with $n=1,2,4$.
If the state is only invariant under $\mathcal{O}_{1z}$ or $\mathcal{O}_{2z}$,
$\psi$ cannot be uniquely determined.
Therefore, we require the state preserving $\mathcal{O}_{4z}(\mathbb{N})$,
resulting in $\phi_j=j\mathbb{N}\pi/2+\rm const$.
Similar to the triangular-lattice phase, there are other states
which are related to the states invariant under $\mathcal{O}_{4z}(\mathbb{N})$
by simply doing a global lattice shift. They may be classified into the same class,
as they have the same lattice structure. The criterion for each class is
determined by the parameter $\Delta \equiv(\phi_1+\phi_{3})-(\phi_0+\phi_{2})$.
The reason is given as follow:
When doing a global lattice shift, the state $\psi$ in Eq. (\ref{square}) changes
with
\begin{eqnarray}
\phi_0+k_gx\rightarrow\phi_0+k_g(x+\delta x),\nonumber\\
\phi_1+k_gy\rightarrow\phi_1+k_g(y+\delta y),\nonumber\\
\phi_2-k_gx\rightarrow\phi_2-k_g(x+\delta x),\nonumber\\
\phi_3-k_gy\rightarrow\phi_3-k_g(y+\delta y).
\end{eqnarray}
Absorbing $\delta x$ and $\delta y$ into $\phi_j$,
we find that there are two invariants $\phi_0+\phi_2$ and $\phi_1+\phi_3$.
Their difference $\Delta$ is also invariant under the global U(1) gauge transformation.
Furthermore, we find that for the states invariant under $\mathcal{O}_{4z}(\mathbb{N})$,
$\Delta=(2\mathbb{M}+1)\pi$ when $\mathbb{N}=1, 3$;
$\Delta=2\mathbb{M}\pi$ when $\mathbb{N}=0, 2$, ($\mathbb{M}\in \mathbb{Z}$).
We then define the criterion for two different types of square-lattice phases with different
lattice structures as (1) $\Delta=(2\mathbb{M}+1)\pi$ and
(2) $\Delta=2\mathbb{M}\pi$.
The states that fall into the same class are connected by a global lattice shift.
By doing a proper lattice shift, the symmetry of the state $\psi$ can be described by
${\rm H}=\{\mathbb{E}, \mathcal{O}_{4z}(\mathbb{N}),\mathcal{O}_{4z}^{2}(\mathbb{N}),\mathcal{O}_{4z}^{3}(\mathbb{N})\}$
with $\mathbb{N}=1, 3$ for the first class and $\mathbb{N}=0, 2$ for the second class.
For the integer-spin cases, the state $\psi$ in the second class
is also invariant under time reversal $\mathcal{T}$ \cite{statement}
by choosing a proper lattice center.
Numerically, we find all such ground states to be integer-spin condensates.
Again, in a harmonic trap, the $\mathcal{O}_{4z}$ symmetry axis does not
always coincide with the trap center to minimize the interaction energy.
The ground states for the case of $\Delta=(2\mathbb{M}+1)\pi$ are found to exist in spin-1 BECs with $\beta<0$,
and spin-2 BECs with $\beta>0$ and $\gamma>0$ \cite{xu2011}.
Figures \ref{fig3}(a) and (b) show the corresponding ground-state density distributions
and order parameters for the spin-1 BECs, with $v'=15$, $\alpha'=0.5$, and $\beta'=-0.1$.
The spin component $M_F=0$ forms a square lattice with lattice constant $\pi/k_g$,
while the spin components $M_F=1$ and $-1$ fill the center of
squares alternatively, both forming a square lattice with lattice constant $\sqrt{2}\pi/k_g$.
The numerically obtained total density distributions are smooth for such ground states,
consistent with the prediction from
the corresponding state $\psi$ in Eq. (\ref{orderparameters}).
Only the component with $M_F=0$ involves vortices, with vortex
cores filled by the $M_F=1$ and $M_F=-1$ components having vorticity $\hbar$ and $-\hbar$, respectively.
Thus, the spin vortex lattice is filled by ferromagnetic vortex cores which are polarized
in the $z$ and $-z$ directions alternatively.
For the $F=2$ case, such phase has already been predicted in our previous work \cite{xu2011}.
We have found that spin components $M_F=2,0,-2$ show the same
density distribution which is different from those of the $M_F=1$ and $-1$ components.
Again, there is no fractional or integer vortex \cite{fractional}.
Another type of square lattice phase with $\Delta=2\mathbb{M}\pi$ appears in the
trapped SO-coupled spin-2 condensates with antiferromagnetic spin-dependent interactions.
Figure \ref{fig3}(c) shows ground-state density distributions of spin components
$M_F=2,1,0$ from left to right, with $v'=15$, $\alpha'=0.5$, $\beta'=0.1$,
and $\gamma'=-0.1$. The spin components $M_F=-1$ and $-2$ show the same density distributions
as $M_F=1$ and $2$, respectively.
In this phase, we find that two physical quantities
$|\langle\vec{F}\rangle|\equiv |\sum_{ij}\psi^*_i\vec{F}_{ij}\psi_j|/\sum_i\psi^*_i\psi_i$ and
$|\langle\Theta\rangle|\equiv|\sum_{ij}(-1)^{i+j}\psi_i^*\psi_{-i}^*\psi_j\psi_{-j}|/\sum_i\psi_i^*\psi_i$
are uniform and are equal to 0 and 1, respectively.
Figure \ref{fig3}(d) illustrates the corresponding order parameter,
showing a spin vortex lattice with uniaxial nematic (UN) and biaxial nematic (BN) vortex cores
aligning alternatively.
\section{Kagome-lattice phase}
The {\it ansatz} in Eq. (\ref{orderparameters}) with $n=6$
can be written as
\begin{eqnarray}
\psi&=&e^{i\phi_0}e^{ik_gx}\zeta_{-F}(0)+e^{i\phi_1}e^{ik_g(x/2+\sqrt{3}y/2)}\zeta_{-F}(\pi/3)\nonumber\\
&+&e^{i\phi_2}e^{ik_g(-x/2+\sqrt{3}y/2)}\zeta_{-F}(2\pi/3)+e^{i\phi_3}e^{-ik_gx}\zeta_{-F}(\pi)\nonumber\\
&+&e^{i\phi_4}e^{ik_g(-x/2-\sqrt{3}y/2)}\zeta_{-F}(4\pi/3)\nonumber\\
&+&e^{i\phi_5}e^{ik_g(x/2-\sqrt{3}y/2)}\zeta_{-F}(5\pi/3),
\label{kagaome}
\end{eqnarray}
where we take $\bar{\varphi}=0$ as an example.
This state can describe kagaome-lattice structures.
Similar to the case of $n=4$,
we note that four independent phases
\begin{eqnarray}
&\Delta_0=\phi_0+\phi_2+\phi_4,\quad \Delta_1=\phi_0+\phi_3,&\nonumber\\
&\Delta_2=\phi_2+\phi_5,\quad \Delta_3=\phi_4+\phi_1,&
\end{eqnarray}
are invariant under a global shift of the lattice.
Besides the one used to describe the global phase change, there
will be three independent phases which determine the structure of the lattice.
Therefore, there are infinite types of lattice structures of the state in Eq. (\ref{kagaome}).
According to symmetry classification scheme, we need to apply specific symmetries
to determine the state $\psi$. Only if the symmetry is high enough,
the state $\psi$ can be uniquely determined.
For the $F=1/2$ case, two high symmetry groups are
generated by operators $\mathcal{O}_{6z}$ and $\mathcal{TO}_{6z}$, respectively,
whereas for the integer-spin cases, high symmetry groups can be generated
by only $\mathcal{O}_{6z}$ or $\{\mathcal{O}_{6z}, \mathcal{T}\}$.
(1) If we require the state is invariant under $\mathcal{O}_{6z}(\mathbb{N})$,
the state $\psi$ is uniquely determined with $\phi_j=j\mathbb{N}\pi/3+\rm const$.
For the integer-spin cases, the state is further invariant under time reversal $\mathcal{T}$ \cite{statement}
if $\mathbb{N}=0, 3$.
(2) For the $F=1/2$ case, there are states that preserve the $\mathcal{TO}_{6z}$ symmetry and break the $\mathcal{O}_{6z}$ symmetry.
To determine such states, we start from the states invariant under $\mathcal{O}_{3z}(\mathbb{N})$ with $\mathbb{N}=0, 1, 2$.
The value of $\phi_j$ is determined as
$\phi_j=j2\mathbb{N}\pi/6+\text{mod}(j,2)\phi+\rm const$, where $\phi$ is arbitrary.
Furthermore, by requiring that the state be invariant under $\mathcal{TC}_{2z}$ \cite{statement},
we obtain $\mathbb{N}=1$ and $\phi=\pi/2+\mathbb{M}\pi$.
We numerically find two distinct classes of kagaome-lattice phases in a spin-1/2 BEC with
$\beta'>0$, whose symmetries are described by the group H generated respectively by (1) $\exp(i2\pi/3)\mathcal{C}_{3z}$ and $\mathcal{TC}_{2z}$
(with $\phi_j= j\pi/3+\bmod(j,2)(1/2+\mathbb{M})\pi +\rm const$)
and (2) $\exp\{i(\mathbb{M}+1/3)\pi\}\mathcal{C}_{6z}$ (with $\phi_j=j(\mathbb{M}+1/3)\pi+\rm const$), where
the ground states with different values of $\exp(i\mathbb{M}\pi)$ in the same class
are time reversal with each other.
Figure \ref{fig4} shows the spin-up, spin-down and total density distributions
for numerically obtained order parameters in a trapped system [Fig. \ref{fig4}(a) and (c)]
and the corresponding {\it ansatz} of Eq. (\ref{kagaome}) [Fig. \ref{fig4}(b) and (d)].
For each spin component, density distributions
show kagome lattice structures.
\section{Fragmented ground states}
Although we have discussed within the mean-field theory so far,
there is a possibility that a fragmented ground state \cite{nozieres,law1998,ho2000,mueller2006},
rather than a mean-field state, arises.
Actually, for the case of SO-coupled spin-1/2 system with an SU(2) symmetry,
we construct a fragmented state
whose energy is degenerate with the mean-field solution
up to the mean-field approximation.
Such a fragmented ground state is expected to arise in a mesoscopic system.
In this section, we discuss fragmented ground states from the point of view of the time-reversal symmetry.
As pointed out in the Sec.\ref{symmetry}, mean-field states
always break time reversal for the $F=1/2$ case because $\mathcal{T}^2=-1$.
In this section, we show that there are also
time-reversal invariant many-body states that are
fragmented in a time-reversal preserving $F=1/2$ bosonic system
with or without SOCs.
For a pseudo spin-1/2 system with $N$ atoms, we have $\mathcal{T}^2=(-)^N$.
When $N$ is odd, all states break time-reversal symmetry.
In contrast, when $N$ is even, there is always a time-reversal
invariant ground state which is fragmented:
if $[\mathcal{T},\mathcal{H}]=0$ and $\mathcal{H}|\psi\rangle=E|\psi\rangle$,
$|\psi_{\mathcal{T}}\rangle=|\psi\rangle+\mathcal{T}|\psi\rangle$ is an eigenstate of $\mathcal{H}$ with the same eigenenergy $E$,
and is invariant under time reversal if $N$ is even.
To check whether $|\psi_{\mathcal{T}}\rangle$ is fragmented or not,
we can diagonalize its single-particle density matrix $\hat{\rho}$ \cite{mueller2006}.
We can regroup the single-particle eigenstates as $\{\Psi_{\mu_i},\Psi_{\nu_i}\}$,
where we use $\mu$ and $\nu$ to distinguish two states by their time reversal as
$\mathcal{T}\Psi_{\mu_i}=\Psi_{\nu_i}$ and $\mathcal{T}\Psi_{\nu_i}=-\Psi_{\mu_i}$.
Time-reversal properties of the corresponding creation operators
$\{\hat{a}^{\dag}_{\mu_i},\hat{a}^{\dag}_{\nu_i}\}$ and annihilation operators
$\{\hat{a}_{\mu_i},\hat{a}_{\nu_i}\}$ for the states $\{\Psi_{\mu_i},\Psi_{\nu_i}\}$ are
\begin{eqnarray}
\mathcal{T}\hat{a}^{\dag}_{\mu_i}\mathcal{T}^{-1}=\hat{a}^{\dag}_{\nu_i},
\quad
\mathcal{T}\hat{a}^{\dag}_{\nu_i}\mathcal{T}^{-1}=-\hat{a}^{\dag}_{\mu_i},
\nonumber\\
\mathcal{T}\hat{a}_{\mu_i}\mathcal{T}^{-1}=\hat{a}_{\nu_i},
\quad
\mathcal{T}\hat{a}_{\nu_i}\mathcal{T}^{-1}=-\hat{a}_{\mu_i}.
\label{timereversal}
\end{eqnarray}
We also have
$\langle \psi_{\mathcal{T}}|\hat{a}_{\mu_i}^{\dag}\hat{a}_{\mu_j}|\psi_{\mathcal{T}}\rangle
=\langle \mathcal{T}\psi_{\mathcal{T}}|\mathcal{T}\hat{a}_{\mu_i}^{\dag}\hat{a}_{\mu_j}|\psi_{\mathcal{T}}\rangle^*
=\langle \psi_{\mathcal{T}}|\hat{a}_{\nu_i}^{\dag}\hat{a}_{\nu_j}|\psi_{\mathcal{T}}\rangle^*$,
where for the first equality we use the fact that $\mathcal{T}$ is an antiunitary operator,
while the second equality is due to Eq. (\ref{timereversal}) and the fact that
$|\psi_{\mathcal{T}}\rangle$ is invariant under time reversal.
Similarly, we obtain
$\langle \psi_{\mathcal{T}}|\hat{a}_{\nu_i}^{\dag}\hat{a}_{\nu_j}|\psi_{\mathcal{T}}\rangle=
\langle \psi_{\mathcal{T}}|\hat{a}_{\mu_i}^{\dag}\hat{a}_{\mu_j}|\psi_{\mathcal{T}}\rangle^*$,
$\langle \psi_{\mathcal{T}}|\hat{a}_{\mu_i}^{\dag}\hat{a}_{\nu_j}|\psi_{\mathcal{T}}\rangle=
-\langle \psi_{\mathcal{T}}|\hat{a}_{\nu_i}^{\dag}\hat{a}_{\mu_j}|\psi_{\mathcal{T}}\rangle^*$,
and
$\langle \psi_{\mathcal{T}}|\hat{a}_{\nu_i}^{\dag}\hat{a}_{\mu_j}|\psi_{\mathcal{T}}\rangle=
-\langle \psi_{\mathcal{T}}|\hat{a}_{\mu_i}^{\dag}\hat{a}_{\nu_j}|\psi_{\mathcal{T}}\rangle^*$.
These equalities imply that the single-particle density matrix is invariant under $\Xi$ as
\begin{eqnarray}
\Xi\hat{\rho}\Xi^{-1}=\hat{\rho},\quad \Xi\equiv\mathcal{K}\prod_i\otimes
\left(
\begin{array}{cc}
0 & 1\\
-1 & 0\\
\end{array}
\right).
\label{densitymatrix}
\end{eqnarray}
Due to this symmetry, we can infer that if $|\Psi\rangle$ is an eigenstate
of $\hat{\rho}$, $\Xi|\Psi\rangle$ is also an eigenstate of $\hat{\rho}$ with
the same eigenenergy but orthorgonal to $|\Psi\rangle$.
Our argument above is also valid for the ground state.
A simple state that preserves time-reversal symmetry is
\begin{eqnarray}
|\psi_{\mathcal{T}}\rangle\propto \left(\hat{a}_{\mu_1}^{\dag}\right)^{N/2}\left(\hat{a}_{\nu_1}^{\dag}\right)^{N/2}|\rm vac\rangle.
\label{twoorbital}
\end{eqnarray}
Using this {\it ansatz} wave function, we numerically minimize the energy functional
$\langle\psi_{\mathcal{T}}|\mathcal{H}|\psi_{\mathcal{T}}\rangle$,
and find that the ground-state energy is the same as that obtained with single-orbital
mean-field approximation for an SU(2)-symmetric system without spin-dependent interactions,
where we take the lowest-energy band approximation by considering only states $\Psi_{0,l}$
with $v'=15$, $\beta'=0$, and $\alpha'$ ranging from 0 to 1.
Before conclusions, we would like to point out that the fragmented ground states
we discuss here are protected by time-reversal symemtry for a spin-1/2 system.
The two-orbital ground states for an SU(2)-symmetric spin-1/2 system we constructed in Eq. (\ref{twoorbital})
are different from that predicted in Refs. \cite{kuklov2002,ashhab2003}
in which the fragmented ground states are produced in a cooling process that conserves the total spin $S$.
For an initial state at high temperature, we have $S\sim\sqrt{N}$.
If all atoms condense into the same orbital state, $S=N/2$.
Therefore, to be compatible with the requirement of the total spin conservation,
atoms should condense at least into two orbitals, resulting in
a fragmented ground state.
\section{Summary}
We have systematically classified ground states of
strong SO-coupled trapped spinor BECs,
including pseudo spin-1/2, spin-1 and spin-2 cases, based
on symmetry analysis. In accordance with breaking of simultaneous SO(2) spin-space rotation symmetry
in favor of discrete symmetries,
there emerge lattice phases showing stripe, triangular,
square, and kagome lattice structures on each spin component.
Imposing symmetries of $\mathcal{T}$, $\mathcal{C}_{nz}$ or $\mathcal{TC}_{nz}$
or combined them with the U(1) gauge transformation
on the order parameter, we predict several lattice phases, some of which are
found to be ground states of a trapped system.
For the spin-1/2 case, the ground states can be classified into
two classes: one breaks time-reversal symmetry
and the other is invariant under time reversal and shows
fragmented properties.
\section{Acknowledgement}
Z.F.X. acknowledges Shunsuke Furukawa for useful discussions,
Shohei Watabe and Nguyen Thanh Phuc for reading the manuscript.
This work was supported by KAKENHI 22340114 and 22740265, a Grant-in-Aid for Scientific Research on Innovation
Areas ``Topological Quantum Phenomena'' (KAKENHI 22103005),
a Global COE Program ``the Physical Sciences Frontier'',
the Photon Frontier Network Program, from MEXT of Japan,
NSFC (No.~91121005 and No.~11004116), and the
research program 2010THZO of Tsinghua University.
Z.F.X. acknowledges the support from JSPS (Grant No. 2301327).
Y.K. acknowledges the support from Inoue Foundation for Science.
|
{
"timestamp": "2012-09-13T02:01:15",
"yymm": "1203",
"arxiv_id": "1203.2005",
"language": "en",
"url": "https://arxiv.org/abs/1203.2005"
}
|
\section{Introduction}
Recently, two hidden-bottom charged meson resonances were observed
by the Belle Collaboration~\cite{Adachi:2011} as two narrow resonance
structures in the $\pi^{\pm}\US(nS)$ ($n=1,2,3$) and $\pi^{\pm}h_b(mP)$
$(m=1,2)$ mass spectra.
They are produced in association with a single
charged pion in $\US(5S)$ decays with the following values of
mass and width:
$M[\zb(1061)]=10608.4\pm 2.0$~MeV,
$\Gamma[\zb(10610)]=15.6\pm 2.5$~MeV,
$M[\zbp(10650)]=10653.2\pm 1.5$~ MeV,
$\Gamma[\zbp(10650)]=14.4\pm 3.2$~MeV.
Analyses of the charged pion angular distributions favor
the $I^G(J^P)=1^+(1^+)$ quantum numbers of the $Z$-states~\cite{Adachi:2011}.
Theoretical structure assignments for these hidden-bottom
meson resonances were proposed immediately after their
observation~\cite{Voloshin:2011qa}-\cite{Richard:2011}, mainly based
on molecular~\cite{Voloshin:2011qa} and
tetra-quark interpretations~\cite{Guo:2011gu,Richard:2011}
using the analogy to the charm sector. Also, in~\cite{Danilkin:2011sh}
the new resonances were identified as a hadro-quarkonium system
based on the channel coupling of light and heavy quarkonia to
intermediate open-flavor heavy-light mesons.
In this paper we analyze the two-body strong decays $\US(nS) \pi^+$
of $\zb$ and $\zbp$ using a phenomenological Lagrangian
approach developed in Refs.~\cite{Faessler:2007gv}-\cite{Dong:2010gu}
which is based on the compositeness
condition~\cite{Weinberg:1962hj,Efimov:1993ei}.
In particular, in~\cite{Faessler:2007gv}--\cite{Dong:2010gu}
recently observed unusual hadron states (like $D_{s0}^*(2317)$,
$D_{s1}(2460)$, $X(3872)$, $Y(3940)$, $Y(4140)$, $Z(4430)$,
$\Lambda_c(2940)$,
$\Sigma_c(2800)$) were analyzed within the structure assumption as hadronic
molecules. The compositeness condition implies that the renormalization
constant of the hadron wave function is set equal to zero or that the hadron
exists as a bound state of its constituents. It was originally applied to the
study of the deuteron as a bound state of proton and
neutron~\cite{Weinberg:1962hj} (see also Ref.~\cite{Dong:2008mt}
for a further application
of this approach to the case of the deuteron). Then it was extensively used
in low--energy hadron phenomenology as the master equation for the treatment
of mesons and baryons as bound states of light and heavy constituent quarks
(see e.g. Refs.~\cite {Efimov:1993ei,Anikin:1995cf,Dubnicka:2010kz}).
By constructing a phenomenological Lagrangian including the couplings
of the bound state to its constituents and the constituents to other final
state particles, we evaluated meson--loop diagrams which describe the
different decay modes of the molecular states
(see details in~\cite{Faessler:2007gv}).
In the present report we proceed as follows. In Sec.~II we briefly review the
basic ideas of our approach. We now consider the two new resonances
$\zb$ and $\zbp$ as the two molecular states of $\bar{B}B^*$ and
$\bar{B}^*B^*$. Then we proceed to estimate their strong two-body
decays $\zb\to \US(nS)+\pi^+$ and $\zbp\to \US(nS)+\pi^+$ where $n=1,2,3$
based on an phenomenological Lagrangian approach.
In Sec.~III we present our numerical results and
a short summary is given in Sec. ~IV.
\section{Phenomenological Lagrangian approach}
Here we briefly discuss the formalism for the study of the composite
(molecular) structure of the $\zb$ and $\zbp$ resonances.
In the following calculation we adopt the spin and parity quantum numbers
$J^P = 1^{+}$ for the two resonances $\zb$ and $\zbp$. We consider
these two new charged hidden-bottom meson resonances as a superposition
of molecular states of $\bar{B}B^{*}$ and $\bar{B}^*B^*$ as
\eq
|Z_b^+(10610)\ra &=&\frac{1}{\sqrt{2}} \Big| B^{*+}\bar{B}^0+\bar{B}^{*0}B^+
\Big\ra,
\nonumber \\
|Z_b^{+'}(10650)\ra &=& |B^{*+}\bar{B}^{*0}\ra.
\en
Our approach is based on an interaction Lagrangian describing
the coupling of the $\zb$ (or $\zbp$) to its constituents. The simplest
forms of such Lagrangians read
\eq\label{Lagr}
\hspace*{-1cm}
{\cal L}_{Z_b}(x)&=&\frac{g_{_{Z_b}}}{\sqrt{2}}\, M_{Z_b} \,
Z_b^{\mu}(x)\int d^4y \, \Phi_{Z_b}(y^2)\Big (B(x+y/2)
\bar{B}^*_{\mu}(x-y/2)\nonumber\\
&+&B^*_{\mu}(x+y/2)\bar{B}(x-y/2)\Big ), \\
\hspace*{-1cm}
{\cal L}_{Z_b'}(x)&=&\frac{g_{_{Z_b'}}}{\sqrt{2}} \,
i \epsilon_{\mu\nu\alpha\beta}
\partial^{\mu}Z_b^{'\nu}(x) \int d^4y \,
\Phi_{Z_b'}(y^2)B^{*\alpha}(x+y/2)\bar{B}^{*\beta}(x-y/2),
\en
where $y$ is a relative Jacobi coordinate,
$g_{_{Z_b}}$ and $g_{_{Z_b^\prime}}$ are the dimensionless
coupling constants of $\zb$ and $\zbp$ to the molecular $\bar{B}B^{*}$ and
$\bar{B}^*B^{*}$
components, respectively. Here $\Phi_{Z_b}(y^2)$ and $\Phi_{Z_b^\prime}(y^2)$
are correlation functions, which describe the distributions of
the constituent mesons in the bound states.
A basic requirement for the
choice of an explicit form of the correlation function $\Phi_H(y^2)$
($H = Z_b, Z_b^\prime$) is that
its Fourier transform vanishes sufficiently fast in the ultraviolet region
of Euclidean space to render the Feynman diagrams ultraviolet finite. We
adopt a Gaussian form for the correlation function. The Fourier transform of
this vertex function is given by
\eq\label{corr_fun}
\tilde\Phi_H(p_E^2/\Lambda^2) \doteq \exp( - p_E^2/\Lambda^2)\,,
\en
where $p_{E}$ is the Euclidean Jacobi momentum.
$\Lambda$ is a size parameter characterizing the distribution of the two
constituent mesons in the $\zb$ and $\zbp$ systems, which also leads to a
regularization of the ultraviolet divergences in the Feynman diagrams.
From our previous analyses of the strong two-body decays of the
$X, Y, Z$ meson resonances and of the $\Lambda_c(2940)$ and $\Sigma_c(2880)$
baryon states we deduced a value of $\Lambda \sim 1$~GeV~\cite{Dong:2009tg}.
For a very loosely bound system like the $X(3872)$ a size parameter of
$\Lambda \sim 0.5$~GeV~\cite{Dong:2009uf} is more suitable.
The coupling constants $g_{_{Z_b}}$ and $g_{_{Z_b^\prime}}$ are then
determined by the compositeness condition~\cite{Weinberg:1962hj,
Efimov:1993ei,Anikin:1995cf, Dong:2009tg,Faessler:2007gv}. It implies
that the renormalization constant of the hadron wave function is set equal
to zero with:
\eq\label{ZLc}
Z_H = 1 - \Sigma_H^\prime(M_H^2) = 0 \,.
\en
Here, $\Sigma_H^\prime$ is the derivative of the transverse part of the
mass operator $\Sigma_H^{\mu\nu}$ of the molecular states (see Fig.1),
which is defined as
\eq
\Sigma_H^{\mu\nu}(p) = g^{\mu\nu}_\perp \, \Sigma_H(p)
+ \frac{p^\mu p^\nu}{p^2}
\Sigma_H^L(p)\,, \quad
g^{\mu\nu}_\perp = g^{\mu\nu} - \frac{p^\mu p^\nu}{p^2} \,.
\en
The compositeness condition~(\ref{ZLc}) gives a constraint
on the choice of the free parameter~$\Lambda$.
Analytical expressions for the couplings $g_{_{Z_b}}$ and $g_{_{Z_b^\prime}}$
are given in Appendix A.
In the calculation the masses of $Z_b$ and $Z_b^\prime$
are expressed in terms of the constituent masses and the binding
energy $\epsilon$ (a variable quantity in our calculations):
\eq
M_{Z_b}=M_{B}+M_{B^*}-\epsilon\,, \quad
M_{Z_b^\prime}=2M_{B^*}-\epsilon\,.
\en
Here we assume bound states for the $Z_b$ and $Z_b^\prime$.
In the calculation of the two-body decays
$\zb\to \US(nS)+\pi^+$ and $\zbp\to \US(nS)+\pi^+$
we include the direct four-point interactions for the $BB^*\US\pi^+$ and
$B^*B^*\US\pi^+$ vertices.
The respective phenomenological Lagrangians take the form
\eq
{\cal L}_{BB^*\US\pi}(x)&=&g_{_{BB^*\US\pi}}\US_{\mu}(x)
\bar{B}^{*\mu}(x)\vec{\pi}(x)\cdot\vec{\tau}B(x) + \mathrm{H.c.}\,,
\label{BBSUPpi}\\
{\cal L}_{B^*B^*\US\pi}(x)&=&i\epsilon_{\mu\nu\alpha\beta}
\Big (g_{_{B^*B^*\US\pi}}\US^{\mu}(x)
\bar{B}^{*\beta}(x)\partial^{\nu}\vec{\pi}(x)
\cdot\vec{\tau}B^{*\alpha}(x)\nonumber\\
&+&f_{_{B^*B^*\US\pi}}\partial^{\nu}\US^{\mu}(x)
\bar{B}^{*\beta}(x)\vec{\pi}(x)
\cdot\vec{\tau}B^{*\alpha}(x)\Big ) \,. \label{BSBSUPpi}
\en
The four-particle coupling constants defined in Eqs.~(\ref{BBSUPpi}) and
(\ref{BSBSUPpi}) are effective, which also include off-shell effects. Such couplings
are obviously different from the strong couplings of molecular states to
their constituents which model their composite structure via vertex function
distibutions. We use effective Lagrangians (using both SU(4) and SU(5)
classification schemes) developed by Ko and collaborators~\cite{Lin:2000ke}
which worked phenomenologically quite successfully.
Corresponding diagrams contributing to the $\zb \to \US(nS)+\pi^+$
and $\zbp\to \US(nS)+\pi^+$ processes are shown in Fig.2.
\begin{figure}
\centering
\includegraphics [scale=0.5]{fig1a.eps}
\hspace{1cm}
\includegraphics [scale=0.5]{fig1b.eps}
\caption{Mass operators of $\zb$ and $\zbp$.}
\vspace*{-1.5cm}
\includegraphics [scale=0.6]{fig2a.eps}
\hspace*{.5cm}
\includegraphics [scale=0.6]{fig2b.eps}
\vspace*{1.cm}
\caption{Two-body decays $\zb \to \Upsilon(nS) + \pi$ and
$\zbp \to \Upsilon(nS) + \pi$.}
\end{figure}
Among the three couplings $g_{_{BB^*\US\pi}}$,
$g_{_{B^*B^*\US\pi}}$ and $f_{_{B^*B^*\US\pi}}$ we have the
relations~\cite{Lin:2000ke}
\eq\label{relations}
g_{_{BB^*\US\pi}} \ = \ \frac{g_{_{\US BB}} \, g_{_{B^*B\pi}}}{2\sqrt{2}}
\,,\quad\quad
g_{_{B^*B^*\US\pi }} \ = \ f_{_{B^*B^*\US\pi}}=\frac{g_{_{BB^*\US\pi}}}
{2\sqrt{M_BM_{B^*}}} \,.
\en
The hadronic couplings $g_{_{\US BB}}$ and $g_{_{B^*B\pi}}$
are defined as~\cite{Dong:2009yp,Dong:2009uf}:
\eq
{\cal L}_{B^\ast B \pi}(x) &=& \frac{g_{_{B^\ast B \pi}}}{\sqrt{2}} \,
\bar B^\ast_\mu(x) i\partial^\mu \vec{\pi}(x) \, \vec{\tau} \,
B(x) + {\rm H.c.} \,, \nonumber\\
{\cal L}_{\US BB}(x) &=& g_{_{\US BB}} \, \US_\mu(x)
\bar B(x) i \partial^\mu B(x) + {\rm H.c.}
\en
The coupling constant $g_{_{\US(nS)BB}}$ is given by
\eq\label{V_universality}
g_{_{\US(nS)BB}}= \frac{M_{\US(nS)}}{f_{\US(nS)}} \,,
\en
where $f_{\US(nS)}$ is
determined from the leptonic decays of the $\US(nS)$ states as
\eq
\Gamma \Big (\US(nS)\to e^+e^-\Big)=
\frac{4\pi\alpha_{_{\rm EM}}^2}{27}\frac{f^2_{\US(nS)}}
{M_{\US(nS)}}\,,
\en
where $\alpha_{_{\rm EM}} = 1/137.036$ is the fine-structure constant.
The relation (\ref{V_universality}) is analogue of the $\rho$-meson
universality
\eq
g_{_{\rho\pi\pi}} = \frac{M_\rho}{f_\rho} = \frac{1}{g_{\rho\gamma}}
\en
extended to the bottom sector in Ref.~\cite{Lin:2000ke}, where
$g_{\rho\gamma}$ is the $\rho\to\gamma$ transition coupling.
Here
For the last couplings we get
$f_{\US(1S)}=715.2$ MeV,
$f_{\US(2S)}=497.5$ MeV,
$f_{\US(3S)}=430.2$ MeV,
where we used the mass values $M_{\US(1s,2s,3s)}=9460.30\pm 0.26~$MeV,
$10023.26 \pm 0.31~$MeV and $10355.2 \pm 0.5~$MeV as well as the results
for the leptonic decay widths of the $\US(nS)$ states
\eq
\Gamma\Big (\US(1S)\to e^+e^-\Big )&=&1.340\pm 0.018~\mathrm{keV}\,,
\nonumber\\
\Gamma\Big (\US(2S)\to e^+e^-\Big )&=&0.612\pm 0.011~\mathrm{keV}\,,
\nonumber\\
\Gamma\Big (\US(3S)\to e^+e^-\Big )&=&0.443\pm 0.008~\mathrm{keV}\, .
\en
Note that we explicitly take into account the $M_{\US(nS)}$ dependence of
the $f_{\US(nS)}$ and $g_{_{\US BB}}$ couplings.
The coupling $g_{_{BB^*\pi}}$ can be related
to the effective coupling constant $\hat{g}=0.44\pm 0.03^{+0.01}_{-0.00}$
determined in a lattice calculation~\cite{Becirevic:2009yb}.
The relation is
\eq
g_{B^*B\pi}=\frac{2\hat{g}}{f_{\pi}}\sqrt{M_BM_{B^*}} \simeq 35.34,
\en
where $f_\pi \simeq 132$ MeV is the pion decay constant.
We should stress that the phenomenological Lagrangians developed
in Refs.~\cite{Lin:2000ke} were successfully applied to different
aspects of heavy flavor physics.
\section{Numerical results}
With the phenomenological Lagrangians introduced and discussed in Sec.~II
one can proceed to determine the widths of the two-body decays
$\zb\to \US(ns)+\pi^+$ and $\zbp\to \US(ns)+\pi^+$ with $n=1,2,3$
(see corresponding diagrams in Fig.2).
The corresponding decay widths are given by:
\eq
\Gamma_{\zb\to\US(nS)\pi^+}&\simeq&
\frac{g_{Z_b\Upsilon(nS)\pi}^2}{16\pi M_{Z_b}}
\lambda^{1/2}(M_{Z_b}^2, M_{\US(nS)}^2, M_{\pi}^2) \,, \nonumber\\
&&\\
\Gamma_{\zbp\to\US(nS)\pi^+}&\simeq&
\frac{g_{Z_b^\prime\Upsilon(nS)\pi}^2}{16 \pi M_{Z_b^\prime}}
\lambda^{1/2}(M_{Z_b^\prime}^2, M_{\US(nS)}^2, M_{\pi}^2) \,, \nonumber
\en
where $\lambda(x,y,z)=x^2+y^2+z^2-2xy-2xz-2yz$
is the K\"allen function.
The decay coupling constants
$g_{Z_b\US(nS)\pi}$ and $g_{Z_b^\prime\US(nS)\pi}$
involve the products
$g_{Z_b\US(nS)\pi} = g_{_{Z_b}} \, g_{_{BB^*\US(nS)\pi}} \, J_1$
and $g_{Z_b^\prime\US(nS)\pi} = g_{_{Z_b^\prime}} \,
g_{_{B^*B^*\US(nS)\pi}} \, M_{Z_b^\prime} \, J_2$ where the loop integrals
$J_1$ and $J_2$ are given in Appendix A.
For our numerical evaluation hadron masses are taken from the compilation of
the PDG~\cite{PDG:2010}. The only free parameter of our calculation is the
dimensional parameter $\Lambda$ entering in the correlation function of
Eq.~(\ref{corr_fun}). As mentioned before, the parameter
$\Lambda$ describes the distributions of $BB^*$ and $B^*B^*$ in the $\zb$
and $\zbp$ bound state systems, respectively.
Tables I and II contain
our estimates for the
decay widths of $\zb\to\US(nS)+\pi^+$ and $\zbp\to\US(nS)+\pi^+$.
We also indicate the values for the couplings $g_{_{Z_b}}$ and
$g_{_{Z_b^\prime}}$ as determined from the compositeness condition.
We find that the data on the strong $\US(nS)\pi$ decays of
$Z_b$ and $Z_b^\prime$ can be
qualitatively described for $\Lambda \simeq 0.5$ GeV. This value is close
to the one used for the molecular system of the $X(3872)$\cite{Dong:2009uf}.
This also means that the $Z_b$ and $Z_b^\prime$ states are considered as
extended molecular states.
For transparency we present the results for several choices of the size
parameter $\Lambda=0.4, 0.45, 0.5, 0.55$~GeV and the binding energy
$\epsilon$.
Note that an increase of $\Lambda$ leads to a larger decay width.
Although the dependence of the decay widths on the binding energy is rather
moderate, a quantitative prediction for these decays strongly depends on
the size of the system. The decay pattern of $Z_b$ and $Z_b^\prime$
to $\US(nS)+\pi^+$, that is the relative importance of the
$\US(nS)+\pi^+$ decay channels for $n=1,2,3$, can be reproduced and
could give further support for the molecular interpretation of these states.
One can see from Tables I and II
that the rates are increased by a factor 2-2.5 which means that the
amplitudes are roughly increased by a factor 1.4-1.6.
Latter value is consistent with a growth of the cutoff parameter of
$0.55/0.4=1.375$. In this respect the calculation is consistent and
we do not see any disagreement. It is also clear that any model based
on cutoff regularization has cutoff-dependent result.
Here the cutoff is related to the size of hadronic molecular compound.
As done in our previous analyses of other heavy hadron molecules
data help to do a fine tuning of the cutoff parameter which is specific
for a particular molecular state.
There is no a universal cutoff for the whole tower of possible hadronic
molecular states. Our previous analyses of molecular states consisting
of two heavy mesons (like the $X(3872)$ state) indicate that the cutoff
parameter in the vertex functions of $Z_b$ and $Z_b'$ states should be
around 0.5 GeV. This is the reason why we do predictions for the valus
of $\Lambda$ close to 0.5 GeV and are also waiting for more precise data
with smaller error bars. Please also note that the ratio of rates is
less dependent on the cutoff and hence a stabile prediction of the model.
Since data are given for the absolute rates we also choose to use this
presentation for our results and not the relative rates.
\begin{center}
{\bf Table I.} $Z_b^+\to \US(nS)+\pi^+$ decay properties.
\vspace*{.15cm}
\hspace*{-1.25cm}
\begin{tabular}{|c|c|c|c|c|}\hline
$\epsilon(\mathrm{MeV})$ &$g_{Z_b}$
&$\Gamma_{1S}(\mathrm{MeV})$
&$\Gamma_{2S}(\mathrm{MeV})$
&$\Gamma_{3S}(\mathrm{MeV})$\\ \hline
1 &3.3, 3.4, 3.6, 3.7 &11.6, 16.4, 22.3, 29.5 &13.7, 19.4, 26.4, 34.9 &7.2, 10.2, 13.9, 18.4\\
\hline
5 &4.0, 4.0, 4.1, 4.2 &11.0, 16.5, 22.4, 29.5 &13.0, 19.4, 26.3, 34.7 &6.7, 10.1, 13.6, 18.0\\
\hline
10 &5.0, 4.9, 4.8, 4.8 &11.7, 16.9, 22.7, 29.7 &13.7, 19.8, 26.6, 34.8 &7.0, 10.1, 13.5, 17.6\\
\hline
20 &7.2, 6.6, 6.3, 6.0 &13.3, 18.3, 24.0, 30.8 &15.4, 21.2, 27.8, 35.7 &7.4, 10.2, 13.4, 17.3\\
\hline
30 &9.4, 8.5, 7.9, 7.4 &14.5, 19.6, 25.4, 32.3 &16.7, 22.5, 29.2, 37.1 &7.7, 10.3, 13.4, 17.0\\
\hline
40 &11.7, 10.4, 9.5, 8.8&15.4, 20.4, 26.7, 33.7 &17.5, 23.2, 30.3, 38.3 &7.5, 10.0, 13.1, 16.5\\
\hline
50 &13.9, 12.3, 11.1, 10.2&15.9, 21.2, 27.6, 34.9&17.8, 23.9, 31.1, 39.2&7.1, 9.6, 12.5, 15.7\\
\hline
Exp. & &$22.9\pm 7.3\pm 2$ &$21.1\pm 4^{+2}_{-3}$ &$12.2\pm 1.7\pm 4$
\\ \hline
\end{tabular}
\end{center}
\vspace*{.15cm}
\begin{center}
{\bf Table II.} $\zbp \to \US(nS) + \pi^+$ decay properties.
\vspace*{.15cm}
\hspace*{-1.25cm}
\begin{tabular}{|c|c|c|c|c|c|}\hline
$\epsilon(\mathrm{MeV})$ &$g_{Z_b^\prime}$
&$\Gamma_{1S}(\mathrm{MeV})$
&$\Gamma_{2S}(\mathrm{MeV})$
&$\Gamma_{3S}(\mathrm{MeV})$\\ \hline
1 &3.2, 3.4, 3.6, 3.7 &12.0, 16.9, 23.0, 30.4 &14.7, 20.8, 28.3, 37.4 &9.0, 12.8, 17.4, 23.0\\
\hline
5 &3.9, 4.0, 4.1, 4.2 &12.1, 17.0, 23.0, 30.3 &14.9, 20.8, 28.2, 37.2 &9.0, 12.6, 17.1, 22.6\\
\hline
10 &5.0, 4.8, 4.8, 4.7 &12.7, 17.4, 23.4, 30.6 &15.4, 21.3, 28.5, 37.3 &9.2, 12.7, 17.1, 22.3\\
\hline
20 &7.2, 6.6, 6.3, 6.0 &14.0, 18.8, 24.6, 31.7 &16.9, 22.7, 29.8, 39.3 &9.8, 13.2, 17.3, 22.3\\
\hline
30 &9.4, 8.5, 7.8, 7.3 &15.1, 20.1, 26.1, 33.1 &18.1, 24.1, 31.3, 39.7 &10.2, 13.5, 17.6, 22.3\\
\hline
40 &11.7, 10.4, 9.4, 8.7&15.8, 21.1, 27.3, 34.5 &18.9, 25.1, 32.4, 41.0 &10.2, 13.6, 17.6, 22.2\\
\hline
50 &13.9, 12.3, 11.0, 10.1&16.2, 21.7, 28.2, 35.6&19.1, 25.5, 33.2, 41.9&9.9, 13.3, 17.3, 21.8\\
\hline
Exp. & &$12\pm 10\pm 3$ &$16.4\pm 3.6^{+4}_{-6}$ &$10.9\pm 2.6^{+4}_{-2}$\\
\hline
\end{tabular}
\end{center}
\section{Summary}
To summarize, we have pursued a hadronic molecular interpretation for the
two recently observed hidden-bottom charged mesons $\zb$ and $\zbp$.
In our calculation we have used the spin-parity assignment $J^P=1^+$ for
the two resonances, which is currently favored
by the experimental decay distributions.
We have studied the consequences for their two-body decays
$\zb \to \US(nS)+\pi^+$ and $\zbp \to \US(nS)+\pi^+$
within a phenomenological Lagrangian approach.
The calculated results for the decay widths (see Tables I and II)
are of the order of MeV and for the most part
qualitatively consistent with the numbers deduced by the Belle collaboration.
Especially the experimental results for $\zb(\zbp) \to \US(nS)+\pi^+$ can
be reproduced taking a value for the free size parameter $\Lambda $
near 0.5 GeV.
\section*{Acknowledgments}
This work is supported by the DFG under Contract No. LY 114/2-1,
National Sciences Foundations of China No.10975146 and 11035006,
Federal Targeted Program "Scientific and scientific-pedagogical personnel
of innovative Russia" Contract No.02.740.11.0238.
One of us (YBD) thanks the Institute of Theoretical Physics,
University of T\"ubingen for the warm hospitality and thanks the support
from the Alexander von Humboldt Foundation.
|
{
"timestamp": "2012-12-03T02:01:43",
"yymm": "1203",
"arxiv_id": "1203.1894",
"language": "en",
"url": "https://arxiv.org/abs/1203.1894"
}
|
\section{Introduction}
There are many problems in which one seeks to develop predictive models
to map between a set of predictor variables and an outcome. Statistical
tools such as multiple regression or neural networks provide mature
methods for computing model parameters when the set of predictive
covariates and the model structure are pre-specified. Furthermore,
recent research is providing new tools for inferring the structural
form of non-linear predictive models, given good input and output
data \cite{Bongard:2007}. However, the task of choosing which potentially
predictive variables to study is largely a qualitative task that requires
substantial domain expertise. For example, a survey designer must
have domain expertise to choose questions that will identify predictive
covariates. An engineer must develop substantial familiarity with
a design in order to determine which variables can be systematically
adjusted in order to optimize performance.
The need for the involvement of domain experts can become a bottleneck to new insights.
However, if the wisdom of crowds could be harnessed to produce insight into difficult problems, one might see exponential rises in the discovery of the causal factors of behavioral outcomes, mirroring the exponential growth on other online collaborative communities.
Thus, the goal of this research was to test an alternative approach to modeling in which the wisdom of crowds is harnessed to both propose potentially predictive variables to study by asking questions, and respond to those questions, in order to develop a predictive model.
\subsection*{Machine science}
Machine science \cite{Evans:2010} is a growing trend that attempts
to automate as many aspects of the scientific method as possible.
Automated generation of models from data has a long history, but recently
robot scientists have been demonstrated that can physically carry
out experiments \cite{King:04,King:09} as well as algorithms that
cycle through hypothesis generation, experimental design, experiment
execution, and hypothesis refutation \cite{Bongard:2006,Bongard:2007}.
However one aspect of the scientific method that has not yet yielded
to automation is the selection of variables for which data should
be collected to evaluate hypotheses. In the case of a prediction problem,
machine science is not yet able to select the independent variables
that might predict an outcome of interest, and for which data collection
is required.
This paper introduces, for the first time, a method by which non domain
experts can be motivated to formulate independent variables as well
as populate enough of these variables for successful modeling. In
short, this is accomplished as follows. Users arrive at a website
in which a behavioral outcome (such as household electricity usage
or body mass index, BMI) is to be modeled. Users provide their own
outcome (such as their own BMI) and then answer questions that may
be predictive of that outcome (such as `how often per week do you
exercise'). Periodically, models are constructed against the growing
data set that predict each user's behavioral outcome. Users may also
pose their own questions that, when answered by other users, become
new independent variables in the modeling process. In essence, the
task of discovering and populating predictive independent variables
is outsourced to the user community.
\subsection*{Crowdsourcing}
The rapid growth in user-generated content on the
Internet is an example of how bottom-up interactions can, under some
circumstances, effectively solve problems that previously required
explicit management by teams of experts \cite{Giles:2005}. Harnessing
the experience and effort of large numbers of individuals is frequently
known as {}``crowdsourcing\textquotedblright{} and has been used
effectively in a number of research and commercial applications \cite{Brabham:2008}.
For an example of how crowdsourcing can be useful, consider Amazon's
Mechanical Turk. In this crowdsourcing tool a human describes a {}``Human
Intelligence Task\textquotedblright{} such as characterizing data
\cite{Sorokin:2008}, transcribing spoken language \cite{Marge:2010},
or creating data visualizations \cite{Kong:2010}. By involving large
groups of humans in many locations it is possible to complete tasks
that are difficult to accomplish with computers alone, and would be
prohibitively expensive to accomplish through traditional expert-driven
processes \cite{Kittur:2008}.
Although arguably not strictly a crowdsourced system, the rapid rise of Wikipedia illustrates how online collaboration can be used to solve difficult problems (the creation of an encyclopedia) without
financial incentives. Ref. \cite{Wightman:2010} reviews several crowdsourcing
tools and argues that direct motivation tasks (tasks in which users
are motivated to perform the task because they find it useful, rather
than for financial motivation) can produce results that are superior
to financially motivated tasks. Similarly, ref. \cite{Wightman:2010}
reports that competition is useful in improving performance on a task
with either direct or indirect motivation.
This paper reports on two tasks with direct motivation:
for the household energy usage task, users are motivated to understand their home energy usage as a means to improve their energy efficiency;
for the body mass index task, users are motivated to understand their lifestyle choices in order to approach a healthy body weight.
Both instantiations include an element of competition by allowing participants to see how they compare with other participants and by ranking the predictive quality of questions that participants provide.
There is substantial evidence in the literature and commercial applications
that laypersons are more willing to respond to surveys and queries
from peers than from authority figures or organizations. For example
within the largest online collaborative project, Wikipedia, article writers
often broadcast a call for specialists to fill in details on a particular
article. The response rates to such peer-generated requests are enormous,
and have led to the overwhelming success of this particular project.
In the open source community, open source software that crashes automatically
generates a debug request from the user. Microsoft adopted this practice
but has found that users tend not to respond to these requests, while
responses to open source crashes are substantially higher \cite{Fitzgerald:2006,Howe:2009}.
Medpedia, a Wikipedia-styled crowdsourced system, increasingly hosts
queries from users as to what combinations of medications work well
for users on similar medication cocktails. The combinatorial explosion
of such cocktails is making it increasingly difficult for health providers
to locate such similar patients for comparison without recourse to
these online tools.
Collaborative systems are generally more scalable than top-down systems.
Wikipedia is now orders-of-magnitude larger than Encyclopedia Britannica.
The climateprediction.net project has produced over 124 million hours
of climate simulation, which compares favorably with the amount of
simulation time produced by supercomputer simulations. User-generated
news content sites often host as many or more readers than conventional
news outlets \cite{Thurman:2008}. Finally, many of the most recent
and most successful crowdsourced systems derive their success from
their viral \cite{DiBona:2005,Leskovec:2007} nature: they are designed
such that selective forces exerted by users lead to an exponential
increase in content, automated elimination of inferior content, and
automated propagation of quality content \cite{Lerman:2006}.
Citizen science \cite{Anderson:2002,Cohn:2008,Silvertown:2009} platforms
are a class of crowdsourcing systems that include non-scientists in
the scientific process. The hope is that participants in such systems
are motivated ideologically, as their contributions forward what they
perceive as a worthy cause. In most citizen science platforms user
contributions are `passive': they contribute computational but not
cognitive resources \cite{Anderson:2002,Beberg:2009}. Some platforms
allow users to actively participate by searching for items of interest
\cite{Lintott:2008} or solve problems through a game interface \cite{Cooper:2010}.
The system proposed here falls into this latter category: users are
challenged to pose new questions that, when answered by enough of
their peers, can be used by a model to predict the outcome of interest.
Finally, problem solving through crowdsourcing can produce novel,
creative solutions that are substantially different from those produced
by experts. An iterative, crowdsourced poem translation task produced
translations that were both surprising and preferable to expert translations
\cite{Kittur:2010}. We conjecture that crowdsourcing the selection
of predictive variables can reveal creative, unexpected predictors
of behavioral outcomes. For problems in which behavioral change is
desirable (such as is the case with obesity or energy efficiency),
identifying new, unexpected predictors of the outcome may be useful
in identifying relatively easy ways for individuals to change their
outcomes.
\section{Methodology}
The system described here wraps a human behavior modeling paradigm
in cyberinfrastructure such that: (1) the investigator defines some
human behavior-based outcome that is to be modeled; (2) data is collected
from human volunteers; (3) models are continually generated automatically;
and (4) the volunteers are motivated to propose new independent variables.
Fig. \ref{FigOverview} illustrates how the investigator, participant
group and modeling engine work together to produce predictive models
of the outcome of interest. The investigator begins by constructing
a web site and defining the human behavior outcome to be modeled (Fig.
\ref{FigOverview}a). In this paper a financial and health outcome
were investigated: the monthly electric energy consumption of an individual
homeowner (Sect. \ref{sectEnergy}), and their body mass index (Sect.
\ref{sectBMI}). The investigator then initializes the site by seeding
it with a small set (one or two) of questions known to correlate with
the outcome of interest (Fig. \ref{FigOverview}b). For example, based
on the suspected link between fast food consumption and obesity \cite{Bowman:2004,Currie:2010},
we seeded the BMI website with the question {}``\textit{How many
times a week do you eat fast food?}''
\begin{figure}[!t]
\centering \includegraphics[width=1.0\columnwidth]{Overview_BMI}
\caption{ \label{FigOverview} \textbf{Overview of the system.}
The investigator (\textbf{a-f}) is responsible for initially creating the web platform, and seeding it with a starting question. Then, as the experiment runs they filter new survey questions generated by the users. Users (\textbf{g-l}) may elect to answer as-yet unanswered survey questions or pose some of their own. The modeling engine (\textbf{m-p}) continually generates predictive models using the survey questions as candidate predictors of the outcome and users' responses as the training data.}
\end{figure}
\begin{figure*}[!t]
\centering \includegraphics[width=1\textwidth]{Screenshot_BMI}
\caption{\label{figScreenBMI}Screenshot from the Body Mass Index (BMI) website
as seen by a user who has responded to all of the available questions.
The user has the option to change their response to a previous question,
pose a new question, or remove themselves automatically from the study.}
\end{figure*}
Users who visit the site first provide their individual value for
the outcome of interest, such as their own BMI (Fig. \ref{FigOverview}g).
Users may then respond to questions found on the site (Fig. \ref{FigOverview}h,i,j).
Their answers are stored in a common data set and made available to
the modeling engine. Periodically the modeling engine wakes up (Fig.
\ref{FigOverview}m) and constructs a matrix $\mathbf{A}\in\Re^{n\times k}$
and outcome vector $\mathbf{b}$ of length $n$ from the collective
responses of $n$ users to $k$ questions (Fig. \ref{FigOverview}n).
Each element $a_{ij}$ in $\mathbf{A}$ indicates the response of
user $i$ to question $j$, and each element $b_{i}$ in $\mathbf{b}$
indicates the outcome of interest as entered by user $i$. In the
work reported here linear regression was used to construct models
of the outcome (Fig. \ref{FigOverview}o), but any model form could
be employed. The modeling process outputs a vector $\mathbf{c}$ of
length $k+1$ that contains the model parameters. It also outputs
a vector $\mathbf{d}$ of length $k$ that stores the predictive power
of each question: $d_{j}$ stores the $r^{2}$ value obtained by regressing
only on column $j$ of $\mathbf{A}$ against the response vector $\mathbf{b}$.
These two outputs are then placed in the data store (Fig. \ref{FigOverview}p).
At any time a user may elect to pose a question of their own devising
(Fig. \ref{FigOverview}k,l). Users could pose questions that required
a yes/no response, a five-level Likert rating, or a number. Users
were not constrained in what kinds of questions to pose. However,
once posed, the question was filtered by the investigator as to its
suitability (Fig. \ref{FigOverview}d). A question was deemed unsuitable
if any of the following conditions were met: (1) the question revealed
the identity of its author (e.g. {}``\textit{Hi, I am John Doe. I
would like to know if...}'') thereby contravening the Institutional
Review Board approval for these experiments; (2) the question contained
profanity or hateful text; (3) the question was inappropriately correlated
with the outcome (e.g. {}``\textit{What is your BMI?}''). If the
question was deemed suitable it was added to the pool of questions
available on the site (Fig. \ref{FigOverview}e); otherwise the question
was discarded (Fig. \ref{FigOverview}f).
Each time a user responded to a question, they were shown a new, unanswered
question as well as additional data devised to maintain interest in
the site and increase their participation in the experiment. Once
a user had answered all available questions, they were shown a listing
of the questions, their responses, and contextual information to indicate
how their responses compared to those of their peers. Fig. \ref{figScreenBMI}
shows the listing that was shown to those users who participated in
the BMI site; the individual elements are explained in more detail
in Sect. \ref{sectBMI}.
The most important datum shown to each user after responding to each
question was the value of their actual outcome as they entered it
($b_{i}$) as well as their outcome as predicted by the current model
($\hat{b}_{i}$). Fig. \ref{figScreenBMI} illustrates that visitors
to the BMI site were shown their actual BMI (as entered by them) and
their predicted BMI. The models were able to predict each user's outcome
before they had responded to every question by substituting in missing
values. Thus after each response from a user
\begin{equation}
\hat{b}_{i}=c_{0}+c_{1}a_{i1}+c_{2}a_{i2}+...+c_{k}a_{ik}+\epsilon_{i}\label{eq:Regression-model}
\end{equation}
where $a_{ij}=0$ if user $i$ has not yet responded to question
$j$ and $a_{ij}$ is set to the user's response otherwise.
\section{Energy efficiency instantiation and results\label{sectEnergy}}
In the first instantiation of this concept, we developed a web-based
social network to model residential electric energy consumption. Because
of policy efforts to increase energy efficiency, many are working
to provide consumers with better information about their energy consumption.
Research on consumer perception of energy efficiency indicates that
electricity customers often misjudge the relative importance of various
activities and devices to reducing energy consumption \cite{Attaria:2010}.
To provide customers with better information, numerous expert-driven
web-based tools have been deployed \cite{ms-hohm,Mills:2008,Allcott:2011}.
In some cases these tools use social pressure as a means of improving
energy efficiency \cite{Petersen:2007,Kaufman:2009}, however the
feedback provided to customers typically comes from a central authority
(i.e., top-down feedback) and research on risk perception \cite{Slovic:1999}
indicates that the public is often skeptical of expert opinions. A
recent industry study \cite{Guthridge:2010} indicates that customers
are notably skeptical of large online service providers (e.g., Google,
Microsoft) and (to a lesser extent) electric utilities as providers
of unbiased information about energy conservation. Therefore, information
generated largely by energy consumers themselves, in a bottom-up fashion,
may have value in terms of motivating energy efficient behavior.
Thus motivated, we designed the {}``EnergyMinder'' website to predict
and provide feedback about monthly household (residential) electricity
consumption. Participants were invited to join the site through notices
in university e-mail networks, a university news letter, and reddit,
a user-generated content news site. The site was launched in July
of 2009, and gradually accumulated a total of 58 registered users
by December of 2009. The site consisted of a simple login page and
five simple, interactive pages. The\emph{ Home Page} (after login)
contained a simple to-do list pointing users to tasks on the site,
such as, enter bill data, answer questions, check their energy efficiency
ranking, etc. The \emph{Energy Input Page} showed a time series trend
of the consumer's monthly electricity consumption and asked the user
to enter the kilowatt hours (kWh) consumed for recent months. This
value became the output variable ($b_{i}$) in the regression model
(Eq. \ref{eq:Regression-model}) for a particular month. The \emph{Ask-A-Question
Page} allowed users to ask questions of the group, such as {}``How
many pets do you have?'' (Question 10, Table \ref{tab:EM-Questions}).
When typing in a new question, users were instructed to specify the
type of answer expected (numeric, yes/no, agree/disagree) and to provide
their own response to the question. The \emph{Answer Page} asked participants
to respond to questions, and provided them with information about
each answered question including the distribution of answers within
the social network. Finally, a \emph{Ranking Page} showed users their
energy consumption, relative to that of others in the group. In addition
the Ranking Page reported the predictive power (the percentage of
explained variance) for each statistically significant question/factor.
This final page was intended to provide information to participants
that might help them in choosing behaviors that would reduce electricity
consumption.
\begin{table*}
\caption{\label{tab:EM-Questions}Questions entered into the EnergyMinder web
site. }
\begin{tabular}{clccccccc}
\hline
& & & \# of & answers & { Model 1{**}} & & { Model 2{**}} & \tabularnewline
& { Question} & { Type} & answers & { in $G$ } & { $c_{i}$} & { $P$ } & { $c_{i}$} & { $P$}\tabularnewline
\hline
{ 1.} & { What is the square footage of your house?{*}} & { numeric} & { 45} & { 22} & { 0} & { 0.52} & { -} & { -}\tabularnewline
{ 2.} & { How many children do you live with?{*}} & { numeric} & { 41} & { 22} & { 109} & { 0.47} & { -} & { -}\tabularnewline
{ 3.} & { How many adults do you live with?} & { numeric} & { 43} & { 22} & { 303} & { 0.03} & { 297} & {{} 0.01 }\tabularnewline
{ 4.} & { How many south facing windows do you have?} & { numeric} & { 42} & { 22} & { -11} & { 0.77} & { -} & { -}\tabularnewline
{ 5.} & { Do you have an electric clothes dryer?} & { yes/no} & { 35} & { 19} & { 430} & { 0.23} & { 240} & {{} 0.28 }\tabularnewline
{ 6.} & { Do you have an electric water heater?} & { yes/no} & { 33} & { 18} & { -577} & { 0.04} & { -535} & {{} 0.01 }\tabularnewline
{ 7.} & { Do you have gas heating?} & { yes/no} & { 34} & { 18} & { 188} & { 0.44} & { -} & { -}\tabularnewline
{ 8.} & { Do you have geothermal heating?} & { yes/no} & { 16} & { 10} & { -} & { -} & { -} & { -}\tabularnewline
{ 9.} & { How many adults are typically home throughout the day?} & { numeric} & { 17} & { 10} & { -} & { -} & { -} & { -}\tabularnewline
{ 10.} & { How many pets do you have?} & { numeric} & { 15} & { 9} & { -} & { -} & { -} & { -}\tabularnewline
\hline
& { $r^{2}$ value for predictive models} & & & & { 0.63} & & { 0.57} & \tabularnewline
\hline
\end{tabular}
{ {*} Questions 1 and 2 were seed questions placed on
the site by the investigators.}{ \par}
{ {*}{*} In Model 1 and Model 2, $c_{i}$ is the parameter
estimate ($\mbox{kWh}\cdot\mbox{month}^{-1}\cdot\mbox{unit}^{-1})$
and $P$ is the significance level of the parameter estimate.}
\end{table*}
\begin{figure}
\includegraphics[width=1\columnwidth]{eminder-fig}
\caption{\label{fig:e-minder}\textbf{EnergyMinder Question Statistics}. Panel
(\textbf{a}) shows the $r^{2}$ value for each question as numbered in table
\ref{tab:EM-Questions}. (\textbf{b}) shows that there is a mild correlation
between the response rate and the $r^{2}$ values. (\textbf{c}) shows the questions
sorted by their $r^{2}$ value, and (\textbf{d}) shows the number of responses
for each question, sorted by the number of responses.}
\end{figure}
In total the site attracted 58 participants, of whom 46 answered one
or more questions, and 33 (57\%) provided energy consumption data.
Eight new questions were generated by the group, after the seed questions
($Q_{1}$ and $Q_{2}$ in Table \ref{tab:EM-Questions}) were placed
there by the investigators. The fact that only about half of the participants
provided energy data was most likely due to the effort associated
with finding one or more electricity bills and entering data into
the site. This low response rate emphasized that the utility of this
approach depends highly on the ease with which the user can access
the outcome data.
Despite the small sample size, this initial trial resulted in a statistically
significant predictive model, and provided insight into the nature
of the method. Of the 33 participants, 24 provided data for the months
of June, July or August. Because this was the largest period for which
common data were available, the mean outcome for these three months
was used as the outcome variable $b_{i}$. One participant reported
kWh values that were far outside of the mean (46,575 kWh per month)
and one did not answer any questions. These two data sets were discarded
as outliers. The $N=22$ that remained comprised the sample-set used
to produce the results that follow.
Table \ref{tab:EM-Questions} shows results from two predictive models.
Model 1 included all questions that had 18 or more answers ($Q_{1}$-$Q_{7}$).
The total explained variance for Model 1 was $r^{2}=0.63$. Model
1 indicated that the number of adults in the home ($Q_{3}$) significantly
increased monthly electricity consumption ($P<0.05$) and the ownership
of a natural gas hot water heater ($Q_{6}$) significantly decreased
electricity consumption ($P<0.05$). Note that this second result
is not consistent with the fact that owning an electric hot water
heater increases electricity consumption. It appears either that this
correlation was due to chance, or that ownership of a gas hot water
heater correlates to some other factor, such as (for example) home
ownership. Model 2 tested the removal of the least significant predictors,
and included only $Q_{3}$, $Q_{5}$, and $Q_{6}$. Model 2 showed
the same pair of statistically significant predictors ($Q_{3}$ and
$Q_{6}$).
Figure \ref{fig:e-minder} shows the relative predictive power of
the 10 questions. The results show that the most highly correlated
factors ($Q_{3}$, $Q_{5}$, and $Q_{6}$) were posed after the initial
two seed questions (Fig. \ref{fig:e-minder}a) and a weak correlation between
the response rate and the $r^{2}$ values, indicating that more answers
to questions would have likely produced improved results.
Panels (c) and (d) show the distributions of $r^2$ values and the number of responses, to facilitate comparison with the BMI results (Fig. \ref{BMI_Fig3}).
While the small sample size in this study limits the generality of
these results, this initial trial provided useful information about
the crowdsourced modeling approach. Firstly, we found that participants
were reluctant or unable to provide accurate outcome data due to the
challenge of finding one's electric bills. Our second experiment corrects
this problem by focusing on an outcome that is readily accessible
to the general public. Secondly, we found that participants were quite
willing to answer questions posed by others in the group. Questions
1-4 were answered by over 70\% of participants. This indicated that
it is possible to produce user-generated questions and answers, and
that a trial with a larger sample size might provide more valuable
insight. Finally, questions that were posed early in the trial gained
a higher response rate, largely because many users did not return
to the site after one or two visits. This emphasizes the importance
of attracting users back to the site to answer questions in order
to produce a statistically useful model.
\section{Body Mass Index instantiation and results\label{sectBMI}}
In order to test this approach with an outcome that was more readily
available to participants a second website was deployed in which models
attempted to predict the body mass index of each participant. Body
mass index (BMI) is calculated as mass(kg) / (height(m))$^{2}$ and,
although it is known to have several limitations \cite{Romero:2008},
is still the most common measure for determining a patient's level
of obesity. Each user's BMI could readily be calculated as all users
know and are thus able to immediately enter their height and weight.
A second motivator for investigating this behavioral outcome is that
obesity has been cited \cite{Barness:2007} as one of the major global
public health challenges to date, it is known to have myriad causes
\cite{Parsons:1999,Wang:2007}, and people with extreme BMI values
are likely to have intuitions as to why they deviate so far from the
norm.
\begin{table*}
\begin{centering}
\caption{\label{tableBMI}Listing of the 20 most predictive questions from
the BMI site.}
\par\end{centering}
\centering{}%
\begin{tabular}{clcc}
\hline
Index & Question & $r^{2}$ & Responses \tabularnewline
\hline
1 & Do you think of yourself as overweight? & 0.5524 & 43 \tabularnewline
2 & How often do you masturbate a month? & 0.3887 & 32 \tabularnewline
3 & What percentage of your job involves sitting? & 0.3369 & 57 \tabularnewline
4 & How many nights a week do you have a meal after midnight? & 0.2670 & 67 \tabularnewline
5 & You would consider your partner/boyfriend/girlfriend/spouse etc to
be overweight? & 0.2655 & 24 \tabularnewline
6 & How many, if any, of your parents are obese? & 0.2311 & 57 \tabularnewline
7 & Are you male? & 0.2212 & 32 \tabularnewline
8 & I am happy with my life & 0.2062 & 31 \tabularnewline
9 & How many times do you cook dinner in an average week? & 0.2005 & 44 \tabularnewline
10 & How many miles do you run a week? & 0.1865 & 28 \tabularnewline
11 & Do you have a college degree? & 0.1699 & 12 \tabularnewline
12 & Do you have a Ph.D. & 0.1699 & 12 \tabularnewline
13 & Would you describe yourself as an emotional person? & 0.1648 & 30 \tabularnewline
14 & How often do you eat (meals + snacks) during a day & 0.1491 & 33 \tabularnewline
15 & How many hours do you work per week? & 0.1478 & 46 \tabularnewline
16 & Do you practice a martial art? & 0.1450 & 31 \tabularnewline
17 & What is your income? & 0.1419 & 55 \tabularnewline
18 & I was popular in high school & 0.1386 & 31 \tabularnewline
19 & Do you ride a bike to work? & 0.1383 & 64 \tabularnewline
20 & What hour expressed in 1-24 on average do you eat your last meal before
going to bed? & 0.1364 & 30 \tabularnewline
\hline
\end{tabular}
\end{table*}
Participants arriving for the first time at the BMI site were asked
to enter their height and weight in feet, inches and pounds respectively,
as most of the visitors to the site resided in the U.S. Participants
were then free to respond to and pose new questions.
In order to further motivate the participants, in addition to displaying
their predicted outcome, users were also shown how their responses
compared to two peer groups. For each user the peer groups were constructed
as follows. The first peer group was composed of 10 other users who
had BMI values as close to but below that of the user; the second
group was composed of 10 other users who had BMI values as close to
but above that of the user. If $N<10$ users could be found
the peer group was composed of those $N$ users. The average BMI for
each of the two peer groups, as well as the user's own BMI, were displayed
(see Fig. \ref{figScreenBMI}). Also, the responses to each question,
within each peer group, were averaged and shown alongside the user's
response to that question. Finally, the `predictive power' of each
question was shown. Predictive power was set equal to the $r^{2}$
obtained when the responses to that question alone were regressed
against the outcome.
The peer group data were meant to help users compare how their lifestyle
choices measured up to their most similar peers who were slightly
more healthy than themselves, and slightly less healthy than themselves.
This approach in effect provides individualized suggestions to each
user as to how slight changes in lifestyle choices may lead to improvements
in the health indicator being measured. Presenting the user with the
predictive power of each question was designed to help them learn
what questions tend to be predictive, and thus motivate them to formulate
new or better questions that might be even more predictive. For example
one user posed the question {}``\emph{How many, if any, of your parents
are obese?}''. Another user may realize that the `predictive power'
of this question (which achieved an $r^{2}$ in the actual experiment
of 0.23 and became the sixth-most predictive question out of a total
of 57) may be due to it serving as an indirect measure of the hereditary
component of obesity. This may cause the user to pose a new question
better tailored to eliciting this information, such as {}``\emph{How
many, if any, of your }\textbf{\emph{biological}}\emph{ parents are
obese?}'' (a question of this form was not posed during the actual
experiment).
The BMI site went live at 3:00pm EST on Friday, November 12, 2010,
stayed live for slightly less than a week, and was discontinued at
10:20am EST on Thursday, November 18, 2010. During that time it attracted
64 users who supplied at least one response. Those users proposed
56 questions (in addition to the original seed question), and together
provided 2021 responses to those questions.
Users were recruited from reddit.com and the social networks of the
principal investigators. Fig. \ref{BMI_Fig1}a shows an initial burst
of new users followed by a plateau during the weekend, and then a
steady rise thereafter until the termination of the experiment. Fig.
\ref{BMI_Fig1}b shows a similar, initially rapid increase in the
number of questions, and no significant increase until one user submits
8 new questions on day 6. Fig. \ref{BMI_Fig1}c shows a relatively
steady rise in the number of responses collected per day. This can
be explained by the fact that although fewer users visit the site
from the third day onward, there are more questions available when
they do and thus, on average, more responses are supplied by later
users than earlier users.
\begin{figure}[!t]
\centering \includegraphics[width=1.0\columnwidth]{BMI_Fig1}
\caption{ \label{BMI_Fig1} \textbf{User behavior on the BMI site.} The BMI
site was maintained for slightly less than seven days. During that
time it attracted 64 users (\textbf{(a)}) who together posted a total
of 57 questions \textbf{(b)} and 2021 responses to those questions
\textbf{(c)}. Every five minutes a regression model was constructed
against the site's data: The quality of these models are shown as
a function of their $R^{2}$ value \textbf{(f)}. }
\end{figure}
This increase is supplemented by a few early users who return to the
site and respond to new questions, as shown in Fig. \ref{BMI_Fig2}.
It shows that of the 100 users who registered, only 57 supplied at
least one response. The triangular form of the matrix is due to the
fact that for the majority of users, they only visited the site once
and answered the questions that were available at that time. This
led to a situation in which questions posted early received disproportionally
more responses than those questions posted later.
\begin{figure}[!t]
\centering \includegraphics[width=1.0\columnwidth]{BMI_Fig2}
\caption{ \label{BMI_Fig2} \textbf{Participation Rate by User of the BMI site.}
Each row corresponds to a user of the BMI site, sorted by time of registration.
Each column corresponds to one of the questions, sorted by time of posting.
A black pixel at row $i$ column $j$ indicates that user $i$ responded to question $j$; a white pixel indicates they did not.}
\end{figure}
For the first several hours of the experiment the modeling engine
(Fig. \ref{FigOverview}m-p) was run once every minute. At 5:30pm
on November 12 the modeling engine was set to run once every five
minutes. With the decrease in site activity the modeling engine was
set to run once an hour starting at 2:20pm on November 16 until the
termination of the experiment. Fig. \ref{BMI_Fig1}d reports the $r^{2}$
value of the regression models as the experiment proceeded. During
the first few hours of the experiment when there were more users than
questions (see Fig. \ref{BMI_Fig1}a,b), the early models had an $r^{2}$
near 1.0, suggesting that overfitting of the data was occurring. However
at the termination of the experiment when there were more users (64)
than questions (57)---and many users had not responded to those questions---the
models were still performing well with an $r^{2}$ near 0.9. There
is still a possibility though that the models overfit the data as
the site was not instrumented with the ability to create a testing
set composed of users whose responses were not regressed against.
\begin{figure}[!t]
\centering \includegraphics[width=1.0\columnwidth]{BMI_Fig3}
\caption{\label{BMI_Fig3} \textbf{BMI Question Statistics.}
(\textbf{a,b}): No relationship was found between questions' time of posting, response rate or predictive power.
However a power law relationship was discovered among questions' predictive power (\textbf{c,d}) but not for their response rate (\textbf{e,f}).
}
\end{figure}
Fig. \ref{BMI_Fig3} reports statistics about the user-posed questions.
Fig. \ref{BMI_Fig3}a shows that there is no correlation between when
a question was posed and how predictive it became: the second- and
fifth-most predictive question were posed as the 35th and 42nd question,
respectively. Similarly, Fig. \ref{BMI_Fig3}b reports the lack of
correlation between the number of responses a question receives and
its final predictive power. Although a slight positive correlation
may exist, several of the most predictive questions (including the
second- and fifth-most) received less than half of all possible responses.
Fig. \ref{BMI_Fig3}c reports the questions sorted in order of decreasing
$r^{2}$, and reveals that this distribution has a long tail: a large
number of questions have low, but non-zero $r^{2}$ when regressed
alone against the outcome. This distribution is replotted in Fig.
\ref{BMI_Fig3}c on a log-log scale. Linear regression was performed
on the 20 most predictive questions (indicated by the line), and the resulting fit was found to be highly correlated, with $r^{2}=0.994$. This finding
suggests that a power law relationship exists among these predictive
questions%
\footnote{The close linear fit for these questions does not guarantee that a
power law exists among these questions, however \cite{Clauset:2009}.
Subsequent work and a larger data set will be required to confirm
if power law relationships do indeed exist among user-generated questions
predictive of a behavioral outcome.%
}. It is possible that the power law exists because of an underlying
power law relationship in the number of responses these questions
attracted. However, Fig. \ref{BMI_Fig3}b indicates there is little
or no correlation between the number of responses a question attracts
and its predictive power. Further, Fig. \ref{BMI_Fig3}e reports the
questions sorted by number of responses, and, when plotted on a log-log
scale (Fig. \ref{BMI_Fig3}f) shows that there is no power law ($r^{2}=0.65$)
among the 20 most responded-to questions. This suggests the power
law relationship among the most predictive questions has some other
cause.
Table \ref{tableBMI} reports the 20 most predictive questions, sorted
by decreasing $r^{2}$. The questions span many of the classes of
factors known (or hypothesized) to influence obesity, including demographic
(q. 7 \cite{Boumtje:2005}), social or economic (qs. 5, 11, 12, 15,
17, 18 \cite{Wang:2007}), genetic (q. 6 \cite{Herbert:2006}), psychological
(qs. 1, 8, 13 \cite{Friedman:1995,vandermerwe:2007}) dietary (qs.
4, 9, 14, 20 \cite{Bonow:2003}), and physical activity-related (qs.
2, 3, 10, 16, 19 \cite{Ewing:2008}). This indicates that although
the majority of participants are unlikely to be experts in the domain
of interest, collectively they uncovered many of the classes of known
correlates of obesity, and responded sufficiently honestly so that
these correlates became predictive of BMI on the site. It could be
argued that the most predictive question should not have been accepted
as it is highly likely that it correlates with the outcome: people
who perceive themselves as overweight are likely to be overweight.
However, it is known that for those suffering from body image disorders
the opposite is often the case: those that perceive themselves incorrectly
as overweight can become extremely underweight \cite{Grogan:2008}.
Separating the auto- and anti-correlated components of this broad
question could be accomplished by supplementing it with more targeted
questions (eg., {}``\textit{Do you think you are overweight but everyone
else tells you the opposite?}'').
Despite the lack of filtering on the site there were only a few cases
of clearly dishonest responses. Fig. \ref{figScreenBMI} indicates
that at least one member of this user's peer group answered the fast
food question dishonestly. It is interesting to note that this dishonest
answer (or answers) was supplied for the seed question, and this question---despite
collecting the most responses (70)---had nearly no individual correlation
($r^{2}=0.054$) and thus contributed negligibly to the predictions
of the models. Questions 3, 4, 6, 9, 15, and 20 as shown in Table
\ref{tableBMI} have maximum possible values (qs. 3 max=100; qs. 4
and 9 max=7; qs. 6 max=2; qs. 15 max=168; qs. 20 max=24), and together
collected 301 responses. Of those responses, none were above the maximum
or below the minimum (min=0 for all qs.) indicating that all responses
were not theoretically impossible. This suggests that clear dishonesty
(defined as supplying a response below or above the theoretical minimum
or maximum, respectively) was quite rare for this experiment. Conversely,
unlike the popular yet corrupted seed question, these questions became
significantly predictive as the experiment progressed. Further investigation
into whether or how the rare cases of clear dishonesty (and the possibly
larger amount of hidden dishonesty) affect modeling in such systems
remains to be investigated.
\section{Discussion/conclusions}
This paper introduced a new approach to social science modeling in which
the participants themselves are motivated to uncover the correlates
of some human behavior outcome, such as homeowner electricity usage
or body mass index. In both cases participants successfully uncovered
at least one statistically significant predictor of the outcome variable.
For the body mass index outcome, the participants successfully formulated
many of the correlates known to predict BMI, and provided sufficiently
honest values for those correlates to become predictive during the
experiment.
While, our instantiations focus on energy and BMI, the proposed method is general, and might, as the method improves, be useful to answer many difficult questions regarding why some outcomes are different than others.
For example, future instantiations might provide new insight into difficult questions like:
"Why do grade point averages or test scores differ so greatly among students?",
"Why do certain drugs work with some populations, but not others?",
"Why do some people with similar skills and experience, and doing similar work, earn more than others?"
Despite this initial success, much work remains to be done to improve
the functioning of the system, and to validate its performance. The
first major challenge is that the number of questions approached the
number of participants on the BMI website. This raises the possibility
that the models may have overfit the data as can occur when the number
of observable features approaches the number of observations of those
features. Nevertheless the main goal of this paper was to demonstrate
a system that enables non domain experts to collectively formulate
many of the known (and possibly unknown) predictors of a behavioral
outcome, and that this system is independent of the outcome of interest.
One method to combat overfitting in future instantiations of the method
would be to dynamically filter the number of questions a user may
respond to: as the number of questions approaches the number of users
this filter would be strengthened such that a new user is only exposed
on a small subset of the possible questions.
\subsection{User Fatigue}
Another challenge for this approach is user fatigue: Fig. \ref{BMI_Fig2}
indicates that many of the later users only answered a small fraction
of the available questions. Thus it is imperative that users be presented
with questions that most require additional responses first. This
raises the issue of how to order the presentation of questions. In
the two instantiations presented here, questions were simply presented
to all users in the same order: the order in which they were posted
to the site. It was possible that this ordering could have caused
a `winner take all' problem in that questions that accrue more responses
compared to other questions would achieve a higher predictive power,
and users would thus be attracted to respond to these more predictive
questions more than the less predictive questions. However, the observed
lack of correlation between response rate and predictive power (Fig.
\ref{BMI_Fig3}b) dispelled this concern.
In future instantiations of the method, question ordering will be
approached in a principled way. Instead of training a single model
$m$, an ensemble of methods $m_{1},...,m_{k}$ will be trained on
different subsets of the data \cite{Skurichina:2002,Lu:2009}. Then,
query by committee \cite{Seung:1992} will be employed to determine
question order: The question that induces maximal disagreement among
the $k$ models as to its predictive power will be presented first,
followed by the question that induces the second largest amount of
disagreement, and so on. In this way questions that may be predictive
would be validated more rapidly than if question ordering is fixed,
or random.
\subsection{User Motivation}
Typically, human subjects play a passive role in social science studies, regardless of whether that study is conducted offline (pen-and-paper questionnaire) or online (web-based survey): They contribute responses to survey questions, but play no role in crafting the questions. This work demonstrates that users can also contribute to the hypothesis-generation component of the discovery process: Users can collectively contribute---and populate---predictors of a behavioral outcome.
It has been shown here that users can be motivated to do this without requiring an explicit reward: The subjects were unpaid for both studies. Much work remains to be done to clarify under what conditions subjects will be \textit{willing} and \textit{able} to contribute predictors.
We hypothesize that \textit{willingness} to generate candidate predictors of a behavioral outcome may be stimulated under several conditions. First, if subjects are incurring a health or financial cost as a result of the outcome under study, they may be motivated to contribute. For example a user that has an above average electricity bill or body mass index, yet has similar lifestyle attributes as his fellow users, may wish to generate additional attributes to explain the discrepancy.
Conversely, a user that posts a superior outcome (i.e. a low electricity bill or very healthy body mass index) may wish to uncover the predictor that contributes to their superior outcome (i.e. a well-insulated house or good exercise regimen) and thus advertise it to their peers. This may act as a form of online `boasting', a well known motivator among online communities.
In the current studies, some participants may have been motivated to contribute because they were part of the authors' social networks. However, a substantial number of users were recruited from online communities outside of the authors' social networks, indicating that some online users are motivated to contribute to such studies even if they do not know those responsible for the study. The exact number of users in these two groups is not clear on account of the anonymity requirements stipulated for these human subject studies.
Similarly, a non domain expert's \textit{ability} to contribute a previously unknown yet explanatory predictor of a behavioral outcome may rely on them suffering or benefiting from a far-from-average outcome. For example consider someone who is extremely underweight yet their outcome is not predicted by the common predictors of diet and exercise: this user has a high caloric intake and does not exercise. This user may be able to generate a predictor that a domain expert may not have though of, yet is predictive for a certain underweight demographic: this user may ask her peers: ``Are you in an abusive relationship?''
Users may also be motivated to contribute to such studies because it provides entertainment value: users may view the website as a competitive game in which the `goal' is to propose the best questions. In a future version we plan to create a dynamically sorted list of user-generated questions: questions bubble up to the top of the list if (1) it is a question that many other users wish to respond to, (2) it is orthogonal to the other questions, and (3) it is found to be predictive of the outcome under study. Users may then compete by generating questions that climb the leaderboard and thus advertise the user's understanding of the outcome under study.
\subsection{Rare Outcomes}
Obesity and electricity usage are well-studied behavioral outcomes. It remains to be seen though how the proposed methodology would work for outcomes that affect a small minority of online users, or for which predictors are not well known.
We hypothesize that for rare outcomes, online users who have experience with this outcome, could be encouraged to participate, and would be intrinsically motivated to contribute. For example if the outcome to be studied were a rare disease, users who suffer from the disease would be attracted to the site. Once there, they may be in a unique position to suggest and collectively discover previously unknown predictors of that disease. Moreover, a user who suffers from the disease is likely to know more people who suffer from that disease and would be motivated to advertise the site to them. Finally, even if a user discovers the site and does not suffer from the disease, he may know someone who does and thus introduce the site to that person. Such a person may serve as a caregiver for someone suffering from the disease, such as a family member. A caregiver may be able to contribute novel predictors that are different from those proposed by the sufferer himself.
Thus, a website that hosts such a rare outcome may serve as a `magnet' for people who exhibit the outcome or know people that do. In future we will study the 'attractive force' of such websites: if such a website experiences increased user traffic as the study goes forward, and the average outcome of users on the site drifts away from the global population's mean value for this outcome, that would indicate that a growing number of people with such an outcome are being attracted to the site.
In closing, this paper has presented a novel contribution to the growing
field of machine science in which the formulation of observables for
a modeling task---and the \emph{populating} of those observables with
values---can be offloaded to the human group being modeled.
\section{Acknowledgement}
The authors acknowledge valuable contributions from three anonymous reviewers, and useful discussions with collaborators in the UVM Complex Systems center.
\bibliographystyle{IEEEtran}
|
{
"timestamp": "2012-03-09T02:03:30",
"yymm": "1203",
"arxiv_id": "1203.1833",
"language": "en",
"url": "https://arxiv.org/abs/1203.1833"
}
|
\section{Introduction}
\label{sec:intro}
Galaxy clusters are the largest gravitationally bound systems
in the universe, and have taken nearly a Hubble time to form.
They therefore have the potential to act as powerful probes of
cosmology if systematic errors can be controlled. Precision
cluster cosmology will require a deep understanding of cluster
astrophysics, particularly as it relates to the hot gas of the
intracluster medium (ICM). The most detailed studies of the
ICM have thus far been performed
by X-ray telescopes, which are sensitive to the bremsstrahlung
emission from the $10^7$-$10^8$~K gas. The Sunyaev-Zel'dovich
(SZ) effect is a complementary probe of the ICM.
The amplitude of the SZ signal depends on the line-of-sight
integral of $n_{\mathrm{e}} T$, while the X-ray surface brightness depends on $n_{\mathrm{e}}^2$,
so sensitive SZ measurements can access tenuous gas outside the cluster core and
directly measure pressure disturbances.
Features found commonly in the outer regions of clusters,
such as shocked gas from mergers, may therefore be easier
to detect using the SZ effect than using X-rays. Moreover,
the combination of X-ray and SZ data can be
used to obtain a more complete picture of the ICM thermodynamics.
To take advantage of these opportunities,
advances in SZ imaging capabilities are needed.
Measurements of the SZ effect have become routine over the
last decade, but the full potential of the SZ effect as
a probe of cluster physics remains largely unexploited due to
technical challenges: the combination of high sensitivity
and large angular dynamic range required for detailed SZ imaging has
proven difficult to achieve with existing instruments.
As a result, the use of the SZ effect has been limited primarily to
studies where resolved imaging is unnecessary.
The small number of higher-resolution SZ images obtained to date
have served to demonstrate the utility of the technique. However,
single-dish measurements such as those by \citet{Korngut11} can suffer
from radio point source contamination and have been limited
to scales $<45^{\prime\prime}$ by the necessity of
filtering out modes contaminated by atmospheric noise.
Multi-dish SZ measurements by arrays such as ATCA \citep[e.g.,][]{Malu10}
can constrain and remove point sources using the inherent
spatial filtering ability of interferometers, but most
millimeter-wave interferometers lack
sensitivity at arcminute angular scales where the
SZ cluster signal is largest.
CARMA is a heterogeneous
interferometric array consisting of 23 antennas with
diameters of 3.5, 6.1, and 10.4~m operating at 1~cm,
3~mm, and 1~mm. This particular combination of antennas and
bands makes CARMA a uniquely powerful SZ instrument: its
3.5~m antennas can be placed in a compact configuration
sensitive to arcminute-scale emission, and its 6.1
and 10.4~m dishes can be used to obtain the sensitivity
necessary to resolve smaller angular scale SZ
features. In this work, we make use of CARMA data from three array
configurations and two bands to obtain an SZ image of the galaxy cluster
RX~J1347.5$-$1145. These data represent the highest-fidelity
picture of a galaxy cluster ever obtained using the SZ effect.
This paper is organized as follows:
Section~\ref{sec:rxj1347} provides background on RX~J1347.5$-$1145,
Section~\ref{sec:obs} describes the observations and data
reduction, and Section~\ref{sec:decon} discusses the
modeling and deconvolution method. We present our
results and compare them to previous measurements
in Section~\ref{sec:results}, and review the conclusions
and discuss prospects for future work
in Section~\ref{sec:conclusions}.
\section{RX~J1347.5$-$1145}
\label{sec:rxj1347}
The target for these observations is the cluster RX~J1347.5$-$1145,
an object that has been characterized extensively using a variety of
techniques. First discovered by the ROSAT all-sky survey \citep{ROSAT},
RX~J1347.5$-$1145 is the most luminous cluster known in the X-ray sky,
and has been measured by several X-ray instruments including
\emph{Chandra} \citep{Allen02} and XMM-\emph{Newton} \citep{Gitti04}.
Optical observations have revealed the presence of two cD galaxies,
one coincident with the X-ray emission peak and one directly to the east.
The system has also been found to host a radio mini-halo \citep{Gitti07}.
Its gravitational potential has been probed using both
strong and weak gravitational lensing \citep[e.g.,][]{Miranda08,Bradac08}.
These multi-wavelength observations indicate that RX~J1347.5$-$1145 is
a massive ($>10^{15}$~M$_\odot$) cluster at redshift $z=0.4510$ which
has recently undergone a merger with a smaller object.
The cluster's SZ signal has been measured using
single-dish \citep{Komatsu00,Kitayama04,Korngut11} and
interferometric \citep{Carlstrom02,Bonamente08,Bonamente12} imaging instruments,
and its spectrum near the thermal SZ null has also been characterized
\citep{Zemcov12}.
The higher angular resolution measurements have revealed
a compact region of very hot ($\sim 20$~keV) gas to the
southeast of the X-ray emission peak, while the low-resolution
data indicate a smaller arcminute-scale SZ signal than
suggested by a spherical fit to the X-ray data.
The existence of a central radio-bright AGN, along with the
limited angular dynamic range of most SZ instruments, has
complicated efforts to bring SZ data to bear on understanding
this system. The CARMA data we present help to overcome both
limitations.
\section{Observations and Data Reduction}
\label{sec:obs}
RX~J1347.5$-$1145 was observed with three different sets of CARMA antennas at two wavelengths:
an 8-element sub-array consisting of 3.5~m antennas at 1~cm (``CARMA-8''), a 15-element
sub-array consisting of 6.1~m and 10.4~m antennas at 3~mm (``CARMA-15''), and the full
23-element array at 3~mm (``CARMA-23'').
The CARMA-8 data were obtained in August 2009 and totaled 25.7~hours of
unflagged, on-source time. The center frequency was 31~GHz with a bandwidth
of 8~GHz, and the target R.A.\ and decl.\ were 13:47:30.7 and \mbox{-11:45:08.6} in
J2000 coordinates. The array was configured with six elements in a
compact array sensitive to arcminute-scale SZ signals, and two
outlying elements providing simultaneous discrimination for compact radio
source emission. The compact array and longer baselines sample
$uv$-spacings of 350$-$1300$~\lambda$ and 2$-$7.5~k$\lambda$,
respectively. The data were reduced using the
Sunyaev-Zel'dovich Array (SZA) pipeline described in \citet{Muchovej07}.
The CARMA-15 data were obtained in July 2010 and totaled
7.4 hours of unflagged, on-source time. The center frequency
was 90~GHz with a bandwidth of 8~GHz, and the target R.A.\ and decl.\ were
13:47:32.0 and \mbox{-11:45:42.0} in J2000 coordinates. The CARMA-15 array was pointed
slightly southeast of the CARMA-8 phase center, directly toward the
region of hot gas. The antennas were in the E configuration,
the most compact standard positions for the 6.1 and 10.4~m antennas.
Data reduction was performed using MIRIAD \citep{MIRIAD}.
We obtained CARMA-23 data as part of a commissioning run in February 2011.
All 23 antennas were operated at a center frequency of 86~GHz and were attached to the CARMA spectral line
correlator, which operated at a reduced bandwidth---4~GHz for the double-sideband receivers
on the 10.4 and 6.1~m antennas, and 2~GHz for the single-sideband receivers
on the 3.5~m antennas. A total of 8.6~hours of unflagged on-source data were obtained;
a relatively high fraction of the 3.5~m data were flagged due to hardware
issues in the commissioning run that have since been corrected. The target R.A.\ and decl.\ were
the same as for the CARMA-8 data, and the array configuration was approximately the
combined CARMA-8 and CARMA-15 configurations. As with the CARMA-15 data,
we reduced the CARMA-23 data using MIRIAD.
For the CARMA-15 and CARMA-23 arrays,
we treat each baseline type (10.4m$\times$10.4m, 10.4m$\times$6.1m, etc.)
separately to properly account for the differing primary beams. We therefore
have ten data sets: one for CARMA-8, three for CAMRA-15, and six for CARMA-23.
We apply a cutoff in $uv$ radius for each data set, using the data beyond the
cutoff only to constrain the point source emission. The cutoff is
chosen to exclude portions of the $uv$ plane which are poorly sampled for a
given baseline type.
\begin{figure*}
\begin{center}
\includegraphics*[height=4in]{uvweight.pdf}
\caption{Normalized data weight distribution in the $uv$ plane for the union of data sets described in Table~\ref{tab:obs}.
Weights are calculated from the inverse variance of each visibility, scaled by the SZ intensity spectrum.
The $uv$-plane extent of each visibility weight is determined from
the cross-correlation of the illumination patterns of the corresponding antennas,
providing a more complete view of the $uv$ sampling in the heterogeneous array. The weights are
well-matched to the cluster signal---which is largest at small $uv$ radius---except for a relatively
under-sampled region around $\sim 2$~k$\lambda$. This region of the $uv$ plane
will be well-measured by the 23-element CARMA array at $1$~cm, which is currently under development.
\label{fig:uv}}
\end{center}
\end{figure*}
\begin{figure*}
\begin{center}
\includegraphics[height=4in]{radbin.pdf}
\end{center}
\caption{Radially binned visibilities from CARMA-8 at 1~cm, CARMA-15 at 3~mm,
and CARMA-23 at 3~mm. The CARMA-8 data measure the cluster signal at large
angular scales, while the other sub-arrays measure the smaller substructure
of the cluster.}
\label{fig:radbin}
\end{figure*}
The outputs of the data reduction pipelines consist of
flagged, calibrated visibilities $V(u,v)$. All absolute flux
calibration is performed using the \citet{Rudy87} Mars model,
and is accurate to 5\%. To combine data from different
bands, we define $y(u,v)$, the Fourier-space counterpart
to the Compton $y$ parameter \citep{Carlstrom02}:
\begin{equation}
\label{eq:yuv}
y(u,v) \equiv \frac{V_{\nu}(u,v)}{g(\nu,\langle T_{\mathrm{e}} \rangle) \, I_0}
\end{equation}
where $\langle T_{\mathrm{e}} \rangle = 12$~keV is the mean ICM electron temperature of the
cluster\footnote{We make this approximation due to our
incomplete of knowledge of the true $T_{\mathrm{e}}(x,y,z)$. The
relative value of $g(\nu,T_{\mathrm{e}})$ between 1~cm and 3~mm is nearly independent
of $T_{\mathrm{e}}$, varying by just 2\% between 5 and 15~keV.}.
Our ten $y(u,v)$ data sets are summarized in Table~\ref{tab:obs}.
The data weight distribution in the $uv$ plane for the combined data
sets is shown in Figure~\ref{fig:uv}. To illustrate the
contributions of each of the three sub-arrays, we show
the measured $y(u,v)$ binned in $uv$ radius in
Figure~\ref{fig:radbin}. CARMA-8 at 1~cm constrains
the cluster at small $uv$ radius (large angular scale)
where the signal is largest, while CARMA-15 and CARMA-23 at
3~mm are sensitive to the large $uv$ radius (small angular scale)
substructure of the cluster.
\section{Modeling and Deconvolution}
\label{sec:decon}
We wish to combine the information in all ten sets of visibility data to
form a single image of the cluster. The first step in this process
is to remove the radio point source emission, which is
accomplished by fitting a point source model to the
visibilities above the $uv$ cutoff.
Since radio source fluxes often vary considerably with time,
identical fluxes should not be expected in the CARMA-15 and CARMA-23 despite their
similar central frequencies. We therefore demand consistency within each individual
sub-array, but allow the source flux to vary between sub-arrays.
We find a best-fit centroid consistent with a source in the
NVSS catalog \citep{NVSS} at an
R.A.\ and decl.\ of 13:47:30.7 and \mbox{-11:45:08.6}. The flux of this source was
found to be 8.9$\pm$0.5~mJy in the 1~cm CARMA-8 data, 4.2$\pm$0.2~mJy in the 3~mm CARMA-23 data,
and 4.0$\pm$0.2~mJy in the 3~mm CARMA-15 data. We subtract the best-fit point source
models from the visibility data sets before proceeding.
\begin{figure*}
\begin{center}
\includegraphics[height=5in]{dirty_yperbeam_1e13_beam.pdf}
\end{center}
\caption{Dirty map of RX~J1347.5$-$1145 using ``robust'' visibility
weighting. The synthesized beam is
shown in grey in the lower left. The map is in units of Compton
$y \times 10^{13}/$beam.}
\label{fig:dirty}
\end{figure*}
We next apply a phase shift to the CARMA-15 visibilities to establish
a single phase center for all ten data sets. At this stage,
all of the visibilities can be combined to form a
single dirty map (Figure~\ref{fig:dirty}). However, this map is
difficult to interpret because it includes data
with dramatically different primary beams, and because noise at
small angular scales masks the arcminute-scale emission.
In order to proceed, we must use some image deconvolution method such as the
Maximum Entropy Method (MEM) or CLEAN. For simplicity, we choose to
build a model of the cluster iteratively using the CLEAN algorithm,
though we note that MEM also holds promise for future work.
Our algorithm successively builds up a single model of the cluster
in image space by CLEANing individual data sets. We begin by making a dirty map
from the first data set $y_1(u,v)$. We then run the H\"{o}gbom CLEAN
algorithm \citep{Hogbom74} on this map with gain 0.05, stopping at
$2.5\sigma$. The CLEAN components are corrected for the appropriate
primary beam, and a restored map is generated. This restored
map (in units of Compton $y$ per pixel) is our initial cluster model.
The cluster model is then multiplied by the primary beam of the next data
set, $y_2(u,v)$, and the result is Fourier transformed and subtracted
from $y_2(u,v)$. From these model-subtracted visibilities, a dirty map
is produced and H\"{o}gbom CLEANed. The CLEAN components are corrected
for the primary beam and restored, and the restored map is added to
the cluster model. We repeat this process on all ten data sets, resulting
in a cluster model that incorporates information from all of our data.
We continue refining the model by making additional passes through all ten
data sets until each model-subtracted dirty map is consistent with noise.
Three iterations are found to be sufficient. Our final cluster model is
then added to the weighted average of the ten model-subtracted dirty maps
to form a CLEAN map (Figure~\ref{fig:clean}).
Though the resolution of the map is not well-defined, the smallest
restoring beam of the ten used to construct the model
is 10\farcs6 $\times$ 16\farcs9.
\begin{figure*}
\begin{center}
\includegraphics[height=4in]{clean_020112.pdf}
\end{center}
\caption{CLEAN map of RX~J1347.5$-$1145. The map is in units of Compton
$y \times 10^{15}$/pixel, where the pixel size is
$0.5^{\prime\prime}\times 0.5^{\prime\prime}$. The smallest CLEAN beam
used to construct the model is $10^{\prime\prime}.6 \times 16^{\prime\prime}.9$}.
\label{fig:clean}
\end{figure*}
\begin{figure*}
\begin{center}
\includegraphics[height=4in]{cleanx_020112.pdf}
\end{center}
\caption{CLEAN map of RX~J1347.5$-$1145 with the scaled relaxed X-ray pressure
profile subtracted from the visibilities. The map is in units of Compton
$y \times 10^{15}$/pixel, where the pixel size is
$0.5^{\prime\prime}\times 0.5^{\prime\prime}$.}
\label{fig:disturbed}
\end{figure*}
To assess the accuracy and flux recovery of our deconvolution technique,
we construct a heuristic model of RX~J1347.5$-$1145. We generate mock
data using this model, repeat the
procedure described above on the simulated visibilities, and compare the
results to the input model.
Modeling the SZ signal requires knowledge of the density $n_{\mathrm{e}}$ and
the temperature $T_{\mathrm{e}}$ of the ICM, both of which can be approximated
using the results of \citet{Allen02} (hereafter \citetalias{Allen02}) derived
from \emph{Chandra} X-ray data.
For the electron density, we use the \citetalias{Allen02} density fit
within the central region with an $\alpha=-2.331$ power law in the outer
region inferred from the \citetalias{Allen02} surface brightness profile fit.
We piece together the inner and outer regions by requiring that $n_{\mathrm{e}}(r)$ be
continuous. For the electron temperature, we use the
\citetalias{Allen02} measured temperature profile out to the outer radius of
their largest bin, and use the average temperature of
$12.0$~keV at larger radius. We assign a temperature of $18.0$~keV
in the shocked region of the southeast quadrant. Finally,
we truncate both the temperature and density profiles at $r_{200}$.
Using these approximations for the electron density and temperature, we
integrate along the line of sight to produce a simulated Compton $y$
map, and Fourier transform to produce simulated visibilities $y_{\mathrm{sim}}(u,v)$.
We then randomize the phases in our visibility data $y(u,v)$ to produce
simulated noise $y_{\mathrm{noise}}(u,v)$ and run our iterative CLEAN algorithm on
$y_{\mathrm{sim}}(u,v) + y_{\mathrm{noise}}(u,v)$. We find that our algorithm accurately
reproduces the morphology of the input model, and that it recovers
69\%, 78\%, and 92\% of the integrated flux within $r_{200}$, $r_{500}$, and $r_{2500}$, where the
$r_\Delta$ values are determined from the \citet{Allen02} NFW model fit. The
flux recovery ratio at larger radii is highly sensitive to the assumed electron
density power law index. This is due to the fact that $r_{200}$ for this
cluster corresponds to an angular scale of 5\farcm75,
comparable to the 10\farcm7 FWHM 3.5~m primary beam at 1~cm.
\section{Results}
\label{sec:results}
The map shown in Figure~\ref{fig:clean} can be understood as a
relaxed cluster SZ signal with additional sub-arcminute structure in the
southeast quadrant imparted by the merger event.
The imaging of both the extended and compact structure in our SZ map
is of significantly higher fidelity than previous measurements
due to the ability of CARMA to remove the central point source
and to the large angular dynamic range of the combined arrays.
In contrast with previous work \citep{Mason10,Komatsu00}, we
find that the peaks of the SZ and X-ray signals are coincident.
Figure~\ref{fig:multiband} shows a comparison between the
CARMA SZ map and the {\it Chandra}\ X-ray pressure map \citep{Bradac08}.
Though a full multi-wavelength reconstruction is beyond the scope
of this paper, it is obvious that the two techniques produce
maps with consistent morphologies.
To separate the relaxed and disturbed components,
we make use of the pressure profile fit to the {\it Chandra}\ X-ray data
described in \citet{Allen08}, in which the southeast quadrant was
excised. We first project the pressure profile
along the line of sight to produce an integrated pressure map. For each sub-array,
we multiply this map by the appropriate primary beam, convert to Compton $y$,
Fourier transform, and subtract the result from the visibility data. We then
iteratively build a model of the remaining SZ signal, following the procedure
described in Section~\ref{sec:decon}. We allow the X-ray pressure profile
to be scaled by a multiplicative constant to compensate for cluster projection effects
and calibration errors. The scale factor is chosen so as to produce
no net CARMA-8 (arcminute-scale) signal in the iteratively-determined model.
Removing this estimate of the relaxed signal from
our visibility data allows us to focus on the sub-arcminute-scale
signal resulting from the merger event. The result is shown in
Figure~\ref{fig:disturbed}. The total Compton $y$ signal recovered in this
map, which corresponds to the fraction of the thermal energy in the cluster
ICM associated with the merger-related substructure, is $9.1$\% of
the total recovered from the map in Figure~\ref{fig:clean}.
The value of the multiplicitive constant by which the X-ray pressure
is scaled can also yield information about the characteristics of the cluster.
Since the scale factor is determined by requiring consistency with the
CARMA-8 data, its value depends upon the three-dimensional morphology
of the ICM on large angular scales.
Due to the different dependencies of the X-ray surface brightness and the
SZ signal on density, a cluster more (less) extended
along the line of sight than in the plane of the sky would yield a scale
factor of greater than (less than) one \citep[see e.g.,][]{Grego04}.
For this cluster, we find a scale factor of 0.58, indicating a
cluster which is strongly compressed along the line-of-sight direction.
A similar ratio is reported in \citet{Bonamente12} using the CARMA-8
data reported here. \citet{Chakra08}, using a combination of X-ray and SZ data,
also find that RX~J1347.5$-$1145 is compressed along the line of sight
with an axis ratio of $\sim 5$.
A simple comparison of the X-ray surface brightness profile and our data,
assuming an isothermal cluster, implies compression along the line of sight by
a factor of roughly three-to-one---a fairly extreme value, though less than
suggested by \citet{Chakra08}.
Clumping of the ICM, i.e., a systematic discrepancy between
$\langle n_{\mathrm{e}}^2(r) \rangle$ and $\langle n_{\mathrm{e}}(r) \rangle^2$,
could also lead to differences between the
SZ and X-ray signals. Clumping is observed in the outskirts of
simulated clusters \citep{Nagai11}, and is implied by observations
of flattened entropy profiles and gas mass fractions apparently in excess of
the cosmic mean in some clusters \citep[e.g.,][]{Simionescu11}. However,
in both cases the clumping occurs at large cluster radii (at least $>r_{500}$),
and is thus unlikely to affect the normalization of the X-ray model used
here (which was fit to data at $r<r_{500}$) at the level required to
explain the offset.
The difference between the CARMA and {\it Chandra}\ pressure
estimates could also arise in principle from calibration or
systematic errors in the SZ or X-ray data. As a cross-check of our CARMA
calibration, we compared our data to previous BIMA SZ measurements
reported in \citet{Bonamente08}, finding consistent integrated $Y$
values and binned visibility data. Since the {\it Chandra}\ calibration
is unlikely to be mistaken at this level, and since
major mergers such as RX~J1347.5$-$1145 are not expected to be spherically
symmetric, we suggest that line-of-sight compression is most likely to be
the dominant effect. However, ICM clumping and calibration errors may
also be contributing to the discrepancy; a more complete explanation will require
a joint analysis of the two data sets.
\begin{table*}
\centering
\caption{CARMA Observations of RX~J1347.5$-$1145.}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|}
\hline
\hline
Array & Ant 1 & Ant 2 & Frequency & $uv$ cutoff & Noise & Minor axis & Major axis & Beam P.A. \\
& & & (GHz) & (k$\lambda$)& mJy/beam & arcsec & arcsec & Degrees \\
\hline
CARMA-8 & 3.5m & 3.5m & 31 & 2000 & 0.16 & 100.1 & 128.8 & -33.4 \\
CARMA-15 & 10.4m & 10.4m & 90 & 10000 & 0.38 & 13.4 & 16.8 & 72.9 \\
CARMA-15 & 10.4m & 6.1m & 90 & 10000 & 0.19 & 12.8 & 16.5 & -44.8 \\
CARMA-15 & 6.1m & 6.1m & 90 & 10000 & 0.30 & 15.1 & 37.4 & 3.0 \\
CARMA-23 & 10.4m & 10.4m & 86 & 10000 & 0.22 & 13.9 & 14.6 & -80.2 \\
CARMA-23 & 10.4m & 6.1m & 86 & 10000 & 0.14 & 14.4 & 16.4 & -65.3 \\
CARMA-23 & 6.1m & 6.1m & 86 & 10000 & 0.26 & 14.9 & 37.3 & 4.4 \\
CARMA-23 & 10.4m & 3.5m & 88 & 10000 & 0.80 & 16.2 & 18.3 & 59.1 \\
CARMA-23 & 6.1m & 3.5m & 88 & 10000 & 0.76 & 10.6 & 16.9 & -81.1 \\
CARMA-23 & 3.5m & 3.5m & 88 & 5000 & 2.64 & 31.4 & 67.5 & -44.3 \\
\hline
\hline
\end{tabular}
\label{tab:obs}
\end{table*}
\begin{figure*}
\begin{center}
\includegraphics*[width=0.9\textwidth]{xraycomp.pdf}
\caption{
The CLEANed CARMA SZ map (left) cropped and scaled for direct
comparison with the published X-ray-derived integrated pressure
map from \citet{Bradac08} (right).
}
\label{fig:multiband}
\end{center}
\end{figure*}
\section{Conclusions and Future Work}
\label{sec:conclusions}
We have demonstrated the ability of CARMA to measure the SZ signature of
galaxy clusters at high sensitivity across a wide range of angular scales.
By combining data from three CARMA configurations and two frequency
bands, we measure the arcminute-scale SZ signal as well as the pressure
substructure of RX~J1347.5$-$1145. The large angular dynamic range of
the CARMA data and the capability to constrain and remove point sources
make our measurement a significant improvement in image fidelity over
previous work \citep[e.g.,][]{Korngut11}.
By comparing our data to the X-ray measurements of \citet{Allen08}, we are
able to determine that $\sim 9$\% of the SZ signal is localized in the
disturbed region of the cluster, and that the system is likely compressed
along the line of sight relative to the plane of the sky.
Although our results demonstrate that CARMA in its current form is a
highly capable SZ instrument, several upgrades will soon bring about
significant enhancements. An effort is currently underway to equip all
antennas with 1~cm receivers and expand the correlator bandwidth to
8~GHz for a 23-element array. The 3~mm CARMA-23 array described in this work
provided sensitivity from $uv$ radii of $\sim 2.0$ to $10$k$\lambda$;
the upgraded array placed in the same configuration but operated at 1~cm
will provide higher-sensitivity coverage from $\sim 0.35$ to $3.3$k$\lambda$.
As shown in Figure~\ref{fig:radbin}, this corresponds to the portion of
the $uv$ plane where the SZ signal is large. A planned upgrade to
more sensitive 3~mm receivers will allow the SZ signal at finer
angular scales to be measured more precisely.
The sensitivity to smaller angular scale SZ structures provided CARMA's
larger telescopes can be directed toward regions of interest, as was
done in this study by pointing them toward the region of hot gas to
the southeast of the cluster center. With the increased sensitivity
enabled by the ongoing upgrades, this technique can be used to search
for shock-enhanced features in the outskirts of clusters. Mosaicking
can also be used to provide sensitivity to small-scale structures
over the entire cluster.
Taken together, these upgrades and observing strategies will
allow CARMA to image clusters precisely and efficiently
over a wide angular dynamic range, making it possible to fully exploit
the power of the SZ effect as a probe of cluster astrophysics and
precision cosmology.
\section*{Acknowledgements}
We thank Evan Million for providing the X-ray pressure data from \citet{Bradac08}.
We also thank Maru{\v s}a Brada{\v c} and Myriam Gitti for useful discussions.
Support for CARMA construction was derived from the Gordon and Betty Moore
Foundation, the Kenneth T. and Eileen L. Norris Foundation, the James
S. McDonnell Foundation, the Associates of the California Institute of
Technology, the University of Chicago, the states of California,
Illinois, and Maryland, and the National Science Foundation. Ongoing
CARMA development and operations are supported by the National Science
Foundation under a cooperative agreement, including grant AST-0838187
at the University of Chicago, and by the CARMA partner universities.
Partial support is provided by NSF Physics Frontier Center grant
PHY-1125897 to the Kavli Institute of Cosmological Physics. D.~P.~M.~was
supported for part of this work by NASA through Hubble Fellowship
grant HST-HF-51259.01.
Finally, we thank the CARMA staff for making
the 23-element commissioning observations possible.
{\it Facilities:} \facility{CARMA}
|
{
"timestamp": "2012-03-12T01:02:48",
"yymm": "1203",
"arxiv_id": "1203.2175",
"language": "en",
"url": "https://arxiv.org/abs/1203.2175"
}
|
\section{Introduction}
The violent stellar explosion that gives rise to long $\gamma$-ray bursts \citep[see e.g.,][for reviews]{2004RvMP...76.1143P, 2009ARA&A..47..567G} and their multi-wavelength afterglows has been firmly related to broad-line supernovae (SNe) of type Ic, and hence star-formation (SF), via the core-collapse of massive stars \citep[e.g.,][]{1998Natur.395..670G, 2003Natur.423..847H, 2003ApJ...591L..17S, 2004ApJ...609L...5M, 2006Natur.442.1011P, 2006Natur.442.1008C}. The GRB's high-energy signature is very luminous, and unaffected by dust and therefore pin-points regions of star-formation irrespective of galaxy brightness, dust obscuration and redshift. GRB-selected galaxies hence provide a sample of high-redshift, star-forming galaxies that is fully complementary to conventional survey studies.
The luminous afterglows furthermore facilitate redshift measurements, and detailed investigation about the chemical composition \citep[e.g.,][]{2003ApJ...585..638S, 2006ApJ...648...95P, 2009ApJ...691L..27P, 2010A&A...513A..42D} and the dust properties of the host \citep[e.g.,][]{2001ApJ...549L.209G, 2006ApJ...641..993K, 2007MNRAS.377..273S, 2010MNRAS.401.2773S, 2011arXiv1102.1469Z}. {GRB hosts can hence be targeted with a known redshift, position and information about the galaxy's interstellar medium (ISM) at hand, providing an independent diagnostic of galaxy evolution and star-formation.}
{Notably at the highest redshifts \citep{2009ApJ...693.1610G, 2009Natur.461.1254T, 2009Natur.461.1258S, 0429BCucc}, GRBs allow us to set observational constraints on the history of star-formation \citep[e.g.,][]{2009ApJ...705L.104K, 2012ApJ...744...95R, 2012arXiv1202.1225E}, the galaxy luminosity function \citep{2012arXiv1201.6074T, 2012arXiv1201.6383B} as well as on the nature of young and star-forming galaxies \citep[e.g.,][]{2004A&A...425..913C, 2009ApJ...691..152C, 2010arXiv1010.1783W} beyond the detection limit of state-of-the-art surveys.}
{To represent a robust tool for cosmology and probe of star-formation, the physical conditions that lead to the formation of the GRB progenitor must be understood. As direct observations of GRB progenitors akin to those of some SNe remain impossible due to the cosmological distances, afterglow sight-line \citep[e.g.,][]{2009ApJS..185..526F}, spatially-resolved \citep[e.g.,][]{2008A&A...490...45C, 2008ApJ...676.1151T, 2011ApJ...739...23L} or galaxy-integrated measurements \citep[e.g.,][]{2009AIPC.1133..269G, 2012MNRAS.419.3039C} provide the most constraining information on the kind of galactic environments GRBs occur in.}
However, the properties of an unbiased sample of long GRBs hosts are still largely unknown, and selection effects due to optically-dark bursts \citep{1998ApJ...493L..27G, 2001A&A...369..373F, 2009AJ....138.1690P} arguably play a crucial role \citep[e.g.,][]{2011arXiv1108.0674K, 2011AAS...21710802P}. Consequently, the conditions for the formation of GRBs, the relation between GRB hosts and field galaxies and the extent to which GRBs trace the cosmic SFR remain highly debated \citep[e.g.,][]{2005MNRAS.362..245J, 2006Natur.441..463F, 2009ApJ...702..377K, 2011arXiv1105.1378C, 2011ApJ...735L...8K}.
{Local galaxies hosting long GRBs tend to be of low stellar mass and metal content with respect to SDSS galaxies \citep{2010AJ....139..694L} as well as the hosts of core-collapse SNe \citep{2006Natur.441..463F, 2008AJ....135.1136M}, which has been interpreted as support for a limited chemical evolution of the GRB host - seemingly in line with metallicity constraints on the GRB progenitor from theoretical calculations based on the collapsar model \citep{1993ApJ...405..273W, 1999ApJ...524..262M}. These properties, however, are not indicative per-se of GRBs preferring low-metallicity environments, but instead could be the result of low-mass, low-metallicity galaxies dominating the local star-formation rate \citep{2011MNRAS.414.1263M}. In fact, at higher redshifts \citep[see e.g.,][]{2006ApJ...647..471L, 2007ApJ...660..504B} and in \textit{Swift} GRB host samples with a better controlled selection a population of red, luminous, high-mass hosts emerges \citep{Rossi2011, 2011ApJ...736L..36H, 2011ApJ...727L..53C, 2011arXiv1109.3167S}.}
A fundamental characteristic of (GRB-selected) galaxies is their gas-phase metallicity, and in particular whether they follow the relation between stellar mass ($M_*$), metallicity ($Z$) and SFR defined by local field galaxies \citep{2010MNRAS.408.2115M, 2010A&A...521L..53L}. However, observational access to the metallicity of GRB hosts remained largely elusive, and robust constraints are only available up to $z\sim1$ \citep{2009ApJ...691..182S, 2010AJ....139..694L}. This is largely due to the absence of efficient NIR spectrographs, as important tracers of metallicity {(such as \nii~($\lambda$6584) and H$\alpha$)}, are redshifted into the NIR wavelength regime above $z\sim0.5$.
Here we present optical/NIR observations of the galaxy hosting GRB 080605 at $z = 1.64$ obtained with the X-shooter spectrograph at the Very Large Telescope (VLT), and NIR imaging {with HST/WFC3 and} LIRIS mounted at the William Herschel Telescope (WHT). The spectroscopic observations probe the rest-frame wavelength range between $1150$ and $8700\,\AA$~and reveal a wealth of emission lines including H$\beta$, [\ion{O}{iii}], H$\alpha$ and \nii~($\lambda$6584).
GRB 080605 \citep{2008GCN..7828....1S} was initially detected by the \textit{Swift} satellite \citep{2004ApJ...611.1005G}, and its optical/NIR afterglow was readily identified \citep{2008GCN..7829....1K, 2008GCN..7834....1C}. Spectroscopy of the afterglow was obtained with FORS2 at the VLT which yields a redshift of $z=1.6403$ \citep{2008GCN..7832....1J, 2009ApJS..185..526F}. The optical/NIR afterglow is characterized by the presence of significant amounts of dust with $A_V \sim 0.5$~mag \citep{2011A&A...526A..30G, 2011arXiv1102.1469Z}, including evidence of a 2175~\AA~feature \citep{2011Zafar}. The 2175~\AA~dust bump is a common characteristic observed along sight-lines through the Milky-Way. It becomes weaker in the Large Magellanic Cloud, and is absent from most sight-lines through the Small Magellanic Cloud. It is only rarely observed towards high-redshift environments such as quasars or absorbing systems, but common along sight-lines to highly extinguished afterglows \citep[e.g.][]{2008ApJ...685..376K, 2009ApJ...697.1725E, 2011arXiv1102.1469Z, 2010arXiv1009.0004P}. The carrier of the bump is currently not fully understood with graphite and polycyclic aromatic hydrocarbons being primary candidates \citep[see e.g.,][for a review]{2003ARA&A..41..241D}.
We adopt the concordance ($\Omega_M=0.27$, $\Omega_{\Lambda}=0.73$, $H_0=71~\rm{km}\,s^{-1}\,\rm{Mpc}^{-1}$) $\Lambda$CDM cosmology. All errors are given at $1\sigma$ confidence levels. All magnitudes are given in the AB system and {are corrected for the Galactic reddening of $E_{B-V}=0.137$~mag \citep{1998ApJ...500..525S}. The solar oxygen abundance is assumed to be $12+\log(\rm{O}/\rm{H}) = 8.69$ \citep{2009ARA&A..47..481A} throughout this work.} Wavelengths are given in vacuum and the redshifts in the heliocentric system.
\section{Observations and data reduction}
\subsection{Space-based imaging}
\begin{figure}
\centering
\includegraphics[width=0.99\columnwidth]{Figs/fig1.pdf}
\caption{{Finding chart ($8\arcsec\,\times\,8\arcsec$) for the host of GRB~080605 as imaged with HST/WFC3. The afterglow position and its uncertainty are indicated by a red circle, and the different components are labeled A and B. The barycenter of each component is indicated by a white cross. The geometry of X-shooter's UVB slit with width of 1\farc{0} is illustrated by dashed black lines. The VIS and NIR slit have the same orientation but a width of 0\farc{9}. Logarithmically spaced contours are shown in white lines.}}
\label{fig:fc}
\end{figure}
{The host of GRB~080605 was observed with the Hubble Space Telescope (HST) and Wide Field Camera 3 (WFC3) as part of a snapshot program targeting GRB hosts (PI: A.~J. Levan, Proposal ID: 12307) on 2012-02-22. HST imaging (see Figure~\ref{fig:fc}) was obtained in the F160W filter in a three-point dither pattern resulting in a total exposure time of 1209~s. Individual images (pixel scale 0\farc{128}/px.) were drizzled to an output image with a pixel scale of 0\farc{08} per pixel. Using several unsaturated stars in the field of view we measure a FWHM of the stellar PSF of $2.6\pm0.1$~px, which is $0\farc{21}\pm0\farc{01}$.}
{To accurately locate the position of the afterglow within its host, we first used a GROND afterglow image from \citet{2011A&A...526A..30G} and calibrated it astrometrically against $\sim80$ sources from the USNO catalog. This sets the absolute astrometric scale with an accuracy of around 0\farc{4} in each coordinate. The uncertainty introduced by centroiding errors of the afterglow is $\approx15$~mas. Afterwards, we registered a deeper GROND host image \citep{2011arXiv1108.0674K} against the afterglow image using common field stars. The mapping uncertainty between the two GROND images is 20~mas. Finally, we used fainter stars from the host image that are unsaturated in the WFC3 frame to tie the space- to the ground-based imaging. In the last step the RMS-scatter of stellar positions is 60~mas in each coordinate, which dominates the total relative accuracy (65~mas) of the position of the afterglow within its host.}
{The host of GRB~080605 is clearly extended in the N/E direction in the HST imaging, and consists of two, somewhat blended components A and B (Fig.~\ref{fig:fc}) with a projected distance of 1\farc{0} between the brightest pixel of each component (corresponding to 8.6~kpc at $z=1.641$). Photometry (see Table \ref{tab:photobs}) was derived using elliptical Kron magnitudes via Sextractor \citep{1996A&AS..117..393B}, an aperture correction of $6\pm4$\% to the total flux \citep{2005PASA...22..118G} and the tabulated HST/WFC3 zeropoints\footnote{\texttt{http://www.stsci.edu/hst/wfc3/phot\_zp\_lbn}} from March 06, 2012. Deblending parameters were set one time to measure the integrated flux of the both components to be comparable to the ground based imaging, and a second time to measure the flux contribution of the individual host components (see also Table~\ref{tab:photobs}). Given the small angular separation, the two components are not resolved in our ground-based imaging.}
\subsection{Ground-based imaging}
The field of the host of GRB 080605 was also imaged with the LIRIS instrument \citep{2004SPIE.5492.1094M} mounted at the 4.2~m WHT. We obtained a total of 0.55~hr of exposures in the $J$ (average FWHM of the stellar PSF is 1\farcs{4}), and 0.70~hr in the $K_s$-band (average FWHM of the stellar PSF is 1\farcs{0}) at airmasses between 1.3 and 2.0. The data were reduced and photometry was performed within pyraf/IRAF \citep{1993ASPC...52..173T} in a standard manner.
Absolute calibration was obtained against roughly $40$ field stars with magnitudes from the 2MASS catalog. This procedure resulted in an absolute photometric accuracy of around 0.05~mag in the $J$, and 0.07~mag in the $K$~band, which is negligible compared to the error introduced by photon statistics. The LIRIS photometry is summarized in Table~\ref{tab:photobs}.
\input{Tabs/seds.tex}
\subsection{X-shooter optical/NIR spectroscopy}
\label{Xsred}
X-shooter \citep{2006SPIE.6269E..98D, 2011arXiv1110.1944V} at the VLT observed the host of GRB 080605 starting at 08:22 UT on 2011-04-26 for a total exposure time of 0.98~hr in the ultra-violet/blue (UVB), 1.01~hr in the visual (VIS), and 1.00~hr in the NIR arm, respectively. Spectroscopy was obtained with slit widths of 1\farcs{0} (UVB), and 0\farcs{9} (VIS and NIR), which results in resolving powers of $\lambda/\Delta\lambda \approx 5100$, 8800 and 5100 for the three arms. The geometry of the slit is illustrated in Figure~\ref{fig:fc}.
Sky conditions were clear with an average seeing of 1\farcs{2}. In total, four nodded exposures in the sequence ABBA were obtained. In each nodding position a single UVB and VIS frame (885 and 910~s exposure time each), and three NIR frames (300~s exposure time each) were taken. {Data were reduced with the X-shooter pipeline {v. 1.5.0} \citep{2006SPIE.6269E..80G} in physical mode, and the spectra were extracted using an optimal, variance-weighted method in IRAF \citep{1993ASPC...52..173T}. }
The wavelength-solution was obtained against ThAr arc-lamp frames leaving residuals of around 0.2~pixel which corresponds to 6 km s$^{-1}$ at 10\,000~\AA. Flux-calibration was performed against the spectro-photometric standard LTT7987\footnote{\texttt{http://www.eso.org/sci/facilities/paranal/\\instruments/xshooter/tools/specphot\_list.html}} observed during the same night at 09:46 UT, immediately after the science exposures.
The stellar continuum of the host of GRB~080605 is detected in the X-shooter spectrum with a S/N $\approx$ 0.3-0.9 per pixel in parts of the UVB (3600~\AA~to 5500~\AA) and VIS (5600~\AA~to 9800~\AA). {Within the NIR arm the continuum is only marginally seen in the $J$ and $H$ bands with S/N $\sim$ 0.1-0.2 per pixel due to X-shooter's lower sensitivity in this wavelength range. The host is undetected in the wavelength range of the $K$-band with a S/N smaller than $\sim 0.1$ per pixel.}
{A robust flux-calibration within the broad wavelength range of X-shooter's sensitivity is challenging.
We hence further corrected the flux-calibrated X-shooter spectrum in the UVB and VIS arms by integrating it over the filter curves of GROND \citep{2008PASP..120..405G} and HST and matching it to the available host photometry \citep{2011arXiv1108.0674K}. This procedure results in scaling factors of around $1.63\pm0.09$ for the $g'$-band in the UVB arm ($\approx 4590~\AA$), and $1.56\pm0.13$, $1.35\pm0.12$ and $1.26\pm0.14$ for the $r^{\prime}$, $i^{\prime}$ and $z^{\prime}$ band at $6220$, $7640$~and $8990~\AA$,~respectively. For the NIR arm, we derive factors of $1.4\pm0.4$ for the $J$-band and $1.4\pm0.2$ for the F160W-band. Due to the non-detection of the continuum in the $K$-band, no correction can be obtained between 18\,000 and 23\,000~\AA, but no emission lines are detected in this wavelength regime.}
{We further tested the absolute flux calibration and its inter- and intra-arm continuity via observations of telluric standard stars taken on the same night. We find that the absolute flux of the telluric is typically recovered within uncertainties of 30\%, while its spectral shape is robust to an accuracy better than 15\% within each arm.}
\section{Results}
\subsection{Host galaxy system and afterglow position}
{The system hosting GRB~080605 consists of two components A and B (see Fig.~\ref{fig:fc}) with barycentric coordinates of RA~(J2000) = 17:28:30.05, decl.~(J2000) = +04:00:56.2 for component A and RA~(J2000) = 17:28:30.02, decl.~(J2000) = +04:00:55.3 for component B, respectively. The half-light radii $r_{\rm{e}}$ in the observed F160W-band (rest-frame $\sim5800\,\AA$) for the two components are marginally resolved ($r^{\rm A}_{\rm{e,5800\,\AA}}\sim0\farc{19}$ or 1.6~kpc, $r^{\rm B}_{\rm{e,5800\,\AA}}\sim0\farc{26}$ or 2.2~kpc). The half-light radius for the total host complex is $r_{\rm{e,5800\,\AA}}=0\farc{41}$ or 3.5~kpc.
}
{The afterglow position coincides with the center of component A (Figure~\ref{fig:fc}). Within our astrometric accuracy of 65~mas, no significant offset is detected and we conclude that the GRB exploded within a projected distance of 900~pc (90\% confidence) to the central region of component A.}
\subsection{Emission line profile}
\begin{figure}
\centering
\includegraphics[width=0.99\columnwidth]{Figs/fig2.pdf}
\caption{{Two-dimensional cutouts of the X-shooter NIR spectrum centered on the observed wavelength of [\ion{O}{iii}]($\lambda 5007)$ and \ha. Skylines are indicated with grey shading. Linearly spaced contours are shown in red lines.}}
\label{fig:twod}
\end{figure}
The X-shooter spectrum of the host galaxy of GRB 080605 covers the wavelength range between $3050$ and $23\,000~\AA$ (rest-frame $1150$ and $8700~\AA$) and is rich in emission lines. The emission lines are identified as the doublets of \oii, [\ion{O}{iii}], \sii, \nii, as well as H$\alpha$, H$\beta$, and \oi. {The significance of the detection of the Balmer lines, [\ion{O}{iii}], \oii~and \nii~($\lambda\,6584$) is $>\, 8\sigma$, while it is between 2 and 4$\sigma$ for \nii~($\lambda\,6548$), the \sii~doublet, and \oi~($\lambda\,6366$).}
{The two emission lines detected at the highest S/N ([\ion{O}{iii}]~($\lambda5007$) and H$\alpha$, see also Section~\ref{EmissionLines}) are marginally tilted, reflecting the contributions of component A and B. Figure~\ref{fig:twod} shows the two-dimensional cutouts centered at the wavelength of the [\ion{O}{iii}]~($\lambda5007$) and \ha~lines. They define heliocentric\footnote{The heliocentric correction in the direction of GRB~080605 is 19~km~s$^{-1}$ for our observations.} redshifts of $z_{\rm A}=1.64104\pm0.00004$ and $z_{\rm B}=1.64083\pm0.00007$ measured from the peak of the emission lines. These values correspond to a separation of $\Delta v \sim 20\,\rm{km\,s^{-1}}$ (Figures~\ref{fig:fc} and \ref{fig:twod}).}
{For the fainter emission lines, we lack signal-to-noise ratio in our X-shooter spectrum and individual contributions of components A and B are strongly blended and can not be resolved. Similar to the ground-based photometry, we will thus report line-fluxes integrated over the complete host galaxy complex (Section~\ref{EmissionLines}) in the following.}
\subsection{Host SED}
\label{SED}
{Fitting the HST and LIRIS NIR photometry of the entire host system together with published broad-band magnitudes \citep[see][for details]{2011arXiv1108.0674K} in LePhare\footnote{\texttt{http://www.cfht.hawaii.edu/$\sim$arnouts/lephare.html}} \citep{1999MNRAS.310..540A, 2006A&A...457..841I} yields the galaxy parameters listed in Table~\ref{tab:hostprop}. Here we assumed models from \citet{2003MNRAS.344.1000B} based on an initial mass function (IMF) from \citet{2003PASP..115..763C} and a Calzetti dust attenuation law \citep{2001PASP..113.1449C}. Given that both method and data are largely unchanged, these values are only slightly refined with respect to those computed by \citet{2011arXiv1108.0674K}.}
\input{Tabs/hostprop.tex}
\subsection{Emission line fluxes}
\label{EmissionLines}
\begin{figure*}
\centering
\includegraphics[width=1.5\columnwidth]{Figs/O3.pdf}\\
\includegraphics[width=1.5\columnwidth]{Figs/N2.pdf}
\caption{Continuum-subtracted emission lines used to determine the gas-phase metallicity in the X-shooter spectrum, as well as the \sii~doublet. The black line shows the raw spectrum including errors, and the red-shaded areas denote the 90\% confidence region of the fit of the emission lines using Gaussians. Grey shaded areas denote wavelength regions that have been omitted in the fitting due to skyline contamination.}
\label{fig:eml}
\end{figure*}
{In the measurement of emission line fluxes (Table~\ref{tab:eml}), the redshift (i.e., line centroids) and line widths were fitted simultaneously by tying weak emission lines to those detected at high S/N. In detail, we linked the parameters of the two components of the \oii~doublet in the visual arm, as well as the Gaussian widths and centroids of the various emission lines in the NIR arm. Although the emission of the forbidden and recombination lines does not necessarily arise from the same physical components, the assumption of a common redshift and line width provides a fair approximation and a good fit to the data (Fig.~\ref{fig:eml}). The robustness of the procedure is further supported by, within errors, unchanged line parameters, fluxes and flux ratios when using different combination of ties (i.e., free FWHM, tying H$\beta$ to [\ion{O}{iii}]$\,(\lambda$5007) or all lines except \oii~to each other) {or allowing for multiple Gaussians components in the individual lines}.}
{In addition, we cross-checked our method by numerically integrating the flux of the emission lines. Here, errors were estimated via Monte-Carlo techniques. This results in values that are consistent with those of the Gaussian fitting at $2 \sigma$ confidence, but is more sensitive to skylines and small-scale irregularities in the data. It further disregards the physical information of a common redshift, and hence results in larger errors than the Gaussian fitting in particular for lines with low S/N, or those affected by skylines.
We thus report fit-based values in Table~\ref{tab:eml}. Our conclusions remain unchanged when using different Gaussian fitting methods or numerical integration techniques for the line flux measurements.}
From the observed FWHM of the \oii~doublet ($\approx$ 5~\AA), and assuming a resolving power of 8800 of X-shooter's VIS arm, we derive a measured velocity dispersion $\sigma$ of around $\sigma \sim $ 50~km~s$^{-1}$ for the host galaxy complex, comparable to star-forming systems of similar mass observed through gravitational lenses \citep[e.g.,][]{2010MNRAS.406.2616C}.
We do not detect significant emission from the resonant Ly$\alpha$ transition. Using the redshift, and assuming an intrinsic FWHM of twice the recombination lines \citep[e.g.,][]{2010MNRAS.408.2128F}, we set a limit on the Ly$\alpha$ flux of $4.7\times10^{-17}\,\rm{erg}\,\rm{cm}^{-2}\,\rm{s}^{-1}$ ($7.6\times10^{-17}\,\rm{erg}\,\rm{cm}^{-2}\,\rm{s}^{-1}$ after matching the spectrum to photometry) at the redshifted Ly$\alpha$ wavelength of $3210~\AA$. It is estimated from an artificial emission line added on top of the sky contribution at the respective wavelength range, folded with the error spectrum and represents the flux that is detected at a combined S/N~$>$~3 in 99\% of all iterations. Similar limits are obtained when allowing for an offset of several ten to few hundreds of $\rm{km}\,s^{-1}$ for Ly$\alpha$ with respect to the recombination lines \citep{Bo2012}. The non-detection of Ly$\alpha$ is further discussed in Section~\ref{LyA}.
In the further analysis, we matched the spectrum to broad-band photometry (see Section \ref{Xsred}), and applied when appropriate the correction for an average stellar Balmer absorption using a rest-frame equivalent width of {1~\AA~\citep{2008ApJ...686...72C, 2011ApJ...730..137ZF}}, and for host galaxy extinction using the Balmer decrement (see Section \ref{Balmer}). The corresponding wavelength-dependent factors are shown in Table~\ref{tab:eml}.
\input{Tabs/eml.tex}
Comparing the emission line ratios of [\ion{O}{iii}]/H$\beta$ versus \nii~($\lambda$6584)/H$\alpha$ against standard diagnostic relations \citep[e.g.,][]{2001ApJ...556..121K, 2003MNRAS.346.1055K}, a significant contribution of an AGN to the host emission of GRB~080605 is readily excluded. Measurements of emission line fluxes and upper limits are reported in Table~\ref{tab:eml}.
\subsection{Balmer decrement}
\label{Balmer}
The ratio between the Balmer lines H$\alpha$ and H$\beta$ is a tracer of the visual extinction towards the \ion{H}{ii} regions. We used the respective photometry-matched and stellar Balmer absorption corrected line fluxes to derive the intrinsic Balmer ratio, which is a direct measure of the selective reddening, or the total visual extinction under the assumption of a specific extinction law (and treating the \ion{H}{ii} regions as point-like). This probes a different physical quantity than the reddening value inferred from fitting the galaxy's SED from Table~\ref{tab:hostprop}, as the SED modeling is sensitive to the attenuation of the stellar component which depends on the topology of the ISM and dust and galaxy geometry \citep[e.g.,][]{2004ApJ...617.1022P}
Under standard assumptions of electron density ($10^2\, \rm{cm}^{-3} \lesssim n_e \lesssim 10^4$~cm$^{-3}$, see also Sect.~\ref{Te}) and temperature ($T_e \sim 10^4$~K) for case B recombination \citep{1989agna.book.....O}, the Balmer ratio indicates an average reddening towards the \ion{H}{ii} regions of $E_{B-V}^{\rm{gas}} = 0.07_{-0.07}^{+0.13}$~mag. This corresponds to $A_{V}^{\rm{gas}} = 0.22_{-0.22}^{+0.40}$~mag when assuming a MW-type extinction law with $R_V = 3.1$ \citep{1989ApJ...345..245C}. The reddening corrections according to the Balmer decrement for all emission lines except Ly$\alpha$ are fairly robust and independent on the assumption of a specific extinction law, as there is little difference within the wavelength range of the Balmer lines between sight-lines through local galaxies \citep[e.g.,][]{1992ApJ...395..130P} or with respect to extra-galactic extinction laws derived from GRB afterglows \citep{2011arXiv1110.3218S}.
\subsection{Electron density}
\label{Te}
The flux ratio between the two components of the \oii~doublet is sensitive to the electron density \citep{2006agna.book.....O}. Individual components are resolved and well detected because of the high spectral resolution of X-shooter (see Fig.~\ref{fig:o2eml}). Assuming an electron temperate $T_e$ of $10^4$~K, we derive an electron density of $n_e \sim 200\,\rm{cm}^{-3}$. This value is typical for Galactic \ion{H}{ii} regions \citep[e.g.,][]{2000A&A...357..621C}. The low significance of the detection of the \sii~doublet (see Fig.~\ref{fig:eml}) prevents meaningful constraints on the electron density based on \sii.
\begin{figure}
\centering
\includegraphics[width=.8\columnwidth]{Figs/O2.pdf}
\caption{The continuum-subtracted \oii~$(\lambda\lambda\, 3726\, 3729)$ doublet. Lines and shadings are the same as in Fig.~\ref{fig:eml}.}
\label{fig:o2eml}
\end{figure}
\subsection{Star-formation rate}
Emission line fluxes of H$\alpha$ and \oii, as well as the UV continuum flux trace the un-obscured star-formation within a galaxy \citep{1998ARA&A..36..189K, 2004AJ....127.2002K}. The H$\alpha$-derived value presents the most reliable optical indicator of a galaxy's SFR, as it is independent of metal abundances, and less sensitive to the uncertainties in the visual extinction than the other methods. SFR values depend quite strongly on the assumption of the IMF. In the following, we report values based on the initial formulation of the SFR from \citet{1998ARA&A..36..189K}, but converted to a Chabrier IMF \citep{2003PASP..115..763C}\footnote{Assuming a Salpeter IMF would increase all SFR estimates by a factor of $\approx 1.7$ \citep[e.g.,][]{2009ApJ...706.1364F}.}. Based on H$\alpha$, the SFR of the host of GRB~080605 is SFR$_{\rm{H}\alpha} = 31^{+12}_{-6}\,M_{\sun}\,\rm{yr}^{-1}$. Here we used correction-factor and its error (see Tab.~\ref{tab:eml}), and thus include the uncertainty in the flux calibration and host-intrinsic reddening. {SFRs from \oii~(SFR$_{\oii} = 55^{+55}_{-22}\,M_{\sun}\,\rm{yr}^{-1}$) and the SED modeling (SFR$_{\rm{SED}} = 49_{-13}^{+26}\,M_{\sun}\,\rm{yr}^{-1}$) are within the larger uncertainties in good agreement with the H$\alpha$-derived value.}
Optically derived SFRs do not provide a full picture of the total (obscured and unobscured) star-formation in a galaxy. Based on sub-mm and radio measurements, there is evidence that the total SFR of few GRB-selected galaxies can be around or even higher than $100 \,M_{\sun}\,\rm{yr}^{-1}$ \citep[e.g.,][]{2003ApJ...588...99B, 2004MNRAS.352.1073T, Michal2012}. The sample of sub-mm detected GRB hosts, however, is still very limited \citep[e.g.,][]{2008ApJ...672..817M}, and a full census of the actual SFR of GRB hosts will have to await the advent of statistically representative samples observed with sensitive far-infrared, sub-mm or radio observatories such as \textit{Herschel}, ALMA or the eVLA.
Despite these limitations, optically-derived SFRs are well-established tools for the characterization of galaxies. We will thus put the host of GRB~080605 into the context of SFRs from field galaxies derived in a similar manner, with the caveat that the reported SFRs might trace only a fraction of the total SFR of a given galaxy.
Together with the stellar mass measurement of $M_* = 8.0^{+1.3}_{-1.6}\times 10^9M_{\sun}$, the specific SFR (sSFR$_{\rm{H}\alpha}$ = SFR$_{\rm{H}\alpha}$/$M_{*}$) and growth timescale $\tau=1/\rm{sSFR_{\rm{H}\alpha}}$ are 4~Gyr$^{-1}$ and 260~Myr, respectively, making the host of GRB~080605 a highly active and star-bursting galaxy.
\subsection{Metallicity}
The gas-phase metallicity of galaxies is typically measured using different diagnostic ratios of emission lines originating in \ion{H}{ii} regions \citep[see e.g.,][and references therein]{2008ApJ...681.1183K}. Most commonly used are the $R_{23}$ calibrator, that requires measurements of line fluxes from \oii, [\ion{O}{iii}]~and H$\beta$ \citep{1979MNRAS.189...95P, 1991ApJ...380..140M, 2004ApJ...617..240K}, the O3N2 and N2 diagnostics which uses ratios of [\ion{O}{iii}], and H$\beta$, and/or \nii~and H$\alpha$ \citep{1979A&A....78..200A, 2004MNRAS.348L..59P}, and the N2O2 indicator via \oii~and \nii~\citep{2002ApJS..142...35K}. For a detailed description on the individual strong-line diagnostics we refer to \citet{2008ApJ...681.1183K}.
The $R_{23}$ method is double-valued but its degeneracy can be broken via the line ratios of {\nii/H$\alpha$ or $\nii/\oii$. In our case, \nii/H$\alpha=0.14\pm0.02 $ and $\nii/\oii =0.10\pm0.04$.}
The significant flux detected in \nii~strongly points to the upper branch solution. Similarly, the N2O2 ratio is only applicable for high metallicities with $\log(\nii/\oii) > -1.2$. Due to the large difference in wavelength of the lines used by N2O2 and $R_{23}$, their values are sensitive to the reddening in the host and wavelength-dependent errors in the flux calibration.
Both the O3N2 (see Eq.~1) and N2 methods, however, use flux ratios of adjacent emission lines, which are relatively close in wavelength space, and in our case are all within the NIR arm. The observed lines of [\ion{O}{iii}]~and H$\beta$ are located in the $J$, and \nii~and H$\alpha$ in the $H$-band. Errors in the flux calibration or systematic uncertainties due to flat-fielding, slit-losses, intrinsic host extinction or reddening in the Galaxy are hence not going to affect the overall metallicity measurement in this case.
Based on O3N2, for example, the oxygen abundance is \citep{2004MNRAS.348L..59P}:
\begin{equation}
12 + \log(\rm{O}/\rm{H}) = 8.73-0.32\times\log\left({\frac{F_{[\ion{O}{iii}](\lambda5007)}/F_{\rm{H}\beta}}{F_{\nii(\lambda6584)}/F_{\rm{H}\alpha}}}\right)
\end{equation}
{which is $12 + \log(\rm{O}/\rm{H}) = 8.31\pm0.02$ for GRB~080605. Using the N2, $R_{23}$ and N2O2 strong-line indicators, the oxygen abundance is $12 + \log(\rm{O}/\rm{H}) = 8.36\pm0.03$ for N2, $12 + \log(\rm{O}/\rm{H}) = 8.45^{+0.09}_{-0.12}$ for $R_{23}$, and $12 + \log(\rm{O}/\rm{H}) = 8.60^{+0.11}_{-0.19}$ for N2O2. Here, all errors are based on the uncertainties of the line flux measurement and the correction factor only (see Table~\ref{tab:eml}), and do not include the systematic error inherent to the calibrator. Errors correctly reflect the larger uncertainty in the strong-line diagnostics that include flux ratios between lines in different wavelength regimes ($R_{23}$ and N2O2). Oxygen abundances based on different indicators are further summarized in Table~\ref{tab:met}. We use the appropriate diagnostics when comparing to literature values.}
\input{Tabs/met.tex}
\section{Discussion}
The metallicity of GRB hosts is measured either directly in absorption using the bright afterglow emission, or, as in this work, in emission via host galaxy spectroscopy and strong-line diagnostics. In the former case, measurements are typically restricted to $z \gtrsim 2$ \citep[e.g.,][]{2006A&A...460L..13J, 2006NJPh....8..195S}, while the latter case requires NIR spectroscopy for $z \gtrsim 1$. Emission line metallicities are calibrated on local samples, and hence depend on the assumption that the physical processes underlying these diagnostic ratios are still valid at high redshift. {There is hence considerable systematic uncertainty between metallicities derived directly in absorption or through emission lines. Furthermore, metallicity measurements at high-redshift via the different techniques are available for only a few objects. They tend to agree reasonably well \citep[see e.g.,][]{2010A&A...510A..26D}, but systematic effects in a direct comparison remain hard to quantify until larger samples of objects with both, ISM as well as gas-phase metallicities, become available.}
Our measurement of the gas-phase metallicity of the host of GRB~080605 represents a first view into the metal abundances of GRB hosts in the redshift range $1 \lesssim z \lesssim 2$ (Fig.~\ref{fig:met}). Generally, the distribution in metallicity as inferred from GRB-DLAs shows a large dispersion with values $0.01 \lesssim Z/Z_{\sun}$ \citep{2004A&A...419..927V, 2008ApJ...685..344P, 2010ApJ...720..862R} to solar or even super-solar \citep{2009ApJ...691L..27P, 2003ApJ...585..638S, 2011arXiv1110.4642S}. A similar spread in metallicities is also found in hydrodynamical solutions of individual sight-lines through GRB hosts \citep{2010MNRAS.402.1523P}. The metallicity derived from afterglow spectroscopy could be dominated by sight-line effects, and large samples might be required to assess the general properties of GRB hosts via afterglow spectroscopy in a statistical approach.
Host-integrated metallicities via emission lines should therefore give a more self-contained picture of the metal-enrichment of the ISM in high-redshift GRB hosts. Galaxy metallicity measurements are however challenging observationally in a stellar mass range around or below $10^{10}\,M_{\sun}$, and thus still sparse, in particular at $z > 1$. Current GRB host samples are furthermore subject to complex selection biases \citep{2011arXiv1108.0674K}, which are only resolved through statistical samples of high completeness \citep[e.g.,][]{2009ApJS..185..526F, 2009ApJ...693.1484C, 2011A&A...526A..30G, 2011arXiv1112.1700S, Jens2012}. Consequently, a consistent picture of the relation between galaxies selected through GRBs and normal field galaxies is not yet reached.
\subsection{The host of GRB~080605 within the sample of GRB hosts}
{With respect to previous GRB host galaxies, the metallicity, stellar mass and star-formation rate of the host of GRB~080605 are relatively high (see Fig.~\ref{fig:met}). With a metallicity around half solar, a stellar mass of $8\times 10^9~M_{\rm{\sun}}$ and SFR $\sim 30~M_{\sun}\,\rm{yr}^{-1}$, it is significantly enriched with metals and vigorously forming stars.} This contradicts the suggestion, that an upper metallicity limit\footnote{Adopted to our reference solar oxygen abundance.} for cosmological, $z \gtrsim 1$, GRBs of $Z \lesssim 0.2\, Z_{\sun}$ exists \citep{2006AcA....56..333S}.
The substantial gas-phase metallicity of the host is even more intriguing, as GRB~080605 itself is energetic. The inferred isotropic-equivalent energy release in $\gamma$-rays is $E_{\gamma,\rm iso} \sim 2.2\times10^{53}$~erg as calculated from the prompt emission data from \citet{2008GCN..7854....1G}. This value puts GRB~080605 within the most-energetic 15\% of all \textit{Swift} bursts \citep{2007ApJ...671..656B, 2010ApJ...711..495B}.
A connection between host metallicity and $\gamma$-ray energy release of the GRB, or a metallicity cut-off might be expected in the collapsar scenario \citep{1993ApJ...405..273W, 1999ApJ...524..262M}, for example. Progenitor stars with lower metallicities are likely to have higher angular momentum due to smaller wind losses, and thus result in a more energetic explosion \citep[e.g.,][]{2005A&A...443..581H, 2005A&A...443..643Y}. An energetic burst such as GRB~080605 would hence be more likely in a low-metallicity environment, in contrast to our observations. Our observations are, however, in line with the work of \citet{2007MNRAS.375.1049W} and \citet{2010ApJ...725.1337L}, who find no correlation between $E_{\gamma, \rm iso}$ and host metallicity in 18 GRBs at $z < 1$.
{The role of metallicity in long GRB progenitors is thus far from being understood. The metallicity distribution of a representative GRB host sample will indirectly also allow us to put constraints on the metal content of the progenitor. For example, complex scenarios of stellar evolution, or binary models for the formation of long GRBs \citep[e.g.,][]{1999ApJ...526..152F} can both relax the constraints on progenitor metallicity. In addition, even within a metal-rich galaxy a metal-poor progenitor could in principle form in specific regions of fairly primordial chemical composition such as gas inflows or in (merging) galaxies with substantial diversity in their metal enrichment.} To first order, however, the gas-phase, i.e., \ion{H}{ii}-region averaged, metallicity should provide a fair representation of the chemical evolution of the galaxy as a whole.
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{Figs/GRBMET.pdf}
\caption{{Metallicity of the host of GRB~080605 (star). Other GRB host metallicities are shown with black circles and upward/downward triangles as compiled and in the scale of \citet{2009ApJ...691..182S} (top panel) and \citet{2011MNRAS.414.1263M} (bottom panel).} Field galaxies are shown as grey dots \citep{2005ApJ...635..260S, 2009A&A...495...73P, 2009MNRAS.398.1915M, 2009ApJ...691..140H, 2011MNRAS.413..643R}, and in similar metallicity scales as the GRB measurements. Absorption metallicities from GRB afterglows \citep[][and references therein]{2010ApJ...720..862R, 2010A&A...523A..36D, 2011arXiv1110.4642S, 2011100219A} and QSOs \citep{2003ApJS..147..227P} are plotted as black and grey diamonds, respectively. {Errorbars for individual events in the comparison samples are omitted to enhance clarity.} The error bars at the bottom right corner of each panel illustrate {uncertainties of 0.2~dex., which are typical for both, GRB-DLA \citep[e.g.,][]{2010ApJ...720..862R, 2011100219A} and GRB host metallicity \citep[e.g.,][]{2011MNRAS.414.1263M} measurements}.}
\label{fig:met}
\end{figure}
The galaxy hosting GRB~080605 has indeed a disturbed morphology, indicative of an early merger or intrinsically clumpy structure. A merger could have also triggered the enhanced star formation of the host of GRB~080605 when compared to GRB hosts at low redshift \citep[e.g.,][]{2009ApJ...691..182S, 2010AJ....139..694L}. GRB hosts with similar SFR, however, might not be uncommon at $z > 1$. A good fraction of X-ray selected GRB hosts at $1 < z < 2$ has observed $R$-band brightnesses (probing the rest frame UV) in a range between $24$~mag and $22.5$~mag \citep{2009AIPC.1111..513M} indicating dust un-corrected SFRs up to 10~$M_{\sun}\rm{yr}^{-1}$. Already mild dust-attenuation in the host can easily increase this to values of 50~$M_{\sun}\rm{yr}^{-1}$ or even higher, illustrating that GRB hosts with SFRs significantly above 10~$M_{\sun}\rm{yr}^{-1}$ are not an exceptionally rare phenomenon at $z > 1$ \citep[see also e.g.,][]{2005ApJ...633..317F, 2011ApJ...727L..53C, 2011arXiv1108.0674K, 2011arXiv1110.4642S}.
\subsection{Afterglow versus host properties}
The substantial gas-phase metallicity of $Z\sim Z_{\sun}/2$ might directly relate to the substantial $A_V \sim 0.5$~mag including the presence of the 2175~\AA~dust feature as observed in the afterglow SED \citep{2011A&A...526A..30G} and spectrum \citep{2011Zafar}. A metallicity of around solar was also inferred from GRB-DLAs for GRBs~070802 and 080607, both of which were substantially reddened, and had 2175~\AA~dust features \citep{2009ApJ...697.1725E, 2009ApJ...691L..27P, 2010arXiv1009.0004P} as well. This seems to support the association between the 2175~\AA~bump and chemically evolved galaxies \citep[e.g.,][]{2009A&A...499...69N}. With only a small handful of such events, however, no strong conclusions can be drawn, yet.
\subsection{The mass-metallicity relation at $z\sim 2$}
Having the key parameters of stellar mass, metallicity and SFR of the host of GRB~080605 at hand, we can now investigate its relation to the mass-metallicity ($M_*$-$Z$) relation at $z\sim2$ \citep[e.g.,][]{2006ApJ...644..813E}. A further basic property is the host's location with respect to the fundamental metallicity relation (FMR) defined by SDSS galaxies in a mass range between $ 9.2 \lesssim \log (M_*) \lesssim 11.4$. The FMR connects $M_*$, $12+\log(\rm{O}/\rm{H})$, and SFR \citep{2010MNRAS.408.2115M} via:
\begin{equation}
\label{fmp}
12+\log(\rm{O}/\rm{H}) = 8.90 + 0.47\times(\mu_{0.32}-10)
\end{equation}
where $\mu_{0.32} = \log (M_*\,[M_{\sun}]) - 0.32 \times \log($SFR$_{\rm H\alpha}\,[M_{\sun}\,\rm{yr}^{-1}])$. {The oxygen abundance for GRB~080605 on the \citet{2010MNRAS.408.2115M, 2011MNRAS.414.1263M} scale is $12+\log(\rm{O}/\rm{H}) = 8.52\pm0.09$. The value derived from $M_*$ and SFR via Eq.~\ref{fmp} is consistent with it ($12+\log(\rm{O}/\rm{H}) = 8.63\pm 0.08$). Errors are again based on the statistical uncertainty of line-flux measurement, correction factor and stellar mass only\footnote{A systematic error on the stellar mass estimate of $\pm0.2$~dex., for example, would translate into additional systematic errors of $\pm 0.12$ on the derived metallicity.}.}
This establishes the host of GRB~080605 as a star-forming galaxy which has no significant deficit of metals with respect to star-forming galaxies at low redshift for its given mass and SFR. Or, conversely, the selection through the energetic GRB~080605 does not lead to its host being metal-poor with respect to field galaxies of comparable stellar mass and SFR.
The host of GRB~080605 hence provides the opportunity to probe the mass-metallicity relation at $z\sim 2$ \citep[e.g.,][]{2006ApJ...644..813E} at lower stellar masses (Fig.~\ref{fig:mmetz}). If populated with more events, GRB hosts can thus provide unique constraints on the low-mass end of the $M_*-Z$ relation \citep[see also e.g.,][]{2011arXiv1107.3841V} similar to measurements via gravitationally lensed objects \citep[e.g.,][]{2012arXiv1202.5267W} but without the need for (and uncertainty of) a detailed lens model (Fig.~\ref{fig:mmetz}).
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{Figs/MZ.pdf}
\caption{The host of GRB 080605 with respect to the mass-metallicity relation at $z\sim2$. Different colored symbols represent the averaged galaxy distribution from \citet{2006ApJ...644..813E} in black, as well as individual sources from \citet{2006ApJ...645.1062F, 2009ApJ...706.1364F} in green, from \citet{2009ApJ...697.2057L} in blue and gravitationally lensed galaxies from \citet{2009ApJ...699L.161Y} and \citet{2010ApJ...719.1168E} in cyan and grey, respectively. Upper limits are shown with downward triangles with the same color-coding. All measurements are in the N2 scale of \citet{2004MNRAS.348L..59P}. The horizontal dashed-dotted line marks the solar oxygen abundance. {The dashed line is the local $M-Z$ relation \citep{2004ApJ...613..898T}, which is also shown shifted (solid line) to the observations at $z\sim 0.7$ \citep{2005ApJ...635..260S}}. Approximate systematic errors on the N2 metallicity scale and the mass determination are indicated in the top left corner.}
\label{fig:mmetz}
\end{figure}
\subsection{The non-detection of Ly$\alpha$}
\label{LyA}
The luminosity-independent selection of star-forming galaxies through GRBs offers a unique probe of the escape fraction ($f_{\rm{esc}}$) of Ly$\alpha$ photons. The path length of resonantly scattered Ly$\alpha$ photons depends on the geometry and kinematics of \ion{H}{i} within a galaxy, and could thus be greatly enhanced as compared to, for example, the path length of photons from recombination lines such as H$\alpha$. The longer path length directly translates into a higher dust absorption probability for Ly$\alpha$ photons and hence $f_{\rm{esc}}$ might end up anywhere below unity \citep[e.g.,][]{2009A&A...506L...1A}.
Ly$\alpha$ emission from GRB hosts was detected in both narrow-band imaging and afterglow/host spectroscopy \citep[e.g.,][]{2003A&A...406L..63F, 2005MNRAS.362..245J, 2010A&A...522A..20D, Bo2012}. The broad wavelength coverage of X-shooter extending down to the UV (Ly$\alpha$ line at $z\sim 1.64$ is redshifted to 3210~\AA) coupled with the tight constraints on the galaxies reddening and extreme luminosity of H$\alpha$, makes the host of GRB~080605 an ideal test case for the escape fraction in a high-redshift environment.
At $f_{\rm{esc}} = 1$, the intrinsic ratio between Ly$\alpha$ and H$\alpha$ is 8.7 \citep{1971MNRAS.153..471B}. Consequently, Ly$\alpha$ is expected to be a factor $12$ more luminous than our non-detection implies. {This corresponds to an escape fraction of $f_{\rm{esc}} < 0.08$, which was estimated in the same way as the flux limit but using the photometry-matched spectrum and its errors as discussed in Section~\ref{Xsred}.}
While the evidence for reddening from the recombination lines and the stellar continuum is weak, the properties of the afterglow \citep{2011Zafar} provide compelling evidence that there is enough dust in the ISM to absorb the scattered Ly$\alpha$ photons efficiently.
Our limit is consistent with previous estimates using narrow-band surveys targeting both Ly$\alpha$ and H$\alpha$ \citep{2010Natur.464..562H} or measured from the column density distribution of GRB-DLAs \citep{2009ApJS..185..526F}. A larger sample of hosts observed in similar fashion can provide competitive constraints on the average escape fraction in high-redshift environments at $1.6 < z < 2.5$. These measurements would be completely independent on conventional selection techniques, and representative of young, star-forming galaxies common in the early Universe. Establishing the average escape fraction at cosmological distances and for typical star-forming galaxies has strong implications for the use of Ly$\alpha$ emission as a tracer of star-formation and luminosity functions derived from Ly$\alpha$ galaxies at the highest redshifts.
\section{Conclusions}
{We presented medium-resolution optical/NIR spectroscopy and ground and space-based imaging of the galaxy selected through GRB~080605 at $z=1.64$. Our HST imaging probes and resolves the large-scale structure of the host, and shows it to be a morphologically complex system that consists of two components separated by 8.6~kpc.} An X-shooter spectrum covering its rest-frame UV-to optical wavelength range (1150 to 8700~\AA) reveals a wealth of emission lines, including \oii, [\ion{O}{iii}], H$\beta$ as well as \nii~and H$\alpha$. These recombination and forbidden lines allow us to put unique constraints on the conditions of the ISM in the host. It is in particular the first robust measurement of the gas-phase metallicity of a GRB host at $z > 1$ using strong-line indicators based on \nii~($\lambda$6584).
The host of GRB~080605 is significantly enriched with metals with an oxygen abundance $12 + \log(\rm{O}/\rm{H})$ between 8.3 and 8.6 ($0.4\,Z_{\sun} < Z < 0.8\,Z_{\sun}$) for several different strong-line diagnostics. In addition, its stellar mass is $M_* = 8.0^{+1.3}_{-1.6} \times 10^9 M_{\sun}$ and the galaxy is extremely star-forming (SFR$_{\rm{H}\alpha} = 31^{+12}_{-6}\,M_{\sun}\,\rm{yr}^{-1}$, sSFR$_{\rm{H}\alpha} = 4$~Gyr$^{-1}$). With a gas-phase metallicity above 40\% of the solar value and luminosity above $L^{*}$ \citep{2011arXiv1108.0674K}, it contrasts many observation of GRBs at lower redshift, which typically showed their hosts to be sub-luminous and metal-poor galaxies. Coupled with the high energy-release in $\gamma$-rays of $E_{\gamma, \rm iso} \sim 2.2\times10^{53}$~erg, it challenges those GRB progenitor models in which the formation of energetic GRBs requires very low metallicities.
{The metallicity measurement of the host of GRB 080605 directly shows that GRB hosts at $z > 1$ are not necessarily metal-poor, both on absolute scales as well as relative to their stellar mass and SFR.} Our detailed spectroscopic observations in fact suggest that the hosts of GRBs in general might provide a fair representation of the high-redshift, SFR-weighted population of ordinary star-forming galaxies.
GRB hosts thus offer a selection of star-forming galaxies at high redshifts, including objects in the low-mass ($M_* \lesssim 10^{10} \,M_{\sun}$) regime, which are challenging to study otherwise. Targeted spectroscopic investigation become feasible through the afterglow's redshift, its sub-arcsec position and the substantial star-formation within GRB-selected galaxies.
Similar data for a representative and statistically significant sample of GRB hosts hold the key for understanding the nature of GRB hosts in particular and give important insights into the high-redshift population of star-forming galaxies in general. Furthermore, they yield the fundamental information to establish GRBs as probes of the star-formation up to the era of re-ionization. With the availability of highly redshift-complete GRB, afterglow and host samples such as TOUGH\footnote{\texttt{http://www.dark-cosmology.dk/TOUGH}} \citep[][Malesani et al., in prep.]{Jens2012, Palli2012, Bo2012, Tom2012, Michal2012} and NIR spectroscopy with X-shooter these studies are now feasible for the first time, and will continue to open the window with respect to the properties of GRB hosts in the previously unexplored redshift range $1 \lesssim z \lesssim 3$.
\begin{acknowledgements}
We thank L. Christensen and S. Savaglio for important insights and valuable discussion, and A. Rau for providing data for Figure 5 in machine-readable format. We also thank the referee for constructive comments, that helped to improve the quality of the manuscript. TK acknowledges support by the European Commission under the Marie Curie Intra-European Fellowship Programme in FP7. JPUF acknowledges support from the ERC-StG grant EGGS-278202. The Dark Cosmology Centre is funded by the Danish National Research Foundation. Based on observations made with the NASA/ESA Hubble Space Telescope, obtained from the data archive at the Space Telescope Institute.
\end{acknowledgements}
\input{bib.tex}
\end{document}
|
{
"timestamp": "2012-09-11T02:04:19",
"yymm": "1203",
"arxiv_id": "1203.1919",
"language": "en",
"url": "https://arxiv.org/abs/1203.1919"
}
|
\section{Introduction}
Gauge symmetries play a crucial role in modern particle physics. However, they also imply massless gauge bosons. To reconcile gauge invariance with massive gauge interaction mediators, one invokes spontaneous symmetry breaking. This is precisely the case with the electroweak interactions of the Standard Model of particle physics. Unfortunately though, the origin of the symmetry breaking has not yet been established experimentally. A leading candidate for the underlying mechanism is
the condensation of a weakly coupled fundamental scalar field,
whose manifestation in nature would be the existence of the Higgs boson. There has been a lot of excitement recently about a possible discovery of a spin-zero particle at the LHC, which might be the Higgs. However, an actual confirmation requires more data. Furthermore, there are other possible explanations for the existence of a (fundamental or composite) scalar with mass in the relevant energy range. Regardless of the experimental status of the Higgs boson, however, the existence of a fundamental scalar would lead to well-know theoretical problems. This motivates the search for alternative explanations of the origin of mass in the Standard Model.
An appealing alternative relies on dynamical effects as the cause of symmetry breaking. In this approach, the role of the Higgs is played by a composite scalar, formed as a result of strongly coupled gauge dynamics. Theories of that kind are known as technicolor theories \cite{Tech}. In fact, to be phenomenologically viable, they have to be characterized by a gauge coupling that varies very slowly in an intermediate energy range. This phenomenon is called walking, in analogy with the more familiar running of couplings. Walking technicolor was first proposed in \cite{WalkTech}. There has been a lot of work on this subject over the years. However, a major hindrance for progress has been the need to study strongly coupled gauge theories. This is clearly, beyond standard perturbative QFT methods. As a result, the computation of electroweak observables from models of dynamical electroweak symmetry breaking has long been a challenge.
In recent years, a new theoretical tool was developed, that addresses precisely the investigation of the non-perturbative regime of gauge theories. This is the gauge/gravity duality, which allows one to study the strongly coupled regime of a gauge theory via a weakly coupled (called dual) gravitational background in a different number of dimensions. Thus non-perturbative QFT problems are mapped to (almost) classical gravitational computations. This powerful method has already produced interesting insights into a variety of phenomena, ranging from hydrodynamics \cite{hydro} to superconductivity \cite{scond}.
Of, perhaps, greatest interest for particle physics is the gravity dual, studied by Sakai and Sugimoto \cite{SS}. This is a dual of a gauge theory that captures the characteristic features of QCD, including chiral symmetry breaking. It arises from a certain D8-$\overline{{\rm D}8}$ brane configuration in a background sourced by a stack of D4 branes in type IIA string theory. The Sakai-Sugimoto model, as well as a wide variety of other similar D-brane configurations in both type IIA and type IIB, have been easily adapted to study QCD-like technicolor \cite{HolTech}.\footnote{We should also note the existence of a huge literature, loosely inspired by AdS/CFT, on bottom-up holographic technicolor models \cite{PhenoHolTech}, which do not arise from any D-brane construction.} In these models one can compute explicitly the $S$ parameter, which is an important electroweak observable \cite{PT}. However, as is to be expected, the answer is not compatible with the known experimental bounds.
In \cite{LA,ASW}, we studied the gravity dual of a model of walking technicolor. This model has two key ingredients. The first is a certain type IIB background \cite{NPP} sourced by D5 branes wrapped on an $S^2$, whose dual is a walking gauge theory. The second ingredient, which introduces the techniflavor degrees of freedom, is a particular D7-$\overline{{\rm D}7}$ probe-branes embedding \cite{LA} in the background of \cite{NPP}. In \cite{ASW}, we showed that the $S$ parameter of this model can be small enough to be compatible with experiment.\footnote{It is, perhaps, worth pointing out that the recent work \cite{LRTZ} finds a lower bound on the $S$ parameter in a large class of bottom-up holographic technicolor models, which is in conflict with observation. However, the considerations in the bottom-up approach rely on certain assumptions, that are not satisfied in the actual supergravity duals arising from appropriate D-brane configurations. Thus, the results of \cite{LRTZ} do not apply to our model.} This demonstrates, by means of an example, that walking technicolor models can, indeed, satisfy the $S$ parameter test for phenomenological viability.
Here we continue the study of the model of \cite{LA,ASW}. In addition to the spectra of vector and axial-vector mesons, that were investigated in \cite{ASW}, there should also be scalar mesons. The latter arise from the fluctuations of the techniflavor D7-$\overline{{\rm D}7}$ probe branes.
As there are two directions transverse to the probe worldvolume, there are two kinds of scalar mesons.
Since the D7-$\overline{{\rm D}7}$ embedding is not supersymmetric, it is an open question whether the scalar spectrum will contain any tachyonic modes or not. In this paper we show that the embedding is in fact stable with respect to such fluctuations. Namely, we find the mass spectrum of scalar mesons explicitly and thus demonstrate that all mass-squareds are positive. Interestingly, it turns out that the scalar meson spectrum is only slightly shifted compared to the spectrum of vector mesons.\footnote{By that we mean that it is shifted only by terms that are subleading in an expansion in a certain small parameter.} This is not a priori clear,
as examplified by the models of \cite{KS,SS} in which the scalar and vector meson spectra differ by order one terms. The near-degeneracy in our case could indicate that the amount of supersymmetry breaking, induced by our D7-$\overline{{\rm D}7}$ embedding, is small. Recall that a massive ${\cal N} = 1$ vector multiplet has the same field content as a massless vector multiplet and a chiral multiplet. Thus, the massive vector and scalar states belong to the same long ${\cal N}=1$ representation. However, in the present case, the two kinds of scalar mesons turn out to have slightly different mass shifts compared to the vector meson spectrum. This suggests that there may be some
additional mechanism at work,
which would be very interesting to understand in the future.
We should point out that the scalar spectrum investigated here is not directly relevant for settling the interesting open question of whether there is a technidilaton. The latter is a light composite scalar\footnote{By light we mean with mass that is (parametrically) much smaller than the masses of the other states in the spectrum (in particular, the technirho meson).}, which is
the would be pseudo-Goldstone boson associated with the spontaneous breaking of
near-conformal invariance of the walking regime. In models, in which the breaking of conformal invariance is entirely due to the flavor probes (as in \cite{KS,KLP}, for example), the technidilaton (if present) should appear in the spectrum of scalar mesons arising from the fluctuations of those probes. Such mesons are exactly the kind of scalar states studied here. However, in our model the technicolor background already encodes conformal symmetry breaking. Hence, in our case the technidilaton should arise predominantly from the technicolor, not the techniflavor, sector.\footnote{It is rather likely that this state will turn out to be related to the light scalar mode found in \cite{ENP}.} Thus the question of whether our model has a technidilaton is beyond the scope of the present study. We will address this problem in another publication.
The present paper is organized as follows. In Section 2, we briefly recall the basic ingredients of the walking technicolor model of \cite{LA,ASW}. We also derive the action for the two kinds of fluctuations of the flavour probe branes, that give rise to the scalar meson spectrum. In Sections 3 and 4 we study the mass spectra of each of the two kinds of fluctuations in detail. More precisely, we show that to leading order in a small parameter expansion they are the same as the vector meson spectrum of \cite{ASW}. Furthermore, we compute the first subleading correction, that gives the mass shift between the scalar and vector boson spectra. Finally, Appendices A and B contain some technical details of the computations in the main text.
\section{Action for scalar mesons}
\setcounter{equation}{0}
In this section we review the basic ingredients of the technicolor model that we will be studying. More precisely, we recall the background that encodes the technicolor degrees of freedom, as well as the D7-$\overline{{\rm D}7}$ probe-branes embedding that introduces the techniflavour ones. Furthermore, we derive the action for the fluctuations of the D7-$\overline{{\rm D}7}$ embedding, which give rise to the spectrum of scalar mesons in this model.
\subsection{Background and flavour probes}
The background we will consider is due to a stack of D5 branes wrapped on an $S^2$.
It consists of nontrivial ten-dimensional metric, RR 3-form flux and string dilaton.
Furthermore, it is a solution of the type IIB equations of motion with ${\cal N}=1$ supersymmetry.
In \cite{NPP}, this solution was found as an expansion in a small parameter. For our purposes, it is enough to consider only the leading order contributions to the background fields. Note that, to this order, the string dilaton is constant. The leading order metric is \cite{NPP}:
\bea \label{BM}
ds^2 &=& A \left[ dx_{1,3}^2 + \frac{cP_1' (\rho)}{8} \left( 4 d\rho^2 + (\omega_3 + \tilde{\omega}_3)^2 \right) \right. \nn \\
&+& \left. \frac{c\,P_1 (\rho) \coth (2 \rho)}{4} \left( d\Omega_2^2 + d\tilde{\Omega}_2^2 + \frac{2}{\cosh ( 2\rho)} (\omega_1 \tilde{\omega}_1 - \omega_2 \tilde{\omega}_2) \right) \right] \, ,
\eea
where
\be \label{AP1}
A=\left( \frac{3}{c^3 \sin^3 \alpha} \right)^{1/4} , \quad P_1' (\rho) =\frac{\pd P_1 (\rho)}{\pd \rho} \,\, , \quad P_1 (\rho) = \left( \cos^3 \alpha + \sin^3 \alpha \left( \sinh (4 \rho) - 4 \rho \right) \right)^{1/3} \, ,
\ee
\bea
\tilde{\omega}_1 &=& \cos \psi d\tilde{\theta} + \sin \psi \sin \tilde{\theta} d \tilde{\varphi} \,\, , \hspace*{2cm} \omega_1 = d \theta \,\, , \nn \\
\tilde{\omega}_2 &=& - \sin \psi d\tilde{\theta} + \cos \psi \sin \tilde{\theta} d \tilde{\varphi} \,\, , \hspace*{1.6cm} \omega_2 = \sin \theta d \varphi \,\, , \nn \\
\tilde{\omega}_3 &=& d \psi + \cos \tilde{\theta} d \tilde{\varphi} \,\, , \hspace*{3.7cm} \omega_3 = \cos \theta d \varphi
\eea
and
\be
d\tilde{\Omega}_2^2 = \tilde{\omega}_1^2 + \tilde{\omega}_2^2 \,\, , \hspace*{2cm} d\Omega_2^2 = \omega_1^2 + \omega_2^2 = d \theta^2 + \sin^2 \theta d \varphi^2 \,\, .
\ee
The constants $c$ and $\alpha$ above are parameters of the solution and, as in \cite{NPP}, we have set $\alpha' = 1 = g_s$. For an explicit expression for the RR flux, see \cite{NPP}.\footnote{Note also that the background described above is the leading order solution in an expansion in $1/c$ with $c>\!\!>1$.}
This background is dual to a color gauge theory with a coupling constant, which exhibits walking behavior. Namely, in a certain intermediate energy range, the gauge coupling is almost constant as a function of the energy. In this walking region, $\rho$ is always of order 1 or larger \cite{NPP}. Hence, $\coth(2\rho) \approx 1$ while $\frac{1}{\cosh(2\rho)}$ is negligible. In addition, the walking region is characterized by \cite{NPP}:
\be
\beta \equiv \sin^3 \alpha <\!\!< 1 \, .
\ee
As a result, to leading order in small $\beta$, one can use the approximations:
\be
P_1 = 1 \qquad , \qquad P_1' = \frac{2}{3} \beta e^{4 \rho} \, .
\ee
Therefore, the metric (\ref{BM}) simplifies to:
\be \label{leadMetric}
ds^2_{{\rm walk}} = A \left[ \eta_{\mu \nu} dx^{\mu} dx^{\nu} + \frac{c}{12} \beta e^{4 \rho} \left( 4 d\rho^2 + \left( \omega_3 + \tilde{\omega}_3 \right)^2 \right) + \frac{c}{4} \left( d \Omega_2^2 + d \tilde{\Omega}_2^2 \right) \right] .
\ee
To add flavor degrees of freedom, one can introduce D7 probe branes in the above background. In fact, as shown in \cite{LA}, there is an embedding of D7 and anti-D7 branes such that the D7 and $\overline{{\rm D}7}$ merge at some finite value, $\rho_0$, of the radial coordinate. This realizes geometrically chiral symmetry breaking and thus enables us to construct a model of walking technicolor. In \cite{ASW}, we studied the spectra of vector and axial-vector mesons in this model. Here we will investigate the spectrum of scalar mesons. The latter arise from the fluctuations of the ${\rm D}7$-$\overline{{\rm D}7}$ embedding. So let us, first, describe this embedding in more detail.
The eight worldvolume directions of the probe branes are $x^{\mu}$, $\rho$, $\psi$, $\tilde{\theta}$ and $\tilde{\varphi}$. The two transverse space coordinates are functions of $\rho$ only: $\theta = \theta (\rho)$, $\varphi = \varphi (\rho)$. The equations of motion for $\theta (\rho)$ and $\varphi (\rho)$, that arise from the DBI action for the D7 branes, can be solved by the following classical configuration \cite{LA,ASW}:
\be \label{clSol}
\theta_{cl} = \frac{\pi}{2} \qquad {\rm and} \qquad \varphi_{\rho}^{cl} = \pm 2 \sqrt{\frac{\beta}{3}} \,e^{2\rho_0} \frac{1}{\left(1 - e^{4(\rho_0 - \rho)} \right)^{1/2}} \,\,\, ,
\ee
where $\varphi_{\rho}^{cl} \equiv \frac{\pd \varphi^{cl}}{\pd \rho}$.\footnote{Of course, we can integrate $\varphi_{\rho} (\rho)$ to obtain $\varphi (\rho)$. However, this will not be necessary or useful, since the function $\varphi (\rho)$ will appear in the relevant action below only via its derivative.} For future use, let us also introduce a new, worldvolume, radial coordinate:
\be \label{ChVar}
z = \pm \sqrt{e^{4 (\rho - \rho_0) } - 1} \, ,
\ee
which runs over both the ${\rm D}7$ and $\overline{{\rm D}7}$ branes and is such that $z=0$ when $\rho = \rho_0$. Note that, up to an overall constant, this is the same worldvolume variable as the one in equation (4.2) of \cite{ASW}. In terms of $z$, the classical solution for the derivative of $\varphi$ is:
\be \label{derVarPhi}
\varphi_z^{cl} = \frac{\pd \rho}{\pd z} \,\,\varphi_{\rho}^{cl} = e^{2\rho_0} \sqrt{\frac{\beta}{3}} \,\frac{1}{\sqrt{1+z^2}} \,\, .
\ee
Note also that at $z=0$ the above expression for $\varphi_z^{cl}$ is regular, unlike $\varphi_{\rho}^{cl}$ in (\ref{clSol}) at $\rho = \rho_0$. This is a key reason to introduce the variable $z$. Namely, since we would like to expand the field $\varphi$ in fluctuations around the configuration $\varphi_{cl}$, an appropriate choice of coordinate should be such that $\varphi_{cl}$ does not diverge anywhere along its profile.
\subsection{Fluctuations of ${\rm D}7$-$\overline{{\rm D}7}$ embedding}
To find the spectrum of scalar mesons in our model, we want to study 4d spacetime-dependent fluctuations of the ${\rm D}7$-$\overline{{\rm D}7}$ embedding around the configuration (\ref{clSol}). So we promote the embedding functions to fields in spacetime and expand them as:
\be \label{tpFluct}
\theta (z,x^{\mu}) = \theta_{cl} + \delta \theta (z,x^{\mu}) \quad , \quad \varphi (z,x^{\mu}) = \varphi_{cl} (z) + \delta \varphi (z,x^{\mu}) \,\, .
\ee
To obtain the action for the fluctuations $\delta \theta$ and $\delta \varphi$, we need to compute
\be
{\cal L}_{DBI}^{{\rm D}7} = - \sqrt{-\det g_{8d}} \,\, ,
\ee
where $g_{8d}$ is the induced metric on the D7 worldvolume, and expand it to second order in fluctuations.\footnote{We have dropped an overall constant, that is due to the constant string dilaton in this background.} To do that, we substitute
\bea\label{d1}
d\theta = \theta_z dz + \theta_{\mu} dx^\mu \quad , \quad d\varphi = \varphi_z dz + \varphi_{\mu} dx^\mu
\eea
in (\ref{leadMetric}) with the change of variable (\ref{ChVar}) performed. Note that $\theta_z$, $\theta_{\mu}$ and $\varphi_{\mu}$ are all first order in the fluctuations, since $\theta_{cl} = const$ and $\pd_{\mu} \varphi_{cl} = 0$. However, $\varphi_z$ does contain a zeroth order contribution from $\varphi_{cl}$. The details of the computation of $\sqrt{-\det g_{8d}}$ can be found in Appendix A. The resulting Lagrangian is the following:
\be\label{Lz}
{\cal L} = const \sqrt{3(1+z^2) \!\left[4\,\theta_z^2+\sin^2 \!\theta \left(4+c\,\theta_\mu^2\right)\varphi_z^2\right]+e^{4 \rho_0}\beta\,z^2\left(4+c\,\theta_\mu^2+c\,\varphi_\mu^2\right)} \, .
\ee
To study the action for the fluctuations to quadratic order, we need to expand the last expression further. First, however, note that all terms under the square root are quadratic in the fields and there is no mixing between the $\theta$ and $\varphi$ contributions to second order; in particular $\sin^2 \!\theta = \sin^2 \!\left( \frac{\pi}{2} + \delta \theta \right) \approx 1 - (\delta \theta)^2$. Hence, the desired action for each of the two fields can be obtained by first turning off the fluctuations of the other field and then expanding to second order. We will do this in turn in the next two sections.
Before concluding the present section, though, let us perform a consistency check on the Lagrangian (\ref{Lz}). Namely, let us verify that it does have as a solution of its equations of motion the same classical configuration as (\ref{clSol}), (\ref{derVarPhi}). Turning off the $x^{\mu}$-dependence, which only arises at the level of fluctuations, we have from the square root in (\ref{Lz}):
\be
L_c = \sqrt{3(1+z^2) \!\left[\theta_z^2+\sin^2 \!\theta \varphi_z^2\right]+e^{4 \rho_0}\beta\,z^2} \,\, ,
\ee
where we have also divided by an overall factor of $2$. The variational equations for $\theta (z)$ and $\varphi (z)$ are then:
\bea
\frac{d}{dz}\left\{\frac{(1+z^2)\sin^2 \!\theta\,\varphi_z}{L_c}\right\} &=& 0 \, ,\label{phivar}\\
\frac{d}{dz}\left\{\frac{(1+z^2)\,\theta_z}{L_c}\right\} - \frac{(1+z^2) \sin(2\theta) (\varphi_z)^2}{L_c} &=& 0 \, . \label{thetavar}
\eea
Equation (\ref{thetavar}) is solved by $\theta = \pi/2$. As a result, (\ref{phivar}) simplifies to:
\be\label{phivar2}
\frac{d}{dz}\left\{\frac{(1+z^2)\,\varphi_z}{\sqrt{3\,(1+z^2)\,\varphi_z^2 + e^{4\rho_0} \beta \,z^2}}\right\}=\,0 \,\, .
\ee
The latter equation is solved by:
\be\label{classical1}
\varphi_z(z)=e^{2\rho_0}\sqrt{\frac{\beta}{3}}\frac{1}{\sqrt{1+z^2}} \,\, ,
\ee
which is the same as (\ref{derVarPhi}). For completeness, let us also note that (\ref{classical1}) implies
\be\label{classicalphi}
\varphi (z) = e^{2\rho_0}\sqrt{\frac{\beta}{3}} \,\,{\rm arcsinh} (z) \, ,
\ee
where we have set the integration constant to zero in order to have $\varphi = 0$ at $z=0$ (in other words, in order to have the ${\rm D}7$ and $\overline{{\rm D}7}$ branches merge at $z=0$). Finally, note that the profile (\ref{classicalphi}) agrees with the one determined by (4.12) of \cite{LA}. Namely, in our approximations the constant $B$ there becomes $B = \frac{\beta}{3}$; in addition, the plus sign there should be taken for $z>0$ while the minus sign for $z<0$.
\section{Spectrum of $\theta$ bosons}
\setcounter{equation}{0}
In this section we begin investigating the spectrum of scalar mesons in our model. The latter arise from the fluctuations of the embedding functions $\theta$ and $\varphi$ around the U-shaped D7-$\overline{{\rm D}7}$ embedding, that realizes chiral symmetry breaking. First, we address the spectrum of $\theta$ bosons. In the next subsection we derive the relevant equation of motion and show that the resulting mass spectrum differs from the one for vector mesons, computed in \cite{ASW}, only by an order $\beta$ correction. Finally, in Subsection \ref{ThetaMass} we find explicitly the ${\cal O} (\beta)$ mass difference between the spectra of $\theta$ bosons and vector mesons.
\subsection{Equation of motion}
Let us now turn to studying the spectrum of fluctuations of the field $\theta(z,x^{\mu})$ around the configuration (\ref{clSol}). To do that we have to substitute (\ref{tpFluct}) in (\ref{Lz}) and expand to second order in the fluctuations. Actually, as already pointed out, we can obtain the relevant $\delta \theta$ Lagrangian by first setting to zero the $\varphi$ fluctuations and then expanding. We also need to substitute (\ref{derVarPhi}). The result is:
\be \label{Lz2}
{\cal L} = const \frac{1}{\sqrt{1+z^2}}\left\{12(1+z^2)(\pd_z \delta\theta)^2+e^{4\rho_0}\beta\left[ c\,(1+z^2)\!\left( \pd_{\mu} \delta\theta \right)^2-4(\delta \theta)^2 \right] \right\}.
\ee
Using that $\pd_{\mu} \pd^{\mu} \delta \theta = m_{\theta}^2 \delta \theta$, we then find the following field equation:
\be \label{eq1}
-(\delta \theta)''(z)-\frac{z}{1+z^2}(\delta \theta)'(z)-\frac{\beta\,e^{4 \rho_0}}{3(1+z^2)}\delta \theta(z)=M^2 \delta \theta (z) \, ,
\ee
where $'\equiv \pd_z$ and $M^2 = \frac{m_{\theta}^2 c \beta e^{4\rho_0}}{12}$; also, we have suppressed the $x^{\mu}$ argument for brevity. To bring (\ref{eq1}) to Schrodinger form, we perform the field redefinition:
\be
\delta \theta(z)= \frac{\Theta(z)}{(1+z^2)^{1/4}} \,\, .
\ee
Then the equation for the mass spectrum becomes:\footnote{Let us make an important remark. Clearly, the potential term in the Schrodinger equation (\ref{eqtheta}) changes sign as $z$ is varied. However, the presence of a negative potential region does not by itself indicate an instability. The configuration we are expanding around would be unstable if the spectrum of solutions of (\ref{eqtheta}) with appropriate boundary conditions were to have a negative mass-squared mode. In the rest of this Section, we will show that this is not the case.}
\be\label{eqtheta}
-\Theta''(z)-\left[\frac{z^2-2}{4(1+z^2)^2}-V_1(z)\right]\Theta(z)=M^2\Theta (z) \, ,
\ee
where
\be \label{V1z}
V_1(z)=- \frac{\beta\,e^{4\rho_0}}{3(1+z^2)} \, .
\ee
Before we turn to solving (\ref{eqtheta}), let us make a few comments on its form. First of all, notice that, except for the $V_1 (z)$ correction term, this equation is exactly the same as (4.4) of \cite{ASW}.\footnote{In that regard, note that the parameter $\lambda$ in (4.5) of \cite{ASW} can be removed by introducing a new variable $\hat{z} = z/\sqrt{\lambda}$ in (4.4) there. In addition, using that $c_0 \equiv c \sqrt{\beta}$ and the expression for $\lambda$ in (4.6) of \cite{ASW}, one can see that the new variable $\hat{z}$ is precisely the same as the variable $z$ in (\ref{ChVar}) here.} The latter determines the spectrum of vector and axial-vector mesons in this model. Therefore, the presence of the term (\ref{V1z}) here implies an ${\cal O} (\beta)$ splitting between the masses of $\theta$ bosons and vector mesons. Actually, at first sight one might think that such a conclusion is premature, as in \cite{ASW} we were working at small $\beta$ as well and keeping only the leading contributions. So it is legitimate to suspect that the potential in (4.4) of \cite{ASW} might have an order $\beta$ correction that we have just not written down. However, we will show now that any such correction would be the same for both the vector bosons and $\theta$ mesons. Hence the correction (\ref{V1z}) does, in fact, give rise precisely to the mass splitting between the $\theta$ and vector meson spectra.
To understand that, it is most instructive to derive the full field equation for $\delta \theta$, without using any of the approximations that led to the simplified walking metric (\ref{leadMetric}). Working with the full metric (\ref{BM}) and introducing the same functions $a(\rho)$ and $b(\rho)$ as in (2.21) of \cite{ASW}, namely
\bea \label{ab}
a(\rho) &=& \frac{1}{A^3} \left[ f(\rho) + g(\rho) \varphi_{\rho}^2 \right]^{1/2} \nn \\
b(\rho) &=& \frac{1}{A^2} \left[ f(\rho) + g(\rho) \varphi_{\rho}^2 \right]^{1/2} g^{\rho \rho}
\eea
with
\be
f(\rho) = \frac{A^8 c^4}{256} P_1^2 (\rho) P_1'^2 (\rho) \coth^2 (2 \rho) \quad \, , \quad \, g(\rho) = \frac{A^8 c^4}{2\times 256} P_1^3 (\rho) P_1' (\rho) \coth(2 \rho)
\ee
\be
\hspace*{-0.1cm}{\rm and} \quad \, g^{\rho \rho} = \frac{A^7 c^3 P_1^2 P_1' \coth^2 (2 \rho)}{128 \left( f + g \varphi_{\rho}^2 \right)} \,\, ,
\ee
we find after a lengthy calculation:
\be\label{rhoeq}
-\partial_\rho\,[\, b(\rho)\, \partial_\rho \,\delta \theta(\rho)\,]-b(\rho)\coth^2(2\rho)[\varphi_{\rho}^{cl}(\rho)]^2 \delta \theta(\rho)=m_{\theta}^2\,a(\rho)\,\delta \theta(\rho) \, .
\ee
The above equation of motion differs from the (axial-)vector meson one, eq. (2.26) of \cite{ASW}, only by the presence of the $[\varphi_{\rho}^{cl}]^2$ term. Although it is by no means easier to try to solve (\ref{rhoeq}) instead of (\ref{eqtheta}), it is instructive to see how (\ref{rhoeq}) reduces to (\ref{eqtheta}) in the approximations that give the walking metric (\ref{leadMetric}). Namely, to leading order in small $\beta$, we find that the functions (\ref{ab}) reduce to:\footnote{To obtain (\ref{abrho}), one also needs to use the relation $c \sqrt{\beta} = \frac{\sqrt{3}}{16}$, valid to leading order in small $\beta$, that was derived in \cite{ASW}.}
\bea \label{abrho}
a(\rho (z)) &=& \frac{3^{1/4} c^{5/4} \beta^{3/4} e^{4\rho_0}}{24} \,\frac{\left( 1+z^2 \right)^{3/2}}{z}\nn \\
b(\rho (z)) &=& \frac{3^{1/4} c^{1/4} \beta^{-1/4}}{8} \,\frac{z}{\sqrt{1+z^2}} \,\, ,
\eea
where we have used (\ref{ChVar}). It is then easy to see that (\ref{rhoeq}) becomes exactly (\ref{eq1}), after the change of variable from $\rho$ to $z$. The important point is that the term $-\frac{\beta e^{4\rho_0}}{3 (1+z^2)} \delta \theta (z)$ in (\ref{eq1}), that gives rise to $V_1 (z)$, comes entirely from (and in fact, to leading order, is the only contribution of) the $[\varphi_{\rho}^{cl}]^2$ term in (\ref{rhoeq}), which is precisely the extra term not present for vector bosons. Hence, whatever other ${\cal O} (\beta)$ contributions might arise from the remaining terms in (\ref{rhoeq}), when subleading orders in $a(z)$ and $b(z)$ are taken into account, they will necessarily be the same for both the vector and $\theta$ mesons. Thus the leading order mass difference between the two spectra is determined precisely by $V_1 (z)$, while any subleading correction to it will come from higher order terms in the small $\beta$ expansion of the $[\varphi_{\rho}^{cl}]^2$ term.
\subsection{Mass spectrum} \label{ThetaMass}
Let us now consider the Schrodinger equation (\ref{eqtheta}) in more detail. Clearly, at leading order in small $\beta$, the $V_1 (z)$ term in the potential can be neglected and so one has exactly the same spectrum as for the vector mesons, studied in \cite{ASW}. To evaluate the mass split between the two spectra due to $V_1 (z)$, we will use perturbation theory. Note also that, since we are studying the perturbative spectrum, we will impose the boundary conditions that at $z=\pm z_{\Lambda}$ the wave function vanish.\footnote{Here, as in \cite{ASW}, we have introduced a physical cut-off $z_{\Lambda}$ at the upper end of the walking region, so that $z\in (-z_{\Lambda}, z_{\Lambda})$. The earlier work \cite{LA} discussed holographic renormalization for the background of interest. However, in \cite{ASW} we pointed out that above the scale $z_{\Lambda}$ the light spectrum of the model should change in order to reflect the degrees of freedom appropriate for extended technicolor. Hence, on physical grounds, our current model should not be viewed as valid to arbitrarily high energies. The proper UV completion is still an open problem; see \cite{CGNPR}, though, for important recent progress in that direction.} Let us denote by $\Theta_n^0$ the zeroth order wave function of the $n^{{\rm th}}$ state, i.e. with $V_1(z)$ neglected. Then, according to standard rules, the correction to the mass-squared $M_n^2$ due to the perturbation $V_1(z)$ is given by:
\be \label{Mn2}
\delta \,[M_n^2] =\frac{1}{N^2}\int_{-z_\Lambda}^{z_\Lambda} [\Theta_n^0 (z)]^2 \,V_1(z)\,dz \, ,
\ee
where the normalization factor is
\be \label{norm}
N^2=\int_{-z_\Lambda}^{z_\Lambda} [\Theta^0_n (z)]^2\,dz \, .
\ee
From the analysis of Section 4 in \cite{ASW}, we know the approximate analytical form of $\Theta_n^0 (z)$. More precisely, one can divide the interval $(0, z_{\Lambda})$ into two regions, in each of which the field equation simplifies and can be solved analytically. The matching of the two then gives the quantization condition $M_n = r_n / z_{\Lambda}$, where $r_n$ is the $n^{{\rm th}}$ root of the Bessel function $J_0$. Up to overall constants, the wave function has the following form in each of the two regions:
\be \label{Tworegions}
\Theta_n^0 (z) = \begin{cases} \Theta_n^{(s)} (z) = (1 + z^2)^{1/4}\,,& \mbox{\,\,\,\,\,for small $z$, \,i.e. $\!z \in (0, z_{\bullet})$ \,,}\\ \Theta_n^{(l)} (z) = \sqrt{z} \,J_0 (M_n z)\,, & \mbox{\,\,\,\,\,for large $z$, \,i.e. $\!z \in (z_{\bullet} , z_{\Lambda})$ \,,}
\end{cases}
\ee
where $1<\!\!< z_{\bullet} <\!\!< z_{\Lambda}$.\footnote{For simplicity, we took in (\ref{Tworegions}) the form of the solution, which is symmetric under $z \rightarrow -z$. The antisymmetric solution, that now gives pseudoscalar mesons, is a bit more involved technically as can be seen from the axial-vector considerations in \cite{ASW}. However, the final outcome is the same, except for substituting $r_n$ with the quantity $\mu_n$ defined in (4.16) of \cite{ASW}.} Actually, it is easy to realize that one can approximate the solution on the entire $0<z<z_\Lambda$ interval (except for a $\Delta z\simeq M^2$ neighborhood of $z=\sqrt{2}$) by the following function:
\be\label{interpolate}
\hat{\Theta}_n^0(z)= (1 + z^2)^{1/4}\,J_0(M_n z) \,\, .
\ee
Clearly, $\hat{\Theta}$ reduces to the first line of (\ref{Tworegions}) at small $z$, while it acquires the form on the second line of (\ref{Tworegions}) at large $z$. To see by how much this approximation fails to be an exact solution, let us substitute $\Theta = \hat{\Theta}$ in (\ref{eqtheta}), with $V_1$ omitted. We obtain:
\be\label{eqtheta0}
-\hat{\Theta}_n^0{}''(z)-\left[\frac{z^2-2}{4(1+z^2)^2}+M_n^2\right]\hat{\Theta}_n^0(z)=-\frac{M_n\,J_1(M_n z)}{z(1+z^2)^{3/4}} \,\, ,
\ee
where the nonvanishing of the right hand side is exactly the measure of the deviation of $\hat{\Theta}$ from being an exact solution. It is easy to see that, aside from a small neighborhood of a finite number of points, this right hand side is of ${\cal O}(z_\Lambda^{-2})$ compared to the individual terms on the left hand side on the whole interval $0<z<z_\Lambda$.
In evaluating the integral in (\ref{norm}), it is easy to realize that the leading contribution in the small $\beta$ expansion will come entirely from the large $z$ region. So, to leading order, we have:
\be
N^2=\int_{-z_\Lambda}^{z_\Lambda} [\Theta^0_n (z)]^2\,dz \simeq\int_{-z_\Lambda}^{z_\Lambda} z[J_0(M_n\,z)]^2\,dz= z_{\Lambda}^2 \, [J_1 (M_n z_{\Lambda})]^2 \, .
\ee
On the other hand, the integral in (\ref{Mn2}) can be estimated by using the approximate solution $\hat{\Theta}_n^0(z)$. To leading order in $z_\Lambda$, one obtains:
\be \label{SmallZ}
\int_{- z_{\Lambda}}^{z_{\Lambda}} [\hat{\Theta}_n^0 (z)]^2 V_1(z) dz = -\frac{2}{3} \beta e^{4 \rho_0} \!\left[\,{\rm arcsinh} (z_{\Lambda}) -\frac{1}{4}r_n^2\,\,{}_3F_4 (1,1,\frac{3}{2};2,2,2,2;-r_n^2)+ {\cal O} (z_\Lambda^{-2})\right]\!,
\ee
where ${}_3F_4 (1,1,\frac{3}{2};2,2,2,2;-r_n^2)$ is the hypergeometric function. The details of this computation are somewhat subtle and we have relegated them to Appendix B. Combining the above results, we find to leading order in $z_\Lambda$:
\be
\delta \,[M_n^2] = -\frac{2}{3} \,\frac{\beta \,e^{4 \rho_0}}{r_n{}^2J_1{}^2(r_n)}\,M_n^2 \left[\,{\rm arcsinh} (z_{\Lambda}) -\frac{1}{4}r_n^2\,\, {}_3F_4 (1,1,\frac{3}{2};2,2,2,2;-r_n^2) +{\cal O} (z_\Lambda^{-2})\right],
\ee
where we have used $z_{\Lambda} = r_n / M_n$. Now, recalling that $M^2 = \frac{m_{\theta}^2 c \beta e^{4\rho_0}}{12}$ and also using that for large $z_{\Lambda}$ the function ${\rm arcsinh} (z_{\Lambda}) = \ln ( z_{\Lambda} + \sqrt{1+ z^2_{\Lambda}} )$ simplifies to $\ln (2 z_{\Lambda})$, we can write the correction to the $\theta$ boson mass spectrum, i.e. $\delta (m_n^{\theta})^2 = (m^{\theta}_n)^2-(m_n^V)^2$, as:
\be\label{shift}
\delta (m ^\theta_n)^2=-\,m^2_n\,\frac{2\,\beta\,e^{4\rho_0}}{3\,[r_n J_1(r_n)]^2} \left[ \,\log(2 z_\Lambda) -\frac{1}{4}r_n^2\,\, {}_3F_4 (1,1,\frac{3}{2};2,2,2,2;-r_n^2) + {\cal O}(z_\Lambda^{-2})\right] ,
\ee
where the unperturbed (i.e., with $V_1 (z)$ neglected) mass of the $n^{{\rm th}}$ state is \cite{ASW}:
\be \label{masssq}
m_n^2= \frac{12}{c_0}\beta^{-1/2} e^{-4\rho_\Lambda} \,r_n^2 \,M_{KK}^2
\ee
with $c_0 = \frac{\sqrt{3}}{16}$ to leading order.
As can be seen from (\ref{shift}), the mass split between the $n^{\rm th}$ vector and $\theta$ boson states is suppressed by a power of $\beta$ compared to the leading contribution. It would be very interesting to understand the origin of this split, or rather the reason for its suppression. We leave this for future work.
\section{Spectrum of $\varphi$ bosons}
\setcounter{equation}{0}
In this section we will study the fluctuations of the field $\varphi (z,x^{\mu})$ around the classical solution (\ref{clSol}), (\ref{derVarPhi}). As before, to obtain the Lagrangian of interest we have to substitute (\ref{tpFluct}) in (\ref{Lz}) and expand to second order in $\delta \varphi (z,x^{\mu})$ and $\delta \theta (z, x^{\mu})$. However, as already pointed out, to quadratic order there is no mixing between the $\theta$ and $\varphi$ fluctuations. So we can safely set $\theta = \pi /2$ from the start. Hence, the Lagrangian (\ref{Lz}) acquires the form:
\be\label{Lz1}
{\cal L} = const \sqrt{12(1+z^2) \left(\varphi'_{cl} + \delta \varphi'\right)^2+e^{4\rho_0}\beta\,z^2\left[ 4+c\, (\pd_{\mu} \delta \varphi \right)^2]} \,\, ,
\ee
where again $' = \pd_z$. Expanding to second order and substituting (\ref{derVarPhi}), we have:
\be\label{Leff1}
{\cal L} = \frac{3 e^{-2\rho_0}}{\sqrt{\beta}}\frac{z^2}{\sqrt{1+z^2}} \, (\delta \varphi')^2+2\sqrt{3}\, \delta \varphi' + \frac{e^{2\rho_0} c \, \sqrt{\beta}}{4} \frac{z^2}{\sqrt{1+z^2}} \, ( \pd_{\mu} \delta \varphi )^2 \, ,
\ee
where for convenience we have dropped the overall constant in front of the square root in (\ref{Lz1}); also, we have omitted the additive term $2 e^{2\rho_0} \sqrt{\beta} \sqrt{1+z^2}$\,, since clearly it drops out of the field equation. Now, varying (\ref{Leff1}) and using $\pd_{\mu} \pd^{\mu} \delta \varphi = m_{\varphi}^2 \delta \varphi$, we obtain the following equation of motion:
\be\label{eq2}
-\,\delta \varphi''(z)-\frac{2+z^2}{z(1+z^2)} \,\delta \varphi'(z)=M_\varphi^2 \,\delta \varphi(z) \, ,
\ee
where $M^2_{\varphi} = m^2_{\varphi} c \beta e^{4 \rho_0} / 12$ and we have suppressed the argument $x^{\mu}$ for brevity.
In line with the previous section (as well as our considerations in \cite{ASW}), it would seem that the strategy to study (\ref{eq2}) invloves first transforming to Schrodinger form. For technical reasons, however, we will use another approach. Despite that, let us briefly discuss for completeness the Schrodinger form of equation (\ref{eq2}). It is achieved via the field redefinition:
\be
\delta \varphi (z) = \frac{(1+z^2)^{1/4}}{z} \,\Phi (z) \, ,
\ee
which leads to:
\be \label{VpSchr}
- \Phi'' (z) - \frac{6+z^2}{4(1+z^2)^2} \,\Phi (z) = M_{\varphi}^2 \,\Phi (z) \, .
\ee
This is rather similar, although not quite the same as the equation for the vector mesons. Recall that the latter is
\be \label{VSchr}
- \Psi'' (z) - \frac{z^2 - 2}{4(1+z^2)^2} \,\Psi (z) = M_V^2 \,\Psi (z) \, ,
\ee
as explained in the previous section. There we found that the difference between the $\theta$ and vector meson cases was given by the order $\beta$ correction to the potential $V_1 (z)$ in (\ref{V1z}). On the other hand, (\ref{VpSchr}) and (\ref{VSchr}) differ by a potential difference $\Delta V = \frac{2}{(1+z^2)^2}$. At large $z$, one has $\Delta V \sim \frac{2}{z^4}$, which is negligible compared to the large $z$ behaviour of each of the two potentials in (\ref{VpSchr}) and (\ref{VSchr}), that is $\frac{1}{4 z^2}$. However, at small $z$ the difference is of the same order as the potentials themselves. So we cannot view $\Delta V$ as a small perturbation compared to the vector meson case. Nevertheless, one may expect that the mass difference between the two cases is small due to the same large $z$ asymptotics they exhibit.
Since we cannot solve exactly either of (\ref{VpSchr}) and (\ref{VSchr}), it turns out that the most efficient way to compare their mass spectra is to take a step back and to not go to Schrodinger form. That is, for the $\varphi$ bosons we will consider (\ref{eq2}); for future use let us also introduce the notation:
\be
H_{\varphi} = - \frac{d^2}{dz^2} - \frac{2+z^2}{z(1+z^2)} \frac{d}{dz}
\ee
for the differential operator on its left-hand side. Clearly, $H_{\varphi}$ can be viewed as a Hamiltonian operator with eigenfunctions $\delta \varphi (z)$ and eigenvalues $M^2_{\varphi}$\,. The analogue of (\ref{eq2}) for the vector bosons is
\be \label{eq3}
- \psi'' (z) - \frac{z}{1+z^2} \,\psi' (z) = M_V^2 \,\psi(z) \, ,
\ee
where
\be \label{psired}
\psi (z) = \frac{\Psi (z)}{(1+z^2)^{1/4}}
\ee
with the same $\Psi (z)$ as in (\ref{VSchr}). Note that in \cite{ASW} we did not write down equation (\ref{eq3}) explicitly. Instead, we obtained directly (\ref{VSchr}), because we combined the change of variables from $\rho$ to $z$ with the redefinition (\ref{psired}). Indeed, the factor $\frac{1}{[a(\rho) b(\rho)]^{1/4}}$ in (4.3) of \cite{ASW} accounts precisely for the factor $\frac{1}{(1+z^2)^{1/4}}$ in (\ref{psired}) here. As a last preparation, let us also introduce the notation:
\be
H_V = - \frac{d^2}{dz^2} - \frac{z}{1+z^2} \frac{d}{dz}
\ee
for the differential operator on the left-hand side of (\ref{eq3}).
Now, our goal will be to compare the Hamiltonians $H_{\varphi}$ and $H_V$. Since, as already pointed out above, the mass difference is expected to be small, we can view $\Delta H = H_{\varphi} - H_V$ as a small perturbation. Of course, once we obtain the answer, it will be easy to verify that this assumption is satisfied. Before proceeding further, it is very useful to notice the following relation:
\be \label{3Hs}
H_{\varphi} - H_V = 2 (H_{\varphi} - H_0) \, ,
\ee
where the operator $H_0$ is
\be
H_0 = - \frac{d^2}{dz^2} - \frac{1}{z} \frac{d}{dz} \,\, .
\ee
The differential equation $H_0 \chi (z) = M^2 \chi(z)$, namely
\be \label{chieq}
- \chi'' (z) - \frac{1}{z} \chi' (z) = M^2 \chi (z) \, ,
\ee
is in fact precisely of the form that both (\ref{eq2}) and (\ref{eq3}) acquire at large $z$. Unlike those equations, though, (\ref{chieq}) can be solved easily. The regular solutions, that vanish at $z=\pm z_{\Lambda}$, are given by:
\be
\chi_n (z) = J_0 \!\left( \frac{r_n z}{z_{\Lambda}} \right) ,
\ee
where $r_n$ is again the $n^{{\rm th}}$ root of the Bessel function $J_0$. The key use of (\ref{3Hs}) is therefore that, since we know the eigenfunctions of $H_0$, we can compute the mass shift, due to $\Delta H$, by using standard first order perturbation theory:
\be \label{DeltaM2}
\Delta M^2_n = \frac{2}{\langle \chi_n | \chi_n \rangle} \,\langle \chi_n | \Delta H_0 | \chi_n \rangle \, ,
\ee
where $\Delta H_0 \equiv H_{\varphi} - H_0$. And, of course, since (\ref{chieq}) is the same as the large $z$ asymptotic form of (\ref{eq2}), the perturbation $\Delta H_0$ is expected to be small just as $\Delta H = H_{\varphi} - H_V$ is. Before turning to the computation of $\Delta M_n^2$ according to (\ref{DeltaM2}), we also need to note the following. Hermiticity of the Hamiltonian $H_0$ requires the use of the measure $|z| dz$. Indeed, one can easily show that
\be
\int_{-z_{\Lambda}}^{z_{\Lambda}} \chi_m (z) \chi_n (z) \,|z| \,dz \,\sim \,\delta_{mn}
\ee
and
\be
\int_{-z_{\Lambda}}^{z_{\Lambda}} \chi_m (z) \,H_0 \,\chi_n (z) \,|z| \,dz \,= \int_{-z_{\Lambda}}^{z_{\Lambda}} \chi_n (z) \,H_0 \,\chi_m (z) \,|z| \,dz \,\sim \,\delta_{mn} \,\, .
\ee
\newline
\indent
Now we are finally ready to compute $\Delta M_n^2 = [M_n^{\varphi}]^2 - [M_n^V]^2$. Substituting the explicit form of $\chi_n (z)$ in (\ref{DeltaM2}), we find:
\be
\left[M_n^\varphi\right]^2-\left[M_n^V\right]^2=-\frac{4 }{N_n^2}\int_0^{z_\Lambda} z\,J_0(M_n\,z)\frac{1}{z(1+z^2)}\frac{d}{dz}J_0(M_n\,z)\,dz \, ,
\ee
where
\be
N_n^2=2\int_0^{z_\Lambda}z \,[J_0(M_n\,z)]^2 dz=z_\Lambda^2[J_1(r_n )]^2 \, .
\ee
On the last line we have used $J_0(M_n z_\Lambda)=0$. After substituting $x= M_n z$\,, we obtain:
\bea \label{Mnvphi}
\left[M_n^\varphi\right]^2-\left[M_n^V\right]^2&=&-\frac{2}{N_n^2}\int_0^{r_n}\frac{M_n^2}{M_n^2+x^2}\frac{d}{dx}[J_0(x)]^2dx\nn\\&=&\frac{2}{N_n^2}M_n^2[\log(1/M_n)+c_n+O(M_n^2\,\log(M_n))] \, ,
\eea
where the constants $c_n$ are determined by:
\be
c_n=\frac{1}{r_n^2}+\log(r_n^2)-\frac{1}{2}-2\int_0^{r_n}\frac{1}{x^3}\left[J_0^2(x)+\frac{x^2}{2}-1\,\right] dx \, .
\ee
Equivalently, we can rewrite (\ref{Mnvphi}) as:
\be\label{phidev}
\Delta (m_n^\varphi)^2 = (m_n^\varphi)^2 - (m_n^V)^2 = m_n^2 \frac{2}{z_\Lambda^2[J_1(r_n )]^2}[\log(z_\Lambda/r_n)+c_n+O(z_\Lambda^{-2}\,\log(z_\Lambda)\,)] \, ,
\ee
where again $m_n^2$ is given by (\ref{masssq}). For example, for the lowest lying state $c_1\simeq 0.619$.
Thus, for $n=1$ we have:
\be
(m_1^{\varphi})^2-(m_1^V)^2=m_1^2 \,\frac{3.86}{z_\Lambda^2}\,[\log(z_\Lambda)+0.887+O(z_\Lambda^{-2}\,\log(z_\Lambda)\,)] \, .
\ee
It is interesting to compare the mass differences between the two kinds of scalar bosons, $\theta$ and $\varphi$, and the vector mesons. First, from (\ref{shift}) and (\ref{phidev}) one can see that $\Delta (m^\theta_n)^2$ and $\Delta (m_n^{\varphi})^2$ have opposite signs. More precisely, the $\theta$ bosons are slightly lighter, while the $\varphi$ mesons are slightly heavier than the vector bosons. Despite that, the magnitudes of $\Delta (m^\theta_n)^2$ and $\Delta (m_n^{\varphi})^2$ are almost comparable. This is because $\beta \sim z_{\Lambda}^{-2}$, which follows from $z_{\Lambda} \equiv z(\rho_{\Lambda}) \approx e^{2 \rho_{\Lambda}}$ as well as the relation $e^{2 \rho_{\Lambda}} \sim \beta^{-1/2}$ that was elaborated on in \cite{ASW}. However, to be more precise, we should write $e^{4 \rho_{\Lambda}} = \sigma \beta^{-1}$, where $\sigma$ is a number of order $10^{-1}$ or $10^{-2}$ \cite{ASW}. Therefore, the ratio $\Delta (m^\theta_n)^2 / \Delta (m_n^{\varphi})^2 \sim \sigma <1$ and thus the $\varphi$ boson shift is somewhat greater than the $\theta$ boson one.
\section{Discussion}
We investigated the spectrum of scalar mesons in the gravity dual of a walking technicolor model, obtained by embedding D7-$\overline{{\rm D}7}$ techniflavor probes \cite{LA,ASW} in the technicolor background of \cite{NPP}. We showed that all mass-squareds are positive and thus there is no tachyonic mode with respect to fluctuations of the probe branes embedding. Furthermore, we found that both types of scalar mesons, arising from those fluctuations, have spectra that are nearly degenerate with the vector meson spectrum, studied in \cite{ASW}.\footnote{Interestingly, this is in line (albeit, more pronounced in our case) with the expectations, based on the use of the Schwinger-Dyson and Bethe-Salpeter equations, that in IR-fixed-point walking theories the masses of scalar and vector mesons at a given level are numerically close to each other \cite{KSh}.} Namely, the mass differences are of higher order, in an expansion in the small parameter $\beta$, compared to the leading contributions to the masses. It would be very interesting to understand the underlying reason for this near-degeneracy. Likely, it will mostly turn out to be due to a small amount of supersymmetry breaking, as the massive vector and scalar states should belong to the same long ${\cal N} = 1$ supermultiplet.\footnote{Such an expectation is supported by the results of \cite{NPR}, where the flavor probe embedding is supersymmetric and the spectra of vector and scalar mesons are exactly degenerate. We thank C. Nunez for pointing this out.} However, the difference between the mass shifts of the two kinds of scalars may indicate a role for an additional factor.
We leave
investigation of this question for future work.
One should keep in mind that the scalar spectrum studied here does not address the question of whether our walking technicolor model has a dilaton or not. The latter is also a scalar field. However, it is a mode that, for us, should arise predominantly from fluctuations of the technicolor background (see the discussion in the introduction). In particular, its distinguishing feature in the effective Lagrangian is a certain coupling to the square of the technicolor field strength.\footnote{Note that \cite{ENP} found evidence for a light scalar in the spectrum of fluctuations of the color background of \cite{NPP}. However, at present, it is unclear whether this state can be identified with the technidilaton.} The phenomenological importance of this field is great, since it is a (potentially) light scalar boson rather similar to the Higgs. To differentiate between the two, one would have to compute their couplings to other fields; for such a study see \cite{GGS}. There is a lot of literature for \cite{Yam,BLL,Sannino,AB} or against \cite{NSY,against,KLP} the existence of a dilaton in walking technicolor.\footnote{It is also worth mentioning the pioneering work of \cite{GW} on Goldstone bosons for broken conformal invariance, albeit in the case of {\it weakly-coupled} gauge theories.} The arguments for are inspired by the approximate conformal symmetry of the walking region, whereas the arguments against rely on the fact that the theory is never exactly conformal. We hope to investigate the issue of the presence of a technicolor dilaton in our model in a future publication.
Other important open problems are the computation of the $T$ and $U$ electroweak observables, as well as the anomalous dimension of the technifermion condensate. Finally, there are various interesting possibilities for extending our model. An important issue, equally relevant for all other D-brane constructions of technicolor models \cite{HolTech}, is to go beyond the flavor probe approximation. This would enable one to study gravity duals of models with comparable numbers of colors and flavors, like the walking technicolor models that rely on the existence of a Banks-Zaks fixed point for the gauge coupling in the infrared region \cite{BZ}. Another interesting direction, relevant for understanding the generation of quark and lepton masses, is to try to build a gravity dual of extended technicolor \cite{extTC}. In that regard, the recent work \cite{CGNPR} might provide valuable insights.
\section*{Acknowledgements}
We are grateful to T. Appelquist, P. Argyres, S. Das, Z. Komargodski, M. Kruczenski, C. Nunez, M. Piai, F. Sannino, G. Semenoff, A. Shapere and R. Shrock for useful discussions. In addition, L.A. thanks the October, 2011, SPOCK meeting in Cincinnati and the 2011 Simons Workshop in Stony Brook for hospitality during various stages of this work. Research at Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Research \& Innovation. L.A. is also supported in part by funding from NSERC. The research of P.S. and R.W. is supported by DOE grant FG02-84-ER40153.
|
{
"timestamp": "2012-05-11T02:00:42",
"yymm": "1203",
"arxiv_id": "1203.1968",
"language": "en",
"url": "https://arxiv.org/abs/1203.1968"
}
|
\section{Introduction}
\label{sec:introduction}
In a previous paper \cite{us} we studied the signal of a shock heated
cosmic string wake in a position space 21cm redshift maps.
We pointed out that cosmic string wakes give rise
to a pronounced signal in such maps, in particular at redshifts larger than
that of conventional reionization. The signal of a single wake is a wedge
which is extended (a degree or larger for redshifts of $z\sim 20$) in both
angular directions and thin in redshift direction.
This effect comes from the fact that wakes form overdensities in baryons already
at very high redshifts, and that these regions then lead to the extra 21cm
emission or absorption.
In a followup paper \cite{us2} we computed the angular power spectrum for
shock heated cosmic string wakes emitting 21 cm radiation. Because of our
interest in shock heating in both \cite{us} and \cite{us2}
we focused our attention on those wakes with the greatest kinetic gas
temperature $T_K$. Thus we concentrated our analyses on the wakes
formed earliest, $z_i \sim 3000$ and emitting 21~cm radiation as late
as possible before reionization, $z_e \sim 20$. This led us to miss the
constraints on $G\mu$ that shock heated wakes formed at later
$z_i$ could provide by considering the absorption of 21~cm radiation.
In this paper we consider these constraints.
Furthermore, even if baryons collapsing onto the primordial string wake
do not shock
heat, the string will nevertheless induce an overdense region of baryons.
This region, however, will be more diffuse than if shock heating occurs.
The wakes will be thicker but the overdensity smaller. In this paper, we
study the 21cm signal of such a ``diffuse" cosmic string wake.
Current constraints on the cosmic string
tension from analyses of the angular power spectrum of CMB anisotropy
maps yield
\be \label{limit1}
G \mu < 1.5 \times 10^{-7}
\ee
using combined data from WMAP and SPT microwave experiments \cite{Dvorkin}.
Limits based only on WMAP data gives a bound larger than this by a factor
of two~\cite{previous}. It must be emphasized that any limits on $G \mu$ coming
from power spectra analyses implicitly depend on parameters
describing the cosmic string scaling solution which are quite
uncertain. Two prime examples of such parameters are the number
$N$ of string segments crossing any Hubble volume, and the
curvature radius $c_1$ of a long string relative to the Hubble radius.
These parameters are known only to within an order of magnitude
since they must be determined via numerical simulations of cosmic
string network evolution, and these simulations are extremely
challenging because of the huge hierarchy of scales which must
be included at the same time.
In our analyses below we show that a cosmic string signal will exceed
the thermal noise of an individual pixel in a future radio telescope
such as the Square Kilometre Array (SKA) for a string tension
\be \label{limit2}
G \mu > 2.5\times 10^{-8} \, .
\ee
Our new limit is
independent of the parameters $N$ and $c_1$.
In addition, since the
signal of a string wake has a very special geometry in position space,
position space shape detection algorithms are likely to be able to
detect string wake signals even if they do not stick out above the
Gaussian noise on a pixel by pixel basis. The situation is similar
to what is encountered when searching for cosmic string signals
in CMB anisotropy maps: since long string segments produce line
discontinuities in the maps, these can be identified by edge
detection algorithms such as the Canny algorithm \cite{Canny}
for values of the tension substantially lower than the value for which
the string signal dominates on a pixel by pixel basis, as has been
studied in detail in \cite{Amsel}.
The paper is organized as follows:
We begin in Section \ref{sec:csreview} with a brief review of cosmic
strings, their wakes, and their 21 cm signatures. Readers familiar
with such results may skip this section.
Section \ref{sec:diffuse-wakes} is the main part of our work. In this
section we extend the analysis of the 21 cm signature of cosmic string
wakes in several ways. First we consider the constraints
on $G\mu$ from the absorptions signal of shock heated wakes laid down
much later than matter radiation equality. Secondly we analyze the
signal of diffuse wakes, that is those wakes in which there is a baryon
overdensity but which have not shock heated. Finally we compare the
size of these signals to the expected thermal noise per pixel
which dominates over the background cosmic gas brightness temperature.
In Section~\ref{sec:conclusion} we present our conclusions and put our
work in context.
\section{Cosmic Strings, Their Wakes and 21 cm Signatures}
\label{sec:csreview}
Many particle physics models beyond the Standard Model predict the
existence of cosmic strings. In particular, cosmic strings are produced
at the end of inflation in many models of brane inflation \cite{Dvali} and
in many supergravity models \cite{Rachel}. Cosmic strings may also survive
in early universe models based on superstring theory such as ``string gas
cosmology" \cite{BV, RHBSGCrev}.
Since the amplitude of cosmological signatures of cosmic strings is proportional
to the string tension, which itself is set by the energy scale of the new
physics, searching for signatures of strings in cosmology is a way of
probing Beyond the Standard Model physics which is complementary to
accelerator probes such as the Large Hadron Collider experiments which
probe the low energy limit of such theories.
Cosmic strings can be used as a cosmological
probe of New Physics because any particle physics model which admits
stable cosmic strings as a solution inevitably results in a a network of
such strings forming during a symmetry breaking phase transition in the
early universe (see \cite{ShellVil, HK, RHBrev} for reviews on cosmic
strings and cosmology).
Such a network will persist to the present time
and approaches a dynamical attractor configuration, the
so-called ``scaling solution" in which the statistical properties of
the string network are independent of time if all lengths are scaled to
the Hubble radius. The cosmic string scaling solution is charaterized by
a random-walk-like network of infinite (or ``long") strings with step length
comparable to the Hubble radius, plus a distribution of cosmic string loops
with radii smaller than the Hubble radius. The typical transverse
velocity of a long string segment is the speed of light. This implies
that at all times $t > t_{eq}$ ($t_{eq}$ being the time of equal matter and
radiation) relevant for structure formation there are
strings which act as seeds for the accretion of matter.
The key parameter which characterizes cosmic strings and their cosmological
consequences is the mass per unit length $\mu$.
It is usually quoted in terms of the dimensionless
number $G \mu$, where $G$ is Newton's gravitational constant.
Both long string segments and string loops lead to nonlinearities in the
matter distribution at all redshifts up to $z_{eq}$,
the redshift of equal matter and radiation. Cosmic string network
simulations \cite{CSsimuls} indicate that the long strings have a larger
effect than the loops, so we will here focus our attention on the effects of
long string segments, as we did in \cite{us, us2} (For a
recent study of the 21cm signal of a cosmic string loop see \cite{Pagano}.).
In the standard $\Lambda$CDM cosmology without strings, nonlinear structures are
negligible until shortly before the time of reionization. Hence, the signals
of cosmic strings should be more and more pronounced against the ``noise"
of effects from structure formation in the $\Lambda$CDM model the
higher the redshift is. 21cm surveys hence appear as an ideal window to probe
for the possible existence of cosmic strings. They probe the
distribution of matter at much higher redshifts than galaxy redshift
surveys since they map out the distribution of neutral
hydrogen in the universe, in particular before the time of reionization.
Compared to CMB maps, 21cm maps have the advantage of yielding
three-dimensional information.
The nonlinearity in the matter distribution induced by a long string
segment at time $t$ takes the form of a ``wake" \cite{CSwake} whose length
is set by the length of the string segment (which will be similar to the
Hubble radius $t$),
whose width is set by $v_s \gamma_s t$, where $v_s$ is the transverse velocity
of the string segment and $\gamma_s$ is the associated relativistic gamma
factor, and whose average thickness is proportional to $G \mu t$. The wake
formation starts with an initial velocity perturbation and then grows by
gravitational accretion.
The velocity perturbation (see Figure~\ref{fig:wakewedge}) stems from
the fact that space perpendicular to a long string segment is conical,
i.e. flat space with a missing
wedge whose deficit angle is $\alpha = 8 \pi G \mu$.
A string moving through a gas of particles with transverse velocity $v_s$
will hence induce a velocity perturbation $\delta v$ of the gas towards
the plane behind the string, whose magnitude is given by
$\delta v = 4 \pi G \mu v_s \gamma_s $.
.
\begin{figure}[htbp]
\includegraphics[height=5cm]{wakewedge}
\caption{Sketch of the mechanism by which a wake behind a moving
string is generated. Consider a string perpendicular to the plane of
the graph moving straight downward. From the point of view of the
frame in which the string is at rest, matter is moving upwards, as
indicated with the arrows in the left panel. From the point of view
of an observer sitting behind the string (relative to the string motion)
matter flowing past the string receives a velocity kick towards the
plane determined by the direction of the string and the velocity
vector (right panel). This velocity kick towards the plane leads
to a wedge-shaped region behind the string with twice the
background density (the shaded region in the right panel).}
\label{fig:wakewedge}
\end{figure}
A cosmic string segment laid down at time $t_i$ (we are interested
in $t_i \geq t_{eq}$) will generate a wake with
physical dimensions:
\be \label{size}
l_1(t_i)\times l_2(t_i)\times w(t_i)~ =~ t_i ~c_1 ~\times~ t_i ~v_s\gamma_s ~\times~ t_i ~4\pi G\mu v_s\gamma_s \, .
\ee
where $c_1$ is a constant of order one. In the above, the first
dimension is the length
in direction of the string, the second is the depth, and the third is the
average width. After being laid down, the wake will grow by gravitational
accretion. This has been studied using the Zel'dovich approximation \cite{Zel}
in a number of works \cite{wakegrowth}. In this method of analysis, we
follow the time evolution of the height above the centre of the wake of mass
shells to determine the comoving distance $q_{nl}$ which ``turns around''
at time $t$, i.e. for which the physical height $h(q, t)$ is maximal. We
obtain that the comoving thickness grows linearly with the scale factor
of the universe. A wake laid down at redshift $z_i$ will have comoving
thickness at redshift $z$ given by
\be \label{qnl}
q_{nl}(z) \, = \,
\frac{16 \pi}{5} {G \mu v_s \gamma_s\over H_o} (z_i + 1)^{1/2}{1 \over (z+1)}
\, .
\ee
The physical height of the shell above the centre of the wake at the time of
``turnaround'' is half what the height would be in the absence of gravitational
accretion, and hence the relative overdensity is a factor of 2. It is then
assumed that the shell collapses to half of the height it had at turnaround,
that it then undergoes shocks and virializes at that distance. Then,
the overdensity will be a factor of 4 greater than the background density.
This approximate analytical analysis has been confirmed in detailed
hydrodynamical simulations \cite{Sorn}.
During the infall onto the string wake, the baryons acquire kinetic energy
which during the process of shock heating transforms to thermal energy.
The kinetic temperature $T_K$ acquired by the hydrogen atoms can be obtained
from the kinetic energy which particles acquire when they hit the shock
front. This is determined
using the Zel'dovich approximation \cite{Zel}.The result obtained in
\cite{us} is
\be
T_K \, \simeq \, [20~{\rm K}] (G \mu)_6^2 (v_s \gamma_s)^2 \frac{z_i + 1}{z + 1} \, ,
\ee
where $(G \mu)_6$ indicates the value of $G \mu$ in units
of $10^{-6}$. The result is expressed in degrees Kelvin.
Note that the wake kinetic temperature increases as $z$ decreases since
the wakes have more time to grow, which in turn leads to a growing
gravitational potential. Since $T_{\gamma}$ decreases in time, then it
is more and more likely that the string wake 21cm signal will be in
emission rather than absorption as $t$ increases. However, it turns
out that for reshifts significantly larger than that of conventional
reionization and for string tensions smaller than the current limit
(\ref{limit1}), $T_K$ is smaller than $T_{\gamma}$ and that hence the string
wake signal will be in absorption.
The 21cm signal of a cosmic string wake has a distinctive geometry.
We will see the string-induced absorption (or emission) regions in
directions of the sky where our past light cone intersects the
cosmic string wake. These are wedges in the sky of angular scale
determined by the comoving Hubble radius at the time that the string
is formed. The string wake grows in thickness, but its planar
dimensions are constant in comoving coordinates. For wakes formed
at a redshift corresponding to recombination this angular scale is
about one degree, for wakes formed at equal matter and radiation slightly
smaller. Since photons emitted at the top and bottom of the wake
undergo slightly different redshifts, the region in 21cm redshift
surveys of extra emission/absorption takes on the form of a thin
wedge in redshift direction with a thickness which is \cite{us}
\be \label{freqwidth}
\left|{\delta z\over z+1}\right| =
\left|\frac{\delta \nu}{\nu}\right|
\, = \, \frac{24 \pi}{15} G \mu v_s \gamma_s
{\bigl( z_i + 1 \bigl)^{1/2}\over \bigl( z+ 1 \bigr)^{1/2}} \, ,
\ee
\section{Diffuse Cosmic String Wakes}
\label{sec:diffuse-wakes}
The analysis in the previous section neglected the temperature of the gas
at the time that the wake was formed. The accretion onto a
string wake further increases the intrinsic gas velocities,
leading to an effective gas temperature which is $2.5 T_g$,
where $T_g$ is the temperature of the background gas. If
\be
T_K \, < \, 2.5 \times T_g \, ,
\ee
then the incoherent velocities due to the thermal motion of
the accreted gas dominate over the coherent velocity induced
by accretion. In this case, there is no shock heating, and
the thickness of the overdense region induced by the wake
is larger than it would be in the absence of thermal motion
of the gas particles. The resulting overdense region we will
call a ``diffuse wake''.
We now estimate the size of the diffuse wake. In
linear cosmological perturbation theory, the total mass
accreted by the string wake is the same both in the
cases with and without shock heating. The difference
is that the large intrinsic thermal gas velocity will
render the wake wider and hence less dense.
The first step is to compute the width $h_w$ of the wake
obtained taking into account the gas temperature $T_g$.
The result is
\be \label{height}
h_w(z)|_{T_K < T_g} \, = \, h_w(z)|_{T_g = 0} ~\Bigl( \frac{T_g}{T_K} \Bigr)
\ee
for $T_K < T_g$. The diffuse wake will thus be thicker by the
ratio of temperatures $T_g / T_K$, but the density of baryons
will be smaller by the same factor. Hence, the width of the
21cm signal induced by a diffuse wake will be thicker in redshift
direction by the above ratio of temperatures compared to what is
given in (\ref{freqwidth}). On the other hand, the 21cm brightness
temperature will be smaller since it is proportional to the baryon
density which is smaller than in the case of a shock-heated wake.
The formula (\ref{height}) for the width of a diffuse wake
can be derived by assuming equipartition of
energy between thermal energy and potential energy which for $T_K \ll T_g$ yields
\be \label{equip}
m_{HI} \delta \Phi \, = {3\over2}T_g \, ,
\ee
where $m_{HI}$ is the mass of a hydrogen atom and $\delta \Phi$ is
the gravitational potential induced by the wake. The latter, in
turn, is
\be \label{pot}
\delta \Phi \, = \, 2 \pi G \sigma |h| \, ,
\ee
where $\sigma$ is the surface density of the wake, and $|h|$ is
the height. Combining (\ref{equip}) with (\ref{pot}) yields
the linear scaling in temperature given in (\ref{height}).
The surface density $\sigma$ of a
diffuse wake is the same as that of a shock heated wake. It is given
by the energy per unit area of matter within a comoving height
$q_{nl}(z)$, the shell which is turning around at redshift $z$.
Thus $\sigma(z) = q_{nl}(z) \rho_0$,
where $\rho_o$ is the current background density. The
overdensity $\Delta \rho$ in the diffuse wake is given by
\be
\Delta \rho(z) \, = \, \frac{\sigma(z)}{h_w(z)} \, = \, \rho_0 ~\Bigl( \frac{T_K}{T_g} \Bigr)\, .
\ee
which results in
\be
{\rho\over\rho_0 } \, = \, \Bigl( 1+\frac{T_K}{T_g} \Bigr)\, .
\ee
The extra baryon density in the wake leads to extra 21~cm
absorption or emission.
The expression for the wake brightness temperature written in terms
of kinetic temperature $T_{Kg}$ and the hydrogen atom number
density $n_{HI}$ is very similar for a shock heated or a diffuse wake. The only difference is that $T_{Kg}$ is the kinetic temperature $T_K$ of the wake when shock heating occurs whereas it should be interpreted as the kinetic temperature of the cosmic gas $T_g$ for diffuse wakes.
\be \label{eq:deltaTb}
\delta T_b(z_e) \ = \, [17~{\rm mK}]\frac{x_c}{1+x_c}\left(1-\frac{T_\gamma}{T_{Kg}}\right)
\frac{n_{HI}^{wake}}{n_{HI}^{bg}}
{(1+z_e)^{1/2}\over2\sin^2\theta}~.
\ee
Here, $z_e$ is the redshift of 21~cm emission or absorption, $T_\gamma$ is
the CMB temperature, $\theta$ is the angle of the 21~cm ray with respect
to the vertical to the wake, and $x_c$ are the collision coefficients whose
values can be obtained from the tables listed in \cite{xc}. Here and
throughout we take the cosmological parameters to be
$H_0=73~{\rm km~s}^{-1}~{\rm Mpc}^{-1}$,
$\Omega_b=0.0425$, $\Omega_m=0.26$.
We work with a matter dominated universe for $z \le 3000$ with the age of
the universe $t_0=4.3\times10^{17}$~s. The origin of the $\sin^{-2}(\theta)$
factor will be discussed in the Appendix, where it will also be shown that
this factor does not lead to any physical divergence.
For a shock heated wake, the ratio of the wake's hydrogen atom number density
to the background density, $n_{HI}/n_{HI}^{bg}$, is 4.
For a diffuse wake the relative brightness temperature is lower
than for a wake which has undergone shock heating since the number
density $n_{HI}$ of hydrogen atoms inside the diffuse wake is smaller,
and thus its ratio in (\ref{eq:deltaTb}) to the background density
$n_{HI}^{bg}$ is smaller. For our analysis below we take:
\bea
{n_{HI}\over n_{HI}^{bg}}
= \Big(1+{T_K\over T_g}\Big) ~~~~~~~~&~ & T_K\le 3 T_g \\
= 4 ~~~~~~~~~~~~~~~~~~~~&~ & T_K \ge 3T_g
\eea
Note that for simplicity we have used $T_K \le 3 T_g$ instead of $T_K \le 2.5 T_g$.
The brightness temperature signal in (\ref{eq:deltaTb}) discussed in
\cite{us,us2} was compared to the average brightness temperature of the
surrounding cosmic gas. Between redshift $z=20$ and 35 the average background
brightness temperature varies from -0.34~mK to -8.6~mK, respectively.
However the analysis of the signal did not consider beam size nor thermal
noise. In fact for $z\sim 20$ it is the thermal noise, and not the
background brightness temperature, that will first limit our ability to
detect a cosmic string wake signal. The opposite is true for $z=35$.
Beam size and thermal noise for the 21~cm brightness temperature
were considered in \cite{Oscar} where their Eq.~(4.19) gives the thermal
noise per pixel:
\be
T_n=
\frac{12 ~\text{mK}~({\rm arcmin}/\theta_{\rm resolution})}{\sqrt{(\tau/10^4{\rm hr}) (A_e / {\rm km^2})}}
\Bigl(\frac{1+z}{21}\Bigr)^{3.85}
\Bigl[\frac{\sqrt{1+z}}{\sqrt{1+z}-1}\Bigr]^{1/2}
\label{Tnoiseperpixel}
\ee
Here $\tau$ is the total observing time and $A_e$ is the effective antenna area.
To derive this noise temperature per pixel in \cite{Oscar} we began with
the thermal noise per visibility as given by Morales~\cite{Morales:2004ca} to arrive
at the thermal noise per pixel for the brightness temperature.
The angular resolution is $\lambda/D$ where $\lambda$ is the observation frequency and $D$ is the baseline.
The angular resolution is assumed to be tuned by a dilution of the array
from being fully compact by a simple scaling of all baseline positions by
a fixed amount. The system temperature is given by
ARCADE 2~\cite{Fixsen:2009xn}. Further details can be found in ~\cite{Oscar}.
The noise per pixel resolution dependence given in eq.~\ref{Tnoiseperpixel}
assumes an isotropic resolution in all three spatial directions. However
for a cosmic wake signal, the width is order $4\pi G\mu(z_i+1)/(z+1)$ smaller
than the two length directions, whereas each length direction is of order the
Hubble size.
If $\theta_{\rm hubble}(z)$ is the angular resolution needed for the wake's
length, then $\theta_{\rm hubble}(z) 4\pi G\mu(z_i+1)/(z+1)$ is the angular
resolution needed for the width. Thus an appropriate estimate of the
resolution in radians needed to calculate the noise per pixel is:
\be
\theta_{\rm resolution} (z) = \Big({4\pi G\mu(z_i+1)\over(z+1)}\Big)^{1/3} ~ \theta_{\rm hubble}(z)
= \Big({4\pi G\mu(z_i+1)\over(z+1)}\Big)^{1/3} {1\over (\sqrt{z+1}-1)}
\ee
Hence, in a three dimensional map with the necessary resolution to detect a
cosmic string wake laid down at $z_i$ and emitting or absorbing radiation at
$z$, the thermal noise per pixel is:
\be
T_n^{\rm wake}=\frac{0.19~{\rm mK}~ (G\mu)_{9}^{-1/3} (z_i+1)^{-1/3} }
{\sqrt{(\tau/10^4{\rm hr}) (A_e / {\rm km^2})}}
\Big({z+1\over21}\Big)^{4.68}(1-1/\sqrt{z+1})^{1/2}
\ee
where $(G\mu)_{9}\equiv 10^{9} G\mu $.
We will consider 10 000 hours of total observing time. Evidently for
the SKA the effective antenna area is 1 sq km. For $z_i=3000$ and $z=20$,
a $G\mu=4\times10^{-8}$ leads to an average noise per pixel
$T_{n}=0.3~ {\rm mK}$ comparable in size to the background brightness
temperature. Smaller $G\mu$ require a smaller resolution and lead to
larger noise per pixel. Hence for $G\mu<4\times10^{-8}$ the thermal
noise dominates over the background brightness temperature for the
wake signal. For example at $z_i=3000$ and $z=20$
a $G\mu= 5\times10^{-9}$ gives a
$\theta_{\rm resolution} (z)=5.8 \times 10^{-3}$ radians, which is 20 minutes
of arc, and hence an average noise per pixel of $T_{n}=0.66~ {\rm mK}$.
For strings laid down at a lower redshift the noise is greater.
The noise is also a rising function of the emission redshift $z$. For a
$z_i=1000$ with $z=20, 25$ and 35, $T_{n}=1.6~ {\rm mK},~2.0~ {\rm mK}$
and 2.7~ mK, respectively.
In Figure~\ref{fig:dTb20-vs-Gu} we show the relative brightness temperature
(vertical axis) as a function of $(G \mu)_6$ (horizontal axis) for various
values of the redshift $z_i $ of wake formation, evaluated
at the redshift $z = 20$. Negative (positive) brightness temperature
means absorption (emission). The two almost horizontal (brown) lines
give the noise per pixel level in an experiment such as the SKA for
the pixel size chosen such as to optimize the search for a string
wake with the respective value of $G \mu$ (see the above discussion).
\begin{figure}[htbp]
\includegraphics[height=6cm]{dTb20-vs-Gu}
\caption{The relative brightness temperature (vertical axis) in degrees
Kelvin as a function of $(G \mu)_6$ (horizontal axis) for various values
of the formation redshift $z_i$, evaluated for an observation redshift of
$z = 20$. The curves from left to right (in the region of low values of
$G \mu$) correspond to $z_i = 3000$ (red curve), $z_i = 2500$ (orange),
$z_i = 2000$ (green), $z_i = 1500$ (light blue) and $z_i = 1000$ (dark blue).
The two brown lines indicate the expected thermal noise per pixel in an
experiment such as the SKA, with a pixel size chosen to depend on
$G \mu$ as described in the text. The black line that is almost
indistinguishable from the x-axis is the brightness temperature of
the background cosmic gas.}
\label{fig:dTb20-vs-Gu}
\end{figure}
\begin{figure}[htbp]
\includegraphics[height=6cm]{dTb20-vs-Gu-Zoom}
\caption{A zoom in of figure~\ref{fig:dTb20-vs-Gu} to the point where
the noise curve (brown) crosses the signal curves (colours red to blue).
The black line at the top of the graph is the brightness temperature of
the background cosmic gas.} \label{fig:dTb20-vs-Gu-Zoom}
\end{figure}
As expected, the larger the value of $z_i $, the easier it is
to detect low values of $G \mu$ since the wakes have had a longer
time to grow in thickness. A feature overlooked in our previous
work~\cite{us, us2} is that for a fixed observational redshift $z$, there
will be a range of values of $G \mu$ for which the wake's brightness
temperature $T_K$ will be so close to the background cosmic gasp temperature
that there will be no signal. In this situation,
we can consider smaller values of $z_i $ which lead to a clean and
large absorption signal.
For each curve in Figure~\ref{fig:dTb20-vs-Gu} it is apparent that
there is a kink in
the brightness curve (in the absorption region). This kink occurs
at the value of $G \mu$ for which $T_K = 3 T_g$, the
transition point between a wake with shock heating (larger values
of $G \mu$) and a diffuse wake. Our previous analysis was only
valid until that point. We see here that with our current work
it is possible to extend the range of values of $G \mu$ which
can be probed by 21cm redshift surveys.
Taking into account that the optimal pixel size will depend on
$G \mu$, then the pixel noise level will also depend on $G \mu$.
In Figure~\ref{fig:dTb20-vs-Gu-Zoom} we zoom in on the small
$G \mu$ region of the previous figure and compare the string
signal to the pixel noise. We see that the absorption
signal of the diffuse wakes is larger than
the noise level for values of $G \mu$ larger than
\be \label{limit3}
G \mu \,>\, 2.5 \times 10^{-8} \,
\ee
for a detection redshift of $z = 20$. This is an
improvement of the current bound~(\ref{limit1})
and demonstrates the potential of 21cm surveys to probe
cosmic strings.
For smaller values of $G \mu$, cosmic string wakes will still
provide a signal. The signal has a particular shape in a three-dimensional
redshift survey and should hence be detectable using shape
detection algorithms for values of $G \mu$ much smaller than the
above value (\ref{limit3}). In work in progress we are exploring
this issue quantitatively. Note that a similar problem arises in
the case of identifying cosmic string signatures in CMB temperature
maps. It was shown \cite{Amsel} that edge detection algorithms provide
the promise of pushing the limits of detection several orders of
magnitude below the level of the pixel noise, and one order of
magnitude lower than limits which can be achieved using CMB angular
power spectra studies. Thus, we are optimistic that 21cm redshift
surveys will provide a very promising arena to probe for the
possible existence of cosmic strings.
\begin{figure}[htbp]
\includegraphics[height=6cm]{dTb20-vs-zi}
\caption{The relative brightness temperature (vertical axis) as a
function of formation redshift $z_i$ (horizontal axis) for various
values of $(G \mu)_6$. The values of $(G \mu)_6$ chosen are
0.05, 0.09, 0.13, 0.17, 0.21, and 0.25
(red, orange, green, light blue, dark blue and purple curves, respectively).
At low values of $z_i$ some of these curves are labelled by an increasing
amplitude of the absorption signal. The black and brown lines near the
x-axis represents the background gas brightness temperature and the noise
temperature, respectively. They are nearly indistinguishable on this scale
for these values of $G\mu$. The noise is plotted for $(G \mu)_6$ = 0.05.
The larger values of $(G \mu)_6$ would give less noise. }
\label{fig:dTb20-vs-zi}
\end{figure}
In Figure~\ref{fig:dTb20-vs-zi} we show how that formation redshift
which leads to the largest amplitude of the absorption signal depends
on $z_i$. If we are looking
for strings of a particular value of $G \mu$ we would be focusing on wakes
formed at the redshift which yields largest absorption amplitude. Since
the formation redshift determines the size of the 21cm wedge which the
corresponding wake produces, we can use the size information to optimize
the search strategy.
\section{Conclusions}
\label{sec:conclusion}
We have extended the analysis of the 21cm signal
of cosmic string wakes to the case when the string tension is too small to
yield shock heating of the wake. Instead, a ``diffuse wake" forms. This wake
is less dense but thicker than a wake which has undergone shock heating.
Hence, the relative 21cm brightness temperature is smaller in amplitude,
but it is extended over a larger redshift interval.
Demanding that the relative brightness temperature exceeds the pixel
thermal noise in a future 21cm experiment such as the Square
Kilometre Array (SKA) (with the pixel size chosen to be able to
identify the wake signal, as described in the text) leads to a
limit of $G \mu > 2.5 \times 10^{-8}$, an improvement over the
current limit of Eq.~(\ref{limit1}) obtained from CMB anisotropy
maps~\cite{Dvorkin}. There are three important comments to make.
First, a limit obtained by direct searches in position space is
independent of some of the unknown parameters characterizing the scaling
solution of strings (for example, it is largely independent of the number of
strings per Hubble volume), whereas a bound obtained from the angular
correlation function of CMB anisotropies depends on these parameters.
Thus, a bound obtained by the methods we describe is more robust.
Secondly with more statistics, even a signal strength below the noise
can be detected.
And finally, since cosmic string wakes produce signals with a distinguished
geometry (namely thin wedges in redshift maps), these signals should
be detectable by clever statistical techniques even if the amplitude is
substantially smaller than the pixel noise. An example of a statistical
tool which could be used to look for the specific string wake signal is
the set of Minkowski functionals, statistics designed to characterize
the topology of the structure of maps. For a preliminary study of
this approach with references to the original literature see \cite{Evan}.
\section{Appendix}
In this appendix we present how the $(\sin{\theta})^{-2}$
factor in the expression for the wake brightness temperature arises from the line profile term $\phi(\nu)$. We also explain why it does not imply any physical singularity in the limit $\theta \rightarrow 0$.
We begin with the formula for the relative brightness temperature of photons (see \cite{us} and references therein)
\be \label{three3}
\delta T_b(\nu) \, \simeq \, {\bigl( T_S - T_{\gamma} \bigr) \over (1+z)}
\tau_{\nu} \, ,
\ee
where the optical depth $\tau_{\nu}$ is given by
\be \label{three5}
\tau_{\nu} \, = \,
\frac{3 c^2 A_{10}}{4 \nu^2} \bigl( \frac{\hbar \nu_{10}}{k_B T_S} \bigr) \frac{N_{HI}}{4} \phi(\nu) \, .
\ee
Here $N_{HI}$ is the column density of HI, and $\phi(\nu)$ is the line profile.
The line profile accounts for the nonzero width of the 21~cm line.
The line profile is a sharply peaked function about $\nu_{21}=$1420 MHz normalized such that
\be
{\int_0^\infty} \phi(\nu) d\nu =1
\ee
For the case at hand the line profile is not a Dirac delta function because the Hubble expansion leads to a velocity gradients along the line of sight. Due to this radial velocity gradient, the emitted 21~cm photons incur a local line broadening which one can interpret as a relative Doppler shift.
\begin{figure}[htbp]
\includegraphics[height=8cm]{invsin2}
\caption{A 21 cm light ray traverses a cosmic string wake of width $w$.}
\label{fig:invsin2}
\end{figure}
In particular, let us consider photons reaching us at an angle $\theta$ relative
to the vertical of the wake having width $w$ (see fig.~\ref{fig:invsin2}). There is a relative Doppler shift $\delta \nu$
between photons from the front and back of the wake along the photon
trajectory. The horizontal distance $d$ along the wake between the back
point of the photon trajectory and the photon trajectory exiting the wake is
given by
\be
d \, = \, w \tan{\theta} \, .
\ee
The relative Doppler shift is due to the Hubble expansion of the planar
dimensions of the wake. Thus, the component of this expansion
velocity in direction of the photon trajectory is given by
\be
v_{//} \, = \, H d \sin{\theta} \, = \,{ H w \sin^2{\theta}\over \cos{\theta}} \, ,
\ee
and we have a frequency width given by
\be
\delta\nu=v_{//} ~ \nu_{21}
\ee
The line profile is
\be \label{three8}
\phi(\nu) \, = \, \frac{1}{\delta \nu} \,\,\, \rm{for} \,\,\,
\nu \, \epsilon \, [\nu_{21} - \frac{\delta \nu}{2}, \nu_{21} + \frac{\delta \nu}{2}] \, ,
\ee
and $\phi(\nu) = 0$ otherwise. For a frequency within the
range of 21cm radiation passing through the wake the relative brightness
temperature is given by (\ref{eq:deltaTb}). This makes the origin
of the $(\sin{\theta})^{-2}$ factor manifest.
Any actual measurement will have a finite frequency resolution. If we integrate
over a frequency interval larger than $\delta \nu$, the $(\sin{\theta})^{-2}$ factor
will cancel as it must. If we have a very small frequency resolution $R$, then
the brightness temperature density at a fixed $\nu$ will increase as $\theta$
decreases until $\delta \nu$ becomes smaller than $R$. From that point
onwards, the result will be independent of $\theta$. Thus there is never any divergence associated with $\theta \rightarrow 0$.
\begin{acknowledgments}
This work is supported in part by a NSERC Discovery Grant, by funds from the
CRC Program, and by the FQRNT Programme de recherche
pour les enseignants de coll\`ege.
\end{acknowledgments}
|
{
"timestamp": "2012-08-01T02:04:46",
"yymm": "1203",
"arxiv_id": "1203.2307",
"language": "en",
"url": "https://arxiv.org/abs/1203.2307"
}
|
\section{Introduction}
\label{sec:intro}
We consider linear stability analyses of general fluid flows, and begin by recalling the main steps of these, before introducing the motivation of the present work, thereby also introducing the necessary notation. We first restrict ourselves to the linear stability analysis of a parallel flow $U_0(z)$ to two-dimensional disturbances (extensions of the work to non-parallel flows and to three-dimensional disturbances are set out in \S\ref{sec:disc}). Such analysis is accomplished by solving the Orr--Sommerfeld (OS) equation~\citep{orr1907}. The solution of the equation at fixed parameter values (such as the Reynolds number $Re$) yields an eigenvalue problem that connects the wave number, the frequency, and the growth rate of the disturbance~\citep{DrazinReidBook}. For two-dimensional disturbances, the perturbation streamfunction has the form $\psi(x,z,t)=\phi(z)\mathrm{e}^{\mathrm{i} \left(\alpha x-\omega t\right)}$,
where $\alpha=\alpha_{\mathrm{r}}+\mathrm{i}\alpha_{\mathrm{i}}$ and $\omega=\omega_{\mathrm{r}}+\mathrm{i}\omega_{\mathrm{i}}$ are complex numbers, $\alpha_{\mathrm{r}}$ is the (real) wave number, $\omega_{\mathrm{r}}$ is the (real) frequency, and $\alpha_{\mathrm{i}}$ and $\omega_{\mathrm{i}}$ are the spatial and temporal growth rates respectively; in this context, the OS equation reads
\begin{equation}
\mathrm{i}\alpha\left(U_0(z)-c\right)\left(\partial_z^2-\alpha^2\right)\phi-\mathrm{i}\alpha U_0''(z)\phi=Re^{-1}\left(\partial_z^2-\alpha^2\right)^2\phi,\qquad c=\omega/\alpha.
\label{eq:os0}
\end{equation}
That the wavenumber $\alpha$ is complex indicates that the instability is spatio-temporal in nature~\citep{huerre90a}; in a \textit{temporal analysis}~\citep{DrazinReidBook}, $\alpha_{\mathrm{i}}=0$ and the eigenvalue problem is solved for $\omega_{\mathrm{i}}$ and $\omega_{\mathrm{r}}$. This results in the following temporal {dispersion relations} for the temporal frequency and growth rate respectively: $\omega_{\mathrm{r}}^{\mathrm{temp}}=\omega_{\mathrm{r}}(\alpha_{\mathrm{r}},\alpha_{\mathrm{i}}=0)$, and $\omega_{\mathrm{i}}^{\mathrm{temp}}=\omega_{\mathrm{i}}(\alpha_{\mathrm{r}},\alpha_{\mathrm{i}}=0)$.
The basic flow is called \textit{unstable} if the growth rate is positive for some $\alpha_{\mathrm{r}}$. It is of interest further to classify unstable flows as convectively or absolutely unstable: the flow is convectively unstable if initially localized pulses are amplified in at least one moving frame of reference but are damped in the laboratory frame; on the other hand, the flow is absolutely unstable if such pulses lead to growing disturbances in the entire domain in the laboratory frame.
\citet{Briggs1964} developed a method to classify unstable plasmas according to this dichotomy. This criterion was extended to fluid dynamics by a number of authors~\citep{Gaster1968,Kupfer1987,Brevdo1988,huerre90a,Lingwood1997}
The approach is based on a {\it local} eigenvalue analysis of the OS equation, which forms the key ingredient in global analyses: (i) a necessary condition for absolute instability is if the imaginary part of the frequency is positive at a saddle point in the complex $\alpha$-plane: $\omega_{\mathrm{i}0}:=\omega_{\mathrm{i}}(\alpha_0)>0$, where $\alpha_0$ solves $d\omega/d\alpha=0$; (ii) to obtain a sufficient condition for absolute instability, the saddle point $\alpha_0$ in the complex $\alpha$-plane must be the result of the coalescence of spatial branches that originate from opposite half-planes at a larger and positive value of $\omega_{\mathrm{i}}$ (this coalescence or `pinching' of the spatial branches is accompanied by the formation of a cusp at $\omega_{\mathrm{i}0}$ in the complex $\omega$ plane).
Typically, the saddle-point and Briggs criteria are checked using a fast numerical eigenvalue solver~\citep{Boyd,TrefethenBook}. Our purpose here is not to supplant such effective computations,
but rather to develop an analytical connection between spatio-temporal and
temporal growth rates to assist in understanding these numerical results. A second
motivation is the possible extension of this analytical approach to globally-unstable
flows. We discuss these motivations in more detail now.
Our purpose within a local description is to develop an analytical connection between spatio-temporal and temporal growth rates. Such a connection is desirable for a number of reasons.
First, in flows where several temporal modes are unstable,
the interpretation of the results of a spatio-temporal stability analysis (obtained numerically, for instance) becomes difficult, as does the mere determination of the convective/absolute (C/A) transitions~\citep{Suslov06}. In this case, $\omega$-plots of the distinct spatio-temporal modes consist of complicated interpenetrating surfaces, whose domain is the entire complex $\alpha$-plane. On the other hand, many theoretical tools exist for characterizing temporal instabilities, including energy-budget analyses (as reviewed by \citet{Boomkamp1996}) and asymptotic analytical solutions~\citep{DrazinReidBook}, so the governing physical mechanisms for growth of temporal modes is often well documented and competing temporal modes can easily be distinguished. A direct link with temporal modes could therefore enable one to distinguish the competing spatio-temporal modes. Furthermore, in parametric studies involving large parameter spaces (e.g., thermal boundary layers or two-phase flows), wherein the C/A transition curves are plotted as a function of the flow parameters, the parametric dependence of the C/A transition curves may be difficult to analyse. For example, for a system with a two-dimensional parameter space $(\mu_1,\mu_2)$, the curve demarcating the transition between convective and absolute instability is given by the generic formula $\mu_2=f(\mu_1)$. Here, knowledge of the physical properties of the system (in particular the familiar temporal stability properties) may be beneficial (if not crucial) to deduce the function $f(\cdot)$.
Although the main focus in this study is on an analytic connection between spatio-temporal and temporal local stability growth rates, the results are also of interest in {\it global} stability analyses. There, the base state can be determined numerically or otherwise (using either an unperturbed solution or a time-averaged perturbed result), in terms of the streamwise coordinate. Assuming this to evolve slowly with the streamwise coordinate the saddle-node frequency $\omega_{\mathrm{i}0}$ can then be determined as a function of the real scaled streamwise coordinate, $X$. But in order to determine global modes (detailed criteria are recalled briefly in \S~\ref{sec:disc}), $\omega_{\mathrm{i}0} (X)$ is required off the real axis, for which analytic continuation is used. In other words, rather than a connection between growth rates for complex and real wave numbers, one between growth rates for complex and real spatial coordinates is needed. An example of
wherein analytical continuation is used for this purpose is the study of~\citet{Hammond1997}. The formulation proposed herein in the context of a local analysis is readily reformulated for use in this setting for a global analysis, as outlined at the end of this paper, and offers the possibility of developing a combination of both.
This work is organized as follows. We derive an exact formula connecting the spatio-temporal growth rate with purely temporal quantities in \S~\ref{sec:exact}. We discuss the two lowest-order truncations of this formula in \S\ref{sec:limiting}: Gaster's formula, and a situation in which the temporal quantities depend only quadratically on the wavenumber $\alpha_{\mathrm{r}}$. This enables the formulation of condition (i) into a succinct equation encoding the competition between \textit{in-situ} growth and convective effects. In \S\ref{sec:singular} we describe the singularities that typically occur in the dispersion relation, and discuss how these can hamper the convergence of our formula. In \S\ref{sec:numerics} we apply our results to a simple OS analysis involving wake instability. In \S\ref{sec:disc} we summarize our arguments and describe further applications of the exact formula and its quadratic approximation, including in a global stability analysis.
\section{An exact formula for the complex frequency, derived from purely temporal quantities}
\label{sec:exact}
The development of the formula starts with the assumption that an eigenvalue analysis of the OS equation yields a complex frequency $\omega$ that depends on the streamwise wave number $\alpha$ as an analytic (holomorphic) function, $\omega=\omega_{\mathrm{r}}(\alpha_{\mathrm{r}},\alpha_{\mathrm{i}})+\mathrm{i} \omega_{\mathrm{i}}(\alpha_{\mathrm{r}},\alpha_{\mathrm{i}})$,
where we have fixed the other system parameters (e.g. Reynolds number). The domain $D$ of analyticity is an open simply-connected subset of $D\subset\mathbb{C}$, containing a part of the real line. Details concerning the domain are discussed further below.
As a consequence of the analyticity of $\omega(\alpha)$, we have the following Cauchy--Riemann conditions:
\begin{subequations}
\vspace{-0.25in}
\begin{center}
\begin{tabular}{p{0.4\textwidth}p{0.4\textwidth}}
{\begin{align}
&\frac{\partial\omega_{\mathrm{r}}}{\partial \alpha_{\mathrm{r}}}=\frac{\partial\omega_{\mathrm{i}}}{\partial\alpha_{\mathrm{i}}},
\label{eq:cr1}
\end{align}}
&
{\begin{align}
& \frac{\partial\omega_{\mathrm{r}}}{\partial \alpha_{\mathrm{i}}}=-\frac{\partial\omega_{\mathrm{i}}}{\partial\alpha_{\mathrm{r}}}.
\label{eq:cr2}
\end{align} }
\end{tabular}
\end{center}
\vspace{-0.2in}
\end{subequations}
We use Equations~\eqref{eq:cr1} and~\eqref{eq:cr2} to derive an expression for $\omega_{\mathrm{i}}(\alpha_{\mathrm{r}},\alpha_{\mathrm{i}})$ as a function of the purely temporal stability properties.
We work at a fixed value of $\alpha_{\mathrm{r}}$ in Equation~\eqref{eq:cr1} and identify the group velocity $c_{\mathrm{g}}(\alpha_{\mathrm{r}})=(\partial\omega_{\mathrm{r}}/\partial\alpha_{\mathrm{r}})_{(\alpha_{\mathrm{r}},0)}$.
Furthermore, we Taylor-expand $\partial\omega_{\mathrm{r}}/\partial\alpha_{\mathrm{r}}$ into the region $D$:
\begin{equation}
\frac{\partial\omega_{\mathrm{r}}}{\partial\alpha_{\mathrm{r}}}=\sum_{n=0}^\infty \frac{1}{n!}c_{\mathrm{g}n}(\alpha_{\mathrm{r}})\alpha_{\mathrm{i}}^n,\qquad
c_{\mathrm{g}n}(\alpha_{\mathrm{r}})=\frac{\partial^n}{\partial\alpha_{\mathrm{i}}^n}\frac{\partial\omega_{\mathrm{r}}}{\partial\alpha_{\mathrm{r}}}\bigg|_{\alpha_{\mathrm{i}}=0}.
\end{equation}
This amounts to a complex Taylor expansion centred at $(\alpha_{\mathrm{r}},0)$, and is therefore valid on a disc of radius $R$ contained entirely in $D$. The magnitude of the radius of convergence $R$ is discussed below.
Each term in this Taylor expansion is available from a purely temporal analysis. We describe this process for the first few terms (for brevity, we use $\partial_{\mathrm{i}}:=\partial/\partial\alpha_{\mathrm{i}}$ and $\partial_{\mathrm{r}}:=\partial/\partial\alpha_{\mathrm{r}}$).
At $n=0$ we have $c_{\mathrm{g}0}=\partial_{\mathrm{r}}\omega_{\mathrm{r}}|_{\alpha_{\mathrm{i}}=0}=c_{\mathrm{g}}(\alpha_{\mathrm{r}})$, the group velocity. At $n=1$ we obtain
\begin{equation}
c_{\mathrm{g}1}=\partial_{\mathrm{i}}\partial_{\mathrm{r}}\omega_{\mathrm{r}}=\partial_{\mathrm{r}}\partial_{\mathrm{i}}\omega_{\mathrm{r}}\overset{\mathrm{C.R.}}=-\partial_{\mathrm{r}}\partialr\omega_{\mathrm{i}}\overset{\alpha_{\mathrm{i}}=0}=-\frac{d^2\omega_{\mathrm{i}}^{\mathrm{temp}}}{d\alpha_{\mathrm{r}}^2},
\end{equation}
where $\omega_{\mathrm{i}}^{\mathrm{temp}}$ is the purely temporal growth rate.
At $n=2$ we have
\begin{equation}
c_{\mathrm{g}2}=\partial_{\mathrm{i}}\partialim\partial_{\mathrm{r}}\omega_{\mathrm{r}}=\partial_{\mathrm{i}}\partial_{\mathrm{r}}\partial_{\mathrm{i}}\omega_{\mathrm{r}}\overset{\mathrm{C.R.}}=-\partial_{\mathrm{i}}\partial_{\mathrm{r}}\partialr\omega_{\mathrm{i}}=-\partial_{\mathrm{r}}\partialr\partial_{\mathrm{i}}\omega_{\mathrm{i}}\overset{\mathrm{C.R.}}=-\partial_{\mathrm{r}}\partialr\partial_{\mathrm{r}}\omega_{\mathrm{r}}
\overset{\alpha_{\mathrm{i}}=0}=-\frac{d^2c_{\mathrm{g}}}{d\alpha_{\mathrm{r}}^2}.
\end{equation}
At $n=3$ we compute
\begin{eqnarray}
c_{\mathrm{g}3}&=&\partial_{\mathrm{i}}\partialim\partial_{\mathrm{i}}\partial_{\mathrm{r}}\omega_{\mathrm{r}}=\partial_{\mathrm{i}}\partialim\partial_{\mathrm{r}}\partial_{\mathrm{i}}\omega_{\mathrm{r}}\overset{\mathrm{C.R.}}=-\partial_{\mathrm{i}}\partialim\partial_{\mathrm{r}}\partialr\omega_{\mathrm{i}}=-\partial_{\mathrm{i}}\partial_{\mathrm{r}}\partialr\partial_{\mathrm{i}}\omega_{\mathrm{i}}\nonumber\\
&\overset{\mathrm{C.R.}}=&-\partial_{\mathrm{i}}\partial_{\mathrm{r}}\partialr\partial_{\mathrm{r}}\omega_{\mathrm{r}}
=-\partial_{\mathrm{r}}\partialr\partial_{\mathrm{r}}\partial_{\mathrm{i}}\omega_{\mathrm{r}}
\overset{\mathrm{C.R.}}=\partial_{\mathrm{r}}\partialr\partial_{\mathrm{r}}\partialr\omega_{\mathrm{i}}
\overset{\alpha_{\mathrm{i}}=0}=\frac{d^4\omega_{\mathrm{i}}^{\mathrm{temp}}}{d\alpha_{\mathrm{r}}^4}.
\end{eqnarray}
Similarly, at $n=4$ we have
\begin{eqnarray}
c_{\mathrm{g}4}&=&\partial_{\mathrm{i}}\partialim\partial_{\mathrm{i}}\partialim\partial_{\mathrm{r}}\omega_{\mathrm{r}}=\partial_{\mathrm{i}}\partialim\partial_{\mathrm{i}}\partial_{\mathrm{r}}\partial_{\mathrm{i}}\omega_{\mathrm{r}}\overset{\mathrm{C.R.}}=-\partial_{\mathrm{i}}\partialim\partial_{\mathrm{i}}\partial_{\mathrm{r}}\partialr\omega_{\mathrm{i}}=
-\partial_{\mathrm{i}}\partialim\partial_{\mathrm{r}}\partialr\partial_{\mathrm{i}}\partial\omega_{\mathrm{i}}\nonumber\\
&\overset{\mathrm{C.R.}}=&-\partial_{\mathrm{i}}\partialim\partial_{\mathrm{r}}\partialr\partial_{\mathrm{r}}\omega_{\mathrm{r}}=
-\partial_{\mathrm{i}}\partial_{\mathrm{r}}\partialr\partial_{\mathrm{r}}\partial_{\mathrm{i}}\omega_{\mathrm{r}}\overset{\mathrm{C.R.}}=\partial_{\mathrm{i}}\partial_{\mathrm{r}}\partialr\partial_{\mathrm{r}}\partialr\omega_{\mathrm{i}}=
\partial_{\mathrm{r}}\partialr\partial_{\mathrm{r}}\partialr\partial_{\mathrm{i}}\omega_{\mathrm{i}}\nonumber\\
&\overset{\mathrm{C.R.}}=&\partial_{\mathrm{r}}\partialr\partial_{\mathrm{r}}\partialr\partial_{\mathrm{r}}\omega_{\mathrm{r}}
\overset{\alpha_{\mathrm{i}}=0}=\frac{d^4c_{\mathrm{g}}}{d\alpha_{\mathrm{r}}^4}.
\end{eqnarray}
We deduce the higher-order terms by inspection. Assembling these results, we obtain an expression for $\partial\omega_{\mathrm{r}}/\partial\alpha_{\mathrm{r}}$, extended into the complex plane:
\begin{equation}
\frac{\partial\omega_{\mathrm{r}}}{\partial\alpha_{\mathrm{r}}}\bigg|_{(\alpha_{\mathrm{r}},\alpha_{\mathrm{i}})}=\sum_{n=0}^\infty \frac{(-1)^n}{(2n)!}\frac{d^{2n}c_{\mathrm{g}}}{d\alpha_{\mathrm{r}}^{2n}}\alpha_{\mathrm{i}}^{2n}+\sum_{n=0}^\infty \frac{(-1)^{n+1}}{(2n+1)!}\frac{d^{2n+2}\omega_{\mathrm{i}}^{\mathrm{temp}}}{d\alpha_{\mathrm{r}}^{2n+2}}\alpha_{\mathrm{i}}^{2n+1}.
\label{eq:omr_taylor}
\end{equation}
Next, we use the Cauchy--Riemann condition~\eqref{eq:cr1} and connect the Taylor expansion in Equation~\eqref{eq:omr_taylor} to an expression for $\omega_{\mathrm{i}}(\alpha_{\mathrm{r}},\alpha_{\mathrm{i}})$, valid in the entire complex half-plane $\alpha_{\mathrm{r}}>0$. We have
\begin{equation}
\frac{\partial\omega_{\mathrm{i}}}{\partial\alpha_{\mathrm{i}}}\overset{\mathrm{C.R.}}=\frac{\partial\omega_{\mathrm{r}}}{\partial\alpha_{\mathrm{r}}}
=\sum_{n=0}^\infty \frac{(-1)^n}{(2n)!}\frac{d^{2n}c_{\mathrm{g}}}{d\alpha_{\mathrm{r}}^{2n}}\alpha_{\mathrm{i}}^{2n}+\sum_{n=0}^\infty \frac{(-1)^{n+1}}{(2n+1)!}\frac{d^{2n+2}\omega_{\mathrm{i}}^{\mathrm{temp}}}{d\alpha_{\mathrm{r}}^{2n+2}}\alpha_{\mathrm{i}}^{2n+1}.
\end{equation}
We integrate this relation along a line perpendicular to the real axis, from $(\alpha_{\mathrm{r}},0)\rightarrow (\alpha_{\mathrm{r}},\alpha_{\mathrm{i}})$ ($\alpha_{\mathrm{i}}$ is arbitrary, i.e. $|\alpha_{\mathrm{i}}|$ is not necessarily small). The result is
\begin{equation}
\omega_{\mathrm{i}}(\alpha_{\mathrm{r}},\alpha_{\mathrm{i}})=\omega_{\mathrm{i}}^{\mathrm{temp}}(\alpha_{\mathrm{r}})+\sum_{n=0}^\infty \frac{(-1)^n}{(2n+1)!}\frac{d^{2n}c_{\mathrm{g}}}{d\alpha_{\mathrm{r}}^{2n}}\alpha_{\mathrm{i}}^{2n+1}
+\sum_{n=0}^\infty \frac{(-1)^{n+1}}{(2n+2)!}\frac{d^{2n+2}\omega_{\mathrm{i}}^{\mathrm{temp}}}{d\alpha_{\mathrm{r}}^{2n+2}}\alpha_{\mathrm{i}}^{2n+2}.
\label{eq:omi_taylor}
\end{equation}
This is an exact formula for the imaginary part of the complex frequency, which depends only on purely temporal quantities. Mathematically, this is a trivial statement: since the complex frequency is analytic,
analytic continuation implies that its behaviour on the real line completely determines its behaviour on $D$. Nevertheless, this formula will help us to create a practical means of predicting the threshold for absolute instability.
The series~\eqref{eq:omi_taylor} amounts to a complex Taylor series centred at the point $(\alpha_{\mathrm{r}},0)$ and therefore converges inside a disc of radius $R$. The radius $R$ is the minimum distance from the point $(\alpha_{\mathrm{r}},0)$ to the nearest singularity of $\omega(\alpha)$. Care must be taken that the point of interest (e.g. the saddle point of $\omega(\alpha)$) lies inside this disc. A detailed discussion of the radius of convergence of Equation~\eqref{eq:omi_taylor} is given in \S\ref{sec:singular}.
\section{Low-order truncations and an approximate criterion for the C/A transition}
\label{sec:limiting}
In this section we investigate the implications of truncating Equation~\eqref{eq:omi_taylor} at linear order, and at quadratic order. At linear order, the spatial analysis of Gaster is recovered; at quadratic order, the complex Ginzburg--Landau model is obtained, albeit with further understanding, in the form of a `balance condition' for the onset of absolute instability.
\subsection{Relation to the analysis of Gaster}
\label{sec:limiting:gaster}
The Gaster transformation~\citep{Gaster1962} concerns purely \textit{spatial} instabilities, which correspond to setting $\omega_{\mathrm{i}}=0$ in the OS eigenvalue analysis, and computing the resulting spatial growth rate $\alpha_{\mathrm{i}}$ as a function of $(\omega_{\mathrm{r}},\alpha_{\mathrm{r}})$. Practically, this corresponds to a localized disturbance in the flow that oscillates at a characteristic frequency $\omega_{\mathrm{r}}$, and grows downstream of the disturbance at a (spatial) rate $-\alpha_{\mathrm{i}}$. Spatial growth therefore corresponds to negative values of $\alpha_{\mathrm{i}}$. Using the fact that the general complex frequency $\omega(\alpha)$ is an analytic function, Gaster integrated the resulting Cauchy--Riemann conditions along a straight-line path of small length, $(\alpha_{\mathrm{r}},0)\rightarrow (\alpha_{\mathrm{r}},\alpha_{\mathrm{i}})$, with $|\alpha_{\mathrm{i}}|\ll 1$. The resulting integrands are thus regarded as constant and, setting $\omega_{\mathrm{i}}=0$, Gaster obtained the following formula for the spatial growth rate:
\begin{equation}
\alpha_{\mathrm{i}}=-\omega_{\mathrm{i}}^{\mathrm{temp}}\Big\slash\frac{d\omega_{\mathrm{r}}^{\mathrm{temp}}}{d\alpha_{\mathrm{r}}}.
\label{eq:gaster0}
\end{equation}
This equation provides a precise connection between the spatial growth rate and more familiar temporal quantities. Nevertheless, it is limited, in the sense that it applies only to spatial growth (as opposed to fully spatio-temporal growth) and is valid only for small values of $|\alpha_{\mathrm{i}}|$.
Equation~\eqref{eq:gaster0} can be recovered directly from the approach in \S\ref{sec:exact}. Consider Equation~\eqref{eq:omi_taylor} with $|\alpha_{\mathrm{i}}|\ll 1$, such that only the lowest-order term gives a contribution to the equation. The result is
\begin{equation}
\omega_{\mathrm{i}}(\alpha_{\mathrm{r}},\alpha_{\mathrm{i}})\sim \omega_{\mathrm{i}}^{\mathrm{temp}}(\alpha_{\mathrm{r}})+c_{\mathrm{g}}(\alpha_{\mathrm{r}})\alpha_{\mathrm{i}},\qquad |\alpha_{\mathrm{i}}|\rightarrow 0.
\label{eq:gaster_reduce1}
\end{equation}
On a spatial branch, we have $\omega_{\mathrm{i}}=0$, hence Equation~\eqref{eq:gaster_reduce1} becomes $$\alpha_{\mathrm{i}}\sim -\omega_{\mathrm{i}}^{\mathrm{temp}}(\alpha_{\mathrm{r}})/c_{\mathrm{g}}(\alpha_{\mathrm{r}}),$$ as $|\alpha_{\mathrm{i}}|\rightarrow 0$,
which is precisely the formula of~\citet{Gaster1962} (Equation~\eqref{eq:gaster0}) for spatial modes.
\subsection{The quadratic approximation }
\label{sec:limiting:quadratic}
We also derive conditions for the C/A transition based on a second-order truncation of the series dispersion relation~\eqref{eq:omi_taylor} -- the quadratic approximation.
In this truncation, the Taylor series~\eqref{eq:omi_taylor} reduces to
\begin{equation}
\omega_{\mathrm{i}}(\alpha_{\mathrm{r}},\alpha_{\mathrm{i}})=\omega_{\mathrm{i}}^{\mathrm{temp}}(\alpha_{\mathrm{r}})+c_{\mathrm{g}}(\alpha_{\mathrm{r}})\alpha_{\mathrm{i}}-\tfrac{1}{2}\frac{d^2\omega_{\mathrm{i}}^{\mathrm{temp}}}{d\alpha_{\mathrm{r}}^2}\alpha_{\mathrm{i}}^2.
\label{eq:omi_approx}
\end{equation}
The necessary saddle-point condition for an absolute instability (as reviewed in \S\ref{sec:intro}) is $\partial\omega_{\mathrm{i}}/\partial\alpha_{\mathrm{r}}=\partial\omega_{\mathrm{i}}/\partial\alpha_{\mathrm{i}}=0$; using Equation~\eqref{eq:omi_approx}, these conditions amount to
\begin{equation}
\frac{d\omega_{\mathrm{i}}^{\mathrm{temp}}}{d\alpha_{\mathrm{r}}}+\frac{dc_{\mathrm{g}}}{d\alpha_{\mathrm{r}}}\alpha_{\mathrm{i}}-\tfrac{1}{2}\frac{d^3\omega_{\mathrm{i}}^{\mathrm{temp}}}{d\alpha_{\mathrm{r}}^3}\alpha_{\mathrm{i}}^2=0,
\qquad
c_{\mathrm{g}}(\alpha_{\mathrm{r}})-\frac{d^2\omega_{\mathrm{i}}^{\mathrm{temp}}}{d\alpha_{\mathrm{r}}^2}\alpha_{\mathrm{i}}=0.
\label{eq:ai_linear}
\end{equation}
Taking the second condition, we get
\begin{equation}
\alpha_{\mathrm{i}}=c_{\mathrm{g}}(\alpha_{\mathrm{r}})\bigg\slash \frac{d^2\omega_{\mathrm{i}}^{\mathrm{temp}}}{d\alpha_{\mathrm{r}}^2}.
\end{equation}
Substitution into the first condition yields
\begin{equation}
c_{\mathrm{g}}(\alpha_{\mathrm{r}})\frac{dc_{\mathrm{g}}}{d\alpha_{\mathrm{r}}}=-\frac{d\omega_{\mathrm{i}}^{\mathrm{temp}}}{d\alpha_{\mathrm{r}}}\frac{d^2\omega_{\mathrm{i}}^{\mathrm{temp}}}{d\alpha_{\mathrm{r}}^2}+\tfrac{1}{2}c_{\mathrm{g}}^2(\alpha_{\mathrm{r}})\left(\frac{d^3\omega_{\mathrm{i}}^{\mathrm{temp}}}{d\alpha_{\mathrm{r}}^3}\bigg\slash\frac{d^2\omega_{\mathrm{i}}^{\mathrm{temp}}}{d\alpha_{\mathrm{r}}^2}\right).
\label{eq:saddle}
\end{equation}
(Note that third-order derivatives appear in this calculation -- but only through the process of finding the saddle-point location, and not in the series expansion of the dispersion relation.)
The existence of a real root of Equation~\eqref{eq:saddle} is a necessary condition for a saddle point to occur.
Once the root of Equation~\eqref{eq:saddle} is extracted, we derive a condition for the $\omega_{\mathrm{i}}$ to vanish at the saddle point, as this is the sign of the transition to absolute instability. Referring to Equation~\eqref{eq:omi_approx}, $\omega_{\mathrm{i}}$ vanishes at the saddle when
\begin{equation}
-\frac{d^2\omega_{\mathrm{i}}^{\mathrm{temp}}}{d\alpha_{\mathrm{r}}^2}\bigg|_{\alpha_{\mathrm{r}}^*}\omega_{\mathrm{i}}^{\mathrm{temp}}(\alpha_{\mathrm{r}}^*)=\tfrac{1}{2}c_{\mathrm{g}}^2(\alpha_{\mathrm{r}}^*).
\label{eq:saddle_sign}
\end{equation}
where $\alpha_{\mathrm{r}}^*$ is the root of Equation~\eqref{eq:saddle}. Knowledge of the saddle-point location $\alpha_{\mathrm{r}}^*$, together with the dispersion relation~\eqref{eq:omi_approx} yield further information that can be used to verify if the saddle point arises as a result of the coalescence of spatial branches ramifying into different halves of the complex $\alpha$-plane (i.e. the necessary and sufficient conditions for the onset of absolute instability).
Finally, we note that the dispersion relation for the model linear complex Ginzburg--Landau equation is quadratic in the wavenumber~\citep{huerre2000,Suslov2004}, and no error is incurred in this case in applying the quadratic truncation of the series dispersion relation~\eqref{eq:omi_taylor}.
\subsection{Higher-order approximations}
For certain flows, a higher-order truncation of the series~\eqref{eq:omi_taylor} may be required, in order to capture the multiplicity of saddle points that can contribute to the spatio-temporal growth in certain anomalous situations~\citep{Brevdo1999}. A straightforward extension of the analysis in \S\ref{sec:limiting}\ref{sec:limiting:quadratic} applies. The reader is referred to the Supplementary Material for details of an application of the series expansion of the dispersion relation for free-surface flow on an inclined plane~\citep{Brevdo1999}.
An advantage of the framework developed in this section (compared with the direct approach) concerns the insight given by Equation~\eqref{eq:saddle_sign} into the balance of competing effects that brings about absolute instability.
Intuitively, we may think of the condition~\eqref{eq:saddle_sign} as a competition between \textit{in-situ} growth on the left and convection of the disturbance on the right. Absolute instability can set in only if the \textit{in-situ} growth equals or exceeds the tendency of the parallel flow to convect disturbances downstream. Similar advantages carry over to situations where high-order truncations are required: the existence of multiple $\alpha_{\mathrm{i}}$-roots in the polynomial truncation of the criterion $d\omega_{\mathrm{i}}/d\alpha_{\mathrm{i}}=0$ corresponds exactly to multiple branches in the spatio-temporal growth rate in a moving frame~\citep{Brevdo1999}.
Finally, we note that although a `quadratic approximation' was pursued in earlier works, the context was different: either in relation to a numerical, iterative procedure for determining the location of the saddle point (which anyway fails in the presence of multiple saddle points)\citep{Deissler1987,Monkey1988}, or in the context of asymptotic theories concerned with reduction of a complicated model to a Complex Ginzburg--Landau equation~\citep{Suslov2004}.
None of these works contains the explicit condition linking the C/A transition to the temporal analysis which is a key result of this section.
\section{Convergence of series for $\omega_{\mathrm{i}}$ in the presence of a bestiary of singularities}
\label{sec:singular}
In this section we review the different kinds of singularity that can occur in the dispersion relation $\omega=\omega(\alpha)$, and discuss the implications of such possible non-analytic behaviour for the convergence of our formula~\eqref{eq:omi_taylor}. Here, the `impulse response function' refers to the solution of the spatiotemporal problem
\begin{multline}
\left[U_0(z)\partial_x+\partial_t\right]\left(\partial_z^2+\partial_x^2\right)\psi(x,z,t)-U_0''(z)\partial_x\psi(x,z,t)=Re^{-1}\left(\partial_z^2+\partial_x^2\right)^2\psi(x,z,t),\\
\psi(x,z,t=0)=\delta(z)\delta(x).
\label{eq:impulse}
\end{multline}
Equation~\eqref{eq:impulse} is typically solved using a Laplace-Fourier decomposition; the dispersion relation of the resulting eigenvalue problem (the OS equation) determines the shape of the impulse response.
\textit{1. Branch cuts in unconfined flows:} In so-called \textit{unconfined flows}, the dispersion relation $\omega=\omega(\alpha)$ may possesses branch cuts along the imaginary axis $\alpha_{\mathrm{r}}=0$~\citep{huerre85,Lingwood1997}. In such systems (e.g. mixing layers, flow past obstacles), both the streamwise and normal dimensions of the fluid container extend to infinity (labelled by coordinates $x$ and $z$ respectively), and the disturbance streamfunction decays to zero as $|z|\rightarrow\infty$. Then, the asymptotic ($|z|\rightarrow\infty$) OS equation reads
\[
\mathrm{i}\alpha\left(U_\infty-c\right)\left(\partial_z^2-\alpha^2\right)\phi=Re^{-1}\left(\partial_z^2-\alpha^2\right)\phi,\qquad c=\omega/\alpha,
\]
for a symmetric base state $U_0(z)$ with
\[
U_\infty=\lim_{z\rightarrow\pm\infty}U_0(z), \text{ and } \lim_{z\rightarrow\pm\infty}U_0''(z)=0.
\]
This asymptotic solution possesses an inviscid mode $\phi\sim\mathrm{e}^{-\mathrm{sign}(\alpha_{\mathrm{r}})\alpha z}$ and a viscous mode $\phi\sim\mathrm{e}^{-\mathrm{sign}(\gamma_\mathrm{r})\gamma z}$, where $\gamma=\sqrt{\alpha^2+\mathrm{i}\alpha Re(U_\infty-c)}$. The inviscid mode induces a branch cut along the imaginary axis~\citep{huerre85}, while the viscous mode induces hyperbolic branch cuts in the $\alpha$-plane~\citep{Aships1990,Schmid2001}. All of these branch cuts (or `continuous spectra'~\citep{Schmid2001}) contribute to the contour integral associated with the impulse-response function. The contour integral of the impulse-response function therefore possesses two contributions: one from the zeros of the dispersion relation $\omega(\alpha)=0$ (`poles'), and the other from the continuous spectrum. However, to diagnose absolute instability, it suffices to consider the discrete part, as the continuous spectrum can produce temporal growth only when the discrete part is absolutely unstable~\citep{Lingwood1997}.
\textit{2. Discrete poles along the imaginary axis in the complex $\alpha$-plane:} \citet{Healey2007,Healey2009} has shown that confining an \textit{inviscid} version of the flow described in (2) between two plates at $z=\pm H$ causes the character of the singularity along the axis $\alpha_{\mathrm{r}}=0$ to switch from a continuous branch cut to a set of discrete poles. For, the inviscid OS (Rayleigh) equation
\[
\mathrm{i}\alpha\left(U_0(z)-c\right)\left(\partial_z^2-\alpha^2\right)\phi-\mathrm{i}\alpha U_0''(z)\phi=0,
\]
can be re-written as
\[
\left(\partial_z^2-\alpha^2\right)\phi=\frac{\mathrm{i}\alpha U_0''(z)\phi}{\mathrm{i}\alpha U_0(z)-\mathrm{i} \omega},
\]
such that the singularity $\omega=\infty$ corresponds to $(\partial_z^2-\alpha^2)\phi=0$. This limiting equation satisfies the appropriate boundary conditions for a confined flow (no-penetration at the walls where confinement is enforced) whenever $\phi$ can be made into an oscillatory function, that is, when $\alpha=\pm n\pi\mathrm{i}/H$, with $n=1,2,\cdots$. These poles induce a series of saddle points near the imaginary axis. These saddle points can dominate over the saddle point associated with the analogous unconfined flow for sufficiently small $H$. Such saddles can even pinch (according to the Briggs criterion) and thus produce absolute instability~\citep{Healey2007,Healey2009}, leading to the conclusion that a confined flow is `more absolutely unstable' than its unconfined analogue.
Since the introduction of viscosity to the problem does not change the large-$\omega$ form of the eigenvalue problem, it is expected that such poles will persist in the viscous case (with some modification of the large-$\omega$ streamfunction arising from the different boundary conditions in the viscous case)~\citep{Healey2012}. This contention is supported by numerical evidence (\S\ref{sec:numerics}).
\textit{3. Further branch cuts in the complex $\alpha$-plane:}
The generic dispersion relation $\mathcal{D}(\alpha,\omega)=0$ of the OS equation naturally evokes consideration of the multivariable function $\mathcal{D}(\alpha,\omega)$ itself. This is an analytic function in each of its variables~\citep{Gaster1968}. Moreover, $\partial \mathcal{D}/\partial\omega$ has at least one zero~\citep{Gaster1968}. In the neighbourhood of this point (labelled $(\alpha_0,\omega_0)$), $\mathcal{D}(\alpha,\omega)$ can be expanded as
\begin{multline*}
\mathcal{D}(\alpha,\omega)=\mathcal{D}(\alpha_0,\omega_0)+\mathcal{D}_\alpha(\alpha_0,\omega_0)(\alpha-\alpha_0)+\tfrac{1}{2}\mathcal{D}_{\omega\omega}(\alpha_0,\omega_0)(\omega-\omega_0)^2\\
+\text{Higher-order terms},
\end{multline*}
where the subscripts $\alpha$ and $\omega$ indicate partial differentiation. If the points $(\alpha,\omega)$ and $(\alpha_0,\omega_0)$ satisfy the dispersion relation $\mathcal{D}=0$, then this equation can be recast as
\begin{equation}
\omega=\omega_0+\left(-\frac{\mathcal{D}_\alpha(\alpha_0,\omega_0)}{\mathcal{D}_{\omega\omega}(\alpha_0,\omega_0)}\right)^{1/2}\left(\alpha-\alpha_0\right)^{1/2},
\label{eq:riemann_surf}
\end{equation}
The dispersion relation $\omega(\alpha)$ is therefore non-differentiable at $\alpha_0$ and possesses a branch cut along a line emanating from the point $\alpha_0$. Equivalently, one may regard the dispersion relation as being multi-valued, taking distinct values on the two distinct sheets of the Riemann surface described by Equation~\eqref{eq:riemann_surf}.
Such singularities can be problematic in a number of ways. First, they limit the radius of convergence of the series dispersion relation~\eqref{eq:omi_taylor}, thus inhibiting the description of a dynamically-relevant saddle point via this route. Examples of this kind are found in the works of~\citet{Brevdo1988,Juniper2006}.
Even graver, if a saddle point of the dispersion relation is sufficiently close the branch cut~\eqref{eq:riemann_surf}, the branch cut can prevent the saddle point from pinching, such that the Briggs criterion is not satisfied (see \S\ref{sec:intro}). Equivalently, in such cases, it is not possible to deform the contour in the impulse-response integral into a steepest-descent path without enclosing the singular point $\alpha_0$. Such saddles do not therefore generate absolute instability~\citep{Lingwood1997}.
It is clear from this list that knowledge of the global topography of the dispersion relation is required to determine absolute instability (on this point, see also~\cite{Lingwood1997}). Equally, such knowledge is necessary to determine when Equation~\eqref{eq:omi_taylor} can be used with confidence. Since Equation~\eqref{eq:omi_taylor} is the imaginary part of a complex-valued power series, it is valid in the disc of convergence of this complex-valued series. The outermost radius $R$ of this disc is given by
\begin{multline}
R=\text{Distance between the point of interest }(\alpha_{\mathrm{r}},0)\\\text{and the nearest singularity in }\omega(\alpha);
\label{eq:roc0}
\end{multline}
a necessary and sufficient condition for the validity of Equation~\eqref{eq:omi_taylor} at the point $(\alpha_{\mathrm{r}},\alpha_{\mathrm{i}})$ is thus $|\alpha_{\mathrm{i}}|<R$.
In practice, the requirement that the global topography of the dispersion relation $\omega(\alpha)$ be known will not limit the potential uses of the local formula~\eqref{eq:omi_taylor}: since we envisage that our formula should be applied to the in-depth analysis of numerical results (for energy-budget analyses and the characterization of mode competition and C/A transition curves), this does not result in any loss of relevance for the formula~\eqref{eq:omi_taylor}.
\section{Application of the formula to a canonical fluid instability}
\label{sec:numerics}
The quadratic approximation holds true exactly only for model dispersion relations such as the linearized complex Ginzburg--Landau equation (\S~\ref{sec:limiting:quadratic}). In more practical systems, the exact series expansion (\ref{eq:omi_taylor}) would have to converge quickly for the quadratic approximation to be useful. The convergence properties have already been discussed in general terms in \S\ref{sec:singular}; here, we discuss convergence for the particular model (dimensionless) velocity field
\begin{equation}
U_0(z)=1-\Lambda+2\Lambda\big\{1+\sinh^{2N}\left[z\sinh^{-1}(1)\right]\big\}^{-1},\qquad \Lambda<0,
\label{eq:uzero}
\end{equation}
where $\Lambda$ and $N$ are dimensionless parameters, and $-\infty<z<\infty$. Equation~\eqref{eq:uzero} models the steady wake profile generated by flow past a bluff body. The quantity $\Lambda=(U_\mathrm{c}-U_\mathrm{max})/(U_\mathrm{c}+U_\mathrm{max})$ is the velocity ratio, where $U_\mathrm{c}$ is the wake centreline velocity and $U_\mathrm{max}$ is the maximum velocity. Furthermore, $N$ is the shape parameter, which controls the ratio between the mixing-layer thickness and the width of the wake. It ranges from $N=1$ (the `$\mathrm{sech}^2$ wake') to $N=\infty$, a `top-hat wake' bounded by two vortex sheets~\citep{Monkey1988}.
The base state~\eqref{eq:uzero} is absolutely unstable~\citep{Monkey1988} (this serves as a model test case for our purposes; for subsequent work on cylinder wakes, see~\citet{Pier2002}). The associated OS equation possesses three independent parameters, $(\Lambda,N,Re)$, where $Re$ is the Reynolds number.
\subsection{Numerical method}
\label{sec:numerics:method}
Practical numerical methods should be confirmed with a notional \textit{ideal method}, which would preserve
the branch-cut singularity in the dispersion relation $\omega(\alpha)$ along the imaginary axis, associated with the unbounded domain in Equation~\eqref{eq:uzero}.
We considered (1) a standard Chebyshev collocation method using Chebyshev polynomials as the basis functions, wherein confinement is introduced at $z=\pm H$, such that $\phi(\pm H)=\phi'(\pm H)=0$; we also considered (2) a Chebyshev collocation method equipped with basis functions appropriate to an infinite domain~\citep{Boyd}.
The streamfunction for Method (2) is expanded in terms of basis functions as follows:
\begin{equation}
\phi(z)=\sum_{n=0}^{N_C}a_n TB_n(z),\qquad TB_n(y)=\cos\left(n\,\mathrm{arccot}\left(\frac{y}{L}\right)\right),
\label{eq:chebydef}
\end{equation}
where $L$ is an adjustable map parameter (typically chosen to be unity, e.g.~\citep{Boyd}) and $y\in(-\infty,\infty)$. Each rational basis function $TB_n(y)$ possesses the asymptotic behaviour $TB_n(y)\sim 1$ as $|y|\rightarrow\infty$. The $a_n$'s are determined (along with the eigenvalue $-\mathrm{i}\alpha c$) by substitution of Equation~\eqref{eq:chebydef} into the OS equation, evalution at collocation points, and solution of the resulting (finite) matrix eigenvalue problem. Convergence of the numerical method is obtained by increasing $N_C$ until no change in the eigenvalue is observed (to within working precision).
One would expect this second approach to capture the continuous singularity along the imaginary axis.
Methods (1) and (2) yield identical results with respect to a purely temporal stability analysis: we have verified the correctness of these methods by computing the critical Reynolds number for the onset of (convective) instability: $Re_\mathrm{c}(N=1,\Lambda=-1)=1.8$, and $Re_\mathrm{c}(N=2,\Lambda=-1)=1.9$, in agreement with the existing literature~\citep{Monkey1988}.
Not surprisingly, given the discussion in \S\ref{sec:singular}, we have found that method (1) produces a sequence of equally-separated poles in the dispersion relation along the imaginary-wavenumber axis.
The value of $H$ is chosen as $H=H_\mathrm{c}$; here $H_\mathrm{c}$ is large in the sense that the linear-stability results are the same for $H=H_\mathrm{c}$ and $H>H_\mathrm{c}$. The only exception is near the imaginary axis, where confinement features occur; these features are well separated from the physical saddle points in the dispersion relation and do not affect the stability results (non-physical singularities and saddle points are identified as they move around upon changing the amount of confinement). In this work, we take $H_\mathrm{c}=8$.
On the other hand -- rather surprisingly -- we have found that method (2) does not produce a continuous spectrum in the dispersion relation along the imaginary-wavenumber axis, but rather a sequence of poles: even with the basis functions appropriate for the infinite domain, the numerical solution artificially interprets the problem as though its geometry were confined.
\subsection{Results}
\label{sec:numerics:results}
Before applying Equation~\eqref{eq:omi_taylor}, we investigate the feasibility of so doing, that is, we examine the topography of $\omega(\alpha)$ and the proximity of any singularities in this function to the point of interest $(\alpha_{\mathrm{r}},0)$ on the real line. Each study is presented using numerical method (1), but further tests were carried out using method (2), producing near-identical results.
A first study is carried at the parameter values $(Re,\Lambda,N)=(100,-1.1,2)$. Because only the sinuous mode produces absolute instability, the varicose mode is projected out of the problem using appropriate boundary conditions at the symmetry line $z=0$~\citep{Monkey1988}. This removes a potential source of mode competition from the complex dispersion relation. The results of these calculations are shown in Figure~\ref{fig:singular0}.
\begin{figure}[htb]
\centering
\subfigure[]{\includegraphics[width=0.32\textwidth]{study1_imag_with_branches}}
\subfigure[]{\includegraphics[width=0.32\textwidth]{study1_imag_with_steepest_descent}}
\subfigure[]{\includegraphics[width=0.32\textwidth]{study1_omega_cusp}}
\caption{The global topography of $\omega_{\mathrm{i}}$ for $(Re,\Lambda,N)=(100,-1.1,2)$. Figure~(c) shows contours of $\alpha_{\mathrm{r}}$ in a complex $\omega$ plane. The $\omega$ cusp corresponds to the $\alpha$-pinchpoint in Figures~(a) and (b).}
\label{fig:singular0}
\end{figure}
We first of all comment on the large peaks along the imaginary axis in Figure~\ref{fig:singular0}. We have confirmed numerically that these are poles (rather than large but finite values of $|\omega_{\mathrm{i}}|$): successively decreasing the resolution in the scan of the complex $\alpha$-space leads to larger numerical maximum values of $(\omega_{\mathrm{r}},\omega_{\mathrm{i}})$. Clearly, these poles are confinement poles (as in the works of~\citet{Juniper2006,Healey2007,Healey2009}).
We investigate the saddles induced by the confinement poles in Figure~\ref{fig:singular0}(a). Here, the spatial branch corresponding both to the confinement saddle and the `physical' saddle are shown (the `physical' saddle is the one that persists in the limit as $H\rightarrow\infty$). Only the physical saddle possesses spatial branches emanating from opposite half-planes and is therefore the only one that satisfies the Briggs criterion for contributing to absolute instability (the $\omega$-cusp at the $\alpha$-pinchpoint is shown in Figure~\ref{fig:singular0}(c)). This result was verified further by a so-called \textit{ray analysis} (e.g. the work by~\citet{Onaraigh2012a}), where the growth-rate $\sigma(V)$ along rays moving with velocity $V$ with respect to the laboratory frame is computed from direct numerical simulation of the linearized equations of motion: again, $\sigma(V)$ possesses only one branch, associated with the physical saddle (the reader is referred to the Supplementary Material for details about such direct numerical simulations). Furthermore, the path of steepest descent is shown in Figure~\ref{fig:singular0}(b). This path follows a line of constant $\omega_{\mathrm{r}}$ and passes through the physical saddle of $\omega_{\mathrm{i}}$ and points from the South West to the North East, and connects at infinity to the real axis.
The dispersion relation for $(Re,\Lambda,N)=(100,-1.1,5)$ was found to contain a branch cut emanating from the point $(0.44,-1.79)$. However, this occurred far below the saddle in the complex plane and was not collocated with the steepest-descent path. Consideration of the growth rate $\sigma(V)$ computed via linearized DNS showed that this singularity did not contribute to the evolution of an initially-localized pulse.
Finally, dispersion relations for a further parameter set $(Re,\Lambda,N)=(100,-1.1,5)$ are shown in Figure~\ref{fig:singular1}. These results are very similar to Figure~\ref{fig:singular0} (in particular, only the physical saddle is found to contribute to the absolute instability) and are not analysed in any further detail.
\begin{figure}[htb]
\centering
\includegraphics[width=0.45\textwidth]{study2_imag}
\caption{The global topography of $\omega_{\mathrm{i}}$ for $(Re,\Lambda,N)=(100,-1.1,5)$.}
\label{fig:singular1}
\end{figure}
In Figures~\ref{fig:singular0}--\ref{fig:singular1} the singularity nearest to the real axis lies at $\alpha_{\mathrm{i}}=-0.18$ for both $N=2$ and $N=5$ (the dispersion relations are generated with confinement at $H=8$).
The physical saddle is located at $(\alpha_{\mathrm{r}},\alpha_{\mathrm{i}})=(0.94,-0.7)$ and $(0.79,-1.0)$ for $N=2$ and $N=5$ respectively (its computed location is the same for numerical methods (1) and (2) and is independent of $H$, for $H$ sufficiently large).
Thus, the radius of convergence defined by Equation~\eqref{eq:roc0} is $R=0.96$ and $R=0.82$ for $N=2$ and $N=5$ respectively.
It would therefore appear that the series~\eqref{eq:omi_taylor} converges for the $N=2$ case and diverges for the $N=5$ case; however, we have found that high-order truncations of the power series for $\omega_{\mathrm{i}}$ (and an analogous series for $\omega_{\mathrm{r}}$)
give good approximations to $(\omega_{\mathrm{r}},\omega_{\mathrm{i}})$ outside of this region (Figure~\ref{fig:singular2}). The numerical differentiation of the $\alpha_{\mathrm{r}}$-derivatives for Figure~\ref{fig:singular2} was been performed by interpolating the temporal dispersion relation on to a Chebyshev collocation grid and then performing analytic differentiation on the Chebyshev basis functions of this interpolation (this is a different Chebyshev collocation grid from the one discussed in \S\ref{sec:numerics}\ref{sec:numerics:method}).
\begin{figure}[htb]
\centering
\subfigure[]{\includegraphics[width=0.48\textwidth]{convergence_N5_imag}}
\subfigure[]{\includegraphics[width=0.48\textwidth]{convergence_N5_real}}
\caption{Successive truncations of (a) the series for $\omega_{\mathrm{i}}$ (Equation~\eqref{eq:omi_taylor}); (b) an analogous series for $\omega_{\mathrm{r}}$, and a comparison with the true functions for $(\omega_{\mathrm{r}},\omega_{\mathrm{i}})$. Here $\alpha_{\mathrm{r}}=0.7729$ (i.e. a point close to the saddle), $(Re,\Lambda,N)=(100,-1.1,2)$. The solid vertical line indicates the radius of convergence, while the broken vertical line indicates the $\alpha_{\mathrm{i}}$-location of the saddle point.}
\label{fig:singular2}
\end{figure}
In certain cases, a truncation of a divergent Taylor series can approximate a function reasonably well. Indeed, for certain cases there exists an `optimal truncation', whereby some finite truncation order $N_{\mathrm{trunc,opt}}<\infty$ minimizes the difference between the generating function and the truncated Taylor series\footnote{For example, the function $f(x)=(1+x)\log (1+x)-x$. On the interval $|x|<1$, this function can be represented as a power series. However, for $x>1$ the associated series diverges, but an optimal truncation can be found that minimizes the difference between $f(x)$ and the truncated series.}. Thus, for physical applications such as the present one, it suffices to determine on a case-by-case basis whether finite truncations of the series for $\omega_{\mathrm{i}}$ are good approximations to the underlying function.
Motivated by the good agreement between the true function $\omega_{\mathrm{i}}(\alpha_{\mathrm{r}},\alpha_{\mathrm{i}})$ and finite truncations of its Taylor series in Figure~\ref{fig:singular2}, we have investigated the purely temporal stability properties of the system in Figure~\ref{fig:singular2} with a view to applying Equation~\eqref{eq:omi_taylor} to the spatiotemporal problem.
\begin{figure}[htb]
\centering
\subfigure[]{\includegraphics[width=0.4\textwidth]{omega_imag_sample_new}}
\subfigure[]{\includegraphics[width=0.4\textwidth]{ddomega_imag_sample_new}}\\
\subfigure[]{\includegraphics[width=0.4\textwidth]{c_group_sample_new}}
\subfigure[]{\includegraphics[width=0.4\textwidth]{dc_group_sample_new}}
\caption{Dispersion curves for two test cases with $(Re,\Lambda)=(100,-1.1)$. Solid line: $N=2$; broken line: $N=5$. The curves for the $N=2$ case are not continued beyond $\alpha_{\mathrm{r}}\approx 2.3$ because of the sharp mode competition that occurs there. However, this is a region where all modes are stable and is therefore not relevant to the dynamics. }
\label{fig:bluff1}
\end{figure}
This is done in Figure~\ref{fig:bluff1}: the growth rate has the basic features of a quadratic equation in $\alpha_{\mathrm{r}}$, while the group velocity depends only weakly on the wavenumber $\alpha_{\mathrm{r}}$. However, the second and first derivatives of these quantities are non-linear and non-constant respectively,
and the assumptions of the quadratic approximation do not apply. We therefore examine both the quadratic approximation and further, higher-order approximations; the latter involve a truncation of Equation~\eqref{eq:omi_taylor} at cubic (or higher) order, together with a procedure analogous to Equations~\eqref{eq:omi_approx}--\eqref{eq:saddle_sign} to obtain the criterion for the onset of absolute instability.
The true boundary between the convective and absolute regions is calculated numerically from a spatio-temporal OS analysis and is identified according to steps (i)--(ii) in \S\ref{sec:intro}.
\begin{figure}[htb]
\centering
\subfigure[]{\includegraphics[width=0.4\textwidth]{ac_transition_quadratic_new}}
\subfigure[]{\includegraphics[width=0.4\textwidth]{ac_transition_cubic_new}}
\caption{Boundary between the absolute and convective regions for different Reynolds numbers, with comparison to the (a) quadratic, and (b) cubic approximations. Curves without symbols correspond to direct OS eigenvalue calculations in the complex $\alpha$-plane and agree exactly with the results by~\citet{Monkey1988}.
Dashed curve: $Re=20$; solid curve: $Re=100$. Circles: approximate curves at $Re=20$; crosses: approximate curves $Re=100$.
}
\label{fig:bluff2}
\end{figure}
The estimated boundary is calculated, according to quadratic and cubic approximations. The quadratic approximation yields a maximum error of $20\%$ with respect to the numerically-generated curves. For the cubic approximation, the $Re=20$ approximate curve and the numerically-generated curve virtually coincide, while the maximum error in the $Re=100$ curve for the cubic approximation is reduced to $7.5\%$ (Figure~\ref{fig:bluff2}). We have verified that going to higher-order truncations in Equation~\eqref{eq:omi_taylor} for the $Re=100$ case leads to near-perfect overlap between these curves (a maximum error of $2.5\%$ in a fifth-order truncation, at $Re=100$).
\subsection{Interpretation of results via the quadratic approximation}
The quadratic approximation can now be used to further analyse the C/A transition. The coefficients in a quadratic approximation for the temporal dispersion relation $\omega_{\mathrm{i}}^{\mathrm{temp}}(\alpha_{\mathrm{r}})$ can be related to the temporally most-dangerous mode $(\alpha_{\mathrm{m}},\omega_{\mathrm{i\,m}}^{\mathrm{temp}})$, such that
\begin{equation}
\omega_{\mathrm{i}}^{\mathrm{temp}}(\alpha_{\mathrm{r}})=2\omega_{\mathrm{i\,m}}^{\mathrm{temp}}\left[\frac{\alpha_{\mathrm{r}}}{\alpha_{\mathrm{m}}}-\tfrac{1}{2}\left(\frac{\alpha_{\mathrm{r}}}{\alpha_{\mathrm{m}}}\right)^2\right],\qquad\text{quadratic approximation}.
\end{equation}
Moreover, we have found in our numerical calculations that the $\alpha_{\mathrm{r}}$-location of the unstable saddle point in the complex dispersion relation almost coincides with the most-dangerous mode, such that (using the notation in \S\ref{sec:limiting}\ref{sec:limiting:quadratic}) $\alpha_{\mathrm{r}}^*\approx \alpha_{\mathrm{m}}$. The criterion~\eqref{eq:saddle_sign} in \S\ref{sec:limiting}(b) describing the onset of absolute instability then reduces to
\begin{equation}
{2\omega_{\mathrm{i\,m}}^{\mathrm{temp}}}\slash{\alpha_{\mathrm{m}}}=c_{\mathrm{g}} (\alpha_{\mathrm{m}}).
\label{eq:CA_QA}
\end{equation}
Both sides of the condition~\eqref{eq:CA_QA} are well represented by expressions of the form $A(N,Re)+B(N,Re)\Lambda$ over the part of parameter space considered in \S\ref{sec:numerics}\ref{sec:numerics:results}. In Figure~\ref{fig:lhscg} the dependencies of these
\begin{figure}[htb]
\centering
\subfigure[]{\includegraphics[width=0.49\textwidth]{lhscoeff}}
\subfigure[]{\includegraphics[width=0.49\textwidth]{cgcoeff}}
\caption{Dependency on $N$ of coefficients in fits linear in $\Lambda$ of the left-hand side of Equation~\eqref{eq:saddle_sign} (a) and the group velocity (b) at $Re=100$ (solid lines) and $Re=20$ (dashed lines); plusses and crosses represent $A$ and $B$, respectively. In (a), the sign of both coefficients has been reversed.
}
\label{fig:lhscg}
\end{figure}
coefficients on $N$ and $Re$ are shown. The most significant of these is the $N-$dependency of the coefficient $B$ for the group velocity: increasing $N$ makes the group velocity approach unity rapidly from below. If we ignore for the time being the other dependencies on $N$ and use for the left-hand side of Equation~\eqref{eq:CA_QA} simple $N-$averaged values ($A\equiv a_0\approx -0.05$ and $B\equiv a_1\approx -0.8$ for $Re=100$), and for the group velocity the curve fit
\begin{equation}
c_g ( \alpha_m ) \approx 1+b_0{\rm exp}(-b_1N)\Lambda
\label{eq:cgCA}
\end{equation}
(with $b_0\approx 0.6$ and $b_1\approx 0.5$ for $Re=100$), we obtain an approximate expression for the C/A transition based on the quadratic approximation:
\begin{equation}
\Lambda_{CA}\approx \frac{a_0-1}{b_0{\rm exp}(-b_1N)-a_1}.
\label{eq:appCA}
\end{equation}
We have found this to represent the full results obtained with the quadratic approximation within the uncertainty of the value of the coefficients. Overall then, the main trend is that in making $\Lambda$ more negative, the group velocity (the right-hand side of Equation~\eqref{eq:CA_QA}) decreases, whereas the left-hand side of Equation~\eqref{eq:CA_QA} increases; these two effects promote absolute instead of convective instability. Thus, the steep drop in the curves at large $N$ in Figure~\ref{fig:bluff2} is associated with the fact that increasing $N$ increases the group velocity.
\begin{figure}[htb]
\centering
\subfigure[]{\includegraphics[width=0.49\textwidth]{ddom}}
\subfigure[]{\includegraphics[width=0.49\textwidth]{om}}
\caption{The dependencies on $N$ and $\Lambda$ of (a) $-d^2\omega_{\mathrm{i}}^{\mathrm{temp}}/d\alpha_{\mathrm{r}}^2$; (b) $\omega_{\mathrm{i}}^{\mathrm{temp}}$, at the most-dangerous temporal mode for $Re=100$. Results are shown for $N=1,1.25,1.5,2,3,4,5,6,7,8$ from top to bottom in (a), and from bottom to top in (b).
}
\label{fig:om_ddom}
\end{figure}
Regarding the dependencies on $Re$, we first note that at low values of $N$, the magnitude of the left-hand side of Equation~\eqref{eq:CA_QA} is small at $Re=20$ compared with $Re=100$, whereas the magnitude of the right-hand side is large; these changes both favour convective instability. At intermediate-to-large values of $N$, both terms decrease when the value of $Re$ is decreased, with the left-hand side more so than the right-hand side. Hence, the C/A transition in Figure~\ref{fig:bluff2} occurs at a more negative value of $\Lambda$ for the lowest value of $Re$ due to effects on both sides in Equation~\eqref{eq:CA_QA}. These effects can be accounted for in the above through the introduction of explicit $Re-$dependencies in the coefficients $a_0,a_1,b_0,b_1$ (even if only representing the $Re-$dependencies of their $N-$averaged values). These dependencies are already visible in Figure~\ref{fig:lhscg}, although their parametrization is beyond the scope of this paper.
These somewhat crude approximations can be refined by representing the dependency of the left-hand-side coefficients $A$ and $B$ on $N$, the causes of which we first briefly investigate here. At values of $N$ near unity, the magnitude of the left-hand side of Equation~\eqref{eq:CA_QA} first increases strongly with $N$ before decreasing beyond $N\approx 3$, such that the region of absolute instability in parameter space is reduced further at large values of $N$. In fact, the results in Figure~\ref{fig:lhscg}(a) tell two different stories, for values below and above $N=3$. Referring back to Equation~\eqref{eq:saddle_sign}, we recall that the left-hand side is the product of the maximum temporal growth rate and minus the curvature of the dispersion relation. Results for these two components are shown in Figure~\ref{fig:om_ddom}. We conclude from Figure~\ref{fig:om_ddom} that $\omega_{\mathrm{i}}^{\mathrm{temp}}(\alpha_{\mathrm{m}})$ increases uniformly with increasing $N$, consistent with the increase in the entire left-hand side of Equation~\eqref{eq:saddle_sign} at low values of $N$, and that the resulting promotion of absolute instability is only reversed by a decrease in $-d^2\omega_{\mathrm{i}}^{\mathrm{temp}}/d\alpha_{\mathrm{r}}^2$ at large values of $N$; at low values of $N$, this curvature term hardly depends on $N$. It is possible to account for these trends in Equation~\eqref{eq:appCA}, by using bilinear fits for $\omega_{\mathrm{i\,m}}^{\mathrm{temp}}$ and $\alpha_{\mathrm{m}}$ (as well as Equation~\eqref{eq:cgCA} for the group velocity):
\begin{equation}
\alpha_{\mathrm{m}} \approx a_\alpha^0+a_\alpha^1N+b_\alpha^0\Lambda+b_\alpha^1N\Lambda,\qquad
\omega_{\mathrm{i\,m}}^{\mathrm{temp}} \approx a_\omega^0+a_\omega^1N+b_\omega^0\Lambda+b_\omega^1N\Lambda.
\label{eq:CA_morefits}
\end{equation}
Substitution of Equation~\eqref{eq:CA_morefits} into Equation~\eqref{eq:CA_QA} then yields a quadratic equation for $\Lambda$ at the C/A transition.
This whole approach can be extended by lifting the approximation $\alpha_{\mathrm{r}}^*\approx \alpha_{\mathrm{m}}$, or by going to higher-order truncations of the dispersion relation; however, these complications obscure the understanding achieved here using simple linear and bilinear fitting, and such extensions are not pursued further here.
\section{Discussion}
\label{sec:disc}
We have presented an analytical connection between spatio-temporal and temporal growth rates in a local linear stability analysis. Specifically, we have derived criteria for a transition between convective and absolute instability, explicitly in terms of the temporal problem, to increasingly refined levels of approximation. The simplest approximation is a quadratic one where the location of the saddle $\alpha_{\mathrm{r}}^*$ is assumed to coincide with the temporally most-dangerous mode; then, the criterion for the onset of absolute instability is given by Equation~\eqref{eq:CA_QA}. At the next level of approximation (still quadratic), one determines $\alpha_{\mathrm{r}}^*$ via Equation~\eqref{eq:saddle}.
Although this approach can be used directly to determine C/A transitions, we anticipate that the main use of the theory will be in analysing results obtained with an efficient computational algorithm for the full OS problem.
We imagine that one would perform such a numerical calculation, diagnose absolute instability (using the saddle-point/pinch-point criteria), and then characterize the instability in detail. Our formula can help in this characterization in the following ways:
\textit{1.} In parametric studies, the dependency of C/A transition curves on model parameters can be difficult to analyse, especially if there are many governing parameters. In \S\ref{sec:numerics} we have seen an example wherein simple approximations based on Equation~\eqref{eq:omi_taylor}, together with insight into the trends exhibited by the dispersion relation in the purely temporal problem, yield a detailed explanation of the behaviour of the C/A transition curves.
\textit{2.} Temporal mode competition is observed in various systems and, as pointed out in the Introduction, disentangling different spatio-temporal modes poses a formidable problem. The present approach then provides a convenient aid: for each temporal branch, an approximation of the corresponding spatio-temporal growth rate in the complex $\alpha$ plane results directly from Equation~\eqref{eq:omi_taylor}. An example is given in Figure~\ref{fig:singular2}. Although only a cross-section for a single value of $\alpha_{\mathrm{r}}$ is shown in Figure~\ref{fig:singular2}, this is readily extended to entire sections of the complex $\alpha$ plane. Indeed, in computing the C/A transition in truncations higher than quadratic, we have found this approach more straightforward rather than the root-finding approach taken for the relatively simple quadratic case (\S\ref{sec:limiting}).
\textit{3.} It follows from the previous point that features in contour plots for a spatio-temporal growth rate (such as saddle points) can be related directly to a specific temporal mode with the present approach. The advantage of this is that such features can then be associated with the physical mechanism that governs that temporal mode.
The application of the approach proposed here to real flows has recently been completed in~\citet{Onaraigh2012a}, where we have studied absolute and convective instability in two-phase gas/liquid flows. There, however, the focus was on the application of the theory in a detailed parameter study and a derivation of the quadratic approximation was not included; in the present work, the focus is on the mathematical derivation of Equation~\eqref{eq:omi_taylor}, discussion of the radius of convergence of the formula, and the investigation of the accuracy of finite truncations of the equation~\eqref{eq:omi_taylor}. Such analysis is important before the theory can be used with confidence in further applications.
Throughout this paper we have considered two-dimensional disturbances, taking $\omega$ as a complex-valued function of a single complex variable (the complex wavenumber). However, this approach could be extended to three-dimensional spatio-temporal disturbances, via a double-Taylor-series expansion of the dispersion relation in the streamwise and spanwise wavenumbers. For cases wherein the spanwise wavenumber is real, the analysis presented herein requires no modification, as the spanwise wavenumber would then only enter as an additional parameter.
We conclude by discussing briefly a further application of this method to weakly globally unstable modes. In nonparallel flows, the properties of the system vary in the streamwise direction, and a standard normal-mode decomposition is no longer possible. The generic linear-stability problem to be solved then reads $\partial_t\psi=\mathcal{L}(\partial_x,\partial_z)\psi$,
where $\mathcal{L}$ is a linear operator with coefficients that depend on the streamwise direction; the normal-mode decomposition $\partial/\partial x \rightarrow \mathrm{i} \alpha$ is no longer possible. We make trial substitution $\psi\rightarrow \psi(x,z)\mathrm{e}^{-\mathrm{i} \omega_G t}$ and obtain the global eigenvalue problem, $\-\mathrm{i} \omega_G\psi(x,z)=\mathcal{L}(\partial_x,\partial_z)\psi(x,z)$.
The system is globally unstable if there exists a global mode for which $\Im(\omega_G)>0$. We introduce a parameter $\varepsilon=\lambda/L$, where $\lambda$ is the typical length scale of the disturbance $\psi$, and $L$ is a measure of the system's extent in the streamwise direction. If $\varepsilon\ll 1$, and if the coefficients of the linear operator $\mathcal{L}$ depend only weakly on space, through the combination $X:=\varepsilon x$, then a WKB approximation~\citep{huerre90a,Chomaz1991,Schmid2001} can be used to demonstrate that the leading-order approximation to the most-dangerous global mode is $\omega_G= \omega_0(X_\mathrm{s})+O(\varepsilon)$,
where $\omega_0(X_\mathrm{s})$ is the saddle-point frequency calculated in the usual manner from a purely \textit{local} stability analysis, at the fixed parameter value $X_\mathrm{s}$, and $X_\mathrm{s}$ is the location of the saddle point of the complex-valued function $\omega_0(X)$ in the complex $X$-plane: $(\partial\omega_0/\partial X)(X_\mathrm{s})=0$.
The flow is therefore globally unstable if $\omega_0(X_\mathrm{s})>0$.
Thus, we are interested in a prescribed complex-valued function of a complex variable, $\omega_0(X)$, and its saddle point(s). If $\omega_0(X)$ is holomorphic, and if $\omega_0(X)$ is known along the real line, then the arguments of \S\ref{sec:exact} apply also to the function $\omega_0(X)$, and the following formula pertains:
\begin{multline}
\omega_{0\mathrm{i}}(X_{\mathrm{r}},X_{\mathrm{i}})=\omega_{0\mathrm{i}}(X_{\mathrm{r}},X_{\mathrm{i}}=0)+\sum_{n=0}^\infty \frac{(-1)^n}{(2n+1)!}\frac{d^{2n+1}}{dX_{\mathrm{r}}^{2n+1}}\left[\omega_{0\mathrm{r}}(X_{\mathrm{r}},X_{\mathrm{i}}=0)\right]X_{\mathrm{i}}^{2n+1}\\
+\sum_{n=0}^\infty \frac{(-1)^{n+1}}{(2n+2)!}\frac{d^{2n+2}}{dX_{\mathrm{r}}^{2n+2}}\left[\omega_{0\mathrm{i}}(X_{\mathrm{r}},X_{\mathrm{i}}=0)\right]X_{\mathrm{i}}^{2n+2}.
\label{eq:omi_taylor_global}
\end{multline}
Furthermore, if $\omega_0(X)$ is quadratic in $X$, then the truncation in \S\ref{sec:limiting}\ref{sec:limiting:quadratic} applies, and the condition for the onset of global instability reads
\begin{equation}
-\left[\frac{d^2\omega_{0\mathrm{i}}(X_{\mathrm{r}},X_{\mathrm{i}}=0)}{dX_{\mathrm{r}}^2}\right]_{X_{\mathrm{r}}^*}\omega_{0\mathrm{i}}(X_{\mathrm{r}}^*,X_{\mathrm{i}}=0)=\tfrac{1}{2}\left[\frac{d\omega_{0\mathrm{r}}(X_{\mathrm{r}},X_{\mathrm{i}}=0)}{dX_{\mathrm{r}}}\right]^2_{X_{\mathrm{r}}^*},
\label{eq:saddle_sign_global}
\end{equation}
where $X_{\mathrm{r}}^*$ satisfies a root-finding condition analogous to Equation~\eqref{eq:saddle}.
A first-order truncation Equation~\eqref{eq:omi_taylor_global} was used in \cite{Hammond1997}, with $\omega_{0\mathrm{i}}(X_{\mathrm{r}},X_{\mathrm{i}}=0)$ determined numerically. The present approach would not only allow for rapid extension to higher order, but also open up the possibility of using Equation~\eqref{eq:saddle_sign_global} in conjunction with for instance the local quadratic approximation~\eqref{eq:saddle_sign} to determine the terms that make up Equation~\eqref{eq:saddle_sign_global}.
\subsection*{Acknowledgements}
This research was supported by the Ulysses-Ireland/France Research Visits Scheme, a programme for research visits between Ireland and France, jointly funded and administered by The Irish Research Council, the Irish Research Council for Science Engineering and Technology and Egide, the French agency for international mobility, with participation from the French Embassy in Ireland and Teagasc.
L. \'O~N. would like to thank R. Smith and C. Boyd for helpful discussion, and for a further helpful and interesting discussion with J. Healey. P.~S. would like to acknowledge useful discussions with his colleague B. Pier.
\bibliographystyle{plainnat}
\section{Inviscid shear flow (Juniper)}
\label{sec:juniper}
The dispersion relation for inviscid shear flow in a confined geometry is studied (see~\citet{Juniper2006}). For the varicose mode, in a non-dimensional framework, this is given in implicit form (\citet{Juniper2006}, p. 174) as
\begin{equation}
S\left(1+\Lambda-\frac{\omega}{k}\right)^2\coth(k)+\left(1-\Lambda-\frac{\omega}{k}\right)^2-k\Sigma=0,
\label{eq:disp_implicit}
\end{equation}
where $(S,\Lambda,h,\Sigma)$ are parameters, $\omega$ is the frequency, and $k$ is the wavenumber. This can be solved to give $\omega$ explicitly as a function of $k$:
\begin{equation}
\omega=\frac{k\left[2\coth(k)\pm \sqrt{[\coth k+\coth (kh)]k-4\coth (k)\coth (kh)}\right]}{\coth k+\coth kh}.
\label{eq:disp_explicit}
\end{equation}
Evidently, this dispersion relation contains simple poles at those values of $k$ that solve $\coth k+\coth kh=0$; this can be solved to give
\begin{equation}
k=\mathrm{i}\frac{n\pi}{1+h},\qquad n=1,2,\cdots
\label{eq:poles}
\end{equation}
(there is a superficial singularity at $n=0$).
Furthermore, the function is non-differentiable when at the point where the radicand is zero: this is the real value $x$ for which
\[
\left[\coth x+\coth (xh)\right]x-4\coth (x)\coth (xh)=0,
\]
with $x=x_0\approx 2.05$ in the right half-plane $\Re(k)>0$ (RHP).
Juniper takes a branch cut in the RHP along the line $[x_0,\infty)$, such that the square root in Equation~\eqref{eq:disp_explicit} is understood in the usual way as having a branch cut that extends from the point of non-differentiability to $+\infty$.
However, further points along the imaginary axis exist where the radicand vanishes: these produce branch cuts that co-exist with the poles~\eqref{eq:poles} along the imaginary axis.
The dispersion relation (e.g. Equation~\eqref{eq:disp_explicit}) possesses the symmetry
\[
\omega(-k^*)=-\omega^*(k),
\]
meaning that $\omega_{\mathrm{i}}(LHP)=\omega_{\mathrm{i}}(RHP)$, and $\omega_{\mathrm{r}}(LHP)=-\omega_{\mathrm{i}}(RHP)$, where `LHP' means `left half-plane'. However,~\citet{Juniper2006} cuts the dispersion relation in two along the imaginary axis, and takes the positive sign in Equation~\eqref{eq:disp_explicit} in $\Re(k)>0$ and the negative sign in the LHP $\Re(k)<0$. This results in a dispersion relation that varies smoothly as the imaginary axis is traversed. The result of this procedure is Figure~2 in the paper of Juniper.
The aim of this section is to study the extent to which the power-series dispersion relation based along the real axis (referred to herein as the Gaster series) reproduces the dynamically-relevant saddle points of the dispersion relation. The analytic dispersion relation is generated in Matlab according to the procedure outlined above. Dynamically relevant saddle points are identified via the Briggs criterion. In the RHP, for the parameters $(S,\Lambda,h,\Sigma)=(1,1,1.3,1)$, only one such saddle exists, at $k\approx (1.72,-0.682)$. The steepest-descent path is therefore that contour of $\omega_{\mathrm{r}}$ that passes through the saddle, along that path along which $\omega_{\mathrm{i}}$ attains a minimum. This corresponds to $\omega_{\mathrm{r}}\approx 1.50564$. The results of this calculation are shown in Figure~\ref{fig:juniper1}.
\begin{figure}[htb]
\centering
\includegraphics[width=0.8\textwidth]{juniper_fig2_new1}
\caption{Reproduction of Figure~2 in the paper of~\citet{Juniper2006}. The steepest-descent part of the contour $\omega_{\mathrm{r}}= 1.50564$ is shown with the arrows. }
\label{fig:juniper1}
\end{figure}
It is of interest to attempt to apply the analytic extension of Gaster's formula to the dynamically-relevant saddle point at $k\approx (1.72,-0.682)$. This amounts to a Taylor series centred at $(1.72,0)$. However, such a series will be hampered by the presence of the branch point at $(2.05,0)$, and its radius of convergence is thus $R\apprle 0.33$. The disc of convergence $D((1.72,0),R=0.33)$ does not include the dynamically-relevant saddle point (indeed, it fails in a substantial way to include this point), and the series will therefore not converge at the saddle point. We examine further whether a Taylor polynomial based on the extension of Gaster's formula (but which does not converge to the generating function) yields a good approximation to the saddle point (Figure~\ref{fig:juniper2}).
\begin{figure}[htb]
\centering
\includegraphics[width=0.5\textwidth]{imag_approx}
\caption{Taylor polynomials based on the extension of Gaster's formula at the saddle-point location.}
\label{fig:juniper2}
\end{figure}
Clearly, the second truncation is best, while all other truncations fail badly (the odd-numbered polynomials do not yield any improvements with respect to the even polynomials and are not included here). However, even the second truncation fails to reproduce the saddle point in any meaningful way. This is not surprising, given the non-convergence of the Gaster series at the saddle point. However, the result is still important, since it highlights a limitation of our formula.
Finally, we highlight a key difference between this model problem and that considered in the main paper: in the main paper, the Gaster series of the dispersion relation produced Taylor polynomials that gave an excellent approximation to the saddle point, in spite of the fact that the saddle was located beyond the disc of convergence. This was because the saddle was just barely outside the disc of convergence. On the other hand, in this example, the saddle is located at a distance greater than $2R$ away from the series centre, where $R$ is the series' radius of convergence. The presence of a branch point on the real axis hinders an application of our formula, or even the truncated version thereof.
\section{Eady model (Brevdo)}
The dispersion relation for a simplified model of baroclinic instability (Eady model) is studied (see~\citet{Brevdo1988}). The exact dispersion relation corresponding to the discrete spectrum is known (e.g. Equation~(4.18) in the paper of~\citet{Brevdo1988}):
\begin{equation}
\omega(k)=\tfrac{1}{2}k\pm \sqrt{\frac{k^2}{\mu^2}\left(\tfrac{1}{2}\mu-\coth \tfrac{1}{2}\mu\right)\left(\tfrac{1}{2}\mu-\tanh\tfrac{1}{2}\mu\right)},\qquad \mu^2=(k^2+\ell_m^2)S
\label{eq:eady1}
\end{equation}
where $(S,\ell_m)$ are parameters taken to be $(0.25,\pi/2)$.
We prove the following result:
{\proposition{The function
\begin{equation}
\Phi(k):=\left(\tfrac{1}{2}\mu-\coth\tfrac{1}{2}\mu\right)\left(\tfrac{1}{2}\mu-\tanh\tfrac{1}{2}\mu\right),\qquad \mu=\sqrt{(k^2+\ell_m^2)S}
\label{eq:phidef}
\end{equation}
contains only isolated singularities. Moreover, $\mu=0$ is not a singular point.}}
\vspace{0.1in}
\noindent {\textbf{Proof}} We re-write $\coth(z)=z^{-1}+zf_C(z)$, where $f_C(z)$ is a function of $z^2$ with a Taylor series around $z=0$; in particular,
\begin{subequations}
\begin{equation}
f_C(z)=\tfrac{1}{3}-\tfrac{1}{45}z^2+\tfrac{2}{945}z^4+\cdots.
\end{equation}
Similarly, write $\tanh (z)=z+z f_T(z)$, where $f_T(z)$ is also a function of $z^2$ with a Taylor series around $z=0$; in particular,
\begin{equation}
f_T(z)=-\tfrac{1}{3}z^2+\tfrac{2}{15}z^4+\cdots.
\end{equation}%
\label{eq:taylorft}%
\end{subequations}%
Then, taking $z=\mu/2$, the function $\Phi(k)$ can be written as $\Phi(k)=f_T(z)(1-z^2)+z^2F_T(z)F_C(z)$, where $\mu^2=(k^2+\ell_m^2)S$. Thus, $k$ appears in $\Phi(k)$ only in the form $\mu^2$. Moreover, the form of Taylor series~\eqref{eq:taylorft} demonstrates that $\Phi(k)\sim -\mu^2/3$ as $\mu\rightarrow 0$. \qed
\vspace{0.1in}
\noindent We now list the analytic properties of Equation~\eqref{eq:eady1}:
\begin{enumerate}
\item Since $\Phi(\mu)\sim -\mu^2/3$ as $\mu\rightarrow 0$, the dispersion relation $\omega(k)=(k/2)\pm \sqrt{k^2\Phi(\mu)/\mu^2}$ admits no pole at $\mu^2=0$.
\item The only possible poles in the dispersion relation therefore occur when $\coth(\mu/2)=\infty$, or when $\tanh(\mu/2)=\infty$. This happens when $\mu/2\mathrm{i}$ is half-integral (the $\tanh$-function), or full-integral (the $\coth$-function), hence
\[
\mu/2=\mathrm{i} (\pi/2,3\pi/2,5\pi,\cdots),\text{ or }\mu/2=\mathrm{i} (\pi,2\pi,3\pi,\cdots).
\]
In other words, the singularities are located at
\begin{equation}
k=\mathrm{i} \sqrt{\frac{4n^2\pi^2}{S}+\ell_m^2},\qquad k=\mathrm{i} \sqrt{\frac{4\pi^2}{S}\left(n+\tfrac{1}{2}\right)^2+\ell_m^2}.
\label{eq:eady_sing}
\end{equation}
(The value $n=0$ is not a singularity, since point (1) confirms that $\mu=0$ is a regular point.)
\item There are branch points along the real axis at $k=x_0\approx \pm 4.543$, corresponding to real $k$-values where $\Phi(\mu)=0$. Further branch points exist along the imaginary axis corresponding also zeros of $\Phi(\mu)$.
\end{enumerate}
Using this information, we construct a dispersion-relation manifold as in the paper of~\citet{Juniper2006} by placing branch cuts along the real axis at $[x_0,\infty)$ and $(-\infty,-x_0]$. Further branch cuts along the imaginary axis are placed in such a way as to connect the singularities in Equation~\eqref{eq:eady_sing}. Finally, again as in the paper of~\citet{Juniper2006}, the manifold is rendered smooth across the axis $\Re(k)=0$ by joining together two sheets corresponding to the $\pm$ sign in Equation~\eqref{eq:eady1} (obviously, the smoothness does not extend to the singularities or branch cuts along the imaginary axis).
This procedure amounts to picking the following solution branch of Equation~\eqref{eq:eady1}:
\begin{equation}
\omega(k)=\tfrac{1}{2}k+ \,\mathrm{sign}[\Re(k)]k \sqrt{\frac{1}{\mu^2}\left(\tfrac{1}{2}\mu-\coth \tfrac{1}{2}\mu\right)\left(\tfrac{1}{2}\mu-\tanh\tfrac{1}{2}\mu\right)},
\label{eq:eady1}
\end{equation}
where the $\sqrt{\cdot}$ function is understood in the usual sense, as having a branch cut that extends from the point of non-differentiability to $+\infty$.
The results of this analysis are plotted in the dispersion relation in Figure~\ref{fig:brevdo1}.
\begin{figure}[htb]
\centering
\includegraphics[width=0.8\textwidth]{brevdo1}
\caption{Complex dispersion relation for the Eady model, $\omega_{\mathrm{i}}(k_\mathrm{r},k_\mathrm{i})$ }
\label{fig:brevdo1}
\end{figure}
There is a family of saddles that arise from the singularities of the dispersion relation along the imaginary axis. As demonstrated by~\cite{Brevdo1988}, none of these saddles satisfies the Briggs criterion, and the flow is therefore only convectively unstable. This is closely analogous to the `confinement saddles' in Section~\ref{sec:juniper}. A further analogy with Section~\ref{sec:juniper} is the branch point on the real axis which hinders the development of a Gaster series to connect with the saddle point at complex $k$. Given our experience in Section~\ref{sec:juniper}, where a branch point on the real axis hindered the development of such a series, we do not attempt to compute such a series here.
\section{Falling films (Brevdo et al.)}
\label{sec:dias}
In the work by~\citet{Brevdo1999}, the stability of a viscous liquid film on an inclined plane is studied with respect to perturbations to the film free surface. The problem contains three independent parameters: the inclination angle $\theta$, the Reynolds number $Re$, and the Weber number $W$. For a wide class of physically-relevant parameters, the system is found only to be convectively unstable. To demonstrate these stability properties, the authors studied a modified dispersion relation
\[
\omega_V(k)=\omega(k)-kV,
\]
where $\omega(k)$ is the usual dispersion relation got by solving the Orr--Sommerfeld equation with the relevant free-surface boundary conditions, and $V$ is a free parameter. Physically, the parameter $V$ is a velocity measured with respect to the laboratory frame: the quantity $\omega_V(k)$ can therefore be thought of as the wave frequency observed in a moving frame of reference. In the presence of a single dominant saddle, the asymptotic growth rate of the pulse is
\[
\sigma(V)=\Im\left[\omega_V(k_*)\right],\qquad \frac{d\omega_V}{dk}\bigg|_{k_*}=0,
\]
provided the saddle point pinches according to the Briggs criterion.
\citet{Brevdo1999} studied the function $\sigma(V)$ for a wide class of parameters $(\theta,Re,W)$ and found $\sigma(0)<0$ always, indicating convective instability. Moreover, they found that the function $\sigma(V)$ has two branches, due to the presence of two equally-dominant saddle points in the complex function $\omega_V(k)$.
A previous approach to computing $\sigma(V)$ was based on the assumption that the modified dispersion relation $\omega_V(k)$ should contain only one dominant saddle~\citep{Monkey1988,Deissler1987}; the findings in the paper of~\citet{Brevdo1999} therefore demonstrate that this approach is limited to situations where only one saddle point is of interest.
It is of interest to know whether a Gaster series can reproduce the anomalous behaviour described in the work of~\citet{Brevdo1999}. For that reason, we implemented an Orr--Sommerfeld analysis of the falling-film equations for parameter values $(\theta,Re,W)=(4.6^{\mathrm{o}},200,14.18)$ (as in Figure~5 in the work of~\citet{Brevdo1999}), and plotted the resulting complex dispersion relation (Figure~\ref{fig:disp_dias} herein).
\begin{figure}[htb]
\centering
\includegraphics[width=0.8\textwidth]{true_landscape_fig5_brevdo}
\caption{The modified dispersion relation $\omega_V(k)$ at $V=1.16$, corresponding to Figure~5 in the paper of~\citet{Brevdo1999}. The bold lines show the collision of the different spatial branches that occur at the two saddles.}
\label{fig:disp_dias}
\end{figure}
The saddles are estimated as in Table~\ref{tab:disp_dias} (a more refined estimate, in agreement with the work of~\cite{Brevdo1999} can be obtained by increasing the resolution of the numerical scan through complex $k$-space in the Orr--Sommerfeld solver). Because the real dispersion relation $\omega(k_\mathrm{r})$ has a cusp at $k_\mathrm{r}=0$, the convergence of the Gaster series is restricted to a disc centred on the point of interest $(k_\mathrm{r},0)$, of radius $k_\mathrm{r}$. Thus, the two saddle points in Table~\ref{tab:disp_dias} lie just barely outside the relevant discs of convergence. However, because we are just barely outside the region of convergence, as in the main paper, a truncated series will give a good approximation of the true saddle.
\begin{table}[htb]
\centering
\begin{tabular}{|p{0.6cm}|p{2.5cm}|p{2.5cm}|p{3cm}|p{3.5cm}|}
\hline
& Numerics, Main Saddle & Series, Main Saddle & Numerics, Secondary Saddle & Series, Secondary Saddle \\
\hline
\hline
$\omega_{\mathrm{i}} $ & 0.0086 & 0.0080 & 0.0078 & 0.0080 \\
$k_{\mathrm{r}} $ & 0.20 & 0.16 & 0.045 & 0.055 \\
$k_{\mathrm{i}}$ & -0.18 & -0.21 & -0.048 & -0.040 \\
\hline
\end{tabular}
\caption{Saddle point at $V=1.16$ computed directly via a solution of the Orr--Sommerfeld equation, and by a Gaster series.}
\label{tab:disp_dias}
\end{table}
This is demonstrated in Tab.~\ref{tab:disp_dias} and in Figure~\ref{fig:approx_landscape}. In these cases, the Gaster series is truncated at cubic order, and the coefficients of the series expansion are computed from the real dispersion relation using centred differences.
\begin{figure}[htb]
\centering
{\includegraphics[width=0.45\textwidth]{approx_landscape_big_saddle}}
{\includegraphics[width=0.45\textwidth]{approx_landscape_small_saddle}}
\caption{Cubic approximation to the saddle points.}
\label{fig:approx_landscape}
\end{figure}
Thus, the Gaster series captures -- albeit in a slightly crude way -- the anomalous existence of two `branches' of the growth-rate curve $\sigma(V)$ described in the work of~\citet{Brevdo1999}.
We anticipate that the entire curve $\sigma(V)$ can be reconstructed -- to a reasonable level of approximation, using truncations of the Gaster series that are `just barely non-converging'. However, we do not pursue this matter here. Instead, we focus finally on an aspect of the cubic truncation of the Gaster dispersion relation that highlights its usefulness as an explanatory tool. In this truncation, the Gaster series described in the main paper reduces to
\begin{equation}
\omega_{\mathrm{i}}(k_{\mathrm{r}},k_{\mathrm{i}})=\omega_{\mathrm{i}}^{\mathrm{temp}}(k_{\mathrm{r}})+c_{\mathrm{g}}(k_{\mathrm{r}})k_{\mathrm{i}}-\tfrac{1}{2}\frac{d^2\omega_{\mathrm{i}}^{\mathrm{temp}}}{dk_{\mathrm{r}}^2}k_{\mathrm{i}}^2-\tfrac{1}{6}\frac{d^2c_{\mathrm{g}}}{dk_{\mathrm{r}}^2}k_{\mathrm{i}}^3.
\label{eq:omi_approx}
\end{equation}
The necessary saddle-point conditions for instability in a moving frame of reference are $\partial\omega_{\mathrm{i}}/\partial k_{\mathrm{i}}=V$ and $\partial\omega_{\mathrm{i}}/\partial k_{\mathrm{r}}=0$; under the approximation~\eqref{eq:omi_approx} these conditions become
\begin{equation}
\frac{d\omega_{\mathrm{i}}^{\mathrm{temp}}}{dk_{\mathrm{r}}}+\frac{dc_{\mathrm{g}}}{dk_{\mathrm{r}}}k_{\mathrm{i}}-\tfrac{1}{2}\frac{d^3\omega_{\mathrm{i}}^{\mathrm{temp}}}{dk_{\mathrm{r}}^3}k_{\mathrm{i}}^2-\tfrac{1}{6}\frac{d^3\omega_{\mathrm{i}}^{\mathrm{temp}}}{dk_{\mathrm{r}}^3}k_{\mathrm{i}}^3=0,
\qquad
c_{\mathrm{g}}(k_{\mathrm{r}})-\frac{d^2\omega_{\mathrm{i}}^{\mathrm{temp}}}{dk_{\mathrm{r}}^2}k_{\mathrm{i}}-\tfrac{1}{2}\frac{d^2c_{\mathrm{g}}}{dk_{\mathrm{r}}^2}k_{\mathrm{i}}^2=V.
\label{eq:ai_linear}
\end{equation}
Taking the second condition, we get
\begin{equation}
k_{\mathrm{i}}=\left[-\frac{d^2\omega_{\mathrm{i}}^{\mathrm{temp}}}{dk_{\mathrm{r}}^2}\pm\sqrt{\left(\frac{d^2\omega_{\mathrm{i}}^{\mathrm{temp}}}{dkr^2}\right)^2-2\frac{d^2c_{\mathrm{g}}}{dk_{\mathrm{r}}^2}(V-c_{\mathrm{g}})}\right]\bigg\slash \frac{d^2c_{\mathrm{g}}}{dk_{\mathrm{r}}^2}:=K_{\pm}(V,k_{\mathrm{r}}),
\label{eq:ai_linear1}
\end{equation}
Substitution into the first condition yields
\begin{equation}
\frac{d\omega_{\mathrm{i}}^{\mathrm{temp}}}{dk_{\mathrm{r}}}+\frac{dc_{\mathrm{g}}}{dk_{\mathrm{r}}}K_{\pm}(V,k_{\mathrm{r}})-\tfrac{1}{2}\frac{d^3\omega_{\mathrm{i}}^{\mathrm{temp}}}{dk_{\mathrm{r}}^3}\left[K_{\pm}(V,k_{\mathrm{r}})\right]^2-\tfrac{1}{6}\frac{d^3\omega_{\mathrm{i}}^{\mathrm{temp}}}{dk_{\mathrm{r}}^3}\left[K_{\pm}(V,k_{\mathrm{r}})\right]^2=0.
\label{eq:saddle}
\end{equation}
For small $V$-values, Equation~\eqref{eq:ai_linear1} has two real roots (`branches' in the terminology of~\citet{Brevdo1999}). This is precisely the region of parameter space where~\citet{Brevdo1999} observed anomalous behaviour in the $\sigma(V)$-function, with the appearance of two branches of the spatio-temporal growth rate. These two branches are substituted into Equation~\eqref{eq:saddle} and the precise location of the saddle point is obtained, corresponding to the $k_{\mathrm{r}}$-root of Equation~\eqref{eq:saddle}. Of course, the computed saddle points should be checked to confirm if they contribute to the impulse response (Briggs criterion). We would therefore advocate (as in the main paper) that full numerical eigenvalue computations be performed on the one hand, and that the series expansion be considered on the other hand, in order to exploit the complementarity of these two approaches. The series expansion would be used to explain the precise origin of the multiple dominant saddle points. Therefore, in conclusion, this twin-track approach demonstrates that the Gaster series approach is applicable to the anomalous $\sigma(V)$-function, and can also be used to explain the source of this anomaly in a straightforward manner.
\section{Spatio-temporal growth rates from direct numerical simulation}
In this section we consider the impulse-response problem
\begin{multline}
\left[U_0(z)\partial_x+\partial_t\right]\left(\partial_z^2+\partial_x^2\right)\psi(x,z,t)-U_0''(z)\partial_x\psi(x,z,t)=Re^{-1}\left(\partial_z^2+\partial_x^2\right)^2\psi(x,z,t),\\
\psi(x,z,t=0)=\delta(z)\delta(x),
\label{eq:impulse}
\end{multline}
with the base state described in Section~5 of the main paper, viz.
\begin{equation}
U_0(z)=1-\Lambda+2\Lambda\big\{1+\sinh^{2N}\left[z\sinh^{-1}(1)\right]\big\}^{-1},\qquad \Lambda<0,
\label{eq:uzero}
\end{equation}
where $\Lambda$ and $N$ are dimensionless parameters, and $-H<z<H$. Here $H$ is chosen to be sufficiently large such that confinement has no effect on the linear-stability results. The purpose of this section is to verify that such a value of $H$ exists. We do this by solving the impulse-response problem via direct numerical simulation (using the method developed by~\citet{Onaraigh2012a}) and by subsequent consideration of the norm
\[
n(x,t)=\left[\int_{-H}^H\left|\psi(x,z,t)\right|^2\,\mathrm{d} z\right]^{1/2}.
\]
The growth rate
\[
\sigma(V)=\lim_{\stackrel{t_1,t_2\rightarrow\infty} {t_2\gg t_1}}\left[\frac{\log n(Vt_2,t_2)-\log n(Vt_1,t_1)+\tfrac{1}{2}\left(\log t_2-\log t_1\right)}{t_2-t_1}\right]
\]
is then extracted (for an explanation of this procedure, see the works by~\citet{Chomaz1998a,Chomaz1998b}). For a system where the instability is determined by a single dominant mode with a single dominant saddle point, $\sigma(V)$ is a paraboloidal curve whose maximum corresponds to the temporally most-dangerouos growth rate. Also, the sign of $\sigma(0)$ determines the absolute/convective dichotomy, with $\sigma(0)>0$ for absolute instability.
This procedure was applied to Equations~\eqref{eq:impulse}--\eqref{eq:uzero} for parameter values
\[
(Re,\Lambda,N)=(100,-1.1,5), \text{ and }H=8.
\]
The results are shown in Figure~\ref{fig:sigmaV}.
\begin{figure}[htb]
\centering
\includegraphics[width=0.6\textwidth]{sigmaV}
\caption{Spatio-temporal growth rate computed via direct numerical simulation of Equations~\eqref{eq:impulse}--\eqref{eq:uzero}. Parameter values: $(Re,\Lambda,N)=(100,-1.1,5)$, and $H=8$. Solid line: The value of $\omega_{\mathrm{i}}$ at the temporally most-dangerous mode; Broken line: the value of $\omega_{\mathrm{i}}$ at the saddle point of $\omega(k)$, in other words, the absolute growth rate. The values used to draw the lines have been obtained via standard (eigenvalue) stability analysis. The agreement between the two methods confirms the correctness of the whole approach.}
\label{fig:sigmaV}
\end{figure}
Clearly, the growth rate possesses only a single branch. This is in contrast to Section~\ref{sec:dias} of this supplementary report, where $\sigma(V)$ possessed two branches coming from the two contributing saddle points of $\omega_V(k)$. Additionally, the growth rate of the temporally-most dangerous mode coincides with the maximum of $\sigma(V)$. Finally, the value of $\omega_{\mathrm{i}}$ at the saddle point of $\omega(k)$ in the complex $k$-plane coincides with $\sigma(0)$. These findings confirm that $\sigma(V$) has a single branch and that this branch corresponds to the most-dangerous mode and its associated saddle in the complex $k$-plane. In other words, the confinement saddles have no influence on $\sigma(V)$ for $H$ sufficiently large.
\section{Radius of convergence and optimal truncation}
\label{sec:roc}
In this section we demonstrate that the usual criterion for the convergence of a complex power series applies to the central result of the work, namely the Gaster series
\begin{equation}
\omega_{\mathrm{i}}(\alpha_{\mathrm{r}},\alpha_{\mathrm{i}})=\omega_{\mathrm{i}}^{\mathrm{temp}}(\alpha_{\mathrm{r}})+\sum_{n=0}^\infty \frac{(-1)^n}{(2n+1)!}\frac{d^{2n}c_{\mathrm{g}}}{d\alpha_{\mathrm{r}}^{2n}}\alpha_{\mathrm{i}}^{2n+1}
+\sum_{n=0}^\infty \frac{(-1)^{n+1}}{(2n+2)!}\frac{d^{2n+2}\omega_{\mathrm{i}}^{\mathrm{temp}}}{d\alpha_{\mathrm{r}}^{2n+2}}\alpha_{\mathrm{i}}^{2n+2}.
\label{eq:omi_taylor}
\end{equation}
To maintain the connection with the main part of the paper, we return the the notation used therein, and denote wavenumbers by the symbol $\alpha$.
It is tempting to conclude that the convergence properties of the result~\eqref{eq:omi_taylor} do not depend on the global topography of the function $\omega(\alpha)$, and that a local analysis (similar to that used in Taylor's theorem in real analysis) applies. We demonstrate here that this conclusion is incorrect. We also discuss the notion of an optimal truncation for a divergent power series, such that a finite truncation of a divergent power series can be used to approximate the generating function of the power series. This result explains the `over prediction' obtained in Section~5 of the main paper.
The starting-point for the convergence analysis is Taylor's theorem applied to the \textit{purely real} Taylor expansion~\eqref{eq:omi_taylor}.
We introduce some notation (see also Figure~\ref{fig:sketch}):
\begin{enumerate}
\item The point $O=(\alpha_{\mathrm{r}0},0)$: The centre of the power series based on the real axis.
\item $(\alpha_{\mathrm{r}0},\alpha_{\mathrm{i}0})$: The point of interest at which the power series is to be evaluated. For our purposes, $\alpha_{\mathrm{i}0}$ will be negative.
\item The point $C=(\alpha_{\mathrm{r}0},\alpha_{\mathrm{i}C})$: A point used to compute the remainder in Taylor's theorem; here $\alpha_{\mathrm{i}C}\in[0,\alpha_{\mathrm{i}0}]$.
\item The point $X=\alpha_{\mathrm{x}}$: The singularity closest to $C$.
\end{enumerate}
\begin{figure}[htb]
\centering
\includegraphics[width=0.6\textwidth]{sketch}
\caption{Figure showing the centre of the power series, the singularity, and the point of interest at which the series is to be evaluated.}
\label{fig:sketch}
\end{figure}
We shall assume that the derivatives $\left(\partial^{k}\omega_{\mathrm{i}}/\partial\alpha_{\mathrm{i}}^k\right)_{\alpha_{\mathrm{r}0}}$ exist for all orders $k\in\{0,1,2,\cdots\}$, on the interval $(0,\alpha_{\mathrm{i}0})$ (the brackets $\left(\cdot,\cdot\right)$ here denote an open interval, and not a point in the complex plane).
Taylor's theorem guarantees that in approximating $\omega_{\mathrm{i}}(\alpha_{\mathrm{r}},\alpha_{\mathrm{i}})$ by a finite truncation based on Equation~\eqref{eq:omi_taylor}, we incur an error
\begin{equation}
R_n=\frac{1}{(n+1)!}\frac{\partial^{n+1}\omega_{\mathrm{i}}}{\partial\alpha_{\mathrm{i}}^{n+1}}\bigg|_{\alpha_{\mathrm{i} C}}\alpha_{\mathrm{i}0}^{n+1},\qquad \alpha_{\mathrm{i}C}\in\left[0,\alpha_{\mathrm{i}0}\right].
\label{eq:rem}
\end{equation}
Using Cauchy's theorem, we have
\begin{equation}
\frac{1}{\mathrm{i}^{n+1}}\frac{\partial^{n+1}\omega}{\partial\alpha_{\mathrm{i}}^{n+1}}\bigg|_{(\alpha_{\mathrm{r}0},\alpha_{\mathrm{i}C})}=\frac{d^{n+1}\omega}{d\alpha^{n+1}}\bigg|_{(\alpha_{\mathrm{r}0},\alpha_{\mathrm{i}C})}=\frac{(n+1)!}{2\pi\mathrm{i}}\oint_\gamma\frac{\omega(\alpha)}{\left[\alpha-(\alpha_{\mathrm{r}0}+\mathrm{i}\alpha_{\mathrm{i}C})\right]^{n+2}}\mathrm{d}\alpha,
\label{eq:cauchy0}
\end{equation}
where $\gamma$ is a circle centred at $(\alpha_{\mathrm{r}0},\alpha_{\mathrm{i}C})$, of radius $R$. Here we have switched from a partial derivative to a standard complex-valued derivative, because the function $\omega(\alpha_{\mathrm{r}},\alpha_{\mathrm{i}})$ is analytic away from singularities, and the limits calculated in respect of the derivative are independent of approach.
The radius $R$ is yet to be determined.
Also,
\begin{equation}
\left|\frac{\partial^{n+1}\omega_{\mathrm{i}}}{\partial\alpha_{\mathrm{i}}^{n+1}}\bigg|_{(\alpha_{\mathrm{r}0},\alpha_{\mathrm{i}C})}\right|\leq
\left|\frac{\partial^{n+1}\omega}{\partial\alpha_{\mathrm{i}}^{n+1}}\bigg|_{(\alpha_{\mathrm{r}0},\alpha_{\mathrm{i}C})}\right|\leq
\frac{(n+1)!}{R}\left(\max_\gamma |\omega(\alpha)|\right).
\label{eq:cauchy1}
\end{equation}
Combining Equations~\eqref{eq:rem} and~\eqref{eq:cauchy1}, we have
\[
|R_n|\leq \left(\max_\gamma |\omega|\right)\left|\frac{\alpha_{\mathrm{i}0}}{R}\right|^{n+1}.
\]
To make $|R_n|\rightarrow 0$ as $n\rightarrow\infty$, we would like to take $R>|\alpha_{\mathrm{i}0}|$. However, we cannot take $R$ as arbitrary, we must have $R<|CX|$, where $|CX|$ is the distance between $C$ and the closest singularity thereto. Because we have no \textit{a priori} knowledge of the location of $C$, we must take account of the worst-case scenario, where $C=O$. Then, for the series to converge, we must have
\begin{equation}
R<|OX|,
\label{eq:app:roc}
\end{equation}
that is, in order for the power series centred at $O$ to converge, the point of interest $(\alpha_{\mathrm{r}0},\alpha_{\mathrm{i}0})$ must be inside a disc of radius $R$, where $R$ is the distance between the power-series centre and the nearest singularity. In other words, the standard criterion for complex power series applies: the power series is valid provided the point of interest lies within the disc of convergence.
Equation~\eqref{eq:app:roc} is a sufficient condition for the Taylor series for $\omega_{\mathrm{i}}(\alpha_{\mathrm{r}},\alpha_{\mathrm{i}})$ to be valid. If we extend this Taylor series using the Cauchy--Riemann conditions and construct a Taylor series in $\alpha_{\mathrm{i}}$ for $\omega_{\mathrm{r}}(\alpha_{\mathrm{r}},\alpha_{\mathrm{i}})$, the standard results of complex-variable theory apply, and Equation~\eqref{eq:app:roc} is both sufficient \textit{and} necessary in order for the Taylor-series pair for $(\omega_{\mathrm{r}},\omega_{\mathrm{i}})$ to converge.
On the other hand, it is possible for the Taylor series of $\omega_{\mathrm{i}}$ to converge beyond the radius of convergence, while at the same time the Taylor series for $\omega_{\mathrm{r}}$ diverges.
However, this rather pathological scenario is unlikely to occur in the applications contained in this paper: for example, Figure~3 in the main paper shows that the finite truncations to the power series for both $\omega_{\mathrm{i}}$ \textit{and} $\omega_{\mathrm{r}}$ are good approximations to the underlying dispersion relation outside of the radius of convergence of the power series. For a more complete description of this pathology, see Section~\ref{sec:roc:path} below.
Our next discussion concerns the notion of \textit{optimal truncation} for divergent power series.
Consider a real-valued function $f(x)$ that admits a power-series expansion, centred at the origin:
\begin{equation}
f_P(x)=\sum_{n=0}^\infty c_n x^n,\qquad c_n=\frac{f^{(n)}(0)}{n!}.
\label{eq:app:examplef}
\end{equation}
Assume moreover that the power series $f_P(x)$ converges in a finite interval $|x|<R$, and that $f_P(x)=f(x)$ on the same interval. If $f_P(x)$ remains finite at $x=R$, then it is possible to use a finite truncation of Equation~\eqref{eq:app:examplef} to approximate $f(x)$ in the region $|x|>R$. In other words, for a given $x$ such that $|x|>R$, there exists a positive integer $K$ such that the difference
\[
\delta_{K}(x):=\left(f(x)-\sum_{n=0}^{K} c_n x^n\right)^2
\]
is minimized.
As an example of this sort, consider the function $f(x)=(1+x)\log (1+x)-x$. On the interval $|x|<1$, this function can be represented as a power series:
\begin{equation}
f(x)=\sum_{n=1}^\infty (-1)^{n+1} \frac{x^{n+1}}{n(n+1)}.
\label{eq:app:example_log}
\end{equation}
It is straightforward to show that the power series evaluated at $x=1$ converges (the limit comparison test applies). Thus, we extend finite truncations of the power series beyond $x=1$ to approximate the underlying function $f(x)$. In doing so, we wish to minimize the difference $\delta_{K}(x)$.
Using elementary facts about geometric progressions, this difference can be re-written as
\[
\delta_{K}(x):=\left(\int_0^x\mathrm{d} x'\int_0^{x'}\mathrm{d} x''\frac{1-(-x'')^{K}}{1+x''}-f(x)\right)^2.
\]
The minimum $K$-value satisfies
\begin{equation}
\frac{\partial}{\partialK}\delta_K(x)=0,\qquad \int_0^x\mathrm{d} x'\int_0^{x'}\mathrm{d} x''\frac{(x'')^{K}\log(x'')}{1+x''}=0.
\label{eq:app:exampleN}
\end{equation}
For $x>1$, Equation~\eqref{eq:app:exampleN} has a solution with positive $K$. We have plotted the optimal value of $K$ in Figure~\ref{fig:app:ot0}(a).
\begin{figure}[htb]
\centering
\subfigure[]{\includegraphics[width=0.49\textwidth]{n0}}
\subfigure[]{\includegraphics[width=0.49\textwidth]{otf}}
\subfigure[]{\includegraphics[width=0.49\textwidth]{t20}}
\caption{Analysis of finite truncations of the divergent power series~\eqref{eq:app:example_log}. (a) The optimal truncation value $K$; (b) The difference between the true function $f(x)=(1+x)\log(1+x)-x$ and the optimal truncation; (c) the relative error in making a uniform truncation.}
\label{fig:app:ot0}
\end{figure}
The difference between the optimal truncation of the divergent power series and $f(x)$ is shown in Figure~\ref{fig:app:ot0}(b). There is little difference between the optimal truncation and a uniform truncation with $K=20$. The relative error between the uniform truncation and the true function $f(x)$ is shown in Figure~\ref{fig:app:ot0}(c): this never exceeds $2\%$ on the interval $x\in(1,1.1]$.
Finally, in Figure~\ref{fig:app:ot} we examine the coefficients of the Taylor series for the model dispersion relation developed in the main paper in Section~5. We consider the case $Re=100$, $N=5$, and $\Lambda=-1.1$. Panel~(a) refers to $\alpha_{\mathrm{r}}=2.1978$, and the singularities on the imaginary axis lie far from the point of interest $(\alpha_{\mathrm{r}}^*,0)$ (here $\alpha_{\mathrm{r}}^*$ refers to the real coordinate of the saddle point). The first eight coefficients in the power series exhibit exponential decay. On the other hand, in panel~(b) (for which $\alpha_{\mathrm{r}}=0.7739$), the coefficients decay algebraically. This case corresponds to a $\alpha_{\mathrm{r}}\approx \alpha_{\mathrm{r}}^*$ (i.e. close to the saddle), for which the radius of convergence is $R=0.80$; in the main part of the paper we extended finite truncations of the series beyond this point, i.e. $|\alpha_{\mathrm{i}}|>R$. That the first eight coefficients of the power series decay algebraically suggests that the continuation described herein is appropriate.
Thus, it is possible (albeit with great caution) to use a finite truncation of a divergent power series to approximate the generating function of the power series.
\begin{figure}[htb]
\centering
\subfigure[]{\includegraphics[width=0.48\textwidth]{coefficients_N5_ar21978_new}}
\subfigure[]{\includegraphics[width=0.48\textwidth]{coefficients_N5_ar7729_new}}
\caption{Coefficients of the Taylor series for~\eqref{eq:omi_taylor} ($A_n$) and for the analogous series in $\omega_{\mathrm{r}}$ ($B_n$), for the model discussed in Section~5 of the main paper. Here, $Re=100$, and $N=5$. Case (a) corresponds to $\alpha_{\mathrm{r}}=2.1978$; case (b) corresponds to $\alpha_{\mathrm{r}}=0.7739$ (i.e. a point close to the saddle, for which the radius of convergence is small ($R=0.80$)). }
\label{fig:app:ot}
\end{figure}
\subsection{The series for the imaginary part of a function over-converges but the series for the real part does not}
\label{sec:roc:path}
The `pathological' example referred in the main part of Section~\ref{sec:roc} is the following:
\[
f(z)=\frac{1}{1-\mathrm{i} z}=u(x,y)+\mathrm{i} v(x,y),
\]
with a simple pole at $z=-\mathrm{i}$. The series representation
\[
f(z)=\frac{1}{1-\mathrm{i} z},\qquad f_T(z)=\sum_{n=0}^\infty (\mathrm{i} z)^n,\qquad f(z)=f_T(z)
\]
therefore converges on the interior of a disc of radius $R=1$ centred at $z=0$. However, taking $z=\mathrm{i} y$, we get
\[
v(0,y)=\Im\left(\frac{1}{1+y}\right)=0,\qquad
\Im\left(f_T(z)\right)=\Im\left[ \sum_{n=0}^\infty \left(-y\right)^n\right]=0.
\]
Thus, the series for $v(0,y)$ is a trivial series whose coefficients are all zero, and which converges beyond the radius of convergence for all $y\neq -1$.
The series $\Im(f_T(z))$ therefore agrees trivially with the function $v(0,y)$ for all $y\neq -1$.
However, at the same time,
\[
u(0,y)=\frac{1}{1+y},\qquad \Re(f_T(z))=\sum_{n=0}^\infty (-y)^n,
\]
and $u(0,y)=\Re(f_T(z))$ only for $|y|<R$. Thus, we have an example where the imaginary part of $f(z)$ has a Taylor series that converges beyond the radius of convergence, on the line $\Re(z)=0$ with the point $\Im(z)=-1$ removed, i.e. the set $\{\Re(z)=0,\Im(z)\neq -1\}$. However, the real part of $f(z)$ has a Taylor series that converges only inside the radius of convergence, along the same line. Clearly, this is a very contrived example and for that reason, it was discussed in rather a dismissive fashion in the paper.
\subsection*{Acknowledgements}
L. \'O~N. would like to thank his R. Smith for devising the example in Section~\ref{sec:roc:path} of this Supplementary Report.
\bibliographystyle{plainnat}
|
{
"timestamp": "2013-03-13T01:01:31",
"yymm": "1203",
"arxiv_id": "1203.1797",
"language": "en",
"url": "https://arxiv.org/abs/1203.1797"
}
|
\section{Introduction}
One of the fundamental tenets of the Big-Bang paradigm
is the adiabatic evolution of the Universe. Early thermal equilibrium
among the different particle species, entropy and photon number conservation
produce a Cosmic Microwave Background (CMB) with a blackbody spectrum.
The CMB temperature was measured to be $T_0=2.725\pm 0.002$K
by the Far Infrared Absolute Spectrometer (FIRAS) of the Cosmic Background
Explorer (COBE) satellite (Mather et al 1999). The adiabatic expansion
of the Universe and photon number conservation imply that the CMB temperature
evolves with redshift as $T(z)=T_0(1+z)$. Establishing observationally this
relation would test our current understanding of the Universe since
models like decaying vacuum energy density and gravitational `adiabatic' photon
creation predict different scaling relations (Overduin \&
Cooperstock, 1998; Matyjasek 1995; Lima et al 2000; Puy 2004;
Jetzer et al 2011, Jetzer \& Tortora 2012). In these models, energy is slowly
injected without producing distortions on the blackbody spectrum,
evading the tight FIRAS constraints. Nevertheless, in these models
the blackbody temperature scales nonlinearly as $T(z)=T_0(1+z)^{1-\alpha}$.
Therefore, measuring the redshift dependence of the CMB black body temperature at
various cosmological epochs can provide strong constraints on physical
theories at the fundamental level.
There are currently two methods to determine $T(z)$ at redshifts $z>0$:
(a) using fine structure lines from interstellar atoms or molecules,
present in quasar spectra, whose transition energies are
excited by the CMB photon bath (Bahcall \& Wolfe, 1968) and (b)
from the thermal Sunyaev-Zeldovich anisotropies (hereafter
TSZ, Sunyaev \& Zel'dovich, 1972, 1980) due to the inverse Compton scattering
of photons by the free electrons within the potential wells of
clusters of galaxies.
Early observations of fine-structure levels of atomic
species like carbon only led to upper limits on $T(z)$ because
the CMB is not the only radiation field populating the energy
levels and collisional excitation is an important contribution.
Assuming the CMB is the only source of excitation,
Songaila et al. (1994) measured $T(z=1.776)=7.4\pm 0.8$K; but
collisional excitation was not negligible and it had to be corrected.
The first unambiguous measurement was only achieved six years later, with
a considerably larger error bar (Srianand et al. 2000).
Lately, Noterdaeme et al (2010)
succeeded in obtaining a direct and precise measurements from the rotational
excitation of CO molecules. They constrained the deviation
from linear scaling to be $\alpha=-0.007\pm 0.027$ at $z\sim 3$.
Battistelli et al. (2002) reported the first observations of $T(z)$
using the TSZ effect of the COMA and A2163 clusters of galaxies
with $\alpha=-0.16^{+0.34}_{-0.32}$.
Luzzi et al. (2009) determined the CMB temperature in the redshift
range $z=0.023-0.546$, from the measurements of 13 clusters.
They restricted their analysis to $\alpha\in[0,1]$ and set up an
upper limit of $\alpha\le 0.079$ at the 68\% confidence level.
No significant deviations from the redshift dependence of the CMB temperature
predicted in the standard model has been found.
While there is interest in doing such observations as
far back as possible (which one can do with spectroscopic methods),
low-redshift measurements play an important role too. First,
the two techniques are complementary with each other since they have different
systematics and probe the adiabatic evolution of the Universe at different redshifts.
Spectroscopic observations probe the matter era, roughly between redshifts
$z=2-4$, while TSZ probes the epoch of dark energy domination,
$z\le 1$. In particular, these measurements can shed light on the
onset of dark energy domination; in many models this is associated with
as a phase transition (Mortonson et al. 2009, Nunes et al. 2009)
which could leave imprints in the $T(z)$ relation. Second,
in models where photon number is not conserved, not only does
the temperature-redshift relation change,
but so does the distance duality relation (Etherington 1933), and these
departures from the standard behavior are not independent. This link
between the two relations requires information at all redshifts
and will, when better datasets become available,
be a powerful consistency test for the standard cosmological paradigm
(Avgoustidis et al 2012).
The Planck mission has been designed to produce a full-sky survey
of the CMB with unprecedented accuracy in temperature and polarization
(Planck Collaboration 2011a). The instrument operates at nine frequencies
logarithmically spaced in the range 30-857GHz. The in-flight
performance of the High and Low Frequency Instruments have been
described by the Planck HFI Core Team (2011a) and Mennella et al (2011).
Due to its large frequency coverage, high resolution and low noise,
is an optimal instrument for blind detection of clusters using the
TSZ effect. The first clusters detected by PLANCK
include 189 cluster candidates with signal-to-noise larger than 6
(Planck Collaboration 2011b). These SZ clusters are mostly at moderate redshifts
(86\% had $z<0.3$) and span over a decade in mass, up to the rarest and most
massive clusters with masses above $10^{15}$M$_\odot$.
In this article we analyze how PLANCK data can be used to test the
standard scaling relation of the CMB temperature with redshift.
We use a two-fold approach: first, our pipeline is tested on simulated clusters
drawn from a full hydrodynamical simulation; second, using a catalog of
$623$ clusters derived from ROSAT data and with well measured X-ray
properties, we predict the accuracy that PLANCK measurements will
reach using those clusters.
In comparison with earlier analysis of Horellou et al (2005), we use a catalog
of X-ray selected clusters and in our simulations, gas evolution is fully
taken into account. Briefly, in Sec 2 we describe our methodology; in Sec 3
we discuss our data and simulations; in Sec 4 we explain our pipeline;
in Sec 5 we present our results and in Sec 6 we summarize our main conclusions.
The final goal of the paper is to forecast the accuracy with which PLANCK
will constraint $\alpha$ for our cluster sample.
\begin{center}
\begin{table}[!h]
\begin{tabular}{|l|rrrrrr|}
\hline
Planck Channel & 1 & 2 & 3 & 4 & 5 & 6 \\
\hline
Central frequency $\nu_0$/GHz & 44 &70 &100 &143 &217 &353\\
Frequency resolution $\Delta\nu$ (FWHM/GHz) & 8.8 & 14 &33 &47 &72 &116 \\
Angular resolution $\Delta\theta$ (FWHM/arcmin) & 26.8 & 13.1 & 9.8& 7.1 & 5.0& 5.0\\
Noise per pixel $\sigma_{noise}/\mu$K (Blue Book) & 51 & 52 & 15 & 12 & 19 & 58\\
Noise per pixel $\sigma_{noise}/\mu$K (in-flight performance)&
109 & 96 & 14 & 9 & 13 & 49 \\
\hline
\end{tabular}
\caption{Technical details of Planck channels used in this study.
The noise per pixel in-flight performance corresponds to 1 year of integration.}
\end{table}
\end{center}
\bigskip
\section{Methodology.}
Compton scattering of CMB photons by the hot Intra-Cluster (IC) gas induces
secondary temperature anisotropies on the CMB radiation in the direction
of clusters of galaxies. There are two components: the thermal (TSZ,
Sunyaev \& Zeldovich 1972) due to the thermal motion of the IC medium with
temperature $T_e$ and the kinematic (KSZ, Sunyaev \& Zeldovich 1980)
due to the motion of the cluster with speed $\vec{v}_{cl}$
respect to the isotropic CMB frame. Neglecting relativistic
corrections, the TSZ and KSZ contributions to the temperature
anisotropy in the direction of a cluster $\hat{n}$ are given by
\begin{equation}
\frac{T(\hat{n})-T_0}{T_0}=\int \left[G(\nu)\frac{k_BT_e}{m_ec^2}+
\frac{\vec{v}_{cl}\hat{n}}{c}\right]d\tau=G(\nu)y_c+\tau\frac{\vec{v}_{cl}\hat{n}}{c}
\label{eq:sz}
\end{equation}
In this expression, $d\tau=\sigma_Tn_edl$ is the cluster optical depth and
$n_e(l)$ the electron density evaluated along the line of sight $l$, $\sigma_T$
is Thomson cross section,
$T_0$ the current CMB mean temperature, $k_B$ the Boltzmann constant, $m_ec^2$ the
electron annihilation temperature, $c$ the speed of light and $\nu$ the frequency
of observation. The Comptonization parameter is defined
as $y_c=(k_B\sigma_T/m_ec^2)\int n_eT_e dl$.
Due to its frequency dependence $G(\nu)$, the TSZ is a distortion of the CMB spectrum.
Its amplitude is independent on the cluster distance, making
it a useful tool to detect clusters at high redshifts. All known astrophysical
foregrounds have a different dependence with frequency so clusters can be
clearly detected in CMB maps with enough frequency coverage.
In the non-relativistic limit, $G(x)= x{\rm coth}(x/2)-4$.
The reduced frequency $x$ is given by $x=h\nu(z)/kT(z)$ with
$\nu(z)$ the frequency of a CMB photon scattered off by the IC gas and
$T(z)$ the black body temperature of the CMB at the cluster location.
If the Universe evolves adiabatically, $T(z)=T_0(1+z)$. Due to the
expansion, the frequency of a photon scattered by the
IC plasma at redshift $z$ is Doppler shifted as: $\nu(z)=\nu_0(1+z)$
and the ratio $x=h\nu(z)/kT(z)=h\nu_0/kT_0=x_0$ is
independent of redshift. If the evolution of the Universe is non-adiabatic,
the temperature-redshift relation would not be constant.
Two functional forms have been considered in the literature:
$T(z)=T_0(1+z)^{1-\alpha}$ (Lima et al 2000) and
$T(z)=T_0(1+bz)$ (LoSecco et al 2001). In both cases, the
photon frequency is assumed to be redshifted as in the standard model:
$\nu(z)=\nu_0(1+z)$. Since the largest fraction of known clusters
of galaxies are at redshifts below $z\le 0.7-1$, the differences
between both redshift dependences are small so we will
only analyze the first model. The reduced frequency varies as
$x=x_0(1+z)^\alpha$ and
the spectral frequency dependence of the TSZ effect, $G(\nu)$ now
depends on $\alpha$: $G(x)=G(\nu,\alpha)$.
Using the TSZ effect, two methods have been proposed to constraint $\alpha$.
Fabbri et al. (1978) proposed to measure
the zero cross frequency of clusters at different redshifts
that, for adiabatic evolution, occurs at $\nu\simeq 217GHz$.
Rephaeli (1980) suggested to use the ratio of the TSZ anisotropy at
different scales, $R(\nu_1,\nu_2,\alpha)= G(\nu_1,\alpha)/G(\nu_2,\alpha)$.
Both methods have different systematics.
By taking ratios, the dependence on the Comptonization parameter
is removed and the need to account for model uncertainties on the gas density
and temperature profile is avoided. At the same time, the analysis is
more complicated since the distribution of temperature anisotropy ratios is
highly non-gaussian (Luzzi et al. 2009). The measurement of the cross over
frequency is also problematic since the TSZ is inherently weak and could
be dominated by uncertain systematics. For this reason, the measurements
carried out thus far (Battistelli et al 2002, Luzzi et al 2009),
based in a small number of clusters, have concentrated
in the ratio method. As an alternative to the zero frequency method,
we will fit the TSZ signal at different frequencies and we will measure
the function $G(\nu,\alpha)$. We shall denote this procedure the {\it Fit Method}.
The function $G(\nu,\alpha)$ characterizes uniquely the TSZ contribution.
At each frequency, PLANCK Low Frequency Instrument (LFI) receivers and
High Frequency Instrument (HFI) bolometers are sensitive to a wide range of
frequencies and the spectral dependence is not $G(\nu)$ but
\begin{equation}
\bar{G}(\nu_0,\alpha)=\int_0^\infty G(\nu,\alpha)e^{(\nu-\nu_0)^2/2\sigma_\nu^2} d\nu
\label{eq:freq}
\end{equation}
Hereafter, we will remove the upper bar and $G(\nu,\alpha)$ will refer
to the averaged frequency dependence of eq~(\ref{eq:freq}).
In Fig~\ref{fig1} we plot the temperature ratio (Fig~\ref{fig1}a)
and frequency dependence (Fig~\ref{fig1}b) for different values of $\alpha$.
In Fig~\ref{fig1}a, the solid line represents the adiabatic evolution model
$\alpha=0$ that is independent of redshift; the dot-dashed lines bound
the region where $\alpha=-1,1$. From top to bottom, the
ratios are $R(\nu,353GHz,\alpha)$ with $\nu=143,100,44$GHz.
In Fig~\ref{fig1}b, we plot the spectral dependence $G(\nu,\alpha)$ for
adiabatic evolution ($\alpha=0$, dashed line) and $\alpha=-1,1$ for a cluster at
redshift $z=0.1$ (dot-dashed line) and $z=0.3$ (solid line).
The null TSZ signal, represented by the dotted line, shows that the
zero cross frequency varies in the range $\nu\sim 170-270$GHz.
To construct a pipeline that implements the ratio or zero cross frequency
tests we need to consider the specifics of the PLANCK data.
The cosmological CMB signal is the dominant
contribution except at the most massive clusters.
Foreground residuals or astrophysical contaminants,
while smaller in amplitude, would induce systematic shifts in
the Comptonization parameter (Aghanim, Hansen \& Lagache, 2005) varying
with frequency and biasing the redshift dependence of the TSZ effect.
To characterize the noise and foreground emission, the HFI and LFI core teams
have constructed maps with the CMB cosmological contribution subtracted off the Time
Ordered Information (TOI). They have used six different component separation
algorithms to remove the primordial CMB signal. The difference
between the six methods provides an estimate the CMB residual
(Planck HFI Core Team 2011b, Zacchei et al 2011).
The resulting maps are dominated by noise and foreground residuals. Due to the
scanning strategy, the noise is rather inhomogeneous, largely dominated by a
white noise component plus a $1/f$ contribution. In the LFI instrument,
the $1/f$ noise is largest at 30GHz. For the HFI, the noise is largest at 545
and 857GHz. Also, those channels have the smallest resolution and we will
not be considered in this work. Later we shall show that this is not
a limitation since the channels with the highest resolution are the ones
with the largest statistical power to determine $\alpha$.
The technical details of the maps considered in this study are listed in Table 1:
central frequency and FWHM of the antenna spectral response function, approximated by
a gaussian with FWHM $\Delta\nu$, angular resolution and noise per pixel.
We indicate the required Blue Book
specifications\footnote{http://www.rssd.esa.int/SA/PLANCK/docs/Bluebook-ESA-SCI(2005)1\_V2.pdf})
and the in-flight measured noise per pixel after one year of integration
(Mennella et al 2011, Planck HFI Core Team 2011a).
With respect to astrophysical contaminants,
WMAP used the K, Ka Differencing Assemblies
and the extinction corrected H$\alpha$
maps (Finkbeiner 2003) to subtract the synchrotron and free-free
emissions and the Finkbeiner et al (1999) map to subtract the dust contribution
(see Gold et al 2009 for details). A similar analysis using
the 30, 545 and 857 GHz channels, and foreground templates could
be used to remove the foreground contribution. Ideally, one would
obtain maps without cosmological signal and with a low level of
foreground residuals.
Maps free of the intrinsic CMB signal and/or foreground residuals
are the most convenient to test the adiabatic evolution of the Universe.
Removing the cosmological CMB signal on the TOI data leaves an unknown level
of CMB residuals, whose distribution and power spectrum are difficult to
model and the final results could be biased in an unmeasurable way.
Alternatively, the intrinsic CMB can be removed by subtracting the highest resolution
map, conveniently degraded, from the other maps. Both techniques have
different systematics and the consistency of the results would be a test
of their validity. Therefore, we shall carry out two type of simulations,
depending on what data sets become available:
(A) CMB subtraction in the TOI plus foreground removal using templates
would produce maps with only instrumental noise, KSZ and TSZ
with some unknown levels of primordial CMB and foreground residuals.
Deviations from adiabatic evolution can be measured by taking ratios
of temperature anisotropies at different frequencies (Fig~\ref{fig1}a)
or by fitting the spectral dependence $G(\alpha,\nu)$ (Fig~\ref{fig1}b)
to the data. (B) The component separation takes place on the final maps.
The primordial CMB is removed using the foreground clean 217GHz map.
We degrade the angular resolution of the 217GHz channel to that
of the other 5 channels before subtracting it from the corresponding map.
We checked that the intrinsic CMB and KSZ anisotropies are
removed exactly but the frequency dependence of the TSZ effect is modified.
In Fig~\ref{fig2}a, we plot the ratio of the CMB-KSZ removed
maps at different frequencies:
$R_{[-217GHz]}(\nu_1,\nu_2,\alpha)=[G(\nu_1,\alpha)-G(217GHz,\alpha)]/
[G(\nu_2,\alpha)-G(217GHz,\alpha)]$. In Fig~\ref{fig2}b we represent
$G_{[-217GHz]}(\nu,\alpha)=G(\nu,\alpha)-G(217GHz,\alpha)$.
The lines follow the same conventions than in Fig.~\ref{fig1}.
Our pipeline will analyze maps using CMB subtraction on the TOI
(method A) and on the maps themselves (method B). We will
compute $\alpha$ with both the ratio and the fit method.
By subtracting the 217GHz, the estimator $R_{[-217GHz]}$
has a much weaker dependence on $\alpha$ than $R$.
With respect to the fit method,
we do not measure $G(\nu,\alpha)$ directly but
$\Delta T(\hat{n})=T_0y_cG(\nu,\alpha)$. To measure
the spectral shape from the data we need an independent
determination of $y_c$ using X-ray data. We will
use a proprietary catalog of X-ray selected clusters
with all the required information (see Sec.~\ref{sec:xray})
and for this catalog we will forecast the constrain to be achieved
with PLANCK.
\bigskip
\bigskip
\section{Cluster templates and final maps.}
To test our systematics,
we will construct two TSZ templates: one based on our catalog
of X-ray selected clusters, the other based on an all-sky hydrodynamical simulation.
The final maps were simulated using the HealPix package (Gorski et al 2005)
with resolution $N_{side}=1024$.
\subsection{Y-map from X-ray selected clusters. \label{sec:xray}}
Our cluster sample contains 623 clusters outside WMAP Kp0 mask.
It was created combining the
ROSAT-ESO Flux Limited X-ray catalog (REFLEX, B\"ohringer et al 2004)
in the southern hemisphere, the extended Brightest Cluster Sample
(eBCS, Ebeling et al 1998, Ebeling et al 2000) in the north, and the
Clusters in the Zone of Avoidance (CIZA, Ebeling, Mullis \& Tully
2002, Kocevski et al 2007) sample along the Galactic plane. All three
surveys are X-ray selected and X-ray flux limited. A detailed description
of the creation of the merged catalog is given in Kocevski \& Ebeling (2006).
The position, flux, X-ray luminosity and angular extent of the region
containing the measured X-ray flux were determined directly from
ROSAT All Sky Survey (RASS).
All clusters have spectroscopically measured redshifts.
The X-ray temperature was derived from the $L_X-T_X$ relation
of White, Jones \& Forman (1997). The central electron densities and core
radii were derived by fitting to the RASS data a spherically symmetric isothermal
$\beta$ model (Cavaliere \& Fusco-Femiano 1976) convolved with the RASS point-spread
function. The $\beta$ was fixed at the canonical value of $2/3$
to reduce the dependence of the $\beta$ model parameters with
the choice of radius over which the model is fit. These data
allows to compute the Comptonization parameter at the center of the cluster.
Atrio-Barandela et al (2008) compared the TSZ predicted from the X-ray data with
the signal present in WMAP 3yr data and found it to be in good agreement
within the X-ray emitting region, where the $\beta$ model is a good
description of the electron distribution. In the cluster outskirts, the
TSZ signal was systematically higher than the measured value. The latter
was consistent with the Komatsu \& Seljak (2002) profile, where baryons
are in hydrostatic equilibrium within a dark matter halo well described
by a Navarro-Frenk-White profile (hereafter NFW, Navarro, Frenk \& White 1997),
as expected in the concordance $\Lambda$CDM model. More recently,
Nagai et al (2007) proposed a scaled 3-dimensional electron pressure
profile $p(x)=P_e(r)/P_{500}$ based on a generalizad NFW profile
\begin{equation}
p(x) = \frac{P_0}{(c_{500}x)^\gamma[1+(c_{500}x)^\alpha]^{(\beta-\gamma)/\alpha}},
\label{eq:universal_profile}
\end{equation}
where $(\gamma,\alpha,\beta)$ are the central, intermediate and outer slopes,
$c_{500}$ characterizes the gas concentration and $x=r/R_{500}$ is the radius at
which the average density of the cluster is 500 times the critical density.
Arnaud et al (2010) derived an average cluster pressure profile from
observations of a sample of 33 local ($z<0.2$) clusters, scaled by mass
and redshift with
\begin{equation}
[P_0,c_{500},\gamma,\alpha,\beta]=[8.403 h_{70}^{-3/2},1.177,0.3081,1.0510,5.4905]
\label{eq:Arn_parameters}
\end{equation}
Later, Plagge et al (2010) showed these parameters to be consistent with
the SZ measurements of 15 massive X-ray clusters observed with the South
Pole Telescope (Plagge et al 2010). We determine the scale $R_{500}$ using
(B\"ohringer et al 2007)
\begin{equation}
R_{500}=\frac{(0.753\pm0.063)h^{-1} \rm{ \ Mpc}}{h(z)}
\left(\frac{L_X}{10^{44}h^{-2}\rm{\ erg\ s}^{-1}}\right)^{0.228\pm 0.015}
\label{eq:r500}
\end{equation}
To test the effect of the cluster profile on the final results, we construct
y-maps from the X-ray cluster catalog (a) using the universal pressure profile
of eq.~(\ref{eq:universal_profile}) with the parameters given in
eq.~(\ref{eq:Arn_parameters}) and (b) using the isothermal $\beta=2/3$ model.
The Comptonization parameter is
computed integrating the electron pressure profile along the line
of sight. Clusters are assumed to be spherically symmetric and extending up to
$R_{200}$, the scale where the cluster overdensity reaches 200 times
the critical density. To determine the effect of the cluster profile,
the central value of $y_c$ is assumed to be the same. Finally, the
cluster templates are convolved with the corresponding antenna beams (see Table 1).
In the Fig.~\ref{fig3}a we show the pressure profile integrated
along of line of sight for the $\beta=2/3$ (solid line)
and universal pressure (dashed line) profiles
convolved with the antenna of the 44GHz map. The cluster is located at
$z=0.094$, of $M_{500}=2.4\times 10^{14}h^{-1}M_\odot$ and
$R_{500}=746h^{-1}$Kpc. For illustration, in Fig.~\ref{fig3}b we plot
the value of Comptonization parameter $y_c$ at the center of all the
clusters in our proprietary cluster catalog, derived
using the measured X-ray parameters,
as a function of cluster mass. The solid line
represents the linear regression fit to the data. The central Comptonization
parameter scales as: $y_c=24.5(M_{500}/10^{14}h^{-1}M_\odot)^{1.35}$.
\subsection{Y-map from simulated clusters. \label{sec:sim}}
As an alternative, we also use the low-redshift all-sky maps and the associated galaxy
cluster catalogues of the {\it hydrodynamic} diffuse and kinetic SZ simulations included
in the {\it pre-launch Planck Sky Model}. The simulations are fully described
in Dellabrouille et al (2011).
The catalogues contain cluster positions, mass and radius for an overdensity
contrast of 200 times the critical density. The maps contain the integrated SZ
signal up to $z\simeq 0.25$, computed from a combination of full hydrodynamic
simulations using the box staking method described in Valente, da Silva
\& Aghanim (2012). According to this method, the Universe around the observer
is generated in concentric layers, each with a comoving thickness of
$100h^{-1}$Mpc, using the outputs of hydrodynamic simulations with periodic
boundary conditions. The light-cone integrations
of the TSZ and KSZ signals are carried out using the formulae in
da Silva et al (2000) and (2001). A total of seven
layers were constructed, up to z=0.25. The innermost layer includes the
local constrained simulation of Dolag et al (2005), whereas all the
other layers were produced from gas snapshots of the $\Lambda$CDM simulation
in De Boni et al (2010). Both these simulations include explicit treatment for
gas cooling, heating by UV, star formation and feedback processes.
The y-map constructed from the X-ray selected clusters assumes clusters to be spherically
symmetric and relaxed while the TSZ and KSZ templates constructed from the hydrosimulation
contains clusters with different dynamical state (relaxed, merging systems, etc),
shape and ellipticity. Also, since the latter are constructed integrating
the signal along the line of sight, the projection effects due to low mass clusters
and groups are included. Therefore, these templates are
very well suited to study the effect of all these systematics and
of the KSZ component in the determination of $\alpha$. For a more realistic
comparison, we select 623 clusters from the simulation according to
the measured selection function of the X-ray cluster sample. In Fig~\ref{fig4}a
we plot the mass and in Fig~\ref{fig4}b the redshift distribution of
all clusters in our simulation (solid line) that fulfill the selection
criteria. The dashed line shows the same distributions of the X-ray clusters.
For a better comparison, the histogram of largest amplitude
was normalized to unity. The main difference between the two samples is that
there are 22 clusters in our proprietary cluster catalog that
have redshifts larger than $z=0.25$, the redshift of the last layer
constructed from the simulation.
\subsection{Final Maps.}
The y-maps described above are multiplied by
$G(\nu,\alpha=0)$ to generate TSZ templates and convolved with
the antenna beam. A KSZ template was added to
the hydrodynamical but not to the X-ray selected cluster template
since their peculiar velocity is not available. Noise maps were constructed
assuming the Blue Book noise levels of Table~1.
When presenting our results, we shall demonstrate that the HFI frequencies
have the largest statistical power to constrain $\alpha$.
We model the noise as homogeneous and uncorrelated white noise since
at the frequencies $44-353$GHz the $1/f$ is both small and does not affect
the angular scales subtended by clusters, $\ell\sim 500$ and above.
To take into account the two different component separation techniques, we
carry out two different set of simulations: in (A) the CMB is removed in the TOI.
Maps will only contain instrumental noise, TSZ and KSZ. In total, six
different maps, one for each frequency of Table~1 are simulated. In
(B) the 217GHz map is used to remove the intrinsic CMB. Then, only five difference
maps will be available for the analysis. Those maps
will contain instrumental noise and TSZ, but the frequency
dependence of the TSZ effect changes. More realistic simulations would include
foreground residuals, noise inhomogeneities with an $1/f$ component that
can only be accurately model once the data becomes available. However, we do not
expect that our results obtained with our simplified maps will change with more
realistic simulations if noise inhomogeneities are uncorrelated with the
cluster distribution. Then, we constructed two set of maps, according to
the specific simulation. In simulation (A) six maps, one for each
frequency of Table~1, are constructed by adding noise to the TSZ and KSZ
templates. The temperature anisotropy at
each pixel is: $\Delta T_A(\nu)=y_cG(\nu,0)+\Delta T_{KSZ}\pm\sigma^A_{noise,\nu}$
In simulation (B) six maps are constructed adding cosmological CMB
signal and noise to the cluster templates. The 217GHz map is used
to subtract the cosmological and KSZ signals. Therefore, only
five different maps are available for the analysis.
We checked the final maps had a power spectrum that was a pure white noise,
with a slightly larger variance $\sigma^B_{noise,\nu}$, sum of the
original map plus the noise of the degraded 217GHz map.
At each pixel, the temperature anisotropy is:
$\Delta T_B(\nu)=y_cG_{[-217GHz]}(\nu,0)\pm\sigma^B_{noise,\nu}$.
We neglect relativistic corrections that are only significant for the
most massive clusters (Nozawa et al 1998).
\bigskip
\bigskip
\section{Data Processing.}
In both simulations A and B, we construct estimators using both the Ratio
and the Fit methods. To simplify the notation, let the index
$I=(A,B)$ denote the type of simulation and let us redefine
$G_A=G(\nu,\alpha)$, $R_A=R(\nu_1,\nu_2,\alpha)$,
$G_B=G_{[-217GHz]}(\nu,\alpha)$ and $R_B=R_{[-217GHz]}(\nu_1,\nu_2,\alpha)$.
At each cluster location, projection effects can yield contributions from
different redshifts altering the frequency dependence. To reduce the effect,
we will take averages over the cluster extent. The temperature anisotropy is then
\begin{equation}
\langle\Delta T_I(\nu_1)\rangle= \bar{y}_cG_I(\nu_1)
\pm\sigma^I_{Noise,\nu_1}/\sqrt{N_{pix}} .
\label{eq:deltatsz}
\end{equation}
where $N_{pix}$ is the number of pixels occupied by the cluster. There will
be an extra KSZ component for simulation A.
\subsection{Ratio method.}
To estimate $\alpha$ we compute the likelihood
\begin{equation}
-2\log{\cal L}=\sum_{\nu_1,\nu_2}\sum_{i=1}^{N_{cl}}\left[
\frac{\langle\Delta T_I(\nu_1)\rangle/\langle\Delta T_I(\nu_2)\rangle-
R_I(\nu_1,\nu_2,\alpha)}{\sigma^I_{ratio,i}} \right]^2
\label{eq:chisq_ratio}
\end{equation}
for different values of $\alpha$. A few examples of
$R_I(\nu_1,\nu_2,\alpha)$ are plotted in Fig~\ref{fig1}a and~\ref{fig2}a.
In eq.~(\ref{eq:chisq_ratio}), we compute $\sigma^I_{ratio,i}$ for each
cluster as the rms deviation of 1,000 simulations of the ratio
$\langle\Delta T_I(\nu_1)\rangle/\langle\Delta T_I(\nu_2)\rangle$
where the TSZ component is held fixed to the actual value at the cluster
location and the noise is drawn from a gaussian distribution with
zero mean and variance $(\sigma^I_{noise,\nu})^2/N_{pix,i}$.
As discussed in Luzzi et al (2009) the distribution of ratios is
dominated by the error on the denominator. Therefore, in our simulations
type (A) we exclude the 217GHz channel from the denominator, where
the TSZ signal is null, to minimize the bias.
\subsection{Frequency fit method.\label{sec:fitmethod}}
Alternatively, we can fit the TSZ signal of each cluster to the spectral
dependence of Fig~\ref{fig1}b and \ref{fig2}b. Similarly, the
likelihood function is
\begin{equation}
-2\log{\cal L}=\sum_\nu\sum_{i=1}^{N_{cl}}
\left[\frac{\langle\Delta T_I(\nu)\rangle-\bar{y}_cG_I(\nu,\alpha)}
{\sigma^I_{noise,\nu,i}}\right]^2 ,
\label{eq:chisq_fit}
\end{equation}
where $\sigma^I_{noise,\nu,i}=\sigma^I_{noise,\nu}/\sqrt{N_{pix,i}}$.
This method requires an independent estimate of $y_c$, introducing another
complication. As is indicated in Table~1, different frequencies have different
resolutions. The cluster anisotropies are diluted by the antenna beam and
the TSZ signal does no scale as $G_I(\nu,0)$. As an example, in
Fig~\ref{fig:convolved} open squares represent the average TSZ amplitude
on the 5 difference maps $\Delta T_{[\nu,-217GHz]}$; the solid line represent their
frequency dependence $G_{[-217GHz]}(\nu,0)$ of Fig~\ref{fig2}b. Due to the antenna,
the measured TSZ signal of the clusters differs from the expected scaling.
The effect is most noticeable at 44GHz since this channel has the smallest
resolution. The amplitude of the effect depends on the cluster profile
and angular extent {\it but does not depend on the scaling of the TSZ signal
with redshift, $G(\nu,\alpha)$}. Fig~\ref{fig:convolved}a corresponds to a cluster
at redshift $z=0.218$, with mass $M_{500}=3.64\times10^{14}M_\odot/h$ and size $9.4'$
while in Fig~\ref{fig:convolved}b the cluster is located at $z=0.058$
with mass $M_{500}=7.7\times10^{14}M_\odot/h$ and size $42'$.
For clusters drawn from a simulation, its size, ellipticity and profile
are known exactly. For such clusters, the deconvolution
factor $F$ can be determined exactly by comparing the average
Comptonization parameter before ($\langle{y}_c\rangle$) and
after $\langle y_c*B(\nu)\rangle$ convolving with the antenna beam $B(\nu)$:
$F=\langle{y}_c\rangle/\langle y_c*B(\nu)\rangle$. This factor would be different for
resolved and unresolved clusters and would depend on the cluster profile and redshift.
For X-ray selected clusters, we know the Comptonization parameter in the cluster
cores but their pressure profile has not being measured.
This will introduce an extra uncertainty when comparing the measured TSZ
effect with the theoretical prediction.
For illustration, in Fig~\ref{fig:resolved}a we represent
$F$ for a sample of 110 clusters in the mass range $M_{500}=5-6\times
10^{14}M_\odot/h$. In Fig~\ref{fig:resolved}b, we plot the deconvolution factors
for all clusters in our simulation with masses $M_{500}\ge 10^{15}M_\odot/h$.
Solid black circles correspond to the 353GHz frequency and open squares to 44GHz.
All clusters are resolved at 353GHz. At 44GHz clusters with redshift $z\ge 0.08$
are unresolved. For clarity the clusters at lower redshift, that would be resolved,
are not shown.
In Fig~\ref{fig:resolved} the solid straight lines correspond to
the linear regression fit to the deconvolution factor
for each cluster mass range and channel. Arrows indicate the deconvolution
factor of the clusters plotted in Fig~\ref{fig:convolved}a,b.
If for each redshift, frequency and mass range, $F_{lin}$ is the deconvolution
factor estimated by the linear regression, $F$ the true deconvolution
factor and $\Delta F$ is the rms dispersion
of the true deconvolution values $F$ around $F_{lin}$ for each mass
bin and antenna, then $F=F_{lin}\pm\Delta F$. If
for a real cluster we use $F_{lin}$ instead of the (unknown) true factor
$F$, the deconvolved signal $(\Delta T_{TSZ}*B)F_{lin}$ would differ from the
the true signal $\Delta T_{TSZ}$ by an amount $(\Delta T_{TSZ}*B)\Delta F$.
This uncertainty is uncorrelated with the instrumental noise at the cluster
location and can be included in the Likelihood analysis of
eq.~\ref{eq:chisq_fit} by adding it in quadrature with the instrumental noise:
$\sigma_{tot,i}^2=\sigma_{noise,i}^2+[(\Delta T_{TSZ}*B)\Delta F]^2$.
On the other hand, the deconvolution coefficient does not scale linearly with
redshift, and $F_{lin}$ underestimates the true deconvolution factor $F$
especially at high redshifts, potentially biasing our estimation of $\alpha$.
We used the y-map computed with clusters drawn from a numerical simulation
to compute the deconvolution factors $F_{lin}$ and its uncertainty $\Delta F$
in three mass bins of equal number of clusters: $M_{500}\le 2\times
10^{14}M_\odot/h$, $M_{500}=2-3.6\times 10^{14}M_\odot/h$ and
$M_{500}\ge\times 3.6\times 10^{14}M_\odot/h$.
The deconvolution factors, that were different for resolved and unresolved
clusters, were used to deconvolve the templates of simulated and of X-ray clusters.
\bigskip
\bigskip
\section{Results and Discussion}
We first tested the ratio and fit methods using the template constructed from
simulations, as described in Sec.~\ref{sec:sim}. The template contained a
subset of $623$ clusters distributed in mass and redshift according to the
cluster catalog selection function (see Fig~\ref{fig4}) and included
both TSZ and KSZ components. Second, we repeated the analysis with the template
of X-ray selected clusters. No KSZ contribution was added in this case.
The first and most important conclusion is that we found no significant
differences from the results computed using both templates, implying that
the effect of KSZ, cluster dynamical state and
deviations from spherical symmetry are averaged out over such a large
cluster sample, effects that were important when analyzing observations
of just a few clusters (Battistelli et al 2002, Luzzi et al 2009).
With respect to the method of analysis, the ratio method performs differently
if the CMB is removed in the TOI (method A) or in the final map (method B)
but the differences are small for the fit method. For the ratio method,
we will only present results obtained with the simulated cluster template
and simulation type (A) (Fig~\ref{fig7}). For the fit method, the results presented
are only using the template constructed from X-ray selected clusters
and simulation type (B) (Fig~\ref{fig8}) and we will discuss all the other
cases. To test the importance of the different contributions,
we define three mass bins $M_{500}=([<0.192],[0.192-0.365],[>0.36])
\times 10^{15} M_\odot/h$ of equal number ($\sim 208$) of clusters,
and three redshift bins $z=([<0.11],[0.11-0.17],[>0.17])$ with mean
redshift $\langle z\rangle=(0.08,0.14,0.20)$, also with the same number
of clusters. We computed the likelihood
(eqs.~[\ref{eq:chisq_ratio}] and [\ref{eq:chisq_fit}])
for the different mass and redshift bins and different frequencies to
determine the data subset with the largest statistical power.
To compute likelihoods, we subdivide the interval $\alpha=[-1,1]$
in 2001 steps. We perform 1,000 Monte-Carlo simulations for each
cluster template, simulation type and method.
\subsection{Ratio method.}
Our results indicate that the ratio method is strongly biased.
The likelihood function is dominated by the few clusters were the TSZ signal
is erased by the noise so $\langle \Delta T(\nu_2)\rangle\sim 0$
in the denominator of eq.~(\ref{eq:chisq_ratio}), biasing the results
towards negative values of $\alpha$. We found this bias is
removed by rejecting all ratios where the denominator is smaller
than $0.2\sigma^A_{noise}$. This constrain rejects 5\% of the data
but reduces the bias to insignificant levels.
In Fig.~\ref{fig7} we present the likelihood function of a single
simulation randomly selected of our ensemble of 1,000 simulations.
In Fig.~\ref{fig7}a we represent the likelihood for
clusters within the three redshift bins given above, marginalized
over cluster mass and 15 frequency ratios. Dashed, dot-dashed and solid lines
correspond to the lower, intermediate and high redshift bins.
In Fig.~\ref{fig7}b we present the results binning clusters
according to mass. Dashed, dot-dashed and solid correspond
to lower, intermediate and high mass bin. As expected,
the most massive clusters dominate the likelihood. The final
value obtained in this single simulation is $\alpha=0.0\pm 0.012$.
In Fig.~\ref{fig7}c we plot the histogram distribution of
the $\alpha$ values measured in 1,000 simulations.
Solid line represents the results for simulation type A
where $\langle{\alpha}\rangle=-0.003\pm 0.011$. Our pipeline is marginally
biased since the mean of our simulations $\langle{\alpha}\rangle=-0.003$
differs from $\alpha=0$ by more than $\sigma_{\alpha}/\sqrt{N_{sim}}=3.5\times 10^{-4}$.
For comparison, in Fig.~\ref{fig7}c the dashed line corresponds to
simulation type B. The value of $\alpha$ averaged over all simulations
is $\langle{\alpha}\rangle=-0.19\pm 0.18$. In this case, the larger uncertainty
reflects the weak dependence of the ratio method with $\alpha$, as shown in
Fig~\ref{fig2}a. Finally, we also checked that ratios using lower resolution
maps have less statistical power to constrain $\alpha$.
In retrospect, this justifies neglecting the 30, 545 and 857GHz channels,
with large noise levels or (1/f) contributions, that would complicate
the analysis without adding more information.
\subsection{Fit method.}
The results of the fit method using a template of X-ray clusters and
simulations type B are presented in Fig.~\ref{fig8}.
The cluster template was constructed using the universal pressure profile of
eq.~\ref{eq:universal_profile}, with the parameters given in
eq.~\ref{eq:Arn_parameters}. In (a) we plot the likelihood function
for three different frequencies: 44GHz (dashed), 100 GHz (solid)
and 343GHz (dot-dashed line). The figure shows that the
100GHz channel is the most restrictive of the three. Like for the
ratio method, the final
likelihood is dominated by the channels that have high resolution
and low noise, in this case the 100 and 143GHz channels.
In Fig.~\ref{fig8}b we represent the likelihood
for the three mass bins given above and marginalized over frequencies;
dashed, dot-dashed and solid lines correspond to the low, intermediate
and high mass intervals. The signal is dominated by the most massive
clusters that, on a flux limited sample, are on average at high
redshift than the lower and intermediate mass samples.
For this particular realization, the estimated value is $\alpha=0.013\pm 0.020$,
compatible with adiabatic evolution at the $1\sigma$ level.
In Fig.~\ref{fig8}c we represent the histograms of 1,000
simulations together with their linear fits for
cluster templates constructed using the universal pressure profile
(dot-dashed line) and the $\beta=2/3$ (solid line).
The mean and rms dispersion of the estimated values are
$\langle{\alpha}\rangle=-0.013\pm 0.016$ for the universal profile and
$\langle{\alpha}\rangle=0.003\pm 0.008$ for the $\beta$-model profile.
When all cluster properties are identical, the TSZ integrated over the
cluster extent will be larger for the $\beta$-model than for the universal
profile (see Fig~\ref{fig3}a) so it must constrain $\alpha$ better,
as shown. We also carried out 1,000 simulations using method A, that does
not include the intrinsic CMB signal, with the hydrodynamical template, that
contains the KSZ component. The result was $\langle{\alpha}\rangle=-0.008\pm 0.015$,
identical to the result with the method B above. Therefore, in the fit method
is not so important to have maps with the cosmological signal removed
as, for example, in the ratio method. Using the 217GHz map to remove the
intrinsic CMB signal alters the TSZ frequency dependence, but the TSZ signal
is still strongly dependent with redshift (see Figs.~\ref{fig1}b and \ref{fig2}b)
what is not the case in the ratio method (compare Figs.~\ref{fig1}a and \ref{fig2}a).
Let us remark that the rms dispersion of $\alpha$ on 1,000 simulations,
$\sigma_\alpha$, is very similar to the error on $\alpha$
in one single realization, both in the ratio and in the fit method,
indicating that our pipelines are efficient. The results obtained
using y-maps constructed with clusters drawn from a hydrodynamical
simulation or from a catalog of X-ray selected clusters show no significant
differences. In the hydro-simulation, the y-map integrates the SZ signal
up to $z\simeq 0.25$ and contains all the projection effects up to that redshift,
not included in the
X-ray selected clusters template, we can conclude that projection effects play no
significant role. This can be understood in the light of the results presented
in Fig~\ref{fig8}b; the full likelihood is dominated by the most massive clusters
for which projection effects are not significant (Valente et al 2012).
\bigskip
\bigskip
\section{Conclusions.}
Planck offers an excellent opportunity to constrain
the evolution history of the CMB blackbody temperature with better precision
than quasar excitation lines using currently available X-ray cluster
catalogs. We have found
that taking the ratio of temperature anisotropies at different frequencies
is strongly biased but this bias can be corrected by rejecting all ratios where
the denominator is much smaller than the noise. Fitting the frequency
dependence provides an equally reliable estimator with
different systematics but requires both an independent determination
of the Comptonization parameter and deconvolution of the antenna beam.
The latter can not be done exactly if the cluster pressure profile is not
known precisely. We have shown that deconvolution using linear fits
introduces an error that can be easily incorporated into the analysis.
We have considered two possible methods to remove foregrounds and the
cosmological CMB signal and the KSZ contribution:
the cosmological signal is removed in the TOI and
the 217GHz map is used to remove the cosmological and KSZ signal exactly.
We have carried out simulations of both methods to investigate
the differences on the final results. We have shown that the ratio method
performs rather well if the cosmological CMB signal is clean in the
TOI but very badly otherwise. The fit method performs equally well
in both data sets, giving results that are only marginally biased.
With both methods, massive clusters and the high resolution/low noise
channels have the largest statistical power to constrain $\alpha$.
We have used a proprietary cluster catalog that contains
spectroscopic redshifts and all the required X-ray
information to estimate the accuracy that would be achieved with Planck data.
We forecast that the final uncertainty will be about $0.011-0.016$ a factor
2-3 better than those obtained from quasar spectra by Noterdaeme et al (2010),
depending on what type of Planck data becomes publicly available.
Since our catalog is restricted to
clusters with $z\le 0.3$, we have not extended our analysis beyond that
redshift. Planck has already detected around 200 clusters with a $S/N\ge 10$,
one at $z\simeq 0.94$ with $M_{500}\simeq 8\times 10^{14} M_\odot$. Adding
more clusters with current or future experiments will help to detect
possible deviations from adiabatic evolution, specially if
clusters are of higher mass and are at a higher redshift
like the one recently reported in Planck Collaboration 2011c.
Once all PLANCK and South Pole cluster candidates have been observed
on the X-ray and their redshift determined, the measurements proposed
will provide much stronger constraints on non-adiabatic evolution,
than those quoted here.
\bigskip
\bigskip
\section{Acknowledgements}
This work was done in the context of the FCT/MICINN cooperation grant
'Cosmology and Fundamental Physics with the Sunyaev-Zel'dovich Effect'
AIC10-D-000443, with
additional support from project PTDC/FIS/111725/2009 from FCT, Portugal
and FIS2009-07238 and CSD 2007-00050 from the Ministerio de Educaci\'on y
Ciencia, Spain. The work of CM is funded by a Ci\^encia2007 Research Contract,
funded by FCT/MCTES (Portugal) and POPH/FSE (EC).
|
{
"timestamp": "2012-03-09T02:03:18",
"yymm": "1203",
"arxiv_id": "1203.1825",
"language": "en",
"url": "https://arxiv.org/abs/1203.1825"
}
|
\section{Motivation: Mixing classical and non-classical logics}
\label{Sec:Intro}
Mixing logical behaviours is a more and more investigated topic in logic.
For instance, labelled deductive systems by D.~M.~Gabbay \cite{Gabbay97}
are used at this aim, and the ``stoup'' mechanism introduced by J-Y.~Girard
in \cite{Girard93} makes intuitionistic and classical deductions interact.
In 1989, P.~A.~Miglioli with his co-authors \cite{Miglioli89a} introduced a
constructive logic with strong negation, called \emph{effective logic zero}
and denoted by $E_{0}$, containing a modal operator
$\mathbf{T}$ such that for any formula $\alpha$ of $E_{0}$, $\mathbf{T} (\alpha)$
means that $\alpha$ is classically valid.
More precisely, given a Hilbert-style calculus for constructive logic
with strong negation (CLSN), also called Nelson logic \cite{Nelson49}, the rules for
{\bf T} are
\begin{center}
$({\sim} \alpha \rightarrow \perp) \rightarrow \mathbf{T}(\alpha)$ \qquad and \qquad
$(\alpha \rightarrow \perp) \rightarrow {\sim} \mathbf{T}(\alpha)$,
\end{center}
where $\sim$ denotes the \emph{strong negation}.
One obtains that $\alpha$ is valid in classical logic (CL) if and only if $\mathbf{T}(\alpha)$ is provable in $E_{0}$.
Therefore, $\mathbf{T}$ acts as an intuitionistic double negation $\neg\neg$ which, in view of the
G\"{o}del-Glivenko theorem, is able to grasp classical validity in the intuitionistic propositional
calculus (INT) by stating that $ \vdash_\mathrm{CL} \, \alpha$ if and only if
$\vdash_\mathrm{INT} {\neg\neg \alpha}$.
However, $\mathbf{T}$ fulfils additional distinct features. Firstly, CLSN is equipped with a weak
negation $\neg$, defined similarly to the intuitionistic negation. But, the combinations
$\neg\neg$, ${\sim}\neg$, or $\neg{\sim}$ are not able to cope with classical tautologies
(see \cite{pagliani2008geometry}, for example).
Secondly, consider the Kuroda formula
$\forall x \, \neg\neg \alpha(x) \to \neg\neg \forall x \, \alpha(x)$.
As noted in \cite{Miglioli89a}, it is an example of the divergence between double negation and
an operator intended to represent classical truth, because the formula
$\forall x \, \mathbf{T} ( \alpha(x) )\to \mathbf{T} ( \forall x \, \alpha(x))$
should be intuitively valid if $\mathbf{T}$ represents classical truth. But the Kuroda formula
is unprovable in intuitionistic predicate calculus, while the above-presented $\mathbf{T}$-translation
(and some other translations, too) are provable even in the the predicative version of $E_0$.
The motivation of the logical system $E_{0}$ was to grasp two distinct aspects of computation in program
synthesis and specification: the algorithmic aspect and data. The latter are supposed to be given, not to be proved or
computed; in fact ``data'' is the Latin plural of ``datum'', which, literally, means ``given''.
A single undifferentiated logic is not a wise choice to cope with both aspects, therefore in $E_{0}$
there are two different
logics at work: a constructive logic, representing algorithms, and classical logic, representing the behaviour of data.
Data are assumed not to be constructively analysable and this is connected to the problem of
the meaning of an atomic formula from a constructive point of view.
Since the meaning of a formula is given by its construction, according to the constructivistic philosophy,
and since its construction depends on the logical structure
of the formula, the meaning of an atomic formula, which as such has no structure, is the atomic formula itself.
This is the solution adopted by Miglioli and others in \cite{Miglioli89b}.
In that paper, it is assumed that atomic formulas cannot have a constructive proof, therefore $p$ and
$\mathbf{T}(p)$ must coincide, that is, an axiom schema
\begin{equation} \label{Eq:Atomic}
p \leftrightarrow \mathbf{T}(p) \tag{$\star$}
\end{equation}
is included for propositional variables.
Axiom \eqref{Eq:Atomic} together with the $\mathbf{T}$-version of the Kreisel-Putnam principle
\cite{KreiselPutnam}, that is,
\begin{equation}\tag{T-KP}\label{Eq:TKP}
(\mathbf{T}(\alpha) \rightarrow (\beta\vee\gamma) )
\rightarrow((\mathbf{T}(\alpha) \rightarrow \beta) \vee (\mathbf{T}(\alpha)
\rightarrow\gamma))
\end{equation}
characterises the logic ${\mathcal F}_{\rm CL}$.
Because of ($\star$) the logic ${\mathcal F}_{\rm CL}$ is not standard in the sense that
it does not enjoy uniform substitution.
However, its \emph{stable part}, that is, the part which is closed under uniform substitution, coincides with
a well-known maximal intermediate constructive logic, namely Medvedev's logic, a faithful interpretation of the
intuitionistic logical principles (see \cite{Medvedev62,Miglioli89b}).
One year later, P.~Pagliani \cite{Pagliani90} was able to exhibit an algebraic model
for $E_0$. It turned out that these models are a special kind of Nelson algebras,
called {\em effective lattices}.
The paper is structured as follows. In the next section we recall some well-known facts
about Heyting algebras, Nelson algebras, and effective lattices. In Section~\ref{Sec:RS_equivalence},
we recollecting some well-known results related to rough sets defined by equivalence relations and
the semi-simple Nelson algebras they determine.
In Section~\ref{Sec:RS_quasiorders}, we recall the fact that rough set systems induced by quasiorders
determine Nelson algebras, and show how these algebras can be obtained by Sendlewski's construction.
We also present a completeness theorem for CLSN in terms of finite rough set-based Nelson algebras.
We give several equivalent conditions under which rough set-based Nelson algebras form effective
lattices, and this enables us to characterize the Nelson algebras which are isomorphic to rough
set-based effective lattices determined by quasiorders. Some concluding remarks of
Section~\ref{Sec:Conclusions} end the work. In particular, it is shown how the logic $E_0$ can be interpreted
in terms of rough sets by following the very philosophy of rough set theory.
\section{Preliminaries: Heyting algebras, Nelson algebras, and \mbox{effective} lattices}
\label{Sec:Preliminary}
A \textit{Kleene algebra} is a structure $(A, \vee, \wedge, {\sim}, 0, 1)$ such that
$A$ is a 0,1-bounded distributive lattice and for all $a,b \in A$:
\begin{enumerate}[({K}1)]
\item ${\sim}\,{\sim}a = a $
\item $a \leq b \text{ if and only if } {\sim}b \leq {\sim}a$
\item $a \wedge {\sim}a \leq b \vee {\sim}b$
\end{enumerate}
A \textit{Nelson algebra}
$(A, \vee, \wedge, \rightarrow, {\sim}, 0, 1)$ is a Kleene algebra
$(A, \vee, \wedge, {\sim}, 0, 1)$ such that for all $a,b,c\in A$:
\begin{enumerate}[({N}1)]
\item $a\wedge c \leq {\sim} a\vee b$ if and only if $c\leq a\rightarrow b$,
\item $(a\wedge b)\rightarrow c = a \rightarrow (b\rightarrow c)$.
\end{enumerate}
In each Nelson algebra, an operation $\neg$ can be defined as
$\neg a = a \to 0$. The operation $\to$ is called \emph{weak relative
pseudocomplementation}, $\sim$ is called \emph{strong negation},
and $\neg$ is called \emph{weak negation}. A Nelson algebra
is \emph{semi-simple} if $a \vee \neg a = 1$ for all $a \in A$. It is well
known that semi-simple Nelson algebras coincide with three-valued {\L}ukasiewicz
algebras and regular double Stone algebras.
An element $a^*$ in a lattice $L$ with $0$ is called a \emph{pseudocomplement}
of $a \in L$, if $a \wedge x =0 \iff x \leq a^*$ for all $x \in L$.
If a pseudocomplement of $a$ exists, then it is unique, and a lattice in which every
element has a pseudocomplement is called a \emph{pseudocomplemented lattice}.
Note that pseudocomplemented lattices are always bounded.
An element $a$ of pseudocomplemented lattice is
\emph{dense} if $a^* = 0$. A \emph{Heyting algebra} $H$ is a lattice with $0$
such that for all $a,b \in H$, there is a greatest element $x$ of $H$ with $a \wedge x \leq b$.
This element is the \emph{relative pseudocomplement} of $a$ with respect to $b$,
and is denoted $a \Rightarrow b$. It is known that a complete lattice is a Heyting algebra
if and only if it satisfies the \emph{join-infinite distributive law}, that is, finite meets distribute
over arbitrary joins. In a Heyting algebra, the \emph{pseudocomplement} of $a$
is $a \Rightarrow 0$. By a \emph{double Heyting algebra} we mean a Heyting algebra
$H$ whose dual $H^d$ is also a Heyting algebra. A \emph{completely distributive lattice} is a complete
lattice in which arbitrary joins distribute over arbitrary meets. Therefore,
completely distributive lattices are double Heyting algebras.
A Heyting algebra $H$ can be viewed either as a partially ordered set $(H,\leq)$, because
the operations $\vee$, $\wedge$, $\Rightarrow$, $0$, $1$ are uniquely determined by the
order $\leq$, or as an algebra $(H,\vee,\wedge,\Rightarrow,0,1)$ of type $(2,2,2,0,0)$.
Congruences on Heyting algebras are equivalences compatible with operations $\vee$, $\wedge$,
and $\Rightarrow$. Next we recall some well-known facts about congruences on Heyting algebras
that can be found \cite{blyth2005lattices}, for instance.
Let $L$ be a distributive lattice and let $F$ be a filter of $L$. The equivalence
\[ \theta(F) = \{ (x,y) \mid (\exists z \in F) \, x \wedge z = y \wedge z \} \]
is a congruence on $L$. It is known that if $H$ is a Heyting algebra,
then $\theta(F)$ is a congruence on $H$, that is, $\theta(F)$ is compatible also
with $\Rightarrow$. Additionally, all congruences on Heyting algebras
are obtained by this construction.
A congruence on a Heyting algebra is said to be a \emph{Boolean congruence}
if its quotient algebra is a Boolean algebra. For a Heyting algebra $H$,
a filter $F$ contains the filter $D$ of all the dense elements of $H$ if and
only if $\theta(F)$ is a Boolean congruence. This means that $\theta(D)$
is the \emph{least} Boolean congruence on $H$, which is known as the \emph{Glivenko congruence}
$\Gamma$, expressed also as
\[
\Gamma = \{ (a,b) \mid a^* = b^* \}.
\]
\begin{lemma} \label{Lem:BooleanCongruencePseudo}
Let $H$ be a Heyting algebra and let $a \leq d$
for all dense elements $d$ of $H$. The equivalence
\[ {\cong_a} = \{ (x,y) \mid x \wedge a = y \wedge a \}
\]
is a Boolean congruence on $H$.
\end{lemma}
\begin{proof}
Let $F_a = \{ x \in H \mid a \leq x\}$ be the principal filter of $a$.
Then $\theta(F_a)$ is a congruence on $H$, and clearly $\cong_a$ is equal to
$\theta(F_a)$ (see e.g. \cite{pagliani2008geometry}).
Because $a \leq d$ for all $d \in D$, we have $D \subseteq F_a$ and so $\cong_a$ is a
Boolean congruence.
\end{proof}
Let $\Theta$ be a Boolean congruence on a Heyting algebra $H$.
As shown by A.~Sendlewski \cite{Sendlewski90}, the set of pairs
\begin{equation}\label{Eq:Nelson-construction}
N_\Theta (H)=
\{(a,b)\in H \times H \mid a\wedge b=0 \text{ and } a\vee b \, \Theta \, 1\}
\end{equation}
can be made into a Nelson algebra, if equipped with the operations:
\begin{align*}
(a,b) \vee(c,d) & = (a \vee c, b \wedge d);\\
(a,b) \wedge(c,d) & = (a \wedge c, b \vee d);\\
(a,b) \to(c,d) & = (a \Rightarrow c, a \wedge d);\\
{\sim}(a,b) & = (b,a).
\end{align*}
Note that $(0,1)$ is the $0$-element, $(1,0)$ is the $1$-element, and in the right-hand
side of the above equations, the operations are those of the Heyting algebra
$H$. This Nelson algebra is denoted by $\mathbb{N}_{\Theta}(H)$.
In \cite{Pagliani90}, Pagliani introduced {\em effective lattices}. They
are special type of Nelson algebras determined by Glivenko congruences on
Heyting algebras, that is, for any Heyting algebra $H$ and its
Glivenko congruence $\Gamma$, the corresponding \emph{effective lattice}
is the Nelson algebra $\mathbb{N}_{\Gamma}(H)$. Note that for all
$x \in H$, $x \, {\Gamma} \, 1$ if and only if $x$ is dense. This
means that $N_\Gamma(H)$ consists of the pairs $(a,b)$ such
that $a \wedge b = 0$ and $a \vee b$ is dense (see Remark~2 in
\cite{Sendlewski90} and Proposition~9 of \cite{Pagliani90}).
Additionally, it is proved in \cite{Pagliani90} that
effective lattices are models for the logic $E_0$, with $\mathbf{T}$ defined by
$\mathbf{T}((a,b)) = (a^{**},b^{**})$. Note that each Heyting algebra
defines exactly one effective lattice, and that all effective lattices
are determined this way.
\section{Rough set theory comes into the picture}
\label{Sec:RS_equivalence}
Rough sets were introduced by Z.~Pawlak \cite{Pawl82} in order to provide a formal approach to deal
with incomplete data. In rough set theory, any set of entities (or points, or objects) comes with
a lower approximation and an upper approximation. These approximations are defined on the basis of
the attributes (or parameters, or properties) through which entities are observed or analysed.
Originally, in rough set theory it was assumed that the set of attributes induces an
equivalence relation $E$ on $U$ such that $x \,E \, y$ means that $x$ and $y$ cannot be
discerned on the basis of the information provided by their attribute values. Approximations
are then defined in terms of an {\em indiscernibility space}, that is,
a relational structure $(U,E)$ such that $E$ is an equivalence relation on $U$.
For a subset $X$ of $U$, the \emph{lower approximation} $X_{E}$ of $X$ consists of all
elements whose $E$-class is included in $X$, while the {\em upper approximation} $X^{E}$
is the set of the elements whose $E $-class has non-empty intersection with $X$. Therefore,
$X_{E}$ can be viewed as the set of elements which \emph{certainly} belong to $X$, and $X^{E}$
is the set of objects that \emph{possibly} are in $X$, when elements are observed
through the knowledge synthesized by $E$.
Since inception, a number of generalisations of the notion of a rough set have been
proposed. A most interesting and useful one is the use of arbitrary binary relations
instead of equivalences. Let us now define approximations $(\cdot)_R$ and $(\cdot)^R$
in a way that is applicable for different types of binary relations considered in this paper,
and introduce also other notions and notation we shall need. It is worth pointing out
that $(\cdot)_R$ and $(\cdot)^R$ can be regarded as ``real'' lower and upper approximation operators,
respectively, only if $R$ is reflexive, because otherwise $X_R \subseteq X$
and $X^R \subseteq X$ may fail to hold.
\begin{definition} \label{Def:BasicNotation}
Let $R$ be a reflexive relation on $U$ and $X \subseteq U$. The set
$R(X) = \{y \in U \mid x \, R\, y \text{ for some $x \in X$}\}$ is
the \emph{$R$-neighbourhood} of $X$. If $X = \{a\}$, then we write $R(a)$ instead of $R(\{a\})$.
The approximations are defined as $X_R = \{x \in U \mid R(x) \subseteq X\}$
and $X^R = \{x \in U \mid R(x) \cap X \neq \emptyset\}$. A set $X \subseteq U$
is called \emph{$R$-closed} if $R(X) = X$, and an element $x \in U$
is \emph{$R$-closed}, if its singleton set $\{x\}$ is $R$-closed.
The set of $R$-closed points is denoted by $S$.
\end{definition}
Let us assume that $(U,E)$ is an indiscernibility space.
The set of lower approximations $\mathcal{B}_{E}(U)=\{ X_{E} \mid X \subseteq U \}$ and
the set of upper approximations $\mathcal{B}^{E}(U)= \{ X^{E} \mid X \subseteq U \}$
coincide, so we denote this set simply by $\mathcal{B}_{E}(U)$. The set $\mathcal{B}_{E}(U)$
is a complete Boolean sublattice of $(\wp(U),\subseteq)$, where $\wp(U)$ denotes the
set of all subsets of $U$. This means that $\mathcal{B}_{E}(U)$ forms a \emph{complete field of sets}.
Complete fields of sets are in one-to-one correspondence with equivalence relations,
meaning that for each complete field of sets $\mathcal{F}$ on $U$, we can can define an equivalence
$E$ such that $B_E(U) = \mathcal{F}$. Note that $S$ and all its subsets belong to
$\mathcal{B}_{E}(U)$, meaning that $\wp(S)$ is a complete sublattice of $\mathcal{B}_{E}(U)$,
and therefore in this sense $S$ can be viewed to consist of \emph{completely defined objects}.
Each object in $S$ can be separated from other points of $U$ by the information
provided by the indiscernibility relation $E$, meaning that
for any $x \in S$ and $X \subseteq U$, $x \in X_E$ if and only if $x \in X^E$.
The {\em rough set} of $X$ is the equivalence class of all $Y\subseteq U$ such that $Y_E=X_E$ and $Y^E=X^E$.
Since each rough set is uniquely determined by the approximation pair, one can represent the rough set of
$X$ as $(X_E, X^E)$ or $(X_E,-X^E)$. We call the former \emph{increasing representation} and the latter
\emph{disjoint representation}. These representations induce the sets
\[
\mathit{IRS}_E(U) = \{(X_E,X^E)\mid X\subseteq U\}
\]
and
\[
\mathit{DRS}_E(U) =\{(X_E,-X^E)\mid X\subseteq U\},
\]
respectively. The set $\mathit{IRS}_E(U)$ can be ordered pointwise
\[(X_E,X^E) \leq (Y_E,Y^E) \iff X_E \subseteq Y_E \text{ and } X^E \subseteq Y^E,\]
and $\mathit{DRS}_E(U)$ is ordered by reversing the order for the second components
of the pairs, that is,
\[(X_E,-X^E) \leq (Y_E,-Y^E) \iff X_E \subseteq Y_E \text{ and } -X^E \supseteq -Y^E.\]
Therefore, $\mathit{IRS}_E(U)$ and $\mathit{DRS}_E(U)$ are order-isomorphic, and they
form completely distributive lattices, thus double Heyting algebras
\cite{Pagliani93,Pagliani97,pagliani2008geometry}.
Every Boolean lattice $B$, where $x'$ denotes the complement of $x \in B$,
is a Heyting algebra such that $x \Rightarrow y = x' \vee y$ for $x,y \in B$.
An element $x \in B$ is dense only if $x' = 0$, that is, $x = 1$. Because it is known that on a
Boolean lattice each lattice-congruence is such that the quotient lattice
is a Boolean lattice, also the congruence $\cong_S$ on $\mathcal{B}_{E}(U)$,
defined by $X \cong_S Y$, if $X \cap S = Y \cap S$, is Boolean when
$\mathcal{B}_{E}(U)$ is interpreted as a Heyting algebra.
In \cite{Pagliani93}, it is shown that disjoint representation of rough sets can be characterized as
\begin{equation}\label{Eq:DRS-construction}
\mathit{DRS}_E(U) = \{(A,B) \in \mathcal{B}_E(U)^2 \mid A\cap B =\emptyset\ \text{ and } A \cup B \cong_{S}U\}.
\end{equation}
Thus, ${DRS}_E(U)$ coincides with the Nelson lattice $N_{\cong_S} (\mathcal{B}_E(U))$.
Since $\mathcal{B}_{E}(U)$ is a Boolean lattice, ${\mathbb N}_{\cong_S} (\mathcal{B}_E(U))$
is a semi-simple Nelson algebra (cf. \cite{Pagliani96}).
As a consequence, we obtain the well-known facts that
rough sets defined by equivalences determine also regular double Stone algebras and three-valued
{\L}ukasiewicz algebras.
In the literature also several representation theorems related to
rough sets induced by equivalences can be found. For instance, in \cite{Pagliani97} it was proved that for any
{\em finite} three-valued \L ukasiewicz algebra $\mathbb A$, there is an indiscernibility space $(U,E)$ such that
$\mathbb{N}_{\cong_S}(\mathcal{B}_{E}(U))$ is isomorphic to $\mathbb{A}$.
This result was extended by L.~Iturrioz \cite{Iturrioz99} by showing that any three-valued
{\L}ukasiewicz algebra is a subalgebra of $\mathit{IRS}_E(U)$ for some indiscernibility space $(U,E)$.
Finally, it has been proved by J.~J\"{a}rvinen and S.~Radeleczki \cite{JarRad} that for any semi-simple
Nelson algebra $\mathbb{A}$ with an underlying algebraic lattice there exists an indiscernibility space
$(U, E)$ such that $\mathbb{A}$ is isomorphic to $\mathbb{N}_{\cong_S}(\mathcal{B}_{E}(U))$.
From the latter representation theorem one obtains that every semi-simple Nelson algebra,
regular double Stone algebra and three-valued {\L}ukasiewicz algebra that are
defined on algebraic lattices can be obtained from an indiscernibility space $(U,E)$ by
using Sendlewski's construction \eqref{Eq:Nelson-construction}.
Note that an \emph{algebraic lattice} is a complete lattice $L$ such that each element $x$ of $L$ is the join of a set of
compact elements of $L$, and thus finite lattices are trivially algebraic.
On $\mathcal{B}_{E}(U)$, the Glivenko congruence is simply
the identity relation. This means that the effective lattice determined by the indiscernibility
space $(U,E)$ is just the collection of all ordered pairs of disjoint elements of $\mathcal{B}_{E}(U)$
such that $X \cup Y = U$. Hence, $\mathbb{N}_{\Gamma}(\mathcal{B}_{E}(U))$ equals the set
of pairs $\{(X,-X) \mid X \in \mathcal{B}_{E}(U)\}$, which trivially is an isomorphic copy of
$\mathcal{B}_{E}(U)$ itself. Therefore, in the case of equivalence relations, effective
lattices do not appear that interesting, because on $\mathbb{N}_{\Gamma}(\mathcal{B}_{E}(U))$
for any formula $\alpha$ we would have ${\bf T}(\llbracket\alpha\rrbracket)=\llbracket\alpha\rrbracket$,
where $\llbracket\alpha\rrbracket$ is the ordered pair interpreting $\alpha$.
Then a question arises: {\em Is there
any generalization which makes it possible to go ahead and develop a full correspondence
between rough set systems and effective lattices?}
\section{Effective lattices and quasiorders}
\label{Sec:RS_quasiorders}
For a quasiorder $R$ on $U$, as in case of equivalences, we may define
the \emph{increasing representation} and the \emph{disjoint representation},
respectively, by
\[
\mathit{IRS}_R(U) = \{(X_R,X^R)\mid X\subseteq U\}
\]
and
\[
\mathit{DRS}_R(U) =\{(X_R,-X^R)\mid X\subseteq U\},
\]
and these sets can be identified by the bijection $(X_R,X^R) \mapsto (X_R,-X^R)$.
As shown by J.~J\"{a}rvinen, S.~Radeleczki, and L.~Veres \cite{JRV09}, $\mathit{IRS}_{R}(U)$ is a complete
sublattice of $\wp(U) \times\wp(U)$ ordered by the pointwise set-inclusion relation, meaning that $\mathit{IRS}_R(U)$
is an algebraic completely distributive lattice such that
\[
\bigwedge\left\{ ( X_{R}, X^{R} ) \mid X\in\mathcal{H} \right\} =
\Big ( \bigcap_{X \in\mathcal{H}} X_{R}, \bigcap_{X\in\mathcal{H}} X^{R}
\Big )
\]
and
\[
\bigvee\left\{ ( X_{R}, X^{R} ) \mid X\in\mathcal{H} \right\} =
\Big ( \bigcup_{X\in\mathcal{H}} X_{R}, \bigcup_{X\in\mathcal{H}} X^{R} \Big )
\]
for all $\mathcal{H} \subseteq \mathit{IRS}_{R}(U)$.
Since $\mathit{IRS}_{R}(U)$ is completely distributive, it is a double Heyting algebra.
J\"{a}rvinen and Radeleczki proved in \cite{JarRad} that the bounded distributive
lattice $\mathit{IRS}_R(U)$ equipped with the operation $\sim$ defined by ${\sim}(X_R,X^R) = (-X^R,-X_R)$
forms a Kleene algebra satisfying the interpolation property of \cite{Cign86}.
It is proved by R.~Cignoli \cite{Cign86} that any Kleene algebra that satisfies this
interpolation property and is such that for each pair $a$ and $b$ of its elements,
the relative pseudocomplement $a \Rightarrow {\sim} a \vee b$ exists, forms a Nelson algebra
in which the operation $\to$ is determined by the rule $a \to b := a \Rightarrow {\sim} a \vee b$.
Therefore, as noted in \cite{JarRad}, for any quasiorder $R$ on $U$, $\mathit{IRS}_R(U)$
together with the operation $\sim$ forms a Nelson algebra. This Nelson algebra
is denoted by $\mathbb{IRS}_R(U)$, and its operations
will be described explicitly in Corollary~\ref{Cor:IRS_Sendlewski}.
In \cite{JarRad}, it is also proved that if $\mathbb{A}$ is a Nelson algebra defined on an algebraic lattice,
then there exists a set $U$ and a quasiorder $R$ on $U$ such that $\mathbb{A}$ and the Nelson algebra
$\mathbb{IRS}_{R}(U)$ are isomorphic. From this, we can deduce the following
completeness result, with the finite model property, for CLSN, whose axiomatisation can be found in
\cite{Rasiowa74,Vaka77}, for example.
\begin{theorem} \label{Thm:CompleteCLSN}
Let $\alpha$ be a formula of CLSN. Then the following conditions are equivalent:
\begin{enumerate}[\rm (a)]
\item $\alpha$ is a theorem,
\item $\alpha$ is valid in every finite rough set-based Nelson algebra determined by a quasiorder.
\end{enumerate}
\end{theorem}
\begin{proof} Suppose that $\alpha$ is a theorem. Then, in the view of the completeness theorem
proved in H.~Rasiowa \cite{Rasiowa74}, $\alpha$ is valid in every Nelson algebra. Particularly,
$\alpha$ is valid in every finite rough set-based Nelson algebra determined by a quasiorder.
Conversely, assume $\alpha$ is not a theorem. Then, there exists a finite
Nelson algebra $\mathbb{A}$ such that $\alpha$ is not valid in that algebra,
that is, its valuation $\llbracket \alpha \rrbracket_\mathbb{A}$ is different from $1_\mathbb{A}$
(see \cite{Vaka77}, for example). Because $\mathbb{A}$ is finite, it is defined on an algebraic lattice. Therefore, there
exists a finite set $U$ and a quasiorder $R$ on $U$ such that $\mathbb{A}$ and the finite rough set-based Nelson algebra
$\mathbb{IRS}_R(U)$ determined by $R$ are isomorphic. We denote here $\mathbb{IRS}_R(U)$ simply by $\mathbb{IRS}$.
Let us denote by $f$ the isomorphism between these
Nelson algebras. The valuation on $\mathbb{IRS}$ can be now defined as
$\llbracket \beta \rrbracket_{\mathbb{IRS}} = f(\llbracket \beta \rrbracket_\mathbb{A})$
for all formulas $\beta$, so $\llbracket \alpha \rrbracket_{\mathbb{IRS}}
= f(\llbracket \alpha \rrbracket_\mathbb{A}) \neq f(1_\mathbb{A}) = 1_{\mathbb{IRS}}$, that is,
$\alpha$ is not valid in $\mathbb{IRS}$.
\end{proof}
An element $x$ of a complete lattice $L$ is \emph{completely join-irreducible}
if for every subset $X$ of $L$, $x = \bigvee X$ implies that $x \in X$.
The set of completely join-irreducible elements of $L$ is
denoted by $\mathcal{J}$. It is shown in \cite{JRV09}
that the set of completely join-irreducible elements of $\mathit{IRS}_{R}(U)$ is
\[
\mathcal{J} = \{(\emptyset,\{x\}^{R}) \mid x \in U \text{ and } |R(x)| \geq2 \} \cup\{( R(x),
R(x)^{R}) \mid x \in U\},
\]
and that every element can be represented as a join of elements in
$\mathcal{J}$.
In a Nelson algebra $\mathbb{A}$ defined on an algebraic lattice $(A,\leq)$,
each element is the join of completely join-irreducible elements $\mathcal{J}$.
We may define for any $j \in \mathcal{J}$ the element
$g(j) = \bigwedge \{ x \in A \mid x \nleq {\sim} j \} \ (\in \mathcal{J})$,
and it is shown in \cite{JarRad} that the map $g \colon \mathcal{J} \to \mathcal{J}$
satisfies the following conditions for all $x,y \in \mathcal{J}$:
\begin{enumerate}[({J}1)]
\item if $x \leq y$, then $g(y) \leq g(x)$,
\item $g(g(x)) = x$,
\item $x \leq g(x)$ or $g(x) \leq x$,
\item if $x,y \leq g(x),g(y)$, there exists $z\in \mathcal{J}$ such that $x,y \leq z \leq g(x),g(y)$.
\end{enumerate}
Conversely, the mapping $g$ determines the strong negation $\sim$ on $\mathbb{A}$ by the equation
${\sim} x = \bigvee \{ j \in \mathcal{J} \mid g(j) \nleq x\}$.
Let $R$ be a quasiorder on $U$ and let $\mathcal{J}$ be the set of the completely join-irreducible
elements of $\mathit{IRS}_R(U)$. In \cite{JarRad} it is proved
that the following equations hold:
\begin{align*}
\{ j \in \mathcal{J} \mid j < g(j) \} & = \{(\emptyset,\{x\}^R) \mid x \in U \text{ and } |R(x)| \geq 2 \}, \\
\{ j \in \mathcal{J} \mid j = g(j) \} & = \{( \{x\}, \{x\}^R) \mid \ x \in S \, \}, \\
\{ j \in \mathcal{J} \mid j > g(j) \} & = \{( R(x), R(x)^R) \mid x \in U \text{ and } |R(x)| \geq 2 \}.
\end{align*}
Therefore, elements in $S$ have a special role, because they are such that the completely
join-irreducible elements corresponding to them are the fixed points of $g$.
It should be also noted that in case of an equivalence $E$, the partially ordered set of completely
join-irreducible elements of $\mathit{IRS}_E(U)$ consists of disjoint chains of 1 and 2 elements.
Differently from an equivalence $E$ that defines one complete field of sets $\mathcal{B}_E(U)$,
a quasiorder $R$ determines two \emph{complete rings of sets}, or equivalently, two
\emph{Alexandrov topologies}
\[
\mathcal{T}_{R}(U) = \{ X_{R} \mid X \subseteq U\} \text{ \ and \ }
\mathcal{T}^{R}(U) = \{ X^{R} \mid X \subseteq U\},
\]
that is, $\mathcal{T}_{R}(U)$ and $\mathcal{T}^{R}(U)$ are closed under arbitrary
unions and intersections. Note that $\mathcal{T}_{R}(U)$ and $\mathcal{T}^{R}(U)$
are intended to be the open sets of these topologies, respectively. The Alexandrov
topologies $\mathcal{T}_{R}(U)$ and $\mathcal{T}^{R}(U)$ are dual in the sense that
$X \in\mathcal{T}_{R}(U)$ if and only if $-X \in\mathcal{T}^{R}(U)$.
The topology $\mathcal{T}_{R}(U)$ consists of all $R$-closed sets.
Therefore, for any $X \subseteq U$, the $R$-neighbourhood
$R(X)$ of $X$ is the smallest open set containing $X$. This actually means that
$\mathcal{T}_{R}(U) = \{ R(X) \mid X \subseteq U\}$, which also implies
$R(X)_R = R(X)$ and $R(X_R) = X_R$ for any $X \subseteq U$.
In addition, for all $X \in \mathcal{T}_R(U)$, $X = \bigcup_{x \in X} R(x)$
(see \cite{Jarv07}, for instance).
Because the points of $S$ are $R$-closed, $\wp(S)$ is a complete sublattice of
$\mathcal{T}_{R}(U)$, as in case of equivalences. Again, each object in $S$ can
be separated from other points of $U$ by the information provided by the
relation $R$, because each element of $S$ is $R$-related only to itself, and
for any $x \in S$ and $X \subseteq U$, $x \in X_R$ if and only if $x \in X^R$.
Therefore, also in case of quasiorders, $S$ can be viewed as the set of
\emph{completely defined objects}.
In the Alexandrov topology $\mathcal{T}_{R}(U)$, the map $(\cdot)^{R} \colon\wp(U) \to\wp(U)$ is the
closure operator and $(\cdot)_{R} \colon\wp(U) \to\wp(U)$ is the interior operator. Because
$\mathcal{T}_{R}(U)$ is closed under arbitrary unions and intersections, it is a
completely distributive lattice and a double Heyting algebra. In particular,
for any $X,Y \in \mathcal{T}_{R}(U)$, the relative pseudocomplement
$X \Rightarrow Y$ equals $(-X \cup Y)_{R}$. Thus, the structure
\begin{equation}\label{Eq:Heyting}
(\mathcal{T}_{R}(U),\cup,\cap,\Rightarrow,\emptyset,U)
\end{equation}
forms a Heyting algebra in which $X^* = (-X)_R = -X^R$. Hence, an
element $X \in \mathcal{T}_{R}(U)$ is dense if and only if $X^R = U$, meaning that $X$ is
\emph{cofinal} in $U$, that is, for any $x \in U$, there exists $y \in X$ such that
$x \, R \, y$ (see \cite{Stone68} for more details on cofinal sets).
Because each increasing rough set pair belongs to $\mathcal{T}_{R}(U) \times\mathcal{T}^{R}(U)$,
our next aim is to present a characterization of $\mathit{IRS}_{R}(U)$
in terms of pairs belonging to $\mathcal{T}_{R}(U) \times\mathcal{T}^{R}(U)$.
The next proposition appeared for the first time in \cite{JarPagRad11}, and also an analogous result is
presented independently in \cite{NagaUma11}.
\begin{proposition}\label{Prop:characterization}
Let $R$ be a quasiorder on $U$. Then,
\[
\mathit{IRS}_R(U) = \{(A,B) \in \mathcal{T}_{R}(U) \times\mathcal{T}^{R}(U)
\mid A \subseteq B \text{ and } S \subseteq A \cup-B \}.
\]
\end{proposition}
\begin{proof} ($\subseteq$): Suppose $(X_R,X^R) \in \mathit{IRS}_R(U)$. Then, $X_R \subseteq X^R$.
Suppose now $x \in S$ and $x \notin X_R \cup -X^R$. Then $x \in X^R \setminus X_R$,
which is clearly impossible because $x \in S$. Thus,
$S \subseteq X_R \cup -X^R$.
\medskip\noindent%
($\supseteq$): Assume that $(A,B) \in \mathcal{T}_R(U) \times \mathcal{T}^R(U)$,
$A \subseteq B$, and $S \subseteq A \cup -B$,
This means that $B \setminus A \subseteq -S$, that is, for any $x \in B \setminus A$, we have $|R(x)| \geq 2$.
For any $x \in B \setminus A$,
the pair $(\emptyset, \{x\}^R)$ is a rough set, because $|R(x)| \geq 2$. Additionally, for any
$x \in A$, the pair $( R(x), R(x)^R)$ is also a rough set.
Let us consider the rough set
\begin{align*}
(C,D) &= \bigvee \{ (\emptyset,\{x\}^R) \mid x \in B \setminus A \} \vee \bigvee \{( R(x), R(x)^R) \mid x \in A\} \\
& = \Big ( \bigcup_{x \in A} R(x), \bigcup \{ \{x\}^R \mid x \in B \setminus A \}
\cup \bigcup \{ R(x)^R \mid x \in A \} \Big).
\end{align*}
Clearly, $C = \bigcup_{x \in A} R(x) = A$ since $A \in \mathcal{T}_R(U)$. In turn,
\[
D = \bigcup \{ \{x\}^R \mid x \in B \setminus A \} \cup \bigcup \{ R(x)^R \mid x \in A \}.
\]
Now, in view of the fact that $A$ is $R$-closed,
and that $B$ is an upper approximation, hence $B^R=B$, we have:
\begin{enumerate}[(i)]
\item If $x \in A$, then $R(x) \subseteq A$, so $R(x)^R \subseteq A^R \subseteq B^R = B$.
\item If $x \in B \setminus A$, then $\{x\}^R \subseteq B^R = B$.
\end{enumerate}
Therefore, $D \subseteq B$. Conversely, let $y \in B$. Then, $y \in A$ or $y \in B \setminus A$.
\begin{enumerate}[(i)]
\item If $y \in A$, then $y \in R(y)^R \subseteq D$.
\item If $y \in B \setminus A$, then $y \in \{y\}^R$ and
$y \in \bigcup \{ \{x\}^R \mid x \in B \setminus A \} \subseteq D$.
\end{enumerate}
Thus, we have shown $B = D$. Therefore, $(A,B) = (C,D)$ is a rough set, that is, $(A,B) \in \mathit{IRS}_R(U)$.
\end{proof}
As in the case of equivalences, it is obvious by Proposition~\ref{Prop:characterization} that
\begin{equation}\label{Eq:R-disjoint}
\mathit{DRS}_R(U) = \{ (A,B) \in \mathcal{T}_{R}(U) \times\mathcal{T}_{R}(U)
\mid A \cap B = \emptyset\text{ and } S \subseteq A \cup B \}.
\end{equation}
We can now connect rough sets defined by quasiorders to Sendlewski's construction
\eqref{Eq:Nelson-construction}. First, we need the following lemma.
\begin{lemma} \label{Lem:S=Dense}
The set $S$ is included in all dense elements of $\mathcal{T}_{R}(U)$.
\end{lemma}
\begin{proof} Suppose that the set $X \in \mathcal{T}_{R}(U)$ is dense, that is,
$X^R = U$. If $S \nsubseteq X$, then there exists $x \in S$ such that $x \notin X$.
Because $x \in X^R$ and $R(x) = \{x\}$, we have $x \in X$, a contradiction.
\end{proof}
Because $\mathcal{T}_{R}(U)$ forms a Heyting algebra \eqref{Eq:Heyting},
by the previous lemma and Lemma~\ref{Lem:BooleanCongruencePseudo},
$\cong_S$ is a Boolean congruence on the Heyting algebra $ \mathcal{T}_{R}(U)$.
It is easy to see that for all $X \in \mathcal{T}_{R}(U)$,
$X \cong_S U$ if and only if $S \subseteq X$. Therefore, by \eqref{Eq:R-disjoint}, we may write
\[
\mathit{DRS}_R(U) = N_{\cong_S}(\mathcal{T}_{R}(U)).
\]
By applying Sendlewski's construction \eqref{Eq:Nelson-construction}, we may now write the
following proposition.
\begin{proposition}\label{Prop:DRS_Sendlewski}
If $R$ is a quasiorder on $U$, then $\mathit{DRS}_R(U)$ forms a Nelson algebra
with the operations:
\begin{align*}
(X_R,-X^R) \vee (Y_R,-Y^R) & = (X_R \cup Y_R, -X^R \cap -Y^R): \\
(X_R,-X^R) \wedge (Y_R,-Y^R) & = (X_R \cap Y_R, -X^R \cup -Y^R); \\
{\sim}(X_R,-X^R) &= (-X^R, X_R);\\
(X_R,-X^R) \to (Y_R,-Y^R) &= ( (-X_R \cup Y_R)_R, X_R \cap -Y^R).
\end{align*}
\end{proposition}
We denote this Nelson algebra on $\mathit{DRS}_R(U)$ by $\mathbb{DRS}_R(U)$.
Because the map $(X_R,X^R) \mapsto (X_R,-X^R)$ is an order-isomorphism between complete lattices
$\mathit{IRS}_R(U)$ and $\mathit{DRS}_R(U)$, we may write the following corollary
describing the operations in the Nelson algebra $\mathbb{IRS}_R(U)$
\begin{corollary}\label{Cor:IRS_Sendlewski}
For a quasiorder $R$ on $U$, the operations of $\mathbb{IRS}_R(U)$ are:
\begin{align*}
(X_R,X^R) \vee (Y_R,Y^R) & = (X_R \cup Y_R, X^R \cup Y^R); \\
(X_R,X^R) \wedge (Y_R,Y^R) & = (X_R \cap Y_R, X^R \cap Y^R); \\
{\sim}(X_R,X^R) &= (-X^R, -X_R);\\
(X_R,X^R) \to (Y_R,Y^R) &= ( (-X_R \cup Y_R)_R, -X_R \cup Y^R).
\end{align*}
\end{corollary}
We are now ready to consider effective lattices determined by rough sets.
Recall that for any Heyting algebra $H$, the corresponding effective lattice is
$\mathbb{N}_\Gamma (H)$, where $\Gamma$ is the Glivenko congruence on $H$. In Section~\ref{Sec:Preliminary}
we showed that every element $a \in H$ which is below all dense elements $D$ determines a Boolean
congruence $\cong_a$. If such an $a$ is itself a dense element, it must
be the least dense element, that is, $a = \bigwedge D \in D$.
Therefore, in this case $\Gamma$ is equal both to $\cong_a$ and to the congruence $\theta(F_a)$
of the principal filter $F_a = \{x \in H \mid a \leq x\} = D$. \label{Page:Dense}
It should be noted that Heyting algebras do not necessarily have a least dense element. For instance, the
Heyting algebra defined on the real interval $[0,1] = \{ x \in \mathbb{R} \mid 0 \leq x \leq 1\}$
is such, because each non-zero element of the algebra is dense. On the contrary, in case of finite Heyting
algebras, there exists always the least dense element $\bigwedge D$, and thus $D$ is
the principal filter of $\bigwedge D$.
By definition, $\mathbb{DRS}_R(U)$ is an effective lattice whenever $\cong_S$
is the least Boolean congruence on the Heyting algebra $\mathcal{T}_R(U)$. Because the
Nelson algebras $\mathbb{DRS}_R(U)$ and $\mathbb{IRS}_R(U)$ are essentially the same,
we say that also $\mathbb{IRS}_R(U)$ is an effective lattice, if $\cong_S$ is the
Glivenko congruence of $\mathcal{T}_R(U)$.
Our next lemma characterizes the conditions under which rough set-based Nelson algebras determined
by quasiordes are effective lattices.
\begin{proposition} \label{Prop:Dense}
Let $R$ be a quasiorder on the set $U$ and let $S$ be the set of $R$-closed elements.
The following statements are equivalent:
\begin{enumerate}[\rm (a)]
\item $S$ is cofinal in $U$,
\item $S$ is a dense element of the Heyting algebra $\mathcal{T}_{R}(U)$,
\item $S$ is the least dense element of the Heyting algebra $\mathcal{T}_{R}(U)$,
\item $\cong_S$ is the least Boolean congruence $\Gamma$ on the Heyting algebra $\mathcal{T}_{R}(U)$,
\item $\mathbb{IRS}_R(U)$ and $\mathbb{DRS}_R(U)$ are effective lattices.
\end{enumerate}
\end{proposition}
\begin{proof} Claims (a) and (b) are equivalent by definition, and the same holds
between (d) and (e). Trivially (c) implies (b), and by Lemma~\ref{Lem:S=Dense},
$S$ is included in each dense element of $\mathcal{T}_R(U)$, hence (b) implies (c).
If\, $\cong_S$ equals $\Gamma$, then $S \cong_S U$ implies
$S \, \Gamma \, U$ and $X^* = U^* = \emptyset$, that is, $X$ is dense and
(d)$\Rightarrow$(b). If $S$ is the least dense set, then, as discussed earlier,
${\cong_S}$ equals $\Gamma$ and (c)$\Rightarrow$(d).
\end{proof}
If ${\cong_S}$ is the Glivenko congruence, then for all elements $A,B \in \mathcal{T}_R(U)$,
$A \cup B \cong_S U \iff A \cup B \text{ is dense } \iff (A \cup B)^R =
A^R \cup B^R = U$. Therefore, we can write the following corollary
characterizing the elements of $\mathbb{DRS}_R(U)$ and $\mathbb{IRS}_R(U)$
in the case they are effective lattices.
\begin{corollary} Let $R$ be a quasiorder on $U$ and assume that $S$ is dense.
Then, the following equations hold:
\begin{enumerate}[\rm (a)]
\item $\mathit{DRS}_R(U) = \{ (A,B) \in \mathcal{T}_{R}(U) \times\mathcal{T}_{R}(U)
\mid A \cap B = \emptyset\text{ and } A^R \cup B^R = U \}$;
\item $\mathit{IRS}_R(U) = \{(A,B) \in \mathcal{T}_{R}(U) \times\mathcal{T}^{R}(U)
\mid A \subseteq B \text{ and } B_R \setminus A^R = \emptyset \}$.
\end{enumerate}
\end{corollary}
Next, we consider shortly the case that $R$ is a partial order.
The well-known \emph{Hausdorff maximal principle} states that
in any partially ordered set, there exists a maximal chain.
\begin{corollary} \label{Cor:BoundenChain}
If $(U,\leq)$ is a partially ordered set such that any maximal chain is
bounded from above, then $\mathbb{IRS}_\leq(U)$ and $\mathbb{DRS}_\leq(U)$
are effective lattices.
\end{corollary}
\begin{proof}
Let $(U,\leq)$ be a partially ordered set and $x \in U$. Let us consider the partially
ordered set $(\{y \mid x \leq y\}, \leq_x)$, where $\leq_x$ is the order $\leq$ restricted to
$\{y \mid x \leq y\}$. Then, by the Hausdorff maximal principle, $\{y \mid x \leq y\}$
has a maximal chain $C$. By our assumption, the chain $C$ is bounded from above by
some element $m$. Because $m$ is a maximal element, it is in $S$ and $x\leq m$.
This implies $S^{\leq} = U$, that is, $S$ is dense. Therefore, $\cong_S$ is the least
Boolean congruence $\Gamma$, and $\mathbb{IRS}_{\leq}(U)$ and $\mathbb{DRS}_{\leq}(U)$
are effective lattices.
\end{proof}
\begin{remark}
Clearly, if $U$ is finite, then Corollary~\ref{Cor:BoundenChain} holds, that is,
all rough set-based Nelson algebras determined by finite partially ordered sets are effective lattices.
\end{remark}
\begin{example}
Let $(U,\leq)$ be a partially ordered set with least element $0$ such that
$U \setminus \{0\}$ is an antichain, that is, all elements in $U \setminus \{0\}$ are incomparable.
Clearly, the set of $\leq$-closed elements is $S = U \setminus \{0\}$ and $S^\leq = U$,
meaning that $S$ is cofinal and, by Lemma~\ref{Prop:Dense}, $\cong_S$ equals $\Gamma$ and
$S$ is the least dense set. Additionally, $\mathcal{T}_\leq(U) = \wp(S) \cup \{U\}$, and
the congruence classes of $\cong_S$ are of the form $\{ X, X \cup \{0\} \}$, where $X \in \mathcal{T}_\leq(U)$.
It is also easy to observe that
\[ \textit{IRS}_\leq(U) = \{ (X,X\cup \{0\}) \mid X \subseteq S \, \}
\cup \{ (\emptyset,\emptyset), (U,U) \}.
\]
So, in this case $\textit{IRS}_\leq(U)$ is order-isomorphic to $\wp(S)$ added with
a top element $\mathbf{1}$ corresponding to $(U,U)$ and a bottom element $\mathbf{0}$
corresponding to $(\emptyset,\emptyset)$, that is, $\textit{IRS}_\leq(U)$ is order-isomorphic
to $\mathbf{0} \oplus \wp(S) \oplus \mathbf{1}$. Note also that
in $\textit{IRS}_\leq(U)$, the pair $(\emptyset,\{0\})$ is the least dense set, which
means that in $\textit{IRS}_\leq(U)$, all elements except $(\emptyset,\emptyset)$ are
dense.
\end{example}
We end this section by a presenting a necessary and sufficient condition under which
Nelson algebras are isomorphic to effective lattices of rough sets determined by quasiorders.
\begin{theorem}\label{Thm:Representation}
Let $\mathbb{A}$ be a Nelson algebra. Then, there exists a set $U$ and a quasiorder
$R$ on $U$ such that $\mathbb{IRS}_R(U)$ is an effective lattice and
$\mathbb{A} \cong \mathbb{IRS}_R(U)$ if and only if $\mathbb{A}$
is defined on an algebraic lattice in which each completely
join-irreducible element is comparable with at least one completely
join-irreducible element which is a fixed point of $g$.
\end{theorem}
\begin{proof}
For a Nelson algebra $\mathbb{A}$, there exists a set $U$ and a quasiorder $R$ on $U$ such that
$\mathbb{A} \cong \mathbb{IRS}_R(U)$ if and only if $\mathbb{A}$ is defined on an
algebraic lattice (for details, see \cite{JarRad}). Additionally, by Lemma~\ref{Prop:Dense},
we know that $\mathbb{IRS}_R(U)$ is an effective lattice if and only
if $S$ is cofinal in $U$.
Assume that there exists a set $U$ and a quasiorder $R$ on $U$ such that
$\mathbb{IRS}_R(U)$ is an effective lattice and $\mathbb{A} \cong \mathbb{IRS}_R(U)$.
Let $\varphi$ be the isomorphism in question. This implies that
$\mathbb{A}$ is defined on an algebraic lattice $A$ and each
element of $A$ can be represented as a join of completely irreducible
elements of $\mathcal{J}$. Note that $\varphi$ preserves also the map $g$, that is, $\varphi(g(j))
= g(\varphi(j))$ for all $j \in \mathcal{J}$; see \cite{JarRad}.
Let $j \in \mathcal{J}$. If $j$ is a fixed point of $g$, then
$\varphi(j) = ( \{y\}, \{y\}^R)$ for some $y \in S$, and
we have nothing to prove since $j$ is trivially comparable with itself.
If $j$ is not a fixed point of $g$, then for $\varphi(j)$ we have two possibilities:
\begin{enumerate}[(i)]
\item $\varphi(j) = (\emptyset,\{x\}^R)$ for some $x \in U$ such that $|R(x)| \geq 2$, or
\item $\varphi(j) = ( R(x), R(x)^R) )$ for some $x \in U$ such that $|R(x)| \geq 2$.
\end{enumerate}
Without a loss of generality we may assume that $j < g(j)$. This means that
there exists $x \in U \setminus S$ such that $\varphi(j) =(\emptyset,\{x\}^R)$
and $\varphi(g(j)) = ( R(x), R(x)^R) )$. Because $S$ is cofinal,
there exists $y \in S$ such that $x \, R \, y$.
Let $k$ be the element of $A$ such that $\varphi(k) = (\{y\},\{y\}^R)$.
Obviously, $k \in \mathcal{J}$ and $g(k) = k$. Because $x \, R \, y$, we have
$\varphi(j) = (\emptyset,\{x\}^R) \leq (\{y\},\{y\}^R) = \varphi(k)$, and
hence $j \leq k$; note that this also means $k \leq g(j)$.
Conversely, assume $\mathbb{A}$ is defined on an algebraic lattice whose each completely
join-irreducible element is comparable with at least one completely
join-irreducible element which is a fixed point of $g$. Because $\mathbb{A}$ is defined
on an algebraic lattice, there exists a set $U$ and a quasiorder $R$ on $U$ such that
$\mathbb{A} \cong \mathbb{IRS}_R(U)$ as Nelson algebras. Let us again denote this
isomorphism by $\varphi$.
We show that $S$ is cofinal, which by Proposition~\ref{Prop:Dense} means that
$\mathbb{IRS}_R(U)$ is an effective lattice. Let $x \in U$. If $x \in S$,
then, by reflexivity, $x \, R \, x \in S$. If $x \notin S$, then $|(R(x)| \geq 2$, and
there are two elements $j_1 <j_2$ in $\mathcal{J}$ such that $g(j_1) = j_2$,
$\varphi(j_1) = (\emptyset,\{x\}^R)$, and $\varphi(j_2) = ( R(x), R(x)^R)$.
Because $j_1$ (or equivalently $j_2$) is comparable with at least one completely
join-irreducible element $k$ which is a fixed point of $g$, this
necessarily means that $j_1 < k < j_2$. Let $\varphi(k) = (\{y\},\{y\}^R)$.
It follows that $y \in S$, and now $\varphi(j_1) = (\emptyset,\{x\}^R)
\leq (\{y\},\{y\}^R) = \varphi(k)$ gives $x \in \{x\}^R \subseteq \{y\}^R$, that is,
$x \, R \, y$. Thus, $S$ is cofinal.
\end{proof}
\section{Concluding remarks} \label{Sec:Conclusions}
The results of this paper have been suggested by the very philosophy of rough set theory.
In an indiscernibility space $(U,E)$ two elements $x,y\in U$ are indiscernible if $E(y)=E(x)=\{x,y, ...\}$.
Indeed in rough set theory, the equivalence relation $E$ of an indiscernibility space $(U, E)$ is
induced by attributes values. Therefore, if $x$ and $y$ are indiscernible, then there is no
property which is able to distinguish $y$ from $x$. But if $E(x)=\{x\}$, then we are given a set of
attributes which are able to single out $x$ from the rest of the domain. In other terms,
$x$ is uniquely determined by the set of attributes. In a sense, about $x$ we have complete information.
It is not surprise, therefore, that on the union $S$ of all the singleton equivalence classes,
Boolean logic applies. That is, $S$ is a Boolean subuniverse within a three-valued universe. The fact that
$S$ is Boolean is expressed in two ways: by saying that $X_E\cup -X^E\supseteq S$ or, equivalently, that
$X_E\cap S=X^E\cap S$, meaning that any subset $X$ is \emph{exactly defined} with respect to $S$.
When we move to quasiorders, we face the same situation. A quasiorder $R$ expresses either a preference relation
(see for instance \cite{GrecoMatarazzoSlowinski}) or an information refinement. The latter notion is embedded
in that of a {\em specialisation preorder} which characterizes Alexandrov topologies (see \cite{Vickers89}).
Thus, if $x \in S$, then $x$ is a most preferred element or a piece of information which is
maximally refined. So, $R$-closed elements decide every formula. Otherwise stated, on $S$ excluded middle is valid.
This is the logico-philosophical link between rough set systems induced by a quasiorder $R$ and effective lattices.
If $\mathbb{IRS}_R(U)$ and $\mathbb{DRS}_R(U)$ are effective lattices, then $\cong_S$ is the Glivenko congruence
and the set $S$ is cofinal, which means that for any $x \in U$, there exist an $R$-closed element $y$ such that
$x \, R \, y$. Indeed, this is the characteristic which distinguishes Miglioli's Kripke models for $E_0$
from Thomason's Kripke models for CLSN.
At the very beginning of the paper, we have seen the reasons why Miglioli's research group introduced the
operator {\bf T} and why this operator requires that for any information state $s$
there is a complete state $s'$ which extends $s$. Here ``complete'' means that
for any atomic formula $p$, either $s'$ forces $p$ or $s'$ forces the strong negation of $p$.
Those reasons were connected to problems in program synthesis and specification.
However we can find a similar issue in other fields.
For instance, S.~Akama \cite{Akama87} considers an equivalent system endowed with modal
operators to face the "frame problem" in knowledge bases.
In that paper, intuitively, it is required that any search for
{\em complete information} must be successfully accomplished.
On the basis of our previous discussion it is easy to understand why
Akama satisfies this request by postulating that each maximal chain
of possible worlds ends with a greatest element
fulfilling a Boolean forcing. Hence the set of these elements is dense.
Another interesting example is given by situation theory \cite{BarwisePerry83}.
Given a situation $s$ and a state of affairs $\sigma$, $s\models\sigma$ means that situation $s$
supports $\sigma$ (or makes $\sigma$ factual). In Situation Theory some assumptions are accepted as
"natural", for any $\sigma$:
\begin{enumerate}[\rm (i)]
\item Some situation will make $\sigma$ or its dual factual:\\
$\exists s(s\models\sigma \text{ or } s\models {\sim} \sigma$).
\item No situation will make both $\sigma$ and its dual factual:\\
$\neg\exists s(s\models\sigma \text{ and } s\models {\sim} \sigma$).
\item\label{incomplete} Some situation will leave the relevant issue unsolved
(it is admitted that for some $s$, $s\nvDash\sigma$ and $s\nvdash {\sim} \sigma$).
\end{enumerate}
In contrast with assumption \eqref{incomplete}, the following, on the contrary, is a controversial thesis:
\begin{quote}
\it There is a largest total situation which resolves all issues.
\end{quote}
It is immediate to see that this thesis is connected to the scenario depicted by logic $E_0$.
\providecommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}\thinspace}
\providecommand{\MR}{\relax\ifhmode\unskip\space\fi MR }
\providecommand{\MRhref}[2]{%
\href{http://www.ams.org/mathscinet-getitem?mr=#1}{#2}
}
\providecommand{\href}[2]{#2}
|
{
"timestamp": "2012-03-12T01:02:09",
"yymm": "1203",
"arxiv_id": "1203.2136",
"language": "en",
"url": "https://arxiv.org/abs/1203.2136"
}
|
\section{Introduction}
The Kepler mission is revolutionizing our understanding of exoplanets \citep[]{borucki}, including among its many highlights the discovery of three terrestrial exoplanets orbiting the M dwarf Kepler Object of Interest (KOI) 961 \citep[]{muirhead}, and a 2.4 $R_\oplus$ exoplanet in the habitable zone of its host star \citep[]{borucki4}. A list of 312 KOIs was published in \citet[]{borucki2}, derived from Q0-1 Kepler time-series data. This was soon followed by a second list of 1235 KOIs in \citet[]{borucki3} derived from Q1-Q5 Kepler data. The first two KOI releases relied on stellar parameters from the Kepler Input Catalog \citep[KIC,][]{brown}. \citet[]{batalha3} (hereafter B13) announced 2321 candidate transiting exoplanet KOIs orbiting 1783 host stars from an improved pipeline analysis of the Q1-Q6 Kepler data, and provided updated stellar parameters. A list of $\sim$2700 candidates from analysis of the Q1-Q8 time-series was recently released by the Kepler team, but does not include new estimates of stellar parameters \citep[]{burke13}. This large ensemble of exoplanet candidates, including many multi-exoplanet systems -- `multis' for short -- beckons for the ensemble analysis of exoplanetary system architectures \citep[]{fabrycky,howard,morton,plavchan,figueira,youdin}.
The ensemble analysis of KOIs and the determination of the frequency of Earth-sized planets ($\eta_\oplus$) in habitable zone orbits relies on the accuracy of the estimated stellar host parameters such as mass, radius and temperature \citep[]{batalha,batalha2,pin,traub}. Recent studies suggest that the KIC mischaracterizes some objects, in particular the surface gravity for stars with effective temperatures less than $\sim$4500 K. For example, \citet[]{mann,ciardi} demonstrate that M ``dwarf'' KIC targets brighter than a Kepler magnitude of 14th are most likely giants. Additionally, \citet[]{muirhead2} report that M dwarf KOI host stars have over-estimated radii. \citet[]{dressing} confirm this trend and find that the typical M dwarf radius in the Kepler sample is over-estimated by $\sim$30\%. B13 acknowledges these limitations of the new and old stellar parameters, given that the observational methods used to characterize Kepler host stars were optimized for FGK stars. In contrast, \citet[]{everett} find that KOI stellar radii are instead under-estimated for 87\% of their 268 sample of faint (Kepler magnitude $>$14) FGK host stars, including factors of more than $\sim$35\% for one-quarter of their sample.
In this paper we use the transit durations of KOIs for two interdependent purposes: one, as a probe of the inferred eccentricity distribution of the candidate exoplanets in comparison to exoplanets discovered with the radial velocity (RV) technique, and two, as a diagnostic of the accuracy of the estimated stellar parameters. The analytic framework for these two investigations is already laid out in \citet[][]{ford} (hereafter F08) and \citet[]{burke}. A comprehensive analysis is already carried out with the \citet[]{borucki3} KOI list by \citet[]{moorhead}. This work presents a follow-up to the analysis in \citet[]{moorhead} for the more recent KOI list in B13. Additionally, \citet[]{dawson} carry out a thorough modeling when the impact parameter can reliably be estimated for Jovian-sized exoplanet KOIs with high S/N transits.
In ${\S}$2, we present the samples of RV exoplanets and KOI candidate exoplanets used for this investigation. In ${\S}$3, we present the parameter we calculate for the anomalous transit duration. In ${\S}$4 we present how we generate a synthetic population of transiting exoplanets from the population of RV-discovered exoplanets. In ${\S}$5, we present our results from a comparison of the synthetic population of transiting exoplanets with the observed KOIs and subsets thereof. In ${\S}$6 we discuss the contribution of detection biases and false positives to our results, we assess the accuracy of KOI stellar parameters from our results, and we comment of the dependence of eccentricity of exoplanet radius and multiplicity. In ${\S}$7 we present our conclusions.
\section {Samples}
\subsection{KOIs}
We use the second and third tabulation of KOI candidate exoplanet and stellar parameters from \citet[]{borucki3} and B13 respectively; the latter is available at the NASA Exoplanet Archive \citep[]{akeson}. We make subsets of terrestrial, Neptune-like, and Jovian KOIs from B13 at exoplanet radii of $R_{pl}<$2, 2$<R_{pl}<$6, and $R_{pl}>$6 $R_\oplus$ earth radii respectively. We also separately consider singles and multis: $\sim$10-35\% of single exoplanet KOIs may be false-positives \citep[]{morton,sophie,colon}, whereas the false positive rate for multis is thought to be very small \citep[]{lissauer}. By comparing these latter two samples, we can assess the role of false positives in our analysis. We treat multiple KOIs orbiting the same host star as independent statistical tests.
The KOIs primarily orbit FGK host stars. We do not filter KOIs by their host mass, temperature, or radius, since we are interested in identifying any systematic trends as a function of stellar parameters. We restrict ourselves to KOIs in B13 with transits detected with S/N$>$10 where previous work has shown the KOI list to be reasonably complete \citep[]{howard,Christiansen}. We also restrict ourselves to KOIs in B13 with planet orbital periods $P<160$ days, corresponding to 3 or more observed transits. Our final sample comprises 2205 of the 2321 KOIs in B13. We consider the impact of KOI detection biases on our analysis in ${\S}$4 \& 6.
\subsection{RV-Discovered Exoplanets}
Next, we use the period and eccentricity values of 164 published RV-discovered exoplanets as listed at the NASA Exoplanet Archive as of March 7th, 2012 \citep[]{akeson}. We exclude RV exoplanets with periods greater than 160 days and M$_p sin$i$>$30 M$_{J}$. The mass constraint for the definition of an ``exoplanet'' is the same loose criteria as adopted by the NASA Exoplanet Archive and other major exoplanet archives \citep[]{wright}. Nearly 500 exoplanets have been discovered with the RV method, but the majority possess orbital periods greater than 160 days and/or no constraints on the eccentricity. The mean eccentricity for this 164 exoplanet sample is 0.18. Whereas the distribution of periastron angles for RV exoplanets is uniform random on a sphere, the distribution of periastron angles for transiting exoplanets will more strongly favor angles perpendicular to the plane of the sky for eccentric planets due to the increased probability of transit \citep[F08,][]{burke}. Thus, we do not use the measured periastron angles for RV planets as in \citet[]{kane} since they have an incorrect distribution for comparison to the transiting planets.
We assume that this RV sample represents a viable control population to compare to KOIs, without reverting to analytic or model planet formation synthesis eccentricity distributions. There are several limitations to this assumption. The stellar mass distribution for RV discovered exoplanets is centrally peaked around one solar mass, which is qualitatively similar but not identical to the stellar mass distribution of KOIs; the RV distribution of stellar masses is relatively broader. Also, only $\sim$4\% of KOIs have radii $>$1 $R_J$, and only 5\% of our RV-discovered sample has masses $<$1 $M_J$. Thus, while we are comparing two sets of exoplanets, the two populations likely possess different bulk densities, composition, and dynamical histories that we do not account for. Next, we ignore differences due to Galactic location -- Kepler systems are at larger distances from the Earth than RV-discovered exoplanets, and neither sample is magnitude nor volume-complete. We discuss additional detection biases inherent in this RV sample in ${\S}$4.
\section{Observed Transit Duration Anomalies}
Equal to $\tau_0$ in F08 in the limit $R_{pl}/R_*\rightarrow 0$, we define the transit duration anomaly dimensionless parameter $\alpha$:
\begin{equation}
\alpha = \frac{a T \pi}{(R_* + R_p)P} = \left(\frac{\pi G M_*}{4 P}\right)^{1/3} \frac{T}{(R_* + R_p)}
\end{equation}
\noindent where $\alpha$ is the ratio of the observed transit duration to the expected transit duration for a circular edge-on orbit, and can be computed from the KOI table parameters in B13. $R_*$ and $R_p$ are the stellar and exoplanet radii, $P$ is the orbital period, $T$ is the duration of the transit, $a$ is the semi-major axis of the exoplanet, $G$ is Newton's gravitational constant, and $M_*$ is the stellar mass. The semi-major axis of the exoplanet $a$ is calculated from the stellar mass $M_*^{1/3}$ and $P$ via Kepler's 3$^{rd}$ law. $P$ and $T$ are quantities measured from the time-series, whereas $R_*$ and $M_*$ (and consequently $a$) are estimated from ancillary observations and models such as observed colors, spectroscopy and theoretical isochrones \citep[]{batalha,batalha2,batalha3,muirhead2}. Thus, the measurement of $\alpha$ depends strongly on the determinations of $T$ and $R_*$, and weakly on $P$, $M_*$, and $R_p$ (since $R_p\ll R_*$). The error budget for $\alpha$ is in turn dominated by uncertainties in the stellar radius. This is because errors in the observed KOI transit durations $T$ are typically $<1\%$, the errors in the observed period $P$ are also generally known to better than 1 part in 10$^3$ (B13), $R_p\ll R_*$, and due to the weak dependence on stellar mass in the semi-major axis $a$. Values for $\alpha$ are tabulated in Table 1 for all KOIs from B13.
From Equation 1, we can see that the observed transit duration in hours subtracted from the expected transit duration in hours for a circular edge-on orbit has a dependence on orbital period -- transit durations are longer at longer orbital periods, all else being the same:
\begin{equation}
\Delta t \mbox{(hrs)} = T_{obs} - T_{circ} = (\alpha -1) \frac{P(R_*+R_p)}{\pi a}\propto \sqrt{a}
\end{equation}
\noindent The upper envelop of $\Delta t$ values in Figure 1 of \citet[]{kane} follows the expected $\sqrt{a}$ dependence as a consequence of Kepler's 3$^{rd}$ Law, and thus $\Delta t$ is a not a valid quantity to investigate eccentricity distributions over a range of orbital periods. Our expression in Equation 1 is dimensionless with no unbalanced dependence on orbital period save for the intrinsic eccentricity distribution.
For an eccentric, non-edge-on orbit, the square of the transit duration anomaly can be written in a similar fashion to the square of Equation 1 in F08 as:
\begin{equation}
\alpha^2 \approx \left(\frac{d_t}{a}\right)^2 \left(\frac{1}{1-e^2}\right)\left(1-\frac{b^2}{(1+R_p/R_*)^2}\right)
\end{equation}
\noindent where $e$ is the eccentricity, $b$ is the impact parameter, and $d_t$ is the exoplanet-star separation during transit as defined and derived in F08. Assuming as in F08 that $a/R_* \gg 1$, $a/R_p \gg 1$, and $T/P \ll 1$, $d_t$ can be assumed to be approximately constant during transit (hence the approximation in Equation 3).
From Equation 3, we can identify potential eccentric exoplanets with values of $\alpha\ll1$ transiting near periastron or $\alpha\gg1$ transiting near apastron. Since $b$ is degenerate with eccentricity for a given KOI, the KOIs with $\alpha\ll1$ may alternatively be grazing transits. We tabulate these candidates in Table 2. KOIs with $\alpha\gg1$ are low-probability occurrences (${\S}$5.1), and we exclude KOIs with KOIs with $\alpha\gg1$ from Table 2 unless $\log [Fe]/[H]>-0.11$ and $A_V < 0.33$ mag (${\S}$6.3).
The four KOIs with the smallest values of $\alpha$ -- KOIs 338.01, 338.02, 977.01, and 1054.01 -- are reported to orbit giant stars with log $g<2.5$ with periods of 1.4--7 days interior to the estimated stellar radii, and may instead be associated with photospheric activity or a blend. Visual inspection of the time-series for KOIs 977 and 1054 support this hypothesis. The KOI 338 system has two candidate exoplanets, and the light curve exhibits clear transit-like dips that are short w/r/t to the transit period, and is specifically mentioned in B13 and \citet[]{fabrycky}. We exclude these four KOIs from Table 2.
\section{Generating A Synthetic Population of Eccentric Transiting Exoplanets}
Equations 1 and 3 do not permit a straightforward computation of a KOI eccentricity from its observed transit duration, since the impact parameter $b$ and periastron angle $\omega$ are generally unconstrained and degenerate with the eccentricity for a given KOI exoplanet. High S/N Jovian KOIs are an exception since their light curves with resolved ingress and egress slopes and durations are amenable to model fitting to accurately constrain $b$ and $\omega$ as in \citet[][]{dawson}. \citet[]{burke} and F08 demonstrate that in the absence of constraints on $b$ and $\omega$, the ensemble distribution of $\alpha$ values can instead be used as a proxy for the ensemble eccentricity distribution. Thus, we convert the known distribution of eccentricities of confirmed RV exoplanets into a simulated distribution of transiting exoplanet $\alpha$ values to compare to the KOI $\alpha$ distribution. We assume that every RV exoplanet ``transits'' with a distribution of $b$ and $\omega$ values that are properly weighted by the transit probability. The angle of periastron cannot be assumed to be uniform random on a sphere as is true for RV discovered exoplanets and as is assumed in \citet[]{kane}. Accurate probability distributions for $b$, $e$ and $\omega$ for transiting planets are given in \citet[][Equations 14--16; Figure 4]{burke}, and we numerically integrate these probabilities over $b$ and $\omega$ using the prescription in \citet[]{burke} combined with the known RV eccentricities.
In principle, we can make a straightforward comparison of the simulated distribution of $\alpha$ values from RV-discovered exoplanets to the empirical distribution of $\alpha$ values for KOIs. If they match, as assessed by a 1-D two sample Kolmogorov-Smirnov (K-S) test, then one can assert that the underlying eccentricity distributions are the same. However, before we proceed we must first attempt to account for uncertainties and detection biases between the two methods. First, the stellar parameter uncertainties dominate the error budget for $\alpha$ in Equation 1 (${\S}$3). To assess the impact of stellar parameter uncertainties on KOI $\alpha$ values, we include 20\% and 50\% Gaussian random errors in our RV simulated distribution of $\alpha$ values, which could represent realistic 20 or 50\% errors in the stellar radius, or alternatively $>$70\% errors in the stellar mass given the weaker one-third dependence of $\alpha$ on stellar mass.
Second, we correct for the difference in detection sensitivity (survey completeness) as a function of orbital period for the transit and RV methods, because the eccentricity distribution of RV-discovered exoplanets has a relatively strong dependence on orbital period due to tidal circularization \citep[]{butler,rasio}. For the RV method, the detection probability falls off as the semi-major axis $a^{-1/2}$, whereas for the transit method the detection probability falls more rapidly as $a^{-1}$. For example, 80\% of our sample of RV-discovered exoplanets have orbital periods $P<$ 73 days, whereas 80\% of KOIs have $P<$ 29 days, a difference of more than a factor of two. We plot the cumulative distribution functions (CDFs) as a function of orbital period $P$ for the RV-discovered exoplanets, KOIs and various subsets thereof in Figure 1.
Instead of generating standard CDFs as a function of $\alpha$, where each increment in the CDF value is 1/N and N is the sample size (N=164 from ${\S}$2.2), we generate weighted CDFs where each increment is $w_i$ and $\sum_{i=1}^N w_i = 1$. We calculate the $w_i$ values to yield a weighted CDF($P$) for the simulated exoplanets that is identical to the standard unweighted CDF($P$) for the KOIs or sub-sets thereof. In practice, this means that shorter orbital period simulated exoplanets and their associated $\alpha$ values are typically given more weight than the longer period simulated exoplanets. The particular weights for a given simulation depend on the particular period distribution of the KOIs and subsets thereof to which the simulation is being compared. For example, KOIs with $R_{pl}<$2 $R_\oplus$ generally have shorter orbital periods than KOIs with $R_{pl}>$6 $R_\oplus$ due to the relative detection incompleteness for the smaller radius KOIs at longer orbital periods. By weighting the $\alpha$ values in this manner, we account for this relative incompleteness as well. Our approach is equivalent to generating a standard CDF from a Monte Carlo simulation where planets are drawn non-randomly from the RV-discovered set to yield the desired CDF($P$), while fixing the number of simulated planets at $N$ to retain the applicability of the K-S test in ${\S}$5. The net result is a simulated and weighted CDF($\alpha$) to compare to the unweighted CDF($\alpha$) for KOIs or a subset thereof, where the underlying period distributions are identical between the two samples.
Third, the RV technique is also biased against detecting highly eccentric planets for a sparse cadence of observations. Correcting for this bias would increase the simulated mean eccentricity. Since the correction would strengthen the results presented in ${\S}$5, it can thus be safely ignored. We also do not correct for the $sin$i degeneracy in exoplanet mass, nor the bias of both transit and RV techniques to preferentially detect higher mass/radius planets, nor the lack of sensitivity of the RV technique to detect terrestrial planets. The dependence of eccentricity on exoplanet mass is not observationally well constrained for RV-discovered planets without reverting to theoretical formation models, and thus it is difficult to correct for a priori.
Finally, we do not correct for the qualitatively similar but distinct distributions as a function of stellar mass between the KOIs and RV-discovered exoplanets; there is no published observational literature presenting a dependence of exoplanet eccentricity on stellar mass, and no trend is apparent to us in our data from the NASA Exoplanet Archive \citep[]{akeson}. To summarize, we assume that the sample of eccentricities of RV-discovered exoplanets serves as an appropriate proxy for a `control sample' to compare to the KOIs, and in particular the KOIs with $R_{pl}>$6 $R_\oplus$, after simulating $\alpha$ values from these eccentricities, accounting for differences in the period distributions, and accounting for stellar parameter uncertainties. We defer the discussion of additional detection biases for the KOIs until ${\S}$6.1.
\section{Results}
In this section, we first present the CDFs of simulated $\alpha$ values from the RV-discovered exoplanets in ${\S}$5.1, which are then used as a benchmark to compare to the empirical CDFs for KOIs in ${\S}$5.2.
\subsection{Simulated Distributions}
In Figure 2, we plot the CDFs of simulated $\alpha$ values from the RV-discovered exoplanet sample. We tabulate the distribution medians $\tilde{\alpha}$ and two-sided standard deviations $\pm\sigma$ in Table 3. Our simulated $\alpha$ distribution -- with an average eccentricity of 0.18, no added random errors and no period distribution correcting weights -- has a median, mean and standard deviation in $\alpha$ of $(\tilde{\alpha},\mu,\sigma)=(0.73,0.69,0.25)$. For comparison, the model in F08 that best matches known RV-discovered exoplanets is a Rayleigh eccentricity distribution with an average eccentricity of $\sim$0.25 and with a Rayleigh parameter R(0.3) \citep[]{juric}. This F08 model produces a distribution in $\alpha$ with $(\mu,\sigma)\sim(0.74,0.29)$ that is comparable to the implementation of our simulations.
As previously mentioned in ${\S}$3, a large value of $\alpha\gg1$ (assuming accurate stellar parameters) requires a high eccentricity and transit near apastron, which is a low probability occurrence. This explains the lack of simulated $\alpha$ values $>$1.2 when no additional random errors are included \citep[see also][Figure 4]{burke}. For the addition of 20\% random errors in measuring $\alpha$, and when the simulated period distribution is weighted to match that of all KOIs in B13, we find 2.4\% of $\alpha$ values $>$1.2 and 0\% with $\alpha>$1.5. For 50\% errors, we find 10.4\% of $\alpha$ values $>$1.2, and 2.4\% with $\alpha>$1.5. Thus, our simulations show that significant random errors $>$50\% in measuring $\alpha$ can produce a false overabundance ($\sim$10\%) of eccentric planets improbably transiting near apastron. However, the median $\tilde{\alpha}$ values of the synthetic distributions are relatively insensitive to the amplitude of the random errors (Table 3). We next compare the simulated distributions to the empirical KOI $\alpha$ distributions.
\subsection{Empirical Distributions}
In Figure 2 we also plot the CDFs of $\alpha$ values for all KOIs from B13 and for KOIs with $R_{pl}<2 R_\oplus$, $2 R_\oplus< R_{pl} <6 R_\oplus$, and $R_{pl}>6 R_\oplus$, for single KOI systems, and multiple KOI systems (`multis') also all from B13. The $\alpha$ distribution for all KOIs is significantly different from the corresponding RV $\alpha$ distribution in ${\S}$5.1, in disagreement with \citet[]{kane}. This is particularly evident when comparing the distribution medians, which can deviate by up to 25\% between the empirical and simulated distributions. This result suggests that there is a systematic under-(over-)estimate of stellar radii (mass), since the transit durations and periods are known to better than 1\% and 0.1\% respectively (Equation 1, ${\S}$3). In particular, the CDFs for KOIs with $R_{pl}<6 R_\oplus$ appear to be unphysical. A population of exoplanets on circular orbits only, and with uniform random impact parameters, produces an $\alpha$ distribution with $\tilde{\alpha}=0.866$. As the average eccentricity is increased from zero, $\tilde{\alpha}$ decreases. Thus, any $\tilde{\alpha}>0.866$ is unphysical, implying the KOIs typically have ``more circular than circular'' orbits with $\tilde{\alpha}>0.9$. To ascertain the validity of these claims, we discuss in ${\S}$6 the relevance of detection biases and false positives in the list of KOIs.
Formally, we evaluate the K-S test statistic between all distributions, and the K-S test probabilities are listed in Table 4. The simulated $\alpha$ distribution from RV-discovered exoplanets, with no additional random errors, has a probability of being drawn from the same parent population as the KOIs in B13 with $R_{pl}>6 R_\oplus$ ($2 R_\oplus<R_{pl}<6 R_\oplus$,$R_{pl}<2 R_\oplus$, all) of $4.48\times10^{-4}$ ($3.91\times10^{-12}$,$3.90\times10^{-23}$,$3.31\times10^{-16}$). Thus, KOIs with $R_{pl}>6 R_\oplus$ most closely resemble the simulated $\alpha$ distribution, but are still statistically distinct at the $>3\:\sigma$ level. For KOIs with $R_{pl}<6 R_\oplus$, these results hold at high statistical significance for additional errors in the stellar radii. For KOIs with $R_{pl}>6 R_\oplus$, the K-S test probability increases to $8.63\times10^{-4}$ for 20\% errors and to $\sim$2.4\% for 50\% errors. From Figure 2, it is apparent that much of the disagreement for KOIs with $R_{pl}>6 R_\oplus$ is for large values of $\alpha$. However, the disagreement in CDFs is noticeable at all values of $\alpha$ for $R_{pl}<6 R_\oplus$.
Next, there is a higher occurrence rate for all KOIs with $\alpha>$1.5 than expected for both a Rayleigh eccentricity distribution (F08) and for our simulated distributions with no added random errors. This is true for all KOI radii, whether in a single or multiple system, although the percentages are smaller for the multis. For all KOIs, we find 15\% have $\alpha>$1.2, and 4\% with $\alpha>$1.5. These observed percentages are larger than but comparable to the $\alpha$ distribution simulated from RV-discovered exoplanets with 50\% Gaussian random errors in measuring $\alpha$ as presented in ${\S}$5.1. For the single KOI systems, the percentages are 16.3\% and 5.7\% for $\alpha>$1.2 and 1.5 respectively. For multis, the corresponding percentages are smaller at 11.6\% and 1.6\% respectively, a difference also noted in \citet[]{moorhead}.
This excess of KOIs with $\alpha>$1.5 is also noted in \citet[]{moorhead} and \citet[]{wang} for the \citet[]{borucki3} tabulation of KOIs. We confirm that this excess is still present for KOIs in B13. If we assume that our RV and KOI samples are drawn from the same population of exoplanets, these KOIs must be possess significant systematic errors in the stellar parameters of $\sim$20--50\%. In ${\S}$6.3, we identify how the stellar radius (mass) could be under-(over-)estimated by factors exceeding 1.5 (1.5$^3$) to correct some of these $\alpha$-values. A blanket rejection of these KOIs is not recommended since that could inadvertently exclude a rare and genuine highly eccentric exoplanet transiting near apastron.
\section{Discussion}
\subsection{Detection Biases}
In ${\S}$4, we correct for differences in the orbital period completeness of KOIs and subsets thereof through the use of weighted CDFs. The effect of this correction is minimal as shown in Figure 2 and Tables 3 and 4. We now discuss two additional detection biases that are specific to KOIs, and we evaluate the impact of these biases on our results in ${\S}$5. We expect a deficiency of transiting KOIs at high impact parameters due to two effects -- the lower S/N for shorter transit durations, and the manual removal of `V'-shaped grazing transits because of the increased probability of false-positives from stellar eclipsing binaries \citep[B13,][]{blender,Christiansen}. It is beyond the scope of this work to obtain an accurate measure of the incompleteness due to these two biases. Such an effort requires a detailed modeling of the Kepler pipeline recovery of injected synthetic transits, which fortunately is undertaken in \citet[]{Christiansen}. \citet[]{Christiansen} find there is no significant bias in the Kepler pipeline recovery of individual transit events (known as Kepler Threshold Crossing Events, or TCEs) as a function of transit duration. However, the \citet[]{Christiansen} result does not account for any human biases introduced in the promotion to a KOI from the Kepler TCEs.
We instead use the impact parameter $b_{circ}$ from B13 to provide a reasonable estimate of incompleteness at short transit durations. $b_{circ}$ is calculated assuming a circular orbit and is determined from the ingress and egress times after accounting for limb-darkening \citep[]{seager}. We plot the distribution of KOIs a function of $b_{circ}$ in Figure 3. The frequency of KOIs with $b_{circ}>0.9$ is $\sim$45\% smaller than KOIs with $0.8<b_{circ}<0.9$, particularly for KOIs with $R_{pl}<6 R_\oplus$. No such decrease would be expected for a uniform random impact parameter for circular orbits, and the number of KOIs with $b_{circ}>0.9$ should in fact slightly increase for a population of eccentric orbits. Thus, there is a noticeable and measurable incompleteness of KOIs with $b_{circ}>0.9$. For reference, a transit duration for a circular orbit at an impact parameter of $b$=(0.5,0.6, 0.7, 0.8, 0.9) is (0.866,0.8,0.71,0.6, 0.44) times as long as an edge on ($b$=0) transit.
For an approximate lower bound, we estimate that 50\% (240) of KOIs are missing with $b_{circ}>0.9$, primarily with $R_{pl}<6 R_\oplus$, by linearly extrapolating from the number of KOIs with $0.7<b_{circ}<0.8$ and $0.8<b_{circ}<0.9$. In other words, we assume for the lower bound that the list of KOIs is complete for $b_{circ}<0.9$ and the trend in $b_{circ}$ frequency is linear from 0.7--1.0. Similarly, for an approximate upper bound, we estimate that 40\% (471) of KOIs are missing with $b_{circ}>0.8$ by linearly extrapolating from the number of KOIs with $0.6<b_{circ}<0.7$ and $0.7<b_{circ}<0.8$. In other words, we assume for the upper bound that the list of KOIs is complete for $b_{circ}<0.8$ only, and the trend in $b_{circ}$ frequency is linear from 0.6--1.0). This fraction of missing high impact parameter KOIs corresponds to $\sim$10--20\% of all KOIs in B13. The missing KOIs would have values for $\alpha$ less than $\sim$0.3 for KOIs with $R_{pl}>6 R_\oplus$ and less than $\sim$0.1 for KOIs with $R_{pl}<2 R_\oplus$ (Equations 1,3). Adding this population of missing KOIs to our sample would effectively compress the existing KOI CDFs plotted in Figure 2 from the vertical range of (0,1) to $\sim$(0.09,1) or (0.167,1) for 10 and 20\% incompleteness respectively. Thus, we can estimate a corrected median value of $\tilde{\alpha}_c$=0.876--0.906 for all KOIs. This scenario remains inconsistent at a statistically significant level with a population of RV exoplanets with a mean eccentricity $\bar{e}=$0.18 and $\tilde{\alpha}$=0.73 from ${\S}$4. It also remains marginally inconsistent in an unphysical fashion with a population of exoplanets on only circular orbits ($\bar{\alpha}=0.866$). Further, the distribution of $b_{circ}$ values in Figure 3 implies an average KOI eccentricity of $e>$0 and thus $\tilde{\alpha}_c<0.866$, since $\bar{b}_{circ}>=0.7$ even before correcting for the incompleteness at $b_{circ}>0.8$, and since an average value of $\bar{b}_{circ}=$0.5 would be expected for a population of exoplanets with only circular orbits. Thus, the estimated incompleteness of KOIs at short transit durations is unable to explain the differences between the simulated and empirical $\alpha$ distributions, in particular accounting for the KOI median values of $\tilde{\alpha}$.
Tackling the question of incompleteness at short transit durations from a different direction, we can ask -- if the median $\tilde{\alpha}$ for KOIs should be equal to the median $\tilde{\alpha}$ for the simulated exoplanets such that they are drawn from the same parent population, what fraction of KOIs with $b_{circ}>0.9$ (0.8) would be missing to account for the observed discrepancies in $\tilde{\alpha}$? For all KOIs, the percentage would be 58.4\% (1288 KOIs), corresponding to $\sim$4.7 (1.9) times the existing number of KOIs with $b_{circ}>0.9$ (0.8), or approximately $\sim$5 (2.7) times our estimated incompleteness in the preceding paragraph. Broken down by KOI subsets, for multis, singles, KOIs with $R_{pl}>6 R_\oplus$, $2 R_\oplus<R_{pl}<6 R_\oplus$, and $R_{pl}<2 R_\oplus$, the corresponding percentages are 65,52,17,54, and 75\% times the total number of KOIs, corresponding to factors of 5.2, 4.1, 1.4, 4.4, and 6 times the existing number of KOIs with $b_{circ}>0.9$ respectively. These incompleteness factors would appear to be inconsistent with the observed distribution for $b_{circ}<$0.9 in Figure 3 for all KOI radii. We can conclude that there are instead likely biased errors present in the stellar parameters.
\subsection{False Positives}
False-positives in the B13 KOI list are reported at the $\sim$10-35\% level \citep[]{morton,sophie}. \citet[]{colon} finds no significant correlation in the false positive rate with exoplanet radius and stellar effective temperature, albeit from a limited sample. Fortunately, multiple exoplanet KOI systems are thought to have a very low false-positive rate of a few percent or less \citep[]{lissauer}. Thus, we can compare single exoplanet KOIs to multis to assess the impact of false-positives on the transit duration anomalies $\alpha$. As can be seen in Figure 2 and Tables 3 and 4, the difference in the median $\tilde{\alpha}$ for singles and multis is marginal -- 0.93 vs 0.94 respectively -- both before and after correcting for differences in the orbital period distributions. Thus the false positive rate is unable to account for the errantly high values of $\tilde{\alpha}$ for KOIs when compared to the simulated distributions.
However, we do see a marked difference between singles and multis in the frequency of KOIs with $\alpha>$1.2 and 1.5 as noted in ${\S}$5.2. Taking the ratio of percentages, false positives can account for $\sim$30\% of KOIs with $\alpha>$1.2, and $\sim$70\% of all KOIs with $\alpha>$1.5. Thus a fraction of KOIs with abnormally large $\alpha$ values can be accounted for by false positives, but not all. The remaining $\sim$12\% and $\sim$2\% of all KOIs with $\alpha>$1.2 and 1.5 respectively must still be accounted for. The statuses of the KOIs with $\alpha>$1.2, particularly for single exoplanet KOI systems, need to be confirmed in future work to enable a scientifically valid comparison of the eccentricity distributions of RV and Kepler planets.
\subsection{Accuracy of Stellar Parameters Inferred from Transit Duration}
The significant differences between the $\alpha$ distributions for RV exoplanets and Kepler candidates raises the question about the validity of the KOI host stellar parameters, which we now turn to discuss. From Figure 2 and ${\S}$5, we find that Gaussian random errors in measuring $\alpha$ of $\sim$20--50\% can explain the overabundance of KOIs with $\alpha>$1.2, after accounting for false positives in ${\S}$6.2. However, 20--50\% Gaussian random errors tend to over-predict the frequency of KOIs with $\alpha<$0.4. Additionally, the median $\tilde{\alpha}$ values of KOIs and subsets thereof are up to 25\% larger than their simulated counterparts, with the exception of Jovian KOIs with $R_{pl}>6 R_\oplus$. Our simulations show that the median $\tilde{\alpha}$ value is roughly independent of the simulated Gaussian random uncertainties. Our analysis in ${\S}$6.1 also shows that neither the correction for the orbital period distributions, nor the estimated correction for the incompleteness at high impact parameters, can fully explain the discrepancies in the median values of $\tilde{\alpha}$. Thus, rather than Gaussian random errors, systematic (meaning one-sided, non-Gaussian) over-estimates in the measurement of $\alpha$ of up to $\sim$25\% on average, and up to 50\% for individual KOIs, are likely required to reconcile the observed KOI $\alpha$ distributions with their simulated counterparts.
Given the relatively small uncertainties in the KOI orbital periods and transit durations of $<$0.1\% and $<$1\% respectively in B13, the errors in measuring $\alpha$ must either be due to errors in the stellar radii or the one-third power of the stellar masses. A $>$20\% under-estimate in the cube root of the stellar mass corresponds to a $>$70\% error in the stellar mass. This would seem to be less plausible than an equivalent 20\% error in stellar radius, since stellar mass errors of $>70\%$ would require correspondingly large spectral type errors. Thus, we conclude that systematic under-estimates of $\sim$20--50\% in the estimated stellar radii are the most likely explanation for the systematic over-estimates in the measurement of $\alpha$. This conclusion is independently confirmed with spectroscopy of faint KOI host stars in \citet[]{everett}, who also find that the KOI stellar radii are systematically under-estimated by up to $\sim$30\%. Thus, some KOIs identified as main sequence stars are likely to be more distant sub-giants.
Can we find additional indicators that there are systematic errors in the stellar radii that can explain our results? In Figures 3--5 we plot the dimensionless $\alpha$ parameter as a function of stellar mass, radius and effective temperature, for both the \citet[]{borucki2} and B13 lists of KOIs, and for singles and multis from B13. We bin the $\alpha$ values for each stellar parameter, with bins of 250 K, 0.1 $M_\odot$, and 0.1 $R_\odot$. Our results are insensitive to adjustments in the bin width. We derive median $\tilde{\alpha}$ values and quartile ranges for each bin, which are over-plotted in Figures 3--5 and show distinct trends as a function of stellar mass and radius, but not stellar effective temperature. To quantify these trends, we perform a linear regression fit to the median values, excluding bins with fewer than 4 KOIs. The linear coefficients and the standard errors from the fits are listed in Table 5. We identify a statistically significant trend for increasing stellar radius (mass) of a -0.27 (-0.28) change in $\tilde{\alpha}$ per $R_\odot$ ($M_\odot$) at the $\sim$5-$\sigma$ (3-$\sigma$) level for both the B13 and \citet[]{borucki2} KOIs, as well as for both single and multi KOIs from B13 albeit at a lower statistical significance.
The trends in $\alpha$ as a function of stellar mass and radius imply one of two scenarios. First, there could be a systematic error in these two parameters for the ensemble of KOIs. Relative to the median $\tilde{\alpha}$ value for 1.0 $R_\odot$ ($M_\odot$) KOIs, the ensemble radii (or masses) would be under- (over-)estimated by an additional $\sim$15\% ($\sim$50\%) at 0.5 $R_\odot$ ($M_\odot$), and the ensemble radii (or masses) would be over- (under-)estimated by $\sim$15\% ($\sim$50\%) at 1.5 $R_\odot$ ($M_\odot$). As previously discussed, it is less plausible that the ensemble stellar masses in B13 and the KIC are in error by $\sim$50\% at the low- and high- mass ends, leaving systematic ensemble errors in the stellar radii as the more plausible explanation for the $\tilde{\alpha}$ trends. Additionally, the trend with stellar radius has a higher formal statistical significance.
The first scenario relies on the assumption that the eccentricity distribution of exoplanets is intrinsically independent of the stellar spectral type. Alternatively, the systematic trend in $\alpha$ could instead be due to a real change in eccentricity distributions as a function of spectral type. A $\sim$15\% change in $\tilde{\alpha}$ between 0.5 and 1.0 $M_\odot$ (or 1.0 and 1.5 $M_\odot$) would correspond to a $>$0.2 change in the average ensemble eccentricity. For example, the trend in $\tilde{\alpha}$ could imply that the average eccentricity is $\tilde{e}\sim$(0,0.25, 0.6) for (0.5,1,1.5) $M_\odot$ host stars respectively as inferred from Figure 9 in F08. However, such a large eccentricity dependence on stellar mass is not reported in the literature for the eccentricities of RV-discovered exoplanets. We again conclude that the trends in the ensemble $\tilde{\alpha}$ values for KOIs as a function of stellar mass and radius are likely primarily due to systematic errors in stellar radii. This result contradicts recent work done for M dwarf KOIs by \citet[]{dressing}, and we are suggesting instead that M dwarf radii for KOI host stars are typically under-estimated rather than over-estimated. The spectroscopic work of \citet[]{everett} is consistent with our result.
Finally, the over-abundance of KOIs with $\alpha>$1.5 identified in ${\S}$5 is independent of stellar mass, radius and temperature, and is only partially explained by false-positives. We compare KOI $\alpha$ values against every property calculated in the KOI tables in B13. We find that KOIs with $\alpha>1.2$ are preferentially found around low metallicity KOIs with $\log [Fe]/[H]<$-0.11 for all planet radii, and for $A_V > 0.33$ mag for $R_{pl}>6 R_\oplus$ as shown in Figure 7. In other words, there is a lack of KOIs with $\alpha>$1.2 at high metallicities and low $A_V$. While the extinction may be a crude proxy for brightness, there is no a priori reason to expect either of these two trends. Thus we conclude that the over-abundance of KOIs with $\alpha>1.2$ is likely due to errors in the calculated stellar metallicities and possibly extinction for these sources, rather than constituting a genuine class of exoplanets with high eccentricities (improbably) transiting far outside of periastron. These errors in metallicity and possibly extinction in turn can readily produce the under-estimates of the stellar radii for these KOIs. These stellar hosts are likely more distant (sub-)giants with larger stellar radii and larger secondary companions, or alternatively are false-positives. This is consistent with the result in \citet[]{colon} that both false-positives they identify are the faintest targets in their sample.
\subsection{Eccentricity as a Function of Planet Radius and Multiplicity}
KOIs with $R_{pl}>6 R_\oplus$ have a statistically significant smaller median $\alpha$ compared to KOIs with $R_{pl}<6 R_\oplus$ (Table 3, Figure 2). The difference is less distinguishable but still present when comparing KOIs with $2 R_\oplus<R_{pl}<6 R_\oplus$ and $R_{pl}<2 R_\oplus$. Similarly, KOIs in multiple systems also have a slightly larger median $\alpha$ value compared to single KOIs. All of these differences become slightly more pronounced after we correct the period distributions of each of these subsets to all match the period distribution of KOIs via a weighted CDF($\alpha$) (${\S}$4), as we show in Figure 8. These results would appear to imply that multis are on more circular orbits than single exoplanets, and that larger exoplanets typically have larger eccentricities. The circularity of the multiple planet systems has been previously reported in \citet[]{fang} and \citet[]{fabrycky} due to the small mutual inclinations of KOIs. Our results are consistent with their more robust conclusions. We must be cautious in interpreting the eccentricity distribution as a function of planet radius as noted in ${\S}$6.1 and 6.2. Stellar parameter systematic errors are still prevalent in the KOI $\alpha$ values, and there is some incompleteness at high impact parameters for the smallest radius KOIs. However, even after accounting for these factors, it will be difficult to reconcile the median $\tilde{\alpha}$ values for KOIs with $R_{pl}>6 R_\oplus$ and $R_{pl}<6 R_\oplus$. It is likely that our result points to a different dynamical origin for exoplanets with $R_{pl}>6 R_\oplus$ and $R_{pl}<6 R_\oplus$, but this needs confirmation from future work with improved stellar parameters, completeness and false positive rejection.
\section{Conclusions}
We have carried out an updated analysis of the transit duration anomalies with the list of Kepler exoplanet candidates in B13. In particular, we looked at the KOI distribution of transit durations compared to what would be expected from the eccentricity distribution of RV-discovered exoplanets as a function of stellar host parameters. We find three related systematic errors in the KOI stellar parameters that preclude a scientifically valid ensemble comparison of the two samples at this time. The systematic biases in stellar parameters impact the inferred distributions of exoplanet properties, including radius and habitability. Thus any determinations of $\eta_\oplus$, the frequency of Earth-sized planets in the habitable zone, etc., must be treated with caution.
First, there is an over-abundance of KOIs with transit durations $>$20\% and $>$50\% longer than expected, implying that most of these KOIs most likely have significantly under-estimated stellar radii. This confirms the result in \citet[]{everett}. We identify that biases in the estimated metallicity and extinction may explain these systems.
Second, we identify that the median transit duration for all spectral types is up to $\sim$25\% too long, a result that is not explainable by sample incompleteness at short transit durations that we estimate to be $\sim$10-20\% of all KOIs. This is again most likely due to the systematic under-estimates of KOI stellar radii.
Third, we identify statistically significant trends in the average transit duration as a function of stellar mass and radius, which again are likely due to errors in stellar radii as a function of spectral type rather than an underlying trend in eccentricity distributions.
We thank the anonymous referee for their constructive input on improving the clarity, presentation and content of this paper. This research has made use of the NASA Exoplanet Archive, which is operated by the California Institute of Technology, under contract with the National Aeronautics and Space Administration under the Exoplanet Exploration Program.
|
{
"timestamp": "2013-11-12T02:14:20",
"yymm": "1203",
"arxiv_id": "1203.1887",
"language": "en",
"url": "https://arxiv.org/abs/1203.1887"
}
|
\section{Introduction and statements of the results}
This work is concerned with the dispersive property of the following class
of Schr\"odinger equations with singular homogeneous electromagnetic
potentials
\begin{equation} \label{prob}
iu_t=\left(-i\nabla+ \dfrac{{\mathbf{A}}\big(\frac{x}{|x|}\big)} {|x|}
\right)^{\!\!2} u+ \dfrac{a\big(\frac{x}{|x|}\big)}{|x|^2}\,u;
\end{equation}
here $u=u(x,t):{\mathbb{R}}^{N+1}\to{\mathbb{C}}$, $N\geq 2$, $a\in
L^{\infty}({\mathbb{S}}^{N-1}, {\mathbb{R}})$, ${\mathbb{S}}^{N-1}$ denotes
the unit $(N-1)$-dimensional sphere, and ${\mathbf{A}}\in C^1({\mathbb{S}}%
^{N-1},{\mathbb{R}}^N)$ satisfies the following transversality condition
\begin{equation} \label{transversality}
{\mathbf{A}}(\theta)\cdot\theta=0 \quad \text{for all }\theta\in {\mathbb{S}}%
^{N-1}.
\end{equation}
We always denote by $r:=|x|$, $\theta=x/|x|$, so that $x=r\theta$.
Equation \eqref{prob} describes the dynamics of a (non relativistic)
particle under the action of a fixed external electromagnetic field $(E,B)$
given by
\begin{equation*}
E(x):=\nabla\left(\dfrac{a\big(\frac{x}{|x|}\big)}{|x|^2}\right)\!, \qquad
B(x)=d\left(\dfrac{{\mathbf{A}}\big(\frac{x}{|x|}\big)} {|x|}\right)\!,
\end{equation*}
where $B$ is the differential of the linear 1-form associated to the vector field $\frac{{\mathbf{A}}(x/|x|)
}{|x|}$. In dimension $N=3$, due to the identification
between 1-forms and 2-forms, the magnetic field $B$ is in fact determined by
the vector field $\mathop{\rm curl}\frac{{\mathbf{A}}(x/|x|)}{|x|}$, in the
sense that
\begin{equation*}
B(x)v=\mathop{\rm curl}\big(\tfrac{{\mathbf{A}}(x/|x|)}{|x|}\big)\times v,
\qquad N=3,
\end{equation*}
for any vector $v\in{\mathbb{R}}^3$, the cross staying for the usual
vectorial product.
Under the transversality condition \eqref{transversality}, the hamiltonian
\begin{equation} \label{eq:hamiltonian}
{\mathcal{L}}_{{\mathbf{A}},a}:= \left(-i\,\nabla+\frac{{\mathbf{A}}\big(%
\frac{x}{|x|}\big)} {|x|}\right)^{\!\!2}+\dfrac{a\big(\frac{x}{|x|}\big)}{%
|x|^2}
\end{equation}
formally acts on functions $f:{\mathbb{R}}^N\to{\mathbb{C}}$ as
\begin{equation*}
{\mathcal{L}}_{{\mathbf{A}},a}f= -\Delta f+\frac{|{\mathbf{A}}\big(\frac{x}{%
|x|}\big)|^2+ a\big(\frac{x}{|x|}\big)-i\dive\nolimits_{{\mathbb{S}}^{N-1}}{%
\mathbf{A}}\big(\frac{x}{|x|}\big)
}{|x|^2}\,f-2i\,\frac{{\mathbf{A}}\big(\frac{x}{|x|}\big)}{|x|}\cdot\nabla
f,
\end{equation*}
where $\dive\nolimits_{{\mathbb{S}}^{N-1}}{\mathbf{A}}$ denotes the
Riemannian divergence of ${\mathbf{A}}$ on the unit sphere ${\mathbb{S}}%
^{N-1}$ endowed with the standard metric.
The free Schr\"odinger equation, i.e. \eqref{prob} with ${\mathbf{A}}\equiv{%
\mathbf{0}}$ and $a\equiv0$, can be somehow considered as the canonical
example of dispersive equation. The unique solution $u\in\mathcal{C}({%
\mathbb{R}};L^2({\mathbb{R}}^N))$ of the Cauchy problem
\begin{equation} \label{eq:schrofree}
\begin{cases}
iu_t=-\Delta u \\
u(x,0)=f(x)\in L^2({\mathbb{R}}^N)%
\end{cases}%
\end{equation}
can be explicitly written as follows:
\begin{equation} \label{eq:free}
u(x,t)=e^{it\Delta}f(x):=\frac{1}{(4\pi it)^{\frac N2}} e^{i\frac{|x|^2}{4t}%
}\int_{{\mathbb{R}}^N} e^{-i\frac{x\cdot y}{2t}} e^{i\frac{|y|^2}{4t}%
}f(y)\,dy.
\end{equation}
This shows that, up to scalings and modulations, $u$ is the Fourier
transform of the initial datum $f$. Formula \eqref{eq:free} contains most
of the relevant informations about the dispersion which arises
along the evolution of the Schr\"odinger flow.
In dimension $N=1$, the evolution of an initial wave packet of the form
\begin{equation*}
f_K(x)=e^{iKx}e^{-\frac{x^2}{2}}, \quad K\in\mathbb{N},
\end{equation*}
gives an important description of the phenomenon.
Inserting $f=f_K$ in \eqref{eq:free} gives in turn the
solution
\begin{equation*}
u_K(x,t)=e^{-\frac{K^2}2}u_0(x-iK,2t); \quad u_0(z,t)=\frac{1}{\sqrt{it+1}}e^{-\frac{z^2}{2(it+1)}}.
\end{equation*}
This shows that each wave travels with a speed which is proportional to
the frequency $K$, and describes both the phenomenon and the terminology of dispersion.
The above property can be quantified in
terms of a priori estimates for solutions to \eqref{eq:schrofree}. A first
consequence of \eqref{eq:free} is the time decay
\begin{equation} \label{eq:decay}
\left\|e^{it\Delta}f(\cdot)\right\|_{L^p}\leq\frac{C}{|t|^{N\left(\frac12-%
\frac1p\right)}} \|f\|_{L^{p^{\prime }}}, \qquad p\geq2; \quad
\frac1p+\frac1{p^{\prime }}=1,
\end{equation}
for some $C=C(p,N)>0$ independent on $t$ and $f$.
In the cases $p=\infty, p=2$, \eqref{eq:decay} can be easily obtained by \eqref{eq:free} and Plancherel; the
rest of the range $2<p<\infty$ follows by Riesz-Thorin interpolation. These inequalities play
a fundamental role in many different fields, including scattering theory,
harmonic analysis and nonlinear analysis. In particular, they standardly imply the
following Strichartz estimates
\begin{equation} \label{eq:stri}
\left\|e^{it\Delta}f\right\|_{L^p_tL^q_x}\leq C\|f\|_{L^2},
\end{equation}
for some $C>0$, where $L^p_tL^q_x:=L^p({%
\mathbb{R}};L^q({\mathbb{R}}^N))$ and the couple $(p,q)$ satisfies the
scaling condition
\begin{equation} \label{eq:admis}
\frac2p+\frac Nq=\frac N2, \qquad p\geq2, \qquad (p,q,N)\neq(2,\infty,2).
\end{equation}
The first result in this style has been obtained by Segal in \cite{Se} for
the wave equation; then it was generalized by Strichartz in \cite{St} in
connection with the Restriction Theorem by Tomas in \cite{T}. Later,
Ginibre and Velo introduced in \cite{GV1} (see also \cite{GV2}) a different
point of view which was extensively used by Yajima in \cite{Y1} to prove a
large amount of inequalities for the linear Schr\"odinger equation.
Finally, Keel and Tao in \cite{KT} completed the picture of estimates %
\eqref{eq:stri}, proving the difficult endpoint estimate $p=2$, via
bilinear techniques, for an abstract propagator verifying a time decay
estimate in the spirit of \eqref{eq:decay}.
Time decay and Strichartz estimates turn out to be a fundamental tool
in the nonlinear applications, and consequently a large literature has been devoted, in the
last years, to obtain them in more general situations, as for example
perturbations of the Schr\"odinger equation with linear lower order terms,
as in \eqref{prob}. In particular, since less regular terms usually arise
in the physically relevant models, a deep effort has been spent in order to
overcome the difficulty deriving from the fact that the Fourier transform
does not fit well with differential operators with rough coefficients. Among these, electromagnetic Schr\"odinger hamiltonians have been object of study in several papers.
An electromagnetic Schr\"odinger equation has the form
\begin{equation} \label{eq:schro2}
iu_t=(-i\nabla+A(x))^2u+V(x)u,
\end{equation}
where $u=u(x,t):{\mathbb{R}}^{N+1}\to{\mathbb{C}}$, $A:{\mathbb{R}}^N\to{%
\mathbb{R}}^N$, and $V:{\mathbb{R}}^N\to{\mathbb{R}}$. Homogeneous
potentials like
\begin{equation} \label{eq:omog}
|A|\sim\frac1{|x|}, \qquad |V|\sim\frac1{|x|^2}
\end{equation}
represent a threshold for the validity of estimates \eqref{eq:decay}
and %
\eqref{eq:stri}, as shown by Goldberg, Vega and Visciglia in
\cite{GVV}, when $A\equiv0$ and later generalized by Fanelli and
Garc\'ia in \cite{FG} for $A\neq0$ (actually, the authors in
\cite{FG,GVV} disprove Strichartz estimates, and a byproduct of this
fact is the failure of the usual time decay estimates). Notice that,
for potentials as the ones in \eqref{eq:omog}%
, equation \eqref{eq:schro2} remains invariant under the usual scaling
$%
u_\lambda(x,t)=u(x/\lambda, t/\lambda^2)$, $\lambda>0$, and this is
why we refer to it as the scaling-critical situation. We also recall
that equation \eqref{eq:schro2} is gauge invariant, namely if $u$ is a
solution to \eqref{eq:schro2}, then $v=e^{i\phi(x)}u$ solves the same
equation, with $A$ replaced by $A+\nabla\phi$ and the same magnetic
field $B$.
In the purely electric case $A\equiv0$, several authors studied the
time dispersion when the potential $V$ is close to the scaling
invariant case \eqref{eq:omog} (see \cite{B, BS, BG, DP, GeVi, GS, JN,
JSS, PSTZ, RS} and the references therein, both for Schr\"odinger
and wave equations, and also the useful survey \cite{S}). A typical
perturbative approach consists in writing the action of the flow
$e^{it(\Delta-V)}$ via spectral theorem, and then reducing matters of
proving the desired estimate to perform a suitable analysis of the
resolvent of $-\Delta+V$, in the Agmon-H\"ormander style. We refer to
the results by Goldberg-Schlag \cite{GS} and Rodnianski-Schlag
\cite{RS}, in which also time dependent potentials are treated, as
standard examples of this technique for Schr\"odinger equations; in
these papers, time decay estimates are obtained under integrability
conditions on $%
V$ which are close to the scaling invariant case \eqref{eq:omog}, but
do not include the critical behavior $1/|x|^2$, due to the
perturbation character of the strategy. Another possible approach
consists in studying the mapping properties of the wave operators in
$L^p$, and obtaining the time decay for the perturbed flow
$e^{it(\Delta-V)}$ as a consequence of \eqref{eq:decay}, via
interwining properties. This point of view was introduced by Yajima in
\cite{Y2, Y3, Y4} and then followed by different authors (see
e.g. \cite%
{DF1, W1, W2}). Since it leads to a much stronger result, the
integrability conditions which are needed for the potential $V$ are
usually far from being optimal in the sense of \eqref{eq:omog}. The
unique situation in which, at our knowledge, the $L^p$-boundedness of
the wave operators is proved under almost sharp assumptions on $V$ is
the 1D-case, as it has been proven in \cite{DF1}. About Strichartz
estimates for $e^{it(\Delta-V)}$, the situation is quite clear, thanks
to the results obtained by Burq, Planchon, Stalker and Tahvildar-Zadeh
in \cite{BPSTZ1, BPSTZ}; the authors can prove a suitable
Morawetz-type estimate for the perturbed resolvent, by multiplier
techniques, which implies, together with its free countepart and free
Strichartz, the Strichartz estimates for a class of potentials $V$
which includes the ones which are critical in the sense of
\eqref{eq:omog}.
The situation in the electromagnetic case $A\neq0, V\neq0$ is quite more
complicated and weaker results are available. Some additional difficulties,
performing the above mentioned approach, come into play, due to
the introduction of a first order term in the equation, which makes more complicate the analysis of the resolvent (see e.g. \cite{DF2}%
). On the other hand, some results are available, both for estimates like %
\eqref{eq:decay} and \eqref{eq:stri}, under suitable conditions on the
potentials in \eqref{eq:schro2}, which as far as we know never permit to
recover the critical cases as in \eqref{eq:omog} (see e.g. \cite{C, CS,
EGS1, EGS2, DF2, DF3, DFVV, GST, MMT, RZ, ST, Ste} and the references
therein).
In view of the above considerations, it should be quite interesting to
produce a tool which might permit to prove the decay estimates %
\eqref{eq:decay} for equation \eqref{prob}, in which the potentials are
scaling-critical. The main goal of this manuscript is to give an explicit
representation formula for solutions to \eqref{prob}, which is in fact a
generalization of \eqref{eq:free}. In the approach we follow in the sequel,
the critical homogeneities and the transversality condition %
\eqref{transversality} play a fundamental role. We are now ready to prepare
the setting of our main results.
A key role in the representation formula we are going to derive in section %
\ref{sec:main-theor-repr} is played by the spectrum of the angular component
of the operator ${\mathcal{L}}_{{\mathbf{A}},a}$ on the unit $(N-1)$%
-dimensional sphere ${\mathbb{S}}^{N-1}$, i.e. of the operator
\begin{align} \label{eq:angular}
L_{{\mathbf{A}},a} & =\big(-i\,\nabla_{\mathbb{S}^{N-1}}+{\mathbf{A}}\big)%
^2+a(\theta) \\
& =-\Delta_{\mathbb{S}^{N-1}}+\big(|{\mathbf{A}}|^2+ a(\theta)-i\,\dive%
\nolimits_{{\mathbb{S}}^{N-1}}{\mathbf{A}}\big)-2i\,{\mathbf{A}}\cdot\nabla_{%
\mathbb{S}^{N-1}}. \notag
\end{align}
By classical spectral theory, $L_{{\mathbf{A}},a}$ admits a diverging
sequence of real eigenvalues with finite multiplicity $\mu_1({\mathbf{A}}%
,a)\leq\mu_2({\mathbf{A}},a)\leq\cdots\leq\mu_k({\mathbf{A}},a)\leq\cdots$,
see \cite[Lemma A.5]{FFT}. To each $k\in{\mathbb{N}}$, $k\geq 1$, we
associate a $L^{2}\big({\mathbb{S}}^{N-1},{\mathbb{C}}\big)$-normalized
eigenfunction $\psi_k$ of the operator $L_{{\mathbf{A}},a}$ on $\mathbb{S}%
^{N-1}$ corresponding to the $k$-th eigenvalue $\mu_{k}({\mathbf{A}},a)$,
i.e. satisfying
\begin{equation} \label{angular}
\begin{cases}
L_{{\mathbf{A}},a}\psi_{k}=\mu_k({\mathbf{A}},a)\,\psi_k(\theta), & \text{in
}{\mathbb{S}}^{N-1}, \\[3pt]
\int_{{\mathbb{S}}^{N-1}}|\psi_k(\theta)|^2\,dS(\theta)=1. &
\end{cases}%
\end{equation}
In the enumeration $\mu_1({\mathbf{A}},a)\leq\mu_2({\mathbf{A}}%
,a)\leq\cdots\leq\mu_k({\mathbf{A}},a)\leq \cdots$ we repeat each eigenvalue
as many times as its multiplicity; thus exactly one eigenfunction $\psi_k$
corresponds to each index $k\in{\mathbb{N}}$. We can choose the functions $%
\psi_k$ in such a way that they form an orthonormal basis of $L^2({\mathbb{S}%
}^{N-1},{\mathbb{C}})$. We also introduce the numbers
\begin{equation} \label{eq:alfabeta}
\alpha_k:=\frac{N-2}{2}-\sqrt{\bigg(\frac{N-2}{2}\bigg)^{\!\!2}+\mu_k({%
\mathbf{A}},a)}, \quad \beta_k:=\sqrt{\left(\frac{N-2}{2}\right)^{\!\!2}+%
\mu_k({\mathbf{A}},a)},
\end{equation}
so that $\beta_{k}=\frac{N-2}{2}-\alpha_{k}$, for $k=1,2,\dots$, which will
come into play in the sequel. Notice that $\alpha_{1}\geq\alpha_{2}\geq%
\alpha_{3}\dots$.
Under the condition
\begin{equation} \label{eq:hardycondition}
\mu_1({\mathbf{A}},a)>-\left(\frac{N-2}{2}\right)^{\!\!2}
\end{equation}
the quadratic form associated to $\mathcal{L}_{{\mathbf{A}},a}$ is positive
definite (see Section \ref{sec:functional-setting} below and the paper \cite%
{FFT}), thus implying that the hamiltonian $\mathcal{L}_{{\mathbf{A}},a}$ is
a symmetric semi-bounded operator on $L^2({\mathbb{R}}^N;{\mathbb{C}})$
which then admits a self-adjoint extension (Friedrichs extension) with the
natural form domain. As a consequence, under assumption %
\eqref{eq:hardycondition} the unitary flow $e^{it\mathcal{L}_{{\mathbf{A}}%
,a}}$ is well defined on the domain of $\mathcal{L}_{{\mathbf{A}},a}$ by
Spectral Theorem; therefore, for every $u_0\in L^2({\mathbb{R}}^N;{\mathbb{C}%
})$, there exists a unique solution $u(\cdot,t):=e^{it\mathcal{L}_{{\mathbf{A%
}},a}}u_0\in \mathcal{C}({\mathbb{R}};L^2({\mathbb{R}}^N))$ to %
\eqref{prob} with $u(x,0)=u_0(x)$.
\begin{remark}\label{rem:t_neg}
We notice that
$$
u(\cdot,-s):=e^{-is\mathcal{L}_{{\mathbf{A
}},a}}u_0= \overline{e^{is\mathcal{L}_{{\mathbf{-A
}},a}}\overline{u_0}};
$$
henceforth, for the sake of simplify and without loss of generality, in the sequel we consider
$u=u(x,t):{\mathbb{R}}^{N}\times[0,+\infty)\to{\mathbb{C}}$.
\end{remark}
The main theorem of the present
paper provides a representation formula for such
solution in terms of the following kernel
\begin{equation} \label{nucleo}
K(x,y)=\sum\limits_{k=1}^{\infty }i^{-\beta _{k}}j_{-\alpha
_{k}}(|x||y|)\psi _{k}\big(\tfrac{x}{|x|}\big)\overline{\psi _{k}\big(\tfrac{%
y}{|y|}\big)},
\end{equation}
where $\alpha _{k},\beta _{k}$ are defined in \eqref{eq:alfabeta} and, for every $\nu \in {\mathbb{R}}$,
\begin{equation*}
j_{\nu }(r):=r^{-\frac{N-2}{2}}J_{\nu +\frac{N-2}{2}}(r)
\end{equation*}%
with $J_{\nu }$ denoting the Bessel function of the first kind
\begin{equation*}
J_{\nu }(t)=\bigg(\frac{t}{2}\bigg)^{\!\!\nu }\sum\limits_{k=0}^{\infty }
\dfrac{(-1)^{k}}{\Gamma (k+1)\Gamma (k+\nu +1)}\bigg(\frac{t}{2}\bigg)
^{\!\!2k}.
\end{equation*}
The following lemma provides uniform convergence on compacts of the
queue of the series in \eqref{nucleo}.
\begin{Lemma}\label{l:queue}
Let $a\in L^{\infty }({\mathbb{S}}^{N-1},{\mathbb{R}})$, ${%
\mathbf{A}}\in C^{1}({\mathbb{S}}^{N-1},{\mathbb{R}}^{N})$ such that
\eqref{transversality} and \eqref{eq:hardycondition} hold. Then
there exists $k_0\geq1$ such that the series
$$
\sum_{k=k_0+1}^{\infty }i^{-\beta _{k}}j_{-\alpha
_{k}}(|x||y|)\psi _{k}\big(\tfrac{x}{|x|}\big)\overline{\psi _{k}\big(\tfrac{%
y}{|y|}\big)}
$$
is uniformly convergent on compacts and
$$
K(x,y)-\sum_{k=1}^{k_0}i^{-\beta _{k}}j_{-\alpha _{k}}(|x||y|)\psi
_{k}\big(\tfrac{x}{|x|}\big)\overline{\psi _{k}\big(\tfrac{%
y}{|y|}\big)}\in L_{\rm loc}^{\infty }({\mathbb R}^{2N},{\mathbb{C}}).
$$
\end{Lemma}
\noindent We can now state the main result of this
paper.
\begin{Theorem}[Representation formula]\label{Main}
Let $a\in L^{\infty }({\mathbb{S}}^{N-1},{\mathbb{R}})$,
${\mathbf{A}}\in C^{1}({\mathbb{S}}^{N-1},{\mathbb{R}}^{N})$ such
that \eqref{transversality} and \eqref{eq:hardycondition} hold. Let
$\mathcal{L}_{ {\mathbf{A}},a}$ as in \eqref{eq:hamiltonian} and
$K$ as in \eqref{nucleo}.
If $u_{0}\in L^{2}({\mathbb{R}}^{N})$ and $u(x,t)=e^{it\mathcal{L}_{{\mathbf{%
A}},a}}u_{0}(x)$, then, for all $t>0$,
\begin{equation}\label{representation}
u(x,t)=\frac{e^{\frac{i|x|^{2}}{4t}}}{i(2t)^{{N}/{2}}}\int_{{\mathbb{R}}%
^{N}}K\bigg(\frac{x}{\sqrt{2t}},\frac{y}{\sqrt{2t}}\bigg)e^{i\frac{|y|^{2}}{%
4t}}u_{0}(y)\,dy.
\end{equation}
\end{Theorem}
\begin{remark}\label{rem:int}
The integral at the right hand side of \eqref{representation} is
understood in the sense the improper multiple integrals, i.e.
\begin{equation*}
u(x,t)=\frac{e^{\frac{i|x|^{2}}{4t}}}{i(2t)^{{N}/{2}}}
\lim_{R\to+\infty}\int_{B_R}K\bigg(\frac{x}{\sqrt{2t}},\frac{y}{\sqrt{2t}}\bigg)e^{i\frac{|y|^{2}}{%
4t}}u_{0}(y)\,dy,
\end{equation*}
where $B_R:=\{y\in{\mathbb R}^N:|y|<R\}$.
\end{remark}
\begin{remark}
\label{rem:free_case} Formula \eqref{representation} is in fact a
generalization of \eqref{eq:free}. Indeed, in the free case, i.e. ${\mathbf{A%
}}\equiv {\mathbf{0}}$ and $a\equiv 0$, the operator $L_{{\mathbf{A}},a}$
reduces to the Laplace Beltrami operator $-\Delta _{\mathbb{S}^{N-1}}$,
whose eigenvalues are given by
\begin{equation*}
\lambda _{\ell }=(N-2+\ell )\ell ,\quad \ell =0,1,2,\dots ,
\end{equation*}%
having the $\ell $-th eigenvalue $\lambda _{\ell }$ multiplicity
\begin{equation*}
m_{\ell }=\frac{(N-3+\ell )!(N+2\ell -2)}{\ell !(N-2)!},
\end{equation*}%
and whose eigenfunctions coincide with the usual spherical harmonics. For
every $\ell \geq 0$, let $\{Y_{\ell ,m}\}_{m=1,2,\dots ,m_{\ell }}$ be a $%
L^{2}(\mathbb{S}^{N-1},{\mathbb{C}}^{N})$-orthonormal basis of the
eigenspace of $-\Delta _{\mathbb{S}^{N-1}}$ associated to $\lambda _{\ell }$
with $Y_{\ell ,m}$ being spherical harmonics of degree $\ell $. Hence we
have that
\begin{align*}
& \text{if }k=1,\text{ then }\mu _{1}({\mathbf{0}},0)=\lambda _{0}=0,\quad
\alpha _{1}=0,\quad \beta _{1}=\tfrac{N-2}{2}, \\
& \text{if }k>1\text{ and }{\textstyle{\sum_{n=0}^{\ell -1}}}m_{n}<k\leq {%
\textstyle{\sum_{n=0}^{\ell }}}m_{n},\text{ then }%
\begin{cases}
\mu _{k}({\mathbf{0}},0)=\lambda _{\ell } \\
\alpha _{k}=-\ell \\
\beta _{k}=\frac{N-2}{2}+\ell
\end{cases}%
, \\
& \{\psi _{k}\}_{k=1,2,\dots }=\{Y_{\ell ,m}\}_{\substack{ \ell =1,2,\dots
\quad \ \ \\ m=1,2,\dots ,m_{\ell }}}.
\end{align*}%
The Jacobi-Anger expansion for plane waves combined with the Addition
Theorem for spherical harmonics (see for example \cite[formula (4.8.3), p.
116]{Ismail} and \cite[Corollary 1]{BeStr}) yields
\begin{equation*}
e^{ix\cdot y}=(2\pi )^{N/2}\big(|x||y|\big)^{-\frac{N-2}{2}}\sum_{\ell
=0}^{\infty }i^{\ell }J_{\ell +\frac{N-2}{2}}\big(|x||y|\big)\bigg(%
\sum_{m=1}^{m_{\ell }}Y_{\ell ,m}\big(\tfrac{x}{|x|}\big)\overline{%
Y_{\ell ,m}\big(\tfrac{y}{|y|}\big)}\bigg)
\end{equation*}%
for all $x,y\in {\mathbb{R}}^{N}$. Then in the free case ${\mathbf{A}}\equiv
{\mathbf{0}}$, $a\equiv 0$, we have that
\begin{equation*}
K(x,y)=\frac{e^{-ix\cdot y}}{(2\pi )^{\frac{N}{2}}i^{\frac{N-2}{2}}},
\end{equation*}%
which, together with \eqref{representation} and taking into account
that if $t=-s$, $s>0,$ then
$u(\cdot,-s)=\overline{e^{is(-\Delta)}\overline{u_0}}$, gives in turn
\eqref{eq:free}.
\end{remark}
\begin{remark}
\label{rem:eq_kernel} We remark that, for every $y\in{\mathbb{R}}^N$ fixed,
the function $K(\cdot,y)$ formally solves the equation
\begin{equation*}
\mathcal{L}_{{\mathbf{A}},a}K(\cdot,y)=|y|^2K(\cdot,y),
\end{equation*}
as one can easily check; in fact this fits with the free case, in which $K$
is the plane wave $e^{-ix\cdot y}$, up to constants.
\end{remark}
Formula \eqref{representation} is not present in the literature, at our
knowledge; moreover, as far as we understand, it should provide a
fundamental tool for several different applications. A first immediate
consequence of the representation formula \eqref{representation} is the
following corollary.
\begin{Corollary}[Time decay]
\label{cor:decay} Let $a\in L^{\infty}({\mathbb{S}}^{N-1},{\mathbb{R}})$, ${%
\mathbf{A}}\in C^1({\mathbb{S}}^{N-1},{\mathbb{R}}^N)$ such that %
\eqref{transversality} and \eqref{eq:hardycondition} hold, and $\mathcal{L}%
_{{\mathbf{A}},a}$ as in \eqref{eq:hamiltonian}. If
\begin{equation} \label{eq:claim}
\sup_{x,y\in{\mathbb{R}}^N}|K(x,y)|<+\infty,
\end{equation}
with $K$ as in \eqref{nucleo}, then the following estimate holds
\begin{equation} \label{eq:decaygen}
\left\|e^{it\mathcal{L}_{{\mathbf{A}},a}}f(\cdot)\right\|_{L^p}\leq \frac{C}{%
|t|^{N\left(\frac12-\frac1p\right)}}\|f\|_{L^{p^{\prime }}}, \quad p\in[%
2,+\infty], \quad \frac1p+\frac1{p^{\prime }}=1,
\end{equation}
for some $C=C({\mathbf{A}}, a, p)>0$ which does not depend on $t$ and $f$.
\end{Corollary}
\begin{proof}
The proof is quite immediate. Formula
\eqref{representation}, \eqref{eq:claim}, and Remark \ref{rem:t_neg} automatically yield
\eqref{eq:decaygen} in the case $p=\infty$. The rest of the range
$p\geq2$ in \eqref{eq:decaygen} then follows by interpolation with
the $L^2$ conservation.
\end{proof}
\begin{remark}
\label{rem:fundamental} Once the matters to prove a time decay estimate are
reduced to the study of the kernel $K$ in \eqref{nucleo}, the behavior of
the spherical Bessel functions $j_{-\alpha_k}$ comes into play. The
crucial fact to notice is that condition \eqref{eq:claim} is strictly
related to the requirement $\alpha_1\leq0$ which in other words, by
\eqref{eq:alfabeta} means $\mu_1(A,a)\geq0$. It is easy to verify that
\eqref{eq:claim} implies that $\mu_1(A,a)\geq0$, while arguing as in
the proof of Lemma \ref{l:queue} we can easily check that $\mu_1(A,a)\geq0$ implies that $K$ is
locally bounded.
We are strongly motivated by
the examples in the sequel to conjecture that in fact conditions
\eqref{eq:claim} and $\mu_1(A,a)\geq0$ are equivalent.
\end{remark}
We now pass to give a couple of relevant examples in which the abstract
assumption \eqref{eq:claim} can be checked by hands, and the optimal time
decay can be obtained by working directly on the representation formula
\eqref{representation}.
\subsection{Application 1: Aharonov-Bohm field}
We start with a 2D example of purely magnetic field, which is given by
potentials associated to thin solenoids: if the radius of the solenoid tends
to zero while the flux through it remains constant, then the particle is
subject to a $\delta$-type magnetic field, which is called \emph{%
Aharonov-Bohm} field. A vector potential associated to the Aharonov-Bohm
magnetic field in ${\mathbb{R}}^2$ has the form
\begin{equation} \label{eq:Bohm}
{\boldsymbol{\mathcal{A}}}(x_1,x_2)=\alpha\bigg(-\frac{x_2}{|x|^2},\frac{x_1%
}{|x|^2}\bigg),\quad (x_1,x_2)\in{\mathbb{R}}^2,
\end{equation}
with $\alpha\in{\mathbb{R}}$ representing the circulation of ${\boldsymbol{%
\mathcal{A}}}$ around the solenoid. Notice that the potential in %
\eqref{eq:Bohm} is singular at $x=0$, homogeneous of degree $-1$ and
satisfies the transversality condition \eqref{transversality}.
This situation corresponds to problem \eqref{prob} with
\begin{equation*}
N=2, \quad {\mathbf{A}}(\theta)={\mathbf{A}}(\cos t,\sin t)=\alpha(-\sin
t,\cos t), \quad a(\theta)=0,
\end{equation*}
so that equation \eqref{prob} takes the form
\begin{equation} \label{eq:AB}
iu_{t}= \left(-i\,\nabla+\alpha\bigg(-\frac{x_2}{|x|^2},\frac{x_1}{|x|^2} %
\bigg)\right)^{\!\!2}u,
\end{equation}
with $x=(x_1,x_2)\in {\mathbb{R}}^2$. In this case, an explicit calculation
yields
\begin{equation*}
\{\mu_k({\mathbf{A}},0):k\in{\mathbb{N}}\setminus\{0\}\}=\{(\alpha-j)^2:j\in{%
\mathbb{Z}}\},
\end{equation*}
(see e.g. \cite{lw} for details), and in particular
\begin{equation*}
\mu_1({\mathbf{A}},0)=\big(\mathop{\rm dist}(\alpha,{\mathbb{Z}})\big)%
^2\geq0.
\end{equation*}
If $\mathop{\rm dist}(\alpha,{\mathbb{Z}})\neq\frac12$, then all the
eigenvalues are simple and the eigenspace associated to the eigenvalue $%
(\alpha-j)^2$ is generated by $\psi(\cos t,\sin t)=e^{-ijt}$. If $%
\mathop{\rm dist}(\alpha,{\mathbb{Z}})=\frac12$, then all the eigenvalues
have multiplicity $2$. The following result is an interesting consequence of
Theorem \ref{Main}.
\begin{Theorem}[Time decay for Aharonov-Bohm]\label{thm:AB}
Let $N=2$, $\alpha\in{\mathbb{R}}$, and define
\begin{equation*}
\mathcal{L}_{\alpha}:=\left(-i\,\nabla+\alpha\bigg(-\frac{x_2}{|x|^2},\frac{%
x_1}{|x|^2} \bigg)\right)^{\!\!2}.
\end{equation*}
Then the following estimate holds
\begin{equation} \label{eq:decayAB}
\left\|e^{it\mathcal{L}_{\alpha}}f(\cdot)\right\|_{L^p}\leq
\frac{C}{ |t|^{2\left(\frac12-\frac1p\right)}}\|f\|_{L^{p^{\prime
}}}, \quad p\in[2,+\infty], \quad \frac1p+\frac1{p^{\prime }}=1,
\end{equation}
for some constant $C=C(\alpha, p)>0$ which does not depend on $t$ and $f$.
\end{Theorem}
\begin{remark}
Since $\text{curl}\,A(x)\equiv0$ if $x\neq0$, the action of the magnetic
field in the Aharonov-Bohm case is concentrated at the origin. However the
potential ${\mathbf{A}}$ cannot be eliminated by gauge transformations;
this is in fact the peculiar property of Aharonov-Bohm fields, which indeed
describe an interesting difference between the classical and quantum
version of the electromagnetic theory. Due to the above remark, estimates %
\eqref{eq:decayAB} are not trivial, as far as we understand, and at our
knowledge they are not known. We finally stress that the algebraic
structure of Aharonov-Bohm potentials is exactly the one which has been
used in \cite{FG} in order to disprove the dispersion (in that case
Strichartz inequalities) in the case of magnetic field which decay less
than $|x|^{-1}$ at infinity, in dimension $N\geq3$. In 2D, counterexamples
as the ones in \cite{FG} are missing, and it is still unclear what might
happen for potentials with less decay than the one in \eqref{eq:AB}.
\end{remark}
\subsection{Application 2: The inverse square potential}
We also present an application of formula \eqref{representation} in the case
of perturbation of the Laplace operator in dimension $N=3$ with an inverse
square electric potential; more precisely, we consider problem \eqref{prob}
with ${\mathbf{A}}=0$ and $a(\theta)=a=\mbox{constant}$, so that the
hamiltonian \eqref{eq:hamiltonian} takes the form
\begin{equation} \label{eq:inverse}
\mathcal{L}_{a}:=-\Delta +\frac{a}{|x|^2},\quad\text{in }{\mathbb{R}}^3,
\end{equation}
and condition (\ref{eq:hardycondition}) reads as $a>-\frac14$. Since in this
case the angular eigenvalue problem \eqref{eq:angular} becomes
\begin{equation} \label{Y}
\left\{
\begin{array}{l}
-\Delta_{\mathbb{S}^{2}}\psi_{k}=(\mu_{k}({\mathbf{0}},a)-a) \psi_{k} ,\text{
in } \mathbb{S} ^{2}, \\
\|\psi_{k}\|_{L^{2}(\mathbb{S}^{2})}=1,%
\end{array}%
\right.
\end{equation}
we have that $\{\psi_{k}\}_{k=1}^{\infty}$ are the well known spherical
harmonics and
\begin{equation*}
\mu_{k}({\mathbf{0}},a)=a + \mu_{k}({\mathbf{0}},0)\,\,.
\end{equation*}
As in Remark \ref{rem:free_case}, for every $\ell\geq0$, let $%
\{Y_{\ell,m}\}_{m=1,2,\dots,2\ell+1}$ be a $L^2(\mathbb{S}^{N-1},{\mathbb{C}}%
^N)$-orthonormal basis for the eigenspace of $-\Delta_{\mathbb{S}^{2}}$
associated to the $\ell$-th eigenvalue $\lambda_\ell=\ell(\ell+1)$ of $%
-\Delta_{\mathbb{S}^{2}}$ (which has multiplicity $m_\ell=2\ell+1$), with $%
Y_{\ell,m}$ being spherical harmonics of degree $\ell$. Hence we have that
\begin{align}\label{eq:alfa1}
&\text{if }k=1, \text{ then }
\mu_1({\mathbf{0}},a)=a,\quad\alpha_1=\tfrac12-%
\sqrt{\tfrac14+a}, \\
\label{eq:alpha>1}&\text{if } k>1\text{ and } \ell^2<k\leq (\ell+1)^2, \text{ then }
\begin{cases}
\mu_k({\mathbf{0}},a)=a+\ell(\ell+1)= \\
\alpha_k=\frac12-\sqrt{\big(\ell+\tfrac12\big)^2+a}%
\end{cases}%
.
\end{align}
Notice that $\alpha_1\leq 0$ if and only if $a\geq0$. We define the well
known zonal functions
\begin{equation} \label{eq:zonal}
Z_{\theta }^{(\ell)}( \theta^{\prime })=\sum _{m=1}^{2\ell+1}
Y_{\ell,m}(\theta) \overline{Y_{\ell,m}(\theta^{\prime })},\quad
\theta,\theta^{\prime }\in {\mathbb{S}}^2, \quad \ell=0,1,2,\dots.
\end{equation}
The study of the kernel in \eqref{nucleo}, which can be rewritten as
\begin{equation} \label{eq:Sinverse}
K(x,y)=\sum_{\ell=0}^{\infty }i^{-\sqrt{(\ell+1/2)^2+a}}\,j_{ -\frac12+\sqrt{%
(\ell+1/2)^2+a}} \big(|x||y|\big)Z_{\frac{x}{|x|} }^{(\ell)}\Big( \tfrac{y}{%
|y|} \Big),
\end{equation}
permits to prove the following result.
\begin{Theorem}[Time decay for inverse square potentials]\label{thm:inversesquare}
Let $N=3$, $a>-\frac14$, and define $\mathcal{L}_a$ by \eqref{eq:inverse}.
\begin{itemize}
\item[i)] If $a\geq0$, then the following estimates hold
\begin{equation} \label{eq:decayinverse}
\left\|e^{it\mathcal{L}_{a}}f(\cdot)\right\|_{L^p}\leq \frac{C}{
|t|^{3\left(\frac12-\frac1p\right)}}\|f\|_{L^{p^{\prime }}}, \quad p\in[
2,+\infty], \quad \frac1p+\frac1{p^{\prime }}=1,
\end{equation}
for some constant $C=C(a, p)>0$ which does not depend on $t$ and $f$.
\item[ii)] If $-\frac14<a<0$, let $\alpha_1$ as in (\ref{eq:alfa1}), and define
\begin{equation*}
\|u\|_{p,\alpha_1}:=\bigg(\int_{{\mathbb{R}}^3}(1+|x|^{-\alpha_1})^{2-p}|u(x)|^p\,dx\bigg)^{\!1/p}, \quad
p\geq1.
\end{equation*}
Then the following estimates hold
\begin{equation} \label{eq:decayinverse2}
\left\|e^{it\mathcal{L}_{a}}f(\cdot)\right\|_{p,\alpha_1}\leq \frac{%
C(1+|t|^{\alpha_1})^{1-\frac{2}{p}}}{|t|^{3\left(\frac12-\frac1p\right)}}%
\|f\|_{p',\alpha_1}, \quad p\geq2, \quad \frac1p+\frac1{p^{\prime
}}=1,
\end{equation}
for some constant $C=C(a, p)>0$ which does not depend on $t$ and $f$.
\end{itemize}
\end{Theorem}
\begin{remark}
As far as we know, the best dispersive results concerning this kind of
operators are about Strichartz estimates, and have been obtained by Burq,
Planchon, Stalker and Tahvildar-Zadeh in \cite{BPSTZ1,BPSTZ}. As a fact,
estimates \eqref{eq:decayinverse} imply the ones obtained in \cite{BPSTZ1},
by the standard Ginibre-Velo and Keel-Tao techniques in \cite{GV1, KT}. On
the other hand, in \cite{BPSTZ} the authors can treat more general
potentials with critical decay, including e.g. the cases in which $%
a=a(x/|x|)$ is a 0-degree homogeneous function; in addition, we think that
the restriction $N=3$ in Theorem \ref{thm:inversesquare} is not in fact a
relevant obstruction. We are motivated to claim that a deeper analysis of
formula \eqref{representation} should permit to prove the analog to Theorem
\ref{thm:inversesquare} in the more general case $a=a(x/|x|)$, but this
will not be treated in the present paper.
Moreover, notice that $\alpha_1>0$, in the range $-\frac14<a<0$, see %
\eqref{eq:alfa1}, so that the decay in \eqref{eq:decayinverse2} is weaker
than the usual one. We find it an interesting phenomenon, since on the
other hand the usual Strichartz estimates are still true in this range, as
proved in \cite{BPSTZ1}. Estimates \eqref{eq:decayinverse2} are presumably sharp and, at our
knowledge, new.
\end{remark}
The rest of the paper is organized as follows. In Section \ref%
{sec:functional-setting}, we describe the functional setting in which we
work, in order to prepare the proof of the main result, Theorem \ref{Main};
Section \ref{sec:spectr-harm-magn} is then devoted to the study of the
spectral properties of a magnetic harmonic oscillator with inverse square
potential, denoted by $T_{{\mathbf{A}},a}$ (see formula \eqref{operator}),
which comes into play when a suitable ansatz (formula \eqref{varphi}) is
stated; finally, Section \ref{sec:main-theor-repr} is devoted to the proof
of Theorem \ref{Main}, while in the last Sections \ref{sec:ahar-bohm-magn}
and \ref{sec:inverse} we prove the applications, Theorems \ref{thm:AB} and %
\ref{thm:inversesquare}.
\section{Functional setting}
\label{sec:functional-setting} Let us define the following Hilbert spaces:
\begin{itemize}
\item the space ${\mathcal{H}}$ as the completion of $C^{\infty}_{\mathrm{c}%
}({\mathbb{R}}^N\setminus\{0\},{\mathbb{C}})$ with respect to the norm
\begin{equation*}
\|\phi\|_{{\mathcal{H}}}=\bigg(\int_{{\mathbb{R}}^N}\bigg(|\nabla\phi(x)|^2+
\Big(|x|^2+\frac1{|x|^2}\Big)|\phi(x)|^2\bigg) \,dx\bigg)^{\!\!1/2};
\end{equation*}
\item {the space $H$} as the completion of $C^{\infty}_{\mathrm{c}}({\mathbb{%
R}}^N,{\mathbb{C}})$ with respect to the norm
\begin{equation*}
\|\phi\|_{H}=\bigg(\int_{{\mathbb{R}}^N}\Big(|\nabla\phi(x)|^2+ \big(|x|^2+1%
\big)|\phi(x)|^2\Big) \,dx\bigg)^{\!\!1/2};
\end{equation*}
\item {the space ${\mathcal{H}}_{{\mathbf{A}}}$} as the completion of $%
C^{\infty}_{\mathrm{c}}({\mathbb{R}}^N\setminus\{0\},{\mathbb{C}})$ with
respect to the norm
\begin{equation*}
\|\phi\|_{{\mathcal{H}}_{\mathbf{A}}}=\bigg(\int_{{\mathbb{R}}^N}\Big(%
|\nabla_{\mathbf{A}}\phi(x)|^2+ \big(|x|^2+1\big)|\phi(x)|^2\Big) \,dx\bigg)%
^{\!\!1/2}
\end{equation*}
with $\nabla_{{\mathbf{A}}}\phi= \nabla\phi+i\,\frac {{\mathbf{A}}(x/|x|)}{%
|x|}\phi$.
\end{itemize}
From the above definition, it follows immediately that
\begin{equation} \label{eq:cont_emb}
\mathcal{H }\hookrightarrow H\quad\text{with continuous embedding}.
\end{equation}
A further comparison between the above defined spaces can be derived from
the well known diamagnetic inequality (see e.g. \cite{LL})
\begin{equation} \label{eq:diamagnetic}
|\nabla |\phi|(x)|\leq \left|\nabla \phi(x)+i\frac{{\mathbf{A}}(x/|x|)}{|x|}%
\phi(x)\right|, \qquad N\geq2,
\end{equation}
which holds for a.e. $x\in{\mathbb{R}}^N$ and for all $\phi\in C^{\infty}_{%
\mathrm{c}}({\mathbb{R}}^N\setminus\{0\},{\mathbb{C}})$, and the classical
Hardy inequality (see e.g. \cite{GP,HLP})
\begin{equation} \label{eq:hardy}
\int_{{\mathbb{R}}^N}|\nabla \phi(x)|^{2}\,dx\geq \bigg(\frac{N-2}{2}\bigg)%
^{\!\!2}\int_{{\mathbb{R}}^N} \frac{|\phi(x)|^2}{|x|^{2}}\,dx,
\end{equation}
which holds for all $\phi\in\mathcal{C}_{\mathrm{c}}^\infty({\mathbb{R}}^N,{%
\mathbb{C}})$ and $N\ge 3$. We notice that the presence of a vector
potential satisfying a suitable non-degeneracy condition allows to recover a
Hardy inequality even for $N=2$. Indeed, if $N=2$, \eqref{transversality}
holds, and
\begin{equation} \label{eq:circuit}
\Phi_{\mathbf{A}}:=\frac1{2\pi}\int_0^{2\pi}\alpha(t)\,dt \not\in{\mathbb{Z}}%
,\quad \text{where }\alpha(t):={\mathbf{A}}(\cos t,\sin t)\cdot(-\sin t,\cos
t),
\end{equation}
then functions in $C^{\infty}_{\mathrm{c}}({\mathbb{R}}^N\setminus\{0\},{%
\mathbb{C}})$ satisfy the following Hardy inequality
\begin{equation} \label{eq:hardyN2}
\Big(\min_{k\in{\mathbb{Z}}}|k-\Phi_{\mathbf{A}}|\Big)^2\int_{{\mathbb{R}}^2}%
\frac{|u(x)|^2}{|x|^2}\,dx \leq \int_{{\mathbb{R}}^2}\bigg|\nabla u(x)+i\,
\frac{{\mathbf{A}}\big({x}/{|x|}\big)} {|x|}\,u(x)\bigg|^2\,dx
\end{equation}
being $\big(\min_{k\in{\mathbb{Z}}}|k-\Phi_{\mathbf{A}}|\big)^2$ the best
constant, as proved in \cite{lw}.
Combining (\ref{eq:hardy}), (\ref{eq:diamagnetic}), and (\ref{eq:hardyN2}),
it is easy to verify that if $N\geq 3$, then $H=\mathcal{H}={\mathcal{H}}_{%
\mathbf{A}}$, being the norms $\|\cdot\|_{H}$, $\|\cdot\|_{{\mathcal{H}}}$
and $\|\cdot\|_{{\mathcal{H}}_{\mathbf{A}}}$ equivalent. If $N=2$ then $%
\mathcal{H}\varsubsetneq H$; on the other hand, if $N=2$ and (\ref%
{transversality}), (\ref{eq:circuit}) hold, from (\ref{eq:diamagnetic}) and (%
\ref{eq:hardyN2}) we deduce that $\mathcal{H}={\mathcal{H}}_{\mathbf{A}}$,
being the norms $\|\cdot\|_{{\mathcal{H}}}$, $\|\cdot\|_{{\mathcal{H}}_{%
\mathbf{A}}}$ equivalent.
From (\ref{eq:cont_emb}) and \cite[Proposition 6.1]{KW}, we also deduce that
\begin{equation} \label{eq:compact_emb}
{\mathcal{H}} \text{ is compactly embedded into }L^{p}({\mathbb{R}}^{N})
\end{equation}
for all
\begin{equation*}
2\leq p<
\begin{cases}
2^{*}=\frac{2N}{N-2}, & \text{if }N\geq3, \\
+\infty, & \text{if }N=2.%
\end{cases}
\end{equation*}
The quadratic form $Q_{{\mathbf{A}},a}$ associated to ${\mathcal{L}}_{{%
\mathbf{A}},a}$, i.e.
\begin{align} \label{eq:qf}
&Q_{{\mathbf{A}},a}:{\mathcal{D}}^{1,2}_{*}({\mathbb{R}}^N,{\mathbb{C}})\to{%
\mathbb{R}}, \\
& Q_{{\mathbf{A}},a}(\phi):=\int_{{\mathbb{R}}^N} \bigg[ \big| \nabla_{%
\mathbf{A}}\phi(x)\big|^2 -\frac{a\big({x}/{|x|}\big)}{|x|^2}|\phi(x)|^2%
\bigg]\,dx, \notag
\end{align}
with ${\mathcal{D}}^{1,2}_{*}({\mathbb{R}}^N,{\mathbb{C}})$ being the
completion of $C^\infty_{\mathrm{c}}({\mathbb{R}}^N\setminus\{0\},{\mathbb{C}%
})$ with respect to the norm
\begin{equation*}
\|u\|_{{\mathcal{D}}^{1,2}_{*}({\mathbb{R}}^N,{\mathbb{C}})}:=\bigg(\int_{{%
\mathbb{R}}^N}\bigg(\big|\nabla u(x)\big|^2+ \frac{|u(x)|^2}{|x|^2}\bigg) %
\,dx\bigg)^{\!\!1/2},
\end{equation*}
is positive definite if and only if (\ref{eq:hardycondition}) holds, see
\cite[Lemma 2.2]{FFT}. In particular, assumption (\ref{eq:hardycondition})
ensures that the operator ${\mathcal{L}}_{{\mathbf{A}},a}$ is semibounded
from below, self-adjoint on $L^2$ with the natural form domain, and that
there exists some constant $C(N,{\mathbf{A}},a)>0$ such that
\begin{equation} \label{eq:posdef}
\int_{{\mathbb{R}}^N} \bigg[ \big| \nabla_{\mathbf{A}}\phi(x)\big|^2 -\frac{a%
\big({x}/{|x|}\big)}{|x|^2}|\phi(x)|^2+\frac{|x|^2}4|\phi(x)|^2 \bigg]\,dx
\geq C(N,{\mathbf{A}},a)\|\phi\|_{\mathcal{H}}^2,
\end{equation}
for all $\phi\in\mathcal{H}$ (see \cite{FFT}).
Up to a \textit{pseudo conformal} change of variable, see \cite{KW},
equation (\ref{prob}) can be rewritten in terms of a quantum harmonic
oscillator with the singular electromagnetic potential, as stated in the
following lemma.
\begin{Lemma}
Let (\ref{eq:hardycondition}) hold and $u\in C({\mathbb{R}}; L^{2}({\mathbb{R%
}}^N))$ be a solution to \eqref{prob}. Then
\begin{equation} \label{varphi}
\varphi(x,t)= (1+t^2)^{\frac{N}{4}}u\big(\sqrt{1+t^2}x,t\big)e^{-it\frac{%
|x|^2}{4}}
\end{equation}
satisfies
\begin{align*}
&\varphi\in C({\mathbb{R}}; L^{2}({\mathbb{R}}^N)),\quad \varphi (x, 0)=
u(x,0), \\
&\|\varphi(\cdot,t)\|_{L^{2}({\mathbb{R}}^N)}=\|u(\cdot,t)\|_{L^{2}({\mathbb{%
R}}^N)} \text{ for all }t\in{\mathbb{R}},
\end{align*}
and
\begin{equation} \label{varphieq}
i\dfrac{d\varphi}{d t}(x,t)= \dfrac{1}{(1+t^2)} \bigg({\mathcal{L}}_{{%
\mathbf{A}},a}\varphi(x,t)+\frac{1}{4}|x|^2 \varphi(x,t)\bigg).
\end{equation}
\end{Lemma}
A representation formula for solutions $u$ to (\ref{prob}) can be found by
expanding the transformed solution $\varphi$ to (\ref{varphieq}) in Fourier
series with respect to an orthonormal basis of $L^2({\mathbb{R}}^N)$
consisting of eigenfunctions of the following quantum harmonic oscillator
operator perturbed with singular homogeneous electromagnetic potentials
\begin{equation} \label{operator}
T_{{\mathbf{A}},a}:{\mathcal{H}}\to {\mathcal{H}}^\star,\quad T_{{\mathbf{A}}%
,a}={\mathcal{L}}_{{\mathbf{A}},a}+\frac{1}{4}|x|^2
\end{equation}
acting as
\begin{multline} \label{operator2}
{}_{{\mathcal{H}}^\star}\langle T_{{\mathbf{A}},a}v,w \rangle_{{\mathcal{H}}}
\\
= \int_{{\mathbb{R}}^N}\bigg(\nabla_{{\mathbf{A}}} v(x)\cdot\overline{%
\nabla_{{\mathbf{A}}} w(x)}-\frac{a(\frac{x}{|x|})}{|x|^2}\,v(x)\overline{%
w(x)} +\frac{|x|^2}{4} v(x) \overline{w(x)}\bigg)\,dx,
\end{multline}
for all $v,w\in{\mathcal{H}}$, where ${\mathcal{H}}^\star$ denotes the dual
space of ${\mathcal{H}}$ and ${}_{{\mathcal{H}}^\star}\langle
\cdot,\cdot\rangle_{{\mathcal{H}}}$ is the corresponding duality product.
\section{The spectrum of $T_{{\mathbf{A}},a}$}
\label{sec:spectr-harm-magn} From (\ref{eq:compact_emb}), (\ref{eq:posdef}),
and classical spectral theory, we can easily deduce the following abstract
description of the spectrum of $T_{{\mathbf{A}},a}$.
\begin{Lemma}
\label{Hilbert} Let ${\mathbf{A}}\in C^1({\mathbb{S}}^{N-1},{\mathbb{R}}^N)$
and $a\in L^{\infty}\big({\mathbb{S}}^{N-1}\big)$ such that %
\eqref{eq:hardycondition} holds. Then the spectrum of the operator $T_{{%
\mathbf{A}},a}$ defined in (\ref{operator}--\ref{operator2}) consists of a
diverging sequence of real eigenvalues with finite multiplicity. Moreover,
there exists an orthonormal basis of $L^{2}({\mathbb{R}}^{N})$ whose
elements belong to ${\mathcal{H}}$ and are eigenfunctions of $T_{{\mathbf{A}%
},a}$.
\end{Lemma}
The following proposition gives a complete description of the spectrum of
the operator $T_{{\mathbf{A}},a}$.
\begin{Proposition}
\label{spectrum} The set of the eigenvalues of the operator $T_{{\mathbf{A}}%
,a}$ is
\begin{equation*}
\big\{ \gamma_{m,k}: k,m\in{\mathbb{N}}, k\geq 1\big\}
\end{equation*}
where
\begin{equation} \label{eigenvalues}
\gamma_{m,k}=2m-\alpha_k+\dfrac N2, \quad \alpha_k=\frac{N-2}{2}-\sqrt{\bigg(%
\frac{N-2}{2}\bigg)^{\!\!2}+\mu_k({\mathbf{A}},a)},
\end{equation}
and $\mu_k({\mathbf{A}},a)$ is the $k$-th eigenvalue of the operator $L_{{%
\mathbf{A}},a}$ on the sphere $\mathbb{S}^{N-1}$. Each eigenvalue $%
\gamma_{m,k}$ has finite multiplicity equal to
\begin{equation*}
\#\bigg\{j\in{\mathbb{N}},j\geq 1: \frac{\gamma_{m,k}}{2}+\frac{\alpha_j}%
2-\frac N4\in{\mathbb{N}}\bigg\}
\end{equation*}
and a basis of the corresponding eigenspace is
\begin{equation*}
\left\{V_{n,j}: j,n\in{\mathbb{N}},j\geq 1,\gamma_{m,k}=2n-\alpha_j+\frac
N2 \right\},
\end{equation*}
where
\begin{equation} \label{eigenvectors}
V_{n,j}(x)= |x|^{-\alpha_j}e^{-\frac{|x|^2}{4}}P_{j,n}\Big(\frac{|x|^2}{2}%
\Big) \psi_j\Big(\frac{x}{|x|}\Big),
\end{equation}
$\psi_j$ is an eigenfunction of the operator $L_{{\mathbf{A}},a}$ on the
sphere $\mathbb{S}^{N-1}$ associated to the $j$-th eigenvalue $\mu_{j}({%
\mathbf{A}},a)$ as in \eqref{angular}, and $P_{j,n}$ is the polynomial of
degree $n$ given by
\begin{equation*}
P_{j,n}(t)=\sum_{i=0}^n \frac{(-n)_i}{\big(\frac{N}2-\alpha_j\big)_i}\,\frac{%
t^i}{i!},
\end{equation*}
denoting as $(s)_i$, for all $s\in{\mathbb{R}}$, the Pochhammer's symbol $%
(s)_i=\prod_{j=0}^{i-1}(s+j)$, $(s)_0=1$.
\end{Proposition}
\proof Assume that $\gamma$ is an eigenvalue of $T_{{\mathbf{A}},a}$ and $%
g\in{\mathcal{H}}\setminus\{0\}$ is a corresponding eigenfunction, so that
\begin{equation} \label{calculo}
\left(-i\,\nabla+\frac{{\mathbf{A}}\big(\frac{x}{|x|}\big)} {|x|}%
\right)^{\!\!2}g(x) +\dfrac{a\big(\frac{x}{|x|}\big)}{|x|^2}g(x) + \frac{%
|x|^2}{4}\, g(x)=\gamma\, g(x)
\end{equation}
in a weak ${\mathcal{H}}$-sense. From classical elliptic regularity theory,
$g\in C^{1,\alpha}_{\mathrm{loc}}({\mathbb{R}}^N\setminus\{0\},{\mathbb{C}})$%
. Hence $g$ can be expanded as
\begin{equation*}
g(x)=g(r\theta)=\sum_{k=1}^\infty\phi_k(r)\psi_k(\theta) \quad \text{in }L^2(%
{\mathbb{S}}^{N-1}),
\end{equation*}
where $r=|x|\in(0,+\infty)$, $\theta=x/|x|\in{{\mathbb{S}}^{N-1}}$, and
\begin{equation*}
\phi_k(r)=\int_{{\mathbb{S}}^{N-1}}g(r\theta) \overline{\psi_k(\theta)}%
\,dS(\theta).
\end{equation*}
Equations \eqref{angular} and \eqref{calculo} imply that, for every $k$,
\begin{equation} \label{ODE}
\phi^{\prime \prime }_{k}+\dfrac{N-1}{r}\phi^{\prime }_{k} +\left(\gamma-%
\dfrac{\mu_k}{r^2}-\dfrac{r^2}{4}\right)\phi_{k}=0 \quad\text{in }%
(0,+\infty).
\end{equation}
Since $g\in {\mathcal{H}}$, we have that
\begin{align}\label{regularidadL2}
\infty>\int_{{\mathbb{R}}^N}g^2(x)\,dx&=\int_{0}^{\infty} \!\bigg(\int_{{%
\mathbb{S}}^{N-1}}g^{2}(r\theta)\,dS(\theta)\bigg) r^{N-1}\,dr\\
&\notag\geq
\int_{0}^{\infty}r^{N-1}\phi_{k}^{2}(r)\,dr
\end{align}
and
\begin{equation} \label{regularidadHardy}
\infty>\int_{{\mathbb{R}}^N}\dfrac{g^2(x)}{|x|^2}\,dx\geq\int_{0}^{%
\infty}r^{N-3}\phi_{k}^2(r)\,dr.
\end{equation}
For all $k=1,2,\dots$ and $t>0$, we define $w_{k}(t)=(2t)^{\frac{\alpha_k}{2}%
} e^{\frac{t}{2}}\phi_k(\sqrt{2t})$, with $\alpha_{k}$ as in (\ref%
{eigenvalues}). From \eqref{ODE}, $w_k$ satisfies
\begin{equation*}
t w_{k}^{\prime \prime }(t)+\left(\frac{N}{2}-\alpha_k-t\right)w^{\prime
}_{k}(t)- \left(\frac N 4-\frac{\alpha_k}{2}-\frac \gamma
2\right)w_{k}(t)=0\quad\text{in }(0,+\infty).
\end{equation*}
Therefore, $w_{k}$ is a solution of the well known Kummer Confluent
Hypergeometric Equation (see \cite{AS} and \cite{MA}). Then there exist $%
A_k,B_k\in{\mathbb{R}}$ such that
\begin{equation*}
w_k(t)=A_k M\Big(\frac N 4-\frac{\alpha_k}{2}-\frac \gamma 2,\frac
N2-\alpha_k,t\Big)
+B_k U\Big(\frac N 4-\frac{\alpha_k}{2}-\frac \gamma 2,\frac N2-\alpha_k,t%
\Big), \quad t\in (0,+\infty).
\end{equation*}
Here $M(c,b,t)$ and, respectively, $U(c,b,t)$ denote the Kummer function (or
confluent hypergeometric function) and, respectively, the Tricomi function
(or confluent hypergeometric function of the second kind); $M(c,b,t)$ and $%
U(c,b,t)$ are two linearly independent solutions to the Kummer Confluent
Hypergeometric Equation
\begin{equation*}
tw^{\prime \prime }(t)+(b-t)w^{\prime }(t)-cw(t)=0,\quad t\in (0,+\infty).
\end{equation*}
Since $\big(\frac N2-\alpha_k\big)>1$, from the well-known asymptotics of $U$
at $0$ (see e.g. \cite{AS}), we have that
\begin{equation*}
U\Big(\frac N 4-\frac{\alpha_k}{2}-\frac \gamma 2,\frac N2-\alpha_k,t\Big)
\sim \text{\textrm{const}}\,t^{1-\frac{N}{2}+\alpha_k} \quad\text{as }t\to
0^+,
\end{equation*}
for some $\text{\textrm{const}}\neq 0$ depending only on $N,\gamma$, and $%
\alpha_k$. On the other hand, $M$ is the sum of the series
\begin{equation*}
M(c,b,t)=\sum_{n=0}^\infty \frac{(c)_n}{(b)_n}\,\frac{t^n}{n!}.
\end{equation*}
We notice that $M$ has a finite limit at $0^+$, while its behavior at $\infty
$ is singular and depends on the value $-c=-\frac N 4+\frac{\alpha_k}{2}%
+\frac \gamma 2$. If $-\frac N 4+\frac{\alpha_k}{2}+\frac \gamma 2=m\in {%
\mathbb{N}}=\{0,1,2,\cdots\}$, then $M\big(\frac N 4-\frac{\alpha_k}{2}%
-\frac \gamma 2,\frac N2-\alpha_k,t\big)$ is a polynomial of degree $m$ in $t
$, which we will denote as $P_{k,m}$, i.e.,
\begin{equation*}
P_{k,m}(t)=M\Big(-m,{\textstyle{\frac N2}}-\alpha_k,t\Big)= \sum_{n=0}^m
\frac{(-m)_n}{\big(\frac{N}2-\alpha_k\big)_n}\,\frac{t^n}{n!}.
\end{equation*}
If $\big(-\frac N 4+\frac{\alpha_k}{2}+\frac \gamma 2\big)\not\in {\mathbb{N}%
}$, then from the well-known asymptotics of $M$ at $\infty$ (see e.g. \cite%
{AS}) we have that
\begin{equation*}
M\Big(\frac N 4-\frac{\alpha_k}{2}-\frac \gamma 2,\frac N2-\alpha_k,t\Big)
\sim \text{\textrm{const}}\,e^tt^{-\frac{N}{4}+\frac{\alpha_k}2-\frac{\gamma%
}{2}} \quad\text{as }t\to +\infty,
\end{equation*}
for some $\text{\textrm{const}}\neq 0$ depending only on $N,\gamma$, and $%
\alpha_k$.
Now, let us fix $k\in{\mathbb{N}}$, $k\geq 1$. From the above description,
we have that
\begin{equation*}
w_k(t)\sim \mathrm{const\,}B_k t^{1-\frac{N}{2}+\alpha_k} \quad\text{as }%
t\to 0^+,
\end{equation*}
for some $\text{\textrm{const}}\neq 0$, and hence
\begin{equation*}
\phi_k(r)=r^{-\alpha_k}e^{-\frac{r^2}{4}}w_k\Big(\frac{r^2}{2}\Big)\sim
\mathrm{const\,}B_k r^{-(N-2)+\alpha_k} \quad\text{as }r\to 0^+,
\end{equation*}
for some $\text{\textrm{const}}\neq 0$. Therefore, condition (\ref%
{regularidadHardy}) can be satisfied only for $B_k=0$. If $\frac{\alpha_k}{2}%
+\frac{\gamma}{2}-\frac{N}{4}\not\in {\mathbb{N}}$, then
\begin{equation*}
w_k(t)\sim \mathrm{const\,}A_ke^t t^{-\frac{N}{4}+\frac{\alpha_k}2-\frac{%
\gamma}{2}} \quad\text{as }t\to +\infty,
\end{equation*}
for some $\text{\textrm{const}}\neq 0$, and hence
\begin{equation*}
\phi_k(r)=r^{-\alpha_k}e^{-\frac{r^2}{4}}w_k\Big(\frac{r^2}{4}\Big)\sim
\mathrm{const\,}A_k r^{-\frac{N}{2}-\gamma}e^{r^2/4} \quad\text{as }r\to
+\infty,
\end{equation*}
for some $\text{\textrm{const}}\neq 0$. Therefore, condition (\ref%
{regularidadL2}) can be satisfied only for $A_k=0$. If $\frac{\alpha_k}{2}+%
\frac{\gamma}{2}-\frac{N}{4}=m\in {\mathbb{N}}$, then $r^{-\alpha_k}e^{-%
\frac{r^2}{4}}P_{k,m}\big(\frac{r^2}{2}\big)$ solves (\ref{ODE}); moreover
the function
\begin{equation*}
V_{m,k}(x)=|x|^{-\alpha_k}e^{-\frac{|x|^2}{4}}P_{k,m}\Big(\frac{|x|^2}{2}%
\Big)
\psi_k\Big(\frac{x}{|x|}\Big)
\end{equation*}
belongs to ${\mathcal{H}}$, thus providing an eigenfunction of $L$. \qed
\begin{remark}
\label{rem:harmonic} If $a(\theta)\equiv0$, ${\mathbf{A}}\equiv{\mathbf{0}}$%
, the spectrum of $L_{{\mathbf{0}},0}$ is described in Remark \ref%
{rem:free_case}, so that the spectrum of $T_{{\mathbf{0}},0}$ is $\frac N2+{%
\mathbb{N}}$. Hence, in this case we recover the eigenvalues of the
Harmonic oscillator operator $-\Delta+\frac{|x|^2}4$ (see e.g. \cite{KW}).
\end{remark}
\begin{remark}
\label{rem:ortho} It is easy to verify that
\begin{equation*}
\text{if }(m_1,k_1)\neq(m_2,k_2)\quad\text{then}\quad V_{m_1,k_1}\text{ and }
V_{m_2,k_2}\text{ are orthogonal in }{\ L^{2}({\mathbb{R}}^N)}.
\end{equation*}
By Lemma \ref{Hilbert}, it follows that
\begin{equation*}
\left\{ \widetilde V_{n,j}= \frac{V_{n,j}}{\|V_{n,j}\|_{L^{2}({\mathbb{R}}%
^N)}}: j,n\in{\mathbb{N}},j\geq 1\right\}
\end{equation*}
is an orthonormal basis of $L^{2}({\mathbb{R}}^N)$.
\end{remark}
\begin{remark}
\label{rem:Laguerre} Denoting by $L_{m}^{\alpha}(t)$ the generalized
Laguerre polynomials
\begin{equation*}
L_{m}^{\alpha}(t)=\sum_{n=0}^m (-1)^{n}{\binom{m+\alpha}{m-n}\,\frac{t^n}{n!}%
},
\end{equation*}
and $\beta_{k}=\sqrt{\big(\frac{N-2}{2}\big)^{2}+\mu_k({\mathbf{A}},a)}$ so
that $\gamma_{m,k}=2m+\beta_k+1$, we can write
\begin{equation*}
P_{k,m}\bigg(\frac{|x|^2}{2}\bigg)= M\Big(-\frac{\gamma_{m,k}}{2}+\frac{%
\beta_k}{2}+\frac 12,1+\beta_{k},\frac{|x|^2}{2}\Big)=\binom{m+\beta_k}{m}%
^{\!\!-1} L_{m}^{\beta_k}\Big(\frac{|x|^2}{2}\Big).
\end{equation*}
From the well known orthogonality relation
\begin{equation*}
\int_{0}^{\infty} x^{\alpha} e^{-x} L_{n}^{\alpha}(x) L_{m}^{\alpha}(X)\,dx=%
\frac{\Gamma(n+\alpha+1)}{n!}\delta_{n,m},
\end{equation*}
where $\delta_{n,m}$ denotes the Kronecker delta, it is easy to check that
\begin{equation*}
\|V_{m,k}\|_{L^{2}({\mathbb{R}}^N)}^{2}= 2^{\beta_k}\Gamma(1+\beta_k)\binom{%
m+\beta_k}{m}^{\!\!-1}.
\end{equation*}
\end{remark}
\section{Proof of Theorem
\protect\ref{Main}}\label{sec:main-theor-repr}
The proof of Theorem \ref{Main} uses Lemma \ref{l:queue}, which is
proved below.
\smallskip\noindent
\begin{pfn}{Lemma \ref{l:queue}}
From Theorem \ref{t:weyl} and Lemma \ref{l:stiautf} in the Appendix,
we deduce that there exist some $k_0\in{\mathbb N}$,
and $C_{i}>0$, $i=1,2$, such that for every $k>k_0$,
\begin{equation}\label{cond1}
-\alpha _{k}>C_{1}k^{\delta _{1}},
\end{equation}
and
\begin{equation}\label{cond2}
\vert \psi _{k}(\theta)\vert <C_{2}k^{\delta
_{2}}\quad\text{for all }\theta\in{\mathbb S}^{N-1},
\end{equation}
with $\delta_1=\frac1{N-1}$ and $\delta_2=\frac{2}{N-1}\lfloor
\frac{N-1}2\rfloor$.
From \eqref{cond1} and \eqref{cond2} it follows that, for all $k>k_0$, the $k$-th term of the
series $K_{k}(x,y)=i^{-\beta _{k}}j_{-\alpha
_{k}}(|x||y|)\psi _{k}\big(\tfrac{x}{|x|}\big)\overline{\psi _{k}\big(\tfrac{%
y}{|y|}\big)}$ belongs to $L_{\rm loc}^{\infty
}({\mathbb R}^{2N},{\mathbb{C}})$. Furthermore, if we fix some compact set $\mathcal
K\Subset{\mathbb R}^N$, there exists $k_0'$ such that
$\big|\frac{e|x||y|}{2(-\alpha _{k}+%
\frac{N}{2})}\big|\leq \frac12$ for every $x,y\in\mathcal K$ and
$k> k_0'$. Therefore, for all $x,y\in\mathcal K$ and
$k> k_0'$, we have that
\begin{eqnarray*}
\left\vert K_{k}(x,y)\right\vert &\leq &\left\vert j_{-\alpha
_{k}}(|x||y|)\right\vert \left\vert \psi _{k}\big(\tfrac{x}{|x|}\big)%
\right\vert \left\vert \psi _{k}\big(\tfrac{y}{|y|}\big)\right\vert \\
&\leq &Ck^{2\delta _{2}}\bigg(\frac{|x||y|}{2}\bigg)^{\!\!-\alpha
_{k}}\sum\limits_{m=0}^{\infty }\dfrac{1}{\Gamma (m+1)\Gamma (m-\alpha _{k}+%
\frac{N-2}{2}+1)}\bigg(\frac{|x||y|}{2}\bigg)^{\!\!2m} \\
&\leq &Ck^{2\delta _{2}}\frac{\big(\frac{|x||y|}{2}\big)^{\!\!-\alpha _{k}}%
}{\Gamma (-\alpha _{k}+\frac{N}{2})}e^{\big(\frac{|x||y|}{2}\big)%
^{\!\!2}}\leq C^{\prime }k^{2\delta _{2}}\bigg(\frac{e|x||y|}{2(-\alpha _{k}+%
\frac{N}{2})}\bigg)^{\!\!-\alpha _{k}}e^{\big(\frac{|x||y|}{2}\big)%
^{\!\!2}} \\
&\leq &C^{\prime \prime }k^{2\delta _{2}}\bigg(\frac{1}{2}\bigg)%
^{\!\!C_{1}k^{\delta _{1}}}\equiv M_{k}
\end{eqnarray*}%
where $C^{\prime \prime }$ depends on $\mathcal K$ but not on $k$. Weierstrass
M-test and convergence of $\sum_{k}M_{k}$ yields then the desired uniform
convergence.
\end{pfn}
We are now ready to prove our main result, the representation formula
given by Theorem \ref{Main}.
\begin{pfn}{Theorem \ref{Main}}
Let us expand the initial datum $u_{0}=u(\cdot
,0)=\varphi (\cdot ,0)$ in Fourier series with
respect to the orthonormal basis of $L^{2}({\mathbb{R}}^{N})$
introduced in Remark \ref{rem:ortho} as
\begin{equation}
u_{0}=\sum\limits_{\substack{ m,k\in {\mathbb{N}} \\ k\geq 1}}c_{m,k}%
\widetilde{V}_{m,k}\quad \text{in }L^{2}({\mathbb{R}}^{N}),\quad \text{where
}c_{m,k}=\int_{{\mathbb{R}}^{N}}u_{0}(x)\overline{\widetilde{V}_{m,k}(x)}%
\,dx, \label{datoinicial}
\end{equation}%
and, for $t>0$, the function $\varphi (\cdot ,t)$ defined
in \eqref{varphi} as
\begin{equation}
\varphi (\cdot ,t)=\sum\limits_{\substack{ m,k\in {\mathbb{N}} \\ k\geq 1}}%
\varphi _{m,k}(t)\widetilde{V}_{m,k}\quad \text{in }L^{2}({\mathbb{R}}^{N}),
\label{eq:exp_varphi}
\end{equation}%
where
\begin{equation*}
\varphi _{m,k}(t)=\int_{{\mathbb{R}}^{N}}\varphi (x,t)\overline{\widetilde{V}%
_{m,k}(x)}\,dx.
\end{equation*}%
Since $\varphi (z,t)$ satisfies \eqref{varphieq}, we obtain that $\varphi
_{m,k}\in C^{1}({\mathbb{R}},{\mathbb{C}})$ and
\begin{equation*}
i\varphi _{m,k}^{\prime }(t)=\dfrac{\gamma _{m,k}}{1+t^{2}}\varphi
_{m,k}(t),\quad \varphi _{m,k}(0)=c_{m,k},
\end{equation*}%
which by integration yields $\varphi _{m,k}(t)=c_{m,k}e^{-i\gamma
_{m,k}\arctan t}$. Hence expansion (\ref{eq:exp_varphi}) can be rewritten as
\begin{equation*}
\varphi (z,t)=\sum\limits_{\substack{ m,k\in {\mathbb{N}} \\ k\geq 1}}%
c_{m,k}e^{-i\gamma _{m,k}\arctan t}\widetilde{V}_{m,k}(z)\quad \text{in }%
L^{2}({\mathbb{R}}^{N}),\quad \text{ for all }t>0.
\end{equation*}%
In view of \eqref{datoinicial}, the above series can be written as
\begin{equation*}
\varphi (z,t)=\sum\limits_{\substack{ m,k\in {\mathbb{N}} \\ k\geq 1}}%
e^{-i\gamma _{m,k}\arctan t}\bigg(\int_{{\mathbb{R}}^{N}}u_{0}(y)\overline{%
\widetilde{V}_{m,k}(y)}\,dy\bigg)\widetilde{V}_{m,k}(z),
\end{equation*}%
in the sense that, for all $t>0$, the above series converges
in $L^{2}({\mathbb{R}}^{N})$. Since $u_{0}(y)$ can be expanded~as
\begin{equation*}
u_{0}(y)=u_{0}\big(|y|\,\tfrac{y}{|y|}\big)=\sum_{j=1}^{\infty
}u_{0,j}(|y|)\psi _{j}\big(\tfrac{y}{|y|}\big)\quad \text{in }L^{2}({\mathbb{%
S}}^{N-1}),
\end{equation*}%
where $u_{0,j}(|y|)=\int_{{\mathbb{S}}^{N-1}}u_{0}(|y|\theta )\overline{\psi
_{j}(\theta )}\,dS(\theta )$, we conclude that
\begin{align*}
\varphi (z,t)& =\sum\limits_{\substack{ m,k\in {\mathbb{N}} \\ k\geq 1}}%
\frac{e^{-i\gamma _{m,k}\arctan t}}{\Vert V_{m,k}\Vert _{L^{2}}^{2}}%
V_{m,k}(z)\bigg(\int_{0}^{\infty }\!\!u_{0,k}(r)r^{N-1-\alpha _{k}}P_{k,m}(%
\tfrac{r^{2}}{2})e^{-\frac{r^{2}}{4}}\,dr\bigg) \\
& =\sum\limits_{k=1}^{\infty }\psi _{k}\big(\tfrac{z}{|z|}\big)\frac{%
e^{-i(\beta _{k}+1)\arctan t}}{2^{\beta _{k}}\Gamma (1+\beta _{k})}\!\left[
\sum\limits_{m=0}^{\infty }\frac{{\textstyle{\binom{m+\beta _{k}}{m}}}}{%
e^{i2m\arctan t}}\times \right. \\
& \ \ \ \left. \times \bigg(\int_{0}^{\infty }\!\frac{u_{0,k}(r)}{%
|rz|^{\alpha _{k}}}P_{k,m}\big(\tfrac{r^{2}}{2}\big)P_{k,m}\big(\tfrac{%
|z|^{2}}{2}\big)e^{-\frac{r^{2}+|z|^{2}}{4}}r^{N-1}dr\!\bigg)\!\right] \!.
\end{align*}%
By \cite{AS} we know that
\begin{equation*}
P_{k,m}\Big(\frac{r^{2}}{2}\Big)=\dfrac{\Gamma (1+\beta _{k})}{\Gamma
(1+\beta _{k}+m)}e^{\frac{r^{2}}{2}}r^{-\beta _{k}}2^{\frac{\beta _{k}}{2}}%
\displaystyle\int_{0}^{\infty }e^{-t}t^{m+\frac{\beta _{k}}{2}}J_{\beta
_{k}}(\sqrt{2}r\sqrt{t})\,dt,
\end{equation*}%
where $J_{\beta _{k}}$ is the Bessel function of the first kind of order $%
\beta _{k}$. Therefore,
\begin{align*}
\varphi (z,t)& =4\sum\limits_{k=1}^{\infty }\psi _{k}\big(\tfrac{z}{|z|}\big)%
\frac{\Gamma (1+\beta _{k})}{e^{i(\beta _{k}+1)\arctan t}}\Bigg[%
\sum\limits_{m=0}^{\infty }\binom{m+\beta _{k}}{m}\frac{e^{-i2m\arctan t}}{%
(\Gamma (1+\beta _{k}+m))^{2}}\times \\
& \ \ \ \times \!\bigg(\!\int_{0}^{\infty }\frac{u_{0,k}(r)}{|rz|^{\alpha
_{k}+\beta _{k}}}e^{\frac{r^{2}+|z|^{2}}{4}}\left( \int_{0}^{\infty
}\!\!\!\int_{0}^{\infty }\!\!e^{-s^{2}-{s^{\prime }}^{2}}\!(ss^{\prime})^{
2m+\beta _{k}+1}\times \right. \\
& \ \ \ \left. \times J_{\beta _{k}}(\sqrt{2}rs)J_{\beta _{k}}(\sqrt{2}%
|z|s^{\prime })\,ds\,ds^{\prime }\right) \!r^{N-1}dr\!\bigg)\Bigg] \\
& =4\sum\limits_{k=1}^{\infty }\psi _{k}\big(\tfrac{z}{|z|}\big)e^{-i(\beta
_{k}+1)\arctan t}\!\Bigg[\int_{0}^{\infty }u_{0,k}(r)|rz|^{-\alpha
_{k}-\beta _{k}}e^{\frac{r^{2}+|z|^{2}}{4}}r^{N-1}\times \\
& \ \ \ \times e^{i(\arctan t+\frac{\pi }{2})\beta _{k}}\bigg(%
\int_{0}^{\infty }\!\!\!\int_{0}^{\infty }\!\!\!\frac{ss^{\prime }}{%
e^{s^{2}+s^{\prime 2}}}\times \\
& \ \ \ \times \bigg(\sum\limits_{m=0}^{\infty }\frac{(-1)^{m}e^{-i(\arctan
t+\frac{\pi }{2})(2m+\beta _{k})}}{\Gamma (1+m)\Gamma (1+\beta _{k}+m)}%
(ss^{\prime})^{ 2m+\beta _{k}}\bigg)\times \\
& \ \ \ \times J_{\beta _{k}}(\sqrt{2}rs)J_{\beta _{k}}(\sqrt{2}|z|s^{\prime
})ds\,ds^{\prime }\!\bigg)dr\Bigg].
\end{align*}%
Then, since $\sum\limits_{m=0}^{\infty }(-1)^{m}\frac{e^{-i(\arctan
t+\frac{%
\pi }{2})(2m+\beta _{k})}}{\Gamma (1+m)\Gamma (1+\beta
_{k}+m)}(ss^{\prime})^{2m+\beta _{k}}=J_{\beta _{k}}(2e^{-i(\arctan
t+\frac{\pi }{2})}ss^{\prime })$%
, we have
\begin{multline}
\varphi (z,t) \label{eq:2} \\
=4\sum\limits_{k=1}^{\infty }\psi _{k}\big(\tfrac{z}{|z|}\big)e^{i(\beta _{k}%
\frac{\pi }{2}-\arctan t)}\!\Bigg[\int_{0}^{\infty }u_{0,k}(r)|rz|^{-\frac{%
N-2}{2}}e^{\frac{r^{2}+|z|^{2}}{4}}{\mathcal{I}}_{k,t}(r,|z|)r^{N-1}dr\Bigg],
\end{multline}%
where
\begin{equation*}
{\mathcal{I}}_{k,t}(r,|z|)=\int_{0}^{\infty }\!\!\!\int_{0}^{\infty
}\!\!ss^{\prime} e^{-s^{2}-s^{\prime 2}}J_{\beta _{k}}(2e^{-i(\arctan t+\frac{%
\pi }{2})}ss^{\prime })J_{\beta _{k}}(\sqrt{2}rs)J_{\beta _{k}}(\sqrt{2}%
|z|s^{\prime })\,ds\,ds^{\prime }.
\end{equation*}%
From \cite[formula (1), p. 395]{watson} (with $t=s^{\prime }$, $p=1$, $a=%
\sqrt{2}|z|$, $b=2e^{-i(\arctan t+\frac{\pi }{2})}s$, $\nu =\beta _{k}$
which satisfy $\Re (\nu )>-1$ and $|\mathop{\rm arg}p|<\frac{\pi }{4}$), we
know that
\begin{align*}
& \int_{0}^{\infty }\!\!s^{\prime} e^{-s^{\prime 2}}J_{\beta
_{k}}(2e^{-i(\arctan t+\frac{\pi }{2})}ss^{\prime })J_{\beta _{k}}(\sqrt{2}%
|z|s^{\prime })\,ds^{\prime } \\
& =\frac{1}{2}e^{-\frac{|z|^{2}+2e^{-i(2\arctan t+\pi )}s^{2}}{2}}I_{\beta
_{k}}\bigg(\frac{\sqrt{2}|z|s}{e^{i(\arctan t+\frac{\pi }{2})}}\bigg),
\end{align*}%
where $I_{\beta _{k}}$ denotes the modified Bessel function of order $\beta
_{k}$. Hence
\begin{align*}
& {\mathcal{I}}_{k,t}(r,|z|) \\
& \ \ =\dfrac{1}{2}\displaystyle\int_{0}^{\infty }se^{-s^{2}}J_{\beta _{k}}(%
\sqrt{2}rs)e^{-\frac{|z|^{2}+2e^{-i(2\arctan t+\pi )}s^{2}}{2}}I_{\beta
_{k}}(\sqrt{2}e^{-i(\arctan t+\frac{\pi }{2})}|z|s)\,ds \\
& \ \ =\dfrac{1}{4}\displaystyle\int_{0}^{\infty }se^{-\frac{s^{2}}{2}%
}J_{\beta _{k}}(rs)e^{-\frac{|z|^{2}+e^{-i(2\arctan t+\pi )}s^{2}}{2}%
}I_{\beta _{k}}(e^{-i(\arctan t+\frac{\pi }{2})}|z|s)\,ds.
\end{align*}%
Since $I_{\nu }(x)=e^{-\frac{1}{2}\nu \pi i}J_{\nu }(xe^{\frac{\pi }{2}i})$
(see e.g. \cite[9.6.3, p. 375]{AS}), we obtain
\begin{equation*}
{\mathcal{I}}_{k,t}(r,|z|)=\frac{1}{4}e^{-\frac{\beta _{k}}{2}\pi i}e^{-
\frac{|z|^{2}}{2}}\!\!\!\int_{0}^{\infty }\!\!\!se^{-\frac{s^{2}}{2}
(e^{-i(2\arctan t+\pi )}+1)}J_{\beta _{k}}(rs)J_{\beta _{k}}(e^{-i\arctan
t}|z|s)\,ds.
\end{equation*}%
Applying \cite[formula (1), p. 395]{watson} (with $t=s$, $p^{2}=\frac{{%
1+e^{-i(2\arctan t+\pi )}}}{2}$, $a=r$, $b=e^{-i\arctan t}|z|$, $\nu =\beta
_{k}$ which satisfy $\Re (\nu )>-1$ and $|\mathop{\rm arg}p|<\frac{\pi }{4}$%
) and \cite[9.6.3, p. 375]{AS}, we obtain
\begin{align}
& {\mathcal{I}}_{k,t}(r,|z|) \label{eq:calI} \\
=\dfrac{1}{4}e^{-\frac{\beta _{k}}{2}\pi i}& e^{-\frac{|z|^{2}}{2}}\dfrac{1}{%
1+e^{-i(2\arctan t+\pi )}}e^{-\frac{r^{2}+|z|^{2}e^{-2i\arctan t}}{%
2(1+e^{-i(2\arctan t+\pi )})}}I_{\beta _{k}}\bigg(\dfrac{r|z|e^{-i\arctan t}%
}{1+e^{-i(2\arctan t+\pi )}}\bigg) \notag \\
=\frac{1}{4}e^{-\beta _{k}\pi i}& e^{-\frac{|z|^{2}}{2}}\dfrac{1}{%
1+e^{-i(2\arctan t+\pi )}}e^{-\frac{r^{2}+|z|^{2}e^{-2i\arctan t}}{%
2(1+e^{-i(2\arctan t+\pi )})}}J_{\beta _{k}}\bigg(i\dfrac{r|z|e^{-i\arctan t}%
}{1+e^{-i(2\arctan t+\pi )}}\bigg). \notag
\end{align}%
Noticing that
\begin{equation*}
e^{-i\arctan t}=-\frac{i(t+i)}{\sqrt{1+t^{2}}},\quad \dfrac{1}{%
1+e^{-i(2\arctan t+\pi )}}=\frac{t-i}{2t},
\end{equation*}%
from \eqref{eq:2} and \eqref{eq:calI} we deduce
\begin{multline}
\varphi (z,t) \label{eq:3} \\
=\frac{e^{-i\arctan t}}{1+e^{-i(2\arctan t+\pi )}}\sum\limits_{k=1}^{\infty
}\psi _{k}\big(\tfrac{z}{|z|}\big)e^{-i\beta _{k}\frac{\pi }{2}}\!\Bigg[%
\int_{0}^{\infty }\frac{u_{0,k}(r)}{|rz|^{\frac{N-2}{2}}}e^{\frac{r^{2}}{2}%
\big(\frac{1}{2}-\frac{1}{1+e^{-i(2\arctan t+\pi )}}\big)}\times \\
\times e^{-\frac{|z|^{2}}{4}\big(1+\frac{2e^{-2i\arctan t}}{1+e^{-i(2\arctan
t+\pi )}}\big)}J_{\beta _{k}}\bigg(i\dfrac{r|z|e^{-i\arctan t}}{%
1+e^{-i(2\arctan t+\pi )}}\bigg)r^{N-1}\,dr\Bigg] \\
\ =\frac{\sqrt{1+t^{2}}}{2ti}\sum\limits_{k=1}^{\infty }\psi _{k}\big(\tfrac{%
z}{|z|}\big)e^{-i\beta _{k}\frac{\pi }{2}}\!\Bigg[\int_{0}^{\infty }\frac{%
u_{0,k}(r)}{|rz|^{\frac{N-2}{2}}}e^{-\frac{r^{2}}{4it}}e^{-\frac{|z|^{2}}{4it%
}}J_{\beta _{k}}\bigg(\dfrac{r|z|\sqrt{1+t^{2}}}{2t}\bigg)r^{N-1}\,dr\Bigg].
\end{multline}%
From (\ref{eq:3}) and \eqref{varphi} we get that for $t>0$,
\begin{align}
& u(x,t)=(1+t^{2})^{-\frac{N}{4}}e^{\frac{it|x|^{2}}{4(1+t^{2})}}\varphi %
\bigg(\frac{x}{\sqrt{1+t^{2}}},t\bigg) \label{serie} \\
=e^{-\frac{|x|^{2}}{4ti}}& \dfrac{1}{2ti}|x|^{-\frac{N-2}{2}%
}\sum\limits_{k=1}^{\infty }\psi _{k}\big(\tfrac{x}{|x|}\big)e^{-i\beta _{k}%
\frac{\pi }{2}}\!\bigg(\int_{0}^{\infty }u_{0,k}(r)e^{-\frac{r^{2}}{4ti}%
}J_{\beta _{k}}\bigg(\dfrac{r|x|}{2t}\bigg)r^{\frac{N}{2}}\,dr\bigg). \notag
\end{align}%
Notice that, by replacing $\int_{0}^{\infty }$ by $\int_{0}^{R}$ in %
\eqref{serie} one obtains the series representation of the solution $%
u_{R}(x,t)$ with initial data $u_{0,R}(x)\equiv \chi _{R}(x)u_{0}(x)$ with $%
\chi _{R}(x)$ the characteristic function of the ball of radius $R$ centered
at the origin. Since the evolution by Schr\"{o}dinger equation is an
isometry in $L^{2}$, we have that for all $t\in{\mathbb R}$ $\left\Vert u-u_{R}\right\Vert
_{L^{2}}(t)=\left\Vert u_{0}-u_{0,R}\right\Vert _{L^{2}}\rightarrow 0$, as $%
R\rightarrow \infty $. Hence $u(\cdot,t)=\lim_{R\rightarrow \infty }u_{R}(\cdot,t)$ in $L^{2}({\mathbb R}^N)$. Since
\begin{equation*}
u_{0,k}(r)=\int_{{\mathbb{S}}^{N-1}}u_{0}(r\theta )\overline{\psi
_{k}(\theta )}\,dS(\theta ),
\end{equation*}%
and, by hypothesis, the queue of the series
\begin{equation*}
K(x,y)=\sum\limits_{k=1}^{\infty }e^{-i\beta _{k}\frac{\pi }{2}}\psi _{k}%
\big(\tfrac{x}{|x|}\big)\overline{\psi _{k}(\theta )}J_{\beta _{k}}\bigg(%
\dfrac{r|x|}{2t}\bigg)
\end{equation*}%
is uniformly convergent on compacts, we can exchange integral and sum and
write
\begin{align*}
u_{R}(x,t)&=
\frac{e^{-\frac{|x|^{2}}{4ti}}}{2ti}|x|^{-\frac{N-2}{2}%
}\!\int_{0}^{R}\!\!\int_{{\mathbb{S}}^{N-1}}u_{0}(r\theta
)r^{\frac{N}{2}
}e^{-\frac{r^{2}}{4ti}}\times \\
& \quad\times\bigg[\sum\limits_{k=1}^{\infty }e^{-i\beta
_{k}\frac{\pi }{2}}\psi _{k}%
\big(\tfrac{x}{|x|}\big)\overline{\psi _{k}(\theta )}J_{\beta
_{k}}\bigg(
\dfrac{r|x|}{2t}\bigg)\bigg]dr\,dS(\theta )\\
&= \frac{e^{\frac{i|x|^{2}}{4t}}}{i(2t)^{{N}/{2}}}
\int_{B_R}K\bigg(\frac{x}{\sqrt{2t}},\frac{y}{\sqrt{2t}}\bigg)e^{i\frac{|y|^{2}}{%
4t}}u_{0}(y)\,dy .
\end{align*}
Letting $R\rightarrow \infty $, we obtain \eqref{representation} thus
completing the proof of Theorem \ref{Main}.
\end{pfn}
\section{Proof of Theorem
\protect\ref{thm:AB}}\label{sec:ahar-bohm-magn}
\noindent
In view of Remark \ref{rem:t_neg}, it is enough to prove the stated
estimate for $t>0$. Moreover, thanks to Corollary \ref{cor:decay}, it is sufficient to prove condition
\eqref{eq:claim}, namely, uniform boundedness of
\begin{equation*}
K(x,y)=\frac{1}{(2\pi)^2}\sum\limits_{j\in {\mathbb{Z}}}
e^{-i|\alpha -j|\frac{\pi }{2}}e^{-ij\arctan \frac{x_{2}}{x_{1}}
}e^{ij\arctan \frac{y_{2}}{y_{1}}}J_{|\alpha -j|}(|x||y|)
\end{equation*}
which can be written as $K(x,y)=\frac{1}{(2\pi)^2}W\big(\arctan
\frac{x_{2}}{x_{1}}-\arctan \frac{y_{2}}{y_{1}},|x||y|\big)$
where
\begin{equation*}
W(z,s)=\sum\limits_{j\in {\mathbb{Z}}}e^{-i|\alpha -j|\frac{\pi }{2}%
}e^{-ijs}J_{|\alpha -j|}\left( z\right).
\end{equation*}
Notice that
\begin{equation*}
\left\vert \alpha -j\right\vert =
\begin{cases}
\alpha -j,&\text{if }j<\alpha, \\
j-\alpha ,& \text{if }j>\left[ \alpha \right] \equiv j_{0},
\end{cases}
\end{equation*}
so that we can write%
\begin{align*}
W(z,s)&= i^{-\alpha }\sum\limits_{j<-\left\vert j_{0}\right\vert+1
}i^{j}e^{-ijs}J_{\alpha -j}\left( z\right) \\
&\quad +\sum\limits_{-\left\vert j_{0}\right\vert+1 }^{\left\vert j_{0}\right\vert+1
}e^{-i|\alpha -j|\frac{\pi }{2}}e^{-ijs}J_{|\alpha -j|}\left( z\right)
+i^{\alpha }\sum\limits_{j>\left\vert j_{0}\right\vert+1
}i^{-j}e^{-ijs}J_{j-\alpha }\left( z\right)\\
&\equiv i^{-\alpha }S_{1}(z,s)+S_{2}(z,s)+i^{\alpha }S_{3}(z,s).
\end{align*}
$S_{2}(z,s)$ is clearly bounded.
By using identity $9.1.27$ in \cite{AS},
\begin{equation*}
J_{\nu }^{\prime }(z)=\frac{1}{2}(J_{\nu -1}(z)-J_{\nu +1}(z)),
\end{equation*}%
we can compute%
\begin{align} \label{iden1}
\frac{d}{dz}&S_{1}(z,s) =\frac{1}{2}\sum\limits_{j>\left\vert j_{0}\right\vert+1
}i^{-j}e^{ijs}(J_{\alpha +j-1}(z)-J_{\alpha +j+1}(z)) \\
&=\frac{1}{2}\sum\limits_{j>\left\vert j_{0}\right\vert}i^{-(j+1)}e^{i(j+1)s}J_{\alpha +j}(z)-\frac{1}{2}\sum\limits_{j>\left%
\vert j_{0}\right\vert +2}i^{-(j-1)}e^{i(j-1)s}J_{\alpha +j}(z) \notag \\
& =\frac{1}{2}\left( i^{-1}e^{is}J_{\alpha +\left\vert j_{0}\right\vert+1
}(z)+J_{\alpha +\left\vert j_{0}\right\vert +2}(z)\right) i^{-\left\vert
j_{0}\right\vert }e^{i\left\vert j_{0}\right\vert s}-\left( i\cos s\right)
S_{1}(z,s), \notag
\end{align}%
\begin{align} \label{iden2}
&\frac{d}{dz} S_{3}(z,s) =\frac{1}{2}\sum\limits_{j>\left\vert j_{0}\right\vert+1
}i^{-j}e^{-ijs}(J_{-\alpha +j-1}(z)-J_{-\alpha +j+1}(z)) \\
& =\frac{1}{2}\sum\limits_{j>\left\vert j_{0}\right\vert
}i^{-(j+1)}e^{-i(j+1)s}J_{-\alpha +j}(z)-\frac{1}{2}\sum\limits_{j>\left%
\vert j_{0}\right\vert +2}i^{-(j-1)}e^{-i(j-1)s}J_{-\alpha +j}(z) \notag \\
& =\frac{1}{2}\left( i^{-1}e^{-is}J_{-\alpha +\left\vert j_{0}\right\vert+1
}(z)+J_{-\alpha +\left\vert j_{0}\right\vert +2}(z)\right) i^{-\left\vert
j_{0}\right\vert }e^{-i\left\vert j_{0}\right\vert s}-\left( i\cos s\right)
S_{3}(z,s). \notag
\end{align}%
Defining%
\begin{equation*}
F(z,s)=i^{-\alpha }S_{1}(z,s)+i^{\alpha }S_{3}(z,s),
\end{equation*}%
we deduce from (\ref{iden1}) and (\ref{iden2}) that, for every $s$,
$F(\cdot,s)$ satisfies the differential equation
\begin{equation}
\frac{d}{dz} F(z,s)+\left( i\cos s\right) F(z,s)=g(z,s) , \label{de1}
\end{equation}%
with%
\begin{align*}
g(z,s)=&\frac{i^{-\alpha }}{2}\left( i^{-1}e^{is}J_{\alpha +\left\vert
j_{0}\right\vert+1 }(z)+J_{\alpha +\left\vert j_{0}\right\vert +2}(z)\right)
i^{-\left\vert j_{0}\right\vert -1}e^{i(\left\vert j_{0}\right\vert+1) s} \\
\\
&+\frac{i^{\alpha }}{2}\left( i^{-1}e^{-is}J_{-\alpha +\left\vert
j_{0}\right\vert+1 }(z)+J_{-\alpha +\left\vert j_{0}\right\vert +2}(z)\right)
i^{-\left\vert j_{0}\right\vert -1}e^{-i(\left\vert j_{0}\right\vert+1) s}.%
\end{align*}
By integration of (\ref{de1}) we obtain that
\begin{equation*}
F(z,s)=e^{-iz\cos s}\bigg(F(0,s)+\int_0^{z}e^{iz^{\prime }\cos
s}g(z^{\prime },s)dz^{\prime }\bigg).
\end{equation*}%
Since $\left\vert j_{0}\right\vert+1 \pm \alpha >0$,
by the asymptotic behavior of Bessel functions close to the
origin (see formula $9.1.7$ in \cite{AS})
\begin{equation*}
J_{\nu }(x)\simeq \frac{1}{\Gamma (\nu +1)}\left( \frac{x}{2}\right) ^{\nu }
\end{equation*}%
we
conclude that
$F(0,s)=0$,
and hence
\begin{equation*}
F(z,s)=\int_{0}^{z}e^{-i(z-z^{\prime })\cos s}g(z^{\prime },s)dz^{\prime }.
\end{equation*}%
Uniform (in $s$ and $z$) boundedness of $F(z,s)$ follows from uniform
boundedness of the function%
\begin{equation*}
f(z,s)\equiv \int_{0}^{z}e^{iz^{\prime }\cos s}\left( i^{-1}e^{is}J_{\sigma
}(z^{\prime })+J_{\sigma +1}(z^{\prime })\right) dz^{\prime }
\end{equation*}%
for any $\sigma >0$. In order to prove it, we use the identity%
\begin{equation}\label{eq:asintinfti}
J_{\sigma }(z)=\sqrt{\frac{2}{\pi z}}\cos \left( z-\frac{\sigma \pi }{2}-%
\frac{\pi }{4}\right) +\xi_{\sigma} (z)
\end{equation}
with%
\begin{equation}
\left\vert \xi_{\sigma} (z)\right\vert \leq \frac{C_{\sigma}}{z^{\frac{1}{2}}(1+z)},
\label{ineqs}
\end{equation}%
which is a simple consequence of the asymptotic behavior of Bessel
functions at infinity (see formula $9.2.1$ in \cite{AS}).
Therefore,
\begin{align*}
f(z,s) =&\int_{0}^{z}\sqrt{\frac{2}{\pi z^{\prime }}}e^{iz^{\prime }\cos
s}\left( e^{-i\frac{\pi }{2}}e^{is}\cos \left( z^{\prime }-\frac{\sigma \pi
}{2}-\frac{\pi }{4}\right) +\sin \left( z^{\prime }-\frac{\sigma \pi }{2}-%
\frac{\pi }{4}\right) \right) dz^{\prime } \\
&+\int_{0}^{z}e^{iz^{\prime }\cos s}(i^{-1}e^{is}\xi_\sigma (z^{\prime })+\xi_{\sigma+1} (z^{\prime }))dz^{\prime } \equiv
I_{1}(z)+I_{2}(z).
\end{align*}
By (\ref{ineqs}), $I_{2}(z)$ is uniformly bounded.
We notice now that
\begin{gather*}
\sqrt{\frac{2}{\pi z'}}e^{iz^{\prime }\cos s}\left( e^{-i\frac{\pi }{2}%
}e^{is}\cos \left( z'-\frac{\sigma \pi }{2}-\frac{\pi }{4}\right) +\sin
\left( z'-\frac{\sigma \pi }{2}-\frac{\pi }{4}\right) \right)\\
=\frac{1}{2i}\sqrt{\frac{2}{\pi z'}}e^{iz^{\prime }\cos s}\left( \left(
e^{is}+1\right) e^{i\left( z'-\frac{\sigma \pi }{2}-\frac{\pi }{4}\right)
}+\left( e^{is}-1\right) e^{-i\left( z'-\frac{\sigma \pi }{2}-\frac{\pi }{4}%
\right) }\right)
\end{gather*}
and since%
\begin{eqnarray*}
\int_{0}^{z}\frac{e^{iz^{\prime }(\cos s+1)}}{\sqrt{z^{\prime }}}dz^{\prime
} &=&\frac{1}{\sqrt{\cos s+1}}\int_{0}^{z(\cos s+1)}\frac{e^{iy}}{\sqrt{y}}dy
\\
\int_{0}^{z}\frac{e^{iz^{\prime }(\cos s-1)}}{\sqrt{z^{\prime }}}dz^{\prime
} &=&\frac{1}{\sqrt{1-\cos s}}\int_{0}^{z(1-\cos s)}\frac{e^{-iy}}{\sqrt{y}}%
dy
\end{eqnarray*}%
and%
\begin{equation*}
\left\vert \frac{\left( e^{is}+1\right) }{\sqrt{\cos s+1}}\right\vert \leq
C,\quad \left\vert \frac{\left( e^{is}-1\right) }{\sqrt{1-\cos s}}\right\vert
\leq C,
\end{equation*}%
we conclude that $I_{1}(z)$ is uniformly bounded.
Therefore, $K(x,y)$ is uniformly
bounded and then inequality \eqref{eq:decayAB} follows by Corollary \ref{cor:decay}. \qed
\section{Proof of Theorem \protect\ref{thm:inversesquare}}\label{sec:inverse}
In view of Remark \ref{rem:t_neg}, it is sufficient to consider the
case $t>0$. Let $N=3$, $a>-\frac14$, and $K$ as in
\eqref{eq:Sinverse}. The proof of the theorem will follow from the
following estimates for $K$:
\begin{align}
\label{eq:estimate_i_K}
&\text{if }\alpha_1<0,\quad\text{then}\quad\sup_{x,y\in{\mathbb R}^3}|K(x,y)|<+\infty,\\
\label{eq:estimate_ii_K}
&\text{if
}\alpha_1>0,\quad\text{then}\quad\sup_{x,y\in{\mathbb R}^3}\frac{|K(x,y)|}{1+(|x||y|)^{-\alpha_1}}<+\infty.
\end{align}
Before proving the above estimates, let us show how
\eqref{eq:estimate_i_K} and \eqref{eq:estimate_ii_K} imply estimates
\eqref{eq:decayinverse} and \eqref{eq:decayinverse2} respectively, thus
proving Theorem \ref{thm:inversesquare}.
We notice that if
$\alpha_1=0$ then $a=0$ and there is nothing to prove since in this
case the result reduces to classical decay estimates for the free
Schr\"{o}dinger equation.
If $\alpha_1<0$, in view of
\eqref{eq:estimate_i_K}
estimate \eqref{eq:decayinverse} directly follows from Corollary \ref{cor:decay}.
If $\alpha _{1}>0$, then \eqref{eq:estimate_ii_K} and
\eqref{representation} imply that, for some $C>0$ (independent of $x$ and $t$),
\begin{align*}
\left\vert u(x,t)\right\vert &\leq
\frac{C}{t^{\frac32}}\int_{{\mathbb R}^3}\bigg(1+\frac{|x|^{-\alpha_1}|y|^{-\alpha_1}}{t^{-\alpha_1}} \bigg)|u_0(y)|\,dy\\
&=\frac{C}{t^{\frac{3}{2}}}\left\Vert
u_{0}\right\Vert _{L^{1}({\mathbb R}^3)}+\frac{C}{t^{\frac{3}{2}-\alpha _{1}}}\frac{1}{
\left\vert x\right\vert ^{\alpha _{1}}}\int_{\mathbb{R}^3} \frac{\left\vert
u_{0}(y)\right\vert }{\left\vert y\right\vert ^{\alpha _{1}}}dy,\
\end{align*}
for a.e. $x\in{\mathbb R}^3$ and all $t\geq0$,
which implies%
\begin{equation}\label{eq:est-w}
\frac{\left\vert x\right\vert ^{\alpha _{1}}}{1+\left\vert x\right\vert
^{\alpha _{1}}}\left\vert u(x,t)\right\vert \leq C\frac{1+t^{\alpha _{1}}}{
t^{\frac{3}{2}}}\int_{\mathbb{R}} \frac{1+\left\vert y\right\vert ^{\alpha
_{1}}}{\left\vert y\right\vert ^{\alpha _{1}}} \left\vert
u_{0}(y)\right\vert dy.
\end{equation}
Let us introduce the weight function $w(y)=\big( \frac{1+| y|^{\alpha _{1}}}{
|y|^{\alpha _{1}}}\big)^{2}$ and the weighted $L^{p}$ norm
\begin{equation*}
\left\Vert v\right\Vert _{L_{w}^{p}}\equiv
\begin{cases}
\left( \int_{{\mathbb R}^3} |v(y)|^{p}w(y)dy\right)^{{1}/{p}
},&\text{if }1\leq p<+\infty,\\
\mathop{\rm ess\,sup}_{y\in{\mathbb R}^3}|v(y)|,&\text{if }p=+\infty.
\end{cases}
\end{equation*}
$L^2$conservation and \eqref{eq:est-w} yield the estimates
\begin{equation*}
\left\Vert \frac{u(\cdot,t)}{\sqrt{w}}\right\Vert _{L_{w}^{2}}
=\left\Vert \frac{u_{0}}{\sqrt w}\right\Vert
_{L_{w}^{2}} ,\quad
\left\Vert \frac{u(\cdot,t)}{\sqrt w}\right\Vert _{L_{w}^{\infty }}
\leq C\,
\frac{1+t^{\alpha _{1}}}{t^{\frac{3}{2}}}\left\Vert
\frac{u_{0}}{\sqrt w}\right\Vert _{L_{w}^{1}}.
\end{equation*}
Then, letting, for all $p>2$,
$$
\theta_p=1-\frac 2p,\quad p'=\frac{p}{p-1},
$$
so that
$$
\theta_p\in(0,1),\quad \frac1{p'}=\frac{1-\theta_p}2+\frac\theta1,\quad
\frac1{p}=\frac{1-\theta_p}2+\frac\theta\infty,
$$
the Riesz-Thorin interpolation theorem yields
\begin{equation*}
\bigg\|\frac{u(\cdot,t)}{\sqrt w}\bigg\|_{L_w^p}\leq
C^{\theta_p}\bigg(
\frac{1+t^{\alpha _{1}}}{t^{\frac{3}{2}}}\bigg)^{\theta_p} \bigg\|\frac{u_0}{\sqrt w}\bigg\|_{L_w^{p'}}
\end{equation*}
i.e.
\begin{multline*}
\bigg(\int_{{\mathbb R}^3}|u(y,t)|^p \big( \tfrac{1+| y|^{\alpha _{1}}}{
|y|^{\alpha _{1}}}\big)^{2-p}dy\bigg)^{\!\!1/p}\\
\leq C^{1-\frac 2p}\bigg( \frac{1+t^{\alpha
_{1}}}{t^{\frac{3}{2}}}\bigg)^{1-\frac 2p}
\bigg(\int_{{\mathbb R}^3}|u_0(y)|^{p'} \big( \tfrac{1+| y|^{\alpha _{1}}}{
|y|^{\alpha _{1}}}\big)^{2-p'}dy\bigg)^{\!\!1/p'}.
\end{multline*}
Hence, inequality \eqref{eq:decayinverse2} in Theorem \ref%
{thm:inversesquare} follows.
Therefore, in order to prove the theorem, it is sufficient to prove
estimates \eqref{eq:estimate_i_K} and \eqref{eq:estimate_ii_K}.
It is well known that the link between plane waves and a
combination of zonal functions is given by the Jacobi-Anger expansion,
combined with the addition theorem for spherical harmonics (see for example
\cite{watson}, \cite{MU} and the references therein). For $N=3$, we get that
\begin{equation} \label{anger}
e^{-i x\cdot y}=4\pi \sqrt{\dfrac{\pi}{2}} \sum_{\ell=0}^\infty i^{-\ell}
j_{\ell}(|x||y|)Z_{x/|x|}^{(\ell)}(y/|y|).
\end{equation}
We need to estimate the kernel $K$ in \eqref{nucleo} which, as
observed in \eqref{eq:Sinverse}, can be written as
$$
K(x,y)=S\Big(|x||y|,\tfrac{x}{|x|},\tfrac{y}{|y|}\Big)
$$
where
\begin{equation*}
S(r,\theta,\theta')
=\sum_{\ell=0}^{\infty
}i^{-b_\ell}\,j_{-a_\ell} (r)Z_{\theta}^{(\ell)}(\theta'),
\end{equation*}
with $b_\ell=\sqrt{(\ell+1/2)^2+a}$,
$a_\ell=\frac12-\sqrt{(\ell+1/2)^2+a}$. We split the sum into two terms
\begin{align}\label{suma}
S(r,\theta ,\theta^{\prime })&=\sum_{\ell=0}^{\ell_{0}-1}i^{-b_{\ell}}
j_{-a_{\ell}}(r)Z_{\theta }^{(\ell)}\left( \theta^{\prime
}\right)+\sum_{\ell=\ell_{0}}^{\infty }i^{-b_{\ell}}j_{-a_{\ell}}(r)Z_{\theta
}^{(\ell)}\left( \theta^{\prime }\right)\\
\notag&=S_{1}(r,\theta ,\theta^{\prime })+S_{2}(r,\theta ,\theta^{\prime }),
\end{align}
with $\ell_{0}\geq0$ such that $a_{\ell}>0$ for any $\ell<\ell_{0}$
and $a_{\ell}<0$ for any $\ell\geq \ell_{0}$
($S_1$ is meant to be zero if $\ell_0=0$).
Our goal is to show that the singularities in the Jacobi-Anger expansion are
described by the first finite sum $S_{1}$ at the right-hand side of (\ref%
{suma}) while the second term $S_{2}$ at the right-hand side is uniformly
bounded. Such boundedness for $S_{2}$ follows from the arguments below. We
have that
\begin{align}\label{suma2}
S_{2}&=\sum\limits_{\ell=\ell_{0}}^{\infty
}i^{-b_{l}}j_{-a_{l}}(r)Z_{\theta }^{(\ell)}
\left( \theta^{\prime }\right) \\
\notag&=\sum\limits_{\ell=\ell_{0}}^{\infty }i^{-(\ell+\frac{1}{2})}
j_{\ell}(r) Z_{\theta }^{(\ell)}\left( \theta^{\prime
}\right)+\sum\limits_{\ell=\ell_{0}}^{\infty }
(i^{-b_{\ell}}j_{-a_{\ell}}(r)-i^{-(\ell+\frac{1}{2})}j_{\ell}(r))Z_{\theta
}^{(\ell)}\left( \theta^{\prime }\right)\\
\notag&=i^{-\frac{1}{2}}\left[(2\pi)^{-\frac32}
e^{-ir\theta\cdot\theta'}-
\sum_{\ell=0}^{\ell_{0}-1}i^{-\ell}j_{\ell}(r)Z_{\theta
}^{(\ell)}\left( \theta^{\prime
}\right)\right] \\
&\notag\qquad+\sum_{\ell=\ell_{0}}^{\infty }
\Big(i^{-b_{l}}j_{-a_{\ell}}(r)-i^{-(\ell+\frac{1}{2})}j_{\ell}(r)\Big)
Z_{\theta }^{(\ell)}\left( \theta^{\prime }\right).
\end{align}
The first term at the right hand side of (\ref{suma2}) is clearly bounded,
since it is the difference between a plane wave and the first $(\ell_{0}-1)$
terms of its Jacobi-Anger expansion.
We first notice that the second term at the right hand side of
(\ref{suma2}) is bounded for $r\leq\delta$ if $\delta>0$ is sufficiently small.
Indeed from the estimates
\begin{align}
\label{eq:stimaJ} &|J_\nu(t)|\leq \frac{1}{\Gamma(1+\nu)}\Big(\frac t2\Big)^\nu
e^{t^2/4},\quad \text{for all }\nu>0,\ t\geq0,\\
\notag &|Z_{\theta }^{(\ell)}\left( \theta^{\prime
}\right)|\leq Z_{\theta }^{(\ell)}(\theta)= \frac{2\ell+1}{4\pi
},\quad \text{for all }\ell\geq0,\ \theta,\theta'\in{\mathbb S}^2,
\end{align}
see for example \cite{MU}, it follows that, if $r\leq\delta$,
\begin{align*}
&\bigg|\sum_{\ell=\ell_{0}}^{\infty
}i^{-b_\ell}j_{-a_{\ell}}(r)Z_{\theta }^{(\ell)}(\eta )\bigg|\leq
\sum_{\ell=\ell_{0}}^{\infty }\dfrac{2\ell+1}{4\pi \Gamma ({b_{\ell}+1})}\dfrac{(
\frac{r}{2})^{b_{\ell}}}{r^{\frac{1}{2}}}e^{r^2/4}\\
&\leq
\frac{e^{\delta^2/4}}{\sqrt2\, 4\pi}
\sum_{\ell=\ell_{0}}^{\infty }\dfrac{2\ell+1}{\Gamma
({b_{\ell}+1})}\Big(\frac r2\Big)^{-a_\ell}
\leq
\frac{e^{\delta^2/4}}{\sqrt2\, 4\pi}\Big(\frac r2\Big)^{-a_{\ell_0}}
\sum_{\ell=\ell_{0}}^{\infty }\dfrac{2\ell+1}{\Gamma
(b_{\ell}+1)}\leq C r^{-a_{\ell_0}}
\end{align*}
for some constant $C>0$ dependent on $\delta$ and $\ell_0$ but
independent of $r,\theta,\theta'$.
Next, for $r>\delta $, we write
\begin{align}\label{suma4}
\sum\limits_{\ell=\ell_{0}}^{\infty }&(i^{-b_{\ell}}j_{-a_{\ell}}(r)-i^{-(\ell+
\frac{1}{2})}j_{\ell}(r))Z_{\theta }^{(\ell)}\left( \theta^{\prime }\right) \\
&\notag=\dfrac{1}{2\pi ir^{\frac{1}{2}}}\displaystyle\int_{\gamma }e^{\frac{r}{2}
\left( z-\frac{1}{z}\right) }\left( \sum\limits_{\ell=\ell_{0}}^{\infty } \left[
(iz)^{\ell+\frac{1}{2}-b _{\ell}}-1\right]\frac{Z_{\theta }^{(\ell)}\left(
\theta^{\prime }\right)}{(iz)^{\ell+\frac{1}{2}}} \right) \frac{dz}{z},
\end{align}
where we have used the following representation for Bessel functions%
\begin{equation*}
J_{\nu }(r)=\frac{1}{2\pi i}\int_{\gamma }e^{\frac{r}{2}\left( z-\frac{1}{z}%
\right) }\frac{dz}{z^{\nu +1}},
\end{equation*}%
with $\gamma $ being the positive oriented contour represented in Figure \ref{fig:pspic} (see
\cite[5.10.7]{lebedev}). We have also exchanged sum and integral, which is allowed for any $r, \theta, \theta'$, as we will see below.
For convenience, we split the integral along $\gamma $ into the integrals $%
I_{1}$, along the circumference of radius $1$ (to be denoted as $\Gamma _{1}$%
), and the integral $I_{2}$, along the lines running between $z=-\infty $
and $z=-1$ (to be denoted as $\Gamma _{2}$):%
\begin{equation*}
\int_{\gamma }=\int_{\Gamma _{1}}+\int_{\Gamma _{2}}\equiv I_{1}+I_{2}.
\end{equation*}%
Notice that, by analyticity of the integrand outside $z=\mathbb{R}%
^{-}+i0^{\pm }$, we can write%
\begin{equation*}
\int_{\gamma }=\int_{\Gamma _{1}^{\varepsilon }}+\int_{\Gamma
_{2}^{\varepsilon }}=\lim_{\varepsilon \rightarrow 0^{+}}\left( \int_{\Gamma
_{1}^{\varepsilon }}+\int_{\Gamma _{2}^{\varepsilon }}\right)
\end{equation*}%
where $\Gamma _{1}^{\varepsilon }$ is the circumference of radius $%
1+\varepsilon $ around the origin and $\Gamma _{2}^{\varepsilon }$ runs
along $\left( -\infty ,-1-\varepsilon \right) +i0^{\pm }$. Notice that, for the integral along $\Gamma_{1}^{\varepsilon }\cup \Gamma_{2}^{\varepsilon }$ since $|z|>1$, one has absolute convergence for any given $r, \theta, \theta'$ and hence the exchange of integral and sum performed in formula \eqref{suma4} is allowed by Fubini's Theorem.
We start
estimating the integral along $\Gamma _{1}$. Taking into account that
\begin{equation*}
b_{\ell }-\big(\ell +\tfrac{1}{2}\big)=\sqrt{\left( \ell +\tfrac{1}{2}%
\right) ^{2}+a}-\tfrac{1}{2}-\ell =\tfrac{a}{2\ell +1}+O(\ell ^{-3}),
\end{equation*}%
we have that
\begin{align}
\left[ (iz)^{(\ell +\frac{1}{2})-b_{\ell }}-1\right] & =-\frac{a}{2\ell +1}%
\log (iz)+\frac{a^{2}}{2}\frac{(\log (iz))^{2}}{(2\ell +1)^{2}}+\frac{O(1)}{%
\ell ^{3}} \label{suma5} \\
& \equiv J_{1,1}(z,\ell )+J_{1,2}(z,\ell )+J_{1,3}(z,\ell ) \nonumber
\end{align}%
as $\ell \rightarrow +\infty $ uniformly with respect to $z\in \Gamma _{1}$.
Since $z^{-b_{\ell }}$ and $z^{(\ell +\frac{1}{2})}$ have a branch-cut at $z\in
\mathbb{R}^{-}$, the function $\log (iz)$ will also have a branch-cut at $z\in
\mathbb{R}^{-}$, as well as the function $(iz)^{\frac{1}{2}}$ that will
appear below. From (\ref{suma5}) the contribution of the right hand side of (
\ref{suma4}) on $\Gamma _{1}$ can be written
\begin{equation*}
I_{1}=\dfrac{1}{2\pi ir^{\frac{1}{2}}}\displaystyle\int_{\Gamma _{1}}e^{%
\frac{r}{2}\left( z-\frac{1}{z}\right) }\left( \sum\limits_{\ell =\ell
_{0}}^{\infty }\Big(J_{1,1}(z,\ell )+J_{1,2}(z,\ell )+J_{1,3}(z,\ell )\Big)%
\frac{Z_{\theta }^{(\ell )}\left( \theta ^{\prime }\right) }{(iz)^{\ell +%
\frac{1}{2}}}\right) \frac{dz}{z}
\end{equation*}%
\begin{equation*}
\equiv \mathcal{J}_{1,1}+\mathcal{J}_{1,2}+\mathcal{J}_{1,3}\ ,
\end{equation*}%
where every summand $\mathcal{J}_{1,i},\,i=1,2,3,$ corresponds to the integrand with the corresponding $J_{1,i}$.
Since on $\Gamma _{1}$ we have that $|z|=1$ and then $\big|e^{\frac{r}{2}
\left( z-\frac{1}{z}\right) }\big|=1$, from the estimate $\big|Z_{\theta
}^{(\ell )}\big(\theta ^{\prime }\big)\big|\leq \frac{2\ell +1}{4\pi }$ we
deduce that, if $r>\delta $,
\begin{equation*}
\left\vert \mathcal{J}_{1,3}\right\vert \leq \mathrm{const\,}
r^{-1/2}\sum\limits_{\ell =\ell _{0}}^{\infty }\frac{2\ell +1}{\ell ^{3}}
\leq \mathrm{const\,}\delta ^{-1/2},
\end{equation*}%
and hence $\left\vert \mathcal{J}_{1,3}\right\vert $ is
bounded. Concerning
$\mathcal{J}_{1,1}$, we notice that
\begin{align}\label{serie1}
-a\log (iz)&\sum_{\ell=0}^{\infty }\frac{Z_{\theta }^{(\ell)}\left(
\theta ^{\prime }\right)
}{2\ell+1}(iz)^{-\ell-\frac{1}{2}}=-\frac{a\log (iz)}{4\pi }
\sum_{\ell=0}^{\infty }P_{\ell}(\theta \cdot \theta ^{\prime })(iz)^{-\ell-\frac{1}{2}}\\
\notag&=-\frac{1}{4\pi }\frac{a(iz)^{-\frac{1}{2}}\log
(iz)}{\sqrt{1+2\theta \cdot \theta ^{\prime
}\frac{i}{z}-\frac{1}{z^{2}}}}=-\frac{1}{4\pi }\frac{a(-iz)^{
\frac{1}{2}}\log (iz)}{\sqrt{z^{2}+2iz(\theta \cdot \theta
^{\prime })-1}}
\end{align}
where we have used the well-known identity (see for example \cite{MU})
\begin{equation}\label{Zonal}
4\pi \frac{Z_{\theta }^{(\ell)}\left( \theta ^{\prime }\right) }{2\ell+1}
=P_{\ell}(\theta \cdot \theta ^{\prime }),
\end{equation}
with $P_{\ell}$ being the Legendre polynomial of index $\ell$, and the identity (see
Formula 22.9.12 in \cite{AS})
\begin{equation}\label{Legendre}
\sum_{\ell=0}^{\infty }P_{\ell}(t)w^{\ell}=\frac{1}{\sqrt{1-2wt+w^{2}}},
\end{equation}
which is valid for $\left\vert w\right\vert <1$. Hence, identity (\ref
{serie1}) is valid for $\left\vert z\right\vert >1$. Therefore
\begin{equation}\label{suma3}
\mathcal{J}_{1,1}\sim (Bounded\,\,\,\,terms)-\frac{a}{8\pi ^{2}ir^{\frac{1}{2
}}}\int_{\Gamma _{1}^{\varepsilon }}e^{\frac{r}{2}\left( z-\frac{1}{z}
\right) }\frac{\log (iz)(-iz)^{\frac{1}{2}}}{\sqrt{z^{2}+2iz(\theta \cdot
\theta ^{\prime })-1}}\frac{dz}{z},
\end{equation}
where the first term at the right-hand side of (\ref{suma3})
represents a finite sum of terms, that are needed to complete the
series (\ref{serie1}) from $\ell=0$ to $\ell=\ell_{0}-1$, and which
are uniformly bounded. Since $|e^{\frac{r}{2}( z-1/{z}) }|=1$ for all
$r>0$ and $z\in\Gamma _{1}$, if $1-(\theta \cdot \theta ^{\prime })^2$
does not approach zero, i.e. if $\theta \cdot \theta ^{\prime }$ stays
far away from $\pm1$, the second term at the right hand side of
(\ref{suma3}) is uniformly bounded with respect to $\e\to0^+$,
$r>\delta$, and $1-(\theta\cdot\theta')^2>\delta$, due to the
integrability of the two square root singularities of the integrand at
$z_{\pm}=-i(\theta \cdot \theta ^{\prime })\pm \sqrt{
1-(\theta\cdot\theta^{\prime })^{2}}$.
When $(\theta \cdot \theta ^{\prime })=\mp1$, the two square root
singularities at $z_{\pm }$ collapse into a stronger singularity at
$z=\pm i$. Let us discuss e.g. the case $(\theta \cdot \theta ^{\prime
})=-1$ (the case $(\theta \cdot \theta ^{\prime
})=1$ can be treated similarly); then
\begin{multline}\label{singint}
\lim_{\varepsilon \rightarrow 0^{+}}\int_{\Gamma _{1}^{\varepsilon
}}e^{ \frac{r}{2}\left( z-\frac{1}{z}\right)
}\frac{(-iz)^{\frac{1}{2}}\log (iz)}{
z-i}\dfrac{dz}{z}\\
=\pi ^{2}ie^{ir}+PV\int_{\Gamma _{1}}e^{\frac{r}{2}\left( z-
\frac{1}{z}\right) }\frac{\log
(iz)}{z-i}(-iz)^{\frac{1}{2}}\dfrac{dz}{z}.
\end{multline}
Equation (\ref{singint}) is simply the Plemelj-Sokhotskyi formula (see
for instance \cite{AF}) for the limit of Cauchy integrals when
approaching a singular point. The first term at the right hand side of
(\ref{singint}) is clearly bounded. The second term at the right hand
side of (\ref{singint}) is a singular integral of the function
$e^{\frac{r}{2}( z-\frac{1}{z})}\frac{\log (iz)(-iz)^{1/2}}{z}$ which
is differentiable for $z=e^{i\theta }$ with $\theta $ in the
neighborhood of $\frac{\pi }{2}$ (remind that the discontinuity of
the argument of $z$ is along the negative real line).
Hence, since the principal value of a Cauchy integral of a
differentiable function is bounded (cf. \cite{AF}), we conclude the
boundedness of $\mathcal{J}_{1,1}$ for any $r>\delta $.
The fact that the principal value integral is bounded for any $r$ does not
exclude the possibility of its diverging as $r\rightarrow \infty $. In order
to exclude this possibility, we consider a neighborhood in $\Gamma _{1}$ of $%
z=i$:
\begin{equation*}
\Gamma _{1}^{s_{0}}=\left\{ z=e^{i\left( \frac{\pi }{2}+s\right) },\
\left\vert s\right\vert <s_{0}\ll 1\right\}
\end{equation*}
and we integrate there for any $r\gg 1$ having into account that
$$
\frac{(\pi +s)e^{i
\frac{s}{2}}}{i(e^{is}-1)}=-\frac{\pi}{s}+ O(1)\quad\text{for } s\thicksim 0.
$$
Hence,
\begin{align*}
PV\displaystyle\int\limits_{-s_{0}}^{s_{0}}e^{ir\cos s}\frac{(\pi
+s)e^{i \frac{s}{2}}}{i(e^{is}-1)}ds&= PV\displaystyle\int
\limits_{-s_{0}}^{s_{0}}e^{ir\cos s} \big(-\frac{\pi}{s}+
O(1)\big)\,ds\\
&=-PV\displaystyle\int\limits_{-s_{0}}^{s_{0}}\pi\frac{ e^{ir\cos
s}}{s}\, ds +PV\displaystyle\int\limits_{-s_{0}}^{s_{0}} O(1)
e^{ir\cos s}\,ds.
\end{align*}
Since $PV\int\limits_{-s_{0}}^{s_{0}}\pi\frac{ e^{ir\cos
s}}{s}\, ds=0$, it follows that
\begin{equation*}
PV\displaystyle\int\limits_{-s_{0}}^{s_{0}}e^{ir\cos s}\frac{(\pi
+s)e^{i \frac{s}{2}}}{i(e^{is}-1)}ds=O(1).
\end{equation*}
Hence, the integral
along $\Gamma _{1}^{s_{0}}$ is uniformly bounded for any $r.$ The
integral over $\Gamma _{1}\backslash \Gamma _{1}^{s_{0}}$ is also
uniformly bounded since $\big\vert e^{\frac{r}{2}\left(
z-\frac{1}{z}\right) }\big\vert =1$. Hence, the principal
value integral over $\Gamma _{1}$ is uniformly bounded.
If one considers the two singularities $z_{\pm }$ sufficiently close,
then, similarly to (\ref{singint}), the integral over $\Gamma _{1}$ can
be written as%
\begin{equation}\label{intdelta}
\int_{\Gamma _{1}}=\int_{Arc(z_{-},z_{+})}+\int_{\Gamma _{1}\backslash
Arc(z_{-},z_{+})}
\end{equation}%
where $Arc(z_{-},z_{+})$ is the small arc of $\Gamma _{1}$ between $z_{+}$
and $z_{-}$. The second integral at the right hand side of (\ref{intdelta})
can be easily estimated just like the principal value above and yields the
same estimates uniformly in $\theta,\theta'$. The first term at the right hand side
of (\ref{intdelta}) is, after writing
\begin{equation*}
z_\pm=e^{i\left(\frac\pi2\pm\overline s\right)}
=-i(\theta\cdot\theta')\pm\sqrt{1-(\theta\cdot\theta')^2},
\end{equation*}
the integral of
\begin{align*}
&
\frac{e^{ir\cos s}(s+\pi)e^{i\frac s2}}
{\sqrt{\left(e^{i\left(s+\frac\pi2\right)}-
e^{i\left(\frac\pi2-\overline s\right)}\right)
\left(e^{i\left(s+\frac\pi2\right)}-
e^{i\left(\overline s+\frac\pi2\right)}\right)}}
\\
&
=
\frac{\pi e^{ir\cos s}}{\sqrt{|s+\overline s||s-\overline s|}}(1+O(s-\overline s)+O(s+\overline s))
\times
\begin{cases}
-1,
\quad\text{if }
s<-\overline s
\\
+1
\quad\text{if }
s>\overline s
\\
-i
\quad\text{if }
-\overline s<s<\overline s
\end{cases}
\\
&
=
\nu(s)
\frac{\pi e^{ir\cos s}}{\sqrt{|s+\overline s||s-\overline s|}}+O\left(\frac1{\sqrt{|s-\overline s|}}\right)
+O\left(\frac1{\sqrt{|s+\overline s|}}\right),
\end{align*}
where $|\nu(s)|=1$. As a consequence,
we can estimate
\begin{align*}
\left|\int_{Arc(z_-,z_+)}\right|
&
=
\left|\int_{-\overline s}^{\overline s}
\frac{\pi e^{ir\cos s}}{\sqrt{\overline s^2-s^2}}ds+O(1)\right|
\\
&
=
\left|\int_{-1}^{1}
\frac{\pi e^{ir\cos(\overline st)}}{\sqrt{1-t^2}}dt+O(1)\right|
\\
&
\leq {\rm const},
\end{align*}
uniformly with respect to $r$ and $\overline s$.
Therefore, we conclude that the integral on $\Gamma_1$ is
uniformly bounded both in $\delta $ and $r$.
Finally, the term $%
J_{1,2}$ in (\ref{suma5}), inserted at the right hand side of (\ref{suma4}),
produces
$$
\dfrac{1}{2\pi ir^{\frac{1}{2}}}\displaystyle\int_{\Gamma _{1}}e^{%
\frac{r}{2}\left( z-\frac{1}{z}\right) }\frac{a^{2}(\log (iz))^{2}}{2}\left( \sum\limits_{\ell =\ell
_{0}}^{\infty }
\frac{Z_{\theta }^{(\ell )}\left( \theta ^{\prime }\right)}{(2\ell +1)^{2}} {(iz)^{-\ell -%
\frac{1}{2}}}\right) \frac{dz}{z}
$$
where the series%
\begin{equation*}
F(\theta,\theta ^{\prime },z)=\sum_{\ell =0}^{\infty }\frac{Z_{\theta }^{(\ell
)}\left( \theta ^{\prime }\right) }{(2\ell +1)^{2}}(iz)^{-\ell -\frac{1}{2}}=\dyle\int g(\theta,\theta', z) \,dz
\end{equation*}%
is the primitive in $z$ of the series%
\begin{equation*}
g(\theta,\theta',z)=-\frac{i}{2}\sum_{\ell =0}^{\infty }\frac{Z_{\theta }^{(\ell )}\left( \theta
^{\prime }\right) }{2\ell +1}(iz)^{-\ell -\frac{3}{2}}.
\end{equation*}%
Thus, using \eqref{Zonal} and \eqref{Legendre}, we conclude that $F(\theta,\theta ^{\prime },z)$ is the primitive of
$$
g(\theta,\theta',z)=-\frac{i}{8\pi }\frac{%
(iz)^{-\frac{3}{2}}}{\sqrt{1+2(\theta \cdot \theta ^{\prime })\frac{i}{z}-%
\frac{1}{z^{2}}}}
$$
and hence, since $g(\theta,\theta',z)$ presents a square root singularity if $\theta \cdot
\theta ^{\prime }\neq -1$ or a $1/(z-i)$ singularity at $z=i$ if $\theta
\cdot \theta ^{\prime }=-1$, we conclude that $F(\theta,\theta',z)$ presents at most a
log-type singularity, which is integrable, and consequently the integral yields a uniformly bounded contribution $\mathcal{J}_{1,2}$.
Therefore, we conclude that $I_{1}$ is uniformly bounded.
We continue estimating $I_{2}$,
$$
I_{2}=\dfrac{1}{2\pi ir^{\frac{1}{2}}}\displaystyle\int_{\Gamma_2 }e^{\frac{r}{2}
\left( z-\frac{1}{z}\right) }\left( \sum\limits_{\ell=\ell_{0}}^{\infty } \left[
(iz)^{\ell+\frac{1}{2}-b _{\ell}}-1\right]\frac{Z_{\theta }^{(\ell)}\left(
\theta^{\prime }\right)}{(iz)^{\ell+\frac{1}{2}}} \right) \frac{dz}{z}.
$$
Introducing the changes of variables, $z=e^{\pm \pi i}e^{t}$, exchanging sum and integral (arguing as above)
and rearranging terms, we rewrite it in the form%
\begin{equation*}
I_{2}=\dfrac{1}{2\pi ir^{\frac{1}{2}}}\sum_{\ell =\ell _{0}}^{\infty }Z_{\theta }^{(\ell )}\left( \theta
^{\prime }\right) (A_{\ell }(r)+B_{\ell }(r))\equiv \dfrac{1}{2\pi ir^{\frac{1}{2}}}\big(\mathcal{J}_{2,1}+%
\mathcal{J}_{2,2})
\end{equation*}%
where
\begin{equation*}
A_{\ell }(r)=-\dfrac{2\sin {(\pi b_{\ell })}}{i^{b_{\ell }-1}}\displaystyle%
\int_{0}^{\infty }e^{-r\sinh t}(e^{-b_{\ell }t}-e^{-(\ell +\frac{1}{2})t})dt
\end{equation*}%
and
\begin{equation*}
B_{\ell }(r)=\displaystyle -2i\int_{0}^{\infty }e^{-r\sinh t}e^{-(\ell +\frac{1%
}{2})t}\bigg(\dfrac{\sin {\pi b_{\ell }}}{i^{b_{\ell }}}-\dfrac{\sin {\pi
(\ell +\frac{1}{2})}}{i^{\ell +\frac{1}{2}}}\bigg)\,dt.
\end{equation*}%
We estimate $A_{\ell }$ by using again that $|Z_{\theta }^{(\ell )}\left(
\theta ^{\prime }\right) |\leq \frac{2\ell +1}{4\pi }$,
\begin{align*}
\left\vert \mathcal{J}_{2,1}\right\vert &=\left\vert \sum_{\ell =\ell
_{0}}^{\infty }Z_{\theta }^{(\ell )}\left( \theta ^{\prime }\right) A_{\ell
}(r)\right\vert \leq C\sum_{\ell =\ell _{0}}^{\infty }(2\ell
+1)\int_{0}^{\infty }e^{-r\sinh t}\left\vert e^{-b_{\ell }t}-e^{-(\ell +
\frac{1}{2})t}\right\vert dt\\
&\leq C\sum_{\ell_{0}}^{\infty }(2\ell+1)\left\vert \frac{1}{b _{\ell}}-\frac{1}{\ell+
\frac{1}{2}}\right\vert \leq C.
\end{align*}
In order to estimate $B_{\ell }$, notice that
\begin{equation*}
\bigg(\dfrac{\sin {\pi b_{\ell }}}{i^{b_{\ell }}}-\dfrac{\sin {\pi (\ell +%
\frac{1}{2})}}{i^{\ell +\frac{1}{2}}}\bigg)
\end{equation*}%
\begin{equation*}
=\frac{1}{i^{{\ell +\frac{1}{2}}}}\left[ \sin {\pi b_{\ell }}-\sin {\pi
(\ell +\frac{1}{2})}\right] -\sin {\pi b_{\ell }}\left[ \frac{1}{i^{{\ell +%
\frac{1}{2}}}}-\frac{1}{i^{b_{\ell }}}\right]
\end{equation*}%
\begin{equation*}
=-\frac{(-1)^{{\ell }}}{i^{{\ell +\frac{1}{2}}}}\frac{1}{2}\left( \frac{a{%
\pi }}{2{\ell +1}}\right) ^{2}-\frac{(-1)^{{\ell }}}{i^{{\ell +\frac{1}{2}}}}%
\left( \frac{a{\pi }}{2(2\ell +1)}i+\frac{a^{2}{\pi }^{2}}{8(2{\ell +1)}^{2}}%
\right)+O(\ell
^{-3})
\end{equation*}%
\begin{equation*}
=-\frac{i^{{\ell +\frac{1}{2}}}}{2{\ell +1}}\frac{a{\pi }}{2}-\frac{5 i^{{\ell -\frac{1}{2}}}}{8} \bigg(\frac{a\pi}{(2{\ell +1)}}\bigg)^2+O(\ell
^{-3}).
\end{equation*}%
By using formula \eqref{Legendre}, it readily follows
\begin{align*}
\left\vert \mathcal{J}_{2,2}\right\vert =&\left\vert \sum_{\ell
=\ell _{0}}^{\infty }Z_{\theta }^{(\ell )}\left( \theta ^{\prime
}\right) B_{\ell }(r)\right\vert \leq \left\vert
\displaystyle\int_{0}^{\infty }e^{-r\sinh
t}e^{-\frac{t}{2}}\times \right.\\
& \times \Bigg[K_{\ell_{0}}(r,\theta,\theta',t)- i^{\frac{1}{2}}\dfrac{a\pi }{4}\dfrac{1}{\sqrt{1-ie^{-t}(\theta \cdot \theta ^{\prime })-e^{-2t}}}\\
&\quad -\sum_{\ell =\ell _{0}}^{\infty }\frac{5a^{2}\pi ^{2}i^{{\ell
-\frac{1}{2}}}e^{-{\ell t} }Z_{\theta }^{(\ell
)}(\theta')}{16}\left( \frac{1}{(2{\ell +1)} ^{2}}+O(\ell
^{-3})\right) \Bigg]\Bigg\vert ,
\end{align*}
where $K_{\ell_{0}}$ accounts for the terms that need to be added in
order to use \eqref{Legendre} and which is uniformly bounded.
Since $\sqrt{|1-ie^{-t}(\theta \cdot \theta ^{\prime })-e^{-2t}|}\geq
\frac{{\rm const\,}\sqrt{t}}{1+\sqrt{t}}$ for some ${\rm const\,}>0$ and, using again that $|Z_{\theta
}^{(\ell )}\left( \theta ^{\prime }\right) |\leq \frac{2\ell +1}{4\pi
}$,
\begin{equation*}
\left\vert \sum_{\ell =\ell _{0}}^{\infty }i^{{\ell }}e^{-{\ell t}%
}Z_{\theta }^{(\ell )}\left(
\theta ^{\prime }\right)\left( \frac{1}{(2{\ell +1)}%
^{2}}+O(\ell ^{-3})\right) \right\vert \leq C\sum_{\ell =\ell _{0}}^{\infty }%
\frac{e^{-{\ell t}}}{{\ell }}\leq 2C\log \left\vert t\right\vert
\end{equation*}%
for some constat $C$ and any $\theta, \theta ^{\prime }$, we conclude
the existence of other constants $C^{\prime },C''$ such that
\begin{equation*}
\left\vert \mathcal{J}_{2,2}\right\vert \leq \frac{C^{\prime }}{4}
\int_{0}^{\infty }e^{-r\sinh t}e^{-\frac{t}{2}}\left[ \dfrac{1+\sqrt{t}}{
\sqrt{t}}+\log \left\vert t\right\vert \right] \leq C''.
\end{equation*}
Hence the uniform boundedness of $\mathcal{J}_{2,2}$ follows. We conclude then
\begin{equation}\label{eq:supS2}
\sup_{\substack{ r\geq 0 \\ \theta ,\theta ^{\prime }\in {\mathbb{S}}^{N-1}}}%
|S_{2}(r,\theta ,\theta ^{\prime })|<+\infty .
\end{equation}
If $\alpha_1=a_0<0$, then $\ell_0=0$. Hence $S_1(r,\theta
,\theta')=0$ and \eqref{eq:estimate_i_K} is proved.
\noindent If $\alpha_1=a_0>0$, then $\ell_0>0$. From \eqref{eq:stimaJ} and the
fact that $a_\ell\leq a_{0}$ for all $\ell\in{\mathbb N}$, we
deduce that
\begin{equation}\label{eq:supS11}
|S_1(r,\theta
,\theta')|\leq {\rm const\,}r^{-a_0}={\rm
const\,}r^{-\alpha_1}\quad\text{for all }r\leq1,\
\theta,\theta'\in{\mathbb S}^{N-1}.
\end{equation}
On the other hand, from \eqref{eq:asintinfti} and \eqref{ineqs} we
easily deduce that
\begin{equation}\label{eq:supS12}
|S_1(r,\theta
,\theta')|\leq {\rm const\,}\quad\text{for all }r\geq1,\
\theta,\theta'\in{\mathbb S}^{N-1}.
\end{equation}
Estimate \eqref{eq:estimate_ii_K} follows from \eqref{suma},
\eqref{eq:supS2}, \eqref{eq:supS11}, and \eqref{eq:supS12}.\qed
\begin{figure}[h]
\begin{pspicture}(-4,-2.5)(4,2.5)
\psset{arrowscale=1.7}
\psline[linewidth=0.02cm](-4,0)(4,0)
\psline[linewidth=0.02cm](0,-2.5)(0,2.5)
\psline[linewidth=0.04cm]{->}(-4,-0.2)(-0.95,-0.2)
\psline[linewidth=0.04cm]{->}(-4,-0.2)(-2,-0.2)
\psline[linewidth=0.04cm]{->}(-4,-0.2)(-3,-0.2)
\psline[linewidth=0.04cm]{<-}(-4,0.2)(-1,0.2)
\psline[linewidth=0.04cm]{<-}(-3,0.2)(-1,0.2)
\psline[linewidth=0.04cm]{<-}(-2,0.2)(-1,0.2)
\usefont{T1}{ptm}{m}{n}
\rput(1,1){{$\bf\Gamma_1$}}
\usefont{T1}{ptm}{m}{n}
\rput(-2.5,0.5){{$\bf\Gamma_2$}}
\psarc[linewidth=0.04]{->}(0,0){1.001998}{190.6}{169.6900423}
\psarc[linewidth=0.04]{->}(0,0){1.001998}{190.6}{90}
\psarc[linewidth=0.04]{->}(0,0){1.001998}{190.6}{0}
\psarc[linewidth=0.04]{->}(0,0){1.001998}{190.6}{-90}
\end{pspicture}
\caption{Integration oriented domain $\gamma$.}
\label{fig:pspic}
\end{figure}
|
{
"timestamp": "2012-03-09T02:02:35",
"yymm": "1203",
"arxiv_id": "1203.1771",
"language": "en",
"url": "https://arxiv.org/abs/1203.1771"
}
|
\section{Introduction}
Since the $B_c$ meson is the lowest bound state of two different
heavy quarks with open flavor, it is stable against strong and
electromagnetic annihilation processes. The $B_c$ meson therefore
decays weakly. Furthermore, the $B_c$ meson has a sufficiently large
mass, thus each of the two heavy quarks can decay individually. It
has rich decay channels, and provides a very good place to study
nonleptonic weak decays of heavy mesons, to test the standard model
and to search for any new physics signals \cite{iiba}. The current
running LHC collider will produce much more $B_c$ mesons than ever
before to make this study a bright future.
Within the standard model (SM), for the double charm decays of
$B_{u,d,s}$ mesons, there are penguin operator contributions as
well as tree operator contributions. Thus the direct CP asymmetry
may be present. However, the double charm decays of $B_{c}$ meson
are pure tree decay modes, which are particularly well suited to
extract the Cabibbo-Kobayashi-Maskawa (CKM) angles due to the
absented interference from penguin contributions. As was pointed out
in ref. \cite{plb286160} and further elaborated in ref.
\cite{prd62057503,jpg301445,plb555189,prd65034016}, the decays
$B_c\rightarrow D_s^+D^0, D_s^+ \bar{D}^0$ are the gold-plated modes
for the extraction of CKM angle $\gamma$ though amplitude relations
because their decay widths are expected to be at the same order of
magnitude. But this needs to be examined by faithful calculations.
Although many investigations on the decays of $B_c$ to double-charm
states have been carried out
\cite{jpg301445,plb555189,prd73054024,prd564133,pan67,prd61034012,prd62014019,prd493399}
in the literature, there are uncontrolled large theoretical errors
with quite different numerical results. In fact, all of these old
calculations are based on naive factorization hypothesis, with
various form factor inputs. Most of them even did not give any
theoretical error estimates because of the non-reliability of these
models. Recently, the theory of non-leptonic B decays has been
improved quite significantly. Factorization has been proved in many
of these decays, thus allow us to give reliable calculations of the
hadronic B decays. It is also shown that the non-factorizable
contributions and annihilation type contributions, which are
neglected in the naive factorization approach, are very important in
these decays \cite{cheng}.
The perturbative QCD
approach (pQCD) \cite{prl744388} is one of the recently developed
theoretical tools based on QCD to deal with the non-leptonic B
decays. Utilizing the $k_T$ factorization instead of collinear
factorization, this approach is free of end-point singularity. Thus
the Feynman diagrams including factorizable, non-factorizable and
annihilation type, are all calculable. Phenomenologically, the pQCD
approach successfully predict the charmless two-body B decays
\cite{plb5046,prd63074009}. For the decays with a single heavy $D$
meson in the final states (the momentum of the $D$ meson is
$\frac{1}{2}m_B(1-r^2)$, with $r=m_D/m_B$), it is also proved
factorization in the soft-collinear effective theory \cite{scet1}.
Phenomenologically the pQCD approach is also demonstrated to be
applicable in the leading order of the $m_D/m_B$ expansion
\cite{09101424,0512347} for this kind of decays. For the double
charm decays of $B_c$ meson, the momentum of the final state $D$
meson is $\frac{1}{2}m_{B_c}(1-2r^2)$, which is only slightly
smaller than that of the decays with a single D meson final state.
The prove of factorization here is thus trivial. The pQCD approach
is applicable to this kind of decays. In fact, the double charm
decays of $B_{u,d,s}$ meson have been studied in the pQCD approach
successfully \cite{dd1,dd2}, with best agreement with experiments.
In this paper, we will extend our study to these $B_c$ decays in
the pQCD approach, in order to give predictions on branching ratios
and polarization fractions for the experiments to test. Since this
study is based on QCD and perturbative expansion, the theoretical
error will be controllable than any of the model calculations.
Our paper is organized as follows: We review the pQCD factorization
approach and then perform the perturbative calculations for these
considered decay channels in Sec.\ref{sec:f-work}. The numerical
results and discussions on the observables are given in
Sec.\ref{sec:result}. The final section is devoted to our
conclusions. Some details of related functions and the decay amplitudes
are given in the Appendix.
\section{ Framework}\label{sec:f-work}
For the double charm decays of $B_c$, only the tree operators of the standard effective weak
Hamiltonian contribute. We can divide them into two groups: CKM favored decays with both
emission and annihilation contributions and pure emission type decays, which are CKM suppressed.
For the former modes, the Hamiltonian is given by:
\begin{eqnarray}\label{eq:H1}
\mathcal {H}_{eff}&=&\frac{G_F}{\sqrt{2}}V_{cb}^*V_{uq}[C_1(\mu)O_1(\mu)+C_2(\mu)O_2(\mu)],\nonumber\\
O_1&=&\bar{b}_{\alpha}\gamma^{\mu}(1-\gamma_5)c_{\beta} \otimes \bar{u}_{\beta}\gamma_{\mu}(1-\gamma_5)q_{\alpha} ,\nonumber\\
O_2&=&\bar{b}_{\alpha}\gamma^{\mu}(1-\gamma_5)c_{\alpha} \otimes \bar{u}_{\beta}\gamma_{\mu}(1-\gamma_5)q_{\beta},
\end{eqnarray}
while the effective Hamiltonian of the latter modes reads
\begin{eqnarray}\label{eq:H2}
\mathcal {H}_{eff}&=&\frac{G_F}{\sqrt{2}}V_{ub}^*V_{cq}[C_1(\mu)O'_1(\mu)+C_2(\mu)O'_2(\mu)],\nonumber\\
O'_1&=&\bar{b}_{\alpha}\gamma^{\mu}(1-\gamma_5)u_{\beta} \otimes \bar{c}_{\beta}\gamma_{\mu}(1-\gamma_5)q_{\alpha} ,\nonumber\\
O'_2&=&\bar{b}_{\alpha}\gamma^{\mu}(1-\gamma_5)u_{\alpha} \otimes \bar{c}_{\beta}\gamma_{\mu}(1-\gamma_5)q_{\beta},
\end{eqnarray}
where $V (q=d,s)$ are the corresponding CKM matrix
elements. $\alpha$, $\beta$ are the color indices. $C_{1,2}$ are Wilson coefficients at renormalization scale $\mu$.
$O_{1,2}$ and $O'_{1,2}$ are the effective four-quark operators.
The factorization theorem allows us to factorize the decay amplitude
into the convolution of the hard subamplitude, the Wilson
coefficient and the meson wave functions, all of which are
well-defined and gauge invariant. It is expressed as
\begin{eqnarray}\label{eq:factorization}
C(t)\otimes H(x,t)\otimes
\Phi(x)\otimes\exp[-s(P,b)-2\int^t_{1/b}\frac{d\mu}{\mu}\gamma_q(\alpha_s(\mu))],
\end{eqnarray}
where $C(t)$ are the corresponding Wilson coefficients of effective
operators defined in eq.(\ref{eq:H1},\ref{eq:H2}). Since the
transverse momentum of quark is kept in the pQCD approach, the large
double logarithm $\ln^2 (Pb)$ (with P denoting the longitudinal
momentum, and b the conjugate variable of the transverse momentum)
to spoil the perturbative expansion. A resummation is thus needed to
give a
Sudakov factor $\exp[-s(P, b)]$ \cite{npb193381}. The term after Sudakov
is from renormalization group running with $\gamma_q=-\alpha_s/\pi$ the
quark anomalous dimension in axial gauge and $t$ the factorization
scale. All non-perturbative components are organized in the form of
hadron wave functions $\Phi(x)$ (with x the longitudinal momentum
fraction of valence quark inside the meson), which can be extracted
from experimental data or other non-perturbative methods. Since the
universal non-perturbative dynamics has been factored out, one can
evaluate all possible Feynman diagrams for the hard subamplitude
$H(x,t)$ straightforwardly, which include both traditional
factorizable and so-called ``non-factorizable" contributions.
Factorizable and non-factorizable annihilation type diagrams are
also calculable without end-point singularity.
\subsection{ Channels with both emission and annihilation contributions}
\begin{figure}[!htbh]
\begin{center}
\vspace{-2cm} \centerline{\epsfxsize=12 cm \epsffile{bctod0b.ps}}
\vspace{-8.4cm} \caption{Feynman diagrams for $B_c\rightarrow
D^+\bar{D}^0$ decays.}
\label{fig:bctod0b}
\end{center}
\end{figure}
At leading order, there are eight kinds of Feynman diagrams
contributing to this type of CKM favored decays according to
eq.(\ref{eq:H1}). Here, we take the decay $B_c\rightarrow
D^+\bar{D}^0$ as an example, whose Feynman diagrams
are shown in Fig.\ref{fig:bctod0b}.
The first line are the emission type diagrams, with the first two
contributing to the usual form factor; the last two so-called
``non-factorizable" diagrams. In fact, the first two diagrams are
the only contributions calculated in the naive factorization
approach. The second line are the annihilation type diagrams, with
the first two factorizable; the last two non-factorizable. The decay
amplitude of factorizable diagrams (a) and (b) in
Fig.\ref{fig:bctod0b} is
\begin{eqnarray}\label{eq:fe}
\mathcal {F}_e&=&-2\sqrt{\frac{2}{3}}C_Ff_Bf_{3}\pi M_B^4
\int_0^1dx_2\int_0^{\infty}b_1b_2db_1db_2\phi_{2}(x_2)\exp(-\frac{b_1^2\omega_B^2}{2})\times\nonumber\\&&\{
[-(r_2-2)r_b+2r_2x_2-x_2]\alpha_s(t_a)h_e(\alpha_e,\beta_a,b_1,b_2)S_t(x_2)\exp[-S_{ab}(t_a)]
\nonumber\\&&+2r_2\alpha_s(t_b)h_e(\alpha_e,\beta_b,b_2,b_1)S_t(x_1)\exp[-S_{ab}(t_b)]\},
\end{eqnarray}
where $r_b=m_b/M_B$, $r_i=m_i/M_B(i=2,3)$ with $m_2,m_3$ are the masses of
the recoiling charmed meson and the emitting charmed meson, respectively;
$C_F=4/3$ is a color factor; $f_3$ is the decay constant of the
charmed meson, which emitted from the weak vertex. The factorization
scales $t_{a,b}$ are chosen as the maximal virtuality of internal
particles in the hard amplitude, in order to suppress the higher
order corrections \cite{prd074004}. The function $h_e$
and the Sudakov factor $\exp[-S]$ are
displayed in the Appendix \ref{sec:b}. $D$ meson distribution amplitude $\phi(x)$ are given in Appendix \ref{sec:c}.
The factor $S_t(x)$ is the jet function resulting from the threshold resummation,
whose definitions can be found in \cite{epjc45711}.
The formula for non-factorizable emission diagrams Fig.
\ref{fig:bctod0b} (c) and (d) contain the kinematics variables of
all the three mesons. Its expression is:
\begin{eqnarray}\label{eq:me}
\mathcal {M}_e&=&-\frac{8}{3}C_Ff_B\pi M_B^4
\int_0^1dx_2dx_3\int_0^{\infty}b_2b_3db_2db_3\phi_{2}(x_2)\phi_{3}(x_3)\exp(-\frac{b_2^2\omega_B^2}{2})\times\nonumber\\&&
\{[1-x_1-x_3-r_2(1-x_2)]\alpha_s(t_c)h_e(\beta_c,\alpha_e,b_3,b_2)\exp[-S_{cd}(t_c)]-
\nonumber\\&&[1-x_1-x_2+x_3-r_2(1-x_2)]\alpha_s(t_d)h_e(\beta_d,\alpha_e,b_3,b_2)\exp[-S_{cd}(t_d)]
\}.
\end{eqnarray}
Generally, for charmless decays of B meson, the non-factorizable
contributions of the emission diagrams are small due to the
cancelation between Fig. \ref{fig:bctod0b} (c) and (d). While for
double charm decays with the light meson replaced by a charmed
meson, since the heavy $\bar{c}$ quark and the light quark is not
symmetric, the non-factorizable emission diagrams ought to give
remarkable contributions. This has been shown in the pQCD
calculation of $B\to D\pi$ decays for a very large branching ratios
of color-suppressed modes \cite{dpi} and proved by the B factory
experiments.
The decay amplitude of factorizable annihilation diagrams Fig.
\ref{fig:bctod0b} (e) and (f) involve only the two final states
charmed meson wave functions, shown as
\begin{eqnarray}
\mathcal {F}_a&=&-8C_Ff_B\pi M_B^4
\int_0^1dx_2dx_3\int_0^{\infty}b_2b_3db_2db_3\phi_{2}(x_2)\phi_{3}(x_3)\times\nonumber\\&&
\{[1-x_2]\alpha_s(t_e)h_e(\alpha_a,\beta_e,b_2,b_3)\exp[-S_{ef}(t_e)]S_t(x_3)-
\nonumber\\&&[1-x_3]
\alpha_s(t_f)h_e(\alpha_a,\beta_f,b_3,b_2)\exp[-S_{ef}(t_f)]S_t(x_2)
\}.
\end{eqnarray}
For the non-factorizable annihilation diagrams Fig.
\ref{fig:bctod0b} (g) and (h), the decay amplitude is
\begin{eqnarray}
\mathcal {M}_a&=&\frac{8}{3}C_Ff_B\pi M_B^4
\int_0^1dx_2dx_3\int_0^{\infty}b_1b_2db_1db_2\phi_{2}(x_2)\phi_{3}(x_3)\exp(-\frac{b_1^2\omega_B^2}{2})\nonumber\\&&\times
\{[x_1+x_3-1-r_c]\alpha_s(t_g)h_e(\beta_g,\alpha_a,b_1,b_2)\exp[-S_{gh}(t_g)]\nonumber\\&&-
[r_b-x_2]\alpha_s(t_h)h_e(\beta_h,\alpha_a,b_1,b_2)\exp[-S_{gh}(t_h)]
\},
\end{eqnarray}
where $r_c=m_c/M_B$, with $m_c$ the mass of c quark in $B_c$
meson. Finally, the total decay amplitude for $B_c\rightarrow
D^+\bar{D}^0$ can be given by
\begin{eqnarray}\label{eq:dpd0b}
\mathcal {A}(B_c\rightarrow D^+\bar{D}^0)&=&V_{cb}^*V_{ud}[a_2\mathcal {F}_e+C_2\mathcal {M}_e+a_1\mathcal {F}_a+C_1\mathcal {M}_a],
\end{eqnarray}
with the combinations of Wilson coefficients $a_1=C_2+C_1/3$ and
$a_2=C_1+C_2/3$, characterizing the color favored contribution
and the color-suppressed contribution in the naive factorization, respectively.
The total decay amplitudes of $B_c\rightarrow D_s^+\bar{D}^0$, $B_c\rightarrow D^+\bar{D}^{*0}$ and $B_c\rightarrow D_s^+\bar{D}^{*0}$
can be obtained from eq.(\ref{eq:dpd0b})
with the following replacement:
\begin{eqnarray}\label{eq:relation1}
\mathcal {A}(B_c\rightarrow D_s^+\bar{D}^0)&=&V_{cb}^*V_{us}[a_2\mathcal {F}_e+C_2\mathcal {M}_e+a_1\mathcal {F}_a+C_1\mathcal {M}_a]|
_{ D^+\rightarrow D^+_s },\nonumber\\
\mathcal {A}(B_c\rightarrow D^+\bar{D}^{*0})&=&V_{cb}^*V_{ud}[a_2\mathcal {F}_e+C_2\mathcal {M}_e+a_1\mathcal {F}_a+C_1\mathcal {M}_a]
|_{\bar{D}^0\rightarrow \bar{D}^{*0} },\nonumber\\
\mathcal {A}(B_c\rightarrow D_s^+\bar{D}^{*0})&=&V_{cb}^*V_{us}[a_2\mathcal {F}_e+C_2\mathcal {M}_e+a_1\mathcal {F}_a+C_1\mathcal {M}_a]
|_{ D^+\rightarrow D^+_s,\bar{D}^0\rightarrow \bar{D}^{*0} }.
\end{eqnarray}
Comparing our eq.(\ref{eq:dpd0b},\ref{eq:relation1}) with the
formulas of previous naive factorization approach
\cite{jpg301445,plb555189,prd73054024,prd564133,pan67,prd61034012},
it is easy to see that only the first term appearing in
eq.(\ref{eq:dpd0b},\ref{eq:relation1}) are calculated in the
previous naive factorization approach. The second, third and fourth
terms in these equations, are the corresponding non-factorizable
emission type contribution, factorizable and non-factorizable
annihilation type contributions, respectively, which are all new
calculations.
In $B_c\rightarrow D^{*+}_{(s)}\bar{D}^{*0}$ decays, the two vector
mesons in the final states have the same helicity due to angular
momentum conservation, therefore only three different polarization
states, one longitudinal and two transverse for both vector mesons,
are possible. The decay amplitude can be decomposed as
\begin{eqnarray}
\mathcal {A}=\mathcal {A}^L+\mathcal {A}^N \epsilon_{2}^T \cdot \epsilon_{3}^T +
i \mathcal {A}^T \epsilon_{\alpha\beta\rho\sigma}n^{\alpha}v^{\beta}\epsilon_{2}^{T \rho}\epsilon_{3}^{T \sigma},
\end{eqnarray}
where $\epsilon_2^T, \epsilon_3^T$ are the transverse polarization
vectors for the two vector charmed mesons, respectively. $\mathcal
{A}^L$ corresponds to the contributions of longitudinal
polarization; $\mathcal {A}^N$ and $\mathcal {A}^T$ corresponds to
the contributions of normal and transverse polarization,
respectively. And the total amplitudes $\mathcal {A}^{L,N,T}$ have
the same structures as eq.(\ref{eq:dpd0b},\ref{eq:relation1}). The
factorization formulae for the longitudinal, normal and transverse
polarizations are listed in Appendix \ref{sec:a}.
For $B_c\rightarrow D^{*+}_{(s)}\bar{D}^0$ decays, only the
longitudinal polarization of $ D^{*+}_{(s)}$ meson will contribute,
due to the angular momentum conservation. We can obtain their decay
amplitudes from the longitudinal polarization amplitudes for the
$B_c\rightarrow D^{*+}_{(s)}\bar{D}^{*0}$ decays with the
replacement $\bar{D}^{*0}\rightarrow \bar{D}^{0}$.
\subsection{ Channels with pure emission type decays}
\begin{figure}[t]
\begin{center}
\vspace{-2cm}\centerline{\epsfxsize=12 cm \epsffile{bctod01.ps}}
\vspace{-12cm}\caption{Color-suppressed emission diagrams contributing to the $B_c\rightarrow D^+D^0$ decays.} \label{fig:bctod01}
\end{center}
\end{figure}
\begin{figure}[t]
\begin{center}
\vspace{-2cm}\centerline{\epsfxsize=12 cm \epsffile{bctod02.ps}}
\vspace{-12cm}\caption{Color-favored emission diagrams contributing to the $B_c\rightarrow D^+D^0$ decays.} \label{fig:bctod02}
\end{center}
\end{figure}
There are also eight kinds of Feynman diagrams contributing to
$B_c\rightarrow D_{(s)}^{(*)+}D^{(*)0}$ decays according to
eq.(\ref{eq:H2}), but all are emission type. Taking the decay
$B_c\rightarrow D^+D^0$ as an example,
Fig. \ref{fig:bctod01} are the color-suppressed emission diagrams while Fig. \ref{fig:bctod02} are the color-favored emission diagrams.
We mark the subscript 2 and 3 to denote the contributions from Fig.
\ref{fig:bctod01} and Fig. \ref{fig:bctod02}, respectively. The
decay amplitude of factorization emission diagrams $\mathcal
{F}_{e2}$, coming from Fig. \ref{fig:bctod01} (a,b), is similar
to eq.(\ref{eq:fe}), but with the replacement $\bar{D}^0\rightarrow
D^0$.
While the decay amplitude of non-factorization emission diagram $\mathcal {M}_{e2}$,
coming from Fig. \ref{fig:bctod01} (c,d), is different from eq.(\ref{eq:me}), since
the heavy c quark and the light anti-quark are not symmetric. The
expression of the non-factorizable emission diagram is
\begin{eqnarray}\label{eq:mep}
\mathcal {M}_{e2}&=&-\frac{8}{3}C_Ff_B\pi M_B^4
\int_0^1dx_2dx_3\int_0^{\infty}b_2b_3db_2db_3\phi_{2}(x_2)\phi_{3}(x_3)\exp(-\frac{b_2^2\omega_B^2}{2})\nonumber\\&&\times
\{[2-x_1-x_2-x_3-r_2(1-x_2)]\alpha_s(t_c)h_e(\beta_c,\alpha_e,b_3,b_2)\exp[-S_{cd}(t_c)]-
\nonumber\\&&[x_3-x_1-r_2(1-x_2)]\alpha_s(t_d)h_e(\beta_d,\alpha_e,b_3,b_2)\exp[-S_{cd}(t_d)]
\}.
\end{eqnarray}
By exchanging the two final states charmed mesons in
Fig.~\ref{fig:bctod01}, one can obtain the corresponding decay
amplitudes formulae $\mathcal {F}_{e3}$ and $\mathcal {M}_{e3}$ for
Fig.~\ref{fig:bctod02}.
The total decay amplitude of $B_c\rightarrow D^+D^0$ decay can be written as
\begin{eqnarray}\label{eq:bctodpd0}
\mathcal {A}(B_c\rightarrow D^+D^0)&=&V_{ub}^*V_{cd}[a_2\mathcal {F}_{e2}+C_2\mathcal {M}_{e2}
+a_1\mathcal {F}_{e3}+C_1\mathcal {M}_{e3}].
\end{eqnarray}
If the final recoiling meson is the vector $D^*$ meson, the decay
amplitudes of factorization emission diagrams and non-factorization
emission diagrams are given as
\begin{eqnarray}\label{eq:dstarfe}
\mathcal {F}^*_{e2}&=&-2\sqrt{\frac{2}{3}}C_Ff_Bf_{3}\pi M_B^4
\int_0^1dx_2\int_0^{\infty}b_1b_2db_1db_2\phi_{2}(x_2)\exp(-\frac{b_1^2\omega_B^2}{2})\nonumber\\&&\times\{
[-(r_2-2)r_b+2r_2x_2-x_2]\alpha_s(t_a)h_e(\alpha_e,\beta_a,b_1,b_2)S_t(x_2)\exp[-S_{ab}(t_a)]
\nonumber\\&&+r^2_2\alpha_s(t_b)h_e(\alpha_e,\beta_b,b_2,b_1)S_t(x_1)\exp[-S_{ab}(t_b)]\},
\end{eqnarray}
\begin{eqnarray}
\mathcal {M}^*_{e2}&=&-\frac{8}{3}C_Ff_B\pi M_B^4
\int_0^1dx_2dx_3\int_0^{\infty}b_2b_3db_2db_3\phi_{2}(x_2)\phi_{3}(x_3)\exp(-\frac{b_2^2\omega_B^2}{2})\nonumber\\&&\times
\{[2-x_1-x_2-x_3-r_2(1-x_2)]\alpha_s(t_c)h_e(\beta_c,\alpha_e,b_3,b_2)\exp[-S_{cd}(t_c)]-
\nonumber\\&&[x_3-x_1+r_2(1-x_2)]\alpha_s(t_d)h_e(\beta_d,\alpha_e,b_3,b_2)\exp[-S_{cd}(t_d)]
\}.\nonumber\\&&
\end{eqnarray}
The total decay amplitudes for other pure emission type decays are then
\begin{eqnarray}\label{eq:bctod0}
\mathcal {A}(B_c\rightarrow D_s^+D^0)&=&V_{ub}^*V_{cs}[a_2\mathcal {F}_{e2}+C_2\mathcal {M}_{e2}
+a_1\mathcal {F}_{e3}+C_1\mathcal {M}_{e3}],\nonumber\\
\mathcal {A}(B_c\rightarrow D^+D^{*0})&=&V_{ub}^*V_{cd}[a_2\mathcal {F}_{e2}+C_2\mathcal {M}_{e2}
+a_1\mathcal {F}^*_{e3}+C_1\mathcal {M}^*_{e3}],\nonumber\\
\mathcal {A}(B_c\rightarrow D^{*+}D^{0})&=&V_{ub}^*V_{cd}[a_2\mathcal {F}^*_{e2}+C_2\mathcal {M}^*_{e2}
+a_1\mathcal {F}_{e3}+C_1\mathcal {M}_{e3}],\nonumber\\
\mathcal {A}(B_c\rightarrow D_s^+D^{*0})&=&V_{ub}^*V_{cs}[a_2\mathcal {F}_{e2}+C_2\mathcal {M}_{e2}
+a_1\mathcal {F}^*_{e3}+C_1\mathcal {M}^*_{e3}],\nonumber\\
\mathcal {A}(B_c\rightarrow D_s^{*+}D^{0})&=&V_{ub}^*V_{cs}[a_2\mathcal {F}^*_{e2}+C_2\mathcal {M}^*_{e2}
+a_1\mathcal {F}_{e3}+C_1\mathcal {M}_{e3}].\nonumber\\
\end{eqnarray}
The $B_c\rightarrow D^{*+}_{(s)}D^{*0}$ decays have a similar
situation to $B_c\rightarrow D^{*+}_{(s)}\bar{D}^{*0}$,
their factorization formulae are also listed in Appendix.\ref{sec:a}.
\section{NUMERICAL RESULTS} \label{sec:result}
In this section, we summarize the numerical results and analysis in
the double charm decays of the $B_c$ meson. Some input parameters
needed in the pQCD calculation are listed in Table
\ref{tab:constant}.
\subsection{The Form Factors}
\begin{table}
\caption{Parameters we used in numerical calculation \cite{npp37}}
\label{tab:constant}
\begin{tabular*}{13.5cm}{@{\extracolsep{\fill}}l|cccc}
\hline\hline
\textbf{Mass(\text{GeV})} & $M_{W}=80.399$ & $M_{B_c}=6.277$ & $m_{b}=4.2$& $m_{c}=1.27$\\[1ex]
\hline
\end{tabular*}
\begin{tabular*}{13.5cm}{@{\extracolsep{\fill}}l|ccc}
\hline
\multirow{2}{*}{{\textbf{CKM}}} & $|V_{ub}|=(3.47^{+0.16}_{-0.12})\times 10^{-3}$
& $|V_{ud}|=0.97428^{+0.00015}_{-0.00015}$ & $|V_{us}|=0.2253^{+0.0007}_{-0.0007}$\\[1ex]
&$|V_{cs}|=0.97345^{+0.00015}_{-0.00016}$ & $|V_{cd}|=0.2252^{+0.0007}_{-0.0007}$
& $|V_{cb}|=0.0410^{+0.0011}_{-0.0007}$ \\[1ex]
\hline
\end{tabular*}
\begin{tabular*}{13.5cm}{@{\extracolsep{\fill}}l|ccc}
\hline
\textbf{Decay constants(MeV)} & $f_{B_c}=489$ & $f_{D}=206.7 \pm 8.9$ & $f_{D_s}=257.5 \pm 6.1$\\[1ex]
\hline
\end{tabular*}
\begin{tabular*}{13.5cm}{@{\extracolsep{\fill}}l|l}
\hline
\textbf{Lifetime} & $\tau_{B_c}=0.453\times 10^{-12}\text{s}$\\[1ex]
\hline\hline
\end{tabular*}
\end{table}
\begin{table}
\caption{The form factors for $B_c\rightarrow D^{(*)}_{(s)}$ at
$q^2=0$ evaluated in the pQCD approach. The uncertainties are from
the hadronic parameters. For comparison, we also cite the
theoretical estimates of other models.} \label{tab:formfactor}
\begin{tabular}[t]{l|c|c|c|c|c|c}
\hline\hline
& This work & Kiselev \cite{jpg301445} \footnotemark[1]& IKP \cite{plb555189} & WSL \cite{prd7905402}
& DSV \cite{jpg35085002}& DW \cite{prd391342}\footnotemark[2] \\ \hline
$F^{B_c\rightarrow D}$ & $0.14^{+0.01}_{-0.02}$ &0.32 [0.29] &0.189 &0.16 &0.075&0.255\\
$F^{B_c\rightarrow D_s}$ & $0.19^{+0.02}_{-0.01}$ &0.45 [0.43] &0.194 &0.28 &0.15 &--\\
$A_0^{B_c\rightarrow D^*}$ & $0.12^{+0.02}_{-0.01}$ &0.35 [0.37] &0.133 &0.09 &0.081 &0.257\\
$A_0^{B_c\rightarrow D_s^*}$ & $0.17^{+0.01}_{-0.01}$ &0.47 [0.52] &0.142 &0.17 &0.16 &--\\
\hline\hline
\end{tabular}
\footnotetext[1]{ The non-bracket (bracketed) results are evaluated
in sum rules (potential model)} \footnotetext[2]{We quote the
result with $\omega=0.7\text{GeV}$}
\end{table}
The diagrams (a) and (b) in Fig.\ref{fig:bctod0b} or
Fig.\ref{fig:bctod02} give the contribution for $B_c\rightarrow
D^{(*)}_{(s)}$ transition form factor at $q^2=0$ point. Our
predictions of the form factors are collected in Table
\ref{tab:formfactor}. The error is from the combined uncertainty
in the hadronic parameters: (1) the shape parameters:
$\omega_B=0.60\pm0.05$ for $B_c$ meson wave function, $a_{D}=(0.5\pm
0.1) \text{GeV}$ for $D^{(*)}$ meson and $a_{D_s}=(0.4\pm 0.1)
\text{GeV}$ for $D_s^{(*)}$ meson wave function \cite{dd1}; (2) the
decay constants in the wave functions of charmed mesons, which are
given in Table \ref{tab:constant}. Since the uncertainties from
decay constants of $D_{(s)}$ and the shape parameters of the wave
functions are very small, the relevant uncertainties to the form
factors are also very small. We can see that the $SU(3)$ symmetry
breaking effects between $B_c$ to $ D^{(*)}$ and $B_c$ to
$D^{(*)}_{s}$ form factors are large, as the decay constant of
$D_s$ is about one-fifth larger than that of the $D$ meson.
In the literature there are already lots of studies on
$B_c\rightarrow D^{(*)}_{(s)}$ transition form factors
\cite{jpg301445,prd7905402,jpg35085002,plb555189,prd391342}, whose
results are collected in Table \ref{tab:formfactor}. Our
results are generally close to the covariant light-front quark model
results of \cite{prd7905402} and the constituent quark model results
of \cite{plb555189}. However, other results collected in Table
\ref{tab:formfactor}, especially for the QCD sum rules (QCDSR)
\cite{jpg301445} and the Bauer, Stech and Wirbel (BSW) model
\cite{jpg35085002} deviate a lot numerically. The predictions of
QCDSR \cite{jpg301445} are larger than those in other works
\cite{prd7905402,jpg35085002,plb555189,prd391342}. The reason is
that they have taken into account the $\alpha_s/v$ corrections and
the form factors are enhanced by 3 times due to the Coulomb
renormalization of the quark-meson vertex for the heavy quarkonium
$B_c$. The results of BSW model \cite{jpg35085002} are quite small
due to the less overlap of the initial and final states wave
functions. Although, the included flavor dependence of the average
transverse quark momentum in the mesons can enhance the form factors
for $B_c\rightarrow D^{*}_{(s)}$ transitions, their predictions are
still smaller than other models. The large differences in different
models can be discriminated by the future LHC experiments.
\subsection{Branching Ratios}
With the decays amplitudes $\mathcal {A}$ obtained in
Sec.\ref{sec:f-work}, the branching ratio $\mathcal {BR}$ reads as
\begin{eqnarray}
\mathcal {BR}=\frac{G_F\tau_{B_c}}{32\pi M_B}\sqrt{(1-(r_2+r_3)^2)(1-(r_2-r_3)^2)}|\mathcal {A}|^2.
\end{eqnarray}
As stated in Sec \ref{sec:f-work}, the contributions from the
penguin operators are absent, since the penguins add an even number
of charmed quarks, while there is already one from the initial
state. There should be no CP violation in these processes. We
tabulate the branching ratios of the considered decays in Table
\ref{tab:branching1} and \ref{tab:branching2}. The processes (1)-(4)
in Table \ref{tab:branching1} have a comparatively large branching
ratios ($10^{-5}$)
with the CKM
factor $V_{cb}^*V_{ud}\sim \lambda^2$. While the branching ratios of other processes are relatively small
due to the CKM factor suppression. Especially for the processes (1)-(4) in Table \ref{tab:branching2},
these channels are suppressed by CKM element $V_{ub}/V_{cb}$ and $V_{cd}/V_{ud}$.
Thus their branching ratios are three order magnitudes smaller.
\begin{table}
\caption {Branching ratios ($10^{-6}$) of the CKM favored decays
with both emission and annihilation contributions, together with
results from other models. The errors for these entries correspond
to the uncertainties in the input hadronic quantities, from the CKM
matrix elements, and the scale dependence, respectively.}
\label{tab:branching1}
\begin{tabular}[t]{l|l|c|ccccc}
\hline\hline
&channels & This work & Kiselev\cite{jpg301445} & IKP\cite{plb555189} & IKS\cite{prd73054024}
&LC\cite{prd564133} & CF\cite{prd61034012} \\ \hline
1&$B_c\rightarrow D^+\bar{D}^0$ & $32^{+6+1+2}_{-6-1-4}$&53 &32&33 &86 &8.4 \\
2&$B_c\rightarrow D^+\bar{D}^{*0}$ &$34^{+7+2+3}_{-6-1-3}$&75 &83 &38 &75&7.5 \\
3&$B_c\rightarrow D^{*+}\bar{D}^0$ & $12^{+3+1+0}_{-3-0-1}$ &49 &17 &9 &30&84 \\
4&$B_c\rightarrow D^{*+}\bar{D}^{*0}$ &$34^{+9+2+0}_{-8-1-0}$ &330 &84&21&55 &140 \\
\hline
5&$B_c\rightarrow D_s^+\bar{D}^0$ & $2.3^{+0.4+0.1+0.2}_{-0.4-0.1-0.2}$ &4.8 &1.7&2.1&4.6 &0.6 \\
6&$B_c\rightarrow D_s^+\bar{D}^{*0}$ &$2.6^{+0.4+0.1+0.1}_{-0.6-0.1-0.2}$ &7.1 &4.3&2.4 &3.9 &0.53 \\
7&$B_c\rightarrow D_s^{*+}\bar{D}^0$ & $0.7^{+0.1+0.0+0.0}_{-0.2-0.0-0.0}$ &4.5&0.95 &0.65 &1.8 &5 \\
8&$B_c\rightarrow D_s^{*+}\bar{D}^{*0}$ & $2.8^{+0.7+0.1+0.1}_{-0.6-0.1-0.0}$ &26 &4.7 &1.6 &3.5&8.4 \\
\hline\hline
\end{tabular}
\end{table}
\begin{table}
\caption{Branching ratios ($10^{-7}$) of the CKM suppressed decays
with pure emission contributions, together with results from other
models. The errors for these entries correspond to the uncertainties
in the input hadronic quantities, from the CKM matrix elements, and
the scale dependence, respectively.} \label{tab:branching2}
\begin{tabular}[t]{l|l|c|ccc}
\hline\hline
&channels & This work & Kiselev\cite{jpg301445} & IKP\cite{plb555189}& IKS\cite{prd73054024}
\\ \hline
1&$B_c\rightarrow D^+D^0$ & $1.0^{+0.2+0.1+0.0}_{-0.1-0.0-0.0}$ &3.2 &1.1&3.1 \\
2&$B_c\rightarrow D^+D^{*0}$ &$0.7^{+0.1+0.1+0.0}_{-0.2-0.0-0.0}$&2.8 &0.25&0.52 \\
3&$B_c\rightarrow D^{*+}D^0$ & $0.9^{+0.1+0.1+0.0}_{-0.2-0.0-0.0}$ &4.0&3.8&4.4\\
4&$B_c\rightarrow D^{*+}D^{*0}$ & $0.8^{+0.2+0.1+0.2}_{-0.1-0.0-0.0}$ &15.9&2.8&2.0 \\
\hline
5&$B_c\rightarrow D_s^+D^0$ & $30^{+5+3+1}_{-4-2-1}$ &66 &25&74 \\
6&$B_c\rightarrow D_s^+D^{*0}$ &$19^{+3+2+0}_{-3-1-1}$ &63 &6&13 \\
7&$B_c\rightarrow D_s^{*+}D^0$ & $25^{+4+2+0}_{-3-2-1}$ &85 &69&93\\
8&$B_c\rightarrow D_s^{*+}D^{*0}$ & $24^{+3+2+1}_{-3-2-1}$ &404&54&45 \\
\hline\hline
\end{tabular}
\end{table}
For comparison, we also cite other theoretical results
\cite{jpg301445,plb555189,prd73054024,prd564133,prd61034012} for the
double charm decays of $B_c$ meson in Tables \ref{tab:branching1}
and \ref{tab:branching2}.
In general, the results of the various model calculations are of
the same order of magnitude for most channels. However the
difference between different model calculations is quite large. This
is expected from the large difference of input parameters,
especially the large difference of form factors shown in
Table~\ref{tab:formfactor}. As stated in the introduction, all the
calculations of these $B_c$ to two D meson decays in the literature
use the same naive factorization approach. Their difference relies
only on the input form factors and decay constants. Therefore the
comparison of results with any of them is straightforward. Larger
branching ratios come always with the larger form factors.
As stated in the previous subsection,
our results of form factors are comparable with the relativistic
constituent quark model (RCQM) \cite{plb555189,prd73054024}, thus
our branching ratios in Table \ref{tab:branching1} are also
comparable with theirs except for the processes $B_c\rightarrow
D^{*+}\bar D^{*0}$ and $B_c\rightarrow D_s^{*+}\bar D^{*0}$. Due to
the sizable contributions of transverse polarization amplitudes, our
branching ratios are larger than those in RCQM model, whose
transverse contribution is negligible.
Since all the previous calculations in the literature are model
calculations, it is difficult for them to give the theoretical error
estimations. In our pQCD approach, the factorization holds at the
leading order expansion of $m_D/m_B$. At this order, we can do the
systematical calculation, so as to the error estimations in the
tables. The first error in these entries is estimated from the
hadronic parameters: (1) the shape parameters:
$\omega_B=0.60\pm0.05$ for $B_c$ meson, $a_{D}=(0.5\pm 0.1)
\text{GeV}$ for $D^{(*)}$ meson and $a_{D_s}=(0.4\pm 0.1)
\text{GeV}$ for $D_s^{(*)}$ meson \cite{dd1}; (2) the decay
constants in the wave functions of charmed mesons, which are given
in Table \ref{tab:constant}.
The second error is from the uncertainty in the CKM matrix elements,
which are also given in Table \ref{tab:constant}. The third error
arises from the hard scale t varying from $0.75t$ to $1.25t$, which
characterizing the size of next-to-leading order QCD contributions.
The not large errors of this type indicate that our perturbative
expansion indeed hold. It is easy to see that the most important
uncertainty in our approach comes from the hadronic parameters. The
total theoretical error is in general around 10\% to 30\% in size.
The eight CKM favored channels (proportional to $|V_{cb}|$) in Table
\ref{tab:branching1} receive contributions from both emission
diagrams and annihilation diagrams. From Fig.\ref{fig:bctod0b}, one
can find that the contributions from the factorizable emission
diagrams are color-suppressed. The naive factorization approach can
not give reliable predictions due to large non-factorizable
contributions \cite{fac}. As was pointed out in
Sec.\ref{sec:f-work}, the non-factorizable emission diagrams give
large contributions in pQCD approach because the asymmetry of the
two quarks in charmed mesons.
Thus, the branching ratios of these decays are dominated by the non-factorizable emission diagrams.
The eight CKM suppressed channels (proportional to $|V_{ub}|$) in
Table \ref{tab:branching2} can occur only via emission type
diagrams. There are two types of emission diagrams in these decays,
one is color-suppressed, one is color favored. It is expected that
the color-favored factorizable amplitude $\mathcal {F}_{e3}$
dominates in eq.(\ref{eq:bctod0}). However, the non-factorizable
contribution $\mathcal {M}_{e2}$, proportional to the large $C_2$,
is enhanced by the Wilson coefficient. Numerically it is indeed
comparable to the color-favored factorizable amplitude. This large
non-factorizable contribution has already been shown in the similar
$B\to D\pi$ decays theoretically and experimentally \cite{dpi}. In
all of these channels the non-factorizable contributions play a very
important role, therefore the branching ratios predicted in
table~\ref{tab:branching1} and \ref{tab:branching2} are not like
the previous naive factorization approach calculations
\cite{jpg301445,plb555189,prd73054024,prd564133,prd61034012}. They
are not simply proportional to the corresponding form factors any
more, but with a very complicated manner, since we have also
additional annihilation type contributions.
From Table III and IV, one can see that as it was expected the
magnitudes of the branching ratios of the decays $B_c\rightarrow
D^+_s \bar{D}^0$ and $B_c\rightarrow D^+_s D^0$ are very close to
each other. In our numerical results, the ratio of the two decay
widths is estimated as $\frac{\Gamma(B_c\rightarrow
D_s^+D^0)}{\Gamma(B_c\rightarrow D_s^+\bar{D}^0)}\approx 1.3$. They
are very suitable for extracting the CKM
angle $\gamma$ though the amplitude relations. Hopefully they will be measured in the
experiments soon.
However, the decays $B_c\rightarrow D^+ \bar{D}^0, D^+D^0$ are problematic from the methodic point of view for
$\mathcal {BR}(B_c\rightarrow D^{+}D^0)\ll \mathcal {BR}(B_c\rightarrow D^{+}\bar{D}^0) $.
The corresponding ratio in $B_c\rightarrow D^+D^0,
D^+\bar{D}^0$ decays is $\frac{\Gamma(B_c\rightarrow
D^+D^0)}{\Gamma(B_c\rightarrow D^+\bar{D}^0)} \sim 10^{-3}$, which
confirm the latter decay modes are not useful to determine the angle
$\gamma$ experimentally.
\begin{table}
\caption{The transverse polarizations fractions ($\%$) for
$B_c\rightarrow VV$.
The errors correspond to the uncertainties in the hadronic parameters and the scale dependence, respectively. }
\label{tab:ratio}
\begin{tabular}[t]{l|c|c|c|c}
\hline\hline
& $B_c\rightarrow D^{*+}\bar{D}^{*0}$ & $B_c\rightarrow D_s^{*+}\bar{D}^{*0}$
& $B_c\rightarrow D^{*+}D^{*0}$ & $B_c\rightarrow D_s^{*+}D^{*0}$\\
\hline
$\mathcal {R}_T$ &$58^{+3+1}_{-3-0}$&$68^{+2+1}_{-2-1}$&$4^{+1+1}_{-1-1}$&$6^{+1+2}_{-0-1}$\\
\hline\hline
\end{tabular}
\end{table}
For the $B_c$ decays to two vector mesons, the decays amplitudes
$\mathcal {A}$ are defined in the helicity basis
\begin{eqnarray}
\mathcal {A}=\sum_{i=0,+,-}|\mathcal {A}_i|^2,\quad
\end{eqnarray}
where the helicity amplitudes $\mathcal {A}_i$ have the following
relationships with $\mathcal {A}^{L,N,T}$
\begin{eqnarray}
\mathcal {A}_0=\mathcal {A}^L,\quad \mathcal {A}_{\pm}=\mathcal
{A}^N \pm \mathcal {A}^T.
\end{eqnarray}
We also calculate the transverse polarization fractions $\mathcal
{R}_T$ of the $B_c\to D_{(s)}^* D^*$ decays, with the definition
given by
\begin{eqnarray}
\mathcal {R}_T=\frac{|\mathcal {A}_+|^2+|\mathcal
{A}_-|^2}{|\mathcal {A}_0|^2+|\mathcal {A}_+|^2+|\mathcal {A}_-|^2}.
\end{eqnarray}
This should be the first time
theoretical predictions in the literature, which are absent in all
the naive factorization calculations. According to the power
counting rules in the factorization assumption, the longitudinal
polarization should be dominant due to the quark helicity analysis.
Our predictions for the transverse polarization fractions of the
decays $B_c\rightarrow D^{*+}_{(s)}D^{*0}$, which are given in Table
\ref{tab:ratio}, are indeed small, since the two transverse
amplitudes are down by a power of $r_2$ or $r_3$ comparing with the
longitudinal amplitudes. However, for $B_c\rightarrow
D^{*+}_{(s)}\bar{D}^{*0}$ decays,
the most
important contributions for these two decay channels are from the
non-factorizable tree diagrams in Fig. \ref{fig:bctod0b}(c) and
\ref{fig:bctod0b}(d). With an additional gluon, the transverse
polarization in the non-factorizable diagrams does not encounter
helicity flip suppression. The transverse polarization is at the
same order as longitudinal polarization. Therefore, we can expect
the transverse polarizations take a larger ratio in the branching
ratios, which can reach $\sim 60\%$. The fact that the
non-factorizable contribution can give large transverse polarization
contribution is also observed in the $B^0 \to \rho^0\rho^0$,
$\omega\omega$ decays \cite{rhorho} and in the $B_c\rightarrow
D_s^{*+}\omega$ decay \cite{11121257}.
\section{ conclusion}
All the previous calculations in the literature for the $B_c$ meson
decays to two charmed mesons are based on the very simple naive
factorization approach. The branching ratios predicted in this kind
of model calculation depend heavily on the input form factors. Since
all of these modes contain dominant or large contributions from
color-suppressed diagrams, the predicted branching ratios are also
not stable due to the large unknown non-factorizable contributions.
In this paper,
we have performed a systematic
analysis of the double charm decays of the $B_c$ meson in the pQCD
approach based on $k_T$ factorization theorem, which is free of
end-point singularities.
All topologies of decay amplitudes are
calculable in the same framework, including the non-factorizable
one and annihilation type. It is found that the non-factorizable
emission diagrams give a remarkable contribution.
There is no CP violation for all these decays within the standard model, since there
are only tree operators contributions.
The predicted branching ratios range from very small numbers of $\mathcal {O}(10^{-8})$ up to the
largest branching fraction of $\mathcal {O}(10^{-5})$.
Since all of the previous naive factorization calculations did not
give the theoretical uncertainty in the numerical results, it is not
easy to compare our results with theirs. The theoretical uncertainty
study in the pQCD approach shows that our numerical results are
reliable, which may be tested in the upcoming experimental
measurements. We predict the transverse polarization fractions of
the $B_c$ decays with two vector $D^*$ mesons in the final states
for the first time. Due to the cancelation of some hadronic
parameters in the ratio, the polarization fractions are predicted
with less theoretical uncertainty. The transverse polarization
fractions are large in some channels, which mainly come from the
non-factorizable emission diagrams.
\begin{acknowledgments}
We thank Hsiang-nan Li and Fusheng Yu for helpful
discussions. This work is partially supported by
National Natural
Science Foundation of China under the Grant No.
11075168; Natural Science Foundation of Zhejiang Province of China,
Grant No. Y606252 and Scientific Research Fund of Zhejiang Provincial Education Department of China, Grant No. 20051357.
\end{acknowledgments}
\begin{appendix}
\section{Factorization formulas for $B_c\rightarrow VV$}\label{sec:a}
In the $B_c$ decays to two vector meson final states, we use the
superscript L, N and T to denote the contributions from longitudinal
polarization, normal polarization and transverse polarization,
respectively. For the CKM favored $B_c\rightarrow
D^{*+}_{(s)}\bar{D}^{*0}$ decays, the decay amplitudes for different
polarizations are
\begin{eqnarray}\label{eq:fel}
\mathcal {F}^L_e&=&-2\sqrt{\frac{2}{3}}C_Ff_Bf_{3}\pi M_B^4
\int_0^1dx_2\int_0^{\infty}b_1b_2db_1db_2\phi_{2}(x_2)\exp(-\frac{b_1^2\omega_B^2}{2})\times\nonumber\\&&\{
[-(r_2-2)r_b+2r_2x_2-x_2]\alpha_s(t_a)h_e(\alpha_e,\beta_a,b_1,b_2)S_t(x_2)\exp[-S_{ab}(t_a)]
\nonumber\\&&+r^2_2\alpha_s(t_b)h_e(\alpha_e,\beta_b,b_2,b_1)S_t(x_1)\exp[-S_{ab}(t_b)]\},
\end{eqnarray}
\begin{eqnarray}\label{eq:mel}
\mathcal {M}^L_e&=&-\frac{8}{3}C_Ff_B\pi M_B^4
\int_0^1dx_2dx_3\int_0^{\infty}b_2b_3db_2db_3\phi_{2}(x_2)\phi_{3}(x_3)\exp(-\frac{b_2^2\omega_B^2}{2})\times\nonumber\\&&
\{[1-x_1-x_3+r_2(1-x_2)]\alpha_s(t_c)h_e(\beta_c,\alpha_e,b_3,b_2)\exp[-S_{cd}(t_c)]-
\nonumber\\&&[1-x_1-x_2+x_3-r_2(1-x_2)]\alpha_s(t_d)h_e(\beta_d,\alpha_e,b_3,b_2)\exp[-S_{cd}(t_d)]
\},
\end{eqnarray}
\begin{eqnarray}
\mathcal {F}^L_a&=&-8C_Ff_B\pi M_B^4
\int_0^1dx_2dx_3\int_0^{\infty}b_2b_3db_2db_3\phi_{2}(x_2)\phi_{3}(x_3)\times\nonumber\\&&
\{[1-x_2]\alpha_s(t_e)h_e(\alpha_a,\beta_e,b_2,b_3)\exp[-S_{ef}(t_e)]S_t(x_3)-
\nonumber\\&&[1-x_3]
\alpha_s(t_f)h_e(\alpha_a,\beta_f,b_3,b_2)\exp[-S_{ef}(t_f)]S_t(x_2)
\},
\end{eqnarray}
\begin{eqnarray}
\mathcal {M}^L_a&=&\frac{8}{3}C_Ff_B\pi M_B^4
\int_0^1dx_2dx_3\int_0^{\infty}b_1b_2db_1db_2\phi_{2}(x_2)\phi_{3}(x_3)\exp(-\frac{b_1^2\omega_B^2}{2})\times\nonumber\\&&
\{[x_1+x_3-1-r_c]\alpha_s(t_g)h_e(\beta_g,\alpha_a,b_1,b_2)\exp[-S_{gh}(t_g)]\nonumber\\&&-
[r_b-x_2]\alpha_s(t_h)h_e(\beta_h,\alpha_a,b_1,b_2)\exp[-S_{gh}(t_h)]
\},
\end{eqnarray}
\begin{eqnarray}\label{eq:fen}
\mathcal {F}^N_e&=&-2\sqrt{\frac{2}{3}}C_Ff_Bf_{3}r_3\pi M_B^4
\int_0^1dx_2\int_0^{\infty}b_1b_2db_1db_2\phi_{2}(x_2)\exp(-\frac{b_1^2\omega_B^2}{2})\times\nonumber\\&&\{
[2-r_b+r_2(4r_b-x_2-1)]\alpha_s(t_a)h_e(\alpha_e,\beta_a,b_1,b_2)S_t(x_2)\exp[-S_{ab}(t_a)]
\nonumber\\&&-r_2\alpha_s(t_b)h_e(\alpha_e,\beta_b,b_2,b_1)S_t(x_1)\exp[-S_{ab}(t_b)]\},
\end{eqnarray}
\begin{eqnarray}
\mathcal {F}^T_e&=&2\sqrt{\frac{2}{3}}C_Ff_Bf_{3}r_3\pi M_B^4
\int_0^1dx_2\int_0^{\infty}b_1b_2db_1db_2\phi_{2}(x_2)\exp(-\frac{b_1^2\omega_B^2}{2})\times\nonumber\\&&\{
[2-r_b-r_2(1-x_2)]\alpha_s(t_a)h_e(\alpha_e,\beta_a,b_1,b_2)S_t(x_2)\exp[-S_{ab}(t_a)]
\nonumber\\&&-r_2\alpha_s(t_b)h_e(\alpha_e,\beta_b,b_2,b_1)S_t(x_1)\exp[-S_{ab}(t_b)]\},
\end{eqnarray}
\begin{eqnarray}
\mathcal {M}^N_e&=&-\mathcal {M}^T_e=\frac{8}{3}C_Ff_B\pi M_B^4r_3
\int_0^1dx_2dx_3\int_0^{\infty}b_2b_3db_2db_3\phi_{2}(x_2)\phi_{3}(x_3)\exp(-\frac{b_2^2\omega_B^2}{2})\nonumber\\&&\times
\{[x_1+x_3-1]\alpha_s(t_c)h_e(\beta_c,\alpha_e,b_3,b_2)\exp[-S_{cd}(t_c)]-
\nonumber\\&&[x_1-x_3]\alpha_s(t_d)h_e(\beta_d,\alpha_e,b_3,b_2)\exp[-S_{cd}(t_d)]\},
\end{eqnarray}
\begin{eqnarray}
\mathcal {F}^N_a&=&-8C_Ff_B\pi M_B^4r_2r_3
\int_0^1dx_2dx_3\int_0^{\infty}b_2b_3db_2db_3\phi_{2}(x_2)\phi_{3}(x_3)\times\nonumber\\&&
\{[2-x_2]\alpha_s(t_e)h_e(\alpha_a,\beta_e,b_2,b_3)\exp[-S_{ef}(t_e)]S_t(x_3)-
\nonumber\\&&[2-x_3]
\alpha_s(t_f)h_e(\alpha_a,\beta_f,b_3,b_2)\exp[-S_{ef}(t_f)]S_t(x_2)\},
\end{eqnarray}
\begin{eqnarray}
\mathcal {F}^T_a&=&-8C_Ff_B\pi M_B^4r_2r_3
\int_0^1dx_2dx_3\int_0^{\infty}b_2b_3db_2db_3\phi_{2}(x_2)\phi_{3}(x_3)\times\nonumber\\&&
\{x_2\alpha_s(t_e)h_e(\alpha_a,\beta_e,b_2,b_3)\exp[-S_{ef}(t_e)]S_t(x_3)+
\nonumber\\&&x_3
\alpha_s(t_f)h_e(\alpha_a,\beta_f,b_3,b_2)\exp[-S_{ef}(t_f)]S_t(x_2)\},
\end{eqnarray}
\begin{eqnarray}
\mathcal {M}^N_a&=&\frac{8}{3}C_Ff_B\pi M_B^4
\int_0^1dx_2dx_3\int_0^{\infty}b_1b_2db_1db_2\phi_{2}(x_2)\phi_{3}(x_3)\exp(-\frac{b_1^2\omega_B^2}{2})\times\nonumber\\&&
\{[r_2^2(x_2-1)+r_3^2(x_3-1)]\alpha_s(t_g)h_e(\beta_g,\alpha_a,b_1,b_2)\exp[-S_{gh}(t_g)]\nonumber\\&&-
[r_2^2x_2+r_3^2x_3-2r_2r_3r_b]\alpha_s(t_h)h_e(\beta_h,\alpha_a,b_1,b_2)\exp[-S_{gh}(t_h)]\},
\end{eqnarray}
\begin{eqnarray}
\mathcal {M}^T_a&=&\frac{8}{3}C_Ff_B\pi M_B^4
\int_0^1dx_2dx_3\int_0^{\infty}b_1b_2db_1db_2\phi_{2}(x_2)\phi_{3}(x_3)\exp(-\frac{b_1^2\omega_B^2}{2})\times\nonumber\\&&
\{[r_2^2(x_2-1)-r_3^2(x_3-1)]\alpha_s(t_g)h_e(\beta_g,\alpha_a,b_1,b_2)\exp[-S_{gh}(t_g)]\nonumber\\&&-
[r_2^2x_2-r_3^2x_3]\alpha_s(t_h)h_e(\beta_h,\alpha_a,b_1,b_2)\exp[-S_{gh}(t_h)]\}.
\end{eqnarray}
For the CKM suppressed $B_c\rightarrow D^{*+}_{(s)}D^{*0}$ decays,
the decay amplitudes for different polarizations are
\begin{eqnarray}
\mathcal {F}^L_{e2}&=&-2\sqrt{\frac{2}{3}}C_Ff_Bf_{3}\pi M_B^4
\int_0^1dx_2\int_0^{\infty}b_1b_2db_1db_2\phi_{2}(x_2)\exp(-\frac{b_1^2\omega_B^2}{2})\times\nonumber\\&&\{
[-(r_2-2)r_b+2r_2x_2-x_2]\alpha_s(t_a)h_e(\alpha_e,\beta_a,b_1,b_2)S_t(x_2)\exp[-S_{ab}(t_a)]
\nonumber\\&&+r^2_2\alpha_s(t_b)h_e(\alpha_e,\beta_b,b_2,b_1)S_t(x_1)\exp[-S_{ab}(t_b)]\},
\end{eqnarray}
\begin{eqnarray}
\mathcal {F}^N_{e2}&=&-2\sqrt{\frac{2}{3}}C_Ff_Bf_{3}r_3\pi M_B^4
\int_0^1dx_2\int_0^{\infty}b_1b_2db_1db_2\phi_{2}(x_2)\exp(-\frac{b_1^2\omega_B^2}{2})\times\nonumber\\&&\{
[2-r_b+r_2(4r_b-x_2-1)]\alpha_s(t_a)h_e(\alpha_e,\beta_a,b_1,b_2)S_t(x_2)\exp[-S_{ab}(t_a)]
\nonumber\\&&-r_2\alpha_s(t_b)h_e(\alpha_e,\beta_b,b_2,b_1)S_t(x_1)\exp[-S_{ab}(t_b)]\},
\end{eqnarray}
\begin{eqnarray}
\mathcal {F}^T_{e2}&=&2\sqrt{\frac{2}{3}}C_Ff_Bf_{3}r_3\pi M_B^4
\int_0^1dx_2\int_0^{\infty}b_1b_2db_1db_2\phi_{2}(x_2)\exp(-\frac{b_1^2\omega_B^2}{2})\times\nonumber\\&&\{
[2-r_b-r_2(1-x_2)]\alpha_s(t_a)h_e(\alpha_e,\beta_a,b_1,b_2)S_t(x_2)\exp[-S_{ab}(t_a)]
\nonumber\\&&-r_2\alpha_s(t_b)h_e(\alpha_e,\beta_b,b_2,b_1)S_t(x_1)\exp[-S_{ab}(t_b)]\},
\end{eqnarray}
\begin{eqnarray}
\mathcal {M}^L_{e2}&=&\frac{8}{3}C_Ff_B\pi M_B^4
\int_0^1dx_2dx_3\int_0^{\infty}b_2b_3db_2db_3\phi_{2}(x_2)\phi_{3}(x_3)\exp(-\frac{b_1^2\omega_B^2}{2})\times\nonumber\\&&
\{[2-x_1-x_2-x_3-r_2(1-x_2)]\alpha_s(t_c)h_e(\beta_c,\alpha_e,b_3,b_2)\exp[-S_{cd}(t_c)]-
\nonumber\\&&[x_3-x_1+r_2(1-x_2)]\alpha_s(t_d)h_e(\beta_d,\alpha_e,b_3,b_2)\exp[-S_{cd}(t_d)]\},
\end{eqnarray}
\begin{eqnarray}
\mathcal {M}^N_{e2}&=&-\mathcal {M}^T_{e2}=\frac{8}{3}C_Ff_B\pi M_B^4
\int_0^1dx_2dx_3\int_0^{\infty}b_2b_3db_2db_3\phi_{2}(x_2)\phi_{3}(x_3)\exp(-\frac{b_1^2\omega_B^2}{2})\nonumber\\&&\times
\{[r_3(x_1-x_3)]\alpha_s(t_c)h_e(\beta_c,\alpha_e,b_3,b_2)\exp[-S_{cd}(t_c)]+
\nonumber\\&&[2r_c-r_3(1-x_1-x_3)]\alpha_s(t_d)h_e(\beta_d,\alpha_e,b_3,b_2)\exp[-S_{cd}(t_d)]\}.
\end{eqnarray}
\section{Scales and related functions in hard kernel }\label{sec:b}
We show here the functions $h_e$, coming from the Fourier transform
of hard kernel,
\begin{eqnarray}
h_e(\alpha,\beta,b_1,b_2)&=&h_1(\alpha,b_1)\times h_2(\beta,b_1,b_2),\nonumber\\
h_1(\alpha,b_1)&=&\left\{\begin{array}{ll}
K_0(\sqrt{\alpha}b_1), & \quad \quad \alpha >0\\
K_0(i\sqrt{-\alpha}b_1),& \quad \quad \alpha<0
\end{array} \right.\nonumber\\
h_2(\beta,b_1,b_2)&=&\left\{\begin{array}{ll}
\theta(b_1-b_2)I_0(\sqrt{\beta}b_2)K_0(\sqrt{\beta}b_1)+(b_1\leftrightarrow b_2), & \quad \beta >0\\
\theta(b_1-b_2)J_0(\sqrt{-\beta}b_2)K_0(i\sqrt{-\beta}b_1)+(b_1\leftrightarrow b_2),& \quad \beta<0
\end{array} \right.
\end{eqnarray}
where $J_0$ is the Bessel function and $K_0$, $I_0$ are modified
Bessel function with $K_0(ix)=\frac{\pi}{2}(-N_0(x)+i J_0(x))$. The
hard scale t is chosen as the maximum virtuality of the internal
momentum transition in the hard amplitudes, including
$1/b_i(i=1,2,3)$:
\begin{eqnarray}
t_a&=&\max(\sqrt{|\alpha_e|},\sqrt{|\beta_a|},1/b_1,1/b_2),\quad t_b=\max(\sqrt{|\alpha_e|},\sqrt{|\beta_b|},1/b_1,1/b_2),\nonumber\\
t_c&=&\max(\sqrt{|\alpha_e|},\sqrt{|\beta_c|},1/b_2,1/b_3),\quad t_d=\max(\sqrt{|\alpha_e|},\sqrt{|\beta_d|},1/b_2,1/b_3),\nonumber\\
t_e&=&\max(\sqrt{|\alpha_a|},\sqrt{|\beta_e|},1/b_2,1/b_3),\quad t_f=\max(\sqrt{|\alpha_a|},\sqrt{|\beta_f|},1/b_2,1/b_3),\nonumber\\
t_g&=&\max(\sqrt{|\alpha_a|},\sqrt{|\beta_g|},1/b_1,1/b_2),\quad t_h=\max(\sqrt{|\alpha_a|},\sqrt{|\beta_h|},1/b_1,1/b_2),
\end{eqnarray}
where
\begin{eqnarray}\label{eq:betai}
\alpha_e&=&(1-x_2)(x_1-r_2^2)(1-r_3^2)M_B^2,\quad \alpha_a=-(1+(r_3^2-1)x_2)(1+(r_2^2-1)x_3)M_B^2,\nonumber\\
\beta_a&=&[r_b^2+(r_2^2-1)(x_2+r_3^2(1-x_2))]M_B^2,\quad \beta_b=(1-r_3^2)(x_1-r_2^2)M_B^2,\nonumber\\
\beta_c&=&[r_c^2-(1-x_2(1-r_3^2))(1-x_1-x_3(1-r_2^2))]]M_B^2,\nonumber\\\quad \beta_d&=&(1-x_2)(1-r_3^2)[x_1-x_3-r_2^2(1-x_3)]M_B^2,\nonumber\\
\beta_e&=&-[1+(r_3^2-1)x_2]M_B^2,\quad \beta_f=-[1+(r_2^2-1)x_3]M_B^2,\nonumber\\
\beta_g&=&[r_c^2+(1-x_2(1-r_3^2))(x_1+x_3-1-r_2^2x_3)]M_B^2,\quad \nonumber\\\beta_h&=&[r_b^2-x_2(r_3^2-1)(x_1-x_3(1-r_2^2))]M_B^2.
\end{eqnarray}
The Sudakov factors used in the text are defined by
\begin{eqnarray}
S_{ab}(t)&=&s(\frac{M_B}{\sqrt{2}}x_1,b_1)+s(\frac{M_B}{\sqrt{2}}x_2,b_2)
+\frac{5}{3}\int_{1/b_1}^t\frac{d\mu}{\mu}\gamma_q(\mu)+2\int_{1/b_2}^t\frac{d\mu}{\mu}\gamma_q(\mu),\nonumber\\
S_{cd}(t)&=&s(\frac{M_B}{\sqrt{2}}x_1,b_2)+s(\frac{M_B}{\sqrt{2}}x_2,b_2)+s(\frac{M_B}{\sqrt{2}}x_3,b_3)
\nonumber\\&&+\frac{11}{3}\int_{1/b_2}^t\frac{d\mu}{\mu}\gamma_q(\mu)+2\int_{1/b_3}^t\frac{d\mu}{\mu}\gamma_q(\mu),\nonumber\\
S_{ef}(t)&=&s(\frac{M_B}{\sqrt{2}}x_2,b_2)+s(\frac{M_B}{\sqrt{2}}x_3,b_3)
+2\int_{1/b_2}^t\frac{d\mu}{\mu}\gamma_q(\mu)+2\int_{1/b_3}^t\frac{d\mu}{\mu}\gamma_q(\mu),\nonumber\\
S_{gh}(t)&=&s(\frac{M_B}{\sqrt{2}}x_1,b_1)+s(\frac{M_B}{\sqrt{2}}x_2,b_2)+
s(\frac{M_B}{\sqrt{2}}x_3,b_2),\nonumber\\&&
+\frac{5}{3}\int_{1/b_1}^t\frac{d\mu}{\mu}\gamma_q(\mu)+4\int_{1/b_2}^t\frac{d\mu}{\mu}\gamma_q(\mu),
\end{eqnarray}
where the functions $s(Q,b)$ are defined in Appendix A of \cite{epjc45711}. $\gamma_q=-\alpha_s/\pi$ is the anomalous dimension of the quark.
\section{Meson Wave functions }\label{sec:c}
In the nonrelativistic limit, the $B_c$ meson wave function can be
written as \cite{prd81014022}
\begin{eqnarray}
\Phi_{B_c}(x)=\frac{if_B}{4N_c}[(\rlap{/}{P}+M_{B_c})\gamma_5\delta(x-r_c)]\exp(-\frac{b^2\omega_B^2}{2}),
\end{eqnarray}
in which the last exponent term represents the $k_T$ distribution.
Here, we only consider the dominant Lorentz structure and
neglect another contribution in our calculation \cite{epjc28515}.
In the heavy quark limit, the two-particle light-cone distribution
amplitudes of $D_{(s)}/D_{(s)}^*$ meson are defined as \cite{prd67054028}
\begin{eqnarray}\label{eq:dwave}
\langle D_{(s)}(P_2)|q_{\alpha}(z)\bar{c}_{\beta}(0)|0\rangle &=&
\frac{i}{\sqrt{2N_c}}\int^1_0dx e^{ixP_2\cdot
z}[\gamma_5(\rlap{/}{P}_2+m_{D_{(s)}})\phi_{D_{(s)}}(x,b)]_{\alpha\beta},
\nonumber\\
\langle D_{(s)}^*(P_2)|q_{\alpha}(z)\bar{c}_{\beta}(0)|0\rangle
&=&-\frac{1}{\sqrt{2N_c}}\int^1_0dx e^{ixP_2\cdot
z}[\rlap{/}{\epsilon}_L(\rlap{/}{P}_2+m_{D_{(s)}^*})\phi^L_{D_{(s)}^*}(x,b)\nonumber\\&&
+\rlap{/}{\epsilon}_T(\rlap{/}{P}_2+m_{D_{(s)}^*})\phi^T_{D_{(s)}^*}(x,b)]_{\alpha\beta},
\end{eqnarray}
with the normalization conditions:
\begin{eqnarray}
\int^1_0dx \phi_{D_{(s)}}(x,0)=\frac{f_{D_{(s)}}}{2\sqrt{2N_c}},\quad \int^1_0dx
\phi^{L}_{D_{(s)}^*}(x,0)= \int^1_0dx
\phi^{T}_{D_{(s)}^*}(x,0)=\frac{f_{D_{(s)}^*}}{2\sqrt{2N_c}},
\end{eqnarray}
where we have assumed $f_{D_{(s)}^*}=f^T_{D_{(s)}^*}$. Note that
equations of motion do not relate $\phi^L_{D_{(s)}^*}$ and
$\phi^T_{D_{(s)}^*}$. We use the following relations derived from
HQET \cite{hqet} to determine $f_{D^*_{(s)}}$
\begin{eqnarray}\label{eq:Ddecayc}
f_{D^*_{(s)}}=\sqrt{\frac{m_{D_{(s)}}}{m_{D_{(s)}^*}}}f_{D_{(s)}}.
\end{eqnarray}
The distribution amplitude $\phi^{(L,T)}_{D_{(s)}^{(*)}}$ is taken as \cite{09101424}
\begin{eqnarray}
\phi^{(L,T)}_{D_{(s)}^{(*)}}=\frac{3}{\sqrt{2N_c}}f_{D^{(*)}_{(s)}}x(1-x)[1+a_{D^{(*)}_{(s)}}(1-2x)]\exp(-\frac{b^2\omega^2_{D_{(s)}}}{2}).
\end{eqnarray}
We use $a_D=0.5 \pm 0.1, \omega_{D}= 0.1 \text{GeV}$ for $D/D^*$ meson and $a_D=0.4 \pm 0.1, \omega_{D_s}= 0.2 \text{GeV}$ for $D_s/D_s^*$ meson,
which are determined in Ref. \cite{dd1} by fitting.
\end{appendix}
|
{
"timestamp": "2012-09-20T02:02:26",
"yymm": "1203",
"arxiv_id": "1203.2303",
"language": "en",
"url": "https://arxiv.org/abs/1203.2303"
}
|
\section*{}
\vspace{-1cm}
\footnotetext{\textit{$^{a}$~Department of Physics, Indian Institute of Science, Bangalore 560 012, India}}
\footnotetext{\textit{$^{b}$~Department of Materials Science and Engineering, Stanford University
Stanford, CA 94305}}
\footnotetext{\textit{$^{c}$~Engineering Mechanics Unit, Jawaharlal Nehru Centre for Advanced Scientific Research, Bangalore 560 064, India. }}
A sphere falling through a Newtonian fluid at low Reynolds number Re
\cite{stokes} reaches a steady terminal velocity through the interplay of
gravity, buoyancy and viscous drag. This simple system, and its extensions to
unsteady high Re flow, are well understood and form the basis of falling ball
viscometers \cite{Chhabra}. In non-Newtonian fluids unsteady settling is
observed \cite{Mollinger, Belmonte} even at low Re. Mollinger \textit{et al.}
\cite{Mollinger} observed periodic stick-slip settling in the beginning of
the fall and investigated the effect of wall confinement on the baseline
settling velocity and the frequency of unsteady events. Jayaraman \textit{et
al.} \cite{Belmonte} found a transition from steady to \textit{irregular
unsteady} settling, remarked that the relevant control parameter was the shear
rate, $\dot{\gamma} = v_{b}/d$ ($v_{b}$ \& $d$ being the baseline velocity and
sphere diameter respectively), and emphasized the importance of the negative
wake behind the particle \cite{Harlen,Hassager-Bisgard}. It was suggested
\cite{Belmonte} that the oscillations were a result of the formation and breakup
of flow-induced structures, known to arise in concentrated micellar systems
\cite{Walker,Rehage,Cates_Fielding_review}.
In this paper we investigate the sedimentation of spherical metallic particles
through an entangled wormlike-micelle solution \cite{Cates_Fielding_review} of
the surfactant CTAT (Cetyl Trimethyl Ammomium p-Toluenesulphonate) in brine. We
report here for the first time that spheres larger than a critical size not only
undergo unsteady motion, but also show sustained, repeated bursts of
oscillations superposed on a constant baseline velocity as shown in Fig.
\ref{bursts}. The intra-burst oscillation period $\tau_{fast} \sim d/v_b$; the
time between bursts $\tau_{slow}$ is at least an order of magnitude larger. We
account for these results through a model incorporating shear-induced coupled
oscillations of viscoelastic stress components and a slow structural relaxation
\cite{ajdarietal2002,aradiancates} giving rise to $\tau_{slow}$. We suggest that
equilibrium fluctuations, i.e., without imposed flow, should show signatures of
the structural variable.
Experiments were conducted on two different samples, Gel A comprising 2 wt$\%$ CTAT $+$ 100 mM NaCl and Gel B with 2.2 wt$\%$ CTAT $+$ 82 mM NaCl. Rheological measurements were carried out on a Paar Physica MCR 300 Rheometer in the cone-plate geometry. The frequency sweep and flow curve of Gel A give an extrapolated zero-shear-rate viscosity of $17.8$ Pa s at 25$^{\circ}$C, onset of shear thinning at a shear rate $\dot{\gamma_c} \sim 0.4 s^{-1}$. The linear viscoelasticity is Maxwellian with a relaxation time $\omega_{co}^{-1} \simeq 1.34$ s. The corresponding numbers for Gel B are $8.5$ Pa s, 1.5 s$^{-1}$ and $0.97$ s respectively. These systems are in the parameter range where shear banding instabilities and spatiotemporal rheochaos are expected above a critical shear rate \cite{Ranjini, Rajesh, RajeshPRE,RajeshJNNFM}.
A transparent cylindrical tube of diameter $D=$ 5.4 cm and height 40 cm was filled with the wormlike-micelle solution. Steel balls of diameters $d$ from 3 mm to 8 mm were introduced one at a time into the solution with zero initial velocity using an electromagnet positioned accurately to ensure that the balls sediment along the axis of the tube. Images of the falling balls were recorded using a Citius C100 Centurio camera capable of capturing 423 frames per second at its full resolution of 1280$\times$1024 pixels. Image analysis was done on the captured frames to extract information about the position and velocity of the ball as a function of time. Experimental runs were separated by at least 30 minutes in order to ensure complete relaxation and homogenization of the fluid prior to each run.
\begin{figure}[h]
\centering
\includegraphics[height=8.5cm]{Fig1.eps}
\caption{Velocity as a function of time: For Gel A; (a) d=0.4 cm, (b) d=0.56 cm and (c) d=0.7 cm. In (d), the velocity variation marked in (c) has been expanded. For Gel B: (e) d=0.5 cm, (f) d=0.7 cm and (g) d=0.8 cm. In (h), the velocity variation marked in (g) has been expanded.}\label{bursts}
\end{figure}
We present first our results for Gel A. Fig. \ref{bursts} shows velocity-time
plots of various diameters $d$. For $d=0.4 cm$ (Fig. \ref{bursts}(a)) the ball
quickly achieves a steady terminal velocity as expected in the Stokesian limit
for a Newtonian fluid. For $d=0.56 cm$ (Fig. \ref{bursts}(b)) a strong
oscillatory transient as in noted by King and Waters \cite{KingWaters} is seen
before terminal velocity sets in. For $d=0.70cm$ (Fig. \ref{bursts}(c)),
oscillations are seen to occur in repeated bursts on top of a constant baseline
velocity $v_{b}$, and are sustained throughout the fall of the particle. The
time between bursts is 4 to 7 sec, at least an order of magnitude larger than
the oscillation period $\sim 0.3$ s within the burst. Fig. \ref{bursts}(d) is an
enlarged version of the bursts highlighted in Fig. \ref{bursts}(c). Within each
burst the oscillations have an exponentially damped sinusoidal form; the total
number of such bursts before the ball reaches the bottom is too small for us to
conclude whether their timing is truly periodic or merely distributed narrowly
about a well-defined mean. The figures also show that the oscillations are
occasionally large enough to result in a small negative velocity; i.e., the ball
sometimes jumps up against gravity. Visual observation shows a strong negative
wake behind the ball during the burst. Gel B shows similar behaviour as seen in
Figs. \ref{bursts}(e-h). Interestingly, it should be noted that the baseline
velocities scale as $d^2$, consistent with Stokes's Law. Jayaraman \textit{et
al}.\cite{Belmonte} remarked that oscillations set in at velocity gradients
$v_{b}/d$ close to the frequency where $G''$ (loss modulus) begins to increase. We do not
observe such a correspondence.
\begin{figure}[h]
\centering
\includegraphics[height=8cm]{Fig2.eps}
\caption{The motion of 0.7 cm ball in CTAT (2wt$\%$)+100 mM NaCl dropped in tubes of various diameters. Note the increase in the baseline velocity, $v_{b}$ in thinner tubes (diameter $D$) followed by decreasing time-period ($T$) of burst oscillations.} \label{WallEffect}
\end{figure}
Fig. 2 shows the effect of wall confinement for a ball of diameter d=0.7 cm. A comparison of baseline velocities $v_{b}$ in tubes of diameter D = 5.4 cm, 3.2 cm and 2.0 cm shows that $v_{b}$ increases as D decreases, an effect opposite to that known in Newtonian fluids \cite{Chhabra}. The effect is observed for smaller diameters as well. We find that within a burst the time-period $T$ (as marked in Fig. \ref{WallEffect}) decreases with decreasing $D$. Local thinning caused by flow alignment of micellar structure along the walls, expected to be more pronounced in narrower tubes, is a likely explanation for these observations.
In order to account for the observation of bursts and oscillations, we now offer
a simple theoretical model based on the interplay of the orientation field of
the micelles and flow generated by the settling particle. We emphasize that our
intent is to offer a rational account of the observed phenomena, not to
reproduce them in full detail.
Further, we will show that the
model, although schematic in nature, nonetheless leads to other predictions that
can be tested in independent experiments. We start by summarizing some features
of the behaviour shown in Fig. \ref{bursts}(c). The phenomenon of primary
interest is the observation of rapidly oscillatory bursts repeated at fairly
regular long intervals. Each burst is damped in a roughly exponential manner,
with a time constant noticeably smaller than the interval between bursts, i.e.,
each successive burst emerges from a flow that has become nearly quiescent. The
baseline settling speed $v_0 \simeq 1.15$ cm/s, with oscillations up to 12 cm/s.
The resulting shear-rate at the scale of a particle (diameter $d \simeq 0.7$ cm)
is $\dot{\gamma} = v_0/d \simeq 1.64$ to $17.1$ s$^{-1}$, corresponding to a
timescale $\dot{\gamma}^{-1} \simeq .06$ s to $0.6$ s for the accumulation of a
strain of unity. The micellar solution has a measured zero-shear-rate viscosity
of $\eta_0 \simeq$ 18.0 Pa s, thinning under shear to about 1.4 Pa s, and a
viscoelastic or orientational relaxation time $\tau_v \simeq 1.34$ s. The
Reynolds number at the particle scale is thus at most $6 \times 10^{-4}$. The
settling particles are steel (density $\rho \simeq 8$ g/cm$^3$) and hence would
reach terminal velocity, if the viscosity were constant, in a time of order
$\tau_i = \rho d^2/18 \eta_0 \simeq$ 1.2 to 15.6 millisec depending on whether
one takes the zero-shear or thinned value for the viscosity. The observed period
\textit{within} the bursts is $T_1 \simeq 0.36$ s, and the period
\textit{between } bursts is $T_2 \simeq 9$ s. $T_1$ and $\tau_v$ are comparable
to $\dot{\gamma}^{-1}$, $T_2 \gg T_1, \tau_v$, and $\tau_i$ is much smaller than
all of these. Both acceleration and advection can thus be neglected in the
Navier-Stokes equation, and we can work in the limit of Stokesian hydrodynamics,
keeping track of stresses arising from the complex nature of the micellar
solution. We now present our model for the coupled dynamics of the particle and
the micellar medium.
Consider a particle settling under gravity in a wormlike-micelle solution with velocity field $\mathbf{u(r)}$ at location $\mathbf{r}$, with solvent viscosity $\eta$ and a contribution $\mathbf{\sigma}^{(m)}$ to the stress tensor from the conformations of the micelles. The steady Stokes equation
\begin{equation}
\label{eq:stokes}
-\eta \nabla^2 \mathbf{u} = -\nabla P + \mathbf{W} -\nabla \cdot \sigma^{(m)}
\end{equation}
balances viscous, pressure, and conformational stresses against the
gravitational force from the buoyant weight $\mathbf{W}$ of the particle.
Inverting (\ref{eq:stokes}) and choosing the particle's centre of mass as
origin, the settling velocity
\begin{equation}
\label{eq:veleq1}
\mathbf{v} = \mathbf{v}_0 + \int d^3 r \, \mathbf{H(r)} \cdot \nabla \cdot \sigma^{(m)}(\mathbf{r})
\end{equation}
where $\mathbf{H}$ is the incompressible Stokesian hydrodynamic propagator and
$\mathbf{v}_0$ is the Stokes settling velocity in the absence of contributions
from micelle conformations, and the second term on the right in
(\ref{eq:veleq1}) contains modifications of the settling speed due to micellar
stresses around the particle. The dynamics of $\sigma^{(m)}$
\begin{equation}
\label{eq:jslike}
{D \sigma^{(m)} \over Dt} = \lambda_0 \mathbf{A} + \lambda_1 [\mathbf{A} \cdot \sigma^{(m)}]_{ST} - {1 \over \tau} \sigma^{(m)}
\end{equation}
combines flow-orientation coupling and relaxation, familiar from a variety of contexts including general rheological models, rheochaos, and nematic hydrodynamics \cite{js,aradiancates,moumitaetalrheochaos,rienaecker}. Here $D/Dt$ is a time derivative comoving and corotating with the fluid, $\mathbf{A}$ is the symmetric deformation-rate tensor, and the subscript $ST$ denotes symmetric traceless part.
It is useful to work with a simple model extracted from (\ref{eq:veleq1}) and
(\ref{eq:jslike}). From (\ref{eq:veleq1}) we suggest that a component $p$ of the
micellar stress, presumably originating in micelle orientation around the
particle alters the settling speed: $v = v_0 + p$ where $v_0$ is the speed in
the absence of contributions from the micellar stress. The form of
(\ref{eq:jslike}) suggests the schematic dynamics $\dot{p} \sim v + v p + vq -
p+ ...$, where production by shear competes with relaxation to local
equilibrium. As in the Johnson-Segalman model \cite{js} or nematic hydrodynamics
\cite{degp,forster,rienaecker,moumitaetalrheochaos} shear, through $v$,
naturally couples $p$ to other components $q$, which in turn have a similar
dynamics. The ellipsis indicates terms of higher order in $p$ and $q$, and we
have assumed that the local shear-rate is proportional to the settling speed
$v$. The structure of these equations raises the possibility of oscillations
with a timescale set by the shear-rate. Such shear-induced mixing of micellar
stress components is very likely the mechanism for the oscillations
\textit{within} a burst, as their frequency, as seen from the numbers reported
above, lies right in the middle of the range of particle-scale shear-rates
$\dot{\gamma}$. The timescale associated with the repeated appearance of the
bursts is far longer, as remarked above, and must arise from a distinct process.
We propose that it enters through the memory mechanism introduced by
Cates and coworkers \cite{ajdarietal2002,aradiancates} that we incorporate and
explain in detail below. Assembling these ingredients we replace the dynamics of
the micellar stress tensor (\ref{eq:jslike}) by the effective model
\begin{equation}
\label{peq}
\dot{p} = \lambda_0 v + \lambda_1 v p + \lambda_2 v q -\zeta p - s w -
\mathcal{P}(p,q),
\end{equation}
\begin{equation}
\label{qeq}
\dot{q} = \mu_0 v + \mu_1 v p + \mu_2 v q - \gamma q - \mathcal{Q}(p,q),
\end{equation}
and (\ref{eq:veleq1}) by the schematic relation $v=v_0+p$ already mentioned
above. The terms $\mathcal{P}$ and $\mathcal{Q}$ in (\ref{peq}), (\ref{qeq})
incorporate nonlinearities that would arise naturally in a local thermodynamic
approach \cite{degp,moumitaetalrheochaos} where $p$ and $q$ are related to the
orientational order parameter. The coefficients $\lambda_i, \, \mu_i$ are
flow-orientation couplings, corresponding to similar terms in (\ref{eq:jslike})
and best understood from nematic hydrodynamics
\cite{forster,degp,rienaecker,starklubensky,moumitaetalrheochaos,lambdafootnote}
. We have allowed the relaxation of $p$ to occur
through two channels: locally in time via the term $\zeta p$ in
(\ref{peq}), and through the intervention of a structural variable\footnote{In
principle $q$ could also relax through such a memory
mechanism. For simplicity we have not explored this possibility.}
\cite{ajdarietal2002,aradiancates,fieldingOlmsted}
\begin{equation}
\label{memeq}
w(t) = \int_{-\infty}^t dt' e^{(t-t')\Gamma} p(t')
\end{equation}
determined by the past history of $p$. Equivalently,
\begin{equation}
\label{weq}
\dot{w}=\Gamma(p-w),
\end{equation}
whence it is clear that $w$ becomes identical to $p$ only for $\Gamma \to
\infty$. We
see formally that $w$ and $p$ stand in the same relation as position and
momentum for an oscillator, even in regard to the relative signs of the
coefficients of $w$ in (\ref{peq}) and $p$ in (\ref{weq}). We emphasize that
this relation has nothing to do with real inertia: our equations are meant to
describe the viscosity-dominated regime. Equation (\ref{weq}) says, plausibly,
that an imposed nonzero local
alignment $p$ shifts the equilibrium value of an underlying structural quantity
such as the Maxwell viscoelastic time of the material
\cite{ajdarietal2002,aradiancates} or the micelle length \cite{fieldingOlmsted}.
As in those works, we assume that the parameters $\Gamma$ and $s$ of this
mechanochemical coupling are properties of the unsheared system; our
observations are consistent with $\Gamma \ll \dot{\gamma}$, the particle-scale
shear
rate.
The parameter space of (\ref{peq}), (\ref{qeq}) and (\ref{weq}) is vast, and our limited exploration finds regimes of spontaneous oscillation. Fig. \ref{OscReproduced} shows $p$ (which is $v$ apart from a constant) as a function
of time for the model equations $\dot{p} = A_1 - A_2 p - A_3q - A_4 w + A_5p^2 - A_6p^3$, $\dot{q} = \alpha(B_1 - B_2 p - q + B_3 p^2)$, $\dot{w} =
\Gamma(p-w)$\footnote{with parameter values $A_1$=3, $A_2$=10, $A_3$=1,
$A_4$=$A_5$=6, $A_6$=1, $B_1$=3, $B_2$=12, $B_3$=6 \& $\Gamma$=0.006}, bringing
in
$q$ gradually by stepping $\alpha = 0,
0.1, 0.15 \& 1.0$.
\begin{figure}[t]
\centering
\includegraphics[height=6cm]{Fig3.eps}
\caption{Oscillations in the settling speed from (\ref{peq}), (\ref{qeq}), (\ref{weq}) for several $\alpha$ as indicated. Note the two distinct timescales, within and between bursts.}\label{OscReproduced}
\end{figure}
Our model, constructed using physically plausible arguments, reproduces key
features of the experimental observation, namely, a rapid oscillation timescale
clearly associated with the shear and a much slower oscillation of distinct
origin. Further, if the contribution of $q$ is diminished by reducing $\alpha$
only the slow oscillation of $p$ against $w$ is seen. Only when $\alpha$ is
large are the rapid oscillations seen, and these disappear on a timescale much
smaller than that associated with the slow oscillation.
As noted in \cite{aradiancates,bulbulprl2011}, equations of the general form
(\ref{peq}) - (\ref{weq}) are widespread in the neuronal modelling literature
\cite{fitz,hindmarsh}. Among the most robust features of such models are
oscillations with a bursty character, and the coexistence of two or more widely
separated timescales. In the neurophysics literature the introduction of an
additional slow ``recovery variable'' was motivated in part by detailed
knowledge of the electrophysiology of the axon \cite{hindmarsh}. The success of
our model in capturing essential features of our experiment encourages us to
suggest independent experimental tests for the coupled dynamics of micellar
stress $p$ and ``structure'' $w$ in the absence of shear. If not for the
structural variable $w$ we should have a conventional Johnson-Segalman or
nematogenic fluid \textit{at rest}, whose dynamical response and spontaneous
fluctuations are overdamped. However, in (\ref{peq}) and (\ref{weq}) the
micellar stress $p$ and the structural variable $w$ act like the momentum and
coordinate of an oscillator. And precisely because we require $w$ to be
\textit{slow}, the damping coefficient $\Gamma \ll s$, the coupled dynamics is
in the
\textit{underdamped} regime, so their linear response to disturbance must be
oscillatory. Further, if we augment the dynamical equations (\ref{peq}),
(\ref{weq}) with thermal noise, the resulting Fourier-transformed dynamic
correlation functions of the system in the absence of the falling ball should
show a peak at a frequency around that of the \textit{slow} oscillation seen in
the falling-ball experiments.This feature seems to be quite robust, relying only
on the assumption that the parameters in the dynamical equation for the
structural variable $w$ depend negligibly on the shear rate. This property
is shared by the models presented by Cates \textit{et al.}
\cite{ajdarietal2002,aradiancates} and emerges naturally from microscopic
kinetic models \cite{Cates_Fielding_review}. However, it is conceivable that
the kinetics in our system is such that $w$ becomes slow only under strong
imposed shear, in which case the \textit{equilibrium}, i.e., unsheared,
fluctuations of the coupled $p-w$ dynamics would be overdamped. A search for
such oscillatory response or correlation in \textit{equilibrium} experiments
would help resolve this issue. This in turn requires a dynamical probe of local
anisotropy; a possible scheme might be to suspend droplets of a strongly
flow-birefringent fluid in the wormlike-micelle solution, and measure the power
spectral density of the spontaneous fluctuations in the light intensity seen
through crossed polars.
We emphasize that our simple model aims primarily to capture the bursts
and the two disparate time-scales in the problem. Within the limitations of this
approach, we have suggested possible independent tests of the model. Further
experiments with different viscoelastic parameters and particle shapes
should lead us to a more quantitative theory, at the level
of reduced models like (\ref{peq}) - (\ref{weq}) or a more complete hydrodynamic
treatment. A better understanding, perhaps in terms of micelle kinetics
\cite{Cates_Fielding_review}, of the slow structural variable in our system is
of particular importance.
To summarise: we have isolated the mechanism for the observed oscillatory
descent of a sphere through a wormlike-micelle solution. The oscillations
produced by (\ref{peq})-(\ref{weq}) show two broadly different timescales, as in
the experiments, although the detailed form of the oscillations is not identical
to those observed. One timescale is associated with the shear, the other with
the coupled dynamics of micellar stress and a structural parameter such as
micellar length or equilibrium Maxwell time. We argue that this second
timescale is likely to appear in the unsheared system in independent
measurements of the autocorrelation of spontaneous fluctuations of stress or
orientation, and suggest experiments to test this possibility.
Flow-birefringence studies \cite{rehage1986,tropea2007} focusing on the velocity
and
orientation fields of the medium around the falling ball will offer further
insight into the detailed mechanism underlying the phenomena presented in this
work. We note that the birefringence contrast as seen by Rehage \textit{et al.}
\cite{rehage1986}
in studies on solutions similar to ours, is small. Such measurements on our
system will therefore require a careful high resolution study, which we defer
to later work.
Numerical studies of the complete hydrodynamic equations for
orientable
fluids, studying the effect of the gravitational settling of a sphere through
the medium, should see these curious oscillations, and will allow us to converge
on the correct effective model of the general type of (\ref{peq}) - (\ref{weq}).
We thank M.E. Cates for useful discussions. NK and SM are grateful to the
University Grants Commission, India for support. AKS and SR
respectively acknowledge a CSIR Bhatnagar fellowship and a DST J.C. Bose
fellowship.
|
{
"timestamp": "2012-03-12T01:02:06",
"yymm": "1203",
"arxiv_id": "1203.2130",
"language": "en",
"url": "https://arxiv.org/abs/1203.2130"
}
|
\section{Introduction}
Density functional theory (DFT) is perhaps one of the most successful
theories in Physics and in Chemistry of the last
half-century~\cite{PhysRev.136.B864,Parr1989,Dreizler90,Fiolhais2003}. It is currently
used to predict the structure and the properties of atoms, molecules,
and solids; it is a key ingredient of the new field of Materials
Design, where one tries to create new materials with specific
properties; it is making its way in Biology as an important tool in
the investigation of proteins, DNA, etc. These are only a few examples
of a discipline that even now, almost 50 years after its birth, is
growing at an exponential rate.
Almost all applications of DFT are performed within the so-called
Kohn-Sham scheme~\cite{PhysRev.140.A1133}, that uses a non-interacting
electronic system to calculate the density of the interacting
system~\cite{vanLeeuwen200325}. The Kohn-Sham scheme leeds to the
following equations. (Hartree atomic units are used throughout the
paper, i.e. $e^2=\hbar=m_e=1$.)
\begin{equation}
\left[ -\frac{\nabla^2}{2} + v_{\rm ext}({\bf r}) + v_{\rm Hartree}[n]({\bf r}) + v_{\rm xc}[n]({\bf r})\right] \psi_i({\bf r})
= \varepsilon_i \psi_i({\bf r})
\,,
\end{equation}
where the first term represents the kinetic energy of the electrons;
the second is the external potential usually generated by a set of
Coulombic point charges (sometimes described by pseudopotentials); the
third term is the Hartree potential that describes the classical
electrostatic repulsion between the electrons,
\begin{equation}
v_{\rm Hartree}[n]({\bf r}) = \int\!\!d^3r'\,\frac{n({\bf r}')}{|{\bf r}-{\bf r}'|}
\,,
\end{equation}
and the exchange-correlation (xc) potential $v_{\rm xc}[n]$ is defined by
\begin{equation}
v_{\rm xc}[n]({\bf r}) = \frac{\delta E_{\rm xc}[n]}{\delta n({\bf r})}
\,.
\end{equation}
$E_{\rm xc}[n]$ is the xc energy functional. Note that by $[n]$ we
denote that the quantity is a {\it functional} of the electronic
density,
\begin{equation}
n({\bf r}) = \sum_i^{\rm occ.} |\psi_i({\bf r})|^2
\,,
\end{equation}
where the sum runs over the occupied states.
The central quantity of this scheme is the xc energy $E_{\rm xc}[n]$
that describes all non-trivial many-body effects. Clearly, the exact
form of this quantity is unknown and it must be approximated in any
practical application of DFT. We emphasize that the precision of any
DFT calculation depends solely on the form of this quantity, as this
is the only real approximation in DFT (neglecting numerical
approximations that are normally controllable).
The first approximation to the exchange energy, the local density
approximation (LDA), was already proposed by Kohn and Sham in the same
paper where they described their Kohn-Sham
scheme~\cite{PhysRev.140.A1133}. It states that the value of xc energy
density at any point in space is simply given by the xc energy density
of a homogeneous electron gas (HEG) with electronic density $n({\bf r})$.
Mathematically, this is written as
\begin{equation}
E^{\rm LDA}_{\rm xc} = \int\!\!d^3r\, n({\bf r}) e^{\rm HEG}_{\rm xc}(n({\bf r}))
\,,
\end{equation}
where $e^{\rm HEG}_{\rm xc}(n)$ is the xc energy {\it per electron} of
the HEG. Note that this quantity is a {\it function} of $n$. While the
exchange contribution $e^{\rm HEG}_{\rm x}(n)$ can be easily
calculated analytically, the correlation contribution is usually taken
from Quantum Monte-Carlo
simulations~\cite{PhysRevLett.45.566,PhysRevB.50.1391}. As defined,
the LDA is unique, but, as we will see in the \ref{app:1}, even such
precise definition can give rise to many different parameterizations.
\begin{figure}[t]
\centering
\begin{pspicture}(12,8)
\psline[linecolor=black](5,1)(5,7)
\psline[linecolor=black](7,1)(7,7)
\psline[linecolor=black](5,2)(7,2)
\psline[linecolor=black](5,3)(7,3)
\psline[linecolor=black](5,4)(7,4)
\psline[linecolor=black](5,5)(7,5)
\psline[linecolor=black](5,6)(7,6)
\rput(6,0.25){Hartree world}
\rput(6,7.5){Chemical accuracy}
\rput[l](7.5,2){LDA}
\rput[l](7.5,3){GGA}
\rput[l](7.5,4){meta-GGA}
\rput[l](7.5,5){EXX with correlation}
\rput[l](7.5,6){EXX with partial exact correlation}
\rput[r](4.5,2){$n({\bf r})$}
\rput[r](4.5,3){$\nabla n({\bf r})$}
\rput[r](4.5,4){$\nabla^2 n({\bf r}), \tau({\bf r})$}
\rput[r](4.5,5){$\psi_i({\bf r})$ (occupied)}
\rput[r](4.5,6){$\psi_i({\bf r})$ (empty)}
\end{pspicture}
\caption{Jacob's ladder of density functional approximations for the
xc energy.}
\label{fig:jacob}
\end{figure}
During the past 50 years, hundreds of different forms
appeared~\cite{Scuseria2005} and they are usually arranged in
families, which have names such as generalized-gradient approximations
(GGAs), meta-GGAs, hybrid functionals, etc. In 2001, John Perdew came
up with a beautiful idea on how to illustrate these families and their
relationship~\cite{perdew:1}. He ordered these families as rungs in a ladder that
leads to the heaven of ``chemical accuracy'', and that he christened
the Jacob's ladder of density functional approximations for the xc
energy (see Fig.~\ref{fig:jacob}). Every rung adds a dependency on
another quantity, thereby increasing the precision of the functional
but also increasing the numerical complexity and the computational
time.
At the bottom of the ladder we find the LDA, a functional that depends
locally on the density only. The second rung is occupied by the GGA
\begin{equation}
E^{\rm GGA}_{\rm xc} = \int\!\!d^3r\, n({\bf r}) e^{\rm GGA}_{\rm xc}(n({\bf r}), \nabla n({\bf r}))
\,.
\end{equation}
As one can see, one now adds the gradient of the density, a semi-local
quantity that depends on an infinitesimal region around ${\bf r}$, as a
parameter to the energy density. Note that there is a considerable
amount of craftsmanship and physical/chemical intuition going into the
creation of the function $e^{\rm GGA}_{\rm xc}(n, \nabla n)$, but also
a fair quantity of arbitrariness. It is therefore not surprising that
many different forms were proposed over the years. The same is true
for the functionals on the next rung, the meta-GGAs
\begin{equation}
E^{\rm mGGA}_{\rm xc} = \int\!\!d^3r\, n({\bf r}) e^{\rm mGGA}_{\rm xc}(n({\bf r}), \nabla n({\bf r}), \nabla^2 n({\bf r}), \tau({\bf r}))
\,.
\end{equation}
This time one adds the Laplacian of the density $\nabla^2 n({\bf r})$ and
also (twice) the kinetic energy density
\begin{equation}
\tau({\bf r}) = \sum_i^{\rm occ}|\nabla \psi_i({\bf r})|^2
\,.
\end{equation}
Note that the meta-GGAs are effectively orbital functionals due to the
dependence in $\tau({\bf r})$.
The forth rung is occupied by functionals that include the
exact-exchange (EXX) contribution to the energy
\begin{equation}
E^{\rm EXX}_{\rm x} = -\frac{1}{2}\int\!\!d^3r\int\!\!d^3r'\,
\frac{\psi_i({\bf r})\psi_i^*({\bf r}')\psi_j({\bf r}')\psi_j^*({\bf r})}{|{\bf r} - {\bf r}'|}
\,.
\end{equation}
These functionals can include the whole $E^{\rm EXX}_{\rm x}$ or only
a fraction of it, and $E^{\rm EXX}_{\rm x}$ can be evaluated with the
bare Coulomb interaction or with a screened version of it.
Furthermore, one can add a (semi-)local xc term or one that depends
on the orbitals. In any case, the fourth rung only includes functionals
that depend on the {\it occupied} orbitals only. Notorious examples of
functionals on this rung are the hybrid functionals
\begin{equation}
E^{\rm Hyb}_{\rm xc} = a_x E^{\rm EXX}_{\rm x} + E^{\rm mGGA}_{\rm xc}[n({\bf r}), \nabla n({\bf r}), \nabla^2 n({\bf r}), \tau({\bf r})]
\,.
\end{equation}
Note that the local part of this functional can be a meta-GGA, a GGA,
or even an LDA.
Finally, on the last rung of Jacob's ladder one finds functionals that
depend on the empty (virtual) Kohn-Sham orbitals. Perhaps the best
known example of these functionals is the random-phase approximation
(RPA).
In a practical DFT calculation one needs typically to evaluate both
$E_{\rm xc}$ and $v_{\rm xc}$. Furthermore, to obtain response
properties higher derivatives of $E_{\rm xc}$ are required. For
example, in first order one can get the electric polarizability, the
magnetic susceptibility, phonon frequencies, etc., and these usually
require the knowledge of the xc kernel
\begin{equation}
f_{\rm xc}({\bf r}, {\bf r}') = \frac{\delta E_{\rm xc}}{\delta n({\bf r}) \delta n({\bf r}')}
\,.
\end{equation}
In second order one can obtain, e.g., hyperpolarizabilities, Raman
tensors, etc., but these calculations usually require
\begin{equation}
k_{\rm xc}({\bf r}, {\bf r}', {\bf r}'') = \frac{\delta E_{\rm xc}}{\delta n({\bf r}) \delta n({\bf r}') \delta n({\bf r}'')}
\,.
\end{equation}
The calculation of these derivatives is fairly straightforward by
using basic functional analysis and the chain rule for functional
derivatives. We give here as an example the case of the xc potential
for a GGA
\begin{multline}
v^{\rm GGA}_{\rm xc}({\bf r}) = e^{\rm GGA}_{\rm xc}(n({\bf r}), \nabla n({\bf r}))
+ \int\!\!d^3r\, n({\bf r}) \left.\frac{\partial e^{\rm GGA}_{\rm xc}}{\partial n}
\right|_{\scriptsize\begin{array}{cc}n=n({\bf r}) \\ \nabla n = \nabla n({\bf r}) \end{array}}
\delta({\bf r}-{\bf r}') \\
+ \int\!\!d^3r\, n({\bf r}) \left.\frac{\partial e^{\rm GGA}_{\rm xc}}{\partial \nabla n}
\right|_{\scriptsize\begin{array}{cc}n=n({\bf r}) \\ \nabla n = \nabla n({\bf r}) \end{array}}
\nabla \delta({\bf r}-{\bf r}')
\,.
\end{multline}
Integrating by parts the third term, one finally arrives at
\begin{multline}
v^{\rm GGA}_{\rm xc}({\bf r}) = e^{\rm GGA}_{\rm xc}(n({\bf r}), \nabla n({\bf r}))
+ n({\bf r}) \left.\frac{\partial e^{\rm GGA}_{\rm xc}}{\partial n}
\right|_{\scriptsize\begin{array}{cc}n=n({\bf r}) \\ \nabla n = \nabla n({\bf r}) \end{array}}
- \nabla \left[n({\bf r}) \left.\frac{\partial e^{\rm GGA}_{\rm xc}}{\partial \nabla n}
\right|_{\scriptsize\begin{array}{cc}n=n({\bf r}) \\ \nabla n = \nabla n({\bf r}) \end{array}}\right]
\,.
\end{multline}
Clearly, higher functional derivatives of $E_{\rm xc}$ involve
higher partial derivatives of $e_{\rm xc}$.
By now it is clear what a code needs to implement for the functionals of
the first three rungs: given $n({\bf r})$, and possibly $\nabla n({\bf r})$,
$\nabla^2 n({\bf r})$, and $\tau ({\bf r})$, one needs the xc energy density
$e_{\rm xc}({\bf r})$ and all relevant partial derivatives of this
quantity. This is indeed the information that is provided by {\sc
Libxc}. For hybrid functionals the library also returns, besides the
semi-local part, also the mixing coefficient $a_x$. Unfortunately, EXX
and other functionals of the forth and fifth rang are too dependent on
the actual numerical representation of the wave-functions and can not
be easily included in a generic library.
Finally, we would like to mention that some applications of DFT do not
use the Kohn-Sham scheme, but try to approximate the kinetic energy
functional directly in terms of the density. This approach follows the
path that was laid down in the 1927 by Thomas and
Fermi~\cite{CambridgeJournals:1732980,Fermi1927}, and is sometimes
referred to as ``orbital-free DFT''~\cite{Chen2008}. In this case, we
also need a functional form (either an LDA or a GGA) for the kinetic
energy, and many of these are present in {\sc Libxc}. A good review of
functionals for the kinetic energy density can be found in
Ref.~\cite{Ludena2002}.
Note that even if the discussion above was restricted to
spin-compensated systems for the sake of simplicity, all functionals
of {\sc Libxc} can also be used with spin-polarization.
\section{Some history}
{\sc Libxc} started as a spin-off project during the initial
development of the (time-dependent) DFT code {\sc
Octopus}~\cite{Marques200360,PSSB:PSSB200642067,andrade:184106,Alberto:2004:1546-1955:231}.
At that point it became clear that the task of evaluation of the xc
functional was completely independent of the main structure of {\sc
Octopus}, and could therefore be transformed into a library. The
first steps into the development of {\sc Libxc} were taken in
September 2006, and the first usable version of the library included a
few of the most popular LDA and GGA functionals.
At the same time, the European Theoretical Spectroscopy Facility was
carrying a coordinate effort to improve code interoperability and
re-usability of its software suite, so efforts were almost immediately
made to interface those codes with {\sc Libxc}. This catalyzed the
development of the library and accelerated its dissemination in the
community.
During the following years, the library was expanded following two
main lines:
(i)~To include as many functionals as possible (by now we include
around 180 --- see \ref{app:1}). This was done for several reasons.
First, we implemented nearly all of the ``old'' functionals that
played an important role in the development of DFT. From this
perspective, we can look at {\sc Libxc} as a ``living museum'' of the
history of this important discipline. It also allows users to
reproduce old results with little effort. Secondly, many of the
functionals that are proposed nowadays are implemented in {\sc Libxc}
within weeks from the moment they are published. In this way new
developments are very rapidly available in several different codes,
allowing for these new functionals to be quickly tested and
benchmarked. Note that we have an agnostic policy, i.e. we try to
include the maximum possible number of functionals, without making any
judgment of value concerning their beauty, elegance, or usefulness.
This judgment is left for the final user to perform.
(ii)~To include derivatives of the xc energy up to high orders. As we
stated before, to perform a standard Kohn-Sham calculation one only
requires the xc energy functional and its first derivatives. However,
higher derivatives are essential in order to obtain response
properties. Of course, these higher derivatives can be calculated
numerically from lower-order derivatives, but this procedure tends to
introduce unnecessary errors and instabilities in the calculations. We
therefore implemented derivatives up to the third order for the LDAs
and up to second order for the other functionals. We note that these
derivatives are hand-coded, and not automatically generated from the
output of symbolic manipulation
software~\cite{Strange2001310,JCC:JCC20758}. Even if the latter
approach is excellent in order to test implementations, the
automatically generated code is often extremely verbose, inefficient,
and unreadable. Note that {\sc Libxc} includes automatic procedures to
check the implementation of the analytic derivatives.
At the beginning, {\sc Libxc} was used exclusively in the code {\sc
Octopus}. However, since then several other codes from both the
Solid-State Physics and Quantum Chemistry communities started to use
this library. The list of codes that use {\sc Libxc} at the time of
writing is as follows (in alphabetical order):
\begin{itemize}
\item {\sc Abinit}~\cite{Gonze20092582,abinit2,Gonze2002478}
(\url{http://www.abinit.org/}) --- This is a general purpose
plane-wave code. Besides the basic functionality, {\sc Abinit}
also includes options to optimize the geometry, to perform
molecular dynamics simulations, or to calculate dynamical
matrices, Born effective charges, dielectric tensors, and many
more properties.
\item {\sc APE}~\cite{Oliveira2008524}
(\url{http://www.tddft.org/programs/APE}) --- The atomic
pseudopotential engine (APE) is a tool for generating atomic
pseudopotentials within DFT. It is distributed under the GPL and
it produces pseudopotential files suitable for use with several
codes.
\item {\sc AtomPAW}~\cite{Holzwarth2001329}
(\url{http://www.wfu.edu/~natalie/papers/pwpaw/man.html}) --- The
computer program {\sc AtomPaw} generates projector and basis
functions which are needed for performing electronic structure
calculations based on the projector augmented wave (PAW)
method. The program is applicable to materials throughout the
periodic table.
\item {\sc Atomistix
ToolKit}~\cite{PhysRevB.65.165401,0953-8984-14-11-302}
(\url{http://quantumwise.com}) --- This is a software package that
uses non-equilibrium Green's functions simulations to study
transport properties like I-V characteristics of nanoelectronic
devices. It uses a powerful combination of DFT, semi-empirical
tight-binding, and classical potentials.
\item {\sc BigDFT}~\cite{Genovese2011149,genovese:014109}
(\url{http://inac.cea.fr/L_Sim/BigDFT/}) --- {\sc BigDFT} is a DFT
massively parallel electronic structure code using a wavelet basis
set. Wavelets form a real space basis set distributed on an
adaptive mesh. Thanks to its Poisson solver based on a Green's
function formalism, periodic systems, surfaces and isolated
systems can be simulated with the proper boundary conditions.
\item {\sc DP}~\cite{QUA:QUA20486} (\url{http://www.dp-code.org/})
--- Dielectric properties (DP) is a Linear Response TDDFT code, in
frequency-reciprocal and frequency-real space, that uses a
plane-wave basis set.
\item {\sc Elk} (\url{http://elk.sourceforge.net/}) --- An
all-electron full-potential linearized aug\-mented-plane wave
(FP-LAPW) code with many advanced features. This code is designed
to be as simple as possible so that new developments in the field
of density functional theory (DFT) can be added quickly and
reliably. The code is freely available under the GNU General
Public License.
\item {\sc ERKALE} (\url{http://erkale.googlecode.com}) --- {\sc
ERKALE} is a quantum chemistry program developed by J. Lehtola
used to solve the electronic structure of atoms, molecules and
molecular clusters. The main use of {\sc ERKALE} is the
computation of X-ray properties, such as ground-state electron
momentum densities and Compton profiles, and core (X-ray
absorption and X-ray Raman scattering) and valence electron
excitation spectra of atoms and molecules.
\item {\sc Exciting}~\cite{PhysRevLett.95.136402,B903676H}
(\url{http://exciting-code.org/}) --- {\sc Exciting} is a
full-potential all-electron DFT package based on the linearized
augmented plane-wave (LAPW) method. It can be applied to all kinds
of materials, irrespective of the atomic species involved, and
also allows for the investigation of the atomic-core region.
\item {\sc GPAW}~\cite{PhysRevB.71.035109,0953-8984-22-25-253202}
(\url{https://wiki.fysik.dtu.dk/gpaw}) --- {\sc GPAW} is a DFT
Python code based on the projector-augmented wave (PAW) method and
the atomic simulation environment (ASE). It uses real-space
uniform grids and multigrid methods or atom-centered
basis-functions.
\item {\sc
Hippo}~\cite{lathiotakis:184103,0295-5075-77-6-67003,PhysRevA.77.032509}
--- This is an electronic structure code, developed by N.
Lathiotakis, that implements Reduced Density Matrix Functionals
for atomic and molecular systems using Gaussian-type orbitals .
\item {\sc
Octopus}~\cite{Marques200360,PSSB:PSSB200642067,andrade:184106,Alberto:2004:1546-1955:231}
(\url{http://www.tddft.org/programs/octopus/}) --- {\sc Octopus}
is a scientific program aimed at the {\it ab initio} virtual
experimentation on a hopefully ever-increasing range of system
types. Electrons are described quantum-mechanically within
density-functional theory (DFT), and in its time-dependent form
(TDDFT) when doing simulations in time. Nuclei are described
classically as point particles. Electron-nucleus interaction is
described within the pseudopotential approximation.
\item {\sc Yambo}~\cite{Marini20091392}
(\url{http://www.yambo-code.org/}) --- {\sc Yambo} is a Fortran/C
code for many-body calculations in solid state and molecular
physics. The code was originally developed in the Condensed Matter
Theoretical Group of the Physics Department at the University of
Rome ``Tor Vergata'' by A. Marini. Previous to its release
under the GPL license, {\sc Yambo} was known as {\sc SELF}.
\end{itemize}
This diversity of codes is extremely important, because in this way a
certain functional can be tested and used with a variety of methods in
different physical situations. For example, a meta-GGA developed within
a certain code to get good band-gaps of solids can be immediately used
in a different code (often after a simple recompilation) to obtain the
ionization potential of molecules.
{\sc Libxc} is freely available from {\tt
http://www.tddft.org/programs/Libxc}, and it is distributed under
the GNU Lesser General Public License v3.0. This license not only
allows everyone to read, modify, and distribute the code, but also
allows {\sc Libxc} to be linked from close-source codes. The reader is
also referred to the web site to obtain more information, updated
documentation, examples, new versions, etc.
As most open source projects, we strongly encourage contributions from
researchers willing to contribute with the implementation of new
functionals, higher derivatives, bug corrections, or even bug reports.
\section{An example}
\subsection{Calling {\sc Libxc}}
Probably the best way to explain the usage of {\sc Libxc} is through
an example. The following small program calculates the xc energy for a
given functional for several values of the density; the available C
bindings can be found in header file {\tt xc.h}. More information and
examples can be found in the manual and in the header files.
\begin{verbatim}
#include <stdio.h>
#include <xc.h>
int main()
{
xc_func_type func;
double rho[5] = {0.1, 0.2, 0.3, 0.4, 0.5};
double sigma[5] = {0.2, 0.3, 0.4, 0.5, 0.6};
double ek[5];
int i, func_id = 1;
/* initialize the functional */
if(xc_func_init(&func, func_id, XC_UNPOLARIZED) != 0) {
fprintf(stderr, "Functional
return 1;
}
/* evaluate the functional */
switch(func.info->family)
{
case XC_FAMILY_LDA:
xc_lda_exc(&func, 5, rho, ek);
break;
case XC_FAMILY_GGA:
case XC_FAMILY_HYB_GGA:
xc_gga_exc(&func, 5, rho, sigma, ek);
break;
}
for(i=0; i<5; i++) {
printf(
}
/* free the functional */
xc_func_end(&func);
}
\end{verbatim}
The functionals are divided in families (LDA, GGA, etc.). Given a
functional identifier, {\tt func\_id}, the functional is
initialized by {\tt xc\_func\_init}, and evaluated by {\tt
xc\_XXX\_exc}, which returns the energy per unit volume ({\tt ek}).
Finally, the function {\tt xc\_func\_end} cleans up. We note that we
follow the convention used in Quantum Chemistry and, instead of
passing the full gradient of the density, we use the variable
\begin{equation}
\sigma({\bf r}) = \nabla n({\bf r}) \cdot \nabla n({\bf r})
\,.
\end{equation}
Converting between partial derivatives with respect to $\sigma$ and
$\nabla n$ is trivially done with the use of the chain rule. All the
quantities passed to and returned by the library are in atomic units.
Fortran 90 bindings are also included in {\sc Libxc}. These can be
found in the file {\tt libxc\_master.F90}. In general, calling {\sc
Libxc} from Fortran is as simple as from C. Here is the previous
example in Fortran:
\begin{verbatim}
program lxctest
use xc_f90_types_m
use xc_f90_lib_m
implicit none
TYPE(xc_f90_pointer_t) :: xc_func
TYPE(xc_f90_pointer_t) :: xc_info
real(8) :: rho(5) = (/0.1, 0.2, 0.3, 0.4, 0.5/)
real(8) :: sigma(5) = (/0.2, 0.3, 0.4, 0.5, 0.6/)
real(8) :: ek(5)
integer :: i, func_id
func_id = 1
! initialize the functional
call xc_f90_func_init(xc_func, xc_info, func_id, XC_UNPOLARIZED)
! evaluate the functional
select case (xc_f90_info_family(xc_info))
case(XC_FAMILY_LDA)
call xc_f90_lda_exc(xc_func, 5, rho(1), ek(1))
case(XC_FAMILY_GGA, XC_FAMILY_HYB_GGA)
call xc_f90_gga_exc(xc_func, 5, rho(1), sigma(1), ek(1))
end select
do i = 1, 5
write(*,"(F8.6,1X,F9.6)") rho(i), ek(i)
end do
! free the functional
call xc_f90_func_end(xc_func)
end program lxctest
\end{verbatim}
\subsection{The info structure}
Besides the mathematical formulas necessary to evaluate the functional
and its derivatives, {\sc Libxc} includes a considerable amount of
metadata that is quite useful for both the calling program and for the
end user. This information is contained for each functional in the
structure {\tt xc\_func\_info\_type}. The relevant part of this structure
for the end user is defined as
\begin{verbatim}
/* flags that can be used in info.flags */
#define XC_FLAGS_HAVE_EXC (1 << 0) /* 1 */
#define XC_FLAGS_HAVE_VXC (1 << 1) /* 2 */
#define XC_FLAGS_HAVE_FXC (1 << 2) /* 4 */
#define XC_FLAGS_HAVE_KXC (1 << 3) /* 8 */
#define XC_FLAGS_HAVE_LXC (1 << 4) /* 16 */
#define XC_FLAGS_1D (1 << 5) /* 32 */
#define XC_FLAGS_2D (1 << 6) /* 64 */
#define XC_FLAGS_3D (1 << 7) /* 128 */
#define XC_FLAGS_STABLE (1 << 9) /* 512 */
#define XC_FLAGS_DEVELOPMENT (1 << 10) /* 1024 */
typedef struct{
int number; /* indentifier number */
int kind; /* XC_EXCHANGE and/or XC_CORRELATION */
char *name; /* name of the functional, e.g. "PBE" */
int family; /* type of the functional, e.g. XC_FAMILY_GGA */
char *refs; /* references */
int flags; /* see above for a list of possible flags */
...
} xc_func_info_type;
\end{verbatim}
For example, for the Slater exchange functional, this structure is defined as
\begin{verbatim}
const XC(func_info_type) XC(func_info_lda_x) = {
XC_LDA_X,
XC_EXCHANGE,
"Slater exchange",
XC_FAMILY_LDA,
"PAM Dirac, Proceedings of the
Cambridge Philosophical Society 26, 376 (1930)\n"
"F Bloch, Zeitschrift fuer Physik 57, 545 (1929)",
XC_FLAGS_3D | XC_FLAGS_HAVE_EXC | XC_FLAGS_HAVE_VXC | XC_FLAGS_HAVE_FXC
| XC_FLAGS_HAVE_KXC,
...
};
\end{verbatim}
Note that the references are separated by a newline. The user of the
library can access the information in the following way:
\begin{verbatim}
#include <stdio.h>
#include <xc.h>
int main()
{
xc_func_type func;
xc_func_init(&func, XC_GGA_X_B88, XC_UNPOLARIZED);
printf("The functional
func.info->name, func.info->refs);
xc_func_end(&func);
}
\end{verbatim}
\section{Conclusions and the future}
{\sc Libxc} is by now seven years old, and the code is quite stable
and in use by hundreds of scientists around the world. We are
committed to continue the development of the library in the future,
mainly by following the two lines mentioned before: include all
functionals, and their derivatives of the highest possible order. By
now essentially all LDA and GGA functionals ever proposed in the
literature are already included in the library. Unfortunately,
important gaps still remain especially in the meta-GGAs and hybrid
functionals. We will fill in those gaps in the near future. It is not
clear if {\sc Libxc} will ever include {\it all} functionals ever
developed (especially, as several new functionals come out every
year), but we will of course try\ldots
\section{Acknowledgements}
{\sc Libxc} profited considerably from other projects dedicated to xc
functionals (such as the density-functional repository in Daresbury of
H.\,J.\,J. van Dam), from generally available code (such as the
library of xc functionals of the Minnesota group --
\url{http://comp.chem.umn.edu/info/dft.htm}, xc routines of {\sc
Abinit}~\cite{Gonze20092582,abinit2,Gonze2002478}, {\sc
Espresso}~\cite{QE-2009}, etc.), and several individuals that
contributed either with code, bug fixes, or even bug reports. To all
of these we would like to express our gratitude.
MJTO thankfully acknowledges financial support from the Portuguese FCT
(contract \#SFRH/BPD/44608/2008).
|
{
"timestamp": "2012-06-29T02:03:31",
"yymm": "1203",
"arxiv_id": "1203.1739",
"language": "en",
"url": "https://arxiv.org/abs/1203.1739"
}
|
\section{Introduction}
Recently, in \cite{Dallaporta/Vu:2011} and \cite{Dalla:2011} the Central Limit Theorem (CLT) for the eigenvalue counting function of {\it Wigner matrices},
that is the number of eigenvalues falling in an interval, was established. This {\it universality result} relies on fine asymptotics of the variance of the eigenvalue counting function, on the Fourth Moment Theorem due to Tao and Vu as well as on recent localization results due to Erd\"os, Yau and Yin. There are many
random matrix ensembles of interest, but to focus our discussion and to clear the exposition we shall restrict ourselves to the most famous model
class of ensembles, the Wigner Hermitian matrix ensembles.
For an integer $n \geq 1$ consider an $n \times n$ Wigner Hermitian matrix $M_n = (Z_{ij})_{1 \leq i,j \leq n}$: Consider a family of jointly independent complex-valued random variables $(Z_{ij})_{1 \leq i, j \leq n}$ with $Z_{ji} = \bar{Z}_{ij}$, in particular the $Z_{ii}$ are real valued. For $1 \leq i < j \leq n$ require that the random variables have mean zero and variance one and the $Z_{ij} \equiv Z$ are identically distributed,
and for $1 \leq i=j \leq n$ require that $Z_{ii} \equiv Z'$ are also identically distributed with mean zero and variance one.
The distributions of $Z$ and $Z'$ are called {\it atom distributions}. An important example of a Wigner Hermitian matrix $M_n$ is the case where the entries are Gaussian, that is $Z_{ij}$ is distributed according
to a complex standard Gaussian $N(0,1)_{\C}$ for $i \not= j$ and $Z_{ii}$ is distributed according to a real standard Gaussian $N(0,1)_{\R}$,
giving rise to the so-called Gaussian Unitary Ensembles (GUE). GUE matrices will be denoted by $M_n'$. In this case, the joint
law of the eigenvalues is known, allowing a good description of their limiting behavior both in the global and local regimes (see \cite{Zeitounibook}). In the
Gaussian case, the distribution of the matrix is invariant by the action of the group $SU(n)$. The eigenvalues of the matrix $M_n$ are independent
of the eigenvectors which are Haar distributed. If $(Z_{i,j})_{1 \leq i <j}$ are real-valued the {\it symmetric Wigner matrix} is defined analogously and
the case of Gaussian variables with $\E Z_{ii}^2=2$ is of particular importance, since their law is invariant under the action of the orthogonal group $SO(n)$, known as Gaussian Orthogonal Ensembles (GOE).
The matrix $W_n := \frac{1}{\sqrt{n}} M_n$ is called the coarse-scale normalized Wigner Hermitian matrix, and $A_n := \sqrt{n} M_n$
is called the fine-scale normalized Wigner Hermitian matrix.
For any $n \times n$ Hermitian matrix $A$ we denote by $\lambda_1(A), \ldots, \lambda_n(A)$ the real eigenvalues of $A$. We introduce the {\it eigenvalue counting function}
$$
N_I(A) := \big| \{ 1 \leq i \leq n : \lambda_i(A) \in I \, \} \big|
$$
for any interval $I \subset \R$. We will consider $N_I(W_n)$ as well as $N_I(A_n)$. Remark that $N_I(W_n) = N_{nI}(A_n)$.
The global {\it Wigner theorem} states that the empirical measure
$\frac 1n \sum_{i=1}^n \delta_{\lambda_i}$
on the eigenvalues of the coarse-scale normalized Wigner Hermitian matrix $W_n$ converges weakly almost surely as $n \to \infty$ to the semicircle law
$$
d \varrho_{sc}(x) = \frac{1}{2 \pi} \sqrt{ 4 - x^2} \, 1_{[-2,2]}(x) \, dx,
$$
(see \cite[Theorem 2.1.21, Theorem 2.2.1]{Zeitounibook}). Consequently, for any interval $I \subset \R$,
$$
\lim_{n \to \infty} \frac{1}{n} N_I(W_n) = \varrho_{sc}(I) := \int_I \varrho_{sc}(y) \, dy
$$
almost surely. At the fluctuation level, it is well known that for the GUE, $W_n' := \frac{1}{\sqrt{n}} M_n'$
satisfies a CLT (see \cite{Soshnikov:2000}): Let $I_n$ be an interval in $\R$. If $\V(N_{I_n}(W_n')) \to \infty$
as $n \to \infty$, then
$$
\frac{N_{I_n}(W_n') - \E [ N_{I_n}(W_n')]}{\sqrt{\V (N_{I_n}(W_n'))}} \to N(0,1)_{\R}
$$
as $n \to \infty$ in distribution.
In \cite{Gustavsson:2005} the asymptotic behavior of the expectation and the variance of the counting function $N_I(W_n')$ for intervals $I=[y, \infty)$ with
$y \in (-2,2)$ strictly in the bulk of the semicircle law was established:
\begin{equation} \label{asymp}
\E [ N_{I}(W_n')] = n \varrho_{sc}(I) + O \bigl( \frac{\log n}{n} \bigr) \,\, \text{and} \,\, \V (N_{I}(W_n')) = \bigl( \frac{1}{2 \pi^2} + o(1) \bigr) \, \log n.
\end{equation}
The proof applied strong asymptotics for orthogonal polynomials with respect to exponential weights, see \cite{Deift/Thomas:1999}.
In particular the CLT holds for $N_I(W_n')$ if $I=[y, \infty)$ with $y \in (-2,2)$, and moreover in this case one obtains with \eqref{asymp}
$$
\frac{N_{I}(W_n') - n \varrho_{sc}(I)}{\sqrt{\frac{1}{2 \pi^2} \log n}} \to N(0,1)_{\R}
$$
as $n \to \infty$ (called the CLT with numerics). These conclusions were extended to non-Gaussian Wigner Hermitian matrices in \cite{Dallaporta/Vu:2011}.
\bigskip
\section{Global moderate deviations at the edge of the spectrum}
Certain deviations results and concentration properties for Wigner matrices were considered. Our aim is to establish moderate
deviation principles. Recall that a sequence of laws $(P_n)_{n \geq 0}$ on a Polish space $\Sigma$ satisfies a large deviation principle (LDP)
with good rate function $I : \Sigma \to \R_+$ and speed $s_n$ going to infinity with $n$ if and only if the level sets $\{x: I(x) \leq M\}$, $0 \leq M < \infty$,
of $I$ are compact and for all closed sets $F$
$$
\limsup_{n \to \infty} s_n^{-1} \log P_n(F) \leq - \inf_{x \in F} I(x)
$$
whereas for all open sets $O$
$$
\liminf_{n \to \infty} s_n^{-1} \log P_n(O) \geq - \inf_{x \in O} I(x).
$$
We say that a sequence of random variables satisfies the LDP when the sequence of measures induced by these variables satisfies the LDP. Formally
a moderate deviation principle is nothing else but the LDP. However, we speak about a moderate deviation principle (MDP) for a sequence of random variables,
whenever the scaling of the corresponding random variables is between that of an ordinary Law of Large Numbers (LLN) and that of a CLT.
Large deviation results for the empirical measures of Wigner matrices are still only known for the Gaussian ensembles since their
proof is based on the explicit joint law of the eigenvalues, see \cite{BenArous/Guionnet:1997} and \cite{Zeitounibook}. A moderate deviation
principle for the empirical measure of the GUE or GOE is also known, see \cite{Dembo/Guionnet/Zeitouni:2003}. This moderate deviations result
does not have yet a fully universal version for Wigner matrices. It has been generalised to Gaussian divisible matrices with a deterministic self-adjoint matrix added with converging empirical measure \cite{Dembo/Guionnet/Zeitouni:2003} and to Bernoulli matrices \cite{DoeringEichelsbacher:2009}.
Recently we proved in \cite{DoeringEichelsbacher:2011} a MDP for the number of eigenvalues of a GUE matrix in an interval.
If $M_n'$ is a GUE matrix and $W_n' := \frac{1}{\sqrt n} M_n'$ and $I_n$ be an interval in $\R$. If $\V(N_{I_n}(W_n')) \to \infty$
for $n \to \infty$, then, for any sequence $(a_n)_n$ of real numbers such that
1 \ll a_n \ll \sqrt{\V (N_{I_n}(W_n'))}
$,
the sequence $(Z_n)_n$ with
$$
Z_n = \frac{N_{I_n}(W_n') - \E [ N_{I_n}(W_n')]}{a_n \, \sqrt{\V (N_{I_n}(W_n'))}}
$$
satisfies a MDP with speed $a_n^2$ and rate function $I(x)=\frac{x^2}{2}$.
Moreover let $I=[y, \infty)$ with $y \in (-2,2)$ strictly in the bulk, then the sequence $(\hat{Z}_n)_n$ with
$
\hat{Z}_n = \frac{N_{I}(W_n') - n \varrho_{sc}(I)}{a_n \, \sqrt{\frac{1}{2 \pi^2} \log n}}
$
satisfies the MDP with the same speed, the same rate function, and in the regime $1 \ll a_n \ll \sqrt{\log n}$ (called the MDP with numerics; see Theorem 1.1
in \cite{DoeringEichelsbacher:2011}). It follows applying \eqref{asymp}.
In \cite{DoeringEichelsbacher:2011}, these conclusions were extended to non-Gaussian Wigner Hermitian matrices.
The first observation in this paper is, that the MDP for $(Z_n)_n$ and $(\hat{Z}_n)_n$, respectively, is not restricted to the bulk of the spectrum.
To state the result, let $\delta >0$ and assume that $y_n \in [-2 + \delta, 2)$ and $n(2 - y_n)^{3/2} \to \infty$ when $n \to \infty$. Then with
\cite[Lemma 2.3]{Gustavsson:2005} the variance of the number of eigenvalues of $W_n'$ in $I_n :=[y_n, \infty)$ satisfies
\begin{equation} \label{edgeasy1}
\V (N_{I_n}(W_n')) = \frac{1}{2 \pi^2} \, \log \bigl( n(2 - y_n)^{3/2} \bigr) \, (1 + \eta(n)),
\end{equation}
where $\eta(n) \to 0$ as $n \to \infty$. Moreover the expected number of eigenvalues of $W_n'$ in $I_n$, when $y_n \to 2^-$, is given by \cite[Lemma 2.2]{Gustavsson:2005}:
\begin{equation} \label{edgeasy2}
\E (N_{I_n}(W_n')) = \frac{2}{3 \pi} n (2 - y_n)^{3/2} + O(1).
\end{equation}
Hence applying Theorem 1.1 in \cite{DoeringEichelsbacher:2011} we immediately obtain:
\begin{theorem} \label{result1}
Let $M_n'$ be a GUE matrix and $W_n' = \frac{1}{\sqrt{n}} M_n$. Let $I_n = [y_n, \infty)$ where $y_n \to 2^-$ for $n \to \infty$. Assume that $y_n \in[-2 + \delta, 2)$ and $n(2 -y_n)^{3/2} \to \infty$ when $n \to \infty$. Then, for any sequence $(a_n)_n$ of real numbers such that
$1 \ll a_n \ll \sqrt{\V (N_{I_n}(W_n'))}$, the sequence
$\frac{N_{I_n}(W_n') - \E [ N_{I_n}(W_n')]}{a_n \, \sqrt{\V (N_{I_n}(W_n'))}}$
satisfies a MDP with speed $a_n^2$ and rate function $I(x)=\frac{x^2}{2}$.
Moreover the sequence
$$
Z_n := \frac{N_{I_n}(W_n') - \frac{2}{3 \pi} n(2 -y_n)^{3/2}}{a_n \, \sqrt{\frac{1}{2 \pi^2} \log (n(2-y_n)^{3/2})}}
$$
satisfies the MDP with the same speed, the same rate function, and in the regime $1 \ll a_n \ll \sqrt{\log (n(2-y_n)^{3/2})}$ (called the MDP with numerics).
\end{theorem}
For symmetry reasons an analogous result could be formulated for the counting function $N_{I_n}(W_n')$ near the left edge of the spectrum.
\bigskip
\section{Local moderate deviations at the edge of the spectrum}
Under certain conditions on $i$ it was proved in \cite{Gustavsson:2005} that the $i$-th eigenvalue $\lambda_i$ of the GUE $W_n'$ satisfies a CLT.
Consider $t(x) \in [-2,2]$ defined for $x \in [0,1]$ by
$$
x = \int_{-2}^{t(x)} \varrho_{sc}(t) \, dt = \frac{1}{2 \pi} \int_{-2}^{t(x)} \sqrt{4 - x^2} \, dx.
$$
Then for $i=i(n)$ such that $i/n \to a \in (0,2)$ as $n \to \infty$ (i.e. $\lambda_i$ is eigenvalue in the bulk), $\lambda_i(W_n')$ satisfies a CLT:
\begin{equation} \label{CLT-Gu}
X_n := \sqrt{\frac{4 - t(i/n)^2}{2}} \frac{\lambda_i(W_n') - t(i/n)}{\frac{\sqrt{\log n}}{n}} \to N(0,1)
\end{equation}
for $n \to \infty$. Remark that $t(i/n)$ is sometimes called the {\it classical or expected location} of the $i$-th eigenvalue. The standard deviation
is $\frac{\sqrt{\log n}}{\pi \sqrt{2}} \, \frac{1}{n \varrho_{sc}(t(i/n))}$. Note that from the semicircular law, the factor $\frac{1}{n \varrho_{sc}(t(i/n))}$ is the mean
eigenvalue spacing. Informally, \eqref{CLT-Gu} asserts in the GUE case, that each eigenvalue $\lambda_i(W_n')$ typically deviates by $O \bigl( \sqrt{\log n}/ (n \varrho(t(i/n))) \bigr)$ around its classical location. This result can be compared with the so called {\it eigenvalue rigidity property} $\lambda_i(W_n') = t(i/n) +
O(n^{-1 + \varepsilon})$ established in \cite{Erdoes/Yau/Yin:2010}, which has a slightly worse bound on the deviation but which holds with overwhelming probability and for
general Wigner ensembles. See also discussions in \cite[Section 3]{TaoVu:2012}. We proved in \cite[Theorem 4.1]{DoeringEichelsbacher:2011}
a MDP for $\bigl(\frac{1}{a_n} X_n\bigr)_n$ with $X_n$ in \eqref{CLT-Gu}, for any $1 \ll a_n \ll \sqrt{\log n}$, with speed $a_n^2$ and rate $x^2/2$. Moreover in
\cite[Theorem 4.2]{DoeringEichelsbacher:2011}, these conclusions were extended to non-Gaussian Wigner Hermitian matrices.
The proofs are achieved by the tight relation between eigenvalues and the counting function
expressed by the elementary equivalence, for $I(y)=[y, \infty)$, $y \in \R$,
\begin{equation} \label{relation}
N_{I(y)}(W_n) \leq n-i \,\, \text{if and only if} \,\, \lambda_i(W_n) \leq y.
\end{equation}
This relation is true for any eigenvalue $\lambda_i(W_n)$, independent of sitting being in the bulk of the spectrum or very close to the right edge
of the spectrum. Hence the next goal is to transport the MDP for the counting function of eigenvalues close to the (right) edge, Theorem \ref{result1}, to a MDP for any singular eigenvalue close to the right edge of the spectrum. Consider
$i=i(n)$ with $i \to \infty$ but $i/n \to 0$ as $n \to \infty$ and define $\lambda_{n-i}(W_n')$ as eigenvalue number $n-i$ in the GUE.
An example is $i(n) = n - \log n$.
In \cite[Theorem 1.2]{Gustavsson:2005} a CLT was proven, which is
\begin{equation} \label{localrv}
Z_{n,i} := \frac{\lambda_{n-i}(W_n') - \bigl( 2 - \bigl( \frac{3 \pi}{2} \frac in \bigr)^{2/3} \bigr)}{ \operatorname{const} \bigl( \frac{ \log i}{i^{2/3} n^{4/3}} \bigr)^{1/2}} \to N(0,1)_{\R}
\end{equation}
in distribution with $\operatorname{const}= \bigl( (3 \pi)^{2/3} 2^{1/3} \bigr)^{-1/2}$. Remark that the formulation in \cite[Theorem 1.2]{Gustavsson:2005}
is different, since first of all the GUE in \cite{Gustavsson:2005} was defined such that the limiting semicircular law has support $[-1,1]$ and, second
the CLT in \cite{Gustavsson:2005} is formulated for $\lambda_{n-i}(M_n')$ instead of $\lambda_{n-i}(W_n')$. The choice of the asymptotic expectation
and variance in \eqref{localrv} can be explained as follows. Let $g(y_n)$ be the expected number of eigenvalues in $I_n = [y_n, \infty)$. Then with \eqref{relation}
$$
P \bigl( \lambda_{n-i}(W_n') \leq y_n \bigr) = P \bigl( N_{I_n}(W_n') \leq i \bigr) = P \biggl( \frac{N_{I_n}(W_n') - g(y_n)}{ \V(N_{I_n}(W_n'))^{1/2}} \leq \frac{i -g(y_n)}{\V(N_{I_n}(W_n'))^{1/2}} \biggr).
$$
Trying to apply the CLT for $N_{I_n}$ is choosing $y_n$ such that $\frac{i -g(y_n)}{\V(N_{I_n}(W_n'))^{1/2}} \to x$ for $n \to \infty$, because this
would imply $P \bigl( \lambda_{n-i}(W_n') \leq y_n \bigr) \to \int_{-\infty}^x \varphi_{0,1}(t) \, dt$, where $\varphi_{0,1}(\cdot)$ denotes the density
of the standard normal distribution. The candidate for $y_n$ can be found as in the proof of \cite[Theorem 1.2]{Gustavsson:2005}, with $g(y_n) = \frac{2}{3 \pi}
n(2 -y_n)^{3/2} + O(1)$ and $h(y_n) = \V(N_{I_n}(W_n'))^{1/2} = \frac{1}{\sqrt{2} \pi} \log^{1/2} (n(2-y_n)^{3/2}) + o(\log^{1/2} (n(2-y_n)^{3/2}))$.
Applying the same heuristic as on page 157 in \cite{Gustavsson:2005}, we obtain
$$
y_n \approx 2 - \biggl( \frac{3 \pi}{2} \frac in \biggr)^{2/3} + x \,\, \biggl( \bigl((3 \pi)^{2/3} 2^{1/3} \bigr)^{-1/2} \, \bigl( \frac{ \log i}{i^{2/3} n^{4/3}} \bigr) \biggr)^{1/2}.
$$
With respect to the statement in Theorem \ref{result1} one might expect a MDP for $\bigl( \frac{1}{a_n} Z_{n,i} \bigr)_n$ for certain growing
sequences $(a_n)_n$. We have
\begin{eqnarray*}
P \bigl( Z_{n,i}/ a_n \leq x \bigr) & = & P \bigl( \lambda_{n-i}(W_n') \leq y_n(a_n) \bigr) = P \bigl( N_{I_n}(W_n') \leq i \bigr) \\
& = & P \biggl( \frac{N_{I_n}(W_n') - \E(N_{I_n}(W_n'))}{ a_n \V(N_{I_n}(W_n'))^{1/2}}
\leq \frac{i -\E(N_{I_n}(W_n'))}{a_n \V(N_{I_n}(W_n'))^{1/2}} \biggr)
\end{eqnarray*}
with
\begin{equation} \label{ynan}
y_n(a_n) := 2 - \biggl( \frac{3 \pi}{2} \frac in \biggr)^{2/3} + a _n \, x \,\, \biggl( \bigl((3 \pi)^{2/3} 2^{1/3} \bigr)^{-1/2} \, \bigl( \frac{ \log i}{i^{2/3} n^{4/3}} \bigr)\biggr)^{1/2}
\end{equation}
and $I_n =[y_n(a_n), \infty)$.
Since $i \to \infty$ and $i/n \to 0$ for $n \to \infty$, we have that $y_n(a_n) \to 2^-$ for every $a_n$ such that $a_n \ll \bigl( \log i\bigr)^{1/2}$.
Hence we can apply \eqref{edgeasy2}, that is $\E (N_{I_n}(W_n')) = \frac{2}{3 \pi} n (2 - y_n(a_n))^{3/2} + O(1)$.
With
$$
2 - y_n(a_n) = \biggl( \frac{3 \pi}{2} \frac in \biggr)^{2/3} \biggl( 1 - \frac{a_n \, x \, \log^{1/2} i}{(3 \pi / \sqrt{2}) i} \biggr)
$$
by Taylor's expansion we obtain
\begin{equation} \label{tay1}
\frac{2}{3 \pi} n (2 - y_n(a_n))^{3/2} = i - \frac{1}{\sqrt{2} \pi} a_n \, x \, \log^{1/2} i + o \bigl( a_n \, x \, \log^{1/2} i \bigl),
\end{equation}
and therefore
$i -\E(N_{I_n}(W_n'))= \frac{1}{\sqrt{2} \pi} a_n \, x \, \log^{1/2} i + o \bigl( a_n \, x \, \log^{1/2} i \bigl) + O(1)$.
From \eqref{tay1} we obtain that $n(2 - y_n(a_n))^{3/2} \to \infty$ for $n \to \infty$ for every $a_n \ll \bigl( \log i \bigr)^{1/2}$.
Hence we can apply \eqref{edgeasy1}, that is $\V (N_{I_n}(W_n')) = \frac{1}{2 \pi^2} \, \log \bigl( n(2 - y_n(a_n))^{3/2} \bigr) \, (1 + o(1))$.
With \eqref{tay1} we get
$$
\V (N_{I_n}(W_n')) = \biggl( \frac{1}{2 \pi^2} \log \bigl( \frac{3 \pi}{2} i \bigr) + \frac{1}{2 \pi^2} \log \biggl( 1 - \frac{a_n \, x \, (\log i)^{1/2}}{\sqrt{2} \pi i} +
o \bigl( \frac{a_n \, x \, (\log i)^{1/2}}{i} \bigl) \biggr) \biggr) (1 + o(1)).
$$
Summarizing we have proven that for any growing sequence $(a_n)_n$ of real numbers such that $1 \ll a_n \ll (\log i)^{1/2}$
$$
\frac{i -\E(N_{I_n}(W_n'))}{a_n \V(N_{I_n}(W_n'))^{1/2}} = x + o(1).
$$
By Theorem \ref{result1} we obtain for every $x < 0$ that
$
\lim_{n \to \infty} \frac{1}{a_n^2} \log P \bigl( Z_{n,i}/ a_n \leq x \bigr) = - \frac{x^2}{2}$.
With $P \bigl( Z_{n,i}/ a_n \geq x \bigr) = P\bigl( N_{I_n}(W_n') \geq i-1 \bigr)$
the same calculations lead, for every $x >0$, to
\begin{equation} \label{soso}
\lim_{n \to \infty} \frac{1}{a_n^2} \log P \bigl( Z_{n,i}/ a_n \geq x \bigr) = - \frac{x^2}{2}.
\end{equation}
Next we choose all open intervals $(a,b)$, where at least one of the endpoints is finite and where none of the endpoints is zero. Denote
the family of such intervals by ${\mathcal U}$. Now it follows for each $U=(a,b) \in {\mathcal U}$,
$$
{\mathcal L}_{U}: = \lim_{n \to \infty} \frac{1}{a_n^2} \log P \bigl( Z_{n,i}/ a_n \in U \bigr) = \left\{ \begin{array}{r@{\quad:\quad}l}
b^2/2 & a < b < 0 \\ 0 & a < 0 < b \\ a^2/2 & 0<a<b \end{array} \right.
$$
By \cite[Theorem 4.1.11]{Dembo/Zeitouni:LargeDeviations}, $(Z_{n,i}/ a_n)_n$ satisfies a weak MDP (see definition in \cite[Section 1.2]{Dembo/Zeitouni:LargeDeviations}) with speed $a_n^2$ and rate function
$t \mapsto \sup_{U \in {\mathcal U}; t \in U} {\mathcal L}_U = \frac{t^2}{2}$.
With \eqref{soso}, it follows that $(Z_{n,i}/ a_n)_n$ is exponentially tight (see definition in \cite[Section 1.2]{Dembo/Zeitouni:LargeDeviations}), hence by Lemma 1.2.18 in \cite{Dembo/Zeitouni:LargeDeviations},
$(Z_{n,i}/ a_n)_n$ satisfies the MDP with the same speed and the same good rate function. Hence we have proven:
\begin{theorem} \label{result2}
Let $M_n'$ be a GUE matrix and $W_n' = \frac{1}{\sqrt{n}} M_n'$. Consider $i=i(n)$ such that $i \to \infty$ but $i/n \to 0$ as $n \to \infty$.
If $\lambda_{n-i}$ denotes the eigenvalue number $n-i$ in the GUE matrix $W_n'$ it holds that for any sequence $(a_n)_n$ of real numbers
such that $1 \ll a_n \ll (\log i)^{1/2}$, the sequence $\bigl( \frac{1}{a_n} Z_{n,i} \bigr)_n$ with $Z_{n,i}$ given by \eqref{localrv}
satisfies a MDP with speed $a_n^2$ and rate function $I(x) = \frac{x^2}{2}$.
\end{theorem}
\bigskip
\section{Universal local moderate deviations near the edge}
Our next goal is to check whether the precise distribution of the atom variables $Z_{ij}$ of a Hermitian random matrix $M_n$ are
relevant for the conclusion of the MDP stated in Theorems \ref{result1} and \ref{result2}, so long as they are normalized to have mean zero and variance one,
and are jointly independent on the upper-triangular portion of $M_n$. It is a remarkable feature of our MDP results that they are {\it universal}, hence
the distribution of the atom variables are irrelevant in some sense. The arguments used above relied heavily on the special structure of the GUE ensemble,
in particular on the determinantal structure of the joint probability distribution (see \cite[Theorem 1.1 and 1.3]{DoeringEichelsbacher:2011}) and
on the fine asymptotics of the expectation and the variance of the eigenvalue counting function of GUE presented in \cite{Gustavsson:2005}. We apply
the swapping method due to Tao and Vu, in which one replaces the entries of one Wigner Hermitian matrix $M_n$ with another matrix $M_n'$ which are close
in the sense of matching moments. This goes back to Lindeberg's exchange strategy for proving the classical CLT, \cite{Lindeberg}, first applied to Wigner matrices in \cite{Chatterjee:2006}. The precise statement of the so called Four Moment Theorem needs some
preparation. We will use the notation as in \cite{TaoVu:2012}.
We say that two complex random variables $\eta_1$ and $\eta_2$ {\it match to order $k$} with $k \in \N$ if
$$
\E \bigl[ \text{Re}(\eta_1)^m \, \text{Im}(\eta_1)^l \bigr] = \E \bigl[ \text{Re}(\eta_2)^m \, \text{Im}(\eta_2)^l \bigr]
$$
for all $m,l \geq 0$ such that $m+l \leq k$. We will consider the case when the real and the imaginary parts of $\eta_1$ or of $\eta_2$
are independent, then the matching moment condition simplifies to the assertion that $E \bigl[ \text{Re}(\eta_1)^m] = E \bigl[ \text{Re}(\eta_2)^m]$
and $E \bigl[ \text{Im}(\eta_1)^l] = E \bigl[ \text{Im}(\eta_2)^l]$ for all $m,l \geq 0$ such that $m+l \leq k$.
We say that the Wigner Hermitian matrix $M_n$
obeys Condition ${\bf(C0)}$ if we have the exponential decay condition
$$
P \bigl( |Z_{ij}| \geq t^C \bigr) \leq e^{-t}
$$
for all $1 \leq i,j \leq n$ and $t \geq C'$, and some constants $C, C'$ independent of $i,j,n$. We say that the Wigner Hermitian matrix $M_n$
obeys Condition ${\bf(C1)}$ with constant $C_0$ if one has
$$
\E |Z_{ij}|^{C_0} \leq C
$$
for some constant $C$ independent of $n$. Of course, Condition ${\bf(C0)}$ implies Condition ${\bf(C1)}$ for any $C_0$, but not conversely.
The statement of the Four Moment Theorem for eigenvalues is:
\begin{theorem}[Four Moment Theorem due to Tao and Vu] \label{taovu}
Let $c_0>0$ be a sufficiently small constant. Let $M_n=(Z_{ij})$ and $M_n'=(Z_{ij}')$ be two $n \times n$ Wigner Hermitian
matrices satisfying Condition ${\bf(C1)}$ for some sufficiently large constant $C_0$. Assume furthermore that for any $1 \leq i <j \leq n$, $Z_{ij}$ and $Z_{ij}'$ match to order 4 and for any $1 \leq i \leq n$, and $Z_{ii}$
and $Z_{ii}'$ match to order 2. Set $A_n :=\sqrt{n} M_n$ and $A_n' := \sqrt{n} M_n'$, let $1 \leq k \leq n^{c_0}$ be an integer,
and let $G : {\fam\Bbbfam \tenBbb R}^k \to {\fam\Bbbfam \tenBbb R}$ be a smooth function obeying the derivative bounds
$|\nabla^jG(x)| \leq n^{c_0}$ for all $0 \leq j \leq 5$ and $x \in {\fam\Bbbfam \tenBbb R}^k$. Then for any $1 \leq i_1 < i_2 \cdots < i_k \leq n$, and
for $n$ sufficiently large we have
\begin{equation} \label{taovu}
|\E \bigl( G(\lambda_{i_1}(A_n), \ldots, \lambda_{i_k}(A_n) ) \bigr) - \E \bigl( G(\lambda_{i_1}(A_n'), \ldots, \lambda_{i_k}(A_n') ) \bigr)| \leq n^{-c_0}.
\end{equation}
\end{theorem}
The preliminary version of this Theorem was first established in the case of bulk eigenvalues and assuming Condition ${\bf(C0)}$, \cite{Tao/Vu:2009}.
Later the restriction to the bulk was removed and the Condition ${\bf(C0)}$ was relaxed to Condition ${\bf(C1)}$
for a sufficiently large value of $C_0$, \cite{Tao/Vu:2010}. Moreover, a natural question is whether the requirement of four matching moments is necessary.
As far as the distribution of individual eigenvalues $\lambda_i(A_n)$ are concerned, the answer is essentially yes. For this see
the discussions in \cite{TaoVu:2012}.
Applying this Theorem for the special case when $M_n'$ is GUE, we obtain the following MDP:
\begin{theorem} \label{result3}
The MDP for $\bigl( \frac{1}{a_n} Z_{n,i} \bigr)_n$, Theorem \ref{result2}, hold for Wigner Hermitian matrices obeying Condition ${\bf(C1)}$ for a sufficiently large $C_0$, and
whose atom distributions match that of GUE to second order on the diagonal and fourth order off the diagonal. Given $i=i(n)$ such that
$i \to \infty$ and $i/n \to 0$ as $n \to \infty$ we have: The sequence $\bigl( \frac{1}{a_n} Z_{n,i} \bigr)_n$ with
\begin{equation} \label{localrv2}
Z_{n,i} := \frac{\lambda_{n-i}(W_n) - \bigl( 2 - \bigl( \frac{3 \pi}{2} \frac in \bigr)^{2/3} \bigr)}{ \operatorname{const} \bigl( \frac{ \log i}{i^{2/3} n^{4/3}} \bigr)^{1/2}}
\end{equation}
satisfies the MDP for any sequence $(a_n)_n$ of real numbers such that $1 \ll a_n \ll (\log i)^{1/2}$ with speed $a_n^2$ and rate function $I(x) = \frac{x^2}{2}$.
\end{theorem}
\begin{proof}
Let $M_n$ be a Wigner Hermitian matrix whose entries satisfy Condition ${\bf(C1)}$ and match the corresponding entries of GUE up to order 4.
Let $i$ be as in the statement of the Theorem, and let $c_0$ be as in Theorem \ref{taovu}. Then \cite[(18)]{Tao/Vu:2009}
says that
\begin{equation} \label{inequ}
P \bigl( \lambda_i(A_n') \in I_{-} \bigr) - n^{-c_0} \leq P \bigl( \lambda_i(A_n) \in I \bigr) \leq P \bigl( \lambda_i(A_n') \in I_{+} \bigr) + n^{-c_0}
\end{equation}
for all intervals $I=[b,c]$, and $n$ sufficiently, where $I_{+} := [b-n^{-c_0/10}, c+n^{-c_0/10}]$
and $I_{-} := [b+n^{-c_0/10}, c-n^{-c_0/10}]$. We present the argument of proof of \eqref{inequ} just to make the presentation more self-contained.
One can find a smooth bump function $G : {\mathbb R} \to {\mathbb R}_+$ which is equal to one on the smaller interval $I$ and vanishes outside
the larger interval $I_+$. It follows that $P \bigl( \lambda_i(A_n) \in I \bigr) \leq \E G(\lambda_i(A_n))$ and
$\E G(\lambda_i(A_n')) \leq P \bigl( \lambda_i(A_n') \in I_+\bigr)$. One can choose $G$ to obey the condition $|\nabla^j G(x)| \leq n^{c_0}$ for $j=0, \ldots, 5$
and hence by Theorem \ref{taovu} one gets
$$
| \E G(\lambda_i(A_n)) - \E G(\lambda_i(A_n'))| \leq n^{-c_0}.
$$
Therefore the second inequality in \eqref{inequ} follows from the triangle inequality. The first inequality is proven similarly using a bump function which is 1 on $I_-$ and vanishes outside $I$.
Now for $n$ sufficiently large we consider the interval $I_n := [b_n, c_n]$ with
$$
b_n := b \, a_n \, n \, \operatorname{const} \bigl( \frac{ \log i}{i^{2/3} n^{4/3}} \bigr)^{1/2} + n \bigl( 2 - \bigl( \frac{3 \pi}{2} \frac in \bigr)^{2/3} \bigr),
$$
$$
c_n := c \, a_n \, n \, \operatorname{const} \bigl( \frac{ \log i}{i^{2/3} n^{4/3}} \bigr)^{1/2} + n \bigl( 2 - \bigl( \frac{3 \pi}{2} \frac in \bigr)^{2/3} \bigr)
$$
with $b,c \in {\mathbb R}$, $b \leq c$ and $\operatorname{const}= \bigl( (3 \pi)^{2/3} 2^{1/3} \bigr)^{-1/2}$.
Then for $\frac{1}{a_n} Z_{n,i}$ defined as in the statement of the Theorem we have
$P \bigl( Z_{n,i}/a_n \in [b,c] \bigr) = P \bigl( \lambda_{n-i}(A_n) \in I_n \bigr)$.
With \eqref{inequ} and \cite[Lemma 1.2.15]{Dembo/Zeitouni:LargeDeviations} we obtain
$$
\limsup_{n \to \infty} \frac{1}{a_n^2} \log P \bigl( Z_{n,i}/a_n \in [b,c] \bigr) \leq \max \biggl( \limsup_{n \to \infty} \frac{1}{a_n^2} \log
P \bigl( \lambda_{n-i}(A_n') \in (I_n)_+ \bigr) ; \limsup_{n \to \infty} \frac{1}{a_n^2} \log n^{-c_0} \biggr).
$$
For the first object we have
$$
\P \bigl( \lambda_{n-i}(A_n') \in (I_n)_+ \bigr) = P \biggl(
\frac{\lambda_{n-i}(A_n') - n \bigl( 2 - \bigl( \frac{3 \pi}{2} \frac in \bigr)^{2/3} \bigr)}{a_n \, n \, \operatorname{const} \bigl( \frac{ \log i}{i^{2/3} n^{4/3}} \bigr)^{1/2}} \in [b - \eta(n), c + \eta(n)] \biggr)
$$
with $\eta(n) = n^{-c_0/10} \bigl( a_n \, n \, \operatorname{const} \bigl( \frac{ \log i}{i^{2/3} n^{4/3}} \bigr)^{1/2} \bigr)^{-1} \to 0$ as $n \to \infty$.
Since $c_0 >0$ and $ \log n / a_n^2 \to \infty$ for $n \to \infty$ by assumption, applying Theorem \ref{result2} we have
$$
\limsup_{n \to \infty} \frac{1}{a_n^2} \log P \bigl( Z_{n,i}/a_n \in [b,c] \bigr) \leq - \inf_{x \in [b,c]} \frac{x^2}{2}.
$$
Applying the first inequality in \eqref{inequ} in the same manner we also obtain the upper bound
$$
\limsup_{n \to \infty} \frac{1}{a_n^2} \log P \bigl( Z_{n,i}/a_n \in [b,c] \bigr) \geq - \inf_{x \in [b,c]} \frac{x^2}{2}.
$$
Finally the argument in the last part of the proof of Theorem \ref{result2} can be repeated to obtain the MDP for $(Z_{n,i}/a_n)_n$.
\end{proof}
\bigskip
\section{Universal global moderate deviations near the edge}
Next we show the MDP for the eigenvalue counting function near the edge of the spectrum
for Wigner Hermitian matrices matching moments with GUE up to order four:
\begin{theorem} \label{result4}
The MDP for $(Z_n)_n$, Theorem \ref{result1}, hold for Wigner Hermitian matrices $M_n$ obeying Condition ${\bf(C1)}$ for a sufficiently large $C_0$, and
whose atom distributions match that of GUE to second order on the diagonal and fourth order off the diagonal.
Let $W_n = \frac{1}{\sqrt{n}} M_n$, let $I_n = [y_n, \infty)$ where $y_n \to 2^-$ for $n \to \infty$. Assume that $y_n \in[-2 + \delta, 2)$ and $n(2 -y_n)^{3/2} \to \infty$ when $n \to \infty$. Then the sequence
$$
Z_n = \frac{N_{I_n}(W_n) - \frac{2}{3 \pi} n(2 -y_n)^{3/2}}{a_n \, \sqrt{\frac{1}{2 \pi^2} \log (n(2-y_n)^{3/2})}}
$$
satisfies the MDP with speed $a_n^2$, rate function $x^2/2$ and in the regime $1 \ll a_n \ll \sqrt{\log (n(2-y_n)^{3/2})}$.
\end{theorem}
\begin{proof}
For every $\xi \in {\mathbb R}$ and $k_n$ defined by
$$
k_n := \xi \, a_n \, \sqrt{\frac{1}{2 \pi^2} \log (n(2-y_n)^{3/2})} + \frac{2}{3 \pi} n(2 -y_n)^{3/2}
$$
we obtain that
$P \bigl( Z_n \leq \xi \bigr) = P \bigl( N_{I_n}(W_n) \leq k_n \bigr)$.
Hence using \eqref{relation} it follows
$$
P \bigl( Z_n \leq \xi \bigr) = P \bigl( \lambda_{n- k_n}(W_n) \leq y_n \bigr) = P \bigl( \lambda_{n- k_n}(A_n) \leq n \, y_n \bigr).
$$
With \eqref{inequ} we have
$
P \bigl( \lambda_{n- k_n}(A_n) \leq n \, y_n \bigr) \leq P \bigl( \lambda_{n- k_n}(A_n') \leq n \, y_n + n^{-c_0/10} \bigr) + n^{-c_0}
$
and
$$
P \bigl( \lambda_{n- k_n}(A_n') \leq n \, y_n + n^{-c_0/10} \bigr) = P \bigl( \lambda_{n- k_n}(W_n') \leq y_n + n^{-1-c_0/10} \bigr) =
P \bigl( N_{J_n}(W_n') \leq k_n \bigr),
$$
where $J_n = [y_n + n^{-1-c_0/10}, \infty)$. With $y_n' := y_n + n^{-1-c_0/10}$ we consider
$$
Z_n' = \frac{N_{J_n}(W_n') - \frac{2}{3 \pi} n(2 -y_n')^{3/2}}{a_n \, \sqrt{\frac{1}{2 \pi^2} \log (n(2-y_n')^{3/2})}}.
$$
In order to apply Theorem \ref{result1} for $(Z_n')_n$, we have to check if $y_n' \to 2^-$ and $n(2 -y_n')^{3/2} \to \infty$ when
$n \to \infty$. For a proof see \cite[Section 2]{Dalla:2011}. We present the arguments just to make the presentation more self-contained. By assumption
we take $y_n \in [-2+ \delta, 2)$ with $y_n \to 2^-$. Suppose that $y_n' > 2$ for some $n$, then $y_n-2 + n^{-1 -c_0/10} > 0$, hence $2-y_n < n^{-1 -c_0/10}$,
which implies $n(2-y_n)^{3/2} < n \, n^{-3/2 - 3c_0/20}$, but the left hand side is growing by assumption, a contradiction. We have proven $y_n' \to 2^-$.
Moreover we have
$$
(2- y_n')^{3/2} = (2-y_n)^{3/2} \biggl( 1 - \frac{n^{-1 -c_0/10}}{2-y_n} \biggr)^{3/2} = (2-y_n)^{3/2} \biggl( 1 - \frac 32 \frac{n^{-1 -c_0/10}}{2-y_n} +o
\biggl( \frac{n^{-1 -c_0/10}}{2-y_n}\biggr) \biggr).
$$
Notice that $\frac{n^{-1 -c_0/10}}{2-y_n}= \frac{n^{-1/3 -c_0/10}}{(n(2-y_n)^{3/2})^{2/3}} \to 0$ and $ n(2 -y_n)^{3/2} \to \infty$ when $n \to \infty$ by assumption.
Hence we can apply Theorem \ref{result1}, which is the MDP for $(Z_n')_n$. Summarizing we have
$$
P \bigl( Z_n \leq \xi \bigr) \leq P \bigl( Z_n' \leq \xi_n \bigr) + n^{-c_0}
$$
with
\begin{eqnarray*}
\xi_n & = & \frac{k_n - \frac{2}{3 \pi} n (2-y_n')^{3/2}}{a_n \sqrt{\frac{1}{2 \pi^2} \log (n(2-y_n')^{3/2})}} \\
& = & \frac{ \frac{2}{3 \pi} n \bigl( (2-y_n)^{3/2} - (2- y_n')^{3/2} \bigr)}{a_n \, \sqrt{\frac{1}{2 \pi^2} \log (n(2-y_n')^{3/2})}} +
\xi \biggl( \frac{ \log (n(2-y_n)^{3/2})}{\log (n(2-y_n')^{3/2})} \biggr)^{1/2}.
\end{eqnarray*}
We will prove that $\xi_n = \xi + o(1)$. Using the preceding representation we have
$$
n \bigl( (2-y_n)^{3/2} - (2- y_n')^{3/2} \bigr) = \frac 32 n^{-c_0/10}(2-y_n)^{1/2} + o(n^{-c_0/10}) \to 0
$$
and $a_n \, \sqrt{\frac{1}{2 \pi^2} \log (n(2-y_n')^{3/2})} \to \infty$ when $n \to \infty$. Moreover
$$
\frac{ \log (n(2-y_n)^{3/2})}{\log (n(2-y_n')^{3/2})} = \frac{ \log (n(2-y_n)^{3/2})}{\log (n(2-y_n)^{3/2}) + \frac 32 \log \bigl( 1 - \frac{n^{-1 -c_0/10}}{2-y_n}\bigr)} \to 1.
$$
Applying Theorem \ref{result1}, it follows that
$
\lim_{n \to \infty} \frac{1}{a_n^2} \log P \bigl( Z_n \leq \xi \bigr) = -\frac{\xi^2}{2}
$
for all $\xi < 0$. Similarly we obtain
for any $\xi >0$ that $\lim_{n \to \infty} \frac{1}{a_n^2} \log P \bigl( Z_n \geq \xi \bigr) = -\frac{\xi^2}{2}$ and the MDP for $(Z_n)_n$
follows along the lines of the proof of Theorem \ref{result1}.
\end{proof}
\begin{remark}
In a next step one could ask whether the statement of Theorem \ref{result4} is true also for the sequence
$$
\frac{N_{I_n}(W_n) - \E [ N_{I_n}(W_n)]}{a_n \, \sqrt{\V (N_{I_n}(W_n))}}.
$$
Hence the question is whether the asymptotic behavior of the expectation and the variance of $N_{I_n}(W_n)$ is identical to the one for GUE
matrices, given in \eqref{edgeasy1} and \eqref{edgeasy2}. The answer is yes, but only for Wigner matrices obeying Condition $(\bf{C0})$.
The reason for is that the Four Moment Theorem \ref{taovu} deals with a finite number of eigenvalues, whereas $N_{I_n}(W_n)$
involves all the eigenvalues of the Wigner matrix $M_n$. Theorem \ref{taovu} does not give the asymptotics \eqref{edgeasy1} and \eqref{edgeasy2}
for Wigner matrices. A recent result of Erd\"os, Yau and Yin \cite{Erdoes/Yau/Yin:2010} describe strong localization of the eigenvalues of Wigner matrices
and this result provides the additional step necessary to obtain \eqref{edgeasy1} and \eqref{edgeasy2} for Wigner matrices $M_n$ obeying Condition $(\bf{C0})$.
The result in \cite{Erdoes/Yau/Yin:2010} is that for $M_n$ being a Wigner Hermitian matrix obeying Condition $(\bf{C0})$, there is a constant $C>0$ such that
for any $i \{1, \ldots, n\}$
$$
P \bigl( |\lambda_i(W_n)-t(i/n)| \geq (\log n)^{C \log \log n} \min(i,n-i+1)^{-1/3} n^{-2/3} \bigr) \leq n^{-3}.
$$
Along the lines of the proof of \cite[Lemma 5]{Dallaporta/Vu:2011} one obtains \eqref{edgeasy1} and \eqref{edgeasy2}. We will not present the details.
\end{remark}
\bigskip
\section{Further random matrix ensembles}
In this section, we indicate how the preceding results for Wigner Hermitian matrices can be stated and proved for {\it real Wigner symmetric}
matrices. Real Wigner matrices are random symmetric matrices $M_n$ of size $n$ such that, for $i<j$, $(M_n)_{ij}$ are i.i.d. with mean zero and
variance one, $(M_n)_{ii}$ are i.i.d. with mean zero and variance 2. The case where the entries are Gaussian is the GOE mentioned in the
introduction. As in the Hermitian case, the main issue is to establish our conclusions for the GOE.
On the level of CLT, this was developed in \cite{Rourke:2010} by means of the famous {\it interlacing formulas}
due to Forrester and Rains, \cite{Forrester/Rains:2001}, that relates the eigenvalues of different matrix ensembles.
\begin{theorem}[Forrester, Rains, 2001] \label{for}
The following relation holds between GUE and GOE matrix ensembles:
\begin{equation} \label{forrai}
{\rm GUE}_n = {\rm even} \bigl( \rm{GOE}_n \cup {\rm GOE}_{n+1} \bigr).
\end{equation}
\end{theorem}
The statement is: Take two independent (!) matrices from the GOE: one of size $n \times n$ and one of size $(n+1) \times (n+1)$. Superimpose
the $2n+1$ eigenvalues on the real line and then take the $n$ even ones. They have the same distribution as the eigenvalues of a $n \times n$
matrix from the GUE. Let $M_n^{\R}$ denote a GOE matrix and let $W_n^{\R} := \frac{1}{\sqrt{n}} M_n^{\R}$. In \cite[Theorem 4.2]{DoeringEichelsbacher:2011} we have proved a MDP for
\begin{equation} \label{GOEZn}
Z_n^{\R} := \frac{N_{I_n}(W_n^{\R}) - \E[N_{I_n}(W_n^{\R})]}{a_n \sqrt{\V(N_{I_n}(W_n^{\R}))}}
\end{equation}
for any $1 \ll a_n \ll \sqrt{\V(N_{I_n}(W_n^{\R}))}$, $I_n$ an interval in ${\mathbb R}$, with speed $a_n^2$ and rate $x^2/2$.
If $M_n^{\C}$ denotes a GUE matrix and $W_n^{\C}$ the corresponding normalized matrix, the nice consequences of \eqref{forrai} were already suitably developed in \cite{Rourke:2010}: applying Cauchy's interlacing theorem one can write
\begin{equation} \label{interl}
N_{I_n}(W_n^{\C}) = \frac 12 \bigl[ N_{I_n}(W_n^{\R}) + N_{I_n}(\widehat{W}_n^{\R}) + \eta_n'(I_n) \bigr],
\end{equation}
where one obtains ${\rm GOE}_n'$ in $N_{I_n}(\widehat{W}_n^{\R})$ from ${\rm GOE}_{n+1}$ by considering the principle sub-matrix of ${\rm GOE}_{n+1}$
and $\eta_n'(I_n)$ takes values in $\{-2,-1,0,1,2\}$. Note that $N_{I_n}(W_n^{\R})$ and $N_{I_n}(\widehat{W}_n^{\R})$ are independent because ${\rm GOE}_{n+1}$\
and ${\rm GOE}_{n}$ denote independent matrices from the GOE. Now the same arguments as in \cite[Section 4]{DoeringEichelsbacher:2011}
and Theorem \ref{result1} lead
to the MDP for $(Z_n^{\R})_n$, if we consider intervals $I_n=[y_n, \infty)$ where $y_n \to 2^-$ for $n \to \infty$. Remark that the interlacing formula \eqref{interl} leads to $2 \V(N_{I_n}(W_n^{\C})) +O(1) = \V(N_{I_n}(W_n^{\R}))$ if $\V(N_{I_n}(W_n^{\C})) \to \infty$. Next the proof of Theorem \ref{result2} can be adapted to obtain an MDP for $\lambda_{n-i}(W_n^{\R})$:
Consider
$$
Z_{n,i}^{\R} := \frac{\lambda_{n-i}(W_n^{\R}) - \bigl( 2 - \bigl( \frac{3 \pi}{2} \frac in \bigr)^{2/3} \bigr)}{ \operatorname{const} \bigl( \frac{ 2 \log i}{i^{2/3} n^{4/3}} \bigr)^{1/2}}.
$$
With $\E[N_{I_n}(W_n^{\R})] = \E[N_{I_n}(W_n^{\C})] + O(1)$ and $2 \V(N_{I_n}(W_n^{\C})) +O(1) =
\V(N_{I_n}(W_n^{\R}))$ if $\V(N_{I_n}(W_n^{\C})) \to \infty$ we get a MDP along the lines of the proof of Theorem \ref{result2}. We omit the details.
The Four Moment Theorem also applies for real symmetric matrices. The proof of the next Theorem is nearly identical to the proofs of Theorem
\ref{result3} and Theorem \ref{result4}.
\begin{theorem} \label{result5}
Consider a real symmetric Wigner matrix $W_n = \frac{1}{\sqrt{n}} M_n$ whose entries satisfy Condition ${\bf (C1)}$ and match the corresponding entries
of GOE up to order 4. Consider $i=i(n)$ such that $i \to \infty$ and $i/n \to 0$ as $n \to \infty$. Denote the $i$th eigenvalue of $W_n$ by
$\lambda_i(W_n)$. Let $(a_n)_n$ be a sequence of real numbers such that $1 \ll a_n \ll \sqrt{\log i}$.
Then the sequence $(Z_{n,i})_n$ with
$$
Z_{n,i} = \frac{\lambda_{n-i}(W_n) - \bigl( 2 - \bigl( \frac{3 \pi}{2} \frac in \bigr)^{2/3} \bigr)}{ \operatorname{const} \bigl( \frac{ 2 \log i}{i^{2/3} n^{4/3}} \bigr)^{1/2}}
$$
universally satisfies a MDP with speed $a_n^2$ and rate function $I(x)=\frac{x^2}{2}$. Moreover the statement of Theorem \ref{result4} can be adapted and proved analogously.
\end{theorem}
Remark that one could consider the Gaussian Symplectic Ensemble (GSE). Quaternion self-dual Wigner Hermitian matrices have not been studied.
Due to Forrester and Rains, the following relation holds between matrix ensembles:
${\rm GSE}_n = {\rm even} \bigl({\rm GOE}_{2n+1} \bigr) \frac{1}{\sqrt{2}}$. The multiplication by $\frac{1}{\sqrt{2}}$ denotes scaling the $(2n+1) \times (2n+1)$ GOE matrix by the factor $\frac{1}{\sqrt{2}}$.
Let $x_1 < x_2 < \cdots < x_n$ denote the ordered eigenvalues of an $n \times n$ matrix from the GSE and let $y_1 <y_2 < \cdots < y_{2n+1}$ denote
the ordered eigenvalues of an $(2n+1) \times (2n+1)$ matrix from the GOE. Then it follows that $x_i = y_{2i}/\sqrt{2}$ in distribution. Hence
the MDP for the $i$-th eigenvalue of the GSE follows from the MDP in the GOE case. We omit formulating the result.
\newcommand{\SortNoop}[1]{}\def$'$} \def\cprime{$'${$'$} \def$'$} \def\cprime{$'${$'$}
\def\polhk#1{\setbox0=\hbox{#1}{\ooalign{\hidewidth
\lower1.5ex\hbox{`}\hidewidth\crcr\unhbox0}}}
\providecommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}\thinspace}
\providecommand{\MR}{\relax\ifhmode\unskip\space\fi MR }
\providecommand{\MRhref}[2]{%
\href{http://www.ams.org/mathscinet-getitem?mr=#1}{#2}
}
\providecommand{\href}[2]{#2}
|
{
"timestamp": "2012-03-13T01:03:51",
"yymm": "1203",
"arxiv_id": "1203.2115",
"language": "en",
"url": "https://arxiv.org/abs/1203.2115"
}
|
\section{Introduction: Properties of Quiet Sun Internetwork Magnetic Fields}
Obtaining a clear picture of the solar magnetism and of the various
physical processes taking place in the solar atmosphere strongly
relies on the precise determination of the magnetic field vector. This
is very true for the tiny magnetic structures that permeate the
quieter areas of the solar photosphere, the so-called internetwork
fields (IN; \citealt{1975BAAS....7..346L,2008ApJ...672.1237L}). Many of the physical
properties of the IN fields have been inferred from the analysis of
steady four-dimensional images (x, y, $\lambda$, p)\footnote{Two
spatial dimensions, wavelength, and four polarization states (Stokes
I, Q, U, and V).} constructed from spectropolarimetric
measurements. For instance, IN observations taken with the
spectropolarimeter (SP; \citealt{2001ASPC..236...33L}) of the 50-cm
telescope onboard the Hinode satellite \citep{2007SoPh..243....3K}
show that IN magnetic fields are weak and have large inclinations to
the local vertical (e.g., \citealt{2012arXiv1203.1440O}). However, even at the best spatial resolution
achieved so far in the IN (0\farcs15 with \textsc{Sunrise}/IMaX,
\citealt{2011SoPh..268....1B,2011SoPh..268...57M}), the amplitude
of the polarization signals are very small and thus exposed to harmful
noise effects. Moreover, the magnetic structures are still partially
unresolved. Recent inversions of Stokes profiles indicate that the
magnetic fill fractions in the IN are about 20\% at 0\farcs3
resolution \citep{2007ApJ...670L..61O}. Also, observations show that
the number and size of magnetic structures increase monotonically as
the resolution improves
\citep{2008ApJ...684.1469D,2010ApJ...723L.149D,
2010ApJ...718L.171I}. On theoretical grounds, it is expected that
magnetic structures will exist up to the resistivity scales of the
solar plasma (a few meters, e.g., \citealt{2009SSRv..144..275D}
and \citealt{2009ApJ...693.1728P}).
In addition, the magnetic fields of the IN are highly dynamic and
evolve through several physical processes such as emergence and
cancellation. However, it is difficult to study their evolution using
high-cadence raster scans from a spectropolarimeter. The reason is
that to one needs exposures of a few seconds per slit position to
achieve a polarimetric sensitivity of, say, $10^{-3} I_{\mathrm{c}}$
(or s/n~$\sim$~1000)\footnote{The signal-to-the-noise ratio (s/n) is
calculated in the continuum of Stokes I.}. Therefore, the measurements
have to be limited to very narrow regions of the solar surface, and
even so the achievable cadence is of the order of minutes. Using this
technique, it has been possible to characterize dynamic properties of
magnetic flux emergence processes (e.g.,
\citealt{2008A&A...481L..33O, 2009ApJ...700.1391M}). However,
most of the dynamic properties of the IN fields have been determined
from filter-based polarimeters which allow the observation of large
fields of view at high cadence, but with far less spectral
purity. From these measurements we know that the mean lifetime of IN
magnetic elements is about 10 minutes, although it is difficult to set
a mean lifetime for the polarization signals at high resolution (e.g.,
\citealt{2010ApJ...723L.149D}). The rms fluctuation of the horizontal
velocity of IN magnetic elements is 1.57~km~s$^{-1}$
\citep{2008ApJ...684.1469D}. A drawback of filter-based instruments is
that the physical information extracted from the observed polarization
signals is often less accurate than that inferred from slit-based
polarimeters because the data are affected by the modest spectral
resolution and coarse wavelength sampling
\citep{2010A&A...522A.101O}.
Finally, the limited spatial resolution impedes the investigation of
small-scale physical processes like reconnections, jets, or
cancellations taking place in the quiet Sun IN. For instance, recent
analyses of cancellation events observed in the IN with the Hinode
spectropolarimeter show evidence of insufficient temporal and spatial
resolution to ``catch'' the important physics behind these events (see
e.g., \citealt{2010ApJ...712.1321K}). An example of a magnetic
cancellation is shown in Fig.~\ref{fig1} using \ion{Na}{I} 589.6~nm
magnetograms acquired with the Narrowband Filter Imager of Hinode at a
spatial resolution of about 0\farcs3. The cancellation takes place in
a very narrow region, right between the two magnetic elements, in the
so-called polarity inversion line. Present spectropolarimetric
measurements lack enough spatial resolution to observe cancellation
boundaries in detail. In summary, the observation of IN fields by
spectropolarimetric means is a very demanding task, but necessary to
determine the physical properties of the quiet Sun.
\begin{figure}[!t]
\centering
\plotone{orozcosuarez_fig1.eps}
\caption{Magnetograms recorded with Hinode's Narrowband Filter Imager
showing the cancellation of two IN magnetic elements (left to right
and top to bottom). The total duration of the sequence is about 22
minutes. The arrow pinpoints the cancellation site or polarity
inversion line that is very small compared to the size of the
canceling magnetic elements. Note how the magnetic elements rotate
clockwise while the amplitude of the signals diminishes, suggesting
that important dynamics may take place during the cancellation. Tick
marks are separated by 1\arcsec.}
\label{fig1}
\end{figure}
Large photon collecting surfaces will improve both the photometric
accuracy and the cadence of the measurements
\citep{2008LNEA....3..113C, 2010AN....331..558D}, as well as the
spatial resolution. Thus, it will be easier to determine the magnetic
field vector from the observed polarization signals because the
changes in the Stokes profile shapes resulting from variations of the
physical quantities will be less influenced by the noise (e.g.,
\citealt{2011IAUS..273...37D}). In addition, the amplitude of the
polarization signals will be greater since the magnetic filling
factors tend to increase as the spatial resolution improves. By the
same token, however, resolution also challenges the observation of
extended areas of the solar surface at high cadence using
spectropolarimeters, because of the large number of steps in the slit
scan direction needed to cover significant fields of view. The
European Solar Telescope (EST; \citealt{2010AN....331..615C}) and the Advanced Technology
Solar Telescope (ATST; \citealt{2010AN....331..609K}) will be equipped with 4m mirrors,
thus facilitating the measurement of quiet Sun IN field. Both will
provide the photons needed to achieve high polarimetric accuracy and
temporal resolution with small exposure times\footnote{For details,
see http://www.est-east.eu and http://atst.nso.edu}.
In this contribution we aim at listing the main requirements for
observing quiet Sun internetwork fields with EST and ATST using the
current understanding about the magnetic and dynamic properties of IN
magnetic elements. We put special attention on the fields of view
needed to get a complete picture of the quiet Sun internetwork
magnetism and of the physical processes taking place there.
\section{Velocities of Internetwork Magnetic Elements and Traveled Distance}
To set up limits on the field of view (FOV) to be observed by EST and
ATST one can use the typical velocities (i.e., proper motions) and
lifetimes (or traveled distances) of IN magnetic elements as a
reference. If we are interested in analyzing individual magnetic
features from birth (emergence/submergence or appearance of flux) to
death (cancellation or disappearance), these two quantities may help
constrain the required FOV because they inform us about the mean
distance traveled by the magnetic elements in the solar surface. Most
measurements concur that IN flux concentrations show two distinct
velocity components: one of random nature, resulting from the
continual buffeting of the fields by granular convection (e.g.,
\citealt{2011A&A...531L...9M}), and a net velocity that transports
the fields from the center of supergranular cells toward their
boundaries, i.e., toward the network. We are interested in the
latter. However, while there exist robust estimations of the random
velocity component, about 1.57~km~s$^{-1}$, the values given in the
literature for the net velocity range from 0.2~km~s$^{-1}$
\citep{2008ApJ...684.1469D} to 0.35~km~s$^{-1}$
\citep{1987SoPh..110..101Z}. Only \cite{1998A&A...338..322Z}
investigated the spatial dependence of the velocity pattern of IN
elements and found a constant radial velocity (measured from the
center of supergranular cells outwards) of about 0.4~km~s$^{-1}$ using
low resolution (1\farcs5) and low cadence (7 min) magnetograms.
\begin{figure*}[!t]
\centering
\epsscale{0.6}
\plotone{orozcosuarez_fig2.eps}
\caption{Right: Quiet Sun magnetogram taken with Hinode's Narrowband
Filter Imager, saturated at $\pm$~30~G. The FOV covers one
supergranular cell. Left: trajectories of some of the several thousand
magnetic elements detected in 13 hours of magnetograms. The
trajectories are more linear in the inner part and more random near
the borders. Colors distinguish individual objects. Axis units are
arcsec.}
\label{fig2}
\end{figure*}
Here, we employ very high spatial resolution (0\farcs3)
\ion{Na}{I} 589.6~nm magnetograms acquired by Hinode to estimate the
net velocity component of IN magnetic elements. The magnetograms have
a cadence of 80 seconds and a FOV of approximately
93\arcsec$\times$82\arcsec, so they cover a few supergranular
cells. The noise is 6.7~G. To analyze the data we first selected a
small area of 50\arcsec$\times$50\arcsec\/ containing one
supergranular cell and the network around it. Then we detected and
tracked the proper motion of each magnetic element in consecutive
magnetograms. In total, we analyzed 13 hours of data. For detecting
and tracking magnetic elements we used the YAFTA\_10
software\footnote{YAFTA\_10 can be downloaded from
http://solarmuri.ssl.berkeley.edu/$\sim$welsch/public/software/YAFTA}
\citep[Yet Another Feature Tracking Algorithm;][]{2007ApJ...666..576D}.
In the detection process we ignored all magnetic elements with fluxes
below 10~G and sizes smaller that 16~px$^2$.
The right panel of Figure \ref{fig2} depicts a single
50\arcsec$\times$50\arcsec\/ snapshot extracted from the magnetogram
time sequence. It contains a supergranular cell. One can clearly see
large circular polarization signals concentrated at the borders of the
image, corresponding to large flux values (i.e., network fields). In
the inner part of the image there are flux elements that show opposite
polarities and much smaller sizes than those associated with the
network. These signals correspond to internetwork magnetic fields.
The left panel shows the paths of some of the several thousand
elements detected and tracked in the magnetograms by YAFTA\_10. The
paths of the elements located in the inner part of the supergranule
are rather linear and radially aligned with respect to the center of
the image. In the network, the paths ``look'' more random and show
little linear trends. This figure alone suggests that quiet Sun
magnetic elements have two distinct velocity components, one random
and one systematic, both weighted differently depending on the
location of the element within the supergranular cell.
Once we have the positions of each magnetic element in each of the
magnetograms, we derive their horizontal velocities $\mathbf{v} =
(v_{\rm x},v_{\rm y})$. Because there is a clear radial symmetry in
the trajectory pattern we decided to translate the horizontal
velocities into the radial $v_{\rm R} = \mathbf{v}\cdot\mathbf{r} /
|\mathbf{r}|$ and transversal $v_{\rm T} =
|\mathbf{r}\times\mathbf{v}| / |\mathbf{r}|$ components with respect
to the center of the supergranular cell. Here the quantity
$\mathbf{r}$ stands for the displacement and $v_{\rm R}$ is measured
positive outwards. Using the network at a reference, the supergranule
center can be located at $(x_{\rm c}, y_{\rm c}) \sim (26\farcs6,
22\farcs1)$, i.e., very close to the center of the magnetogram
displayed in Figure~\ref{fig2}. Next we calculated the azimuthally
averaged values of the radial velocity component as a function of
distance from the center. We also calculated the mean linear distance
$\overline{|\mathbf{r}|}$ traveled by the magnetic elements detected
by the tracking algorithm.
\begin{figure}[!t]
\centering
\plottwo{orozcosuarez_fig3a.eps}{orozcosuarez_fig3b.eps}
\caption{Left: histogram of the mean linear distance traveled by
internetwork magnetic elements. The average value is 4\farcs1. Right:
Variation of the azimuthally averaged radial velocity component of the
magnetic elements as a function of distance (position) from the center
of the supergranule.}
\label{fig3}
\end{figure}
The results are shown in Fig.~\ref{fig3} for the average radial
velocity (right panel) and the mean traveled distance (left
panel). Interestingly, the radial component of the velocity shows a
dependence with the distance to the supergranular cell center. The
magnetic elements close to the center are almost at rest. The radial
velocity increases outwards until it reaches a maximum value of
0.32~km~s$^{-1}$ at 9\arcsec\/ from the supergranular center. Then,
it decreases monotonically. We find a mean radial velocity of
0.15~km~s~$^{-1}$, slightly smaller than previous estimations
\citep{2008ApJ...684.1469D,1987SoPh..110..101Z,1998A&A...338..322Z}. Overall,
the magnetic elements move toward the network. These results indicate
that the dynamic properties of the quiet Sun magnetic elements are
different depending on their location within the supergranular cell.
The histogram for the mean traveled distance shows a peak at small
values and has an extended tail that reaches 20\arcsec. This implies
that most magnetic elements travel small distances, although some of
them can get as far as half the diameter of a typical supergranular
cell. The mean traveled distance is 4\farcs1. It is important to keep
in mind that during the tracking process we did not take into account
the fact that small magnetic elements tend to collide with others
during their trip to the network. Such collisions give rise to
splitting of signals, merging of signals of the same polarity, and
cancellation of signals of opposite polarity. Such interactions create
``new'' elements in the sense that the signals change their properties
enough to make the tracking algorithm interpret them as new magnetic
elements. Hence, the mean distance is being underestimated and the
value given here should be considered a lower limit.
\section{Summary: Minimum Requirements to Observe IN Magnetic Fields}
In this section we compile the minimum requirements for observing
quiet Sun IN fields with ATST and EST. With their 4m mirrors, these
telescopes will collect 64 times more photons than 50-cm telescopes
(if we assume the same photon efficiency), which will alleviate most
of the current limitations encountered in the study of quiet Sun IN
fields.
\begin{itemize}
\item[-] \emph{Spatial resolution}: On average, we know that the fill
fraction is about 20\% for 120~km (0\farcs16) pixel size (e.g.,
\citealt{2007ApJ...670L..61O}). Thus, we need to increase the spatial
resolution up to 30 km (the diffraction limit of ATST and EST at
500~nm) to spatially resolve them.
\item[-] \emph{Temporal resolution}: a minimum time cadence may be
given by the rms velocity of the magnetic elements and the spatial
resolution. Assuming a rms velocity of 1.5~km~s$^{-1}$ (measured
with a 1-meter class telescope) and a spatial resolution of 30~km, the
magnetic elements would take about 20 seconds to move from one pixel
to the next. We shall set half of this time as the minimum cadence for
critical sampling of the motions in the case of diffraction-limited
observations. Note that since the measured rms velocity increases with
spatial resolution, a cadence of 10 seconds may be severely
underestimated. In addition, the rms velocity of the magnetic elements
also poses limitations for the exposure times that should be much
smaller than 10 seconds to prevent image degradation. If we take the
sound speed in the photosphere (about 8~km~s$^{-1}$) as a reference,
then all measurements have to be performed about five times faster,
i.e., in about two seconds. The high cadence will allow the study of
fast physical phenomena like cancellation events.
\item[-] \emph{Polarization sensitivity}: The number of photons collected
per resolution element is independent of the telescope diameter at the
diffraction limit (see e.g., \cite{2003SPIE.4843..100K}). For this reason, 4-meter telescopes will have the same
polarization sensitivity as 50-centimeter telescopes, provided the
photon efficiency is maintained. Thus, taking the exposure times of
the Hinode spectropolarimeter as a reference, at the diffraction limit
ATST and EST will reach a s/n ratio of $\sim$~1000 with integration
times of five seconds. For the quiet Sun internetwork we need to go
beyond noise levels of $10^{-3} I_{\rm c}$. Thus, there are two
options: increase the exposure time or downgrade the spatial
resolution. For instance, we can reach $2.5
\times 10^{-4} \, I_{\rm c}$ with five second exposure times and
0\farcs1 resolution. Note that since it is expected that the
polarization signals will be larger because of the increased
resolution, it may be possible to fully characterize the IN fields
with s/n~$\sim$~1000. Thus, a better option may be to achieve a
s/n of 1000 in less than a few tenths of a second at 0\farcs1 spatial
resolution.
\item[-] \emph{Field of view}: Which FOV is most appropriate for the study
of quiet Sun magnetic fields? To investigate the dynamics and
interactions between magnetic elements, a FOV covering a minimum area
of 4\arcsec seems necessary. However, we have seen that there exist
clear connections between the dynamics and the spatial location within
supergranular cells. Therefore, the observation of a complete
supergranule (about 30\arcsec\/ diameter) would be highly desirable.
\end{itemize}
Is it possible for ATST and EST to meet all the above requirements?
With slit-based instruments only, the answer is negative. The reason
is that the spatial resolution of the observations should be kept
close to the diffraction limit of ATST and EST to advance our
understanding of IN magnetic fields. Hinode observations have shown
that a spatial resolution of 0\farcs3 is not enough to characterize
the magnetic processes taking place in the IN. But, at the diffraction
limit of a 4-meter telescope ($\sim$~0\farcs05), it takes more than
six minutes for a spectrograph to scan a 4\arcsec-wide area with a
s/n~$\sim$~1000. Six minutes is far above the minimum requirements set
by the current observations. One can make a compromise and reduce the
spatial resolution to about 0\farcs1, but it would still take about
1.5 minutes to scan the same area maintaining the s/n.
Therefore, systems capable of scanning a given FOV while maintaining
the polarimetric sensitivity and temporal resolution seem necessary if
one wants to fulfill the minimum requirements to characterize
internetwork fields, study their dynamics, and analyze the physical
processes in which they participate. These are, for instance,
multi-slit configurations, image slicing instruments, or fiber optics
arrays (see e.g., \citealt{2010AN....331..581S}). In particular, the
latter concept has been successfully implemented in the
SpectroPolarimetric Imager for the Energetic Sun (SPIES), an
instrument which observes 2D FOVs without compromising the spatial,
temporal, or spectral resolution (see Lin, this volume).
Another option is to use tunable filters instead of
spectrographs. However, although filtergraph observations can achieve
noise levels close to 10$^{-3}$ (e.g., \citealt{2011SoPh..268...57M})
and the determination of the magnetic field vector is possible from
such observations (e.g., \citealt{2010A&A...522A.101O}), the
uncertainties of the physical quantities inferred from these
instruments are larger than those derived from spectropolarimetric
measurements. Among other reasons, this is due to the fact that the
shapes of the spectral lines are strongly distorted by the passband of
filter-based polarimeters and the long times needed to scan the
spectral line.
\acknowledgements D.O.S. thanks the Japan Society for the Promotion of Science
(JSPS) for financial support through the postdoctoral fellowship
program for foreign researchers. Hinode is a Japanese mission developed and launched by ISAS/JAXA, with NAOJ as domestic partner and NASA and STFC (UK) as international partners. It is operated by these agencies in co-operation with ESA and NSC (Norway).
\bibliographystyle{asp2010}
|
{
"timestamp": "2012-03-13T01:00:08",
"yymm": "1203",
"arxiv_id": "1203.2185",
"language": "en",
"url": "https://arxiv.org/abs/1203.2185"
}
|
\section{Introduction}
Recent numerical studies of gravitational collapse in isotropic coordinates, both in $4 +1$ dimensions (5D) \cite{Khlebnikov, C-E} and 4D \cite{C-E}, have provided numerical evidence that the negative of the total Lagrangian at late times of the collapse process approaches the Helmholtz free energy $E\m TS$ of a Schwarzschild black hole, where $E$, $T$ and $S$ are the mass, temperature and entropy of the black hole. For a stationary black hole, $T\eq \hbar \kappa/2\pi$ and $S\eq A/4\hbar$ where $\kappa$ is the surface gravity at the horizon and $A$ is the horizon area. Though $T$ and $S$ both contain $\hbar$, their product $TS$ does not and this opens the possibility for a classical investigation of black hole thermodynamics via the free energy \cite{Khlebnikov}. There is in fact an argument based on the Euclidean action that relates the negative of the total Lagrangian to the free energy of a stationary black hole \cite{Khlebnikov,Hawking2}. This association can be tested numerically during gravitational collapse by tracking the Lagrangian and comparing its numerical value at late times to the expected free energy from standard black hole thermodynamics. This was carried out numerically for the first time for the collapse of a 5D Yang-Mills instanton \cite{Khlebnikov} to a Schwarzschild black hole in isotropic coordinates. A numerical study in isotropic coordinates was then carried out for the collapse of a 4D and 5D massless scalar field to a Schwarzschild black hole \cite{C-E}. These works constituted a classical numerical study of black hole thermodynamics; quantum mechanics does not enter the picture and Hawking radiation is not observed. Prior studies of thermodynamics during gravitational collapse have consisted mainly of analytical or semi-analytical work on black hole entropy and Hawking radiation during shell or dust collapse \cite{Gao}-\cite{Vaz1} and the approaches have been quantum mechanical or semi-classical.
For the Reissner-Nordstr\"om (RN) black hole, the relevant thermodynamic potential is the Gibbs free energy $G\eq E\m TS\m \Phi_H\,Q$ where $\Phi_H$ is the electrostatic potential at the horizon and $Q$ is the charge of the black hole \cite{Hawking2, Kiritsis}. In this paper, we investigate numerically in isotropic coordinates the thermodynamics during the collapse of a charged scalar field to a RN black hole. Among other things, we study the association between the total Lagrangian and the Gibbs free energy. Charged collapse leads to a large outgoing matter wave and the exterior takes a long time to settle to a static state. However, the interior spacetime (inside the outer horizon) is hardly affected by this wave and settles more quickly than the exterior. One can show analytically that the interior Gibbs free energy $G_{int}$ is equal to $-TS$. The interior contribution of the negative of the total Lagrangian can be obtained numerically and then compared to $G_{int}$. At late times of the collapse, we find that the matter Lagrangian tends towards zero in the interior; the interior Lagrangian stems entirely from the gravitational sector. It is negative and accumulates just inside the horizon where the spacetime is not static. In short, the entropy of the charged black hole is gravitational in origin and stems from the dynamical interior near the horizon. That the entropy stems from the interior is in accord with analytical work on the canonical quantization of the RN black hole \cite{Vaz2} where the entropy was obtained via explicit counting of microstates in the dynamical interior (see \cite{Vaz1} for the uncharged case). Our numerical results for the negative of the gravitational Lagrangian in the interior agrees with the analytical value of $G_{int}$ to within $10\!-\!13\%$ depending on the profile of the initial state. There are sharp changes in the gradients of the metric and matter functions in the near-horizon region and this places limits on how far in time one can evolve before the usual monitors of the code, such as the ADM mass, begin to deviate from their conserved values. The numerical graph representing the Lagrangian and the analytical graph representing the free energy approach each other with time and it is clear that given more time evolution the difference between them would continue to decrease. Of course, to increase the time evolution one needs to increase the resolution and one reaches a limit where the computing time is no longer practical. As a consistency check, we also implement a procedure for prolonging the evolution time in the exterior region. The outgoing matter wave disperses and the metric in the exterior is observed to approach the RN static exterior metric. Unlike Schwarzschild, the matter Lagrangian now makes a contribution to the exterior due to the presence of a static electric field.
The coordinate time $t$ in isotropic coordinates coincides with the time measured by an asymptotic observer at rest. The viewpoint of the static asymptotic observer is appropriate for studying black hole thermodynamics. The temperature of a stationary black hole is the temperature as seen by an observer at rest at spatial infinity \cite{Lindesay} and its entropy represents a measure of an external observer's ignorance of the internal configurations hidden behind the event horizon \cite{Bekenstein, E-C}. In particular, the association between the total Lagrangian and the free energy of a black hole that is found in numerical studies of collapse in isotropic coordinates will in general not hold in other coordinate systems. Unlike the action, the Lagrangian depends on the choice of time coordinate i.e. on the foliation of the spacetime. For example, one should not expect the Lagrangian in numerical studies of gravitational collapse in Painlev\'{e}-Gullstrand (PG) coordinates \cite{Kunstatter} to be associated with the free energy of the black hole. The reason is that in PG coordinates the coordinate time represents the proper time of freely-falling observers \cite{Poisson} not asymptotic observers. This is also true of numerical studies of charged collapse that have been carried out in ``Eddington-Finkelstein" \cite{Brady} and double-null coordinates \cite{Hod,Piran} as there is again no coordinate representing the proper time of an asymptotic observer in these cases.
Our paper proceeds as follows. We first express the exterior RN metric in isotropic coordinates. We then derive the equations of motion in those coordinates: the wave equation governing the complex (charged) scalar field, Maxwell's equations for the gauge field and Einstein's equations governing the metric field. We then discuss how the initial states are obtained using the shooting method. Integral and differential expressions in isotropic coordinates are then derived for the conserved charge $Q$ and mass $M$. During the evolution these quantities should remain constant and this allows one to monitor the accuracy of the code at each time step. Expressions for the gravitational and matter Lagrangian as well as the interior Gibbs free energy are then derived. We finally present the thermodynamic results from the numerical simulation.
We work in geometric units $G\eq c\eq 1$ where energy, mass and time have units of length. Coulomb's constant is set to unity so that electric charge has units of length also. The metric has signature $(-1,1,1,1)$. Spacetime indices are in Greek and run from $0$ to $3$.
\subsection{Reissner-Nordstr\"{o}m metric in isotropic coordinates}
During our numerical simulation in isotropic coordinates, we expect the metric to settle to the Reissner-Nordstr\"{o}m (RN) metric at late times. We therefore need to express the RN metric in isotropic coordinates. A general spherically symmetric time-dependent $4D$ metric in isotropic coordinates takes the form \cite{Finelli1,Finelli2,B-E}:
\beq
ds^2= -N(r,t)^2 dt^2 +\psi(r,t)^4 (dr^2 + r^2 d\Omega^2)
\eeq{isometric}
where $\psi(r,t)$ is referred to as the conformal factor and $N(r,t)$ is called the lapse function. Note that the isotropic radial coordinate $r$ is not the areal radius. We assume asymptotic flatness so that $N$ and $\psi$ are both unity at infinity. Note that the coordinate time $t$ coincides with the time measured by a clock at rest at infinity. Our goal is to find the analytical expressions for $N$ and $\psi$ that correspond to the RN metric. In standard coordinates, the RN metric is given by \cite{Poisson}
\beq
ds^2= -\Big(1-\dfrac{2M}{r'} + \dfrac{Q^2}{r'^{\,2}}\Big)\,dt^2 + \Big(1-\dfrac{2M}{r'}+\dfrac{Q^2}{r'^{\,2}}\Big)^{-1}\, dr'^{\,2} +r'^{\,2}\,d\Omega^2
\eeq{RN}
where $M$ and $Q$ are the mass and charge of the black hole and $r'$ is the areal radius. The function $f(r')\eq 1-2M/r' + Q^2/r'^{\,2}$ has zeroes at $r'_{\pm}= M \pm \sqrt{M^2-Q^2}$ where $r'_+$ and $r'_{-}$ are referred to as the outer and inner horizon respectively. Matching the metric \reff{isometric} to the metric \reff{RN} yields $\psi^4= r'^{\,2}/r^2$, $N^2 =f(r')$ and the equation $\tfrac{dr^2}{r^2}= \tfrac{1}{f(r')}\tfrac{dr'^{\,2}}{r'^{\,2}}$. With the condition that $\psi$ and $N$ are unity asymptotically, the latter has solution $r'\eq r+ M + (M^2-Q^2)/4r$ so that the {\it exterior} region of the RN metric in isotropic coordinates is given by
\beq
ds^2=- \dfrac{\Big(1-\dfrac{M^2-Q^2}{4\,r^2}\Big)^2}{\Big(1 +\dfrac{M}{r} +\dfrac{M^2-Q^2}{4r^2}\Big)^2} \,\,dt^2 + \Big[1 +\dfrac{M}{r} +\dfrac{M^2-Q^2}{4r^2}\Big]^2 \Big( dr^2 + r^2\,d\Omega^2\Big) \,.
\eeq{IsoRN}
The outer horizon in isotropic coordinates is situated at $r_+\eq \sqrt{M^2-Q^2}/2$. At this location, the lapse function is zero: $N(r_+)\eq0$. The metric \reff{IsoRN} does not cover the interior region of the RN black hole; it covers the static exterior region twice. The minimum value of $r'\eq r+ M + (M^2-Q^2)/4r\,$ is $\,r'_+\eq M+\sqrt{M^2-Q^2}$ ; in metric \reff{IsoRN} both the region $r\ge r_+$ and $r \le r_+$ correspond to the exterior region $r'\ge r'_+$. The Killing vectors in the interior region between the two horizons of the RN black hole are all spacelike \cite{C-E} and the spacetime is nonstationary in this region. To cover the interior region of the RN black hole in isotropic coordinates the functions $N$ and $\psi$ in \reff{isometric} must be time-dependent. In standard coordinates, the spacetime becomes nonstationary when one crosses the outer horizon into the interior because the function $f(r')$ switches sign and the radial coordinate becomes timelike and the time coordinate becomes spacelike. In isotropic coordinates, the metric coefficients in \reff{isometric} are positive-definite; they do not switch sign but instead become time-dependent in the interior. During our numerical simulation, the metric functions $N(r,t)$ and $\psi(r,t)$ in the isotropic metric \reff{isometric} are nonstationary in the region $r\!<\!r_+$, reflecting the true nature of the interior spacetime. It is only in the exterior region $r\!>\!r_+$ that the metric approaches the static form \reff{IsoRN} at late times.
We do not observe the second (inner) horizon or timelike singularity associated with the RN metric \reff{RN}; we observe one horizon at $r_+$ (where $N(r_+)=0$) and a spacelike singularity as in the Schwarzschild case. This is in accord with the findings of previous numerical work on charged collapse \cite{Brady,Hod,Piran}. The inner horizon of the RN metric \reff{RN} is an artifact of exact staticity (and exact spherical symmetry) \cite{Poisson} and it has been known since pioneering work in the 90's \cite{Poisson-Israel}, that the inner (Cauchy) horizon is unstable to perturbations.
\section{Evolution and constraint equations in isotropic coordinates}
\subsection{Matter sector}
For matter, we consider a complex (charged) scalar field coupled to an electromagnetic field $A_{\mu}$. The matter Lagrangian density $\mathcal{L}_{m}$ has a local $U(1)$ gauge symmetry and is given by \cite{Schroeder}
\begin{equation}\label{L_Matter}
\mathcal{L}_{m}=-\frac{1}{2}\left(\chi_{;\,\mu}+ieA_\mu\chi\right)g^{\mu\nu}\left(\overline{\chi}_{;\,\nu}-
ieA_\nu\overline{\chi}\right)-\dfrac{1}{16\pi}F_{\mu\nu}\,F^{\mu\nu}
\end{equation}
where a semi-colon denotes covariant differentiation evaluated with metric \reff{isometric}, a bar denotes complex conjugation and $F_{\mu\nu}\!\equiv \!A_{\nu;\,\mu}-A_{\mu;\,\nu}$ is the electromagnetic field tensor. Spherical symmetry reduces the number of gauge components from four to two: only $A_t=A_{0}$ and $A_{r}=A_{1}$ are non-zero. Gauge freedom allows one to further eliminate $A_r$. This leaves $A_t$ as the only non-zero component and for simplicity we label it $a$. The matter fields are therefore $\chi\eq \chi(r,t)$ and $a\eq a(r,t)$. Lagrange's equations of motion for matter are
\begin{equation}\label{Poisson}
\nabla_\alpha\frac{\partial\mathcal{L}_m}{\partial q_{;\,\alpha}}-\frac{\partial\mathcal{L}_m}{\partial q}=0
\end{equation}
where $q$ is a generic field.
\subsubsection{Equations of motion for scalar field $\chi$}
Applying \reff{Poisson} to the scalar field $\chi$ yields the wave equation
\begin{equation}
\chi_{;\,\mu\nu}\,g^{\mu\nu}+ieA_\mu\,g^{\mu\nu}\left(2\chi_{;\,\nu}+ieA_{\nu}\chi\right)+ieA_{\mu;\,\nu}\,g^{\mu\nu}\chi=0 \,.
\label{KG}\end{equation}
With spherical symmetry, the three terms in the above equation reduce to
\begin{align*}
&\chi_{;\,\mu\nu}\,g^{\mu\nu}=\dfrac{1}{\sqrt{-g}}\partial_\mu\left(\sqrt{-g}\,g^{\mu\nu}\partial_\nu\chi\right)
=-\dfrac{1}{N\psi^6}\partial_t\left(\dfrac{\psi^6\dot{\chi}}{N}\right)
+\dfrac{1}{N\psi^6r^2}\partial_r\left(N\psi^2r^2\chi'\right)\\
&ieA_\mu \,g^{\mu\nu}\left(2\chi_{;\,\nu}+ieA_{\nu}\chi\right)=-\dfrac{iea}{N^2}\left(2\dot{\chi}+iea\chi\right)\\
&ieA_{\mu;\,\nu}\,g^{\mu\nu}\chi = -\frac{1}{N\psi^6}\partial_t\left(\frac{iea\chi\psi^6}{N}\right)+\frac{iea\dot{\chi}}{N^2}\,
\end{align*}
where $\dot{\chi}\equiv \partial\chi/\partial t$ and $\chi'\equiv \partial \chi/\partial r$.
Equation \reff{KG} now reads
\beq
-\dfrac{1}{N\psi^6}\,\partial_t\left[\dfrac{\psi^6}{N}\left(\dot{\chi} +iea\chi\right)\right] + \dfrac{1}{N\psi^6r^2}\,\partial_r\left(N\psi^2r^2\chi'\right) -\dfrac{iea}{N^2}(\dot{\chi} +iea\chi)=0\,.
\eeq{KG22}
For numerical purposes, we split the above second order equation into two first order evolution equations. To this end we define the quantity
\begin{equation}\label{p}
p\equiv\frac{\psi^6}{N}\left(\dot{\chi}+iea\chi\right)\,.
\end{equation}
In terms of $p$, equation \reff{KG22} simplifies to
\begin{equation}\label{wave}
\dot{p}=\frac{1}{r^2}\,\partial_r\left(N\psi^2r^2\chi'\right)-ieap.
\end{equation}
Rearranging \reff{p} we obtain
\begin{equation}\label{e2}
\dot{\chi}=\frac{N}{\psi^6}\left(p-\frac{iea\psi^6\chi}{N}\right).
\end{equation}
Equations \reff{wave} and \reff{e2} are the evolution equations for $p$ and $\chi$ respectively.
\subsubsection{Equations of motion for gauge field $a$}
Applying \reff{Poisson} to the gauge field $A_{\mu}$ yields Maxwell's equations:
\begin{equation}
\frac{1}{2\pi}F_{\mu\nu;\rho}\,g^{\nu\rho}+ie\chi\left(\overline{\chi}_{;\mu}-ieA_\mu\overline{\chi}\right)
-ie\overline{\chi}\left(\chi_{;\nu}+ieA_\mu\chi\right)=0.
\end{equation}
Spherical symmetry reduces the above to two equations, one for $\mu=t$ and $\mu=r$. The equation for $\mu=t$ is
\begin{equation}
-\frac{a''}{\psi^4}+\frac{N'a'}{N\psi^4}-\frac{2\psi'a'}{\psi^5}-\frac{2a'}{\psi^4r}
+\frac{2\pi ieN}{\psi^6}\left(\chi\,\overline{p}-\overline{\chi}\,p\right)=0
\end{equation}
where $p$ was defined in \reff{p}. The above can be rewritten as
\begin{equation}\label{M1}
\frac{1}{r^2}\partial_r\left(\frac{\psi^2a'r^2}{N}\right)=2\pi ie\,\left(\chi\,\overline{p}-\overline{\chi}\,p\right)\,.
\end{equation}
The above is a constraint equation for the gauge field $a$ (one can view it as Poisson's equation).
Maxwell's equation for $\mu=r$ yields
\begin{equation}
-\frac{\dot{a}'}{N^2}+\frac{\dot{N}a'}{N^3}-\frac{2\dot{\psi}a'}{N^2\psi}
+2\pi ie\,\chi\,\overline{\chi}'-2\pi ie\,\overline{\chi}\,\chi'=0
\end{equation}
which can be expressed as
\begin{equation}\label{M2}
\dot{g}=2\pi ieN\psi^2\left(\chi\,\overline{\chi}'-\overline{\chi}\,\chi'\right)
\end{equation}
with \begin{equation}\label{g}
g\equiv\frac{a'\psi^2}{N}\,.
\end{equation}
Equation \reff{M2} is the evolution equation for $g$. The equation governing the gauge field $a$ is not an evolution equation but an ordinary differential equation: $a'= g\,N/ \psi^2$.
\subsection{Gravitational sector}
\subsubsection{Stress-energy tensor}
The stress-energy tensor $T_{\mu\nu}$ appearing in Einstein's field equations can be calculated from the matter Lagrangian \reff{L_Matter}. We first express the latter in terms of the new quantities $p$ and $g$ defined in \reff{p} and \reff{g} respectively:
\begin{equation}
\mathcal{L}_m=\frac{p\overline{p}}{2\psi^{12}}-\dfrac{\chi'\,\overline{\chi}'}{2\psi^4}+\dfrac{g^2}{8\pi\psi^8}.
\label{L_Matter2}\end{equation}
The energy-momentum tensor is given by
\begin{equation}
T_{\alpha\beta}=-2\dfrac{\partial\mathcal{L}_m}{\partial g^{\alpha\beta}}+\mathcal{L}_m\,g_{\alpha\beta}
\end{equation}
and its non-zero components are
\begin{equation}
\begin{aligned}
T_{\theta\theta} &=\dfrac{1}{2}\left(\dfrac{p\overline{p}}{\psi^{8}}-\chi'\,\overline{\chi}'
+\dfrac{g^2}{4\pi\psi^4}\right)r^2
&T_{\phi\phi} & =T_{\theta\theta}\sin^2\theta\\
T_{rr} &=\dfrac{1}{2}\left(\dfrac{p\overline{p}}{\psi^{8}}+\chi'\,\overline{\chi}'
-\dfrac{g^2}{4\pi\psi^4}\right)
&T_{rt} & =\dfrac{N}{2\psi^6}\left(\chi'\overline{p}+\overline{\chi}'p\right)\\
T_{tt} &=\dfrac{N^2}{2}\left(\frac{p\overline{p}}{\psi^{12}}+\dfrac{\chi'\overline{\chi}'}{\psi^4}
+\dfrac{g^2}{4\pi\,\psi^8}\right).
\end{aligned}
\end{equation}
\subsubsection{Field equations}
Einstein's field equations are given by $G_{\mu\nu}=\kappa^2\,T_{\mu\nu}$
where $\kappa^2\equiv8\pi\,G \eq 8\pi$ and $G_{\mu\nu}$ is the Einstein tensor evaluated with metric \reff{isometric}.
The $G_{rr}$ equation yields evolution equations for the conformal factor $\psi$ while $G_{tt}$ and $G_{rt}$ yield constraint equations. $G_{\theta\theta}$ yields an ordinary differential equation for the lapse function $N$ ($G_{\phi\phi}$ yields the same equation).
The $G_{rr}$ equation is
\begin{equation}\label{Grr}
\begin{aligned} &\frac{2}{r\psi^2N^3}\left(2r\psi'^2N^3-4r\dot{\psi}^2N\psi^4+2\psi'N^3\psi+2r\dot{N}\dot{\psi}\psi^5-2r\ddot{\psi}N\psi^5
+2rN'\psi'N^2\psi\right.\\
&\left.+N'N^2\psi^2\right)=\dfrac{\kappa^2}{2}\left(\frac{p\overline{p}}{\psi^{8}}
+\chi'\overline{\chi}'-\dfrac{g^2}{4\pi \psi^4}\right).
\end{aligned}
\end{equation}
We split the above second order equation into two first order evolution equations. For this purpose we define
\begin{equation}\label{K}
K\equiv -\dfrac{6 \dot{\psi}}{N\psi}\,.
\end{equation}
$K$ is in fact the negative of the trace of the extrinsic curvature for spacelike hypersurfaces at constant time $t$. Equation \reff{Grr} can now be expressed as
\begin{equation}\label{kev}
\begin{aligned}
\frac{\dot{K}}{N}&=\dfrac{K^2}{2}-\dfrac{6\psi'}{\psi^5}\left(\dfrac{\psi'}{\psi}+\frac{1}{r}\right)
-\dfrac{3N'}{N\psi^4}\left(\dfrac{2\psi'}{\psi}+\dfrac{1}{r}\right)\\
&+\dfrac{3\kappa^2}{4}\left(\dfrac{\chi'\overline{\chi}'}{\psi^4}+\dfrac{p\overline{p}}{\psi^{12}}
-\dfrac{g^2}{4\pi \psi^8}\right)\,.
\end{aligned}
\end{equation}
Equations \reff{K} and \reff{kev} are evolution equations for $\psi$ and $K$ respectively.
The $G_{tt}$ equation is
\begin{equation*}
\dfrac{4}{r\psi^5}\left(3\dot{\psi}^2\psi^3r-2\psi'N^2-N^2r\psi''\right)=
\dfrac{\kappa^2N^2}{2}\left(\frac{p\overline{p}}{\psi^{12}}+\frac{\chi'\overline{\chi}'}{\psi^4}
+\dfrac{g^2}{4\pi \psi^8}\right)
\end{equation*}
and using $K$ can be cast in the form
\begin{equation}\label{c1}
-\dfrac{4}{\psi^5r}\left(2\psi'+r\psi''\right)
=\dfrac{\kappa^2}{2}\left(\dfrac{\chi'\overline{\chi}'}{\psi^4}+\dfrac{p\overline{p}}{\psi^{12}}
+\dfrac{g^2}{4\pi\psi^8}\right)
-\dfrac{K^2}{3}.
\end{equation}
Equation \reff{c1} is the energy constraint.
The $G_{rt}$ equation is
\begin{equation*}
\dfrac{4}{\psi^2N}\left(\dot{\psi}\psi'N-\dot{\psi}'N\psi+\dot{\psi}N'\psi\right)
=\kappa^2\frac{N}{2\psi^6}\left(\chi'\overline{p}+\overline{\chi}'p\right)\,
\end{equation*}
which can be expressed as
\begin{equation}\label{Grt}
\frac{K'}{3}=\frac{\kappa^2}{4\psi^6}\left(\chi'\overline{p}+\overline{\chi}'p\right)\,.
\end{equation}
Equation \reff{Grt} is the momentum constraint.
The $G_{\theta\theta}$ equation gives
\begin{equation*}
\begin{aligned}
&\dfrac{r}{\psi^2N^3}\left(4r\psi^5\dot{\psi}\dot{N}-8r\psi^4\dot{\psi}^2N-4r\psi^5\ddot{\psi}N+N'\psi^2N^2
-2r\psi'^2N^3+2r\psi'N^3\psi\right.\\
&\left.+2rN^3\psi\psi''+r\psi^2N^2N''\right)
=\dfrac{\kappa^2}{2}\left(\dfrac{p\overline{p}}{\psi^{8}}-\chi'\overline{\chi}'
+\dfrac{g^2}{4\pi \psi^4}\right)r^2\,.
\end{aligned}
\end{equation*}
Eliminating the time derivatives using the $G_{rr}$ equation \reff{Grr}, the above equation reduces to an ordinary differential equation for $N$
\begin{equation}\label{Gthetatheta}
\dfrac{2r}{\psi^2}\partial_r\left(\frac{\psi'}{r\psi^3}\right)+\dfrac{r}{N}\partial_r\left(\frac{N'}{r\psi^4}\right)
=-\dfrac{\kappa^2\chi'\overline{\chi}'}{\psi^4}+\dfrac{\kappa^2g^2}{4\pi\psi^8}\,
\end{equation}
The evolution equation \reff{kev} for $K$ can be made more numerically stable by combining it with \reff{c1} to obtain
\begin{equation}
\begin{aligned}
\frac{\dot{K}}{N}&=K^2-\frac{6\psi'}{\psi^5}\left(\dfrac{\psi'}{\psi}+\frac{1}{r}\right)
-\dfrac{3N'}{N\psi^4}\left(\dfrac{2\psi'}{\psi}+\dfrac{1}{r}\right)\\
&-\dfrac{6}{r\psi^5}\left(2\psi'+r\psi''\right)-\dfrac{3\kappa^2g^2}{8\pi\psi^8}.
\end{aligned}
\label{kev2} \end{equation}
To summarize, the evolution equations for the matter sector are \reff{e2} and \reff{p} for the pair ($\chi$,$p$) and \reff{M2} for $g$ respectively. The evolution equations for the gravitational sector are \reff{K} and \reff{kev2} for the pair ($\psi,K$). The gauge field $a$ and lapse function $N$ obey the ordinary differential equations \reff{g} and \reff{Gthetatheta} respectively. There is one constraint equation in the matter sector, namely ``Poisson's" equation \reff{M1}. In the gravitational sector, the energy and momentum constraint are given by equations \reff{c1} and \reff{Grt} respectively. Once the initial states and boundary conditions are specified, the evolution of the fields is unique.
\section{Initial states}
Let $\chi_1$ and $\chi_2$ be the real and imaginary part of the complex scalar field $\chi$. We begin by choosing the initial configuration for $\chi_1$, $\chi_2$, $\dot{\chi_1}$ and $\dot{\chi_2}$. It is convenient to pick a configuration such that the momentum constraint \reff{Grt} is trivially satisfied i.e. where the right hand side of \reff{Grt} is zero so that $K'=0$ initially. Since $K=0$ asymptotically, this implies that $K$ is initially zero everywhere. We use three different profiles that satisfy this property:
\begin{center}
\begin{tabular}{ p{1 cm} p{6 cm} p{4 cm} }
I:
&$\chi_1=\chi_2=\frac{8\lambda_1r^4}{\left(\lambda_1^2+r^2\right)^4}$
&$\dot{\chi_1}=-\dot{\chi_2}=\frac{\lambda_2\chi_1}{8\lambda_1}$\\[0.3 cm]
II:
&$\chi_1=\chi_2=\lambda_2\left(\text{tanh}\left(\lambda_1-r\right)+1\right)$
&$\dot{\chi_1}=-\dot{\chi_2}=\frac{\lambda_3\chi_1}{\lambda_2}$\\[0.3 cm]
III:
&$\chi_1=\frac{8\lambda_1r^4}{\left(\lambda_1^2+r^2\right)^4}\:\:\:\:;\:\:\:\:\chi_2=0$
&$\dot{\chi_1}=\dot{\chi_2}=0.$\\
\end{tabular}
\end{center}
Profile $III$ represents a real scalar field with no charge.
One is free to choose the values of the different $\lambda_i$ and these in turn determine the values of the conserved mass $M$ and charge $Q$. The initial states for the metric functions $\psi$ and $N$ and for the gauge field $a$ are obtained by solving three coupled second order differential equations: the energy constraint \eqref{c1}, the ordinary differential equation \eqref{Gthetatheta} and ``Poisson's" equation \eqref{M1}. These can be split into six first order differential equations:
\begin{equation}\label{SixEq}
\begin{aligned}
&A'=-\frac{\psi^5r^2\kappa^2}{8}\left(\frac{\chi'\overline{\chi}'}{\psi^4}
+\frac{\left(\dot{\chi}+iea\chi\right)\left(\dot{\overline{\chi}}-iea\overline{\chi}\right)}{N^2}
+\frac{C^2}{4\pi r^4\psi^8}\right)\\
&B'=\frac{N}{r}\left(\frac{6A}{r^3\psi^5}\left(1+\frac{A}{r\psi}\right)
+\frac{\kappa^2\left(\dot{\chi}+iea\chi\right)\left(\dot{\overline{\chi}}-iea\overline{\chi}\right)}{4N^2}
-\frac{\kappa^2}{4\psi^4}\left(3\chi'\overline{\chi}'-\frac{5C^2}{4\pi r^4\psi^4}\right)\right)\\
&C'=\frac{2\pi ir^2\psi^6}{N}\left(\chi\dot{\overline{\chi}}-\overline{\chi}\dot{\chi}-2iea\chi\overline{\chi}\right)\\
&\psi'=\frac{A}{r^2}\\
&N'=r\psi^4B\\
&a'=\frac{CN}{r^2\psi^2}
\end{aligned}
\end{equation}
where the last three equations introduce the new variables $A,B$ and $C$:
\begin{equation}
A\equiv r^2\psi' \quad; \quad B\equiv \frac{N'}{r\psi^4}\quad ; \quad C\equiv\frac{r^2\psi^2a'}{N}\,.
\end{equation}
The functions range from 0 to a large distance $R$, the outer computational boundary representing ``infinity". The spacetime is asymptotically flat and boundary conditions at $r\eq R$ that are consistent with this are $N(R)\eq 1$, $\psi(R)\eq 1$ and $B(R)\eq 0$. Gauge invariance allows us to set $a(R)=0$. The initial configuration is non-singular at $r\eq 0$ and this implies that $A(0)\eq 0$ and $C(0)\eq 0$. To solve equations \reff{SixEq} with the above boundary conditions, we use a shooting method with a point located between the origin and $R$. From the boundary conditions, two fields are known at the origin and four fields are known at $R$. We make an educated guess as to the value of each field at the missing end (a total of six numbers.) We then evolve the fields from both ends toward the middle point using fourth order Runge-Kutta. The goal is to modify the six unknown numbers until the fields match in the middle. This can be achieved in a rather straightforward way using a six dimensional Newton method \cite{Press}. The system of linear algebraic equations can be solved using an lower/upper matrix decomposition \cite{Press}. A first estimate for the fields can be obtained by solving the equations for $\dot{\chi}=0$ which applies to a scalar field. The fields can be made to fit at the middle point to within a few parts in $10^{15}$. We then increase $\dot{\chi}$ by a small amount and use the preceding values of the fields to estimate the new ones. We repeat the procedure until a high enough charge to mass ratio has been obtained. It is worthwhile to note that if the fields match at the middle point, it can be shown that their derivatives also match.
\section{Expressions for the conserved charge $Q_{tot}$ and mass $M_{tot}$\\ in isotropic coordinates}
In this section we obtain expressions in isotropic coordinates for the total (conserved) charge $Q_{tot}$ and mass $M_{tot}$. During the simulation, these quantities should remain constant and this is used to monitor the simulation. The mass $M$ and charge $Q$ of the black hole itself is significantly less (magnitude wise) than $M_{tot}$ and $Q_{tot}$ respectively because a considerable amount of charge and mass is expelled during the collapse process. $M$ and $Q$ will be evaluated in the results section with a different expression. For the analytical stationary RN metric \reff{RN} or \reff{IsoRN}, there is no outgoing matter wave so that $M$ and $Q$ are equal to $M_{tot}$ and $Q_{tot}$ respectively. We leave them expressed in terms of $M$ and $Q$ because in the results section they are evaluated with our black hole values of $M$ and $Q$ (not with our values of $M_{tot}$ or $Q_{tot}$ as these do not correspond to the mass and charge of the black hole respectively).
The total conserved charge $Q_{tot}$ is given by the general formula \cite{Poisson}
\beq
Q_{tot}=\dfrac{1}{8\pi}\oint_S F^{\alpha\beta}\,dS_{\alpha\beta}
\eeq{EQ1}
where $S$ is a closed two-surface bounding the charge distribution and $dS_{\alpha\beta}$ is the two dimensional surface element given by \cite{Poisson}
\begin{equation}
dS_{\alpha\beta}=-2n_{[\alpha}r_{\beta]}\sqrt{\sigma}d^2\theta \,.
\end{equation}
$n^{\alpha}$ and $r^{\alpha}$ are the timelike and spacelike normals to the two-surface and the square brackets denote antisymmetrization: $n_{[\alpha}r_{\beta]}= (n_{\alpha}r_{\beta}-n_{\beta}r_{\alpha})/2$. The induced surface element on $S$ is $\sqrt{\sigma} \,d^2\theta$ where $\sigma$= det($\sigma_{AB}$) where $\sigma_{AB}$ is the induced metric on $S$ (A,B is either $\theta$ or $\phi$). For the metric \reff{isometric}, we obtain $n_\alpha=\left(-N,0,0,0\right)$ and $r_\beta=\left(0,\psi^2,0,0\right)$. The only non-zero components of $F^{\alpha\beta}$ are
$F^{tr}=-F^{rt}= \tfrac{a'}{N^2\,\psi^4}$. The only non-zero components of $dS_{\alpha\beta}$ are $dS_{tr}\eq -dS_{rt}\eq N\,\psi^6 r^2 \sin\theta d\theta d\phi$. Putting these results together we obtain an expression for the total charge:
\begin{equation}
Q_{tot}=\dfrac{\psi^2r^2a'}{N}\big|_{r=R}=a'\,R^2\big|_{r=R}
\label{QQ}\end{equation}
where we used the fact that asymptotically, in the large $R$ (infinite) limit, $N$ and $\psi$ approach unity. Using the constraint equation \reff{M1}, one can express the charge $Q_{tot}$ in the integral form
\beq
Q_{tot}= \int_0^R 2\pi ie\,\left(\chi\,\overline{p}-\overline{\chi}\,p\right) \,r^2 dr \,.
\eeq{QInt}
From the above expression, one can define an effective charge density
\beq
\rho= \dfrac{ie}{2}\,\left(\chi\,\overline{p}-\overline{\chi}\,p\right)\,.
\eeq{rho2}
The total (conserved) ADM mass $M_{tot}$ for an asymptotically flat spacetime is given by \cite{Poisson}
\beq
M_{tot} = -\frac{1}{8\pi} \oint_{S} (k-k_0) \sqrt{\sigma}\,d^2\theta
\eeq{ADMmass}
where $S$ is the two-sphere at spatial infinity. $k$ is the trace of the extrinsic curvature of $S$ embedded in $\Sigma_t$, the three-dimensional spacelike hypersurface at constant $t$. $k_0$ is the trace of the extrinsic curvature of $S$ embedded in flat spacetime. $k$ and $k_0$ can be readily evaluated and one obtains $k-k_0 =4\psi'/\psi^3$. The mass $M_{tot}$ is equal to
\beq
M_{tot} = -2 r^2 \psi' \,\psi \big|_{r=R} = -2 R^2 \,\psi' \big|_{r=R}
\eeq{ADMMass}
where we used that $\psi=1$ asymptotically.
Using \reff{c1} one can express the mass in the integral form
\begin{equation}
M_{tot} =\dfrac{\kappa^2}{4}\int_0^R\left(\dfrac{\chi'\overline{\chi}'}{\psi^4}+\dfrac{p\overline{p}}{\psi^{12}}
+\dfrac{g^2}{4\pi\psi^8}-\dfrac{2K^2}{3\kappa^2}\right)r^2\psi^5dr.
\end{equation}
\section{Free energy}
There are arguments, based on the Euclidean action \cite{Hawking2}, for relating the negative of the total Lagrangian of a stationary black hole to its free energy. As far as we know, to date, there is no direct analytical proof of this; it has only been tested numerically during gravitational collapse to a Schwarzschild black hole \cite{Khlebnikov, C-E}. One of the goals of this paper is to test this association numerically for the case of charged collapse where the relevant thermodynamic potential is the Gibbs free energy \cite{Hawking2, Kiritsis}. To do so, we first need to determine an expression for the Lagrangian.
\subsection{Gravitational and Matter Lagrangian}
The gravitational action $S_G[g]$ is a sum of the Einstein-Hilbert term $S_H[g]$, a boundary term $S_B[g]$ and a nondynamical term $S_0$ \cite{Poisson}:
\beq
S_{G}[g] = S_H[g]+ S_B[g] + S_0 = \int L_G \,dt
\eeq{SG}
where $L_G$ is the gravitational Lagrangian and
\beq
S_H[g]= \frac{1}{16\pi} \int \sqrt{-g}\,R \, d^3x\, dt \,.\\
\eeq{SH}
The Ricci scalar in isotropic coordinates is given by
\beqa
R &=& \frac{2}{rN^3\psi^5} \big( 18r\dot{\psi}^2\,N\psi^3 + 6r\ddot{\psi}N\psi^4 - 8\psi'N^3 - 4r\psi''N^3 \\
& & {} - 6r\dot{N}\dot{\psi}\psi^4 - rN''N^2\psi - 2rN'\psi'N^2 - 2N'N^2\psi \big)\nonumber\,.
\eea
The Ricci scalar is not the Lagrangian density as it contains second derivatives of the metric tensor. Besides the Hilbert action $S_H[g]$, one also requires the boundary term $S_B[g]$ to obtain Einstein's field equations \cite{Poisson}. $S_0$ ensures that the action $S_G$ is finite and equal to zero in flat spacetime. We do not need to spell out $S_B$ (or $S_0$) to determine the gravitational Lagrangian. As in \cite{Khlebnikov}, $L_G$ can be obtained simply by integrating out by parts the second derivative terms appearing in $R$ and this yields
\beq
L_{G} = \frac{1}{4} \int_0^R \Big( 8r^2\psi'N'\psi + 8r^2N\psi'^2 - \frac{24r^2\dot{\psi}^2\psi^4}{N} \Big)\, dr.
\eeq{LLG}
Note that $L_G$ contains only first derivatives of the metric functions. The matter Lagrangian is given by the integral of \reff{L_Matter2}:
\beq
L_m =\int \mathcal{L}_m \sqrt{-g}\,d^3x = 4\,\pi \int_0^R \left(\frac{p\overline{p}}{2\psi^{12}}-\dfrac{\chi'\,\overline{\chi}'}{2\psi^4}+\dfrac{g^2}{8\pi\psi^8}\right) \psi^6 \,|\!N\!| \,r^2 \,dr\,.
\eeq{Lm22}
The total Lagrangian is the sum $L_m +L_G$.
\subsection{Gibbs free energy of the Reissnner-Nordstr\"{o}m black hole}
We can easily calculate the Gibbs free energy $G \eq E\m TS\m \Phi_H Q$ of the RN black hole ($E$ is the ADM mass $M$, $T$ and $S$ are the temperature and entropy respectively and $\Phi_H$ the electrostatic potential at the horizon). The temperature is given by $T=\hbar\,\kappa/2\pi$ where $\kappa$ is the surface gravity of the black hole and $S= A/4\hbar$ where $A$ is the area of the event horizon. For a RN black hole, $\kappa= \sqrt{M^2-Q^2}/{r'_+}^2$ and the area of the horizon is $A\eq 4\pi {r'_+}^2$, where $r'_{+}= M + \sqrt{M^2-Q^2}$ is the (areal) radius of the outer horizon. Therefore the product $T\,S$ is equal to $\sqrt{M^2-Q^2}/2$. The electrostatic potential at the horizon is given by $\Phi_H=Q\slash r'_{+}$ \cite{Hawking2, Kiritsis}. The Gibbs free energy of the RN black hole is equal to
\begin{equation}
\begin{split}
G&=M-\dfrac{1}{2}\sqrt{M^2-Q^2}- \dfrac{Q^2}{M+\sqrt{M^2-Q^2}}\\
&=\dfrac{1}{2}\sqrt{M^2-Q^2}.
\end{split}
\eeq{F_RN}
For $Q\eq 0$, expression \reff{F_RN} reduces to the (Helmholtz) free energy $M/2$ of a Schwarzschild black hole \cite{Khlebnikov, C-E}. Note that though $\hbar$ appears in $T$ and $S$, it does not appear in the product $TS$ and hence does not appear in the expression \reff{F_RN} for the Gibbs free energy.
\subsubsection{Analytical formulas for the exterior and interior contribution to the free energy}
There is an interior and exterior contribution to the gravitational Lagrangian $L_G$, the integral given by \reff{LLG}. The analytical expressions for $\psi$ and $N$ in the exterior static region can be extracted from metric \reff{IsoRN}:
\beq
N= \dfrac{\Big(1-\dfrac{M^2-Q^2}{4\,r^2}\Big)}{\Big(1 +\dfrac{M}{r} +\dfrac{M^2-Q^2}{4r^2}\Big)}\quad;\quad \psi=\Big[1 +\dfrac{M}{r} +\dfrac{M^2-Q^2}{4r^2}\Big]^{1/2} \,.
\eeq{NPsi}
Inserting the above functions into \reff{LLG} and integrating from an isotropic radial coordinate of zero to the outer horizon $r_+= \sqrt{M^2-Q^2}/2$ yields the exterior contribution $L_{G_{ext}}$ to the gravitational Lagrangian
\beq
L_{G_{ext}}= \frac{1}{4} \int_{r_+}^{\infty} \Big( 8r^2\psi'N'\psi + 8r^2N\psi'^2 \Big)\, dr = -\dfrac{1}{2}(M+\sqrt{M^2-Q^2})\,.
\eeq{Lext}
There is no charge residing in the exterior of the RN black hole; there is an electric field but no scalar field. The contribution to the matter Lagrangian \reff{Lm22} in the exterior stems entirely from the electromagnetic part,
\beq
L_{M_{ext}}= \dfrac{1}{2} \int_{r_+}^{\infty}\frac{g^2}{\psi^2}N r^2 dr\,=\dfrac{Q^2}{2(M+\sqrt{M^2-Q^2})}\,.
\eeq{Lext}
The exterior contribution to the Gibbs free energy is the negative of the total exterior Lagrangian,
\beq
G_{ext}= - L_{G_{ext}} - L_{M_{ext}} = \sqrt{M^2-Q^2}\,.
\eeq{Gext}
The interior contribution is therefore equal to
\beq
G_{int}= G- G_{ext}= -\dfrac{1}{2}\,\sqrt{M^2-Q^2}=-TS=-r_+
\eeq{Fint}
where $G$ is the Gibbs free energy of the RN black hole given by \reff{F_RN}. As in the Schwarzschild black hole \cite{Khlebnikov, C-E}, the interior contribution to the free energy is equal to the negative of the product of temperature and entropy i.e. $G_{int}=-T\,S$. We will see that the negative dip in the interior occurs in a thin shell near the radius of the horizon, as in the Schwarzschild case \cite{B-E}. Our numerical results in the interior will be compared to the analytical expression \reff{Fint}.
\section{Numerical Results}
\subsection{Code and initial state}
We work in geometrized units where $G\eq c \eq 1$. We also set Coulomb's constant to unity. In these units radius, mass, energy, time and charge have units of length. The coupling constant $e$ has units of inverse length. Its value is arbitrary (there is no experimental value for it in our model). We set it equal to unity and leave the scale unspecified (one can work in centimetres, metres, etc.). One unit of time is the time it takes light to travel a radial distance of one unit in flat spacetime\footnote{If the scale were specified to be $3\times 10^{5}$km, then one unit of time would correspond to one second.}. For the spatial grid, we introduce a new coordinate $x$ such that $r=2x/(1-x)$. We use a constant grid in $x$, with $x$ ranging between $0$ and $1$. This maintains a larger concentration of points in the interior than the exterior. The spatial and time increment are taken to be $\Delta x \eq \Delta t \eq 5\times 10^{-5}$. The fields that are governed by an evolution equation are evolved via a fourth order Adams-Bashforth-Moulton (ABM) scheme \cite{Press}. The fields $N$ and $a$ are governed by an ordinary differential equation and are obtained by iterating backwards starting from infinity using equations \reff{Gthetatheta} and \reff{M1} and their boundary conditions respectively. This is performed at each time step using again ABM. The simulation was carried out for the three different initial profiles discussed in the section on the initial states. The results for profiles I, II and III are presented in tables 1, 2 and 3 respectively. $M$ and $Q$ represent the mass and charge of the black hole itself, whereas $M_{tot}$ and $Q_{tot}$ represent the total (conserved) mass and charge. We will see shortly how $M$ and $Q$ are evaluated. $-L_{int}$ and $G_{int}$ represent the interior Lagrangian and Gibbs free energy at late times respectively. The discussion that follows will focus on one initial state with profile $I\!I$. This will allow us to highlight the essential features of the thermodynamics that are common to all cases. The initial state for profile $I\!I$ amounts to specifying the values of three constants $\lambda_1$, $\lambda_2$ and $\lambda_3$. We will discuss the middle case in table 2: $\lambda_1 \eq 2$, $\lambda_2 \eq 0.11$ and $\lambda_3 \eq 0.032$. The total (conserved) ADM mass and charge corresponding to this initial state is $M_{total}\eq 1.241$ and $Q_{total}\eq -0.842$ respectively\footnote{Switching the sign of $\chi_1$ and $\chi_2$ in the initial state leads to a positive charge. The sign has no bearing on the thermodynamics.}. Note that this differs considerably from the mass $M=0.721$ and charge $Q=-0.215$ of the black hole itself because a significant amount of mass and charge are expelled in the collapse process. The total mass and charge are useful for monitoring the accuracy of the code and are checked at each time step. The ADM mass begins to deviate from its original value before the total charge. For the above initial state, the numerical results and plots are obtained up to an evolution time of $t\eq 23.5$, just below $5\%$ ADM mass deviation. The ADM mass is known to be very sensitive to tiny deviations in the metric and matter functions \cite{C-E} i.e. the functions can be evolving well even when the ADM mass begins to deviate from its original value. For example, the numerical curve for the metric function $\psi$ continues to approach the analytical curve of the RN exterior metric even at $5\%$ deviation from the ADM mass.
\subsection{Plots of metric and matter functions, mass and charge of black hole}
We made plots of all the relevant functions. The evolution of the conformal factor $\psi$ and lapse $N$ are shown in figures~\ref{psi} and~\ref{n} respectively. We denote $r_+(t)$ as the isotropic radius where the lapse $N$ crosses zero at a time $t$. Black hole formation is taken to coincide with $N$ crossing zero \cite{Finelli1, Finelli2, Khlebnikov} since radially outgoing (or ingoing) null geodesics in the interior ($r<r_+(t)$) do not cross the $N\eq 0$ spacelike two-surface as it evolves (see figure~\ref{geodesics}). At very late times the $N=0$ spacelike two-surface is identified as the apparent horizon of the stationary RN black-hole (for more details see \cite{B-E}). The radius $r_+(t)$ increases with time and has a numerical value of $0.344$ at $t=23.5$; this is the radius of the apparent horizon. In the interior at late times, the lapse $N$ shows almost no dependence on $r$ and approaches zero asymptotically from below (i.e. $\dot{N} \to 0$ as $N\to 0$). The conformal factor $\psi$ has its peak value at $r_{+}$. In the interior, it also has no spatial dependence and approaches zero asymptotically from above i.e. $\dot{\psi} \to 0$ as $\psi\to 0$ (see figure~\ref{psidot}). As in \cite{Brady, Hod, Piran} we do not observe an inner horizon; the spacetime structure in the interior resembles that of the uncharged case \cite{C-E}. The absolute value of the complex scalar field $\chi$ is shown in figure~\ref{chi}. It approaches a constant value inside $r_{+}$ and the only region where the time or radial derivatives are high is near the radius $r_{+}$. Note the outgoing matter wave.
In the RN solution, the electromagnetic-field tensor is given by $F^{tr'}=Q\slash r'^2$ \cite{Poisson} where $r'$ is the areal radius. A short calculation yields $g(r)=Q\slash r^2$ where $g$ is given by \reff{g} and $r$ is the isotropic radius. This, in combination with the relation ${r_+}^2=(M^2-Q^2)\slash 4$, allows to determine the charge and mass of the black hole:
\beq
\begin{aligned}
&Q=r_{+}^2\,g(r_{+})\\
&M=\sqrt{4{r_{+}}^2+Q^2}\,.
\end{aligned}
\eeq{MQ}
The above are evaluated at the latest time.
\subsection{Electrostatic potential and charged shell}
The field $a$ is shown in figure~\ref{a}. Inside $r_{+}$, it is almost perfectly constant in space and its value decreases with time. Between $r_{+}$ and the position of the outgoing wave, it decreases smoothly as $r$ increases. On the other side of the outgoing wave, it decreases faster and this is consistent with the fact that the outgoing wave contains charge. This is what would be expected from basic classical electromagnetism for a situation where two charged shells are concentric and the outside one is moving outward. It can be seen from figure~\ref{rho} that the charge density of the black hole becomes increasingly concentrated near $r_{+}$ as time progresses. At late times it is concentrated just inside the apparent horizon (located at isotropic radius $r=0.344$). In short, at late times one has a charged shell \cite{B-E} with a constant potential $a$ inside. The value of $-a$ extracted at the apparent horizon in our numerical simulation at late times is $-0.174$. The value of $\Phi_H=Q/r'_+$ is $-0.153$ where $r'_+ = M+\sqrt{M^2-Q^2}$ is the areal radius at the horizon. The percentage difference between $-a$ and $\Phi_H$ is around $12\%$, which is explained by the fact that the outgoing wave hasn't reached infinity yet. Note that $\Phi_H$ should be compared to $-a$ since $\Phi_H\equiv-A_{\mu}\,\xi^{\mu}|_H=-A_t|_H$ \cite{Poisson} where $|_H$ means evaluated at the horizon. Here $\xi^{\mu}$ is the Killing vector of the stationary RN black hole and in our coordinate system it is given by $\xi^{\mu}=(1,0,0,0)$. Recall that $a=A_t$ in our numerical simulation. What is important here is that the charge accumulates in a shell at the radial location of the horizon \cite{B-E} and has an electrostatic potential which is numerically close to $\Phi_H$. This provides us with a clear physical interpretation of the work term $\delta W=\Phi_H \delta Q$ in the first law of black hole thermodynamics: it is the work needed to bring a charge $\delta Q$ to the charged shell located at the horizon. Note that though the charged shell is located at the radius of the horizon, the proper radial distance between the shell and the origin decreases towards zero with time because the conformal factor $\psi$ approaches zero in the interior. The charged shell is effectively ``collapsing towards the center" (see \cite{B-E} for details).
\subsection{Lagrangian and free energy}
\subsubsection{The interior region}
Our goal here is to compare the negative of the total interior Lagrangian $-L$ (gravity + matter) to the interior Gibbs free energy $G_{int}$. The first thing to note is that the matter Lagrangian, plotted in figure~\ref{lm}, makes basically zero contribution to the interior at late times. The interior Lagrangian is entirely gravitational. The integral of the negative of the gravitational Lagrangian from the origin to a given $r$ is shown in figure~\ref{lg}. There is a clear negative dip just inside the apparent horizon that becomes thinner and closer to the horizon with time. Since $G_{int}=-TS$ this suggests that the entropy of the charged black hole accumulates at the horizon and that it is gravitational in origin. In the interior, the metric is not static near the horizon (see figure~\ref{psidot}) but dynamical so that the entropy is associated with the dynamical interior in accord with some previous analytical studies \cite{Vaz1,Vaz2, E-C}. This was also observed numerically for the Schwarzschild case \cite{Khlebnikov,C-E}. The numerical value of the minimum in figure~\ref{lg} is plotted in figure~\ref{lgmin} as a function of time together with the interior Gibbs free energy $G_{int}=-r_+$. Note how the two graphs approach each other with time. At the late time $t=23.5$, the gravitational minimum (which we label $-L_{int}$) is $-0.306$ and $G_{int}=-0.344$, for a difference of $10.9\%$ (see table 2). It is clear from the graphs that the two values would continue to approach each other if the code could evolve further in time. The ADM mass has deviated by $5\%$ at this point. This is due to sharp changes in the gradients in the near-horizon region.
We verified that the grid size determines how far the simulation runs before the ADM mass begins to deviate from its original value. This is shown in table 4. We used the following sequence for the number of points: $\{1250,2500,5000,10000,20000\}$ with the corresponding step sizes shown in table 4. The times at which the ADM mass deviated by $1\%$ and the times by which it deviated by $5\%$ are listed. The table shows clearly that decreasing the step size increases the time evolution. The numerical results and plots presented above are for the highest resolution $\Delta x =5.0 \times 10^{-5}$ and are taken at $t=23.5$ just below $5\%$ ADM mass deviation. Going beyond this resolution increases significantly the computing time so that it no longer becomes practical.
The results for all three profiles with different initial states are shown in tables 1, 2 and 3. Note that it is possible to have the total charge $|Q_{tot}|$ be greater than the total mass $M_{tot}$ (see table 1). However, in all cases, $|Q|$ is less than the mass $M$ and there are no naked singularities. The discrepancy between $-L_{int}$ and $G_{int}$ is shown for all cases. As one can see, the discrepancy depends slightly on the initial state. However, in all probability, this difference is not due to fundamental physical reasons. It is simply that the initial state affects when the ADM mass starts to deviate from its original value. A higher resolution would decrease the discrepancy for all initial states and it is expected that all should yield the same thermodynamics. The matter Lagrangian tends to zero in the interior in all cases.
\subsubsection{Extending time evolution in the exterior region: matter Lagrangian and exterior RN metric}
The results discussed so far are the robust numerical results of this paper. The exterior region has a large outgoing matter wave which can clearly be seen in figure~\ref{lm}. This limits the ability to extract unambiguous numerical results in the exterior. Nonetheless, we still want to check two things in the exterior, at least qualitatively. We would like to verify that due to the presence of the electric field in the exterior, the matter Lagrangian in the exterior does not settle to zero in contrast to the Schwarzschild case \cite{Khlebnikov,C-E}. Secondly, we know the analytical form of the RN metric in the exterior. As a consistency check, we would like to verify that numerically the metric in the exterior approaches it.
Unlike the near-horizon region, the exterior does not contain regions with very sharp changes in gradients. It is therefore possible to extend the evolution time in the exterior by considering only the points outside of a cutoff where one sets some boundary conditions. The cutoff is chosen here to be beyond $r_+$ at $r=0.5$. The outside evolution starts at $t=23.5$ and we can now relax the time step to $\Delta t =10^{-4}$. The functions that need to be specified at the cutoff are $\psi$, $g$ and the real and imaginary part of $\chi$. However, $\psi$ and $g$ had already plateaued to a constant at the cutoff at $t=23.5$; we therefore set them equal to these respective constants on the boundary. However, $\chi$ is still evolving at $t=23.5$. As time increases, the outgoing wave dissipates and we expect the outside to reach the exterior of the RN black hole which contains no charge density and a zero stress-energy tensor for the scalar field (there is a non-zero stress-energy tensor due to the electric field). This implies that $\chi'$ should tend to zero outside. Since $\chi$ is zero at infinity, the value of $\chi$ at the cutoff should tend to zero. All we know is that it should approach zero but not how it approaches zero as a function of time. Fortunately, the metric is mostly affected by the outgoing wave which is not sensitive to the boundary condition at the cutoff. We therefore impose some time-dependent boundary condition for $\chi$ at the cutoff that leads to a smooth evolution. Figure~\ref{lmat2} shows the matter Lagrangian in the exterior at very large times. It is important to read this graph properly. The wavy part consists of the matter wave which is moving outwards and decreasing in amplitude. What is important is the portion before the wave. From $t=40$ to $t=120$ it remains constant; the matter Lagrangian is clearly plateauing towards a non-zero value. This is due to the external electric field. The conformal factor $\psi$ is plotted in figure~\ref{psiE} together with the analytical form of $\psi$ in the exterior RN metric. At $t=120$, the two curves are very close to each other confirming that the exterior is indeed approaching the Reissner-Nordstr\"{o}m solution.
\section{Conclusion}
In this paper, we showed that the interior Gibbs free energy during charged collapse stems entirely from the gravitational sector and accumulates in the dynamical region inside and near the horizon. Numerically, we showed that the interior Lagrangian agrees with the analytical expectation $G_{int}$ for the interior Gibbs free energy to within roughly $10\%$ depending on the initial state. We observe the formation of a charged shell just inside the horizon with a constant electrostatic potential inside that matches $\Phi_H$ to within $12\%$. The work term $\delta W=\Phi_H\delta Q$ in the first law of black hole thermodynamics can be interpreted as the work required to bring a charge $\delta Q$ to the charged shell at the horizon.
A consistent dynamical picture of black hole thermodynamics is now emerging from numerical studies of gravitational collapse in isotropic coordinates. First, the Helmholtz free energy $F\eq E \m TS$ in the Schwarzschild case \cite{Khlebnikov, C-E} makes an interior contribution of $F_{int}\eq -TS$ and the Gibbs free energy $G\eq E\m TS -\Phi_H Q$ in the charged case makes also an interior contribution $G_{int}\eq -TS$. The interior contribution in both cases is equal to $-TS$ even though we are dealing with different free energies. Secondly, the interior contribution in both cases stems from the gravitational sector. Thirdly, in both cases it accumulates inside near the horizon. Fourthly, the region where it accumulates is dynamical/non-static. This strongly suggests the following: black hole entropy is gravitational entropy and accumulates in the dynamical interior near the horizon.
It is now important to check if gravitational collapse to a Kerr black hole obeys these features. Such a study would be a major numerical undertaking. The number of equations to evolve would increase but more importantly, a two dimensional spatial grid would now be required, increasing massively computation time. The excision technique used in numerical simulations of collapse of a rotating neutron star to a Kerr black hole \cite{Baiotti, Hawke} would not be useful in our case. The interior region where the sharp changes in gradients occur in our simulations is precisely the region where we need to extract pertinent numerical results.
Besides the Kerr black hole, it would be interesting to investigate the thermodynamics of the black holes recently obtained during numerical studies of collapse of a k-essence scalar field \cite{Garfinkle,Gabor}. These fields are exotic in the sense that they have a non-canonical kinetic term \cite{Mukhanov1,Mukhanov2}. It is therefore worthwhile to investigate whether such black holes obey the thermodynamic features discussed above. The equations of motion for k-essence scalar fields coupled to gravity follow from a Lagrangian and this implies that a thermodynamic study should be possible. In particular, one would like to determine if the matter Lagrangian tends towards zero in the interior during collapse as with all previous types of matter investigated to date.
\clearpage
\begin{table}[ht!]
\begin{center}
\begin{tabular}{ p{1.0 cm} p{1.0 cm} p{1.3 cm} p{1.3 cm} p{1.3 cm} p{1.3 cm} p{1.3 cm} p{1.3 cm} p{2.0 cm} }
$\lambda_1$ & $\lambda_2$ & $M_{tot}$ & $Q_{tot}$ & $M$ & $Q$ & $-L_{int}$& $G_{int}$ &Percentage discrepancy\\ \hline
1.85 & 8 & 1.364 & -1.234 & 0.644 & -0.170 & -0.271 & -0.311 & 12.6 \\
1.85 & 12 & 1.669 & -1.959 & 0.638 & -0.237 & -0.258 & -0.296 & 12.8 \\
1.8 & 12 & 1.897 & -2.191 & 0.754 & -0.280 & -0.304 & -0.350 & 13.1 \\
\end{tabular}
\caption{Results for profile I}
\end{center}
\end{table}
\begin{table}[ht!]
\begin{center}
\begin{tabular}{ p{0.7 cm} p{0.7 cm} p{0.7 cm} p{1.2 cm} p{1.2 cm} p{1.2 cm} p{1.2 cm} p{1.2 cm} p{1.2 cm} p{2.0 cm} }
$\lambda_1$ & $\lambda_2$ & $\lambda_3$ & $M_{tot}$ & $Q_{tot}$ & $M$ & $Q$ & $-L_{int}$ & $G_{int}$ & Percentage discrepancy\\ \hline
2 & 0.11 & 0.016 & 1.105 & -0.403 & 0.706 & -0.110 & -0.315 & -0.349 & 9.7 \\
2 & 0.11 & 0.032 & 1.241 & -0.842 & 0.721 & -0.215 & -0.306 & -0.344 & 10.9 \\
2 & 0.11 & 0.048 & 1.488 & -1.357 & 0.754 & -0.307 & -0.295 & -0.345 & 14.5 \\
\end{tabular}
\caption{Results for profile II}
\end{center}
\end{table}
\begin{table}[ht!]
\begin{center}
\begin{tabular}{ p{1.0 cm} p{1.3 cm} p{1.3 cm} p{1.3 cm} p{1.3 cm} p{2.0 cm} }
$\lambda_1$ & $M_{tot}$ & $M$ & $-L_{int}$ & $G_{int}$ & Percentage discrepancy \\ \hline
1.50 & 1.102 & 0.334 & -0.295 & -0.334 & 11.5 \\
1.55 & 0.974 & 0.279 & -0.244 & -0.279 & 12.5 \\
1.60 & 0.867 & 0.236 & -0.203 & -0.236 & 14.0 \\
\end{tabular}
\caption{Results for profile III}
\end{center}
\end{table}
\begin{table}[ht!]
\begin{center}
\begin{tabular}{ p{3.2 cm} p{2.21 cm} p{3.3 cm} p{3.3 cm} }
Number of spatial steps &$\Delta t$ &Time of deviation of 1\% of $M_{tot}$&Time of deviation of 5\% of $M_{tot}$\\
\hline
1250 & 0.0008 & 14.3 & 17.9 \\
2500 & 0.0004 & 16.4 & 19.4 \\
5000 & 0.0002 & 18.3 & 20.9 \\
10000 & 0.0001 & 20.1 & 22.3 \\
20000 & 0.00005 & 21.7 & 23.7 \\
\end{tabular}
\caption{Evolution time as a function of gridsize for profile $II$ with $\lambda_1=2$, $\lambda_2=0.11$ and $\lambda_3=0.032$}
\end{center}
\end{table}
\clearpage
\begin{figure}[tbp]
\begin{center}
\includegraphics[scale=.38, draft=false, trim=1.5cm 1.5cm 2.5cm 2cm, clip=true]{geo.eps}
\end{center}
\caption{\label{geodesics} The position of outgoing radial light-like geodesics is shown as a function of time. They are released at $t=12$. The radius $r_+$ where $N=0$ is also shown as a function of time (the top black line). The null geodesics are unable to cross $r_+$ (see also \cite{B-E}).}
\end{figure}
\clearpage
\begin{figure}[tbp]
\begin{center}
\includegraphics[scale=.38, draft=false, trim=1.5cm 1.5cm 2.5cm 2cm, clip=true]{psi.eps}
\end{center}
\caption{\label{psi} Space profile of the metric field $\psi$ for different times.}
\end{figure}
\clearpage
\begin{figure}[tbp]
\begin{center}
\includegraphics[scale=0.38, draft=false, trim=1.5cm 1.5cm 2.5cm 2cm, clip=true]{N.eps}
\end{center}
\caption{\label{n} Space profile of the lapse function $N$ for different times. $r_{+}$ is the radius where $N$ crosses zero at late times. $N$ is negative in the interior and approaches zero from below.}
\end{figure}
\clearpage
\begin{figure}[tbp]
\begin{center}
\includegraphics[scale=.37, draft=false, trim=1.5cm 1.5cm 2.5cm 2cm, clip=true]{chi.eps}
\end{center}
\caption{\label{chi} Space profile of the absolute value of the matter field $\chi$ for different times. Note the propagation of an outgoing matter wave in the exterior region.}
\end{figure}
\clearpage
\begin{figure}[tbp]
\begin{center}
\includegraphics[scale=.38, draft=false, trim=1.5cm 1.5cm 2.5cm 2cm, clip=true]{a.eps}
\end{center}
\caption{\label{a} Potential $a$ for different times. Inside the horizon, $a$ is almost constant, consistent with a charged shell configuration at the apparent horizon. The change at large $r$ in the profile of $a$ corresponds to the position of the outgoing wave.}
\end{figure}
\clearpage
\begin{figure}[tbp]
\begin{center}
\includegraphics[scale=.38, draft=false, trim=1.5cm 1.5cm 2.5cm 2cm, clip=true]{rho.eps}
\end{center}
\caption{\label{rho} The charge density at different times. As the system evolves, it becomes more and more concentrated towards the inside of the apparent horizon. At late times one has a charged shell in the vicinity of the horizon.}
\end{figure}
\clearpage
\begin{figure}[tbp]
\begin{center}
\includegraphics[scale=.38, draft=false, trim=1.5cm 1.5cm 2.5cm 2cm, clip=true]{lg.eps}
\end{center}
\caption{\label{lg} The gravitational Lagrangian accumulation at different times. Note the negative dip in the thin shell just inside and near the apparent horizon. There is a disturbance in the outgoing wave region.}
\end{figure}
\clearpage
\begin{figure}[tbp]
\begin{center}
\includegraphics[scale=.38, draft=false, trim=1.5cm 1.5cm 2.5cm 2cm, clip=true]{psidot.eps}
\end{center}
\caption{\label{psidot} The time-derivative of the metric function $\psi$. $\psi$ is static in the exterior and its time-derivative is decreasing towards zero in the interior volume. However, it is not static inside and near the horizon, precisely the region where the interior free energy accumulates.}
\end{figure}
\clearpage
\begin{figure}[tbp]
\begin{center}
\includegraphics[scale=.38, draft=false, trim=1.5cm 1.5cm 2.5cm 2cm, clip=true]{lm.eps}
\end{center}
\caption{\label{lm} The matter Lagrangian accumulation at different times. Note that there is no disturbance at the apparent horizon at late times. The disturbance is around the outgoing wave.}
\end{figure}
\clearpage
\begin{figure}[tbp]
\begin{center}
\includegraphics[scale=.38, draft=false, trim=1.5cm 1.5cm 2.5cm 2cm, clip=true]{lgmin.eps}
\end{center}
\caption{\label{lgmin} The minimum in the Lagrangian accumulation and the interior Gibbs free energy $G_{int}$ as a function of time. The interior Gibbs free energy is given by $-r_{+}$.}
\end{figure}
\clearpage
\clearpage
\begin{figure}[tbp]
\begin{center}
\includegraphics[scale=.38, draft=false, trim=1.5cm 1.5cm 2.5cm 2cm, clip=true]{lmat2.eps}
\end{center}
\caption{\label{lmat2} The matter Lagrangian accumulation in the exterior plotted at different times.}
\end{figure}
\clearpage
\begin{figure}[tbp]
\begin{center}
\includegraphics[scale=.38, draft=false, trim=1.5cm 1.5cm 2.5cm 2cm, clip=true]{psiext.eps}
\end{center}
\caption{\label{psiE} The conformal factor $\psi$ at $t=120$ is compared to its theoretical expectation in the exterior.}
\end{figure}
\clearpage
\section*{Acknowledgments}
A.E. acknowledges support from a discovery grant of the National
Science and Engineering Research Council of Canada (NSERC). H.B.
acknowledges financial support from a Bishop's Senate Research Grant.
|
{
"timestamp": "2012-05-22T02:02:03",
"yymm": "1203",
"arxiv_id": "1203.2279",
"language": "en",
"url": "https://arxiv.org/abs/1203.2279"
}
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.